![2 SIGNALS: ANALOG, DISCRETE, AND DIGITAL
The development of electromagnetic theory and the growth of electrical and
electronic communications technologies began to divide these sciences. The
functions of mathematics came to be studied as bearing information, requiring
modi$cation to be useful, suitable for interpretation, and having a meaning. The life
story of this new discipline—signal processing, communications, signal analysis,
and information theory—would follow a curious and ironic path. Electromagnetic
waves consist of coupled electric and magnetic $elds that oscillate in a sinusoidal
pattern and are perpendicular to one another and to their direction of propagation.
Fourier discovered that very general classes of functions, even those containing dis-
continuities, could be represented by sums of sinusoidal functions, now called a
Fourier series [1]. This surprising insight, together with the great advances in analog
communication methods at the beginning of the twentieth century, captured the
most attention from scientists and engineers.
Research efforts into discrete techniques were producing important results, even
as the analog age of signal processing and communication technology charged
ahead. Discrete Fourier series calculations were widely understood, but seldom car-
ried out; they demanded quite a bit of labor with pencil and paper. The $rst theoret-
ical links between analog and discrete signals were found in the 1920s by Nyquist,1
in the course of research on optimal telegraphic transmission mechanisms [2].
Shannon2
built upon Nyquist’s discovery with his famous sampling theorem [3]. He
also proved something to be feasible that no one else even thought possible: error-
free digital communication over noisy channels. Soon thereafter, in the late 1940s,
digital computers began to appear. These early monsters were capable of perform-
ing signal processing operations, but their speed remained too slow for some of the
most important computations in signal processing—the discrete versions of the
Fourier series. All this changed two decades later when Cooley and Tukey disclosed
their fast Fourier transform (FFT) algorithm to an eager computing public [4–6].
Digital computations of Fourier’s series were now practical on real-time signal data,
and in the following years digital methods would proliferate. At the present time,
digital systems have supplanted much analog circuitry, and they are the core of
almost all signal processing and analysis systems. Analog techniques handle only
the early signal input, output, and conditioning chores.
There are a variety of texts available covering signal processing. Modern intro-
ductory systems and signal processing texts cover both analog and discrete theory
[7–11]. Many re#ect the shift to discrete methods that began with the discovery of
the FFT and was fueled by the ever-increasing power of computing machines. These
often concentrate on discrete techniques and presuppose a background in analog
1As a teenager, Harry Nyquist (1887–1976) emigrated from Sweden to the United States. Among his
many contributions to signal and communication theory, he studied the relationship between analog sig-
nals and discrete signals extracted from them. The term Nyquist rate refers to the sampling frequency
necessary for reconstructing an analog signal from its discrete samples.
2Claude E. Shannon (1916–2001) founded the modern discipline of information theory. He detailed the
af$nity between Boolean logic and electrical circuits in his 1937 Masters thesis at the Massachusetts
Institute of Technology. Later, at Bell Laboratories, he developed the theory of reliable communication,
of which the sampling theorem remains a cornerstone.](https://image.slidesharecdn.com/445484-250518164906-f49cc5dc/85/Signal-Analysis-Time-Frequency-Scale-And-Structure-Ronald-L-Allen-31-320.jpg)
![3
signal processing [12–15]. Again, there is a distinction between discrete and digital
signals. Discrete signals are theoretical entities, derived by taking instantaneous—
and therefore exact—samples from analog signals. They might assume irrational
values at some time instants, and the range of their values might be in$nite. Hence,
a digital computer, whose memory elements only hold limited precision values, can
only process those discrete signals whose values are $nite in number and $nite in
their precision—digital signals. Early texts on discrete signal processing sometimes
blurred the distinction between the two types of signals, though some further
editions have adopted the more precise terminology. Noteworthy, however, are the
burgeoning applications of digital signal processing integrated circuits: digital tele-
phony, modems, mobile radio, digital control systems, and digital video to name a
few. The $rst high-de$nition television (HDTV) systems were analog; but later,
superior HDTV technologies have relied upon digital techniques. This technology
has created a true digital signal processing literature, comprised of the technical
manuals for various DSP chips, their application notes, and general treatments on
fast algorithms for real-time signal processing and analysis applications on digital
signal processors [16–21]. Some of our later examples and applications offer some
observations on architectures appropriate for signal processing, special instruction
sets, and fast algorithms suitable for DSP implementation.
This chapter introduces signals and the mathematical tools needed to work with
them. Everyone should review this chapter’s $rst six sections. This $rst chapter com-
bines discussions of analog signals, discrete signals, digital signals, and the methods
to transition from one of these realms to another. All that it requires of the reader is
a familiarity with calculus. There are a wide variety of examples. They illustrate
basic signal concepts, $ltering methods, and some easily understood, albeit limited,
techniques for signal interpretation. The $rst section introduces the terminology of
signal processing, the conventional architecture of signal processing systems, and
the notions of analog, discrete, and digital signals. It describes signals in terms of
mathematical models—functions of a single real or integral variable. A speci$cation
of a sequence of numerical values ordered by time or some other spatial dimension
is a time domain description of a signal. There are other approaches to signal
description: the frequency and scale domains, as well as some—relatively recent—
methods for combining them with the time domain description. Sections 1.2 and 1.3
cover the two basic signal families: analog and discrete, respectively. Many of the
signals used as examples come from conventional algebra and analysis.
The discussion gets progressively more formal. Section 1.4 covers sampling and
interpolation. Sampling picks a discrete signal from an analog source, and interpo-
lation works the other way, restoring the gaps between discrete samples to fashion
an analog signal from a discrete signal. By way of these operations, signals pass
from the analog world into the discrete world and vice versa. Section 1.5 covers
periodicity, and foremost among these signals is the class of sinusoids. These sig-
nals are the fundamental tools for constructing a frequency domain description of a
signal. There are many special classes of signals that we need to consider, and Sec-
tion 1.6 quickly collects them and discusses their properties. We will of course
expand upon and deepen our understanding of these special types of signals
SIGNALS: ANALOG, DISCRETE, AND DIGITAL](https://image.slidesharecdn.com/445484-250518164906-f49cc5dc/85/Signal-Analysis-Time-Frequency-Scale-And-Structure-Ronald-L-Allen-32-320.jpg)
![INTRODUCTION TO SIGNALS 5
striking the eye. Some of these signal values are quanti$able; the phenomenon is a
measurable quantity, and its evolution is ordered by time or distance. Thus, a resi-
dential telephone signal’s value is known by measuring the voltage across the pair
of wires that comprise the circuit. Sound waves are longitudinal and produce
minute, but measurable, pressure variations on a listener’s eardrum. On the other
hand, some signals appear to have a representation that is at root not quanti$able,
but rather symbolic. Thus, most people would grant that sign language gestures,
maritime signal #ags, and even ASCII text could be considered signals, albeit of a
symbolic nature.
Let us for the moment concentrate on signals with quanti$able values. These are
the traditional mathematical signal models, and a rich mathematical theory is avail-
able for studying them. We will consider signals that assume symbolic values, too,
but, unlike signals with quanti$able values, these entities are better described by
relational mathematical structures, such as graphs.
Now, if the signal is a continuously occurring phenomenon, then we can repre-
sent it as a function of a time variable t; thus, x(t) is the value of signal x at time t.
We understand the units of measurement of x(t) implicitly. The signal might vary
with some other spatial dimension other than time, but in any case, we can suppose
that its domain is a subset of the real numbers. We then say that x(t) is an analog
signal. Analog signal values are read from conventional indicating devices or sci-
enti$c instruments, such as oscilloscopes, dial gauges, strip charts, and so forth.
An example of an analog signal is the seismogram, which records the shaking
motion of the ground during an earthquake. A precision instrument, called a seismo-
graph, measures ground displacements on the order of a micron (106
m) and pro-
duces the seismogram on a paper strip chart attached to a rotating drum. Figure 1.1
shows the record of the Loma Prieta earthquake, centered in the Santa Cruz moun-
tains of northern California, which struck the San Francisco Bay area on 18 October
1989.
Seismologists analyze such a signal in several ways. The total de#ection of the
pen across the chart is useful in determining the temblor’s magnitude. Seismograms
register three important types of waves: the primary, or P waves; the secondary, or S
waves; and the surface waves. P waves arrive $rst, and they are compressive, so
their direction of motion aligns with the wave front propagation [22]. The transverse
S waves follow. They oscillate perpendicular to the direction of propagation.
Finally, the large, sweeping surface waves appear on the trace.
This simple example illustrates processing and analysis concepts. Processing the
seismogram signal is useful to remove noise. Noise can be minute ground motions
from human activity (construction activity, heavy machinery, vehicles, and the like),
or it may arise from natural processes, such as waves hitting the beach. Whatever
the source, an important signal processing operation is to smooth out these minute
ripples in the seismogram trace so as to better detect the occurrence of the initial
indications of a seismic event, the P waves. They typically manifest themselves as
seismometer needle motions above some threshold value. Then the analysis prob-
lem of $nding when the S waves begin is posed. Figure 1.1 shows the result of a sig-
nal analysis; it slices the Loma Prieta seismogram into its three constituent wave](https://image.slidesharecdn.com/445484-250518164906-f49cc5dc/85/Signal-Analysis-Time-Frequency-Scale-And-Structure-Ronald-L-Allen-34-320.jpg)
![6 SIGNALS: ANALOG, DISCRETE, AND DIGITAL
trains. This type of signal analysis can be performed by inspection on analog seis-
mograms.
Now, the time interval between the arrival of the P and S waves is critical. These
undulations are simultaneously created at the earthquake’s epicenter; however, they
travel at different, but known, average speeds through the earth. Thus, if an analysis
of the seismogram can reveal the time that these distinct wave trains arrive, then the
time difference can be used to measure the distance from the instrument to the earth-
quake’s epicenter. Reports from three separate seismological stations are suf$cient
to locate the epicenter. Analyzing smaller earthquakes is also important. Their loca-
tion and the frequency of their occurrence may foretell a larger temblor [23]. Fur-
ther, soundings in the earth are indicative of the underlying geological strata;
seismologists use such methods to locate oil deposits, for example [24]. Other simi-
lar applications include the detection of nuclear arms detonations and avalanches.
For all of these reasons—scienti$c, economic, and public safety—seismic signal
intepretation is one of the most important areas in signal analysis and one of the
areas in which new methods of signal analysis have been pioneered. These further
signal interpretation tasks are more troublesome for human interpreters. The signal
behavior that distinguishes a small earthquake from a distant nuclear detonation is
not apparent. This demands thorough computerized analysis.
Fig. 1.1. Seismogram of the magnitude 7.1 Loma Prieta earthquake, recorded by a seis-
mometer at Kevo, Finland. The $rst wiggle—some eight minutes after the actual event—
marks the beginning of the low-magnitude P waves. The S waves arrive at approximately t =
1200 s, and the large sweeping surface waves begin near t = 2000 s.](https://image.slidesharecdn.com/445484-250518164906-f49cc5dc/85/Signal-Analysis-Time-Frequency-Scale-And-Structure-Ronald-L-Allen-35-320.jpg)
![INTRODUCTION TO SIGNALS 9
they are all functions of the same independent variable with the same domain. Alter-
natively, it may be technically useful to maintain the multiple quantities together as
a vector. This is called a multichannel signal. We use boldface letters to denote mul-
tichannel signals. Thus, if x is analog and has N channels, then x(t) = (x1(t),
x2(t), …, xN(t)), where the analog xi(t) are called the component or channel signals.
Similarly, if x is discrete and has N channels, then x(n) = (x1(n), x2(n), …, xN(n)).
One biomedical signal that is useful in diagnosing brain injuries, mental illness,
and conditions such as Alzheimer’s disease is the electroencephalogram (EEG)
[25], a multichannel signal. It records electrical potential differences, or voltages,
that arise from the interactions of massive numbers of neurons in different parts of
the brain. For an EEG, 19 electrodes are attached from the front to the back of the
scalp, in a two–$ve–$ve–$ve–two arrangement (Figure 1.3).
The EEG traces in Figure 1.3 are in fact digital signals, acquired one sample
every 7.8 ms, or at a sampling frequency of 128 Hz. The signal appears to be conti-
nuous in nature, but this is due to the close spacing of the samples and linear inter-
polation by the plotting package.
Another variation on the nature of signals is that they may be functions
of more than one independent variable. For example, we might measure air
0 200 400 600 800 1000 1200
-40
-20
0
20
40
(c)
0 200 400 600 800 1000 1200
-40
-20
0
20
40
(d)
Fig. 1.3 (Continued)](https://image.slidesharecdn.com/445484-250518164906-f49cc5dc/85/Signal-Analysis-Time-Frequency-Scale-And-Structure-Ronald-L-Allen-38-320.jpg)
![10 SIGNALS: ANALOG, DISCRETE, AND DIGITAL
temperature as a function of height: T(h) is an analog signal. But if we con-
sider that the variation may occur along a north-to-south line as well, then the
temperature depends upon a distance measure x as well: T(x, h). Finally, over an
area with location coordinates (x, y), the air temperature is a continuous function
of three variables T(x, y, h). When a signal has more than one independent vari-
able, then it is a multidimensional signal. We usually think of an “image” as
recording light intesity measurements of a scene, but multidimensional signals—
especially those with two or three independent variables—are usually called
images. Images may be discrete too. Temperature readings taken at kilometer
intervals on the ground and in the air produce a discrete signal T(m, n, k). A dis-
crete signal is a sequence of numerical values, whereas an image is an array of
numerical values. Two-dimensional image elements, especially those that repre-
sent light intensity values, are called pixels, an acronym for picture elements.
Occasionally, one encounters the term voxel, which is a three-dimensional signal
value, or a volume element.
An area of multidimensional signal processing and analysis of considerable
importance is the intepretation of images of landscapes acquired by satellites and
high altitude aircraft. Figure 1.4. shows some examples. Typical tasks are to
automatically distinguish land from sea; determine the amount and extent of sea
ice; distinguish agricultural land, urban areas, and forests; and, within the
agricultural regions, recognize various crop types. These are remote sensing
applications.
Processing two-dimensional signals is more commonly called picture or image
processing, and the task of interpreting an image is called image analysis or com-
puter vision. Many researchers are involved in robotics, where their efforts couple
computer vision ideas with manipulation of the environment by a vision-based
machine. Consequently, there is a vast, overlapping literature on image processing
[26–28], computer vision [29–31], and robotics [32].
Our subject, signal analysis, concentrates on the mathematical foundations, pro-
cessing, and especially the intepretation of one-dimensional, single-valued signals.
Generally, we may select a single channel of a multichannel signal for consider-
ation; but we do not tackle problems speci$c to multichannel signal interpretation.
Likewise, we do not delve deeply into image processing and analysis. Certain
images do arise, so it turns out, in several important techniques for analyzing sig-
nals. Sometimes a daunting one-dimensional problem can be turned into a tractable
two-dimensional task. Thus, we prefer to pursue the one-dimensional problem into
the multidimensional realm only to the point of acknowledging that a straightfor-
ward image analysis will produce the intepretation we seek.
