![. In speech synthesis
. In the equalization of transmission lines and cables
. In the design of artificial cochleas
Typically, an electrical filter receives an input signal or excitation and produces an output signal or
response. The frequency spectrum of the output signal is related to that of the input by some rule of
correspondence. Depending on the type of input, output, and internal operating signals, three general
types of filters can be identified, namely, continuous-time, sampled-data, and discrete-time filters.
A continuous-time signal is one that is defined at each and every instant of time. It can be represented
by a function x(t) whose domain is a range of numbers (t1, t2), where 1 t1 and t2 1. A sampled-
data or impulse-modulated signal is one that is defined in terms of an infinite summation of continuous-
time impulses (see Ref. [1, Chapter 6]). It can be represented by a function
^
x(t) ¼
X
1
n¼1
x(nT)d(t nT)
where d(t) is the impulse function. The value of the signal at any instant in the range nT t (n þ 1)T is
zero. The frequency spectrum of a continuous-time or sampled-data signal is given by the Fourier
transform.*
A discrete-time signal is one that is defined at discrete instants of time. It can be represented by a
function x(nT), where T is a constant and n is an integer in the range (n1, n2) such that 1 n1 and
n2 1. The value of the signal at any instant in the range nT t (n þ 1)T can be zero, constant, or
undefined depending on the application. The frequency spectrum in this case is obtained by evaluating
the z transform on the unit circle jzj ¼ 1 of the z plane.
Depending on the format of the input, output, and internal operating signals, filters can be classified
either as analog or digital filters. In analog filters the operating signals are varying voltages and currents,
whereas in digital filters they are encoded in some binary format. Continuous-time and sampled-data
filters are always analog filters. However, discrete-time filters can be analog or digital.
Analog filters can be classified on the basis of their constituent components as
. Passive RLC filters
. Crystal filters
. Mechanical filters
. Microwave filters
. Active RC filters
. Switched-capacitor filters
Passive RLC filters comprise resistors, inductors, and capacitors. Crystal filters are made of piezoelectric
resonators that can be modeled by resonant circuits. Mechanical filters are made of mechanical reson-
ators. Microwave filters consist of microwave resonators and cavities that can be represented by resonant
circuits. Active RC filters comprise resistors, capacitors, and amplifiers; in these filters, the performance of
resonant circuits is simulated through the use of feedback or by supplying energy to a passive circuit.
Switched-capacitor filters comprise resistors, capacitors, amplifiers, and switches. These are discrete-time
filters that operate like active filters but through the use of switches the capacitance values can be kept
very small. As a result, switched-capacitor filters are amenable to VLSI implementation.
This section provides an introduction to the characteristics of analog filters. Their basic characteriza-
tion in terms of a differential equation is reviewed in Section 1.2 and by applying the Laplace transform,
an algebraic equation is deduced that leads to the s-domain representation of a filter. The representation
of analog filters in terms of the transfer function is then developed. Using the transfer function, one can
* See Chapter 4 of Fundamentals of Circuits and Filters.
1-2 Passive, Active, and Digital Filters](https://image.slidesharecdn.com/11053098-250525235716-29db5586/85/Passive-Active-And-Digital-Filters-2nd-Edition-Chen-Waikai-23-320.jpg)
![. Kirchhoff ’s voltage and current laws
. Matrix methods
. Flow graphs
. Mason’s gain formula
. State-space methods
A transfer function is said to be realizable if it characterizes a stable and causal network. Such a transfer
function must satisfy the following constraints:
1. It must be a rational function of s with real coefficients.
2. Its poles must lie in the left-half s plane.
3. The degree of the numerator polynomial must be equal to or less than that of the denominator
polynomial.
A transfer function may represent a network comprising elements with real parameters only if its
coefficients are real. The poles must be in the left-half s plane to ensure that the network is stable and
the numerator degree must not exceed the denominator degree to assure the existence of a causal network.
1.3 Time-Domain Response
From Equation 1.1,
Y(s) ¼ H(s)X(s)
Therefore, the time-domain response of a filter to some arbitrary excitation can be deduced by obtaining
the inverse Laplace transform of Y(s), i.e.,
y(t) ¼ +1
H(s)X(s)
f g
1.3.1 General Inversion Formula
If
1. the singularities of Y(s) in the finite plane are poles,* and
2. Y(s) ! 0 uniformly with respect to the angle of s as jsj ! 1 with s C, where C is a positive
constant, then [2]
y(t) ¼
0 for t 0
1
2pj
Ð
Cþj1
Cj1
Y(s)est
ds ¼ 1
2pj
Ð
G
Y(s)est
ds for t 0
8
:
(1:2)
where G is a contour in the counterclockwise sense make up of the part of the circle s ¼ Re ju
to the left of
line s ¼ C and the segment of the line s ¼ C that overlaps the circle, as depicted in Figure 1.1; C and R are
sufficiently large to ensure that G encloses all the finite poles of Y(s).
From the residue theorem [3] and Equation 1.2, we have
y(t) ¼
0 for t 0
1
2pj
Ð
G
Y(s)est
ds ¼
P
K
i¼1
res
s¼pi
Y0(s) for t 0
8
:
* Such a function is said to be meromorphic [2,3].
1-6 Passive, Active, and Digital Filters](https://image.slidesharecdn.com/11053098-250525235716-29db5586/85/Passive-Active-And-Digital-Filters-2nd-Edition-Chen-Waikai-27-320.jpg)
![where Y0(s) ¼ Y(s)est
and K is the number of poles in
Y(s). If Y0(s) has a pole pi of order mi, the residue can
be obtained by using the general formula [3]
res
z¼pi
Y0(s) ¼
1
mi 1
ð Þ!
lim
s!pi
dmi1
dsmi1
(s pi)mi
Y0(s)
½
Note that complex poles yield complex residues. Hence,
like the poles of Y0(s), its residues occur in complex–
conjugate pairs. For this reason, y(t) is found to be a real
function of t, as can be easily verified.
Condition 1 listed previously may not be satisfied
sometimes, for example, if
lim
s!1
Y(s) ¼ A0
where A0 is a constant. In such a case, we can express
Y(s) as
Y(s) ¼ A0 þ Y0
(s)
where Y0
(s) satisfies conditions 1 and 2. Thus,
y(t) ¼ A0d(t) þ +1
Y0
(s)
The inverse Laplace transform of Y0
(s) can now be obtained by using the inversion formula.
1.3.2 Inverse by Using Partial Fractions
The simplest way to obtain the time-domain response of a filter is to express H(s)X(s) as a partial-fraction
expansion and then invert the resulting fractions individually. If Y(s) has simple poles, we can write
Y(s) ¼ A0 þ
X
K
i¼1
Ai
s pi
where A0 is a constant and
Ai ¼ lim
s!pi
s pi
ð ÞY(s)
½
is the residue of pole s ¼ pi. On applying the general inversion formula to each partial fraction, we obtain
y(t) ¼ A0d(t) þ u(t)
X
K
i¼1
Aiepit
where d(t) and u(t) are the impulse function and unit step, respectively.
jω
R
C
Γ
σ
×
×
×
×
×
×
×
×
R ∞
s plane
FIGURE 1.1 Contour G for the evaluation of the
inverse Laplace transform.
General Characteristics of Filters 1-7](https://image.slidesharecdn.com/11053098-250525235716-29db5586/85/Passive-Active-And-Digital-Filters-2nd-Edition-Chen-Waikai-28-320.jpg)
![1.3.3 Impulse and Step Responses
The response of a filter to an impulse d(t) designated as
y(t) ¼ 5d(t) h(t)
where 5 is an operator, is of considerable importance. Its absolute integrability guarantees the stability of
the filter* and its Laplace transform, namely, H(s), is the transfer function as has been shown in the
section on the transfer function.
For an Nth-order, causal, linear, and time-invariant filter
H(s) ¼
a0 þ a1s þ a2s2
þ þ aMsM
b0 þ b1s þ b2s2 þ þ bN sN
where M N.
The step (or unit-step) response is the output of a filter to the signal
u(t) ¼
1 for t 0
0 for t 0
The Laplace transform of u(t) is 1=s. Hence, the step response of an arbitrary filter is obtained as
y(t) ¼ 5u(t) yu(t) ¼ +1 H(s)
s
1.3.4 Overshoot, Delay Time, and Rise Time
Three time-domain parameters of a filter are usually associated with the step response [4], namely, the
overshoot, delay time, and rise time. The overshoot g is the difference between the peak value and
the asymptotic value of the step response in percent as t ! 1. The delay time td is the time required for
the step response to reach 50% of the asymptotic value. The rise time tr is the time required for the step
response to increase from 10% to 90% of the asymptotic value. These three parameters are illustrated in
Figure 1.2, where K ¼ a0=b0 is a scaling constant that normalizes the asymptotic value of the step response
as t ! 1 to unity.
The delay and rise times defined in terms of the step response entail quite a bit of computation.
Alternative definitions of these parameters that are easier to use have been proposed by Elmore [4]. These
are based on the impulse response and give accurate results if the overshoot is small. The delay time is
defined as
tD ¼
ð
1
0
th(t)dt
and the rise time assumes the form
tR ¼ 2p
ð
1
0
t tD
ð Þ2
h(t)dt
2
4
3
5
1=2
¼
ffiffiffiffiffiffi
2p
p ð
1
0
t2
h(t)dt t2
D
2
4
3
5
1=2
* See Section 22.1 of Fundamentals of Circuits and Filters.
1-8 Passive, Active, and Digital Filters](https://image.slidesharecdn.com/11053098-250525235716-29db5586/85/Passive-Active-And-Digital-Filters-2nd-Edition-Chen-Waikai-29-320.jpg)
![1.4.1 Sinusoidal Response
Consider an Nth-order analog filter characterized by a transfer function H(s). The sinusoidal response of
such a filter is
y(t) ¼ +1
[H(s)X(s)]
where
X(s) ¼ +[u(t) sin vt] ¼
v
(s þ jv)(s jv)
(1:6)
The product H(s)X(s) satisfies conditions 1 and 2 imposed on the general inversion formula of Equation
1.2. Hence, for t 0, we have
y(t) ¼
1
2pj
ð
G
Y(s)est
ds ¼
X
res H(s)X(s)est
½ (1:7)
where G is a contour enclosing the poles of H(s) and X(s) as in Figure 1.1.
Assuming simple poles for the transfer function, Equations 1.6 and 1.7 give
y(t) ¼
X
N
i¼1
X pi
ð Þepit
res
s¼pi
H(s) þ
1
2j
H( jv)ejvt
H(jv)ejvt
(1:8)
If the filter is assumed to be stable, then the poles are in the left-half s plane, i.e., pi ¼ si þ jvi with si 0.*
As a consequence
lim
t!1
epit
¼ lim
t!1
esit
ejvit
¼ 0
and since the residues of H(s) are finite, the steady-state sinusoidal response is obtained from
Equation 1.8 as
~
y(t) ¼ lim
t!1
y(t) ¼
1
2j
H( jv)ejvt
H(jv)ejvt
(1:9)
Equation 1.9 was deduced on the assumption that the poles of the transfer function are simple. However,
it also applies for transfer functions with higher-order poles.
Now from the definition of the Laplace transform
H(s) ¼
ð
1
1
h(t)est
dt
and hence
H(jv) ¼
ð
1
1
h(t)ejvt
dt ¼
ð
1
1
h(t)ejvt
dt
2
4
3
5
*
¼ H*( jv) (1:10)
* See Chapter 22.1 of Fundamentals of Circuits and Filters.
General Characteristics of Filters 1-11](https://image.slidesharecdn.com/11053098-250525235716-29db5586/85/Passive-Active-And-Digital-Filters-2nd-Edition-Chen-Waikai-32-320.jpg)
![If we write
H( jv) ¼ M(v)eju(v)
(1:11)
where
M(v) ¼ H( jv)
j j and u(v) ¼ arg H( jv) (1:12)
the steady-state sinusoidal response of the filter is obtained from Equations 1.9 through 1.12 as
~
y(t) ¼
1
2j
M(v)eju(v)
ejvt
M(v)eju(v)
ejvt
¼ M(v)
1
2j
ej[vtþu(v)]
ej[vtþu(v)]
¼ M(v) sin[vt þ u(v)]
The preceding analysis has shown that the steady-state response of an analog filter to a sinusoid of unit
amplitude is a sinusoid of amplitude M(v), shifted by an angle u(v). In effect, for a given frequency v,
the filter introduces a gain M(v) and a phase shift u(v).
As functions of frequency, M(v) and u(v) are known as the amplitude (or magnitude) response and
phase response of the filter, respectively. The transfer function evaluated on the imaginary axis, namely,
H(jv) is the frequency response and, as was shown, its magnitude and angle are the amplitude response
and phase response, respectively.
Two other quantities of a filter, which are of significant interest, are its phase and group delays These
are defined as
tp(v) ¼
u(v)
v
and tg(v) ¼
du(v)
dv
respectively. For filters, the group delay is the more important of the two. As a function of frequency,
tg(v) is usually referred to as the delay characteristic.
1.4.2 Graphical Construction
Consider a filter characterized by a transfer function of the form
H(s) ¼ H0
N(s)
D(s)
¼ H0
QM
i¼1 (s zi)
QN
i¼1 (s pi)mi
(1:13)
where H0 is a constant. The frequency response of the filter is obtained as
H( jv) ¼ M(v)eju(v)
¼
H0
QM
i¼1 ( jv zi)
QN
i¼1 (jv pi)mi
1-12 Passive, Active, and Digital Filters](https://image.slidesharecdn.com/11053098-250525235716-29db5586/85/Passive-Active-And-Digital-Filters-2nd-Edition-Chen-Waikai-33-320.jpg)
![with arbitrary amplitude and phase response characteristics (see Ref. [1, Chapter 14]) for the application of
these methods for the design of digital filters). Some classical closed-form solutions are the so-called
Butterworth, Chebyshev, and elliptic* approximations to be described in Chapter 2 by A.M. Davis.
In general, the designer is interested in simple and reliable approximation methods that yield precise
designs with the minimum amount of computation.
1.8.2 The Realization Step
The synthesis of a filter is the process of converting some characterization of the filter into a network. The
process of converting the transfer function into a network is said to be the realization step and the
network obtained is sometimes called the realization.
The realization of a transfer function can be accomplished by expressing it in some form that allows
the identification of an interconnection of elemental filter subnetworks and=or elements. Many realiza-
tion methods have been proposed in the past that lead to structures of varying complexity and properties.
In general, the designer is interested in realizations that are economical in terms of the number
of elements, do not require expensive components, and are not seriously affected by variations in
the element values such as may be caused by variations in temperature and humidity, and drift due to
element aging.
1.8.3 Study of Imperfections
During the approximation step, the coefficients of the transfer function are determined to a high degree
of precision and the realization is obtained on the assumption that elements are ideal, i.e., capacitors are
lossless, inductors are free of winding capacitances, amplifiers have infinite bandwidths, and so on.
In practice, however, the filter is implemented with nonideal elements that have finite tolerances and are
often nonlinear. Consequently, once a realization is obtained, sometimes referred to as a paper design,
the designer must embark on the study of the effects of element imperfections. Several types of analysis
are usually called for ranging from tolerance analysis, study of parasitics, time-domain analysis, sensi-
tivity analysis, noise analysis, etc. Tight tolerances result in high-precision filters but the cost per unit
would be high. Hence the designer is obliged to determine the highest tolerance that can be tolerated
without violating the specifications of the filter throughout its working life. Sensitivity analysis is a related
study that will ascertain the degree of dependence of a filter parameter, e.g., the dependence of the
amplitude response on a specific element. If the loss characteristic of a filter is not very sensitive to certain
capacitance, then the designer would be able to use a less precise and cheaper capacitor, which would, of
course, decrease the cost of the unit.
1.8.4 Implementation
Once the filter is thoroughly analyzed and found to meet the desired specifications under ideal
conditions, a prototype is constructed and tested. Decisions to be made involve the type of components
and packaging, and the methods are to be used for the manufacture, testing, and tuning of the filter.
Problems may often surface at the implementation stage that may call for one or more modifications in
the paper design. Then the realization and possibly the approximation may have to be redone.
* To be precise, the elliptic approximation is not a closed-form method, since the transfer function coefficients are given in
terms of certain infinite series. However, these series converge very rapidly and can be treated as closed-form formulas for
most practical purposes (see Ref. [1, Chapter 5]).
1-22 Passive, Active, and Digital Filters](https://image.slidesharecdn.com/11053098-250525235716-29db5586/85/Passive-Active-And-Digital-Filters-2nd-Edition-Chen-Waikai-43-320.jpg)
![1.9 Introduction to Realization
Realization tends to depend heavily on the type of filter required. The realization of passive RLC filters
differs quite significantly from that of active filters which, in turn, is entirely different from the realization
of microwave filters.
