z transforms

SIGNAL
► Signal is a physical quantity that varies with
respect to time , space or any other
independent variable
Eg x(t)= sin t.
► the major classifications of the signal
are:
(i) Discrete time signal
(ii) Continuous time signal

Unit Step &Unit Impulse
 Discrete time Unit impulse is defined as
δ [n]= {0, n≠ 0
{1, n=0
Unit impulse is also known as unit sample.
Discrete time unit step signal is defined by
U[n]={0,n=0
{1,n>= 0
Continuous time unit impulse is defined as
δ (t)={1, t=0
{0, t ≠ 0
Continuous time Unit step signal is defined as
U(t)={0, t<0
{1, t≥0

SIGNAL

► Periodic Signal & Aperiodic Signal
 A signal is said to be periodic ,if it exhibits periodicity.i.e.,
X(t +T)=x(t), for all values of t. Periodic signal has the
property that it is unchanged by a time shift of T. A
signal that does not satisfy the above periodicity property
is called an aperiodic signal
► even and odd signal ?
 A discrete time signal is said to be even when, x[-n]=x[n].
The continuous time signal is said to be even when, x(-
t)= x(t) For example,Cosωn is an even signal.

Energy and power signal

► A signal is said to be energy signal if it
have finite energy and zero power.
► A signal is said to be power signal if it
have infinite energy and finite power.
► If the above two conditions are not
satisfied then the signal is said to be
neigther energy nor power signal

Fourier Series
The Fourier series represents a periodic signal in terms of
frequency components:
p −1 ∞
ikω0n ikω0 t
x(n) = ∑ Xk e x( t ) = ∑ Xk e
k =0 k = −∞
We get the Fourier series coefficients as follows:

1 p −1 1p −ikω0 t
Xk = ∑ x(n)e −ikω0n Xk = ∫ x( t )e dt
p n =0 p0
The complex exponential Fourier coefficients are a sequence of
complex numbers representing the frequency component ω0k.

Fourier series

► Fourier series: a complicated waveform analyzed into a
number of harmonically related sine and cosine functions

► A continuous periodic signal x(t) with a period T0 may be
represented by:
 X(t)=Σ∞k=1 (Ak cos kω t + Bk sin kω t)+ A0

► Dirichlet conditions must be placed on x(t) for the series
to be valid: the integral of the magnitude of x(t) over a
complete period must be finite, and the signal can only
have a finite number of discontinuities in any finite
interval

Trigonometric form for Fourier series
► If the two fundamental components of a
periodic signal areB1cosω0t and C1sinω0t,
then their sum is expressed by trigonometric
identities:
► X(t)= A0 + Σ∞k=1 ( Bk 2+ Ak 2)1/2 (Ck cos kω t- φk)
or
► X(t)= A0 + Σ∞k=1 ( Bk 2+ Ak 2)1/2 (Ck sin kω t+ φk)

Fourier Transform

► Viewed periodic functions in terms of frequency components (Fourier
series) as well as ordinary functions of time
► Viewed LTI systems in terms of what they do to frequency
components (frequency response)
► Viewed LTI systems in terms of what they do to time-domain signals
(convolution with impulse response)
► View aperiodic functions in terms of frequency components via
Fourier transform
► Define (continuous-time) Fourier transform and DTFT
► Gain insight into the meaning of Fourier transform through
comparison with Fourier series

The Fourier Transform
►A transform takes one function (or signal)
and turns it into another function (or signal)
► Continuous Fourier Transform:

∞
H ( f ) = ∫ h ( t ) e 2πift dt
−∞
∞
h ( t ) = ∫ H ( f ) e −2πift df
−∞

Continuous Time Fourier Transform
We can extend the formula for continuous-time Fourier series
coefficients for a periodic signal

1p −ikω0 t 1 p/2
Xk = ∫ x( t )e dt = ∫ x( t )e −ikω0 t dt
p0 p −p / 2
to aperiodic signals as well. The continuous-time Fourier
series is not defined for aperiodic signals, but we call the
formula
∞
X(ω) = ∫ x( t )e −iωt dt
the (continuous time)
−∞ Fourier transform.

Inverse Transforms
If we have the full sequence of Fourier coefficients for a periodic
signal, we can reconstruct it by multiplying the complex
sinusoids of frequency ω0k by the weights Xk and summing:
p −1 ∞
ikω0n ikω0 t
x(n) = ∑ Xk e x( t ) = ∑ Xk e
k =0 k = −∞

We can perform a similar reconstruction for aperiodic signals
1 π 1 ∞
x(n) = ∫ X(ω)eiωn dω x( t ) = ∫ X(ω)eiωt dω
2π − π 2π − ∞
These are called the inverse transforms.