So far we have introduced the basic concepts of signal theory, and we have
considered some examples: analog, discrete, multichannel, and multidimensional
signals. In each case we describe the signals as sequences of numerical values, or
as a function of an independent time or other spatial dimension variable. This con-
stitutes a time-domain description of a signal. From this perspective, we can dis-
play a signal, process it to produce another signal, and describe its signi$cant
features.](https://image.slidesharecdn.com/445484-250518164906-f49cc5dc/85/Signal-Analysis-Time-Frequency-Scale-And-Structure-Ronald-L-Allen-39-320.jpg)
![12 SIGNALS: ANALOG, DISCRETE, AND DIGITAL
Note that a precise time-domain description may elude us, and it may not even be
possible to specify a signal’s values. A fundamentally unknowable or random pro-
cess is the source of such signals. It is important to develop methods for handling
the randomness inherent in signals. Techniques that presuppose a theory of signal
randomness are the topic of the $nal section of the chapter.
Next we look further into two application areas we have already touched upon:
biophysical and geophysical signals. Signals from representative applications in
these two areas readily illustrate the time-domain description of signals.
1.1.2.1 Electrocardiogram Interpretation. Electrocardiology is one of the
earliest techniques in biomedicine. It also remains one of the most important. The
excitation and recovery of the heart muscles cause small electrical potentials, or volt-
ages, on the order of a millivolt, within the body and measurable on the skin. Cardio-
logists observe the regularity and shape of this voltage signal to diagnose heart con-
ditions resulting from disease, abnormality, or injury. Examples include cardiac
dysrhythmia and $brillation, narrowing of the coronary arteries, and enlargement of
the heart [33]. Automatic interpretation of ECGs is useful for many aspects of clini-
cal and emergency medicine: remote monitoring, as a diagnostic aid when skilled
cardiac care personnel are unavailable, and as a surgical decision support tool.
A modern electrocardiogram (ECG or EKG) contains traces of the voltages from
12 leads, which in biomedical parlance refers to a con$guration of electrodes
attached to the body [34]. Refer to Figure 1.5. The voltage between the arms is Lead I,
Lead II is the potential between the right arm and left leg, and Lead III reads between
the left arm and leg. The WCT is a common point that is formed by connecting the
three limb electrodes through weighting resistors. Lead aVL measures potential
difference between the left arm and the WCT. Similarly, lead aVR is the voltage
between the right arm and the WCT. Lead aVF is between the left leg and the WCT.
Finally, six more electrodes are $xed upon the chest, around the heart. Leads V1
through V6 measure the voltages between these sensors and the WCT. This circuit
Fig. 1.5. The standard ECG con$guration produces 12 signals from various electrodes
attached to the subject’s chest, arms, and leg.](https://image.slidesharecdn.com/445484-250518164906-f49cc5dc/85/Signal-Analysis-Time-Frequency-Scale-And-Structure-Ronald-L-Allen-41-320.jpg)
![INTRODUCTION TO SIGNALS 13
arrangement is complicated; in fact, it is redundant. Redundancy provides for situa-
tions where a lead produces a poor signal and allows some cross-checking of the
readings. Interpretation of 12-lead ECGs requires considerable training, experience,
and expert judgment.
What does an ECG trace look like? Figure 1.6 shows an ECG trace from a single
lead. Generally, an ECG has three discernible pulses: the P wave, the QRS complex,
and the T wave. The P wave occurs upon excitation of the auricles of the heart, when
they draw in blood from the body and lungs. The large-magnitude QRS complex
occurs during the contraction of the vertricles as they contract to pump blood out of
the heart. The Q and S waves are negative pulses, and the R wave is a positive pulse.
The T wave arises during repolarization of the ventricles. The ECG signal is origi-
nally analog in nature; it is the continuous record of voltages produce across the var-
ious leads supported by the instrument. We could attach a millivoltmeter across an
electrode pair and watch the needle jerk back and forth. Visualizing the signal’s shape
is easier with an oscilloscope, of course, because the instrument records the trace on
its cathode ray tube. Both of these instruments display analog waveforms. If we could
read the oscilloscope’s output at regular time instants with perfect precision, then we
would have—in principle, at least—a discrete representation of the ECG. But for
computer display and automatic interpretation, the analog signal must be converted
to digital form. In fact, Figure 1.6 is the result of such a digitization. The signal v(n)
appears continuous due to the large number of samples and the interpolating lines
drawn by the graphics package that produced the illustration.
Interpreting ECGs is often dif$cult, especially in abnormal traces. A wide litera-
ture describing the 12-lead ECG exists. There are many guides to help technicians,
nurses, and physicians use it to diagnose heart conditions. Signal processing and
analysis of ECGs is a very active research area. Reports on new techniques, algo-
rithms, and comparison studies continue to appear in the biomedical engineering
and signal analysis literature [35].
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
−2000
−1500
−1000
−500
0
500
1000
1500
2000
Electrocardiogram (ECG) record from a human (male) in a supine position
n
x(n)
Fig. 1.6. One lead of an ECG: A human male in supine position. The sampling rate is 1 kHz,
and the samples are digitized at 12 bits per sample. The irregularity of the heartbeat is evident.](https://image.slidesharecdn.com/445484-250518164906-f49cc5dc/85/Signal-Analysis-Time-Frequency-Scale-And-Structure-Ronald-L-Allen-42-320.jpg)
![INTRODUCTION TO SIGNALS 15
detectable in the ECG, provided that it employs a suf$ciently high sampling rate, is
splintering of the QRS complex. In this abnormal condition, the QRS consists of
many closely spaced positive and negative transitions rather than a single, strong
pulse. Note that in any ECG, there is a signi$cant amount of signal noise. This too
is clearly visible in the present example. Good peak detection and pulse location,
especially for the smaller P and T waves, often require some data smoothing
method. Averaging the signal values produces a smoother signal w(n):
. (1.1)
The particular formula (1.1) for processing the raw ECG signal to produce a
less noisy w(n) is called moving average smoothing or moving average $ltering.
This is a typical, almost ubiquitous signal processing operation. Equation (1.1)
performs averaging within a symmetric window of width three about v(n). Wider
windows are possible and often useful. A window that is too wide can destroy signal
features that bear on interpretation. Making a robust application requires judgment
and experimentation.
Real-time smoothing operations require asymmetric windows. The underlying
reason is that a symmetric smoothing window supposes knowledge of future signal
values, such as v(n + 1). To wit, as the computer monitoring system acquires each
new ECG value v(n), it can calculate the average of the last three values:
; (1.2)
but at time instant n, it cannot possibly know the value of v(n +1), which is neces-
sary for calculating (1.1). If the smoothing operation occurs of#ine, after the entire
set of signal values of interest has already been acquired and stored, then the whole
range of signal values is accessible by the computer, and calculation (1.1) is, of
course, feasible. When smoothing operations must procede in lockstep with acquisi-
tion operations, however, smoothing windows that look backward in time (1.2) must
be applied.
Yet another method from removing noise from signals is to produce a signal
whose values are the median of a window of raw input values. Thus, we might
assign
(1.3)
so that w(n) is the input value that lies closest to the middle of the range of $ve
values around v(n). A median $lter tends to be superior to a moving average $lter
when the task is to remove isolated, large-magnitude spikes from a source signal.
There are many variants. In general, smoothing is a common early processing step
in signal analysis systems. In the present application, smoothing reduces the
jagged noise in the ECG trace and improves the estimate of the QRS peak’s
location.
w n
( )
1
3
--
- v n 1
–
( ) v n
( ) v n 1
+
( )
+ +
[ ]
=
w n
( )
1
3
--
- v n 2
–
( ) v n 1
–
( ) v n
( )
+ +
[ ]
=
w n
( ) Median v n 2
–
( ) v n 1
–
( ) v n
( ) v n 1
+
( ) v n 2
+
( )
, , , ,
{ }
=](https://image.slidesharecdn.com/445484-250518164906-f49cc5dc/85/Signal-Analysis-Time-Frequency-Scale-And-Structure-Ronald-L-Allen-44-320.jpg)
![INTRODUCTION TO SIGNALS 17
signal—the geothermal gradient—that is a function of distance, not time. It ramps
up about 10°C per kilometer of depth and is a primary indicator for geothermal
prospecting. In general, the geothermal gradient is higher for oceanic than for conti-
nental crust. Some 5% of the area of the United States has a gradient in the neighbor-
hood of 40°C per kilometer of depth and has potential for use in geothermal power
generation.
Mathematically, the geothermal gradient is the derivative of the signal with
respect to its independent variable, which in this case measures depth into the earth.
A very steep overall gradient may promise a geothermal energy source. A localized
large magnitude gradient, or edge, in the temperature pro$le marks a geological
artifact, such as a fracture zone. An example of the variation in ground temperature
as one digs into the earth is shown in Figure 1.8.
The above data come from the second of four wells drilled on the Georgia–South
Carolina border, in the eastern United States, in 1985 [36]. The temperature $rst
declines with depth, which is typical, and then warmth from the earth’s interior
appears. Notice the large-magnitude positive gradients at approximately 80 and
175 m; these correspond to fracture zones. Large magnitude deviations often repre-
sent physically signi$cant phenomena, and therein lies the importance of reliable
methods for detecting, locating, and interpreting signal edges. Finding such large
deviations in signal values is once again a time-domain signal analysis problem.
Suppose the analog ground temperature signal is g(s), where s is depth into the
earth. We seek large values of the derivative Approximating the
derivative is possible once the data are digitized. We select a sampling interval D
0 and set x(n) = g(nD); then approximates the geother-
mal gradient at depth nD meters. It is further necessary to identify a threshold M for
what constitutes a signi$cant geothermal gradient. Threshold selection may rely
upon expert scienti$c knowledge. A geophysicist might suggest signi$cant gradients
0 50 100 150 200 250 300 350
14.5
15
15.5
16
16.5
17
17.5
18
18.5
19
19.5
depth (m)
temperature,
deg
C Temperature change versus depth
Fig. 1.8. A geothermal signal. The earth’s temperature is sampled at various depths to pro-
duce a discrete signal with a spatially independent variable.
g′ s
( ) dg/ds.
=
x′ n
( ) x n 1
+
( ) x n 1
–
( )
–
=](https://image.slidesharecdn.com/445484-250518164906-f49cc5dc/85/Signal-Analysis-Time-Frequency-Scale-And-Structure-Ronald-L-Allen-46-320.jpg)
![18 SIGNALS: ANALOG, DISCRETE, AND DIGITAL
for the region. If we collect some statistics on temperature gradients, then the outly-
ing values may be candidates for threshold selection. Again, there are local variations
in the temperature pro$le, and noise does intrude into the signal acquisition appara-
tus. Hence, preliminary signal smoothing may once again be useful. Toward this end,
we may also employ discrete derivative formulas that use more signal values:
. (1.4)
Standard numerical analysis texts provide many alternatives [37]. Among the prob-
lems at the chapter’s end are several edge detection applications. They weigh some
of the alternatives for $ltering, threshold selection, and $nding extrema.
For now, let us remark that the edges in the ECG signal (Figure 1.6) are far steeper
than the edges in the geothermal trace (Figure 1.8). The upshot is that the signal ana-
lyst must tailor the discrete derivative methods to the data at hand. Developing meth-
ods for edge detection that are robust with respect to sharp local variation of the signal
features proves to be a formidable task. Time-domain methods, such as we consider
here, are usually appropriate for edge detection problems. There comes a point, none-
theless, when the variety of edge shapes, the background noise in the source signals,
and the diverse gradients cause problems for simple time domain techniques. In recent
years, researchers have turned to edge detection algorithms that incorporate a notion
of the size or scale of the signal features. Chapter 4 has more to say about time domain
signal analysis and edge detection, in particular. The later chapters round out the story.
1.1.3 Analysis in the Time-Frequency Plane
What about signals whose values are symbolic rather than numeric? In ordinary
usage, we consider sequences of signs to be signals. Thus, we deem the display of
#ags on a ship’s mast, a series of hand gestures between baseball players, DNA
codes, and, in general, any sequence of codes to all be “signals.” We have already
taken note of such usages. And this is an important idea, but we shall not call such a
symbolic sequence a signal, reserving for that term a narrow scienti$c de$nition as
an ordered set of numbers. Instead, we shall de$ne a sequence of abstract symbols
to be a structural interpretation of a signal.
It is in fact the conversion of an ordered set of numerical values into a sequence
of symbols that constitutes a signal interpretation or analysis. Thus, a microphone
receives a logitudinal compressive sound wave and converts it into electrical
impulses, thereby creating an analog signal. If the analog speech signal is digitized,
processed, and analyzed by a speech recognition engine, then the output in the form
of ASCII text characters is a symbolic sequence that interprets, analyzes, or assigns
meaning to the signal. The $nal result may be just the words that were uttered. But,
more likely, the speech interpretation algorithms will generate a variety of interme-
diate representations of the signal’s structure. It is common to build a large hierar-
chy of interpretations: isolated utterances; candidate individual word sounds within
the utterances; possible word recognition results; re$nements from grammatical
rules and application context; and, $nally, a structural result.
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![24 SIGNALS: ANALOG, DISCRETE, AND DIGITAL
functions representing analog signals may be more convenient to sketch rather than
specify mathematically.
1.2.2.1 Polynomial, Rational, and Algebraic Signals. Consider, for exa-
mple, the polynomial signal,
. (1.5)
x(t) has derivatives of all orders and is continuous, along with all of its derivatives. It
is quite unlike any of nature’s signals, since its magnitude, |x(t)|, will approach
in$nity as |t| becomes large. These signals are familiar from elementary algebra,
where students $nd their roots and plot their graphs in the Cartesian plane. The
domain of a polynomial p(t) can be divided into disjoint regions of concavity: con-
cave upward, where the second derivative is positive; concave downward, where the
second derivative is negative; and regions of no concavity, where the second deriva-
tive is zero, and p(t) is therefore a line. If the domain of a polynomial p(t) contain-
san interval a t b where for all , then p(t) is a line.
However familiar and natural the polynomials may be, they are not the signal family
with which we are most intimately concerned in signal processing. Their behavior
for large |t| is the problem. We prefer mathematical functions that more closely
resemble the kind of signals that occur in nature: Signals x(t) which, as |t| gets large,
the signal either approaches a constant, oscillates, or decays to zero. Indeed, we
expend quite an effort in Chapter 2 to discover signal families—called function or
signal spaces—which are faithful models of natural signals.
The concavity of a signal is a very important concept in certain signal analysis
applications. Years ago, the psychologist F. Attneave [38] noted that a scattering of
simple curves suf$ces to convey the idea of a complex shape—for instance, a cat.
Later, computer vision researchers developed the idea of assemblages of simple,
oriented edges into complete theories of low-level image understanding [39–41].
Perhaps the most in#uential among them was David Marr, who conjectured that
understanding a scene depends upon the extraction of edge information [39] over a
range of visual resolutions from coarse to $ne multiple scales. Marr challenged
computer vision researchers to $nd processing and analysis paradigms within bio-
logical vision and apply them to machine vision. Researchers investigated the appli-
cations of concavity and convexity information at many different scales. Thus, an
intricate shape might resolve into an intricate pattern at a $ne scale, but at a coarser
scale might appear to be just a tree. How this can be done, and how signals can be
smoothed into larger regions of convexity and concavity without increasing the
number of differently curved regions, is the topic of scale-space analysis [42,43].