1.9.1 Passive Filters
Passive RLC filters have been the mainstay of communications since the 1920s and, furthermore, they
continue to be of considerable importance today for frequencies in the 100–500 kHz range.
The realization of passive RLC filters has received considerable attention through the years and it is, as
a consequence, highly developed and sophisticated. It can be accomplished by using available filter-
design packages such as FILSYN [5] and FILTOR [6]. In addition, several filter-design handbooks and
published design tables are available [7–10].
The realization of passive RLC filters starts with a resistively terminated LC two-port network such as
that in Figure 1.8. Then through one of several approaches, the transfer function is used to generate
expressions for the z or y parameters of the LC two-port. The realization of the LC two-port is achieved
by realizing the z or y parameters. The realization of passive filters is considered in Section I.
1.9.2 Active Filters
Since the reactance of an inductor is vL, increased inductance values are required to achieve reason-
able reactance values at low frequencies. For example, an inductance of 1 mH which will present a
reactance of 6.28 kV at 1 MHz will present only 0.628 V at 100 Hz. Thus, as the frequency range of
interest is reduced, the inductance values must be increased if a specified impedance level is to be
maintained. This can be done by increasing the number of turns on the inductor coil and to some extent
by using ferromagnetic cores of high permeability. Increasing the number of turns increases the
resistance, the size, and the cost of the inductor. The resistance is increased because the length of
the wire is increased [R ¼ (r 3 length)=Area], and hence the Q factor is reduced. The cost goes up
because the cost of materials as well as the cost of labor go up, since an inductor must be individually
wound. For these reasons, inductors are generally incompatible with miniaturization or microcircuit
implementation.
The preceding physical problem has led to the invention and development of a class of inductorless
filters known collectively as active filters. Sensitivity considerations, which will be examined in Chapter 4
by I. Filanovsky, have led to two basic approaches to the design of active filters. In one approach, the
active filter is obtained by simulating the inductances in a passive RLC filter or by realizing a signal flow
graph of the passive RLC filter. In another approach, the active filter is obtained by cascading a number of
low-order filter sections of some type, as depicted in Figure 1.9a where Zo0 is the output impedance of the
signal source.
R1
R2
Vi Vo
LC network
FIGURE 1.8 Passive RLC filter.
General Characteristics of Filters 1-23](https://image.slidesharecdn.com/11053098-250525235716-29db5586/85/Passive-Active-And-Digital-Filters-2nd-Edition-Chen-Waikai-44-320.jpg)
![2
Approximation
Artice M. Davis
San Jose State University
2.1 Introduction ................................................................................ 2-1
2.2 Butterworth LPP Approximation........................................... 2-8
2.3 Chebyshev LPP Approximation ........................................... 2-12
2.4 Bessel–Thompson LPP Approximation.............................. 2-18
2.5 Elliptic Approximation........................................................... 2-23
References............................................................................................ 2-33
2.1 Introduction
The approximation problem for filters is illustrated in Figure 2.1. A filter is often desired to produce a
given slope of gain over one or more frequency intervals, to remain constant over other intervals, and to
completely reject signals having frequencies contained in still other intervals. Thus, in the example shown
in the figure, the desired gain is zero for very low and very high frequencies. The centerline, shown
dashed, is the nominal behavior and the shaded band shows the permissible variation in the gain
characteristic. Realizable circuits must always generate smooth curves and so cannot exactly meet the
piecewise linear specification represented by the centerline. Thus, the realizable behavior is shown by the
smooth, dark curve that lies entirely within the shaded tolerance band.
What type of frequency response function can be postulated that will meet the required specifications
and, at the same time, be realizable: constructible with a specified catalog of elements? The answer
depends upon the types of elements allowed. For instance, if one allows pure delays with a common delay
time, summers, and scalar multipliers, a trigonometric polynomial will work; this, however, will cause the
gain function to be repeated in a periodic manner. If this is permissible, one can then realize the filter in
the form of an FIR digital filter or as a commensurate transmission line filter, and in fact, it can be
realized in such a fashion that the resulting phase behavior is precisely linear. If one fits the required
behavior with a rational trigonometric function, a function that is the ratio of two trigonometric
polynomials, an economy of hardware will result. The phase, however, will unfortunately no longer be
linear. These issues are discussed at greater length in Ref. [1].
Another option would be to select an ordinary polynomial in v as the approximating function.
Polynomials, however, behave badly at infinity. They approach infinity as v ! 1, a highly undesirable
solution. For this reason, one must discard polynomials. A rational function of v, however, will work
nicely for the ratio of two polynomials will approach zero as v ! 1 if the degree of the numerator
polynomial is selected to be of lower degree than that of the denominator. Furthermore, by the
Weierstrass theorem, such a function can approximate any continuous function arbitrarily closely over
any closed interval of finite length [2]. Thus, one sees that the rational functions in v offer a suitable
approximation for analog filter design and, in fact, do not have the repetitive nature of the trigonometric
rational functions.
2-1](https://image.slidesharecdn.com/11053098-250525235716-29db5586/85/Passive-Active-And-Digital-Filters-2nd-Edition-Chen-Waikai-52-320.jpg)
![Suppose, therefore, that the gain function is of the form
A(v) ¼
N(v)
D(v)
¼
a0 þ a1v þ a2v2
þ þ arvr
b0 þ b1v þ b2v2 þ þ bqvq
(2:1)
where r q for reasons mentioned above. Assuming that the filter to be realized is constrained to be
constructable with real* elements, one must require that A(v) ¼ A(v), that is, that the gain be an event
function of frequency. But then, as it is straightforward to show, one must require that all the odd
coefficients of both numerator and denominator be zero. This means that the gain is a function of v2
:
A(v) ¼
N(v2
)
D(v2)
¼
a0 þ a1v2
þ þ amv2m
b0 þ b1v2 þ þ bnv2n
¼ A(v2
) (2:2)
The expression has been reindexed and the constants redefined in an obvious manner. The net result is
that one must approximate the desired characteristic by the ratio of two polynomials in v2
; the objective
is to determine the numerator and denominator coefficients to meet the stated specifications. Once this is
accomplished one must compute the filter transfer function G(s) in order to synthesize the filter [4,5].
Assuming that G(s) is real (has real coefficients), then its complex conjugate satisfies G*(s) ¼ G(s*), from
which it follows that G(s) is related to A(v2
) by the relationship
[G(s)G(s)]s¼jv ¼ G( jv)G*( jv) ¼ G( jv)
j j2
¼ A2
(v2
) (2:3)
In fact, it is more straightforward to simply cast the original approximation problem in terms of A2
(v2
),
rather than in terms of A(v). In this case, Equation 2.2 becomes
A2
(v2
) ¼
N(v2
)
D(v2)
¼
a0 þ a1v2
þ þ amv2m
b0 þ b1v2 þ þ bnv2n
(2:4)
Thus, one can assume that the approximation process produces A2
(v2
) as the ratio of two real
polynomials in v2
. Since Equation 2.3 requires the substitutions s ! jv, one also has s2
! v2
, and
conversely. Thus, Equation 2.3 becomes
G(s)G(s) ¼ A2
(s2
) (2:5)
Though this has been shown to hold only on the imaginary axis, it continues to hold for other complex
values of s as well by analytic continuation.y
A(ω)
ω
FIGURE 2.1 General approximation problem.
* Complex filters are quite possible to construct, as recent work [3] shows.
y
A function analytic in a region is completely determined by its values along any line segment in that region—in this case, by
its value along the jv axis.
2-2 Passive, Active, and Digital Filters](https://image.slidesharecdn.com/11053098-250525235716-29db5586/85/Passive-Active-And-Digital-Filters-2nd-Edition-Chen-Waikai-53-320.jpg)
![The problem now is to compute G(s) from Equation 2.5, a process known as the factorization problem.
The solution is not unique; in fact, the phase is arbitrary—subject only to certain realizability conditions.
To see this, just let G(jv) ¼ A(v)ejf(v)
, where fv is an arbitrary phase function. Then, Equation 2.3
implies that
G( jv)G*( jv) ¼ A(v)ejf(v)
A(v)ejf(v)
¼ A2
(v) (2:6)
If the resulting structure is to have the property of minimum phase [6], the phase function is determined
completely by the gain function. If not, one can simply perform the factorization and accept whatever
phase function results from the particular process chosen. As has been pointed out earlier in this chapter,
it is often desirable that the phase be a linear function of frequency. In this case, one must follow the
filter designed by the above process with a phase equalization filter, one that has constant gain and a
phase characteristic that, when summed with that of the first filter, produces linear phase. As it happens,
the human ear is insensitive to phase nonlinearity, so the phase is not of much importance for filters
designed to operate in the audio range. For those intended for video applications, however, it is vitally
important. Nonlinear phase produces, for instance, the phenomenon of multiple edges in a reproduced
picture.
If completely arbitrary gain characteristics are desired, computer optimization is necessary [6]. Indeed,
if phase is of great significance, computer algorithms are available for the simultaneous approximation of
both gain and phase. These are complex and unwieldy to use, however, so for more modest applications
the above approach relying upon gain approximation only suffices. In fact, the approach arose histor-
ically in the telephone industry in its earlier days in which voice transmission was the only concern, data
and video transmission being unforeseen at the time. Furthermore, the frequency division multiplexing
of voice signals was the primary concern; hence, a number of standard desired shapes of frequency
response were generated: low pass, high pass, bandpass, and bandreject (or notch). Typical but stylized
specification curves are shown in Figure 2.2. This figure serves to define the following parameters: the
minimum passband gain Ap, the maximum stopband gain As, the passband cutoff frequency vp,
the stopband cutoff frequency vs (the last two parameters are for low-pass and high-pass filters only),
the center frequency vo, upper passband and stopband cutoff frequencies vpu and vsu, and lower
passband and stopband cutoff frequencies vpl and vsl (the last four parameters are for the bandpass
A(ω) A(ω)
A(ω)
A(ω)
Ap
1
Ap
Ap
1
1
Ap
1
ωp ωp
ωo ωo
ωpu ωsu ωsu ωpu
ωs1ωp1 ωp1ωs1
ωs ωs
ω ω
ω
ω
As As
As
As As
(a) (b)
(d)
(c)
FIGURE 2.2 Catalog of basic filter types: (a) low pass, (b) high pass, (c) bandpass, and (d) bandreject.
Approximation 2-3](https://image.slidesharecdn.com/11053098-250525235716-29db5586/85/Passive-Active-And-Digital-Filters-2nd-Edition-Chen-Waikai-54-320.jpg)
Passive Active And Digital Filters 2nd Edition Chen Waikai
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- 7. The Circuits and Filters Handbook Third Edition Fundamentals of Circuits and Filters Feedback, Nonlinear, and Distributed Circuits Analog and VLSI Circuits Computer Aided Design and Design Automation Passive, Active, and Digital Filters Edited by Wai-Kai Chen
- 8. Edited by Wai-Kai Chen University of Illinois Chicago, U. S. A. The Circuits and Filters Handbook Third Edition Passive, Active, and Digital Filters
- 9. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-5885-7 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti- lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy- ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For orga- nizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Passive, active, and digital filters / edited by Wai-Kai Chen. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4200-5885-7 ISBN-10: 1-4200-5885-1 1. Electric filters, Digital. 2. Electric filters, Passive. 3. Electric filters, Active. I. Chen, Wai-Kai, 1936- II. Title. TK7872.F5P375 2009 621.3815’324--dc22 2008048117 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 10. Contents Preface ................................................................................................................................................... ix Editor-in-Chief .................................................................................................................................... xi Contributors ......................................................................................................................................xiii SECTION I Passive Filters 1 General Characteristics of Filters ...................................................................................... 1-1 Andreas Antoniou 2 Approximation....................................................................................................................... 2-1 Artice M. Davis 3 Frequency Transformations ................................................................................................ 3-1 Jaime Ramirez-Angulo 4 Sensitivity and Selectivity .................................................................................................... 4-1 Igor M. Filanovsky 5 Passive Immittances and Positive-Real Functions......................................................... 5-1 Wai-Kai Chen 6 Passive Cascade Synthesis ................................................................................................... 6-1 Wai-Kai Chen 7 Synthesis of LCM and RC One-Port Networks ............................................................. 7-1 Wai-Kai Chen 8 Two-Part Synthesis by Ladder Development ................................................................. 8-1 Wai-Kai Chen 9 Design of Resistively Terminated Networks ................................................................... 9-1 Wai-Kai Chen 10 Design of Broadband Matching Networks.................................................................... 10-1 Wai-Kai Chen v
- 11. SECTION II Active Filters 11 Low-Gain Active Filters ..................................................................................................... 11-1 Phillip E. Allen, Benjamin J. Blalock, and Stephen W. Milam 12 Single-Amplifier Multiple-Feedback Filters .................................................................. 12-1 F. William Stephenson 13 Multiple-Amplifier Biquads .............................................................................................. 13-1 Norbert J. Fliege 14 The Current Generalized Immittance Converter Biquads ........................................ 14-1 Wasfy B. Mikhael 15 High-Order Filters .............................................................................................................. 15-1 Rolf Schaumann 16 Continuous-Time Integrated Filters ............................................................................... 16-1 Rolf Schaumann 17 Switched-Capacitor Filters ................................................................................................ 17-1 Jose Silva-Martinez and Edgar Sánchez-Sinencio SECTION III Digital Filters 18 FIR Filters ............................................................................................................................. 18-1 M. H. Er, Andreas Antoniou, Yong Ching Lim, and Tapio Saramäki 19 IIR Filters .............................................................................................................................. 19-1 Sawasd Tantaratana, Chalie Charoenlarpnopparut, Phakphoom Boonyanant, and Yong Ching Lim 20 Finite Wordlength Effects ................................................................................................. 20-1 Bruce W. Bomar 21 VLSI Implementation of Digital Filters ......................................................................... 21-1 Joseph B. Evans and Timothy R. Newman 22 Two-Dimensional FIR Filters ........................................................................................... 22-1 Rashid Ansari and A. Enis Cetin 23 Two-Dimensional IIR Filters............................................................................................ 23-1 A. G. Constantinides and Xiaojian Xu 24 1-D Multirate Filter Banks ................................................................................................ 24-1 Nick G. Kingsbury and David B. H. Tay 25 Directional Filter Banks ..................................................................................................... 25-1 Jose Gerardo Rosiles and Mark J. T. Smith 26 Nonlinear Filtering Using Statistical Signal Models ................................................... 26-1 Kenneth E. Barner, Tuncer C. Aysal, and Gonzalo R. Arce vi Contents
- 12. 27 Nonlinear Filtering for Image Denoising ...................................................................... 27-1 Nasir M. Rajpoot, Zhen Yao, and Roland G. Wilson 28 Video Demosaicking Filters .............................................................................................. 28-1 Bahadir K. Gunturk and Yucel Altunbasak Index ................................................................................................................................................IN-1 Contents vii