Fourier Transform of Impulse Functions
Find the Fourier transform of the Dirac delta function:
∞ ∞
X(ω) = ∫ x( t )e −iωt dt = ∫ δ( t )e −iωt dt = e −iω0 = 1
−∞ −∞
Find the DTFT of the Kronecker delta function:
∞ ∞
−iωn −iωn −iω0
X( ω) = ∑ x(n)e = ∑ δ(n)e =e =1
n = −∞ n = −∞
The delta functions contain all frequencies at equal amplitudes.
Roughly speaking, that’s why the system response to an impulse
input is important: it tests the system at all frequencies.

Laplace Transform
► Lapalce transform is a generalization of the Fourier transform in the sense
that it allows “complex frequency” whereas Fourier analysis can only
handle “real frequency”. Like Fourier transform, Lapalce transform allows
us to analyze a “linear circuit” problem, no matter how complicated the
circuit is, in the frequency domain in stead of in he time domain.
► Mathematically, it produces the benefit of converting a set of differential
equations into a corresponding set of algebraic equations, which are much
easier to solve. Physically, it produces more insight of the circuit and
allows us to know the bandwidth, phase, and transfer characteristics
important for circuit analysis and design.
► Most importantly, Laplace transform lifts the limit of Fourier analysis to
allow us to find both the steady-state and “transient” responses of a linear
circuit. Using Fourier transform, one can only deal with he steady state
behavior (i.e. circuit response under indefinite sinusoidal excitation).
► Using Laplace transform, one can find the response under any types of
excitation (e.g. switching on and off at any given time(s), sinusoidal,
impulse, square wave excitations, etc.

Application of Laplace Transform to
Circuit Analysis

system

►• A system is an operation that transforms
input signal x into output signal y.

LTI Digital Systems
► Linear Time Invariant
• Linearity/Superposition:
► If a system has an input that can be
expressed as a sum of signals, then the
response of the system can be expressed as
a sum of the individual responses to the
respective systems.
► LTI

Time-Invariance &Causality

► Ifyou delay the input, response is just a delayed
version of original response.

► X(n-k) y(n-k)

► Causality could also be loosely defined by “there is
no output signal as long as there is no input
signal” or “output at current time does not depend
on future values of the input”.

Convolution
► The input and output signals for LTI
systems have special relationship in terms
of convolution sum and integrals.

► Y(t)=x(t)*h(t) Y[n]=x[n]*h[n]

Sampling theory
► The theory of taking discrete sample values (grid of color
pixels) from functions defined over continuous domains
(incident radiance defined over the film plane) and then
using those samples to reconstruct new functions that are
similar to the original (reconstruction).
► Sampler: selects sample points on the image plane
► Filter: blends multiple samples together

Sampling theory
► For band limited function, we can just
increase the sampling rate
► • However, few of interesting functions in
computer graphics are band limited, in
particular, functions with discontinuities.
► • It is because the discontinuity always falls
between two samples and the samples
provides no information of the discontinuity.

Z-transforms
► Fordiscrete-time systems, z-transforms play
the same role of Laplace transforms do in
continuous-time systems
Bilateral Forward z-transform Bilateral Inverse z-transform
∞
1
∑ h[ n] z − n
2 π j ∫R
H [ z] = h[n] = H [ z ] z − n +1dz
n = −∞

► Aswith the Laplace transform, we compute
forward and inverse z-transforms by use of
transforms pairs and properties

Region of Convergence
► Region of the complex ► Four possibilities (z=0
z-plane for which is a special case and
forward z-transform may or may not be
converges Im{z} included) Im{z}

Entire Disk
plane Re{z} Re{z}

Im{z} Im{z}

Intersection
Complement of a disk and
of a disk Re{z} complement Re{z}

of a disk

Z-transform Pairs
► h[n] = δ[n] ► h[n] = an u[n]
∞ 0

∑ δ [ n] z = ∑ δ [ n] z − n = 1
∞

∑ a u[ n] z
−n
H [ z] = H [ z] = n −n
n = −∞ n =0
n = −∞
Region of convergence: ∞
a ∞ n