We have already touched upon some of these ideas in our discussion of edges of the
QRS complex of an electrocardiogram trace and in our discussion of the geothermal
gradient. There the scale of an edge corresponded to the number of points incorpo-
rated in the discrete derivative computation. This is precisely the notion we are
x t
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Signal Analysis Time Frequency Scale And Structure Ronald L Allen
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- 6. SIGNAL ANALYSIS TIME, FREQUENCY, SCALE, AND STRUCTURE Ronald L. Allen Duncan W. Mills A John Wiley & Sons, Inc., Publication
- 9. IEEE Press 445 Hoes Lane Piscataway, NJ 08854 IEEE Press Editorial Board Stamatios V. Kartalopoulos, Editor in Chief M. Akay M. E. El-Hawary M. Padgett J. B. Anderson R. J. Herrick W. D. Reeve R. J. Baker D. Kirk S. Tewksbury J. E. Brewer R. Leonardi G. Zobrist M. S. Newman Kenneth Moore, Director of IEEE Press Catherine Faduska, Senior Acquisitions Editor John Griffin, Acquisitions Editor Tony VenGraitis, Project Editor
- 10. SIGNAL ANALYSIS TIME, FREQUENCY, SCALE, AND STRUCTURE Ronald L. Allen Duncan W. Mills A John Wiley & Sons, Inc., Publication
- 11. Copyright © 2004 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data is available. ISBN: 0-471-23441-9 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 fax 978-646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission
- 12. v To Beverley and to the memory of my parents, Mary and R.L. (Kelley). R.L.A. To those yet born, who will in some manner—large or small—benefit from the technology and principles described here. To the reader, who will contribute to making this happen. D.W.M.
- 14. vii CONTENT Preface xvii Acknowledgments xxi 1 Signals: Analog, Discrete, and Digital 1 1.1 Introduction to Signals 4 1.1.1 Basic Concepts 4 1.1.2 Time-Domain Description of Signals 11 1.1.3 Analysis in the Time-Frequency Plane 18 1.1.4 Other Domains: Frequency and Scale 20 1.2 Analog Signals 21 1.2.1 Definitions and Notation 22 1.2.2 Examples 23 1.2.3 Special Analog Signals 32 1.3 Discrete Signals 35 1.3.1 Definitions and Notation 35 1.3.2 Examples 37 1.3.3 Special Discrete Signals 39 1.4 Sampling and Interpolation 40 1.4.1 Introduction 40 1.4.2 Sampling Sinusoidal Signals 42 1.4.3 Interpolation 42 1.4.4 Cubic Splines 46 1.5 Periodic Signals 51 1.5.1 Fundamental Period and Frequency 51 1.5.2 Discrete Signal Frequency 55 1.5.3 Frequency Domain 56 1.5.4 Time and Frequency Combined 62 1.6 Special Signal Classes 63 1.6.1 Basic Classes 63 1.6.2 Summable and Integrable Signals 65
- 15. viii CONTENTS 1.6.3 Finite Energy Signals 66 1.6.4 Scale Description 67 1.6.5 Scale and Structure 67 1.7 Signals and Complex Numbers 70 1.7.1 Introduction 70 1.7.2 Analytic Functions 71 1.7.3 Complex Integration 75 1.8 Random Signals and Noise 78 1.8.1 Probability Theory 79 1.8.2 Random Variables 84 1.8.3 Random Signals 91 1.9 Summary 92 1.9.1 Historical Notes 93 1.9.2 Resources 95 1.9.3 Looking Forward 96 1.9.4 Guide to Problems 96 References 97 Problems 100 2 Discrete Systems and Signal Spaces 109 2.1 Operations on Signals 110 2.1.1 Operations on Signals and Discrete Systems 111 2.1.2 Operations on Systems 121 2.1.3 Types of Systems 121 2.2 Linear Systems 122 2.2.1 Properties 124 2.2.2 Decomposition 125 2.3 Translation Invariant Systems 127 2.4 Convolutional Systems 128 2.4.1 Linear, Translation-Invariant Systems 128 2.4.2 Systems Defined by Difference Equations 130 2.4.3 Convolution Properties 131 2.4.4 Application: Echo Cancellation in Digital Telephony 133 2.5 The l p Signal Spaces 136 2.5.1 lp Signals 137 2.5.2 Stable Systems 138
- 16. CONTENTS ix 2.5.3 Toward Abstract Signal Spaces 139 2.5.4 Normed Spaces 142 2.5.5 Banach Spaces 147 2.6 Inner Product Spaces 149 2.6.1 Definitions and Examples 149 2.6.2 Norm and Metric 151 2.6.3 Orthogonality 153 2.7 Hilbert Spaces 158 2.7.1 Definitions and Examples 158 2.7.2 Decomposition and Direct Sums 159 2.7.3 Orthonormal Bases 163 2.8 Summary 168 References 169 Problems 170 3 Analog Systems and Signal Spaces 173 3.1 Analog Systems 174 3.1.1 Operations on Analog Signals 174 3.1.2 Extensions to the Analog World 174 3.1.3 Cross-Correlation, Autocorrelation, and Convolution 175 3.1.4 Miscellaneous Operations 176 3.2 Convolution and Analog LTI Systems 177 3.2.1 Linearity and Translation-Invariance 177 3.2.2 LTI Systems, Impulse Response, and Convolution 179 3.2.3 Convolution Properties 184 3.2.4 Dirac Delta Properties 186 3.2.5 Splines 188 3.3 Analog Signal Spaces 191 3.3.1 Lp Spaces 191 3.3.2 Inner Product and Hilbert Spaces 205 3.3.3 Orthonormal Bases 211 3.3.4 Frames 216 3.4 Modern Integration Theory 225 3.4.1 Measure Theory 226 3.4.2 Lebesgue Integration 232
- 17. x CONTENTS 3.5 Distributions 241 3.5.1 From Function to Functional 241 3.5.2 From Functional to Distribution 242 3.5.3 The Dirac Delta 247 3.5.4 Distributions and Convolution 250 3.5.5 Distributions as a Limit of a Sequence 252 3.6 Summary 259 3.6.1 Historical Notes 260 3.6.2 Looking Forward 260 3.6.3 Guide to Problems 260 References 261 Problems 263 4 Time-Domain Signal Analysis 273 4.1 Segmentation 277 4.1.1 Basic Concepts 278 4.1.2 Examples 280 4.1.3 Classification 283 4.1.4 Region Merging and Splitting 286 4.2 Thresholding 288 4.2.1 Global Methods 289 4.2.2 Histograms 289 4.2.3 Optimal Thresholding 292 4.2.4 Local Thresholding 299 4.3 Texture 300 4.3.1 Statistical Measures 301 4.3.2 Spectral Methods 308 4.3.3 Structural Approaches 314 4.4 Filtering and Enhancement 314 4.4.1 Convolutional Smoothing 314 4.4.2 Optimal Filtering 316 4.4.3 Nonlinear Filters 321 4.5 Edge Detection 326 4.5.1 Edge Detection on a Simple Step Edge 328 4.5.2 Signal Derivatives and Edges 332 4.5.3 Conditions for Optimality 334 4.5.4 Retrospective 337
- 18. CONTENTS xi 4.6 Pattern Detection 338 4.6.1 Signal Correlation 338 4.6.2 Structural Pattern Recognition 342 4.6.3 Statistical Pattern Recognition 346 4.7 Scale Space 351 4.7.1 Signal Shape, Concavity, and Scale 354 4.7.2 Gaussian Smoothing 357 4.8 Summary 369 References 369 Problems 375 5 Fourier Transforms of Analog Signals 383 5.1 Fourier Series 385 5.1.1 Exponential Fourier Series 387 5.1.2 Fourier Series Convergence 391 5.1.3 Trigonometric Fourier Series 397 5.2 Fourier Transform 403 5.2.1 Motivation and Definition 403 5.2.2 Inverse Fourier Transform 408 5.2.3 Properties 411 5.2.4 Symmetry Properties 420 5.3 Extension to L2 (R) 424 5.3.1 Fourier Transforms in L1(R) ∩ L2(R) 425 5.3.2 Definition 427 5.3.3 Isometry 429 5.4 Summary 432 5.4.1 Historical Notes 432 5.4.2 Looking Forward 433 References 433 Problems 434 6 Generalized Fourier Transforms of Analog Signals 440 6.1 Distribution Theory and Fourier Transforms 440 6.1.1 Examples 442 6.1.2 The Generalized Inverse Fourier Transform 443 6.1.3 Generalized Transform Properties 444
- 19. xii CONTENTS 6.2 Generalized Functions and Fourier Series Coefficients 451 6.2.1 Dirac Comb: A Fourier Series Expansion 452 6.2.2 Evaluating the Fourier Coefficients: Examples 454 6.3 Linear Systems in the Frequency Domain 459 6.3.1 Convolution Theorem 460 6.3.2 Modulation Theorem 461 6.4 Introduction to Filters 462 6.4.1 Ideal Low-pass Filter 465 6.4.2 Ideal High-pass Filter 465 6.4.3 Ideal Bandpass Filter 465 6.5 Modulation 468 6.5.1 Frequency Translation and Amplitude Modulation 469 6.5.2 Baseband Signal Recovery 470 6.5.3 Angle Modulation 471 6.6 Summary 475 References 476 Problems 477 7 Discrete Fourier Transforms 482 7.1 Discrete Fourier Transform 483 7.1.1 Introduction 484 7.1.2 The DFT’s Analog Frequency-Domain Roots 495 7.1.3 Properties 497 7.1.4 Fast Fourier Transform 501 7.2 Discrete-Time Fourier Transform 510 7.2.1 Introduction 510 7.2.2 Properties 529 7.2.3 LTI Systems and the DTFT 534 7.3 The Sampling Theorem 538 7.3.1 Band-Limited Signals 538 7.3.2 Recovering Analog Signals from Their Samples 540 7.3.3 Reconstruction 543 7.3.4 Uncertainty Principle 545 7.4 Summary 547 References 548 Problems 549
- 20. CONTENTS xiii 8 The z-Transform 554 8.1 Conceptual Foundations 555 8.1.1 Definition and Basic Examples 555 8.1.2 Existence 557 8.1.3 Properties 561 8.2 Inversion Methods 566 8.2.1 Contour Integration 566 8.2.2 Direct Laurent Series Computation 567 8.2.3 Properties and z-Transform Table Lookup 569 8.2.4 Application: Systems Governed by Difference Equations 571 8.3 Related Transforms 573 8.3.1 Chirp z-Transform 573 8.3.2 Zak Transform 575 8.4 Summary 577 8.4.1 Historical Notes 578 8.4.2 Guide to Problems 578 References 578 Problems 580 9 Frequency-Domain Signal Analysis 585 9.1 Narrowband Signal Analysis 586 9.1.1 Single Oscillatory Component: Sinusoidal Signals 587 9.1.2 Application: Digital Telephony DTMF 588 9.1.3 Filter Frequency Response 604 9.1.4 Delay 605 9.2 Frequency and Phase Estimation 608 9.2.1 Windowing 609 9.2.2 Windowing Methods 611 9.2.3 Power Spectrum Estimation 613 9.2.4 Application: Interferometry 618 9.3 Discrete filter design and implementation 620 9.3.1 Ideal Filters 621 9.3.2 Design Using Window Functions 623 9.3.3 Approximation 624 9.3.4 Z-Transform Design Techniques 626 9.3.5 Low-Pass Filter Design 632
- 21. xiv CONTENTS 9.3.6 Frequency Transformations 639 9.3.7 Linear Phase 640 9.4 Wideband Signal Analysis 643 9.4.1 Chirp Detection 643 9.4.2 Speech Analysis 646 9.4.3 Problematic Examples 650 9.5 Analog Filters 650 9.5.1 Introduction 651 9.5.2 Basic Low-Pass Filters 652 9.5.3 Butterworth 654 9.5.4 Chebyshev 664 9.5.5 Inverse Chebyshev 670 9.5.6 Elliptic Filters 676 9.5.7 Application: Optimal Filters 685 9.6 Specialized Frequency-Domain Techniques 686 9.6.1 Chirp-z Transform Application 687 9.6.2 Hilbert Transform 688 9.6.3 Perfect Reconstruction Filter Banks 694 9.7 Summary 700 References 701 Problems 704 10 Time-Frequency Signal Transforms 712 10.1 Gabor Transforms 713 10.1.1 Introduction 715 10.1.2 Interpretations 717 10.1.3 Gabor Elementary Functions 718 10.1.4 Inversion 723 10.1.5 Applications 730 10.1.6 Properties 735 10.2 Short-Time Fourier Transforms 736 10.2.1 Window Functions 736 10.2.2 Transforming with a General Window 738 10.2.3 Properties 740 10.2.4 Time-Frequency Localization 741
- 22. CONTENTS xv 10.3 Discretization 747 10.3.1 Transforming Discrete Signals 747 10.3.2 Sampling the Short-Time Fourier Transform 749 10.3.3 Extracting Signal Structure 751 10.3.4 A Fundamental Limitation 754 10.3.5 Frames of Windowed Fourier Atoms 757 10.3.6 Status of Gabor’s Problem 759 10.4 Quadratic Time-Frequency Transforms 760 10.4.1 Spectrogram 761 10.4.2 Wigner–Ville Distribution 761 10.4.3 Ambiguity Function 769 10.4.4 Cross-Term Problems 769 10.4.5 Kernel Construction Method 770 10.5 The Balian–Low Theorem 771 10.5.1 Orthonormal Basis Decomposition 772 10.5.2 Frame Decomposition 777 10.5.3 Avoiding the Balian–Low Trap 787 10.6 Summary 787 10.6.1 Historical Notes 789 10.6.2 Resources 790 10.6.3 Looking Forward 791 References 791 Problems 794 11 Time-Scale Signal Transforms 802 11.1 Signal Scale 803 11.2 Continuous Wavelet Transforms 803 11.2.1 An Unlikely Discovery 804 11.2.2 Basic Theory 804 11.2.3 Examples 815 11.3 Frames 821 11.3.1 Discretization 822 11.3.2 Conditions on Wavelet Frames 824 11.3.3 Constructing Wavelet Frames 825 11.3.4 Better Localization 829 11.4 Multiresolution Analysis and Orthogonal Wavelets 832 11.4.1 Multiresolution Analysis 835
- 23. xvi CONTENTS 11.4.2 Scaling Function 847 11.4.3 Discrete Low-Pass Filter 852 11.4.4 Orthonormal Wavelet 857 11.5 Summary 863 References 865 Problems 867 12 Mixed-Domain Signal Analysis 873 12.1 Wavelet Methods for Signal Structure 873 12.1.1 Discrete Wavelet Transform 874 12.1.2 Wavelet Pyramid Decomposition 875 12.1.3 Application: Multiresolution Shape Recognition 883 12.2 Mixed-Domain Signal Processing 893 12.2.1 Filtering Methods 895 12.2.2 Enhancement Techniques 897 12.3 Biophysical Applications 900 12.3.1 David Marr’s Program 900 12.3.2 Psychophysics 900 12.4 Discovering Signal Structure 904 12.4.1 Edge Detection 905 12.4.2 Local Frequency Detection 908 12.4.3 Texture Analysis 912 12.5 Pattern Recognition Networks 913 12.5.1 Coarse-to-Fine Methods 913 12.5.2 Pattern Recognition Networks 915 12.5.3 Neural Networks 916 12.5.4 Application: Process Control 916 12.6 Signal Modeling and Matching 917 12.6.1 Hidden Markov Models 917 12.6.2 Matching Pursuit 918 12.6.3 Applications 918 12.7 Afterword 918 References 919 Problems 925 Index 929
- 24. xvii PREFACE PREFACE This text provides a complete introduction to signal analysis. Inclusion of funda- mental ideas—analog and discrete signals, linear systems, Fourier transforms, and sampling theory—makes it suitable for introductory courses, self-study, and refreshers in the discipline. But along with these basics, Signal Analysis: Time, Frequency, Scale, and Structure gives a running tutorial on functional analysis—the mathematical concepts that generalize linear algebra and underlie signal theory. While the advanced mathematics can be skimmed, readers who absorb the material will be prepared for latter chapters that explain modern mixed-domain signal analy- sis: Short-time Fourier (Gabor) and wavelet transforms. Quite early in the presentation, Signal Analysis surveys methods for edge detec- tion, segmentation, texture identification, template matching, and pattern recogni- tion. Typically, these are only covered in image processing or computer vision books. Indeed, the fourth chapter might seem like a detour to some readers. But the techniques are essential to one-dimensional signal analysis as well. Soon after learning the rudiments of systems and convolutions, students are invited to apply the ideas to make a computer understand a signal. Does it contain anything significant, expected, or unanticipated? Where are the significant parts of the signal? What are its local features, where are their boundaries, and what is their structure? The diffi- culties inherent in understanding a signal become apparent, as does the need for a comprehensive approach to signal frequency. This leads to the chapters on the fre- quency domain. Various continous and discrete Fourier transforms make their appearance. Their application, in turn, proves to be problematic for signals with transients, localized frequency components, and features of varying scale. The text delves into the new analytical tools—some discovered only in the last 20 years—for such signals. Time-frequency and time-scale transforms, their underlying mathe- matical theory, their limitations, how they differently reveal signal structure, and their promising applications complete the book. So the highlights of this book are: • The signal analysis perspective; • The tutorial material on advanced mathematics—in particular function spaces, cast in signal processing terms; • The coverage of the latest mixed domain analysis methods. We thought that there is a clear need for a text that begins at a basic level while taking a signal analysis as opposed to signal processing perspective on applications.