- 14. Preface As circuit complexity continues to increase, the microelectronic industry must possess the ability to quickly adapt to the market changes and new technology through automation and simulations. The purpose of this book is to provide in a single volume a comprehensive reference work covering the broad spectrum of filter design from passive, active, to digital. The book is written and developed for the practicing electrical engineers and computer scientists in industry, government, and academia. The goal is to provide the most up-to-date information in the field. Over the years, the fundamentals of the field have evolved to include a wide range of topics and a broad range of practice. To encompass such a wide range of knowledge, this book focuses on the key concepts, models, and equations that enable the design engineer to analyze, design, and predict the behavior of large-scale systems that employ various types of filters. While design formulas and tables are listed, emphasis is placed on the key concepts and theories underlying the processes. This book stresses fundamental theory behind professional applications and uses several examples to reinforce this point. Extensive development of theory and details of proofs have been omitted. The reader is assumed to have a certain degree of sophistication and experience. However, brief reviews of theories, principles, and mathematics of some subject areas are given. These reviews have been done concisely with perception. The compilation of this book would not have been possible without the dedication and efforts of Professor Rashid Ansari and Dr. A. Enis Cetin, and most of all the contributing authors. I wish to thank them all. Wai-Kai Chen ix
- 16. Editor-in-Chief Wai-Kai Chen is a professor and head emeritus of the Department of Electrical Engineering and Computer Science at the University of Illinois at Chicago. He received his BS and MS in electrical engin- eering at Ohio University, where he was later recognized as a distinguished professor. He earned his PhD in electrical engineering at the University of Illinois at Urbana–Champaign. Professor Chen has extensive experience in education and indus- try and is very active professionally in the fields of circuits and systems. He has served as a visiting professor at Purdue University, the University of Hawaii at Manoa, and Chuo University in Tokyo, Japan. He was the editor-in-chief of the IEEE Transactions on Circuits and Systems, Series I and II, the president of the IEEE Circuits and Systems Society, and is the founding editor and the editor-in-chief of the Journal of Circuits, Systems and Computers. He received the Lester R. Ford Award from the Mathematical Association of America; the Alexander von Humboldt Award from Germany; the JSPS Fellowship Award from the Japan Society for the Promotion of Science; the National Taipei University of Science and Technology Distinguished Alumnus Award; the Ohio University Alumni Medal of Merit for Distinguished Achievement in Engineering Education; the Senior University Scholar Award and the 2000 Faculty Research Award from the University of Illinois at Chicago; and the Distinguished Alumnus Award from the University of Illinois at Urbana–Champaign. He is the recipient of the Golden Jubilee Medal, the Education Award, and the Meritorious Service Award from the IEEE Circuits and Systems Society, and the Third Millennium Medal from the IEEE. He has also received more than a dozen honorary professorship awards from major institutions in Taiwan and China. A fellow of the Institute of Electrical and Electronics Engineers (IEEE) and the American Association for the Advancement of Science (AAAS), Professor Chen is widely known in the profession for the following works: Applied Graph Theory (North-Holland), Theory and Design of Broadband Matching Networks (Pergamon Press), Active Network and Feedback Amplifier Theory (McGraw-Hill), Linear Networks and Systems (Brooks=Cole), Passive and Active Filters: Theory and Implements (John Wiley), Theory of Nets: Flows in Networks (Wiley-Interscience), The Electrical Engineering Handbook (Academic Press), and The VLSI Handbook (CRC Press). xi
- 18. Contributors Phillip E. Allen School of Electrical Engineering Georgia Institute of Technology Atlanta, Georgia Yucel Altunbasak School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, Georgia Rashid Ansari Department of Electrical and Computer Engineering University of Illinois at Chicago Chicago, Illinois Andreas Antoniou Department of Electrical and Computer Engineering University of Victoria Victoria, British Columbia, Canada Gonzalo R. Arce Electrical and Computer Engineering Department University of Delaware Newark, Delaware Tuncer C. Aysal School of Electrical and Computer Engineering Cornell University Ithaca, New York Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Newark, Delaware Benjamin J. Blalock Department of Electrical Engineering and Computer Science The University of Tennessee Knoxville, Tennessee Bruce W. Bomar Department of Electrical and Computer Engineering The University of Tennessee Space Institute Tullahoma, Tennessee Phakphoom Boonyanant National Electronics and Computer Technology Center Pathumthani, Thailand A. Enis Cetin Department of Electrical and Electronics Engineering Bilkent University Ankara, Turkey Chalie Charoenlarpnopparut Sirindhorn International Institute of Technology Thammasat University Pathumthani, Thailand Wai-Kai Chen Department of Electrical and Computer Engineering University of Illinois at Chicago Chicago, Illinois A. G. Constantinides Department of Electrical and Electronic Engineering Imperial College of Science, Technology and Medicine London, England Artice M. Davis Department of Electrical Engineering San Jose State University San Jose, California M. H. Er School of Electrical and Electronic Engineering Nanyang Technological University Singapore Joseph B. Evans Department of Electrical Engineering and Computer Science University of Kansas Lawrence, Kansas xiii
- 19. Igor M. Filanovsky Department of Electrical Engineering and Computer Technology University of Alberta Edmonton, Alberta, Canada Norbert J. Fliege Department of Electrical Engineering and Computer Engineering University of Mannheim Mannheim, Germany Bahadir K. Gunturk Department of Electrical and Computer Engineering Louisiana State University Baton Rouge, Louisiana Nick G. Kingsbury Department of Engineering Trinity College University of Cambridge Cambridge, United Kingdom Yong Ching Lim School of Electrical and Electronic Engineering Nanyang Technological University Singapore Wasfy B. Mikhael Department of Electrical Engineering and Computer Science University of Central Florida Orlando, Florida Stephen W. Milam RF Micro-Devices Greensboro, North Carolina Timothy R. Newman Department of Electrical Engineering and Computer Science University of Kansas Lawrence, Kansas Nasir M. Rajpoot Department of Computer Science University of Warwick Coventry, United Kingdom Jaime Ramirez-Angulo Klipsch School of Electrical and Computer Engineering New Mexico State University Las Cruces, New Mexico Jose Gerardo Rosiles Electrical and Computer Engineering Department The University of Texas at El Paso El Paso, Texas Edgar Sánchez-Sinencio Department of Electrical and Computer Engineering Texas A&M University College Station, Texas Tapio Saramäki Institute of Signal Processing Tampere University of Technology Tampere, Finland Rolf Schaumann Department of Electrical Engineering Portland State University Portland, Oregon Jose Silva-Martinez Department of Electrical and Computer Engineering Texas A&M University College Station, Texas Mark J. T. Smith School of Electrical and Computer Engineering Purdue University West Lafayette, Indiana F. William Stephenson Department of Electrical and Computer Engineering Virginia Polytechnic Institute and State University Blacksburg, Virginia Sawasd Tantaratana Sirindhorn International Institute of Technology Thammasat University Pathumthani, Thailand David B. H. Tay Department of Electronic Engineering Latrobe University Bundoora, Victoria, Australia Roland G. Wilson Department of Computer Science University of Warwick Coventry, United Kingdom Xiaojian Xu School of Electronic and Information Engineering Beihang University Beijing, China Zhen Yao Department of Computer Science University of Warwick Coventry, United Kingdom xiv Contributors
- 20. I Passive Filters Wai-Kai Chen University of Illinois at Chicago 1 General Characteristics of Filters Andreas Antoniou ....................................................... 1-1 Introduction . Characterization . Time-Domain Response . Frequency-Domain Analysis . Ideal and Practical Filters . Amplitude and Delay Distortion . Minimum-Phase, Nonminimum-Phase, and Allpass Filters . Introduction to the Design Process . Introduction to Realization . References 2 Approximation Artice M. Davis ............................................................................................. 2-1 Introduction . Butterworth LPP Approximation . Chebyshev LPP Approximation . Bessel–Thompson LPP Approximation . Elliptic Approximation . References 3 Frequency Transformations Jaime Ramirez-Angulo ......................................................... 3-1 Low-Pass Prototype . Frequency and Impedance Scaling . Low-Pass to High-Pass Transformation . Low-Pass to Bandpass Transformation . Low-Pass to Band-Reject Transformation 4 Sensitivity and Selectivity Igor M. Filanovsky .................................................................... 4-1 Introduction . Definitions of Sensitivity . Function Sensitivity to One Variable . Coefficient Sensitivity . Root Sensitivities . Statistical Model for One Variable . Multiparameter Sensitivities and Sensitivity Measures . Sensitivity Invariants . Sensitivity Bounds . Remarks on the Sensitivity Applications . Sensitivity Computations Using the Adjoint Network . General Methods of Reducing Sensitivity . Cascaded Realization of Active Filters . Simulation of Doubly Terminated Matched Lossless Filters . Sensitivity of Active RC Filters . Errors in Sensitivity Comparisons . References 5 Passive Immittances and Positive-Real Functions Wai-Kai Chen ............................... 5-1 References 6 Passive Cascade Synthesis Wai-Kai Chen............................................................................ 6-1 Introduction . Type-E Section . Richards Section . Darlington Type-D Section . References 7 Synthesis of LCM and RC One-Port Networks Wai-Kai Chen .................................... 7-1 Introduction . LCM One-Port Networks . RC One-Port Networks . References 8 Two-Part Synthesis by Ladder Development Wai-Kai Chen......................................... 8-1 Introduction . LC Ladder . RC Ladder . Parallel or Series Ladders . Reference I-1
- 21. 9 Design of Resistively Terminated Networks Wai-Kai Chen .......................................... 9-1 Introduction . Double-Terminated Butterworth Networks . Double-Terminated Chebyshev Networks . References 10 Design of Broadband Matching Networks Wai-Kai Chen........................................... 10-1 Introduction . Basic Coefficient Constraints . Design Procedure . Explicit Formulas for the RLC Load . References I-2 Passive, Active, and Digital Filters
- 22. 1 General Characteristics of Filters Andreas Antoniou University of Victoria 1.1 Introduction ................................................................................ 1-1 1.2 Characterization......................................................................... 1-3 Laplace Transform . Transfer Function 1.3 Time-Domain Response........................................................... 1-6 General Inversion Formula . Inverse by Using Partial Fractions . Impulse and Step Responses . Overshoot, Delay Time, and Rise Time 1.4 Frequency-Domain Analysis................................................. 1-10 Sinusoidal Response . Graphical Construction . Loss Function 1.5 Ideal and Practical Filters....................................................... 1-15 1.6 Amplitude and Delay Distortion.......................................... 1-16 1.7 Minimum-Phase, Nonminimum-Phase, and Allpass Filters ................................................................... 1-17 Minimum-Phase Filters . Allpass Filters . Decomposition of Nonminimum-Phase Transfer Functions 1.8 Introduction to the Design Process..................................... 1-21 The Approximation Step . The Realization Step . Study of Imperfections . Implementation 1.9 Introduction to Realization.................................................... 1-23 Passive Filters . Active Filters . Biquads . Types of Basic Filter Sections References............................................................................................ 1-29 1.1 Introduction An electrical filter is a system that can be used to modify, reshape, or manipulate the frequency spectrum of an electrical signal according to some prescribed requirements. For example, a filter may be used to amplify or attenuate a range of frequency components, reject or isolate one specific frequency compon- ent, and so on. The applications of electrical filters are numerous, for example, . To eliminate signal contamination such as noise in communication systems . To separate relevant from irrelevant frequency components . To detect signals in radios and TV’s . To demodulate signals . To bandlimit signals before sampling . To convert sampled signals into continuous-time signals . To improve the quality of audio equipment, e.g., loudspeakers . In time-division to frequency-division multiplex systems 1-1
- 23. . In speech synthesis . In the equalization of transmission lines and cables . In the design of artificial cochleas Typically, an electrical filter receives an input signal or excitation and produces an output signal or response. The frequency spectrum of the output signal is related to that of the input by some rule of correspondence. Depending on the type of input, output, and internal operating signals, three general types of filters can be identified, namely, continuous-time, sampled-data, and discrete-time filters. A continuous-time signal is one that is defined at each and every instant of time. It can be represented by a function x(t) whose domain is a range of numbers (t1, t2), where 1 t1 and t2 1. A sampled- data or impulse-modulated signal is one that is defined in terms of an infinite summation of continuous- time impulses (see Ref. [1, Chapter 6]). It can be represented by a function ^ x(t) ¼ X 1 n¼1 x(nT)d(t nT) where d(t) is the impulse function. The value of the signal at any instant in the range nT t (n þ 1)T is zero. The frequency spectrum of a continuous-time or sampled-data signal is given by the Fourier transform.* A discrete-time signal is one that is defined at discrete instants of time. It can be represented by a function x(nT), where T is a constant and n is an integer in the range (n1, n2) such that 1 n1 and n2 1. The value of the signal at any instant in the range nT t (n þ 1)T can be zero, constant, or undefined depending on the application. The frequency spectrum in this case is obtained by evaluating the z transform on the unit circle jzj ¼ 1 of the z plane. Depending on the format of the input, output, and internal operating signals, filters can be classified either as analog or digital filters. In analog filters the operating signals are varying voltages and currents, whereas in digital filters they are encoded in some binary format. Continuous-time and sampled-data filters are always analog filters. However, discrete-time filters can be analog or digital. Analog filters can be classified on the basis of their constituent components as . Passive RLC filters . Crystal filters . Mechanical filters . Microwave filters . Active RC filters . Switched-capacitor filters Passive RLC filters comprise resistors, inductors, and capacitors. Crystal filters are made of piezoelectric resonators that can be modeled by resonant circuits. Mechanical filters are made of mechanical reson- ators. Microwave filters consist of microwave resonators and cavities that can be represented by resonant circuits. Active RC filters comprise resistors, capacitors, and amplifiers; in these filters, the performance of resonant circuits is simulated through the use of feedback or by supplying energy to a passive circuit. Switched-capacitor filters comprise resistors, capacitors, amplifiers, and switches. These are discrete-time filters that operate like active filters but through the use of switches the capacitance values can be kept very small. As a result, switched-capacitor filters are amenable to VLSI implementation. This section provides an introduction to the characteristics of analog filters. Their basic characteriza- tion in terms of a differential equation is reviewed in Section 1.2 and by applying the Laplace transform, an algebraic equation is deduced that leads to the s-domain representation of a filter. The representation of analog filters in terms of the transfer function is then developed. Using the transfer function, one can * See Chapter 4 of Fundamentals of Circuits and Filters. 1-2 Passive, Active, and Digital Filters