= ∑ a z = ∑ n −n

entire z-plane n =0 n =0  z 

1 a
= if <1
► h[n] ∞= δ[n-1]1 1−
a z
H [ z ] = ∑ δ [ n − 1] z −n = ∑ δ [ n − 1] z −n = z −1 z
n = −∞ n =1

Region of convergence: Region of
entire z-plane convergence: |z|
> |a| which is
h[n-1] ⇔ z-1 H[z]
the complement

Stability
► Rule #1: For a causal sequence, poles are
inside the unit circle (applies to z-transform
functions that are ratios of two polynomials)
► Rule #2: More generally, unit circle is
included in region of convergence. (In
continuous-time, the imaginary axis would
be in the region of convergence of the
Z 1
a u[ n] ↔
Laplace transform.)a z for z > a
n
−1
1−

 This is stable if |a| < 1 by rule #1.

Inverse z-transform
c + j∞
1
f [ n] = F [ z ] z n −1dz
2πj c∫ j∞
−

► Yuk! Using the definition requires a contour
integration in the complex z-plane.
► Fortunately, we tend to be interested in only
a few basic signals (pulse, step, etc.)
 Virtually all of the signals we’ll see can be built
up from these basic signals.
 For these common signals, the z-transform pairs
have been tabulated (see Lathi, Table 5.1)

Example
z 2 + 2z +1 ► Ratio of polynomial z-
X [ z] =
3 1
z2 − z +
2 2
domain functions
1 + 2 z −1 + z −2
► Divide through by the
X [ z] =
3 1
1 − z −1 + z − 2 highest power of z
2 2
► Factor denominator into
1 + 2 z −1 + z −2
X [ z] =
first-order factors
1 − z (1 − z )
 1 −1  −1

 2  ► Use partial fraction
X [ z ] = B0 +
A1
+
A2 decomposition to get
1 −1 1 − z −1
1− z
2
first-order terms

Example (con’t)
2
1 − 2 3 −1
z − z + 1 z − 2 + 2 z −1 + 1 ► FindB0 by
2 2
z − 2 − 3 z −1 + 2 polynomial division
5 z −1 − 1
− 1 + 5 z −1
X [ z] = 2 +
► Express in terms of
 1 −1 
1 − z  1 − z
−1
( )
 2 
B0
1 + 2 z −1 + z −2 1+ 4 + 4
A1 = = = −9
1 − z −1 z −1 = 2
1− 2

1 + 2 z −1 + z − 2 1+ 2 +1
► Solve for A1 and A2
A2 = = =8
1 1
1 − z −1
2 z −1 =1 2

Example (con’t)
► Express X[z] in terms of B0, A1, and A2
9 8
X [ z] = 2 − +
1 −1 1 − z −1
1− z
2
► Use table to obtain inverse z-transform
n
1  
x[ n] = 2 δ [ n] − 9   u[ n] + 8 u[ n]
 2

► With the unilateral z-transform, or the
bilateral z-transform with region of
convergence, the inverse z-transform is
unique

Z-transform Properties
► Linearity

a1 f1 [ n] + a2 f 2 [ n] ⇔ a1 F1 [ z ] + a2 F2 [ z ]

► Right shift (delay)
f [ n − m] u[ n − m] ⇔ z − m F [ z ]
 m 
f [ n − m] u[ n] ⇔ z F [ z ] + z  ∑ f [ − n] z − n 
−m −m

 n =1 

Z-transform Properties
► Convolution definition
∞
f1 [ n] ∗ f 2 [ n] = ∑ f [ m] f [ n − m]
m = −∞
1 2
► Take z-transform
 ∞ 
Z { f1 [ n] ∗ f 2 [ n]} = Z  ∑ f1 [ m] f 2 [ n − m] 
m = −∞ 
► Z-transform definition
∞
 ∞ 
= ∑  ∑ f1 [ m] f 2 [ n − m]  z − n ► Interchange summation
n = −∞  m = −∞ 
∞ ∞
= ∑ f [ m] ∑ f [ n − m] z
1 2
−n ► Substitute r = n - m
m = −∞ n = −∞
∞ ∞
f1 [ m] ∑ f 2 [ r ] z −( r + m )
► Z-transform definition
= ∑
m = −∞ r = −∞

 ∞
− m 
∞

=  ∑ f1 [ m]z  ∑ f 2 [ r ] z − r 
 m = −∞  r = −∞ 
= F1 [ z ] F2 [ z ]

Introduction
► Impulse response h[n] can fully characterize a LTI
system, and we can have the output of LTI system as

y[ n] = x[ n] ∗ h[ n]
► The z-transform of impulse response is called transfer or

Y ( z ) = X ( z ) H ( z ).
system function H(z).