- 25. xviii PREFACE The goal of signal analysis is to arrive at a structural description of a signal so that later high-level algorithms can interpret its content. This differs from signal pro- cessing per se, which only seeks to modify the input signal, without changing its fundamental nature as a one-dimensional sequence of numerical values. From this viewpoint, signal analysis stands within the scope of artificial intelligence. Many modern technologies demand its skills. Human–computer interaction, voice recog- nition, industrial process control, seismology, bioinformatics, and medicine are examples. Signal Analysis provides the abstract mathematics and functional analysis which is missing from the backgrounds of many readers, especially undergraduate science and engineering students and professional engineers. The reader can begin comfort- ably with the basic ideas. The book gradually dispenses the mathematics of Hilbert spaces, complex analysis, disributions, modern integration theory, random signals, and analog Fourier transforms; the less mathematically adept reader is not over- whelmed with hard analysis. There has been no easy route from standard signal pro- cessing texts to the latest treatises on wavelets, Gabor transforms, and the like. The gap must be spanned with knowledge of advanced mathematics. And this has been a problem for too many engineering students, classically-educated applied research- ers, and practising engineers. We hope that Signal Analysis removes the obstacles. It has the signal processing fundamentals, the signal analysis perspective, the mathe- matics, and the bridge from all of these to crucial developments that began in the mid-1980s. The last three chapters of this book cover the latest mixed-domain transform methods: Gabor transforms, wavelets, multiresolution analysis, frames, and their applications. Researchers who need to keep abreast of the advances that are revolu- tionizing their discipline will find a complete introductory treatment of time- frequency and time-scale transforms in the book. We prove the Balian-Low theorem, which pinpoints a limitation on short-time Fourier representations. We had envisioned a much wider scope for mixed-domain applications. Ultimately, the publication schedule and the explosive growth of the field prevented us from achieving a thorough coverage of all principal algorithms and applications—what might have been a fourth highlight of the book. The last chapter explains briefly how to use the new methods in applications, contrasts them with time domain tactics, and contains further refer- ences to the research literature. Enough material exists for a year-long university course in signal processing and analysis. Instructors who have students captive for two semesters may cover the chapters in order. When a single semester must suffice, Chapters 1–3, 5, 7, 8, and 9 comprise the core ideas. We recommend at least the sections on segmenta- tion and thresholding in Chapter 4. After some programming experiments, the stu- dents will see how hard it is to make computers do what we humans take for granted. The instructor should adjust the pace according to the students’ prepara- tion. For instance, if a system theory course is prerequisite—as is typical in the undergraduate engineering curriculum—then the theoretical treatments of signal spaces, the Dirac delta, and the Fourier transforms are appropriate. An advanced course can pick up the mathematical theory, the pattern recognition material in
- 26. PREFACE xix Chapter 4, the generalized Fourier transform in Chapter 6, and the analog filter designs in Chapter 9. But the second semester work should move quickly to and concentrate upon Chapters 10–12. This equips the students for reading the research literature. RONALD L. ALLEN San José, California DUNCAN W. MILLS Mountain View, California
- 28. xxi ACKNOWLEDGMENTS We would like to thank the editorial and production staffs on John Wiley and Sons and IEEE Press for their efficiency, courtesy, patience, and professionalism while we wrote this book. We are especially grateful to Marilyn G. Catis and Anthony VenGraitis of IEEE Press for handling incremental submissions, managing reviews, and providing general support over the years. We are grateful to Beverley Andalora for help with the figures, to William Parker of Philips Speech Recognition Systems for providing digital speech samples, and to KLA-Tencor Corporation for reflecto- metry and scanning electron microscopy data samples. RONALD L. ALLEN DUNCAN W. MILLS
- 30. 1 Signal Analysis: Time, Frequency, Scale, and Structure, by Ronald L. Allen and Duncan W. Mills ISBN: 0-471-23441-9 Copyright © 2004 by Institute of Electrical and Electronics Engineers, Inc. CHAPTER 1 Signals: Analog, Discrete, and Digital Analog, discrete, and digital signals are the raw material of signal processing and analysis. Natural processes, whether dependent upon or independent of human con- trol, generate analog signals; they occur in a continuous fashion over an interval of time or space. The mathematical model of an analog signal is a function de$ned over a part of the real number line. Analog signal conditioning uses conventional electronic circuitry to acquire, amplify, $lter, and transmit these signals. At some point, digital processing may take place; today, this is almost always necessary. Per- haps the application requires superior noise immunity. Intricate processing steps are also easier to implement on digital computers. Furthermore, it is easier to improve and correct computerized algorithms than systems comprised of hard-wired analog components. Whatever the rationale for digital processing, the analog signal is cap- tured, stored momentarily, and then converted to digital form. In contrast to an ana- log signal, a discrete signal has values only at isolated points. Its mathematical representation is a function on the integers; this is a fundamental difference. When the signal values are of $nite precision, so that they can be stored in the registers of a computer, then the discrete signal is more precisely known as a digital signal. Digital signals thus come from sampling an analog signal, and—although there is such a thing as an analog computer—nowadays digital machines perform almost all analytical computations on discrete signal data. This has not, of course, always been the case; only recently have discrete tech- niques come to dominate signal processing. The reasons for this are both theoretical and practical. On the practical side, nineteenth century inventions for transmitting words, the telegraph and the telephone—written and spoken language, respectively—mark the beginnings of engineered signal generation and interpretation technologies. Mathe- matics that supports signal processing began long ago, of course. But only in the nineteenth century did signal theory begin to distinguish itself as a technical, engi- neering, and scienti$c pursuit separate from pure mathematics. Until then, scientists did not see mathematical entities—polynomials, sinusoids, and exponential func- tions, for example—as sequences of symbols or carriers of information. They were envisioned instead as ideal shapes, motions, patterns, or models of natural processes.
- 31. 2 SIGNALS: ANALOG, DISCRETE, AND DIGITAL The development of electromagnetic theory and the growth of electrical and electronic communications technologies began to divide these sciences. The functions of mathematics came to be studied as bearing information, requiring modi$cation to be useful, suitable for interpretation, and having a meaning. The life story of this new discipline—signal processing, communications, signal analysis, and information theory—would follow a curious and ironic path. Electromagnetic waves consist of coupled electric and magnetic $elds that oscillate in a sinusoidal pattern and are perpendicular to one another and to their direction of propagation. Fourier discovered that very general classes of functions, even those containing dis- continuities, could be represented by sums of sinusoidal functions, now called a Fourier series [1]. This surprising insight, together with the great advances in analog communication methods at the beginning of the twentieth century, captured the most attention from scientists and engineers. Research efforts into discrete techniques were producing important results, even as the analog age of signal processing and communication technology charged ahead. Discrete Fourier series calculations were widely understood, but seldom car- ried out; they demanded quite a bit of labor with pencil and paper. The $rst theoret- ical links between analog and discrete signals were found in the 1920s by Nyquist,1 in the course of research on optimal telegraphic transmission mechanisms [2]. Shannon2 built upon Nyquist’s discovery with his famous sampling theorem [3]. He also proved something to be feasible that no one else even thought possible: error- free digital communication over noisy channels. Soon thereafter, in the late 1940s, digital computers began to appear. These early monsters were capable of perform- ing signal processing operations, but their speed remained too slow for some of the most important computations in signal processing—the discrete versions of the Fourier series. All this changed two decades later when Cooley and Tukey disclosed their fast Fourier transform (FFT) algorithm to an eager computing public [4–6]. Digital computations of Fourier’s series were now practical on real-time signal data, and in the following years digital methods would proliferate. At the present time, digital systems have supplanted much analog circuitry, and they are the core of almost all signal processing and analysis systems. Analog techniques handle only the early signal input, output, and conditioning chores. There are a variety of texts available covering signal processing. Modern intro- ductory systems and signal processing texts cover both analog and discrete theory [7–11]. Many re#ect the shift to discrete methods that began with the discovery of the FFT and was fueled by the ever-increasing power of computing machines. These often concentrate on discrete techniques and presuppose a background in analog 1As a teenager, Harry Nyquist (1887–1976) emigrated from Sweden to the United States. Among his many contributions to signal and communication theory, he studied the relationship between analog sig- nals and discrete signals extracted from them. The term Nyquist rate refers to the sampling frequency necessary for reconstructing an analog signal from its discrete samples. 2Claude E. Shannon (1916–2001) founded the modern discipline of information theory. He detailed the af$nity between Boolean logic and electrical circuits in his 1937 Masters thesis at the Massachusetts Institute of Technology. Later, at Bell Laboratories, he developed the theory of reliable communication, of which the sampling theorem remains a cornerstone.
- 32. 3 signal processing [12–15]. Again, there is a distinction between discrete and digital signals. Discrete signals are theoretical entities, derived by taking instantaneous— and therefore exact—samples from analog signals. They might assume irrational values at some time instants, and the range of their values might be in$nite. Hence, a digital computer, whose memory elements only hold limited precision values, can only process those discrete signals whose values are $nite in number and $nite in their precision—digital signals. Early texts on discrete signal processing sometimes blurred the distinction between the two types of signals, though some further editions have adopted the more precise terminology. Noteworthy, however, are the burgeoning applications of digital signal processing integrated circuits: digital tele- phony, modems, mobile radio, digital control systems, and digital video to name a few. The $rst high-de$nition television (HDTV) systems were analog; but later, superior HDTV technologies have relied upon digital techniques. This technology has created a true digital signal processing literature, comprised of the technical manuals for various DSP chips, their application notes, and general treatments on fast algorithms for real-time signal processing and analysis applications on digital signal processors [16–21]. Some of our later examples and applications offer some observations on architectures appropriate for signal processing, special instruction sets, and fast algorithms suitable for DSP implementation. This chapter introduces signals and the mathematical tools needed to work with them. Everyone should review this chapter’s $rst six sections. This $rst chapter com- bines discussions of analog signals, discrete signals, digital signals, and the methods to transition from one of these realms to another. All that it requires of the reader is a familiarity with calculus. There are a wide variety of examples. They illustrate basic signal concepts, $ltering methods, and some easily understood, albeit limited, techniques for signal interpretation. The $rst section introduces the terminology of signal processing, the conventional architecture of signal processing systems, and the notions of analog, discrete, and digital signals. It describes signals in terms of mathematical models—functions of a single real or integral variable. A speci$cation of a sequence of numerical values ordered by time or some other spatial dimension is a time domain description of a signal. There are other approaches to signal description: the frequency and scale domains, as well as some—relatively recent— methods for combining them with the time domain description. Sections 1.2 and 1.3 cover the two basic signal families: analog and discrete, respectively. Many of the signals used as examples come from conventional algebra and analysis. The discussion gets progressively more formal. Section 1.4 covers sampling and interpolation. Sampling picks a discrete signal from an analog source, and interpo- lation works the other way, restoring the gaps between discrete samples to fashion an analog signal from a discrete signal. By way of these operations, signals pass from the analog world into the discrete world and vice versa. Section 1.5 covers periodicity, and foremost among these signals is the class of sinusoids. These sig- nals are the fundamental tools for constructing a frequency domain description of a signal. There are many special classes of signals that we need to consider, and Sec- tion 1.6 quickly collects them and discusses their properties. We will of course expand upon and deepen our understanding of these special types of signals SIGNALS: ANALOG, DISCRETE, AND DIGITAL
- 33. 4 SIGNALS: ANALOG, DISCRETE, AND DIGITAL throughout the book. Readers with signal processing backgrounds may quickly scan this material; however, those with little prior work in this area might well linger over these parts. The last two sections cover some of the mathematics that arises in the detailed study of signals. The complex number system is essential for characterizing the tim- ing relationships in signals and their frequency content. Section 1.7 explains why complex numbers are useful for signal processing and exposes some of their unique properties. Random signals are described in Section 1.8. Their application is to model the unpredictability in natural signals, both analog and discrete. Readers with a strong mathematics background may wish to skim the chapter for the special sig- nal processing terminology and skip Sections 1.7 and 1.8. These sections can also be omitted from a $rst reading of the text. A summary, a list of references, and a problem set complete the chapter. The sum- mary provides supplemental historical notes. It also identi$es some software resources and publicly available data sets. The references point out other introductory texts, reviews, and surveys from periodicals, as well as some of the recent research. 1.1 INTRODUCTION TO SIGNALS There are several standpoints from which to study signal analysis problems: empiri- cal, technical, and theoretical. This chapter uses all of them. We present lots of examples, and we will return to them often as we continue to develop methods for their processing and interpretation. After practical applications of signal processing and analysis, we introduce some basic terminology, goals, and strategies. Our early methods will be largely experimental. It will be often be dif$cult to decide upon the best approach in an application; this is the limitation of an intuitive approach. But there will also be opportunities for making technical observations about the right mathematical tool or technique when engaged in a practical signal analysis problem. Mathematical tools for describing signals and their characteristics will continue to illuminate this technical side to our work. Finally, some abstract considerations will arise at the end of the chapter when we consider complex num- bers and random signal theory. Right now, however, we seek only to spotlight some practical and technical issues related to signal processing and analysis applications. This will provide the motivation for building a signi$cant theoretical apparatus in the sequel. 1.1.1 Basic Concepts Signals are symbols or values that appear in some order, and they are familiar enti- ties from science, technology, society, and life. Examples $t easily into these cate- gories: radio-frequency emissions from a distant quasar; telegraph, telephone, and television transmissions; people speaking to one another, using hand gestures; rais- ing a sequence of #ags upon a ship’s mast; the echolocation chirp of animals such as bats and dolphins; nerve impulses to muscles; and the sensation of light patterns
- 34. INTRODUCTION TO SIGNALS 5 striking the eye. Some of these signal values are quanti$able; the phenomenon is a measurable quantity, and its evolution is ordered by time or distance. Thus, a resi- dential telephone signal’s value is known by measuring the voltage across the pair of wires that comprise the circuit. Sound waves are longitudinal and produce minute, but measurable, pressure variations on a listener’s eardrum. On the other hand, some signals appear to have a representation that is at root not quanti$able, but rather symbolic. Thus, most people would grant that sign language gestures, maritime signal #ags, and even ASCII text could be considered signals, albeit of a symbolic nature. Let us for the moment concentrate on signals with quanti$able values. These are the traditional mathematical signal models, and a rich mathematical theory is avail- able for studying them. We will consider signals that assume symbolic values, too, but, unlike signals with quanti$able values, these entities are better described by relational mathematical structures, such as graphs. Now, if the signal is a continuously occurring phenomenon, then we can repre- sent it as a function of a time variable t; thus, x(t) is the value of signal x at time t. We understand the units of measurement of x(t) implicitly. The signal might vary with some other spatial dimension other than time, but in any case, we can suppose that its domain is a subset of the real numbers. We then say that x(t) is an analog signal. Analog signal values are read from conventional indicating devices or sci- enti$c instruments, such as oscilloscopes, dial gauges, strip charts, and so forth. An example of an analog signal is the seismogram, which records the shaking motion of the ground during an earthquake. A precision instrument, called a seismo- graph, measures ground displacements on the order of a micron (106 m) and pro- duces the seismogram on a paper strip chart attached to a rotating drum. Figure 1.1 shows the record of the Loma Prieta earthquake, centered in the Santa Cruz moun- tains of northern California, which struck the San Francisco Bay area on 18 October 1989. Seismologists analyze such a signal in several ways. The total de#ection of the pen across the chart is useful in determining the temblor’s magnitude. Seismograms register three important types of waves: the primary, or P waves; the secondary, or S waves; and the surface waves. P waves arrive $rst, and they are compressive, so their direction of motion aligns with the wave front propagation [22]. The transverse S waves follow. They oscillate perpendicular to the direction of propagation. Finally, the large, sweeping surface waves appear on the trace. This simple example illustrates processing and analysis concepts. Processing the seismogram signal is useful to remove noise. Noise can be minute ground motions from human activity (construction activity, heavy machinery, vehicles, and the like), or it may arise from natural processes, such as waves hitting the beach. Whatever the source, an important signal processing operation is to smooth out these minute ripples in the seismogram trace so as to better detect the occurrence of the initial indications of a seismic event, the P waves. They typically manifest themselves as seismometer needle motions above some threshold value. Then the analysis prob- lem of $nding when the S waves begin is posed. Figure 1.1 shows the result of a sig- nal analysis; it slices the Loma Prieta seismogram into its three constituent wave
- 35. 6 SIGNALS: ANALOG, DISCRETE, AND DIGITAL trains. This type of signal analysis can be performed by inspection on analog seis- mograms. Now, the time interval between the arrival of the P and S waves is critical. These undulations are simultaneously created at the earthquake’s epicenter; however, they travel at different, but known, average speeds through the earth. Thus, if an analysis of the seismogram can reveal the time that these distinct wave trains arrive, then the time difference can be used to measure the distance from the instrument to the earth- quake’s epicenter. Reports from three separate seismological stations are suf$cient to locate the epicenter. Analyzing smaller earthquakes is also important. Their loca- tion and the frequency of their occurrence may foretell a larger temblor [23]. Fur- ther, soundings in the earth are indicative of the underlying geological strata; seismologists use such methods to locate oil deposits, for example [24]. Other simi- lar applications include the detection of nuclear arms detonations and avalanches. For all of these reasons—scienti$c, economic, and public safety—seismic signal intepretation is one of the most important areas in signal analysis and one of the areas in which new methods of signal analysis have been pioneered. These further signal interpretation tasks are more troublesome for human interpreters. The signal behavior that distinguishes a small earthquake from a distant nuclear detonation is not apparent. This demands thorough computerized analysis. Fig. 1.1. Seismogram of the magnitude 7.1 Loma Prieta earthquake, recorded by a seis- mometer at Kevo, Finland. The $rst wiggle—some eight minutes after the actual event— marks the beginning of the low-magnitude P waves. The S waves arrive at approximately t = 1200 s, and the large sweeping surface waves begin near t = 2000 s.