- 24. obtain the time-domain response of a filter to an arbitrary excitation, as shown in Section 1.3. Some important time-domain responses, i.e., the impulse and step responses, are examined. Certain filter parameters related to the step response, namely, the overshoot, delay time, and rise time, are then considered. The response of a filter to a sinusoidal excitation is examined in Section 1.4 and is then used to deduce the basic frequency-domain representations of a filter, namely, its frequency response and loss characteristic. Some idealized filter characteristics are then identified and the differences between idealized and practical filters are delineated in Section 1.5. Practical filters tend to introduce signal degradation through amplitude and=or delay distortion. The causes of these types of distortion are examined in Section 1.6. In Section 1.7, certain special classes of filters, e.g., minimum-phase and allpass filters, are identified and their applications mentioned. This chapter concludes with a review of the design process and the tasks that need to be undertaken to translate a set of filter specifications into a working prototype. 1.2 Characterization A linear causal analog filter with input x(t) and output y(t) can be characterized by a differential equation of the form bn dn y(t) dtn þ bn1 dn1 y(t) dtn1 þ þ b0y(t) ¼ an dn x(t) dtn þ an1 dn1 x(t) dtn1 þ þ a0x(t) The coefficients a0, a1, . . . , an and b0, b1, . . . , bn are functions of the element values and are real if the parameters of the filter (e.g., resistances, inductances, etc.) are real. If they are independent of time, the filter is time invariant. The input x(t) and output y(t) can be either voltages or currents. The order of the differential equation is said to be the order of the filter. An analog filter must of necessity incorporate reactive elements that can store energy. Consequently, the filter can produce an output even in the absence of an input. The output on such an occasion is caused by the initial conditions of the filter, namely, dn1 y(t) dtn1 t¼0 , dn2 y(t) dtn2 t¼0 , . . . , y(0) The response in such a case is said to be the zero-input response. The response obtained if the initial conditions are zero is sometimes called the zero-state response. 1.2.1 Laplace Transform The most important mathematical tool in the analysis and design of analog filters is the Laplace transform. It owes its widespread application to the fact that it transforms differential into algebraic equations that are a lot easier to manipulate. The Laplace transform of x(t) is defined as* X(s) ¼ ð 1 1 x(t)est dt where s is a complex variable of the form s ¼ s þ jv. Signal x(t) can be recovered from X(s) by applying the inverse Laplace transform, which is given by * See Chapter 3 by J. R. Deller, Jr. in Fundamentals of Circuits and Filters. General Characteristics of Filters 1-3
- 25. x(t) ¼ 1 2pj ð Cþj1 Cj1 X(s)est ds where C is a positive constant. A shorthand notation of the Laplace transform and its inverse are X(s) ¼ +x(t) and x(t) ¼ +1 X(s) Alternatively, X(s) $ x(t) A common practice in the choice of symbols for the Laplace transform and its inverse is to use upper case for the s domain and lower case for the time domain. On applying the Laplace transform to the nth derivative of some function of time y(t), we find that + dn y(t) dtn ¼ sn Y(s) sn1 y(0) sn2 dy(t) dt t¼0 dn1 y(t) dtn1 t¼0 Now, on applying the Laplace transform to an nth-order differential equation with constant coefficients, we obtain bnsn þ bn1sn1 þ þ b0 Y(s) þ Cy(s) ¼ ansn þ an1sn1 þ þ a0 X(s) þ Cx(s) where X(s) and Y(s) are the Laplace transforms of the input and output, respectively Cx(s) and Cy(s) are functions that combine all the initial-condition terms that depend on x(t) and y(t), respectively 1.2.2 Transfer Function An important s-domain characterization of an analog filter is its transfer function, as for any other linear system. This is defined as the ratio of the Laplace transform of the response to the Laplace transform of the excitation. An arbitrary linear, time-invariant, continuous-time filter, which may or may not be causal, can be represented by the convolution integral y(t) ¼ ð 1 1 h(t t)x(t)dt ¼ ð 1 1 h(t)x(t t)dt where h(t) is the impulse response of the filter. The Laplace transform yields Y(s) ¼ ð 1 1 ð 1 1 h(t t)x(t)dt 2 4 3 5est dt ¼ ð 1 1 ð 1 1 h(t t)est x(t)dt dt ¼ ð 1 1 ð 1 1 h(t t)est est est x(t)dt dt 1-4 Passive, Active, and Digital Filters
- 26. Changing the order of integration, we obtain Y(s) ¼ ð 1 1 ð 1 1 h(t t)es(tt) x(t)est dt dt ¼ ð 1 1 ð 1 1 h(t t)es(tt) dt x(t)est dt Now, if we let t ¼ t0 þ t, then dt=dt0 ¼ 1 and t t ¼ t0 ; hence, Y(s) ¼ ð 1 1 ð 1 1 h(t0 )est0 dt0 x(t)est dt ¼ ð 1 1 h(t0 )est0 dt0 ð 1 1 x(t)est dt ¼ H(s)X(s) Therefore, the transfer function is given by H(s) ¼ Y(s) X(s) ¼ +h(t) (1:1) In effect, the transfer function is equal to the Laplace transform of the impulse response. Some authors define the transfer function as the Laplace transform of the impulse response. Then through the use of the convolution integral, they show that the transfer function is equal to the ratio of the Laplace transform of the response to the Laplace transform of the excitation. The two definitions are, of course, equivalent. Typically, in analog filters the input and output are voltages, e.g., x(t) þ vi(t) and y(t) þ vo(t). In such a case the transfer function is given by Vo(s) Vi(s) ¼ HV (s) or simply by Vo Vi ¼ HV (s) However, on occasion the input and output are currents, in which case Io(s) Ii(s) Io Ii ¼ HI(s) The transfer function can be obtained through network analysis using one of several classical methods,* e.g., by using * See Chapters 18 through 27 of this volume. General Characteristics of Filters 1-5
- 27. . Kirchhoff ’s voltage and current laws . Matrix methods . Flow graphs . Mason’s gain formula . State-space methods A transfer function is said to be realizable if it characterizes a stable and causal network. Such a transfer function must satisfy the following constraints: 1. It must be a rational function of s with real coefficients. 2. Its poles must lie in the left-half s plane. 3. The degree of the numerator polynomial must be equal to or less than that of the denominator polynomial. A transfer function may represent a network comprising elements with real parameters only if its coefficients are real. The poles must be in the left-half s plane to ensure that the network is stable and the numerator degree must not exceed the denominator degree to assure the existence of a causal network. 1.3 Time-Domain Response From Equation 1.1, Y(s) ¼ H(s)X(s) Therefore, the time-domain response of a filter to some arbitrary excitation can be deduced by obtaining the inverse Laplace transform of Y(s), i.e., y(t) ¼ +1 H(s)X(s) f g 1.3.1 General Inversion Formula If 1. the singularities of Y(s) in the finite plane are poles,* and 2. Y(s) ! 0 uniformly with respect to the angle of s as jsj ! 1 with s C, where C is a positive constant, then [2] y(t) ¼ 0 for t 0 1 2pj Ð Cþj1 Cj1 Y(s)est ds ¼ 1 2pj Ð G Y(s)est ds for t 0 8 : (1:2) where G is a contour in the counterclockwise sense make up of the part of the circle s ¼ Re ju to the left of line s ¼ C and the segment of the line s ¼ C that overlaps the circle, as depicted in Figure 1.1; C and R are sufficiently large to ensure that G encloses all the finite poles of Y(s). From the residue theorem [3] and Equation 1.2, we have y(t) ¼ 0 for t 0 1 2pj Ð G Y(s)est ds ¼ P K i¼1 res s¼pi Y0(s) for t 0 8 : * Such a function is said to be meromorphic [2,3]. 1-6 Passive, Active, and Digital Filters
- 28. where Y0(s) ¼ Y(s)est and K is the number of poles in Y(s). If Y0(s) has a pole pi of order mi, the residue can be obtained by using the general formula [3] res z¼pi Y0(s) ¼ 1 mi 1 ð Þ! lim s!pi dmi1 dsmi1 (s pi)mi Y0(s) ½ Note that complex poles yield complex residues. Hence, like the poles of Y0(s), its residues occur in complex– conjugate pairs. For this reason, y(t) is found to be a real function of t, as can be easily verified. Condition 1 listed previously may not be satisfied sometimes, for example, if lim s!1 Y(s) ¼ A0 where A0 is a constant. In such a case, we can express Y(s) as Y(s) ¼ A0 þ Y0 (s) where Y0 (s) satisfies conditions 1 and 2. Thus, y(t) ¼ A0d(t) þ +1 Y0 (s) The inverse Laplace transform of Y0 (s) can now be obtained by using the inversion formula. 1.3.2 Inverse by Using Partial Fractions The simplest way to obtain the time-domain response of a filter is to express H(s)X(s) as a partial-fraction expansion and then invert the resulting fractions individually. If Y(s) has simple poles, we can write Y(s) ¼ A0 þ X K i¼1 Ai s pi where A0 is a constant and Ai ¼ lim s!pi s pi ð ÞY(s) ½ is the residue of pole s ¼ pi. On applying the general inversion formula to each partial fraction, we obtain y(t) ¼ A0d(t) þ u(t) X K i¼1 Aiepit where d(t) and u(t) are the impulse function and unit step, respectively. jω R C Γ σ × × × × × × × × R ∞ s plane FIGURE 1.1 Contour G for the evaluation of the inverse Laplace transform. General Characteristics of Filters 1-7
- 29. 1.3.3 Impulse and Step Responses The response of a filter to an impulse d(t) designated as y(t) ¼ 5d(t) h(t) where 5 is an operator, is of considerable importance. Its absolute integrability guarantees the stability of the filter* and its Laplace transform, namely, H(s), is the transfer function as has been shown in the section on the transfer function. For an Nth-order, causal, linear, and time-invariant filter H(s) ¼ a0 þ a1s þ a2s2 þ þ aMsM b0 þ b1s þ b2s2 þ þ bN sN where M N. The step (or unit-step) response is the output of a filter to the signal u(t) ¼ 1 for t 0 0 for t 0 The Laplace transform of u(t) is 1=s. Hence, the step response of an arbitrary filter is obtained as y(t) ¼ 5u(t) yu(t) ¼ +1 H(s) s 1.3.4 Overshoot, Delay Time, and Rise Time Three time-domain parameters of a filter are usually associated with the step response [4], namely, the overshoot, delay time, and rise time. The overshoot g is the difference between the peak value and the asymptotic value of the step response in percent as t ! 1. The delay time td is the time required for the step response to reach 50% of the asymptotic value. The rise time tr is the time required for the step response to increase from 10% to 90% of the asymptotic value. These three parameters are illustrated in Figure 1.2, where K ¼ a0=b0 is a scaling constant that normalizes the asymptotic value of the step response as t ! 1 to unity. The delay and rise times defined in terms of the step response entail quite a bit of computation. Alternative definitions of these parameters that are easier to use have been proposed by Elmore [4]. These are based on the impulse response and give accurate results if the overshoot is small. The delay time is defined as tD ¼ ð 1 0 th(t)dt and the rise time assumes the form tR ¼ 2p ð 1 0 t tD ð Þ2 h(t)dt 2 4 3 5 1=2 ¼ ffiffiffiffiffiffi 2p p ð 1 0 t2 h(t)dt t2 D 2 4 3 5 1=2 * See Section 22.1 of Fundamentals of Circuits and Filters. 1-8 Passive, Active, and Digital Filters
- 30. The physical interpretation of these parameters is illustrated in Figure 1.3a and b. If the overshoot is small, say less than 1%, then tD td and tR tr The simplification brought about by Elmore’s definitions can be easily demonstrated. Consider a filter whose step response approaches unity as t ! 1. Such a filter has a transfer function of the form H(s) ¼ 1 þ a1s þ a2s2 þ þ aMsM 1 þ b1s þ b2s2 þ þ bNsN (1:3) y u (t)/K t1 t t2 τr τd γ 1.0 0.9 0.5 0.1 FIGURE 1.2 Overshoot, delay time, and rise time. h(t) y u (t) 1.0 0 t t Area = 1 (a) (b) τD τD τR τR FIGURE 1.3 Physical interpretation of Elmore’s definitions of delay and rise times: (a) impulse response h(t) and (b) unit-step response yu(t). General Characteristics of Filters 1-9
- 31. that is, a0 ¼ b0 ¼ 1. From the definition of the Laplace transform, H(s) ¼ ð 1 0 h(t)est dt ¼ ð 1 0 h(t) 1 st þ s2 t2 2! dt ¼ ð 1 0 h(t)dt s ð 1 0 th(t)dt þ s2 2! ð 1 0 t2 h(t)dt ¼ ð 1 0 h(t)dt stD þ s2 2! t2 R 2p þ t2 D (1:4) Alternatively, from Equation 1.3, direct division gives H(s) ¼ 1 b1 a1 ð Þs þ b2 1 a1b1 þ a2 b2 s2 þ (1:5) Now by comparing Equations 1.4 and 1.5, we deduce ð 1 0 h(t)dt ¼ 1, tD ¼ b1 a1 and tR ¼ 2p b2 1 a2 1 þ 2 a2 b2 ð Þ 1=2 The previous definitions are based on the assumption that the unit-step response approaches unity as t ! 1. If this is not the case, i.e., coefficients a0 and b0 are not equal to unity, then we can write H(s) ¼ KH0 (s) where K ¼ a0=b0 and H0 (s) ¼ 1 þ a0 1s þ a0 2s2 þ þ a0 MsM 1 þ b0 1s þ b0 2s2 þ þ b0 N sN Using the coefficients of H0 (s) in the formulas for tD and tR yields approximate values for the delay time and rise time, since these parameters are independent of the absolute value of the step response. 1.4 Frequency-Domain Analysis The frequency response of an analog filter is deduced by finding its steady-state sinusoidal response, as we shall now demonstrate. 1-10 Passive, Active, and Digital Filters
- 32. 1.4.1 Sinusoidal Response Consider an Nth-order analog filter characterized by a transfer function H(s). The sinusoidal response of such a filter is y(t) ¼ +1 [H(s)X(s)] where X(s) ¼ +[u(t) sin vt] ¼ v (s þ jv)(s jv) (1:6) The product H(s)X(s) satisfies conditions 1 and 2 imposed on the general inversion formula of Equation 1.2. Hence, for t 0, we have y(t) ¼ 1 2pj ð G Y(s)est ds ¼ X res H(s)X(s)est ½ (1:7) where G is a contour enclosing the poles of H(s) and X(s) as in Figure 1.1. Assuming simple poles for the transfer function, Equations 1.6 and 1.7 give y(t) ¼ X N i¼1 X pi ð Þepit res s¼pi H(s) þ 1 2j H( jv)ejvt H(jv)ejvt (1:8) If the filter is assumed to be stable, then the poles are in the left-half s plane, i.e., pi ¼ si þ jvi with si 0.* As a consequence lim t!1 epit ¼ lim t!1 esit ejvit ¼ 0 and since the residues of H(s) are finite, the steady-state sinusoidal response is obtained from Equation 1.8 as ~ y(t) ¼ lim t!1 y(t) ¼ 1 2j H( jv)ejvt H(jv)ejvt (1:9) Equation 1.9 was deduced on the assumption that the poles of the transfer function are simple. However, it also applies for transfer functions with higher-order poles. Now from the definition of the Laplace transform H(s) ¼ ð 1 1 h(t)est dt and hence H(jv) ¼ ð 1 1 h(t)ejvt dt ¼ ð 1 1 h(t)ejvt dt 2 4 3 5 * ¼ H*( jv) (1:10) * See Chapter 22.1 of Fundamentals of Circuits and Filters. General Characteristics of Filters 1-11
- 33. If we write H( jv) ¼ M(v)eju(v) (1:11) where M(v) ¼ H( jv) j j and u(v) ¼ arg H( jv) (1:12) the steady-state sinusoidal response of the filter is obtained from Equations 1.9 through 1.12 as ~ y(t) ¼ 1 2j M(v)eju(v) ejvt M(v)eju(v) ejvt ¼ M(v) 1 2j ej[vtþu(v)] ej[vtþu(v)] ¼ M(v) sin[vt þ u(v)] The preceding analysis has shown that the steady-state response of an analog filter to a sinusoid of unit amplitude is a sinusoid of amplitude M(v), shifted by an angle u(v). In effect, for a given frequency v, the filter introduces a gain M(v) and a phase shift u(v). As functions of frequency, M(v) and u(v) are known as the amplitude (or magnitude) response and phase response of the filter, respectively. The transfer function evaluated on the imaginary axis, namely, H(jv) is the frequency response and, as was shown, its magnitude and angle are the amplitude response and phase response, respectively. Two other quantities of a filter, which are of significant interest, are its phase and group delays These are defined as tp(v) ¼ u(v) v and tg(v) ¼ du(v) dv respectively. For filters, the group delay is the more important of the two. As a function of frequency, tg(v) is usually referred to as the delay characteristic. 1.4.2 Graphical Construction Consider a filter characterized by a transfer function of the form H(s) ¼ H0 N(s) D(s) ¼ H0 QM i¼1 (s zi) QN i¼1 (s pi)mi (1:13) where H0 is a constant. The frequency response of the filter is obtained as H( jv) ¼ M(v)eju(v) ¼ H0 QM i¼1 ( jv zi) QN i¼1 (jv pi)mi 1-12 Passive, Active, and Digital Filters
- 34. By letting jv zi ¼ Mzi ejczi (1:14) jv pi ¼ Mpi ejcpi (1:15) we obtain M(v) ¼ H0 j j QM i¼1 Mzi QN i¼1 Mmi pi (1:16) and u(v) ¼ arg H0 þ X M i¼1 czi X N i¼1 micpi (1:17) where arg H0 ¼ p if H0 is negative. The gain and phase shift M(v) and u(v) for some frequency v ¼ vi can be determined graphically by using the following procedure: 1. Mark the zeros and poles of the filter in the s plane. 2. Draw the phasor s ¼ jvi, where vi is the frequency of interest. 3. Draw a phasor of the type in Equation 1.14 for each simple zero of H(s). 4. Draw mi phasor of the type in Equation 1.15 for each pole of order mi. 5. Measure the magnitudes and angles of the phasors in steps 3 and 4 and use them in Equations 1.16 and 1.17 to calculate the gain M(vi) and phase shift u(vi), respectively. The amplitude and phase responses of a filter can be determined by repeating the preceding procedure for frequencies v ¼ v1, v2, . . . , in the range 0 to 1. The procedure is illustrated in Figure 1.4. It should be mentioned that the modern approach for the analysis of filters is through the use of the many circuit analysis programs such as SPICE.* Nevertheless, the above graphical method is of interest and merits consideration for two reasons. First, it illustrates some of the fundamental properties of filters. Second, it provides a certain degree of intuition about the expected amplitude or phase response of a filter. For example, if a filter has pole close to the jv axis, then as v approaches the neighborhood of the pole, the magnitude of the phasor from the pole to the jv axis decreases rapidly to a very small value and then increases as v increases above this value. As a result, the amplitude response will exhibit a large peak in the frequency range close to the pole. On the other hand, a zero close to or on the jv axis will lead to a notch in the amplitude response when v is in the neighborhood of the zero. Other situations are of interest, for example, if the poles of a filter are located in a band of the s plane below the horizontal line s ¼ vc and its zeros are located above this line, then the filter will pass low-frequency and attenuate high-frequency components since Mzi Mpi if v vc for all i. Such a filter is said to be a low-pass filter. If the zeros are located below the line s ¼ vc and the poles above it, then the filter will pass high-frequency and attenuate low-frequency components, i.e., the filter will be a high-pass one. * See Chapter 8 of Computer Aided Design and Design Automation, contribution of J.G. Rollins. General Characteristics of Filters 1-13
- 35. 1.4.3 Loss Function Quite often, it is desirable to represent a filter in terms of its loss function. Consider a filter represented by the voltage transfer function Vo(s) Vi(s) ¼ H(s) ¼ N(s) D(s) where Vi(s) and Vo(s) are the Laplace transforms of the input and output voltages, respectively N(s) and D(s) are polynomials in s The loss (or attenuation) of the filter in decibels is defined as A(v) ¼ 20 log Vi( jv) Vo( jv) ¼ 20 log 1 H( jv) j j ¼ 10 log L v2 (1:18) where L v2 ¼ 1 H( jv)H(jv) A(v) as a function of v is the loss characteristic. jω ψp3 ψz1 Mp2 p1 p3 p2 z1 s plane z2 jωi σ jωi – p1 = Mp1 ejψ p1 jωi – z1 = Mz1 ejψ z1 FIGURE 1.4 Graphical method for the evaluation of the frequency response. 1-14 Passive, Active, and Digital Filters