► ( )
Frequency response at H e = H ( z )
jω
z =1
is valid if
z =
ROC includes 1, and
Y ( e jω ) = X ( e jω ) H ( e jω )

5.1 Frequency Response of LIT
System
jω jω j∠X ( e jω ) jω jω j∠H ( e jω )
► Consider X (e ) = X (e ) e and H (e ) = H (e ) e
, then
 magnitude
Y ( e jω ) = X ( e jω ) H ( e jω )

 phase
jω jω jω
∠Y ( e ) = ∠X ( e ) + ∠H ( e )

► We will model and analyze LTI systems based on the
magnitude and phase responses.

System Function

► General formNof LCCDE M

∑ a y [ n − k ] = ∑ b x[ n − k ]
k =0
k
k =0
k

N M

∑ ak z Y ( z ) = ∑ bk z −k X ( z )
−k

k =0 k =0
► Compute the z-transform
M

Y ( z) ∑ bk z −k
H ( z) = = k =0
X ( z) N

∑ ak z − k
k =0

System Function: Pole/zero
Factorization
► Stability requirement can be verified.
► Choice of ROC determines causality.
► Location of zeros and poles determines the
frequency response1 )and phase : c , c ,..., c .
M
(1 − c z −
b0 ∏ k
zeros 1 2 M

H ( z) = k =1

∏ (1 − d z )
N
a0 −1 poles : d1 , d 2 ,..., d N .
k
k =1

Second-order System
► Suppose the system function of a LTI system is
(1 + z −1 )2
H ( z) = .
1 −1 3 −1
(1 − z )(1 + z )
2 4

► Tofind the difference equation that is satisfied by
the input and out of this system
(1 + z −1 )2 1 + 2 z −1 + z −2 Y ( z)
H ( z) = = =
1 3 1 3
(1 − z −1 )(1 + z −1 ) 1 + z −1 − z −2 X ( z )
2 4 4 8
1 3
y[n ] + y[n − 1] − y[n − 2] = x[n ] + 2 x[n − 1] + 2 x[n − 2]
4 8
► Can we know the impulse response?

System Function: Stability
► Stability of LTI system:
∞

∑ h[n] < ∞
n = −∞

► This condition is identical to the condition
∞
that ∑ h[n ]z −n < ∞ when z = 1.
n = −∞

 The stability condition is equivalent to the
condition that the ROC of H(z) includes the unit
circle.

System Function: Causality
► If the system is causal, it follows that h[n] must be a right-
sided sequence. The ROC of H(z) must be outside the
outermost pole.
► If the system is anti-causal, it follows that h[n] must be a
left-sided sequence. The ROC of H(z) must be inside the
innermost pole.
Im Im Im

a 1 a 1 Re a b
Re Re

Right-sided Left-sided Two-sided
(causal) (anti-causal) (non-causal)

Determining the ROC
► Consider the LTI system
5
y[n ] − y[n − 1] + y[n − 2] = x[n ]
2

► The system1 function is obtained as
H ( z) =
5 −1 −2
1− z +z
2
1
=
1 −1
(1 − z )(1 − 2 z −1 )
2

System Function: Inverse Systems
H i ( z) H ( z)
► is an inverse system for , if

G ( z ) = H ( z ) H i ( z ) = 1 ⇔ g [ n ] = h[ n ] ∗ hi [ n ] = δ [ n ]

1 1
Hi ( z) = jω
⇔ H i (e ) =
H ( z) H ( e jω )
► The ROCs of H ( z ) and H i ( z ) overlap.
must
► Useful for canceling the effects of another system
► See the discussion in Sec.5.2.2 regarding ROC

All-pass System
►A system of the form ∗(or cascade of these)
−1
z −a
H Ap ( Z ) = pole : a = re jθ
1 − az −1 zero : 1 / a* = r −1e jθ

− jω ∗ jω
e −a − jω 1 − a * e
H Ap ( e ) =
jω
− jω
=e − jω
1 − ae 1 − ae

H Ap ( e jω ) = 1

All-pass System: General Form
► In general, all pass systems have form
Mr
z −1 − d k M c ( z −1 − ek )( z −1 − ek )
*
H Ap ( z ) = ∏ −1 ∏ −1 * −1
k =1 1 − d k z k =1 (1 − ek z )(1 − ek z )

real poles complex poles

Causal/stable: ek , d k < 1

All-Pass System Example
Im
Unit
circle z-plane
M r = 2 and M c = 1

0.8
Re
4 3 0.5
− − 2
3 4

pole : re jθ reciprocal & conjugate → zero : r −1e jθ
 

This all - pass system has M = N = 2 M c + M r = 4 poles and zeros.