- 36. INTRODUCTION TO SIGNALS 7 Suppose, therefore, that the signal is a discrete phenomenon, so that it occurs only at separate time instants or distance intervals and not continuously. Then we represent it as a function on a subset of the integers x(n) and we identify x(n) as a discrete signal. Furthermore, some discrete signals may have only a limited range of values. Their measurable values can be stored in the memory cells of a digital com- puter. The discrete signals that satisfy this further constraint are called digital signals. Each of these three types of signals occurs at some stage in a conventional com- puterized signal acquisition system (Figure 1.2). Analog signals arise from some quanti$able, real-world process. The signal arrives at an interface card attached to the computer’s input–output bus. There are generally some signal ampli$cation and conditioning components, all analog, at the system’s front end. At the sample and hold circuit, a momentary stor- age component—a capacitor, for example—holds the signal value for a small time interval. The sampling occurs at regular intervals, which are set by a timer. Thus, the sequence of quantities appearing in the sample and hold device represents the dis- crete form of the signal. While the measurable quantity remains in the sample and hold unit, a digitization device composes its binary representation. The extracted value is moved into a digital acquisition register of $nite length, thereby completing the analog-to-digital conversion process. The computer’s signal processing software or its input–output driver reads the digital signal value out of the acquisition regis- ter, across the input–output bus, and into main memory. The computer itself may be a conventional general-purpose machine, such as a personal computer, an engineer- ing workstation, or a mainframe computer. Or the processor may be one of the many special purpose digital signal processors (DSPs) now available. These are now a popular design choice in signal processing and analysis systems, especially those with strict execution time constraints. Some natural processes generate more than one measurable quantity as a func- tion of time. Each such quantity can be regarded as a separate signal, in which case Sensor Amplifier Filter Analog Discrete Timing Digital Sample and Hold 12 1 2 3 4 5 6 7 8 9 10 11 Conversion Fig. 1.2. Signal acquisition into a computer. Analog, discrete, and digital signals each occur—at least in principle—within such a system.
- 37. 8 SIGNALS: ANALOG, DISCRETE, AND DIGITAL (a) (b) 200 205 210 215 220 225 230 235 240 245 250 -15 -10 -5 0 5 10 15 20 Discrete channel 1 signal Fig. 1.3. A multichannel signal: The electroencephalogram (EEG) taken from a healthy young person, with eyes open. The standard EEG sensor arrangement consists of 19 elec- trodes (a). Discrete data points of channel one (b). Panels (c) and (d) show the complete traces for the $rst two channels, x1(n) and x2(n). These traces span an eight second time inter- val: 1024 samples. Note the jaggedness superimposed on gentler wavy patterns. The EEG varies according to whether the patient’s eyes are open and according to the health of the individual; markedly different EEG traces typify, for example, Alzheimer’s disease.
- 38. INTRODUCTION TO SIGNALS 9 they are all functions of the same independent variable with the same domain. Alter- natively, it may be technically useful to maintain the multiple quantities together as a vector. This is called a multichannel signal. We use boldface letters to denote mul- tichannel signals. Thus, if x is analog and has N channels, then x(t) = (x1(t), x2(t), …, xN(t)), where the analog xi(t) are called the component or channel signals. Similarly, if x is discrete and has N channels, then x(n) = (x1(n), x2(n), …, xN(n)). One biomedical signal that is useful in diagnosing brain injuries, mental illness, and conditions such as Alzheimer’s disease is the electroencephalogram (EEG) [25], a multichannel signal. It records electrical potential differences, or voltages, that arise from the interactions of massive numbers of neurons in different parts of the brain. For an EEG, 19 electrodes are attached from the front to the back of the scalp, in a two–$ve–$ve–$ve–two arrangement (Figure 1.3). The EEG traces in Figure 1.3 are in fact digital signals, acquired one sample every 7.8 ms, or at a sampling frequency of 128 Hz. The signal appears to be conti- nuous in nature, but this is due to the close spacing of the samples and linear inter- polation by the plotting package. Another variation on the nature of signals is that they may be functions of more than one independent variable. For example, we might measure air 0 200 400 600 800 1000 1200 -40 -20 0 20 40 (c) 0 200 400 600 800 1000 1200 -40 -20 0 20 40 (d) Fig. 1.3 (Continued)
- 39. 10 SIGNALS: ANALOG, DISCRETE, AND DIGITAL temperature as a function of height: T(h) is an analog signal. But if we con- sider that the variation may occur along a north-to-south line as well, then the temperature depends upon a distance measure x as well: T(x, h). Finally, over an area with location coordinates (x, y), the air temperature is a continuous function of three variables T(x, y, h). When a signal has more than one independent vari- able, then it is a multidimensional signal. We usually think of an “image” as recording light intesity measurements of a scene, but multidimensional signals— especially those with two or three independent variables—are usually called images. Images may be discrete too. Temperature readings taken at kilometer intervals on the ground and in the air produce a discrete signal T(m, n, k). A dis- crete signal is a sequence of numerical values, whereas an image is an array of numerical values. Two-dimensional image elements, especially those that repre- sent light intensity values, are called pixels, an acronym for picture elements. Occasionally, one encounters the term voxel, which is a three-dimensional signal value, or a volume element. An area of multidimensional signal processing and analysis of considerable importance is the intepretation of images of landscapes acquired by satellites and high altitude aircraft. Figure 1.4. shows some examples. Typical tasks are to automatically distinguish land from sea; determine the amount and extent of sea ice; distinguish agricultural land, urban areas, and forests; and, within the agricultural regions, recognize various crop types. These are remote sensing applications. Processing two-dimensional signals is more commonly called picture or image processing, and the task of interpreting an image is called image analysis or com- puter vision. Many researchers are involved in robotics, where their efforts couple computer vision ideas with manipulation of the environment by a vision-based machine. Consequently, there is a vast, overlapping literature on image processing [26–28], computer vision [29–31], and robotics [32]. Our subject, signal analysis, concentrates on the mathematical foundations, pro- cessing, and especially the intepretation of one-dimensional, single-valued signals. Generally, we may select a single channel of a multichannel signal for consider- ation; but we do not tackle problems speci$c to multichannel signal interpretation. Likewise, we do not delve deeply into image processing and analysis. Certain images do arise, so it turns out, in several important techniques for analyzing sig- nals. Sometimes a daunting one-dimensional problem can be turned into a tractable two-dimensional task. Thus, we prefer to pursue the one-dimensional problem into the multidimensional realm only to the point of acknowledging that a straightfor- ward image analysis will produce the intepretation we seek. So far we have introduced the basic concepts of signal theory, and we have considered some examples: analog, discrete, multichannel, and multidimensional signals. In each case we describe the signals as sequences of numerical values, or as a function of an independent time or other spatial dimension variable. This con- stitutes a time-domain description of a signal. From this perspective, we can dis- play a signal, process it to produce another signal, and describe its signi$cant features.
- 40. INTRODUCTION TO SIGNALS 11 1.1.2 Time-Domain Description of Signals Since time #ows continuously and irreversibly, it is natural to describe sequential signal values as given by a time ordering. This is often, but not always, the case; many signals depend upon a distance measure. It is also possible, and sometimes a very important analytical step, to consider signals as given by order of a salient event. Conceiving the signal this way makes the dependent variable—the signal value—a function of time, distance, or some other quantity indicated between successive events. Whether the independent variable is time, some other spatial dimension, or a counting of events, when we represent and discuss a signal in terms of its ordered values, we call this the time-domain description of a signal. Fig. 1.4. Aerial scenes. Distinguishing terrain types is a typical problem of image analysis, the interpretation of two-dimensional signals. Some problems, however, admit a one-dimen- sional solution. A sample line through an image is in fact a signal, and it is therefore suitable for one-dimensional techniques. (a) Agricultural area. (b) Forested region. (c) Ice at sea. (d) Urban area.
- 41. 12 SIGNALS: ANALOG, DISCRETE, AND DIGITAL Note that a precise time-domain description may elude us, and it may not even be possible to specify a signal’s values. A fundamentally unknowable or random pro- cess is the source of such signals. It is important to develop methods for handling the randomness inherent in signals. Techniques that presuppose a theory of signal randomness are the topic of the $nal section of the chapter. Next we look further into two application areas we have already touched upon: biophysical and geophysical signals. Signals from representative applications in these two areas readily illustrate the time-domain description of signals. 1.1.2.1 Electrocardiogram Interpretation. Electrocardiology is one of the earliest techniques in biomedicine. It also remains one of the most important. The excitation and recovery of the heart muscles cause small electrical potentials, or volt- ages, on the order of a millivolt, within the body and measurable on the skin. Cardio- logists observe the regularity and shape of this voltage signal to diagnose heart con- ditions resulting from disease, abnormality, or injury. Examples include cardiac dysrhythmia and $brillation, narrowing of the coronary arteries, and enlargement of the heart [33]. Automatic interpretation of ECGs is useful for many aspects of clini- cal and emergency medicine: remote monitoring, as a diagnostic aid when skilled cardiac care personnel are unavailable, and as a surgical decision support tool. A modern electrocardiogram (ECG or EKG) contains traces of the voltages from 12 leads, which in biomedical parlance refers to a con$guration of electrodes attached to the body [34]. Refer to Figure 1.5. The voltage between the arms is Lead I, Lead II is the potential between the right arm and left leg, and Lead III reads between the left arm and leg. The WCT is a common point that is formed by connecting the three limb electrodes through weighting resistors. Lead aVL measures potential difference between the left arm and the WCT. Similarly, lead aVR is the voltage between the right arm and the WCT. Lead aVF is between the left leg and the WCT. Finally, six more electrodes are $xed upon the chest, around the heart. Leads V1 through V6 measure the voltages between these sensors and the WCT. This circuit Fig. 1.5. The standard ECG con$guration produces 12 signals from various electrodes attached to the subject’s chest, arms, and leg.
- 42. INTRODUCTION TO SIGNALS 13 arrangement is complicated; in fact, it is redundant. Redundancy provides for situa- tions where a lead produces a poor signal and allows some cross-checking of the readings. Interpretation of 12-lead ECGs requires considerable training, experience, and expert judgment. What does an ECG trace look like? Figure 1.6 shows an ECG trace from a single lead. Generally, an ECG has three discernible pulses: the P wave, the QRS complex, and the T wave. The P wave occurs upon excitation of the auricles of the heart, when they draw in blood from the body and lungs. The large-magnitude QRS complex occurs during the contraction of the vertricles as they contract to pump blood out of the heart. The Q and S waves are negative pulses, and the R wave is a positive pulse. The T wave arises during repolarization of the ventricles. The ECG signal is origi- nally analog in nature; it is the continuous record of voltages produce across the var- ious leads supported by the instrument. We could attach a millivoltmeter across an electrode pair and watch the needle jerk back and forth. Visualizing the signal’s shape is easier with an oscilloscope, of course, because the instrument records the trace on its cathode ray tube. Both of these instruments display analog waveforms. If we could read the oscilloscope’s output at regular time instants with perfect precision, then we would have—in principle, at least—a discrete representation of the ECG. But for computer display and automatic interpretation, the analog signal must be converted to digital form. In fact, Figure 1.6 is the result of such a digitization. The signal v(n) appears continuous due to the large number of samples and the interpolating lines drawn by the graphics package that produced the illustration. Interpreting ECGs is often dif$cult, especially in abnormal traces. A wide litera- ture describing the 12-lead ECG exists. There are many guides to help technicians, nurses, and physicians use it to diagnose heart conditions. Signal processing and analysis of ECGs is a very active research area. Reports on new techniques, algo- rithms, and comparison studies continue to appear in the biomedical engineering and signal analysis literature [35]. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 −2000 −1500 −1000 −500 0 500 1000 1500 2000 Electrocardiogram (ECG) record from a human (male) in a supine position n x(n) Fig. 1.6. One lead of an ECG: A human male in supine position. The sampling rate is 1 kHz, and the samples are digitized at 12 bits per sample. The irregularity of the heartbeat is evident.