- 36. With v ¼ s=j in Equation 1.18, the function L s2 ¼ D(s)D(s) N(s)N(s) can be formed. This is called the loss function of the filter and, as is evident, its zeros are the poles of H(s) and their negatives, whereas its poles are the zeros of H(s) and their negatives. 1.5 Ideal and Practical Filters An ideal low-pass filter is one that will pass only low-frequency components. Its loss characteristic is given by A(v) ¼ 0 for 0 v vc 1 for vc v 1 The frequency ranges 0 to vc and vc to 1 are the passband and stopband, respectively. The boundary between the passband and stopband, namely, vc, is the cutoff frequency. An ideal high-pass filter will pass all components with frequencies above the cutoff frequency and reject all components with frequencies below the cutoff frequency, i.e., A(v) ¼ 1 for 0 v vc 0 for vc v 1 Idealized loss characteristics can similarly be identified for bandpass and bandstop filters as A(v) ¼ 1 for 0 v vc1 0 for vc1 v vc2 1 for vc2 v 1 8 : and A(v) ¼ 0 for 0 v vc1 1 for vc1 v vc2 0 for vc2 v 1 8 : respectively. Practical filters differ from ideal ones in that the passband loss is not zero, the stopband loss is not infinite, and the transition between passband and stopband is gradual. Practical loss characteristics for low-pass, high-pass, bandpass, and bandstop filters assume the forms ALP(v) Ap for 0 v vp Aa for va v 1 ( AHP(v) Aa for 0 v va Ap for vp v 1 ( ABP(v) Aa for 0 v va1 Ap for vp1 v vp2 Aa for va2 v 1 8 : General Characteristics of Filters 1-15
- 37. and ABS(v) ¼ Ap for 0 v vp1 Aa for va1 v va2 Ap for vp2 v 1 8 : respectively, where vp, vp1, and vp2 are passband edges, va, va1, and va2 are stopband edges, Ap is the maximum passband loss, and Aa is the minimum stopband loss. In practice, Ap is determined from the allowable amplitude distortion (see Section 1.6) and Aa is dictated by the allowable adjacent channel interference and the desirable signal-to-noise ratio. It should be mentioned that in practical filters the cutoff frequency vc is not a very precise term. It is often used to identify some hypothetical boundary between passband and stopband such as the 3 dB frequency in Butterworth filters, the passband edge in Chebyshev filters, the stopband edge in inverse- Chebyshev filters, or the geometric mean of the passband and stopband edges in elliptic filters. If a filter is required to have a piecewise constant loss characteristic (or amplitude response) and the shape of the phase response is not critical, the filter can be fully specified by its band edges, the minimum passband and maximum stopband losses Ap and Aa, respectively. 1.6 Amplitude and Delay Distortion In practice, a filter can distort the information content of the signal. Consider a filter characterized by a transfer function H(s) and assume that its input and output signal are vi(t) and vo(t). The frequency response of the filter is given by H( jv) ¼ M(v)eju(v) where M(v) and u(v) are the amplitude and phase responses, respectively. The frequency spectrum of vi(t) is its Fourier transform, namely, Vi(jv). Assume that the information content of vi(t) is concentrated in frequency band B given by B ¼ v: vL v vH f g and that its frequency spectrum is zero elsewhere. Let us assume that the amplitude response is constant with respect to band B, i.e., M(v) ¼ G0 for v 2 B (1:19) and that the phase response is linear, i.e., u(v) ¼ tgv þ uo for v 2 B (1:20) where tg is a constant. This implies that the group delay is constant with respect to band B, i.e., t(v) ¼ du(v) dv ¼ tg for v 2 B 1-16 Passive, Active, and Digital Filters
- 38. The frequency spectrum of the output signal vo(t) can be obtained from Equations 1.19 and 1.20 as Vo( jv) ¼ H( jv)Vi( jv) ¼ M(v)eju(v) Vi( jv) ¼ G0ejvtg þju0 Vi( jv) ¼ G0eju0 ejvtg Vi( jv) and from the time-shifting theorem of the Fourier transform vo(t) ¼ G0eju0 vi t tg We conclude that the amplitude response of the filter is flat and its phase response is a linear function of v (i.e., the delay characteristic is flat) in band B, then the output signal is a delayed replica of the input signal except that a gain Go and a constant phase shift u0 are introduced. If the amplitude response of the filter is not flat in band B, then amplitude distortion will be introduced since different frequency components of the signal will be amplified by different amounts. If the delay characteristic is not flat in band B, then delay (or phase) distortion will be introduced since different frequency components will be delayed by different amounts. Amplitude distortion can be quite objectionable in practice and, consequently, in each frequency band that carries information, the amplitude response is required to be constant to within a prescribed tolerance. The amount of amplitude distortion allowed determines the maximum passband loss Ap. If the ultimate receiver of the signal is the human ear, e.g., when a speech or music signal is to be processed, delay distortion is quite tolerable. However, in other applications it can be as objectionable as amplitude distortion and the delay characteristic is required to be fairly flat. Applications of this type include data transmission, where the signal is to be interpreted by digital hardware, and image process- ing, where the signal is used to reconstruct an image that is to be interpreted by the human eye. The allowable delay distortion dictates the degree of flatness in the delay characteristic. 1.7 Minimum-Phase, Nonminimum-Phase, and Allpass Filters Filters satisfying prescribed loss specifications for applications where delay distortion is unimportant can be readily designed with transfer functions whose zeros are on the jv axis or in the left-half s plane. Such transfer functions are said to be minimum-phase since the phase response at a given frequency v is increased if any one of the zeros is moved into the right-half s plane, as will now be demonstrated. 1.7.1 Minimum-Phase Filters Consider a filter where the zeros zi for i ¼ 1, 2, . . . , M are replaced by their mirror images and let the new zeros be located at z ¼ zi, where Re zi ¼ Re zi and Im zi ¼ Im zi as depicted in Figure 1.5. From the geometry of the new zero-pole plot, the magnitude and angle of each phasor jv zi are given by General Characteristics of Filters 1-17
- 39. M zi ¼ Mzi and c zi ¼ p czi respectively. The amplitude response of the modified filter is obtained from Equation 1.16 as M(v) ¼ H0 j j QM i¼1 M zi QN i¼1 Mmi pi ¼ H0 j j QM i¼1 Mzi QN i¼1 Mmi pi ¼ M(v) Therefore, replacing the zeros of the transfer function by their mirror images leaves the amplitude response unchanged. The phase response of the original filter is given by Equation 1.17 as u(v) ¼ arg H0 þ X M i¼1 czi X N i¼1 micpi (1:21) and since c zi ¼ p czi , the phase response of the modifier filter is given by u(v) ¼ arg H0 þ X M i¼1 c zi X N i¼1 micpi ¼ arg H0 þ X M i¼1 (p czi ) X N i¼1 micpi (1:22) that is, the phase response of the modified filter is different from that of the original filter. Furthermore, from Equations 1.21 and 1.22 u(v) u(v) ¼ X M i¼1 (p 2czi ) and since p=2 czi p=2, we have u(v) u(v) 0 jω – zi s plane jω zi ψzi σ jω jω – zi zi ψzi σ FIGURE 1.5 Zero-pole plots of minimum-phase and corresponding nonminimum-phase filter. 1-18 Passive, Active, and Digital Filters
- 40. or u(v) u(v) As a consequence, the phase response of the modified filter is equal to or greater than that of the original filter for all v. A frequently encountered requirement in the design of filters is that the delay characteristic be flat to within a certain tolerance within the passband(s) in order to achieve tolerable delay distortion, as was demonstrated in Section 1.6. In these and other filters in which the specifications include constraints on the phase response or delay characteristic, a nonminimum-phase transfer function is almost always required. 1.7.2 Allpass Filters An allpass filter is one that has a constant amplitude response. Consider a transfer function of the type given by Equation 1.13. From Equation 1.10, H(jv) is the complex conjugate of H(jv), and hence a constant amplitude response can be achieved if M2 (v) ¼ H(s)H(s)js¼jv ¼ H2 0 N(s) D(s) N(s) D(s) s¼jv ¼ H2 0 Hence, an allpass filter can be obtained if N(s) ¼ D(s) that is, the zeros of such a filter must be the mirror images of the poles and vice versa. A typical zero-pole plot for an allpass filter is illustrated in Figure 1.6. A second-order allpass transfer function is given by HAP(s) ¼ s2 bs þ c s2 þ bs þ c where b 0 for stability. As described previously, we can write M2 (v) ¼ HAP(s)HAP(s)js¼jv ¼ s2 bs þ c s2 þ bs þ c s2 þ bs þ c s2 bs þ c s¼jv ¼ 1 Allpass filters can be used to modify the phase responses of filters without changing their amplitude responses. Hence, they are used along with minimum- phase filters to obtain nonminimum-phase filters that satisfy amplitude and phase response specifications simultaneously. jω s plane FIGURE 1.6 Typical zero-pole plot of an allpass filter. General Characteristics of Filters 1-19
- 41. 1.7.3 Decomposition of Nonminimum-Phase Transfer Functions Some methods for the design of filters satisfying amplitude and phase response specifications, usually methods based on optimization, yield a nonminimum-phase transfer function. Such a transfer function can be easily decomposed into a product of a minimum-phase and an allpass transfer function, i.e., HN(s) ¼ HM(s)HAP(s) Consequently, a nonminimum-phase filter can be implemented as a cascade arrangement of a minimum- phase and an allpass filter. The preceding decomposition can be obtained by using the following procedure: 1. For each zero in the right-half s plane, augment the transfer function by a zero and a pole at the mirror image position of the zero. 2. Assign the left-half s-plane zeros and the original poles to the minimum-phase transfer function HM(s). 3. Assign the right-half s-plane zeros and the left-hand s-plane poles generated in step 1 to the allpass transfer function HAP(s). This procedure is illustrated in Figure 1.7. For example, if Minimum-phase filter Nonminimum-phase filter s plane Allpass filter FIGURE 1.7 Decomposition of nonminimum-phase transfer function. 1-20 Passive, Active, and Digital Filters
- 42. HN(s) ¼ s2 þ 4s þ 5 ð Þ s2 3s þ 7 ð Þ(s 5) s2 þ 2s þ 6 ð Þ s2 þ 4s þ 9 ð Þ(s þ 2) then, we can write HN(s) ¼ s2 þ 4s þ 5 ð Þ s2 3s þ 7 ð Þ(s 5) s2 þ 2s þ 6 ð Þ s2 þ 4s þ 9 ð Þ(s þ 2) s2 þ 3s þ 7 ð Þ(s þ 5) s2 þ 3s þ 7 ð Þ(s þ 5) Hence, HN(s) ¼ s2 þ 4s þ 5 ð Þ s2 þ 3s þ 7 ð Þ(s þ 5) s2 þ 2s þ 6 ð Þ s2 þ 4s þ 9 ð Þ(s þ 2) s2 3s þ 7 ð Þ(s 5) s2 þ 3s þ 7 ð Þ(s þ 5) or HN(s) ¼ HM(s)HAP(s) where HM(s) ¼ s2 þ 4s þ 5 ð Þ s2 þ 3s þ 7 ð Þ(s þ 5) s2 þ 2s þ 6 ð Þ s2 þ 4s þ 9 ð Þ(s þ 2) HAP(s) ¼ s2 3s þ 7 ð Þ(s 5) s2 þ 3s þ 7 ð Þ(s þ 5) 1.8 Introduction to the Design Process The design of filters starts with a set of specifications and ends with the implementation of a prototype. It comprises four general steps, as follows: 1. Approximation 2. Realization 3. Study of imperfections 4. Implementation 1.8.1 The Approximation Step The approximation step is the process of generating a transfer function that satisfies the desired specifications, which may concern the amplitude, phase, and possibly the time-domain response of the filter. The available methods for the solution of the approximation problem can be classified as closed-form or iterative. In closed-form methods, the problem is solved through a small number of design steps using a set of closed-form formulas or transformations. In iterative methods, an initial solution is assumed and, through the application of optimization methods, a series of progressively improved solutions are obtained until some design criterion is satisfied. Closed-form solutions are very precise and entail a minimal amount of computation. However, the available solutions are useful in applications where the loss characteristic is required to be piecewise constant to within some prescribed tolerances. Iterative methods, on the other hand, entail a considerable amount of computation but can be used to design filters General Characteristics of Filters 1-21
- 43. with arbitrary amplitude and phase response characteristics (see Ref. [1, Chapter 14]) for the application of these methods for the design of digital filters). Some classical closed-form solutions are the so-called Butterworth, Chebyshev, and elliptic* approximations to be described in Chapter 2 by A.M. Davis. In general, the designer is interested in simple and reliable approximation methods that yield precise designs with the minimum amount of computation. 1.8.2 The Realization Step The synthesis of a filter is the process of converting some characterization of the filter into a network. The process of converting the transfer function into a network is said to be the realization step and the network obtained is sometimes called the realization. The realization of a transfer function can be accomplished by expressing it in some form that allows the identification of an interconnection of elemental filter subnetworks and=or elements. Many realiza- tion methods have been proposed in the past that lead to structures of varying complexity and properties. In general, the designer is interested in realizations that are economical in terms of the number of elements, do not require expensive components, and are not seriously affected by variations in the element values such as may be caused by variations in temperature and humidity, and drift due to element aging. 1.8.3 Study of Imperfections During the approximation step, the coefficients of the transfer function are determined to a high degree of precision and the realization is obtained on the assumption that elements are ideal, i.e., capacitors are lossless, inductors are free of winding capacitances, amplifiers have infinite bandwidths, and so on. In practice, however, the filter is implemented with nonideal elements that have finite tolerances and are often nonlinear. Consequently, once a realization is obtained, sometimes referred to as a paper design, the designer must embark on the study of the effects of element imperfections. Several types of analysis are usually called for ranging from tolerance analysis, study of parasitics, time-domain analysis, sensi- tivity analysis, noise analysis, etc. Tight tolerances result in high-precision filters but the cost per unit would be high. Hence the designer is obliged to determine the highest tolerance that can be tolerated without violating the specifications of the filter throughout its working life. Sensitivity analysis is a related study that will ascertain the degree of dependence of a filter parameter, e.g., the dependence of the amplitude response on a specific element. If the loss characteristic of a filter is not very sensitive to certain capacitance, then the designer would be able to use a less precise and cheaper capacitor, which would, of course, decrease the cost of the unit. 1.8.4 Implementation Once the filter is thoroughly analyzed and found to meet the desired specifications under ideal conditions, a prototype is constructed and tested. Decisions to be made involve the type of components and packaging, and the methods are to be used for the manufacture, testing, and tuning of the filter. Problems may often surface at the implementation stage that may call for one or more modifications in the paper design. Then the realization and possibly the approximation may have to be redone. * To be precise, the elliptic approximation is not a closed-form method, since the transfer function coefficients are given in terms of certain infinite series. However, these series converge very rapidly and can be treated as closed-form formulas for most practical purposes (see Ref. [1, Chapter 5]). 1-22 Passive, Active, and Digital Filters
- 44. 1.9 Introduction to Realization Realization tends to depend heavily on the type of filter required. The realization of passive RLC filters differs quite significantly from that of active filters which, in turn, is entirely different from the realization of microwave filters. 1.9.1 Passive Filters Passive RLC filters have been the mainstay of communications since the 1920s and, furthermore, they continue to be of considerable importance today for frequencies in the 100–500 kHz range. The realization of passive RLC filters has received considerable attention through the years and it is, as a consequence, highly developed and sophisticated. It can be accomplished by using available filter- design packages such as FILSYN [5] and FILTOR [6]. In addition, several filter-design handbooks and published design tables are available [7–10]. The realization of passive RLC filters starts with a resistively terminated LC two-port network such as that in Figure 1.8. Then through one of several approaches, the transfer function is used to generate expressions for the z or y parameters of the LC two-port. The realization of the LC two-port is achieved by realizing the z or y parameters. The realization of passive filters is considered in Section I. 1.9.2 Active Filters Since the reactance of an inductor is vL, increased inductance values are required to achieve reason- able reactance values at low frequencies. For example, an inductance of 1 mH which will present a reactance of 6.28 kV at 1 MHz will present only 0.628 V at 100 Hz. Thus, as the frequency range of interest is reduced, the inductance values must be increased if a specified impedance level is to be maintained. This can be done by increasing the number of turns on the inductor coil and to some extent by using ferromagnetic cores of high permeability. Increasing the number of turns increases the resistance, the size, and the cost of the inductor. The resistance is increased because the length of the wire is increased [R ¼ (r 3 length)=Area], and hence the Q factor is reduced. The cost goes up because the cost of materials as well as the cost of labor go up, since an inductor must be individually wound. For these reasons, inductors are generally incompatible with miniaturization or microcircuit implementation. The preceding physical problem has led to the invention and development of a class of inductorless filters known collectively as active filters. Sensitivity considerations, which will be examined in Chapter 4 by I. Filanovsky, have led to two basic approaches to the design of active filters. In one approach, the active filter is obtained by simulating the inductances in a passive RLC filter or by realizing a signal flow graph of the passive RLC filter. In another approach, the active filter is obtained by cascading a number of low-order filter sections of some type, as depicted in Figure 1.9a where Zo0 is the output impedance of the signal source. R1 R2 Vi Vo LC network FIGURE 1.8 Passive RLC filter. General Characteristics of Filters 1-23