Minimum-Phase System
► Minimum-phase system: all zeros and all poles are
inside the unit circle.
► The name minimum-phase comes from a property of the
phase response (minimum phase-lag/group-delay).
► Minimum-phase systems have some special properties.
► When we design a filter, we may have multiple choices to
satisfy the certain requirements. Usually, we prefer the
minimum phase which is unique.
► All systems can be represented as a minimum-phase
system and an all-pass system.

Example
y[n] = a1y[n − 1] + a2 y[n − 2] + b0x[n]
► Block diagram representation of

Block Diagram Representation
► LTI systems with
rational system
function can be
represented as
constant-coefficient
difference equation
► The implementation of
difference equations
requires delayed
values of the

Direct Form I
N M

∑ ˆ y[n − k ] = ∑ b x[n − k ]
a
k =0
k
ˆ
k =0
k

► General form of difference equation
N M
y[n] − ∑ a y[n − k ] = ∑ b x[n − k ]
k k
k =1 k =0

► Alternative equivalent form

Direct Form I M

∑ bk z −k
H( z ) = k =0
N
1 − ∑ ak z −k
► Transfer function can be written as
 
k =1

  M
H( z ) = H2 ( z )H1 ( z ) =  1  b z −k 
∑ 
 N  k = 0 k
−k 


 1 − ∑ ak z 
 k =1 
 M −k 
V ( z ) = H1 ( z ) X( z ) =  ∑ bk z X( z ) M

 k =0

 v[n] = ∑ b x[n − k ]
k

► Direct Form I Represents  
k =0
N

1
 y[n] = ∑ a y[n − k ] + v[n]
Y ( z ) = H2 ( z ) V ( z ) =  V ( z ) k
k =1
 N 
 1 − ∑ ak z 
−k

 k =1 

Alternative Representation
► Replace order of cascade LTI systems




 M
 1
H( z ) = H ( z )H ( z ) =  ∑ b z 
 
−k 
1 2

k
 1−  N
k =0


∑a z 
 k =1
k
−k

N
w[n] = ∑ a w[n − k ] + x[n]
  k
  k =1
1
W( z ) = H2 ( z ) X( z ) =  X ( z ) M

 N  y[n] = ∑ b w[n − k ]
k
 1 − ∑ ak z 
−k
k =0
 k =1 
 M 
Y ( z ) = H1 ( z ) W( z ) =  ∑ bk z −k W( z )
 
 k =0 

Alternative Block Diagram
► We can change the order of the cascade
systems
N
w[n] = ∑ a w[n − k ] + x[n]
k
k =1
M
y[n] = ∑ b w[n − k ]
k
k =0

Direct Form II
► No need to store the same data
twice in previous system
► So we can collapse the delay
elements into one chain
► This is called Direct Form II or
the Canonical Form
► Theoretically no difference
between Direct Form I and II
► Implementation wise
 Less memory in Direct II
 Difference when using
finite-precision arithmetic

Signal Flow Graph Representation
► Similar to block diagram representation
 Notational differences
►A network of directed branches connected
at nodes

Example
► Representation of Direct Form II with signal
flow graphs w1 [n] = aw4 [n] + x[n]
w2 [n] = w1 [n]
w3 [n] = b0w2 [n] + b1w4 [n]
w4 [n] = w2 [n − 1]
y[n] = w3 [n]

w1 [n] = aw1 [n − 1] + x[n]
y[n] = b0w1 [n] + b1w1 [n − 1]