- 43. 14 SIGNALS: ANALOG, DISCRETE, AND DIGITAL One technical problem in ECG interpretation is to assess the regularity of the heart beat. As a time-domain signal description problem, this involves $nding the separation between peaks of the QRS complex (Figure 1.6). Large time variations between peaks indicates dysrhythmia. If the time difference between two peaks, v(n1) and v(n0), is , then the instantaneous heart rate becomes beats/m. For the sample in Figure 1.6, this crude computation will, how- ever, produce a wildly varying value of doubtful diagnostic use. The application calls for some kind of averaging and summary statistics, such as a report of the stan- dard deviation of the running heart rate, to monitor the dysrhythmia. There remains the technical problem of how to $nd the time location of QRS peaks. For an ideal QRS pulse, this is not too dif$cult, but the signal analysis algo- rithms must handle noise in the ECG trace. Now, because of the noise in the ECG signal, there are many local extrema. Evidently, the QRS complexes represent sig- nal features that have inordinately high magnitudes; they are mountains above the forest of small-scale artifacts. So, to locate the peak of a QRS pulse, we might select a threshold M that is bigger than the small artifacts and smaller than the QRS peaks. We then deem any maximal, contiguous set of values S = {(n, v(n)): v(n) M} to be a QRS complex. Such regions will be disjoint. After $nding the maximal value inside each such QRS complex, we can calculate between each pair of maxima and give a running heart rate estimate. The task of dividing the signal up into dis- joint regions, such as for the QRS pulses, is called signal segmentation. Chapter 4 explores this time domain procedure more thoroughly. When there is poor heart rhythm, the QRS pulses may be jagged, misshapen, truncated, or irregulary spaced. A close inspection of the trace in Figure 1.7 seems to reveal this very phenomenon. In fact, one type of ventricular disorder that is ∆T n1 n0 – = 60 ∆T ( ) 1 – ∆T 1.35 1.36 1.37 1.38 1.39 1.4 1.41 1.42 1.43 1.44 1.45 x 10 4 −2000 −1500 −1000 −500 0 500 1000 1500 2000 P QRS T x(n) ECG values in one second interval about n = 14000 n Fig. 1.7. Electrocardiogram of a human male, showing the fundametal waves. The 1-s time span around sample n = 14,000 is shown for the ECG of Figure 1.6. Note the locations of the P wave, the QRS complex, and—possibly—the T wave. Is there a broken P wave and a mis- sing QRS pulse near the central time instant?
- 44. INTRODUCTION TO SIGNALS 15 detectable in the ECG, provided that it employs a suf$ciently high sampling rate, is splintering of the QRS complex. In this abnormal condition, the QRS consists of many closely spaced positive and negative transitions rather than a single, strong pulse. Note that in any ECG, there is a signi$cant amount of signal noise. This too is clearly visible in the present example. Good peak detection and pulse location, especially for the smaller P and T waves, often require some data smoothing method. Averaging the signal values produces a smoother signal w(n): . (1.1) The particular formula (1.1) for processing the raw ECG signal to produce a less noisy w(n) is called moving average smoothing or moving average $ltering. This is a typical, almost ubiquitous signal processing operation. Equation (1.1) performs averaging within a symmetric window of width three about v(n). Wider windows are possible and often useful. A window that is too wide can destroy signal features that bear on interpretation. Making a robust application requires judgment and experimentation. Real-time smoothing operations require asymmetric windows. The underlying reason is that a symmetric smoothing window supposes knowledge of future signal values, such as v(n + 1). To wit, as the computer monitoring system acquires each new ECG value v(n), it can calculate the average of the last three values: ; (1.2) but at time instant n, it cannot possibly know the value of v(n +1), which is neces- sary for calculating (1.1). If the smoothing operation occurs of#ine, after the entire set of signal values of interest has already been acquired and stored, then the whole range of signal values is accessible by the computer, and calculation (1.1) is, of course, feasible. When smoothing operations must procede in lockstep with acquisi- tion operations, however, smoothing windows that look backward in time (1.2) must be applied. Yet another method from removing noise from signals is to produce a signal whose values are the median of a window of raw input values. Thus, we might assign (1.3) so that w(n) is the input value that lies closest to the middle of the range of $ve values around v(n). A median $lter tends to be superior to a moving average $lter when the task is to remove isolated, large-magnitude spikes from a source signal. There are many variants. In general, smoothing is a common early processing step in signal analysis systems. In the present application, smoothing reduces the jagged noise in the ECG trace and improves the estimate of the QRS peak’s location. w n ( ) 1 3 -- - v n 1 – ( ) v n ( ) v n 1 + ( ) + + [ ] = w n ( ) 1 3 -- - v n 2 – ( ) v n 1 – ( ) v n ( ) + + [ ] = w n ( ) Median v n 2 – ( ) v n 1 – ( ) v n ( ) v n 1 + ( ) v n 2 + ( ) , , , , { } =
- 45. 16 SIGNALS: ANALOG, DISCRETE, AND DIGITAL Contemplating the above algorithms for $nding QRS peaks, smoothing the raw data, and estimating the instantaneous heart rate, we can note a variety of design choices. For example, how many values should we average to smooth the data? A span too small will fail to blur the jagged, noisy regions of the signal. A span too large may erode some of the QRS peaks. How should the threshold for segmenting QRS pulses be chosen? Again, an algorithm using values too small will falsely identify noisy bumps as QRS pulses. On the other hand, if the threshold values chosen are too large, then valid QRS complexes will be missed. Either circumstance will cause the application to fail. Can the thresholds be chosen automatically? The chemistry of the subject’s skin could change while the leads are attached. This can cause the signal as a whole to trend up or down over time, with the result that the original threshold no longer works. Is there a way to adapt the threshold as the signal average changes so that QRS pulses remain detectable? These are but a few of the problems and tradeoffs involved in time domain signal processing and analysis. Now we have illustrated some of the fundamental concepts of signal theory and, through the present example, have clari$ed the distinction between signal processing and analysis. Filtering for noise removal is a processing task. Signal averaging may serve our purposes, but it tends to smear isolated transients into what may be a quite different overall signal trend. Evidently, one aberrent upward spike can, after smoothing, assume the shape of a QRS pulse. An alternative that addresses this concern is median $ltering. In either case—moving average or median $ltering—the algorithm designer must still decide how wide to make the $lters and discover the proper numerical values for thresholding the smoothed sig- nal. Despite the analytical obstacles posed by signal noise and jagged shape, because of its prominence, the QRS complex is easier to characterize than the P and T waves. There are alternative signal features that can serve as indicators of QRS complex location. We can locate the positive or negative transitions of QRS pulses, for exam- ple. Then the midpoint between the edges marks the center of each pulse, and the distance between these centers determines the instantaneous heart rate. This changes the technical problem from one of $nding a local signal maximum to one of $nding the positive- or negative-transition edges that bound the QRS complexes. Signal analysis, in fact, often revolves around edge detection. A useful indicator of edge presence is the discrete derivative, and a simple threshold operation identi$es the signi$cant changes. 1.1.2.2 Geothermal Measurements. Let us investigate an edge detection problem from geophysics. Ground temperature generally increases with depth. This variation is not as pronounced as the air temperature #uctuations or biophysical sig- nals, to be sure, but local differences emerge due to the geological and volcanic his- tory of the spot, thermal conductivity of the underlying rock strata, and even the amount of radioactivity. Mapping changes in ground termperature are important in the search for geothermal energy resources and are a supplementary indication of the underlying geological structures. If we plot temperature versus depth, we have a
- 46. INTRODUCTION TO SIGNALS 17 signal—the geothermal gradient—that is a function of distance, not time. It ramps up about 10°C per kilometer of depth and is a primary indicator for geothermal prospecting. In general, the geothermal gradient is higher for oceanic than for conti- nental crust. Some 5% of the area of the United States has a gradient in the neighbor- hood of 40°C per kilometer of depth and has potential for use in geothermal power generation. Mathematically, the geothermal gradient is the derivative of the signal with respect to its independent variable, which in this case measures depth into the earth. A very steep overall gradient may promise a geothermal energy source. A localized large magnitude gradient, or edge, in the temperature pro$le marks a geological artifact, such as a fracture zone. An example of the variation in ground temperature as one digs into the earth is shown in Figure 1.8. The above data come from the second of four wells drilled on the Georgia–South Carolina border, in the eastern United States, in 1985 [36]. The temperature $rst declines with depth, which is typical, and then warmth from the earth’s interior appears. Notice the large-magnitude positive gradients at approximately 80 and 175 m; these correspond to fracture zones. Large magnitude deviations often repre- sent physically signi$cant phenomena, and therein lies the importance of reliable methods for detecting, locating, and interpreting signal edges. Finding such large deviations in signal values is once again a time-domain signal analysis problem. Suppose the analog ground temperature signal is g(s), where s is depth into the earth. We seek large values of the derivative Approximating the derivative is possible once the data are digitized. We select a sampling interval D 0 and set x(n) = g(nD); then approximates the geother- mal gradient at depth nD meters. It is further necessary to identify a threshold M for what constitutes a signi$cant geothermal gradient. Threshold selection may rely upon expert scienti$c knowledge. A geophysicist might suggest signi$cant gradients 0 50 100 150 200 250 300 350 14.5 15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 depth (m) temperature, deg C Temperature change versus depth Fig. 1.8. A geothermal signal. The earth’s temperature is sampled at various depths to pro- duce a discrete signal with a spatially independent variable. g′ s ( ) dg/ds. = x′ n ( ) x n 1 + ( ) x n 1 – ( ) – =
- 47. 18 SIGNALS: ANALOG, DISCRETE, AND DIGITAL for the region. If we collect some statistics on temperature gradients, then the outly- ing values may be candidates for threshold selection. Again, there are local variations in the temperature pro$le, and noise does intrude into the signal acquisition appara- tus. Hence, preliminary signal smoothing may once again be useful. Toward this end, we may also employ discrete derivative formulas that use more signal values: . (1.4) Standard numerical analysis texts provide many alternatives [37]. Among the prob- lems at the chapter’s end are several edge detection applications. They weigh some of the alternatives for $ltering, threshold selection, and $nding extrema. For now, let us remark that the edges in the ECG signal (Figure 1.6) are far steeper than the edges in the geothermal trace (Figure 1.8). The upshot is that the signal ana- lyst must tailor the discrete derivative methods to the data at hand. Developing meth- ods for edge detection that are robust with respect to sharp local variation of the signal features proves to be a formidable task. Time-domain methods, such as we consider here, are usually appropriate for edge detection problems. There comes a point, none- theless, when the variety of edge shapes, the background noise in the source signals, and the diverse gradients cause problems for simple time domain techniques. In recent years, researchers have turned to edge detection algorithms that incorporate a notion of the size or scale of the signal features. Chapter 4 has more to say about time domain signal analysis and edge detection, in particular. The later chapters round out the story. 1.1.3 Analysis in the Time-Frequency Plane What about signals whose values are symbolic rather than numeric? In ordinary usage, we consider sequences of signs to be signals. Thus, we deem the display of #ags on a ship’s mast, a series of hand gestures between baseball players, DNA codes, and, in general, any sequence of codes to all be “signals.” We have already taken note of such usages. And this is an important idea, but we shall not call such a symbolic sequence a signal, reserving for that term a narrow scienti$c de$nition as an ordered set of numbers. Instead, we shall de$ne a sequence of abstract symbols to be a structural interpretation of a signal. It is in fact the conversion of an ordered set of numerical values into a sequence of symbols that constitutes a signal interpretation or analysis. Thus, a microphone receives a logitudinal compressive sound wave and converts it into electrical impulses, thereby creating an analog signal. If the analog speech signal is digitized, processed, and analyzed by a speech recognition engine, then the output in the form of ASCII text characters is a symbolic sequence that interprets, analyzes, or assigns meaning to the signal. The $nal result may be just the words that were uttered. But, more likely, the speech interpretation algorithms will generate a variety of interme- diate representations of the signal’s structure. It is common to build a large hierar- chy of interpretations: isolated utterances; candidate individual word sounds within the utterances; possible word recognition results; re$nements from grammatical rules and application context; and, $nally, a structural result. x′ n ( ) 1 12 ----- - x n 2 – ( ) 8x n 1 – ( ) – 8x n 1 + ( ) x n 2 + ( ) – + [ ] =
- 48. INTRODUCTION TO SIGNALS 19 This framework applies to the applications covered in this section. A simple sequence of symbols representing the seismometer background, P waves, S waves, and surface waves may be the outcome of a structural analysis of a seismic signal (Figure 1.9). The nodes of such a structure may have further information attached to them. For instance, the time-domain extent of the region, a con$dence measure, or other ana- lytical signal features can be inserted into the node data structure. Finding signal edges is often the prelude to a structural description of a signal. Figure 1.10 Fig. 1.9. Elementary graph structure for seismograms. One key analytical parameter is the time interval between the P waves and the S waves. Fig. 1.10. Hypothetical geothermal signal structure. The root note of the interpretive struc- ture represents the entire time-domain signal. Surface strata exhibit a cooling trend. There- after, geothermal heating effects are evident. Edges within the geothermal heating region indicate narrow fracture zones.