- 45. Each filter section is made up of an interconnection of resistors, capacitors, and active elements, and by Thévenin’s theorem, it can be represented by its input impedance, open-circuit voltage transfer function, and output impedance as shown in Figure 1.9b. The voltage transfer function of the configuration is given by H(s) ¼ Vo Vi and since the input voltage of section k is equal to the output voltage of section k 1, i.e., Vik ¼ Vo(k1) for k ¼ 2, 3, . . . , K, and Vo ¼ VoK we can write H(s) ¼ Vo Vi ¼ Vi1 Vi Vo1 Vi1 Vo2 Vi2 VoK ViK (1:23) where Vi1 Vi ¼ Zi1 Zo0 þ Zi1 (1:24) and Vok Vik ¼ Zi(kþ1) Zok þ Zi(kþ1) Hk(s) (1:25) is the transfer function of the kth section. From Equations 1.23 through 1.25, we obtain H(s) ¼ Vo Vi ¼ Zi1 Zo0 þ Zi1 Y K k¼1 Zi(kþ1) Zok þ Zi(kþ1) Hk(s) Now if Zik j j Zo(k1) j j Zo0 Vi Vo1 Vo2 ViK Vik Zik Zok Vok VoK = Vo Vi2 (a) (b) Hk(s) Vik Vi1 H2(s) H1(s) HK(s) FIGURE 1.9 (a) Cascade realization and (b) Thévenin equivalent circuit of filter section. 1-24 Passive, Active, and Digital Filters
- 46. for k ¼ 1, 2, . . . , K, then the loading effect produced by section k þ 1 on section k can be neglected and hence H(s) ¼ Vo Vi ¼ Y K k1 Hk(s) Evidently, a highly desirable property in active filter sections is that the magnitude of the input impedance be large and=or that of the output impedance be small since in such a case the transfer function of the cascade structure is equal to the product of the transfer functions of the individual sections. An arbitrary Nth-order transfer function obtained by using the Butterworth, Bessel, Chebyshev, inverse-Chebyshev, or elliptic approximation can be expressed as H(s) ¼ H0(s) Y K k¼1 a2ks2 þ a1ks þ a0k s2 þ b1ks þ b0k where H0(s) ¼ a10sþa00 b10sþb00 for odd N 1 for even N The first-order transfer function H0(s) for the case of an odd-order can be readily realized using the RC network of Figure 1.10. 1.9.3 Biquads From the above analysis, we note that all we need to be able to realize an arbitrary transfer function is a circuit that realizes the biquadratic transfer function HBQ(s) ¼ a2s2 þ a1s þ a0 s2 þ b1s þ b0 ¼ a2 s þ z1 ð Þ s þ z2 ð Þ s þ p1 ð Þ s þ p2 ð Þ (1:26) where zeros and poles occur in complex conjugate pairs, i.e., z2 ¼ z1* and p2 ¼ p1*. Such a circuit is commonly referred to as a biquad. Vi Vo C1 C2 G2 G1 FIGURE 1.10 First-order RC network. General Characteristics of Filters 1-25
- 47. After some manipulation, the transfer function in Equation 1.26 can be expressed as HBQ(s) ¼ K s2 þ 2Re z1 ð Þs þ Re z1 ð Þ2 þ Im z1 ð Þ2 s2 þ 2Re p1 ð Þs þ Re p1 ð Þ2 þ Im p1 ð Þ2 ¼ K s2 þ vz=Qz ð Þs þ v2 z s2 þ vp=Qp s þ v2 p where K ¼ a2 vz and vp are the zero and pole frequencies, respectively Qz and Qp are the zero and pole quality factors (or Q factors for short), respectively The formulas for the various parameters are as follows: vz ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Re z1 ð Þ2 þ Im z1 ð Þ2 q vp ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Re p1 ð Þ2 þ Im p1 ð Þ2 q Qz ¼ vz 2 Re z1 Qp ¼ vp 2 Re p1 The zero and pole frequencies are approximately equal to the frequencies of minimum gain and maximum gain, respectively. The zero and pole Q factors have to do with the selectivity of the filter. A high zero Q factor results in a deep notch in the amplitude response, whereas a high pole Q factor results in a very peaky amplitude response. The dc gain and the gain as v ! 1 in decibels are given by M0 ¼ 20 log HBQ(0) j j ¼ 20 log K v2 z v2 p ! and M1 ¼ 20 log HBQ( j1) j j ¼ 20 log K respectively. 1.9.4 Types of Basic Filter Sections Depending on the values of the transfer function coefficients, five basic types of filter sections can be identified, namely, low-pass, high-pass, bandpass, notch (sometimes referred to as bandreject), and allpass. These sections can serve as building blocks for the design of filters that can satisfy arbitrary specifications. They are actually sufficient for the design of all the standard types of filters, namely, Butterworth, Chebyshev, inverse-Chebyshev, and elliptic filters. 1.9.4.1 Low-Pass Section In a low-pass section, we have a2 ¼ a1 ¼ 0 and a0 ¼ Kv2 p. Hence, the transfer function assumes the form HLP(s) ¼ a0 s2 þ b1s þ b0 ¼ Kv2 p s2 þ vp=Qp s þ v2 p (see Figure 1.11a) 1-26 Passive, Active, and Digital Filters
- 48. 1.9.4.2 High-Pass Section In a high-pass section, we have a2 ¼ K and a1 ¼ a0 ¼ 0. Hence, the transfer function assumes the form HHP(s) ¼ a2s2 s2 þ b1s þ b0 ¼ Ks2 s2 þ vp=Qp s þ v2 p (see Figure 1.11b) 1.9.4.3 Bandpass Section In a bandpass section, we have a1 ¼ Kvp=Qp and a2 ¼ a0 ¼ 0. Hence the transfer function assumes the form HBP(s) ¼ a1s s2 þ b1s þ b0 ¼ K vp=Qp s s2 þ vp=Qp þ v2 p (see Figure 1.11c) s plane |H LP ( jω)| |H LP ( jω)| |H BP ( jω)| M K ωp ωM ωM ωp ωp ω ω M K 2 (a) (b) (c) M Qp 1 √2 K FIGURE 1.11 Basic second-order filter sections: (a) low-pass, (b) high-pass, (c) bandpass, (continued) General Characteristics of Filters 1-27
- 49. 1.9.4.4 Notch Section In a notch section, we have a2 ¼ K, a1 ¼ 0, and a0 ¼ Kv2 p. Hence, the transfer function assumes the form HN(s) ¼ a2s2 þ a0 s2 þ b1s þ b0 ¼ K s2 þ v2 z s2 þ vp=Qp s þ v2 p (see Figure 1.11d) 1.9.4.5 Allpass Section In an allpass section, we have a2 ¼ K, a1 ¼ Kwp=Qp, and a0 ¼ Kv2 p. Hence the transfer function assumes the form M2 M K K (d) (e) K K |HN(jω)| |HN(jω)| |HN(jω)| ≈ωM ≈ωM ωp = ωz ωz ωz ω2 p ωz ωp Qp ω ω ω K ω2 z ω2 p K √2 FIGURE 1.11 (continued) (d) notch, and (e) allpass. 1-28 Passive, Active, and Digital Filters
- 50. HAP(s) ¼ a2s2 þ a1s þ a0 s2 þ b1s þ b0 ¼ K s2 vp=Qp s þ v2 p h i s2 þ vp=Qp s þ v2 p (see Figure 1.11e) The design of active and switched-capacitor filters is treated in some detail in Section II. References 1. A. Antoniou, Digital Filters: Analysis, Design, and Applications, 2nd ed. New York: McGraw-Hill, 1993. 2. R. J. Schwarz and B. Friedland, Linear Systems, New York: McGraw-Hill, 1965. 3. E. Kreyszig, Advanced Engineering Mathematics, 3rd ed. New York: Wiley, 1972. 4. R. Schaumann, M. S. Ghausi, and K. R. Laker, Design of Analog Filters, Englewood Cliffs, NJ: Prentice Hall, 1990. 5. G. Szentirmai, FILSYN—A general purpose filter synthesis program, Proc. IEEE, 65, 1443–1458, Oct. 1977. 6. A. S. Sedra and P. O. Brackett, Filter Theory and Design: Active and Passive, Portland, OR: Matrix, 1978. 7. J. K. Skwirzynski, Design Theory and Data for Electrical Filters, London: Van Nostrand, 1965. 8. R. Saal, Handbook of Filter Design, Backnang: AEG Telefunken, 1979. 9. A. I. Zverev, Handbook of Filter Synthesis, New York: Wiley, 1967. 10. E. Chirlian, LC Filters: Design, Testing, and Manufacturing, New York: Wiley, 1983. General Characteristics of Filters 1-29
- 52. 2 Approximation Artice M. Davis San Jose State University 2.1 Introduction ................................................................................ 2-1 2.2 Butterworth LPP Approximation........................................... 2-8 2.3 Chebyshev LPP Approximation ........................................... 2-12 2.4 Bessel–Thompson LPP Approximation.............................. 2-18 2.5 Elliptic Approximation........................................................... 2-23 References............................................................................................ 2-33 2.1 Introduction The approximation problem for filters is illustrated in Figure 2.1. A filter is often desired to produce a given slope of gain over one or more frequency intervals, to remain constant over other intervals, and to completely reject signals having frequencies contained in still other intervals. Thus, in the example shown in the figure, the desired gain is zero for very low and very high frequencies. The centerline, shown dashed, is the nominal behavior and the shaded band shows the permissible variation in the gain characteristic. Realizable circuits must always generate smooth curves and so cannot exactly meet the piecewise linear specification represented by the centerline. Thus, the realizable behavior is shown by the smooth, dark curve that lies entirely within the shaded tolerance band. What type of frequency response function can be postulated that will meet the required specifications and, at the same time, be realizable: constructible with a specified catalog of elements? The answer depends upon the types of elements allowed. For instance, if one allows pure delays with a common delay time, summers, and scalar multipliers, a trigonometric polynomial will work; this, however, will cause the gain function to be repeated in a periodic manner. If this is permissible, one can then realize the filter in the form of an FIR digital filter or as a commensurate transmission line filter, and in fact, it can be realized in such a fashion that the resulting phase behavior is precisely linear. If one fits the required behavior with a rational trigonometric function, a function that is the ratio of two trigonometric polynomials, an economy of hardware will result. The phase, however, will unfortunately no longer be linear. These issues are discussed at greater length in Ref. [1]. Another option would be to select an ordinary polynomial in v as the approximating function. Polynomials, however, behave badly at infinity. They approach infinity as v ! 1, a highly undesirable solution. For this reason, one must discard polynomials. A rational function of v, however, will work nicely for the ratio of two polynomials will approach zero as v ! 1 if the degree of the numerator polynomial is selected to be of lower degree than that of the denominator. Furthermore, by the Weierstrass theorem, such a function can approximate any continuous function arbitrarily closely over any closed interval of finite length [2]. Thus, one sees that the rational functions in v offer a suitable approximation for analog filter design and, in fact, do not have the repetitive nature of the trigonometric rational functions. 2-1
- 53. Suppose, therefore, that the gain function is of the form A(v) ¼ N(v) D(v) ¼ a0 þ a1v þ a2v2 þ þ arvr b0 þ b1v þ b2v2 þ þ bqvq (2:1) where r q for reasons mentioned above. Assuming that the filter to be realized is constrained to be constructable with real* elements, one must require that A(v) ¼ A(v), that is, that the gain be an event function of frequency. But then, as it is straightforward to show, one must require that all the odd coefficients of both numerator and denominator be zero. This means that the gain is a function of v2 : A(v) ¼ N(v2 ) D(v2) ¼ a0 þ a1v2 þ þ amv2m b0 þ b1v2 þ þ bnv2n ¼ A(v2 ) (2:2) The expression has been reindexed and the constants redefined in an obvious manner. The net result is that one must approximate the desired characteristic by the ratio of two polynomials in v2 ; the objective is to determine the numerator and denominator coefficients to meet the stated specifications. Once this is accomplished one must compute the filter transfer function G(s) in order to synthesize the filter [4,5]. Assuming that G(s) is real (has real coefficients), then its complex conjugate satisfies G*(s) ¼ G(s*), from which it follows that G(s) is related to A(v2 ) by the relationship [G(s)G(s)]s¼jv ¼ G( jv)G*( jv) ¼ G( jv) j j2 ¼ A2 (v2 ) (2:3) In fact, it is more straightforward to simply cast the original approximation problem in terms of A2 (v2 ), rather than in terms of A(v). In this case, Equation 2.2 becomes A2 (v2 ) ¼ N(v2 ) D(v2) ¼ a0 þ a1v2 þ þ amv2m b0 þ b1v2 þ þ bnv2n (2:4) Thus, one can assume that the approximation process produces A2 (v2 ) as the ratio of two real polynomials in v2 . Since Equation 2.3 requires the substitutions s ! jv, one also has s2 ! v2 , and conversely. Thus, Equation 2.3 becomes G(s)G(s) ¼ A2 (s2 ) (2:5) Though this has been shown to hold only on the imaginary axis, it continues to hold for other complex values of s as well by analytic continuation.y A(ω) ω FIGURE 2.1 General approximation problem. * Complex filters are quite possible to construct, as recent work [3] shows. y A function analytic in a region is completely determined by its values along any line segment in that region—in this case, by its value along the jv axis. 2-2 Passive, Active, and Digital Filters
- 54. The problem now is to compute G(s) from Equation 2.5, a process known as the factorization problem. The solution is not unique; in fact, the phase is arbitrary—subject only to certain realizability conditions. To see this, just let G(jv) ¼ A(v)ejf(v) , where fv is an arbitrary phase function. Then, Equation 2.3 implies that G( jv)G*( jv) ¼ A(v)ejf(v) A(v)ejf(v) ¼ A2 (v) (2:6) If the resulting structure is to have the property of minimum phase [6], the phase function is determined completely by the gain function. If not, one can simply perform the factorization and accept whatever phase function results from the particular process chosen. As has been pointed out earlier in this chapter, it is often desirable that the phase be a linear function of frequency. In this case, one must follow the filter designed by the above process with a phase equalization filter, one that has constant gain and a phase characteristic that, when summed with that of the first filter, produces linear phase. As it happens, the human ear is insensitive to phase nonlinearity, so the phase is not of much importance for filters designed to operate in the audio range. For those intended for video applications, however, it is vitally important. Nonlinear phase produces, for instance, the phenomenon of multiple edges in a reproduced picture. If completely arbitrary gain characteristics are desired, computer optimization is necessary [6]. Indeed, if phase is of great significance, computer algorithms are available for the simultaneous approximation of both gain and phase. These are complex and unwieldy to use, however, so for more modest applications the above approach relying upon gain approximation only suffices. In fact, the approach arose histor- ically in the telephone industry in its earlier days in which voice transmission was the only concern, data and video transmission being unforeseen at the time. Furthermore, the frequency division multiplexing of voice signals was the primary concern; hence, a number of standard desired shapes of frequency response were generated: low pass, high pass, bandpass, and bandreject (or notch). Typical but stylized specification curves are shown in Figure 2.2. This figure serves to define the following parameters: the minimum passband gain Ap, the maximum stopband gain As, the passband cutoff frequency vp, the stopband cutoff frequency vs (the last two parameters are for low-pass and high-pass filters only), the center frequency vo, upper passband and stopband cutoff frequencies vpu and vsu, and lower passband and stopband cutoff frequencies vpl and vsl (the last four parameters are for the bandpass A(ω) A(ω) A(ω) A(ω) Ap 1 Ap Ap 1 1 Ap 1 ωp ωp ωo ωo ωpu ωsu ωsu ωpu ωs1ωp1 ωp1ωs1 ωs ωs ω ω ω ω As As As As As (a) (b) (d) (c) FIGURE 2.2 Catalog of basic filter types: (a) low pass, (b) high pass, (c) bandpass, and (d) bandreject. Approximation 2-3
- 55. Exploring the Variety of Random Documents with Different Content
- 56. De Blenau, however, was not long kept in suspense; for, in a few minutes, the door on the other side of the room opened, and the Swedish Ambassador passed out. The door shut behind him, but in a moment after an attendant entered, and although several others had been waiting before him, De Blenau was the first summoned to the presence of the Cardinal. He could not help feeling as if he wronged those he left still in doubt as to their fate: but following the officer through an ante-room, he entered the audience closet, and immediately perceived Richelieu seated at a table, over which were strewed a multitude of papers of different dimensions, some of which he was busily engaged in examining;—reading them he was not, for his eye glanced so rapidly over their contents, that his knowledge of each could be but general. He paused for a moment as De Blenau entered, bowed his head, pointed to a seat, and resumed his employment. When he had done, he signed the papers, and gave them to a dull-looking personage, in a black silk pourpoint, who stood behind his chair. “Take these three death-warrants,” said he, “to Monsieur Lafemas, and then these others to Poterie at the Bastille. But no—stop,” he continued after a moment’s thought; “you had better go to the Bastille first, for Poterie can put Caply to the torture, while you are gone to Lafemas; and you can bring me back his confession as you return.” De Blenau shuddered at the sang froid with which the Minister commanded those things that make one’s blood curdle even to imagine. But the attendant was practised in such commissions; and taking the packets, as a mere matter of course, he bowed in silence, and disappearing by a door on the other side, left De Blenau alone with the Cardinal. “Well, Monsieur de Blenau,” said Richelieu, looking up with a frank smile, “your pardon for having detained you. There are many things upon which I have long wished to speak to you, and this caused me to desire your company. But I have no doubt that we shall part perfectly satisfied with each other.” The Cardinal paused, as if for a reply. “I hope so too, my Lord,” said De Blenau. “I can, of course, have no cause to be dissatisfied with your Eminence; and for my own part, I feel my bosom to be clear.” “I doubt it not, Monsieur le Comte,” replied the Minister, with a gracious inclination of the head—“I doubt it not; I know your spirit to be too frank and noble to mingle in petty faction and treasonable cabal. No one more