Determination of System
Function from Flow Graph
w1 [n] = w4 [n] − x[n]
w2 [n] = αw1 [n]
w3 [n] = w2 [n] + x[n]
w4 [n] = w3 [n − 1]
y[n] = w2 [n] + w4 [n]
W1 ( z ) = W4 ( z ) − X( z ) (
αX( z ) z −1 − 1 )
W2 ( z ) =
W2 ( z ) = αW1 ( z ) 1 − αz −1 Y ( z) z −1 − α
H( z ) = =
W3 ( z ) = W2 ( z ) + X( z ) X( z ) z −1 (1 − α ) X( z ) 1 − αz −1
W4 ( z ) =
W4 ( z ) = W3 ( z ) z −1 1 − αz −1 h[n] = αn −1u[n − 1] − αn +1u[n]
Y ( z ) = W2 ( z ) + W4 ( z ) Y ( z ) = W2 ( z ) + W4 ( z )

Basic Structures for IIR Systems:
Direct Form I

Direct Form II

Cascade Form
► General form for cascade implementation
M1 M2

∏ (1 − f z )∏ (1 − g z )(1 − g z )
k
−1
k
−1 ∗
k
−1

H( z ) = A k =1
N1
k =1
N2

∏ (1 − c z )∏ (1 − d z )(1 − d z )
k =1
k
−1

k =1
k
−1 ∗
k
−1

M1
b0k + b1k z −1 − b2k z −2
H( z ) = ∏
k =1 1 − a1k z −1 − a2k z −2
► More practical form in 2nd order systems

Example
H( z ) =
1 + 2z + z −1
(1 + z
−2
=
−1
)(1 + z )
−1

1 − 0.75z −1
+ 0.125z −2
(1 − 0.5z −1
)(1 − 0.25z )
−1

=
(1 + z )
−1
(1 + z )
−1

(1 − 0.5z ) (1 − 0.25z )
−1 −1

► Cascade of Direct Form I subsections

► Cascade of Direct Form II subsections

Parallel Form
► Represent system function using partial fraction expansion

H( z ) =
NP NP
Ak NP
(
Bk 1 − ek z −1 )
∑ Ck z − k
k =0
+∑
k =1 1 − ck z
−1
+∑
(
k =1 1 − dk z
−1
)(
1 − dk z −1
∗
)
NS
NP
e0k + e1k z −1
H( z ) = ∑C z k
−k
+∑
k =0 k =1 1 − a1k z −1 − a2k z −2
► Or by pairingthe real poles

Example
► Partial Fraction Expansion
1 + 2z −1 + z −2 18 25
H( z) = =8+ −
−1
1 − 0.75z + 0.125z −2
(
1 − 0.5z ) (
−1
1 − 0.25z −1 )

► Combine poles to get
− 7 + 8z −1
H( z ) = 8 +
1 − 0.75z −1 + 0.125z −2

Transposed Forms
► Linear signal flow graph property:
 Transposing doesn’t change the input-output
relation 1
H( z ) = −1
► Transposing:1 − az
 Reverse directions of all branches
 Interchange input and output nodes
► Example:

Example
Transpose

y[n] = a1y[n − 1] + a2y[n − 2] + b0x[n] + b1x[n − 1] + b2x[n − 2]

► Both have the same system function or
difference equation

Basic Structures for FIR Systems: Direct Form
► Special cases of IIR direct form structures

► Transpose of direct form I gives direct form II
► Both forms are equal for FIR systems
► Tapped delay line

Basic Structures for FIR Systems:
Cascade Form
► Obtainedby factoring the polynomial
system function
MS

∏ (b )
M
H( z ) = ∑ h[n]z
n=0
−n
=
k =1
0k + b1k z −1 + b2k z −2

Structures for Linear-Phase FIR
Systems
► Causal FIR system with generalized linear phase are
symmetric:
h[M − n] = h[n] n = 0,1,..., M (type I or III)
h[M − n] = −h[n] n = 0,1,..., M (type II or IV)
► Symmetry means we can half the number of
multiplications
► Example: For even M and type I or type III systems:
M M / 2 −1 M
y[n] = ∑ h[k ]x[n − k ] = ∑ h[k ]x[n − k ] + h[M / 2]x[n − M / 2] + ∑ h[k ]x[n − k ]
k =0 k =0 k = M / 2 +1
M / 2 −1 M / 2 −1
= ∑ h[k ]x[n − k ] + h[M / 2]x[n − M / 2] + ∑ h[M − k ]x[n − M + k ]
k =0 k =0
M / 2 −1
= ∑ h[k ]( x[n − k ] + x[n − M + k ] ) + h[M / 2]x[n − M / 2]
k =0

Structures for Linear-Phase FIR
Systems
► Structure for even M

► Structure for odd M

z transforms

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to z transforms (20)

Recently uploaded (20)

z transforms