- 49. 20 SIGNALS: ANALOG, DISCRETE, AND DIGITAL illustrates the decomposition of the geothermal pro$le from Figure 1.8 into a rela- tional structure. For many signal analysis problems, more or less #at relational structures that divide the signal domain into distinct regions are suf$cient. Applications such as natural language understanding require more complicated, often hierarchical graph structures. Root nodes describe the coarse features and general subregions of the signal. Applying specialized algorithms to these distinct regions decomposes them further. Some regions may be deleted, further subdivided, or merged with their neighbors. Finally, the resulting graph structure can be compared with existing structural models or passed on to higher-level arti$cial intelligence applications. 1.1.4 Other Domains: Frequency and Scale While we can achieve some success in processing and analyzing signals with ele- mentary time-domain techniques, applied scientists regularly encounter applica- tions demanding more sophisticated treatment. Thinking for a moment about the seismogram examples, we considered one aspect of their interpretation: $nding the time difference between the arrival of the P and S waves. But how can one distin- guish between the two wave sets? The distinction between them, which analysis algorithms must $nd, is in their oscillatory behavior and the magnitude of the oscil- lations. There is no monotone edge, such as characterized the geothermal signal. Rather, there is a change in the repetitiveness and the sweep of the seismograph needle’s wiggling. When the oscillatory nature of a signal concerns us, then we are interested in its periodicity—or in other words, the reciprocal of period, the frequency. Frequency-domain signal descriptions decompose the source signals into sinuso- idal components. This strategy does improve upon pure time domain methods, given the appropriate application. A frequency-domain description uses some set of sinusoidal signals as a basis for describing a signal. The frequency of the sinusoid that most closely matches the signal is the principal frequency component of the signal. We can delete this principal frequency component from the source signal to get a difference signal. Then, we iterate. The $rst difference signal is further fre- quency analyzed to get a secondary periodic component and, of course, a second difference signal. The sinusoidal component identi$cation and extraction continue until the difference signal consists of nothing but small magnitude, patternless, ran- dom perturbations—noise. This is a familiar procedure. It is just like the elementary linear algebra problem of $nding the expansion coef$cients of a given vector in terms of a basis set. Thus, a frequency-domain approach is suitable for distinguishing the P waves from the S waves in seismogram interpretation. But, there is a caveat. We cannot apply the sinusoidal signal extraction to the whole signal, but rather only to small pieces of the signal. When the frequency components change radically on the sepa- rate, incoming small signal pieces, then the onset of the S waves must be at hand. The subtlety is to decide how to size the small signal pieces that will be subject to frequency analysis. If the seismographic station is far away, then the time interval
- 50. ANALOG SIGNALS 21 between the initial P waves and the later S waves is large, and fairly large subinter- vals should suf$ce. If the seismographic station is close to the earthquake epicenter, on the other hand, then the algorithm must use very small pieces, or it will miss the short P wave region of the motion entirely. But if the pieces are made too small, then they may contain too few discrete samples for us to perform a frequency analy- sis. There is no way to know whether a temblor that has not happened yet will be close or far away. And the dilemma is how to size the signal subintervals in order to analyze all earthquakes, near and far, and all possible frequency ranges for the S and P waves. It turns out that although such a frequency-domain approach as we describe is adequate for seismic signals, the strategy has proven to be problematic for the inter- pretation of electrocardiograms. The waves in abnormal ECGs are sometimes too variable for successful frequency-domain description and analysis. Enter the notion of a scale-domain signal description. A scale-domain descrip- tion of a signal breaks it into similarly shaped signal fragments of varying sizes. Problems that involve the time-domain size of signal features tend to favor this type of representation. For example, a scale-based analysis can offer improvements in electrocardiogram analysis; in this $eld it is a popular redoubt for researchers that have experimented with time domain methods, then frequency-domain methods, and still $nd only partial success in interpreting ECGs. We shall also illustrate the ideas of frequency- and scale-domain descriptions in this $rst chapter. A complete understanding of the methods of frequency- and scale- domain descriptions requires a considerable mathematical expertise. The next two sections provide some formal de$nitions and a variety of mathematical examples of signals. The kinds of functions that one normally studies in algebra, calculus, and mathematical analysis are quite different from the ones at the center of signal the- ory. Functions representing signals are often discontinuous; they tend to be irregu- larly shaped, blocky, spiky, and altogether more ragged than the smooth and elegant entities of pure mathematics. 1.2 ANALOG SIGNALS At the scale of objects immediately present to human consciousness and at the macroscopic scale of conventional science and technology, measurable phenomena tend to be continuous in nature. Hence, the raw signals that issue from nature— temperatures, pressures, voltages, #ows, velocities, and so on—are commonly mea- sured through analog instruments. In order to study such real-world signals, engi- neers and scientists model them with mathematical functions of a real variable. This strategy brings the power and precision of mathematical analysis to bear on engi- neering questions and problems that concern the acquisition, transmission, interpre- tation, and utilization of natural streams of numbers (i.e., signals). Now, at a very small scale, in contrast to our perceived macroscopic world, natu- ral processes are more discrete and quantized. The energy of electromagnetic radia- tion exists in the form of individual quanta with energy , where h is E h λ ⁄ =
- 51. 22 SIGNALS: ANALOG, DISCRETE, AND DIGITAL Planck’s constant,3 and λ is the wavelength of the radiation. Phenomena that we normally conceive of possessing wave properties exhibit certain particle-like behav- iors. On the other hand, elementary bits of matter, electrons for instance, may also reveal certain wave-like aspects. The quantization of nature at the subatomic and atomic levels leads to discrete interactions at the molecular level. Lumping ever greater numbers of discretized interactions together, overall statistics take priority over particular interactions, and the continuous nature of the laws of nature at a large scale then become apparent.4 Though nature is indeed discrete at the microlevel, the historical beginnings of common sense, engineering, and scienti$c endeavor involve reasoning with continuously measurable phenomena. Only recently, within the last century have the quantized nature of the interactions of mat- ter and energy become known. And only quite recently, within our own lifetimes, have machines become available to us—digital computers—that require for their application the discretization of their continuous input data. 1.2.1 Definitions and Notation Analog signal theory proceeds directly from the analysis of functions of a real vari- able. This material is familiar from introductory calculus courses. Historically, it also precedes the development of discrete signal theory. And this is a curious cir- cumstance, because the formal development of analog signal theory is far more subtle—some would no doubt insist the right term is perilous—than discrete time signal processing and analysis. De$nition (Analog Signals). An analog signal is a function , where R is the set of real numbers, and x(t) is the signal value at time t. A complex-valued analog signal is a function . Thus, , where xr(t) is the real part of x(t); xi(t) is the imaginary part of x(t); both of these are real-valued signals; and . Thus, we simply identify analog signals with functions of a real variable. Ordi- narily, analog signals, such the temperature of an oven varying over time, take on real values. In other cases, where signal timing relationships come into question, or the frequency content of signals is an issue, complex-valued signals are often used. We will work with both real- and complex-valued signals in this section. Section 1.7 considers the complex number system, complex-valued signals, and the mathemat- ics of complex numbers in more detail. Complex-valued signals arise primarily in the study of signal frequency. 3To account for the observation that the maximum velocity of electrons dislodged from materials depended on the frequency of incident light, Max Planck (1858–1947) conjectured that radiant energy consists of discrete packets, called photons or quanta, thus discovering the quantum theory. 4This process draws the attention of philosophers (N. Hartmann, New Ways of Ontology, translator R. C. Kuhn, Chicago: Henry Regnery, 1953) and scientists alike (W. Zurek, “Decoherence and the transition from quantum to classical,” Physics Today, vol. 44, no. 10, pp. 36–44, October 1991). x:R R → x:R C → x t ( ) xr t ( ) jxi t ( ) + = j 2 1 – =
- 52. ANALOG SIGNALS 23 Of course, the independent variable of an analog signal does not have to be a time variable. The pneumatic valve of a bicycle tire follows a sinusoidal course in height above ground as the rider moves down the street. In this case the analog sig- nal is a function of distance ridden rather than time passed. And the geothermal gra- dient noted in the previous section is an example of a signal that is a function of depth in the earth’s crust. It is possible to generalize the above de$nition to include multichannel signals that take values in Rn , n ≥ 2. This is a straightforward generalization for all of the theory that we develop. Another way to generalize to higher dimensionality is to consider signals with domains contained in Rn , n ≥ 2. This is the discipline of image processing, at least for n = 2, 3, and 4. As a generalization of signal processing, it is not so straightforward as multichannel theory; the extra dimension in the indepen- dent signal variable leads to complications in signal interpretation and imposes severe memory and execution time burdens for computer-based applications. We should like to point out that modeling natural signals with mathematical functions is an inherently #awed step; many functions do not correspond to any real-world signal. Mathematical functions can have nonzero values for arbitrarily large values of their independent variable, whereas in reality, such signals are impossible; every signal must have a $nite past and eventually decay to nothing. To suppose otherwise would imply that the natural phenomenon giving rise to the sig- nal could supply energy inde$nitely. We can further imagine that some natural sig- nals containing random noise cannot be exactly characterized by a mathematical rule associating one independent variable with another dependent variable. But, is it acceptable to model real-world signals with mathematical models that eventually diminish to zero? This seems unsatisfactory. A real-world signal may decay at such a slow rate that in choosing a function for its mathematical model we are not sure where to say the function’s values are all zero. Thus, we should prefer a theory of signals that allows signals to continue forever, perhaps diminishing at an allowable rate. If our signal theory accomodates such models, then we have every assurance that it can account for the wildest natural signal that the real world can offer. We will indeed pursue this goal, beginning in this $rst chapter. With persis- tence, we shall see that natural signals do have mathematical models that re#ect the essential nature of the real-world phenomenon and yet are not limited to be zero within $nite intervals. We shall $nd as well that the notion of randomness within a real-world signal can be accommodated within a mathematical framework. 1.2.2 Examples The basic functions of mathematical analysis, known from algebra and calculus, furnish many elementary signal models. Because of this, it is common to mix the terms “signal” and “function.” We may specify an analog signal from a formula that relates independent variable values with dependent variable values. Sometimes the formula can be given in closed form as a single equation de$ning the signal values. We may also specify other signals by de$ning them piecewise on their domain. Some functions may best be described by a geometric de$nition. Still other
- 53. 24 SIGNALS: ANALOG, DISCRETE, AND DIGITAL functions representing analog signals may be more convenient to sketch rather than specify mathematically. 1.2.2.1 Polynomial, Rational, and Algebraic Signals. Consider, for exa- mple, the polynomial signal, . (1.5) x(t) has derivatives of all orders and is continuous, along with all of its derivatives. It is quite unlike any of nature’s signals, since its magnitude, |x(t)|, will approach in$nity as |t| becomes large. These signals are familiar from elementary algebra, where students $nd their roots and plot their graphs in the Cartesian plane. The domain of a polynomial p(t) can be divided into disjoint regions of concavity: con- cave upward, where the second derivative is positive; concave downward, where the second derivative is negative; and regions of no concavity, where the second deriva- tive is zero, and p(t) is therefore a line. If the domain of a polynomial p(t) contain- san interval a t b where for all , then p(t) is a line. However familiar and natural the polynomials may be, they are not the signal family with which we are most intimately concerned in signal processing. Their behavior for large |t| is the problem. We prefer mathematical functions that more closely resemble the kind of signals that occur in nature: Signals x(t) which, as |t| gets large, the signal either approaches a constant, oscillates, or decays to zero. Indeed, we expend quite an effort in Chapter 2 to discover signal families—called function or signal spaces—which are faithful models of natural signals. The concavity of a signal is a very important concept in certain signal analysis applications. Years ago, the psychologist F. Attneave [38] noted that a scattering of simple curves suf$ces to convey the idea of a complex shape—for instance, a cat. Later, computer vision researchers developed the idea of assemblages of simple, oriented edges into complete theories of low-level image understanding [39–41]. Perhaps the most in#uential among them was David Marr, who conjectured that understanding a scene depends upon the extraction of edge information [39] over a range of visual resolutions from coarse to $ne multiple scales. Marr challenged computer vision researchers to $nd processing and analysis paradigms within bio- logical vision and apply them to machine vision. Researchers investigated the appli- cations of concavity and convexity information at many different scales. Thus, an intricate shape might resolve into an intricate pattern at a $ne scale, but at a coarser scale might appear to be just a tree. How this can be done, and how signals can be smoothed into larger regions of convexity and concavity without increasing the number of differently curved regions, is the topic of scale-space analysis [42,43]. We have already touched upon some of these ideas in our discussion of edges of the QRS complex of an electrocardiogram trace and in our discussion of the geothermal gradient. There the scale of an edge corresponded to the number of points incorpo- rated in the discrete derivative computation. This is precisely the notion we are x t ( ) akt k k 0 = N ∑ = t2 2 d d p t ( ) 0 = t a b , ( ) ∈
- 54. ANALOG SIGNALS 25 trying to illustrate, since the scale of an edge is a measure of its time-domain extent. Describing signal features by their scale is most satisfactorily accomplished using special classes of signals (Section 1.6). At the root of all of this deep theory, how- ever, are the basic calculus notion of the sign of the second derivative and the intui- tive and simple polynomial examples. Besides motivating the notions of convexity and concavity as component build- ing blocks for more complicated shapes, polynomials are also useful in signal the- ory as interpolating functions. The theory of splines generalizes linear interpolation. It is one approach to the modern theory of wavelet transforms. Interpolating the val- ues of a discrete signal with continuous polynomial sections—connecting the dots, so to speak—is the opposite process to sampling a continuous-domain signal. If p(t) and q(t) are polynomials, then x(t) = p(t)/q(t) is a rational function. Sig- nals modeled by rational functions need to have provisions made in their de$nitions for the times t0 when q(t0) = 0. If, when this is the case, p(t0) = 0 also, then it is pos- sible that the limit, (1.6) exists and can be taken to be x(t0). This limit does exist when the order of the zero of p(t) at t = t0 is at least the order of the zero of q(t) at t = t0. Signals that involve a rational exponent of the time variable, such as x(t) = t1/2 , are called algebraic signals. There are often problems with the domains of such sig- nals; to the point, t1/2 does not take values on the negative real numbers. Conse- quently, we must usually partition the domain of such signals and de$ne the signal piecewise. One tool for this is the upcoming unit step signal u(t). 1.2.2.2 Sinusoids. A more real-to-life example is a sinusoidal signal, such as sin(t) or cos(t). Of course, the mathematician’s sinusoidal signals are synthetic, ideal creations. They undulate forever, whereas natural periodic motion eventually deteriorates. Both sin(t) and cos(t) are differentiable: and . From this it follows that both have derivatives of all orders and have Taylor5 series expansions about the origin: (1.7a) . (1.7b) 5The idea is due to Brook Taylor (1685–1731), an English mathematician, who—together with many others of his day—sought to provide rigorous underpinnings for Newton’s calculus. p t ( ) q t ( ) --------- t t0 → lim r0 x t0 ( ), = = t d d t ( ) sin t ( ) cos = t d d t ( ) cos t ( ) sin – = t ( ) sin t t 3 3! ---- - – t 5 5! ---- - t 7 7! ---- - – … + + = t ( ) cos 1 t 2 2! ---- - – t 4 4! ---- - t 6 6! ---- - – … + + =
- 55. Exploring the Variety of Random Documents with Different Content
- 56. gold from somewhere to make the prospectus, and also enough to make a brooch for the manager's wife; and no doubt they would have got much more in course of time, but something failed--the water in the English Channel was a bit off, or some other natural cause--and my father said it would have been far better for everybody concerned if the works had been put up in the Isle of Skye, or perhaps in Norway, or in the West Indies, or the Fiji Islands, where conditions might have been better suited to success. But gold was none the less made for my father and one or two others, though not from the sea, as my father said thoughtfully when discussing the winding up of the affair. There is another and even higher branch of the financier's art-- the loftiest of all in fact. This consists in floating loans for hard-up monarchs, and it is absolutely the biggest thing the financier does. It wants great skill and delicacy. You can also float loans for hard-up nations if you understand how to do it, but there are hundreds of financiers who never reach these dizzy heights of the profession, just as there are hundreds-- you may say millions--of soldiers who never get above being colonels, and thousands of clergymen who fall short of becoming bishops. My father, of course, understood these high branches of his profession, and once even went so far as to be interested in a loan for a South American Republic; but before the thing was matured, one side of the Republic was destroyed by a volcano and the other side by insurgents, who shot the President and all his best friends; and these events so shook investors in general that they would not
- 57. subscribe to that loan, though the Republic, in its financial extremities, offered fabulous rates of interest. I mention my father at such great length just to show the man he was and to explain my own bent of mind, which lay in the same direction. He said once, in a genial mood, that no man had ever made more bricks without straw than he had. It seemed to me a very dignified and original profession, because you are on your own, so to say, and you go out into the world single-handed, and by simple force of a brilliant imagination and hard work, win to yourself an honourable position. You may even get knighted or baroneted, if your financial genius is crowned with sufficient success to give away a few tons of money to a hospital, or the party chest, whatever that is. So, understanding all these things fairly well, it was natural that I took the line I did in the affair of Protheroe minimus and young Mayne. And, whatever the Doctor thought, my father didn't see any objection to the operation; and, of course, his opinion was the only one I cared about. It was like this. Young Mayne, though very poor, had a most amazing knack of prize-winning. He was in a class where all the chaps were a year older than him, and yet he always beat them with the greatest ease. He was good all round, and thought nothing of raking in prizes term after term. In fact, it seemed a thousand pities, seeing that he was very poor and the only son of a lawyer's clerk, that his great prize- winning powers were not yielding a better return. For, not to put too fine a point upon it, as they say, the prizes at Merivale were piffle of
- 58. the deepest dye, and of no money value worth mentioning. Dr. Dunston went on getting the same books term after term, and simply unreadable slush was all you could call them. The few things that were good were all back numbers, like Robinson Crusoe--all right in themselves, but nobody wants to read them twice; and then there were school stories that would have made angels weep, especially one called St. Winifred's, in which boys behaved like girls and blushed if anybody said something dashing. Then there were books about birds and animals and insects, and for the Lower School the Doctor used to sink to Peter Parley and the Peep of Day, and such-like absolute mess of a bygone age. These things were all bound in blue leather and had a gold owl stamped upon them, which was the badge of Merivale. I believe the owl was supposed to be the bird of Athena, and stood for wisdom, or some such rot. Anyhow, it wasn't a bad idea in its way, for a more owlish sort of school than Merivale I never was at. And young Mayne got more of these books than anybody; but to him they were as grass, and he thought nothing of them. Whereas Protheroe minimus had never won a prize in his life, and wanted one fearfully--not for itself, but for the valuable effect it would have on his mother. She was a widow and loved Protheroe minimus best of her three sons. The others had taken prizes and were fair fliers at school; but Protheroe min. was useless except at running. So, woman-like, just because he couldn't get a prize anyhow, his mother was set on his doing so, and promised him rare rewards if he would
- 59. only work extra hard, or be extra good, or extra something, and so scare up a blue book with a gold owl at any cost. Well, if you have a financial mind, you will see at a glance that here was a possible opportunity. At least, so it looked to me. Because on the one hand was young Mayne, always fearfully hard up and always getting prizes at the end of each term as a matter of course; while on the other hand was Protheroe min., never hard up but never a scholastic success, so to say, from the beginning of the term to the end--and, of course, never even within sight of a prize of any sort. Here it seemed to me was the whole problem of supply and demand in a nutshell; and the financier instinct cried out in me, as it were, that I ought to be up and doing. So I went to young Mayne and said that I thought it was a frightful pity all his great skill was being chucked away, and bringing no return more important than the mournful things that he won as prizes. And he said: A time will come, Mitchell. And then I told him that a time had come. I know you sell your prizes for a few bob at home, and that you think nothing of them, I said. But I had a bit of a yarn with that kid Protheroe yesterday, and it seems that what is nothing to you would be a perfect godsend to him. You may not believe it, but his mother, who is a bit dotty on him, has promised him five pounds if he will bring home a prize. Five pounds! said Mayne. The best prize old Dun ever gave wasn't worth five bob.