- 57. admires your brave and independent bearing than myself. You must remember that I have marked you from your youth. You have been educated, as it were, under my own eye; and were it now necessary to trust the welfare of the State to the honour of any one man, I would confide it to the honour of De Blenau.” “To what, in the name of Heaven, can this lead?” thought De Blenau; but he bowed without reply, and the Cardinal proceeded. “I have, for some time past,” he continued, “been thinking of placing you in one of those high stations, to which your rank and consideration entitle you to aspire. At present, none are vacant; but as a forerunner to such advancement, I propose to call you to the Council, and to give you the government of Poitou.” De Blenau was now, indeed, astonished. The Cardinal was not a man to jest: and yet what he proposed, as a mere preliminary, was an offer that the first noble in France might have accepted with gladness. The Count was about to speak. But Richelieu paused only for a moment, to observe the effect of what he said upon his auditor; and perhaps over-rating the ambition of De Blenau, he proceeded more boldly. “I do not pretend to say, notwithstanding my sense of your high merit, and my almost parental feelings towards you, that I am wholly moved to this by my individual regard; but the truth is, that the State requires, at this moment, the services of one, who joins to high talents a thorough knowledge of the affairs of Spain.” “So!” thought De Blenau, “I have it now. The government of Poitou, and a seat at the Council, provided I betray the Queen and sell my own honour.” Richelieu seemed to wait an answer, and De Blenau replied: “If your Eminence means to attribute such knowledge to me, some one must have greatly misled you. I possess no information on the affairs of Spain whatever, except from the common reports and journals of the time.” This reply did not seem to affect Richelieu’s intentions. “Well, well, Monsieur de Blenau,” said he, with a smile, “you will take your seat at the Council, and will, of course, as a good subject and an honourable man, communicate to us whatever information you possess, on those points which concern the good of the State. We do not expect all at once; and every thing shall be done to smooth your way, and facilitate your views. Then, perhaps, if Richelieu live to execute the plans he has formed, you,
- 58. Monsieur de Blenau, following his path, and sharing his confidence, may be ready to take his place, when death shall at length call him from it.” The Cardinal counted somewhat too much on De Blenau’s ambition, and not sufficiently on his knowledge of the world; and imagining that he had, the evening before, discovered the weak point in the character of the young Count, he thought to lead him to any thing, by holding out to him extravagant prospects of future greatness. The dish, however, was somewhat too highly flavoured; and De Blenau replied, with a smile,— “Your Eminence is exceeding good to think at all of me, in the vast and more important projects which occupy your mind. But, alas! my Lord, De Blenau would prove but a poor successor to Richelieu.—No, my Lord Cardinal,” he continued, “I have no ambition; that is a passion which should be reserved for such great and comprehensive minds as yours. I am contented as I am. High stations are always stations of danger.” “I had heard that the Count de Blenau was no way fearful,” said Richelieu, fixing on him a keen and almost scornful glance. “Was the report a mistake? or is it lately he has become afraid of danger?” De Blenau was piqued, and lost temper. “Of personal danger, my Lord, I am never afraid,” replied he. “But when along with risk to myself is involved danger to my friends, danger to my country, danger to my honour, and danger to my soul,” and he returned the Cardinal’s glance full as proudly as it had been given, “then, my Lord Cardinal, I would say, it were no cowardice, but true courage to fly from such peril—unless,” he added, remembering the folly of opposing the irritable and unscrupulous Minister, and thinking that his words had, perhaps, been already too warm—“unless, indeed, one felt within one’s breast the mind of a Richelieu.” While De Blenau spoke, the Cardinal’s brow knitted into a frown. A flush too came over his cheek; and untying the ribbon which served as a fastening, he took off the velvet cap he generally wore, as if to give himself air. He heard him, however, to the end, and then answered drily, “You speak well, Monsieur de Blenau, and, I doubt not, feel what you say. But am I to understand you, that you refuse to aid us at the Council with your information and advice?” “So far, your Eminence is right,” replied the Count, who saw that the storm was now about to break upon his head; “I must, indeed, decline the honours which you offer with so bountiful a hand. But do not suppose that I
- 59. do so from unwillingness to yield you any information; for, truly, I have none to give. I have never meddled with politics. I have never turned my attention to State affairs; and therefore still less could I yield you any advice. Your Eminence would be woefully disappointed, when you expected to find a man well acquainted with the arts of government, and deep read in the designs of foreign states, to meet with one, whose best knowledge is to range a battalion, or to pierce a boar; a soldier, and not a diplomatist; a hunter, and not a statesman. And as to the government of Poitou, my Lord, its only good would be the emolument, and already my revenues are far more than adequate to my wants.” “You refuse my kindness, Sir,” replied the Cardinal, with an air of deep determined haughtiness, very different from the urbanity with which he had at first received De Blenau; “I must now speak to you in another tone. And let me warn you to beware of what you say; for be assured, that I already possess sufficient information to confound you if you should prevaricate.” “My Lord Cardinal,” replied De Blenau, somewhat hastily, “I am not accustomed to prevaricate. Ask any questions you please, and, so long as my honour and my duty go with them, I will answer you.” “Then there are questions,” said the Cardinal, “that you would think against your duty to answer?” “I said not so, your Eminence,” replied De Blenau. “In the examination I find I am to undergo, give my words their full meaning, if you please, but no more than their meaning.” “Well then, Sir, answer me as a man of honour and a French noble,” said the Cardinal—“Are you not aware of a correspondence that has been, and is now, carried on between Anne of Austria and Don Francisco de Mello, Governor of the Low Countries?” “I know not whom you mean, Sir, by Anne of Austria,” replied De Blenau. “If it be her Majesty, your Queen and mine, that you so designate, I reply at once that I know of no such correspondence, nor do I believe that it exists.” “Do you mean to say, Monsieur de Blenau,” demanded the Cardinal, fixing his keen sunken eyes upon the young Count with that basilisk glance for which he was famous—“Do you mean to say, that you yourself have not forwarded letters from the Queen to Madame de Chevreuse, and Don Francisco de Mello, by a private channel?—Pause, Monsieur de Blenau,
- 60. before you answer, and be well assured that I am acquainted with every particular of your conduct.” “Your Eminence is, no doubt, acquainted with much more intricate subjects than any of my actions,” replied the Count. “With regard to Madame de Chevreuse, her Majesty has no need to conceal a correspondence with her, which has been fully permitted and sanctioned, both by your Eminence and the still higher authority of the King; and I may add, that to my certain knowledge, letters have gone to that lady by your own courier. On the other point, I have answered already; and have only to say once more, that I know of no such correspondence, nor would I, assuredly, lend myself to any such measures, which I should conceive to be treasonable.” “I have always hitherto supposed you to be a man of honour,” said the Cardinal coolly; “but what must I conceive now, Monsieur le Comte, when I tell you that I have those very letters in my possession?” “You may conceive what you please, Sir,” replied De Blenau, giving way to his indignation; “but I will dare any man to lay before me a letter from her Majesty to the person you mention, which has passed through the hands of De Blenau.” The Cardinal did not reply, but opening an ebony cabinet, which stood on his right hand, he took from one of the compartments a small bundle of papers, from which he selected one, and laid it on the table before the Count, who had hitherto looked on with no small wonder and expectation. “Do you know that writing, Sir?” demanded the Cardinal, still keeping his hand upon the paper, in such a manner as to allow only a word or two to be visible. De Blenau examined the line which the Cardinal suffered to appear, and replied—“From what little I can see, I should imagine it to be the hand- writing of her Majesty. But that does not show that I have any thing to do with it.” “But there is that in it which does,” answered Richelieu, folding down a line or two of the letter, and pointing out to the Count a sentence which said, “This will be conveyed to you by the Count de Blenau, who you know never fails.” “Now, Sir!” continued the Cardinal, “once more let me advise you to give me all you possess upon this subject. From a feeling of personal
- 61. regard, I have had too much patience with you already.” “All I can reply to your Eminence,” answered the Count, not a little embarrassed, “is, that no letter whatever has been conveyed by me, knowingly, to the Governor of the Low Countries.” De Blenau’s eyes naturally fixed on the paper, which still lay on the table, and from which the Cardinal had by this time withdrawn his hand; and feeling that both life and honour depended upon that document, he resolved to ascertain its authenticity, of which he entertained some doubt. “Stop,” said he hastily, “let me look at the superscription,” and before Richelieu could reply, he had raised it from the table and turned to the address. One glance was enough to satisfy him, and he returned it to the Cardinal with a cool and meaning smile, repeating the words—“To Madame de Chevreuse.” At first the Cardinal had instinctively stretched out his hand to stop De Blenau in his purpose, but he instantly recovered himself, nor did his countenance betray the least change of feeling. “Well, Sir,” replied he, “you said that you would dare any one to lay before you a letter from the Queen to the person I mentioned. Did I not mention Madame de Chevreuse, and is not there the letter?” “Your Eminence has mistaken me,” replied De Blenau, bowing his head, and smiling at the Minister’s art; “I meant, Don Francisco de Mello. I had answered what you said in regard to Madame de Chevreuse, before.” “I did mistake you then, Sir,” said the Cardinal; “but it was from the ambiguity of your own words. However, passing over your boldness, in raising that letter without my permission; I will show you that I know more of your proceedings than you suspect. I will tell you the very terms of the message you sent to the Queen, after you were wounded in the wood of Mantes, conveying to her, that you had not lost the packet with which you were charged. Did not Seguin tell her, on your part, that though the wound was in your side, your heart was not injured?” “I dare say he did, my Lord,” replied De Blenau, coolly; “and the event has proved that he was quite right, for your Eminence must perceive that I am quite recovered, which, of course, could not have been the case, had any vital part been hurt. But I hope, your Eminence, that there is no offence, in your eyes, either in having sent the Queen, my mistress, an account of my health, or in having escaped the attack of assassins.”
- 62. A slight flush passed over Richelieu’s cheek. “You may chance to fall into less scrupulous hands than even their’s,” replied he. “I am certainly informed, Sir, that you, on the part of the Queen, have been carrying on a treasonable intercourse with Spain—a country at war with France, to whose crown you are a born subject and vassal; and I have to tell you, that the punishment of such a crime is death. Yes, Sir, you may knit your brow. But no consideration shall stay me from visiting, with the full severity of the law, such as do so offend; and though the information I want be but small, depend upon it, I shall not hesitate to employ the most powerful means to wring it from you.” De Blenau had no difficulty in comprehending the nature of those means, to which the Cardinal alluded; but his mind was made up to suffer the worst. “My Lord Cardinal,” replied he, “what your intentions are, I know not; but be sure, that to whatever extremes you may go, you can wring nothing from me but what you have already heard. I once more assure you, that I know of no treasonable correspondence whatsoever; and firm in my own innocence, I equally despise all attempts to bribe or to intimidate me.” “Sir, you are insolent!” replied the Cardinal rising: “Use no such language to me!—Are you not an insect I can sweep from my path in an instant? Ho, a guard there without! We shall soon see, whether you know aught of Philip of Spain.” Had the Cardinal’s glance been directed towards De Blenau, he would have seen, that at the name of Philip of Spain, a degree of paleness came over his cheek; but another object had caught Richelieu’s eye, and he did not observe it. It was the entrance of the attendant whom he had despatched with the death-warrants, which now drew his notice; and well pleased to show De Blenau the dreadful means he so unscrupulously employed to extort confession from those he suspected, he eagerly demanded, “What news?” “May it please your Eminence,” said the attendant, “Caply died under the torture. In truth, it was soon over with him, for he did not bear it above ten minutes.” “But the confession, the confession!” exclaimed Richelieu. “Where is the procès verbal?”
- 63. “He made no confession, Sir,” replied the man. “He protested, to the last, his innocence, and that he knew nothing.” “Pshaw!” said Richelieu; “they let him die too soon; they should have given him wine to keep him up. Foolish idiot,” he continued, as if meditating over the death of his victim; “had he but told what he was commanded, he would have saved himself from a death of horror. Such is the meed of obstinacy.” “Such,” thought De Blenau, “is, unhappily, often the reward of firmness and integrity. But such a death is honourable in itself.” No one could better read in the face what was passing in the mind than Richelieu, and it is probable that he easily saw in the countenance of De Blenau, the feelings excited by what had just passed. He remembered also the promise given by Chavigni; and if, when he called the Guard, he had ever seriously proposed to arrest De Blenau, he abandoned his intention for the moment. Not that the high tone of the young Count’s language was either unfelt, or forgiven, for Richelieu never pardoned; but it was as easy to arrest De Blenau at St. Germain’s as in Paris; and the wily Minister calculated, that by giving him a little liberty, and throwing him off his guard, he might be tempted to do those things which would put him more completely in the power of the government, and give the means of punishing him for his pride and obstinacy, as it was internally termed by a man long unaccustomed to any opposition. De Blenau was principally obnoxious to the Cardinal, as the confidant of the Queen, and from being the chief of her adherents both by his rank, wealth, and reputation. Anne of Austria having now become the only apparent object which could cloud the sky of Richelieu’s political power, he had resolved either to destroy her, by driving her to some criminal act, or so to entangle her in his snares, as to reduce her to become a mere instrument in his hands and for his purposes. To arrest De Blenau would put the Queen upon her guard; and therefore, the Minister, without hesitation, resolved to dissemble his resentment, and allow the Count to depart in peace; reserving for another time the vengeance he had determined should overtake him at last. Nor was his dissembling of that weak nature which those employ, who have all the will to deceive, without the art of deceiving. Richelieu walked rapidly up and down the closet for a moment, as if striving to repress some strong emotion, then stopped, and turning to De
- 64. Blenau with some frankness of manner, “Monsieur le Comte,” said he, “I will own that you have heated me,—perhaps I have given way to it too much. But you ought to be more careful of your words, Sir, and remember that with men whose power you cannot resist, it is sometimes dangerous even to be in the right, much more to make them feel it rudely. However, it is all past, and I will now detain you no longer; trusting to your word, that the information which I have received, is without foundation. Let me only add, that you might have raised yourself this day to a height which few men in France would not struggle to attain. But that is past also, and may, perhaps, never return.” “I am most grateful, believe me,” replied De Blenau, “for all the favours your Eminence intended me; and I have no doubt, that you will soon find some other person, on whom to bestow them, much more worthy of them than myself.” Richelieu bowed low, and fixed his eyes upon the Count without reply— a signal that the audience was over, which was not lost upon De Blenau, who very gladly took his leave of the Minister, hoping most devoutly never to see his face again. The ambiguity of his last sentence, however, had not escaped the Cardinal. “So, Monsieur de Blenau!” said he, as soon as the Count had left him, “you can make speeches with a double meaning also! Can you so? You may rue it though, for I will find means to bend your proud spirit, or to break it; and that before three days be over. Is every thing prepared for my passage to Chantilly?” he continued, turning to the attendant. “All is prepared, please your Eminence,” replied the man; “and as I passed, I saw Monsieur de Chavigni getting into his chaise to set out.” “We will let him be an hour or two in advance,” said the Cardinal. “Send in the Marquis de Goumont;” and he again applied himself to other affairs.
- 65. CHAPTER XII. “An entire new comedy, with new scenery, dresses, and decorations.” THE little village of Mesnil St. Loup, all insignificant as it is, was at the time of my tale a place of even less consequence than it appears now-a- days, when nine people out of ten have scarcely ever heard of its existence. It was, nevertheless, a pretty-looking place; and had its little auberge, on the same scale and in the same style as the village to which it belonged,— small, neat, and picturesque, with its high pole before the door, crowned with a gay garland of flowers, which served both for sign and inscription to the inn; being fully as comprehensible an intimation to the peasantry of the day, that “Bon vin et bonne chère” were to be obtained within, as the most artful flourish of a modern sign-painter. True it is, that the little cabaret of Mesnil St. Loup was seldom troubled with the presence of a traveller; but there the country people would congregate after the labours of the day, and enjoy their simple sports with a relish that luxury knows not. The high road from Paris to Troyes passed quite in another direction; and a stranger in Mesnil St. Loup was a far greater stranger than he could possibly have been anywhere else, except perhaps in newly discovered America. For there was nothing to excite either interest or curiosity; except it were the little church, which had seen many a century pass over its primitive walls, remaining still unaltered, while five or six old trees, which had been its companions for time out of mind, began to show strong signs of decay, in their rifted bark and falling branches, but still formed a picturesque group, with a great stone cross and fountain underneath them, and a seat for the weary traveller to rest himself in their shade. Thus, Mesnil St. Loup was little known to strangers, for its simplicity had no attractions for the many. Nevertheless, on one fine evening, somewhere about the beginning of September, the phenomenon of a new face showed itself at Mesnil St. Loup. The personage to whom it appertained, was a horseman of small mean appearance, who, having passed by the church, rode through the village to the auberge, and having raised his eyes to the garland over the door, he divined from it, that he
- 66. himself would find there good Champagne wine, and his horse would meet with entertainment equally adapted to his peculiar taste. Thereupon, the stranger alighted and entered the place of public reception, without making any of that bustle about himself, which the landlord seemed well inclined to do for him; but on the contrary sat himself down in the most shady corner, ordered his bottle of wine, and inquired what means the house afforded of satisfying his hunger, in a low quiet tone of voice, which reached no farther than the person he addressed. “As for wine,” the host replied, “Monsieur should have such wine that the first merchant of Epernay might prick his ears at it; and in regard to eatables, what could be better than stewed eels, out of the river hard by, and a civet de lievre?—Monsieur need not be afraid,” he added; “it was a real hare he had snared that morning himself, in the forest under the hill. Some dishonourable innkeepers,” he observed—“innkeepers unworthy of the name, would dress up cats and rats, and such animals, in the form of hares and rabbits; even as the Devil had been known to assume the appearance of an Angel of light; but he scorned such practices, and could not only show his hare’s skin, but his hare in the skin. Farther, he would give Monsieur an ortolan in a vine leaf, and a dish of stewed sorrel.” The stranger underwent the innkeeper’s oration with most exemplary patience, signified his approbation of the proposed dinner, without attacking the hare’s reputation; and when at length it was placed before him, he ate his meal and drank his wine, in profound silence, without a word of praise or blame to either one or the other. The landlord, with all his sturdy loquacity, failed in more than one attempt to draw him into conversation; and the hostess, though none of the oldest or ugliest, could scarce win a syllable from his lips, even by asking if he were pleased with his fare. The taciturn stranger merely bowed his head, and seemed little inclined to exert his oratorical powers, more than by the simple demand of what he wanted; so that both mine host and hostess gave him up in despair—the one concluding that he was “an odd one,” and the other declaring that he was as stupid as he was ugly. This lasted some time, till one villager after another, having exhausted every excuse for staying to hear whether the stranger would open his lips, dropped away in his turn, and left the apartment vacant. It was then, and not till then, that mine host was somewhat surprised, by hearing the silent traveller pronounce in a most audible and imperative manner, “Gaultier,
- 67. come here.” The first cause of astonishment was to hear him speak at all; and the next to find his own proper name of Gaultier so familiar to the stranger, forgetting that it had been vociferated at least one hundred times that night in his presence. However, Gaultier obeyed the summons with all speed, and approaching the stranger with a low reverence, begged to know his good will and pleasure. “Your wine is good, Gaultier,” said the stranger, raising his clear grey eyes to the rosy round of Gaultier’s physiognomy. Even an innkeeper is susceptible of flattery; and Gaultier bent his head down towards the ground, as if he were going to do kou-tou. “Gaultier, bring me another bottle,” said the stranger. This phrase was better than the former; that sort of substantial flattery that goes straight to an innkeeper’s heart. Truly, it is a pity that innkeepers are such selfish beings. And yet it is natural too;—so rapidly does mankind pass by them, that theirs can be, at best, but a stage-coach sort of affection for their fellow-creatures —The coachman shuts the door—Drive on!—and it is all over. Thus, my dear Sir, the gaieties, the care, and the bustle in which you and I live, render our hearts but as an inn, where many a traveller stays for an hour, pays his score, and is forgotten.—I am resolved to let mine upon lease.—— The bottle of wine was not long in making its appearance; and as Gaultier set it on the table before the stranger, he asked if he could serve him farther. “Can you show me the way to the old Chateau of St. Loup?” demanded the stranger. “Surely, I can, Sir,” replied the innkeeper; “that is to say, as far as knowing where it is. But I hope Monsieur does not mean to-night.” “Indeed do I,” answered the stranger; “and pray why not? The night is the same as the day to an honest man.” “No doubt, no doubt!” exclaimed Gaultier, with the greatest doubt in the world in his own mind.—“No doubt! But, Holy Virgin! Jesu preserve us!”—and he signed the cross most devoutly—“we all know that there are spirits, and demons, and astrologers, and the Devil, and all those sort of things; and I would not go through the Grove where old Père Le Rouge, the sorcerer, was burnt alive, not to be prime minister, or the Cardinal de Richelieu, or any other great man,—that is to say, after nightfall. In the day I would go anywhere, or do any thing,—I am no coward, Sir,—I dare do
- 68. any thing. My father served in the blessed League against the cursed Huguenots—so I am no coward;—but bless you, Sir, I will tell you how it happened, and then you will see—” “I know all about it,” replied the stranger, in a voice that made the innkeeper start, and look over his left shoulder; “I know all about it; but sit down and drink with me, to keep your spirits up, for you must show me the way this very night. Père Le Rouge was a dear friend of mine, and before he was burnt for a sorcerer, we had made a solemn compact to meet once every ten years. Now, if you remember aright, it is just ten years, this very day, since he was executed; and there is no bond in Hell fast enough to hold him from meeting me to-night at the old chateau. So sit you down and drink!”— And he poured out a full cup of wine for the innkeeper, who looked aghast at the portentous compact between the stranger and Père Le Rouge. However, whether it was that Gaultier was too much afraid to refuse, or had too much esprit de corps not to drink with any one who would drink with him, can hardly be determined now; but so it was, that sitting down, according to the stranger’s desire, he poured the whole goblet of wine over his throat at one draught, and, as he afterwards averred, could not help thinking that the stranger must have enchanted the liquor, for no sooner had he swallowed it, than all his fears of Père Le Rouge began to die away, like morning dreams. However, when the goblet was drained, Gaultier began more justly to estimate the danger of drinking with a sorcerer; and that the stranger was such, a Champenois aubergiste of 1642 could never be supposed to doubt, after the diabolical compact so unscrupulously confessed. Under this impression, he continued rolling his empty cup about upon the table, revolving at the same time his own critical situation, and endeavouring to determine what might be his duty to his King and Country under such perilous circumstances. Rolling the cup to the right—he resolved instantly to denounce this malignant enchanter to the proper authorities, and have him forthwith burnt alive, and sent to join Père Le Rouge in the other world, by virtue of the humane and charitable laws in that case especially made and provided. Then rolling the cup to the other side—his eye glanced towards the stranger’s bottle, and resting upon the vacuum which their united thirst had therein occasioned, his heart over- flowed with the milk of human kindness, and he pitied from his soul that perverted taste which could lead any human being from good liquor, comfortable lodging, and the society of an innkeeper, to a dark wood and a
- 69. ruined castle, an old roasted sorcerer, and the Devil perhaps into the bargain. “Would you choose another bottle, Sir?” demanded Gaultier; and as his companion nodded his head in token of assent, was about to proceed on this errand—with the laudable intention also of sharing all his newly arisen doubts and fears with his gentle help-mate, who, for her part, was busily engaged in the soft domestic duties of scolding the stable-boy and boxing the maid’s ears. But the stranger stopped him, perhaps divining, and not very much approving, the aforesaid communication. He exclaimed, “La Bourgeoise!” in a tone of voice which overpowered all other noises: the abuse of the dame herself—the tears of the maid—the exculpation of the stable-boy—the cackle of the cocks and hens, which were on a visit in the parlour—and the barking of a prick-eared cur included. The fresh bottle soon stood upon the table; and while the hostess returned to her former tender avocations, the stranger, whose clear grey eye seemed reading deeply into Gaultier’s heart, continued to drink from the scanty remains of his own bottle, leaving mine host to fill from that which was hitherto uncontaminated by any other touch than his own. This Gaultier did not fail to do, till such time as the last rays of the sun, which had continued to linger fondly amidst a flight of light feathery clouds overhead, had entirely left the sky, and all was grey. At that moment the stranger drew forth his purse, let it fall upon the table with a heavy sort of clinking sound, showing that the louis-d’ors within had hardly room to jostle against each other. It was a sound of comfortable plenty, which had something in it irresistibly attractive to the ears of Gaultier; and as he stood watching while the stranger insinuated his finger and thumb into the little leathern bag, drawing forth first one broad piece and then another, so splendid did the stranger’s traffic with the Devil begin to appear in the eyes of the innkeeper, that he almost began to wish that he had been brought up a sorcerer also. The stranger quietly pushed the two pieces of gold across the table till they got within the innkeeper’s sphere of attraction, when they became suddenly hurried towards him, with irresistible velocity, and were plunged into the abyss of a large pocket on his left side, close upon his heart. The stranger looked on with philosophic composure, as if considering some natural phenomenon, till such time as the operation was complete.
- 70. “Now, Gaultier,” cried he, “put on your beaver, and lead to the beginning of the Grove. I will find my way through it alone. But hark ye, say no word to your wife.” Gaultier was all complaisance, and having placed his hat on his head, he opened the door of the auberge, and brought forth the stranger’s horse, fancying that what with a bottle of wine, and two pieces of gold, he could meet Beelzebub himself, or any other of those gentlemen of the lower house, with whom the Curé used to frighten the little boys and girls when they went to their first communion. However, the stranger had scarcely passed the horse’s bridle over his arm, and led him a step or two on the way, when the cool air and reflection made the innkeeper begin to think differently of the Devil, and be more inclined to keep at a respectful distance from so grave and antique a gentleman. A few steps more made him as frightened as ever; and before they had got to the end of the village, Gaultier fell hard to work, crossing himself most laboriously, and trembling every time he remembered that he was conducting one sorcerer to meet another, long dead and delivered over in form, with fire and fagot, into the hands of Satan. It is probable that he would have run, but the stranger was close behind, and cut off his retreat. At about a mile and a half from the little village of Mesnil, stood the old Chateau of St. Loup, situated upon an abrupt eminence, commanding a view of almost all the country round. The valley at its foot, and the slope of the hill up to its very walls, were covered with thick wood, through which passed the narrow deserted road from Mesnil, winding in and out with a thousand turns and divarications, and twice completely encircling the hill itself, before it reached the castle gate, which once, in the hospitable pride of former days, had rested constantly open for the reception equally of the friend and the stranger, but which now only gave entrance to the winds and tempests—rude guests, that contributed, even more than Time himself, the great destroyer, to bring ruin and desolation on the deserted mansion. Hard by, in a little cemetery, attached to the Chapel, lay many of the gay hearts that had once beat there, now quiet in the still cold earth. There, mouldering like the walls that overshadowed them, were the last sons of the brave and noble race of Mesnil, without one scion left to dwell in the halls of their forefathers, or to grieve over the desolation of their heritage. There, too, lay the vassals, bowed to the will of a sterner Lord, and held in the surer
- 71. bondage of the tomb; and yet perhaps, in life, they had passed on, happier than their chief, without his proud anxiety and splendid cares; and now, in death, his bed was surely made as low, and the equal wind that whispered over the grave of the one, offered no greater flattery to the monument of the other. But, beyond all these, and removed without the precincts of consecrated ground, was a heap of shards and flints—the Sorcerer’s grave! Above it, some pious hand had raised the symbol of salvation—a deed of charity, truly, in those days, when eternal mercy was farmed by the Church, like a turnpike on the high road, and none could pass but such as paid toll. But, however, there it rose,—a tall white cross, standing, as that symbol should always stand, high above every surrounding object, and full in view of all who sought it. As the aubergiste and his companion climbed the hill, which, leading from the village of Mesnil, commanded a full prospect of the rich woody valley below, and overhung that spot which, since the tragedy of poor Père Le Rouge, had acquired the name of the Sorcerer’s Grove, it was this tall white cross that first caught their attention. It stood upon the opposite eminence, distinctly marked on the back-ground of the evening sky, catching every ray of light that remained, while behind it, pile upon pile, lay the thick clouds of a coming storm. “There, Monsieur,” cried Gaultier, “there is the cross upon the Sorcerer’s grave!” And the fear which agitated him while he spoke, made the stranger’s lip curl into a smile of bitter contempt. But as they turned the side of the hill, which had hitherto concealed the castle itself from their sight, the teeth of Gaultier actually chattered in his head, when he beheld a bright light shining from several windows of the deserted building. “There!” exclaimed the stranger, “there, you see how well Père Le Rouge keeps his appointment. I am waited for, and want you no farther. I can now find my way alone. I would not expose you, my friend, to the dangers of that Grove.” The innkeeper’s heart melted at the stranger’s words, and he was filled with compassionate zeal upon the occasion. “Pray don’t go,” cried Gaultier, almost blubbering betwixt fear and tender-heartedness; “pray don’t go! Have pity upon your precious soul! You’ll go to the Devil, indeed you will! —or at least to purgatory for a hundred thousand years, and be burnt up like an overdone rabbit. You are committing murder, and conspiracy, and
- 72. treason,”—the stranger started, but Gaultier went on—“and heresy, and pleurisy, and sorcery, and you will go to the Devil, indeed you will—and then you’ll remember what I told you.” “What is fated, is fated!” replied the stranger, in a solemn voice, though Gaultier’s speech had produced that sort of tremulous tone, excited by an inclination either to laugh or to cry. “I have promised, and I must go. But let me warn you,” he continued, sternly, “never to mention one word of what has passed to-night, if you would live till I come again. For if you reveal one word, even to your wife, the ninth night after you have done so, Père Le Rouge will stand on one side of your bed, and I on the other, and Satan at your feet, and we will carry you away body and soul, so that you shall never be heard of again.” When he had concluded, the stranger waited for no reply, but sprang upon his horse, and galloped down into the wood. In the mean time, the landlord climbed to a point of the hill, from whence he could see both his own village, and the ruins of the castle. There, the sight of the church steeple gave him courage, and he paused to examine the extraordinary light which proceeded from the ruin. In a few minutes, he saw several figures flit across the windows, and cast a momentary obscurity over the red glare which was streaming forth from them upon the darkness of the night. “There they are!” cried he, “Père Le Rouge, and his pot companion!—and surely the Devil must be with them, for I see more than two, and one of them has certainly a tail—Lord have mercy upon us!” As he spoke, a vivid flash of lightning burst from the clouds, followed instantly by a tremendous peal of thunder. The terrified innkeeper startled at the sound, and more than ever convinced that man’s enemy was on earth, took to his heels, nor ceased running till he reached his own door, and met his better angel of a wife, who boxed his ears for his absence, and vowed he had been gallanting. END OF THE FIRST VOLUME. LONDON: PRINTED BY S. AND R. BENTLEY, Dorset Street, Fleet Street.
- 73. Typographical errors corrected by the etext transcriber: shas ent to inquire= has sent to inquire {pg 115} Frontrailles= Fontrailles {pg 163} Gualtier= Gaultier {pg 283}
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