- 60. She doesn't want to sell it--she wants to keep it for the honour and glory of Protheroe min., I explained. And the idea in my mind in bringing you chaps together for your mutual advantage was, firstly, that you should let Protheroe have one of your prizes to take home in triumph to his mother; and, secondly, that he should give you a document swearing to let you have two pounds of his five pounds at the beginning of next term. Mayne was much interested at this suggestion, and, knowing that he must be a snip for at least two prizes, if not three, at the end of the summer term, he had no difficulty whatever in falling in with my scheme. We were allowed to walk in the playing-fields on Sunday after chapel before dinner, and then Mayne and Protheroe minimus and myself discussed the details. Funnily enough, they were so full of it between themselves that they did not exactly realize where I came in; so I had to remind Protheroe that it was I who had arranged the supply when I heard about his demand; and I had also to remind him he had certainly said that if anybody could put him in the way of a prize, he would give that person a clear pound at the beginning of next term. I also had to remind Mayne that he had promised me ten shillings on delivery of his two pounds. In fact, before the day was done I got them both to sign documents; because, as I say, when they once got together over it, they seemed rather to forget me. So I explained to them that my part was simply that of a financier, and that many men made their whole living in that way, arranging supplies for demands and bringing capitalists together in a friendly spirit. But not for nothing.
- 61. They quite saw it, but thought I asked too much. However, I was older than they were, and speedily convinced them that I had not. There was only one difficulty in the way after this, and Protheroe came to me about it, and I helped him over it free of charge. He said: When I take home the prize, what shall I say it's for? You know what my school reports are like. There's never a loophole for a prize of any kind. You might say good conduct, I suggested; but Protheroe min. scorned the thought. That would give away the whole show at once, he said. Because even my mother wouldn't be deceived. It's no good taking back a prize for good conduct when the report will be sure to read as usual--'No attempt at any improvement,' which is how it always goes. Everything I suggested, Protheroe scoffed at in the same way, so I could see the prize would have to be for something not mentioned at all in the school report. Of course, you don't get book prizes for cricket, or footer, or running, which--especially the latter--were the only things that Protheroe min. could have hoped honestly to get a prize for. But I stuck to the problem, and had a very happy idea three nights before the end of the term. I then advised Protheroe to say the prize was for calisthenics. There are no prizes for calisthenics at Merivale; but it sounded rather a likely subject, especially as he was a dab at it. And, anyway, he thought it would satisfy his mother and be all right.
- 62. So that was settled, and it only remained for Mayne to get his lawful prizes and hand over the least important to Protheroe min. It all went exceedingly well--at the start--and young Mayne got the prizes and gave Protheroe the second, which was for literature. The thing was composed entirely of poems--Longfellow, or Southey, or some such blighter--and Protheroe said that his mother would fairly revel to think that he had won it. He packed it in his box after breaking up, and we exchanged our agreements; and it came out, when all was over, that young Mayne was to have two pounds out of Protheroe's five, and I was to have ten bob from Mayne and a pound from Protheroe--thirty shillings in all; and Protheroe would have the prize and two pounds, not to mention other pickings, which would doubtless be given to him by his proud and grateful mother. You might have thought that nothing could go wrong with a sound financial scheme of that sort. I put any amount of time and thought into the transaction, and as it was my first introduction into the world of business, so to speak, and I stood to net a clear thirty shillings, naturally I left no stone unturned, as they say, to make it a brilliant and successful affair. And yet it all went to utter and hopeless smash, though it was no fault of mine. And you certainly couldn't blame Protheroe min. or Mayne either. In fact, Protheroe must have carried it off very well when he got home, and the calisthenics went down all right; and Mayne, when his people asked how it was that he hadn't got more than one prize, was ingenious enough to say that he'd suffered from hay fever all the term and been too off colour to make his usual haul.
- 63. So everything would have been perfection but for the idiotic and footling behaviour of Protheroe min.'s mother. This excitable and weak-minded woman was not content with just quietly taking the prize and putting it in a glass case with the prizes won in the past by Protheroe's brothers. She must go fluttering about telling his wretched relations what he'd done; and, as if that was not enough, she got altogether above herself and wrote to Dr. Dunston about it. She said how glad and happy it had made her, and that success in the gymnasium was something to begin with, and that she hoped and prayed that it would lead to better things, and that they would live to be proud of Protheroe minimus yet, and such-like truck! Well, the result was a knock-down blow to us all, as you may imagine, and the Doctor showed himself both wily and beastly, as usual. For he merely asked Protheroe's mother to send back the prize at the beginning of the term, as he fancied there might have been some mistake; but he begged her not to mention the matter to Protheroe minimus. So when Protheroe and Mayne and myself all arrived again for the arduous toil of the winter term, and Mayne and I were eager for the financial disimbursements to begin, we heard the shattering news that, at the last moment, Protheroe hadn't got his fiver. It was to have been given to him on the day that he came back to school; but instead his mother had merely told him that she feared there was a little mistake somewhere, and that she couldn't give him his hard-earned cash till Dr. Dunston had cleared the matter up.
- 64. Needless to say that Dunston did clear it up with all the brutality of which he was capable. As for myself, when the crash came, I hoped it would happen to me as it often does to professional financiers in real life, and that I should escape, as it were. Not, of course, that I had done anything that in fairness made it necessary for me to escape, because to take advantage of supply and demand is a natural law of self- preservation, and everybody does it as a matter of course, not only financiers. But, much to my annoyance, the common-sense view of the thing was not taken, and I found myself in the cart, as they say, with young Mayne and Protheroe minimus. The Doctor, on examining Protheroe's prize for calisthenics, instantly perceived that it was in reality young Mayne's prize for literature. But evidently anything like strategy of this kind was very distasteful to the Doctor. In fact, he took a prejudiced view from the first, and as young Mayne was only eleven and Protheroe min. merely ten and a half, it instantly jumped to Dunston's hateful and suspicious mind that somebody must have helped them in what he called a nefarious project. And, by dint of some very unmanly cross-questioning, he got my name out of Mayne. I never blamed Mayne; in fact, I quite believed him when he swore that it only slipped out under the treacherous questions of the Doctor; but the result was, of course, unsatisfactory in every way for me. I was immediately sent for, and had no course open to me but to explain the whole nature of financial operations to Dr. Dunston,
- 65. and try to make him see that I had simply fallen in with the iron laws of supply and demand. Needless to say, I failed, for he was in one of his fiery and snorting conditions and above all appeal to reason. It was an ordinary sort of transaction, sir, I said, and I don't see that anybody was hurt by it. In fact, everybody was pleased, including Mrs. Protheroe. This made him simply foam at the mouth. I had never been what you may call a great success with him, and now to hear sound business views from one still at the early age of sixteen, fairly shook him up. He ordered me to go back to my class, and when I had gone, he flogged young Mayne and Protheroe minimus. He then forgave them and told them to go and sin no more; and the same day, doubtless after the old fool had cooled down a bit, he wrote to my father and put the case before him--though not quite fairly--and said that, apparently, I had no moral sense, and a lot of other insulting and vulgar things. In conclusion, he asked my father to remove me, that I might find another sphere for my activities. And my father did. He never took my view of the matter exactly; but he certainly did not take Dr. Dunston's view either. He seemed to be more amused than anything, and was by no means in such a wax with Dr. Dunston as I should have expected. He said that the scholastic point of view was rather stuffy and lacked humour; and then he explained that I had certainly not acted quite on the straight, but had been a deceitful and cunning little bounder.
- 66. I was a good deal hurt at this view, and when he found a billet for me in the firm of Messrs. Martin Moss, Stock Brokers, I felt very glad indeed to go into it and shake off the dust of school from my feet, as they say. It is a good and a busy firm, and I have been here a fortnight now. Ten days ago, happening to pass Mr. Martin's door, and catching my name, I naturally stood and listened and heard an old clerk tell Mr. Martin that I was taking to the work like a duck takes to water. I am writing this account of the business at Merivale on sheets of the best correspondence paper of Messrs. Martin Moss! They would not like it if they knew. But they won't know. THE END Printed in the United States of America. * * * * * * * * The following pages contain advertisements of Macmillan books by the same author
- 67. BY THE SAME AUTHOR Old Delabole BY EDEN PHILLPOTTS Author of Brunei's Tower, etc. Cloth, 12mo, $1.50 A critic in reviewing Brunei's Tower remarked that it would seem that Eden Phillpotts was now doing the best work of his career. There was sufficient argument for this contention in the novel then under consideration and further demonstration of its truth is found in Old Delabole, which, because of its cheerful and wise philosophy and its splendid feeling for nature and man's relation to it, will perhaps ultimately take its place as its author's best. The scene is laid in Cornwall. Delabole is a slate mining town and the tale which Mr. Phillpotts tells against it as a background, one in which a matter of honor or of conscience is the pivot, is dramatic in situation and doubly interesting because of the moral problem which it presents. Mr. Phillpotts's artistry and keen perception of those motives which actuate conduct have never been better exhibited. Another good story from an able hand.--New York Sun. A novel of large significance.--Boston Herald. A more effective piece of dramatic description could scarcely be put into print.--North American (Philadelphia).
- 68. Besides being a good story, richly peopled, and brimful of human nature in its finer aspects, the book is seasoned with quiet humor and a deal of mellow wisdom.--New York Times. Brunei's Tower BY EDEN PHILLPOTTS Cloth, 12mo, $1.50 The regeneration of a faulty character through association with dignified, honest work and simple, sincere people is the theme which Mr. Phillpotts has chosen for his latest novel. Always an artist, he has, in this book, made what will perhaps prove to be his most notable contribution to literature. Humor and a genuine sympathetic understanding of the human soul are reflected throughout it. The scene is largely laid in a pottery, and the reader is introduced in the course of the action to the various processes in the art. The central figure is a lad who, having escaped from a reform school, has sought shelter and work in the pottery. Under the influence of the gentle, kindly folk of the community he comes in a measure to realize himself. It touches lightly upon love, upon the pathos of old age, upon the workman's passion for his work, upon the artist's worship of his art, upon an infinite variety of human ways and moods, and it is filled to its depths with reflections upon life that are very near to life itself. It is Mr. Phillpotts at his characteristic best.--Boston Transcript.
- 69. The daily bread of life is in this book ... magnificently written, ... absorbingly interesting, and holds that element of surprise which is never lacking in the work of the true story teller. It is a book for which to be frankly grateful, for it holds matter for many hours' enjoyment.--New York Times. BY THE SAME AUTHOR Faith Tresilion Decorated Cloth, 12mo, $1.35 Its movement is brisk, and the development of its plot is emphasized at certain steps with sudden surprises--all of which contribute toward holding the reader's attention.--New York Times. A rousing story, having about it the tang and flavor of the sea, and with the sound of trumpets ringing through.... Mr. Phillpotts has chosen a period of thrills for his story and has succeeded very well in putting across the bracing atmosphere of perilous times. His portrayal of the coast folk of Daleham rings true and refreshing.-- Kansas City Star. A tale picturesque in its scenes and rich in its character.-- Boston Transcript. Mr. Phillpotts may be congratulated upon having written a remarkable book in which there is not a dull page.--Philadelphia Ledger. A book that is distinctly interesting.--New York Herald. No character that Mr. Phillpotts has created can surpass that of Emma Tresilion.--Boston Times.
- 70. It is a very readable story.--The Outlook. A book of stirring adventure and sensational experiences.-- Literary Digest. Never has Eden Phillpotts written so swinging a romance.-- Bellman. A rattling good story.--Los Angeles Times. OTHER OF MR. EDEN PHILLPOTTS' NOVELS The Three Brothers Cloth, 12mo, $1.50 'The Three Brothers' seems to us the best yet of the long series of these remarkable Dartmoor tales. If Shakespeare had written novels we can think that some of his pages would have been like some of these.... The book is full of a very moving interest, and it is agreeable and beautiful.--New York Sun. Knock at a Venture Cloth, 12mo, $1.50 Sketches of the rustic life of Devon, rich in racy, quaint, and humorous touches. The Portreeve
- 71. Cloth, 12mo, $1.50 Twice, at least, he has reached and even surpassed the standard of his first notable work. Once was in 'The Secret Woman.' The second time is in 'The Portreeve.' In sheer mastery of technique it is the finest thing he has done. From the beginning to the end the author's touch is assured and unfaltering. There is nothing superfluous, nothing unfinished.... And the characters, even to the least important, have the breath of life in them.--The Providence Journal. My Devon Year Cloth, 8vo, $2.00, with 38 Monotint plates, $2.00 One of the most charming nature books recently published.... This book will inspire such persons and many others to get back to Mother Earth and see her wonders with a new eye. To those who know the rich Devon country it describes 'My Devon Year' is a delight from beginning to end, but this knowledge is not essential to its thorough enjoyment.--New York Mail. THE MACMILLAN COMPANY Publishers -- 64-66 Fifth Avenue -- New York
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