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Unformatted text preview: Chapter Fourier Analysis ANALYZING SOUND IS A SPECIAL CASE of the general problem of ana—
lyzing complex patterns. As a rule, we select a variable, for example
local sound pressure, and we graph the measured values of this
variable as a function of either location or time or both. How can
we characterize any given graph that we might obtain, and how do
we compare two different graphs? Suppose we record two different
sounds and obtain the corresponding plots of pressure versus time
(Figure 3.1). We can clearly see that these two sounds have different
waveforms, but how does one go about providing a quantified.
comparison? If we had three such sounds, how would we decide
whether two of the sounds were more similar to each other than
either was to the third? If the sounds were simple sine waves, there
would be no problem; we could specify the frequency, peak ampli—
tude, and relative phases of the two sounds and they would be
completely described. In this case, neither of the sounds is a pure
sine wave. 41 42 Chapter 3 Sound ‘1 Sound 2 Ambient pressure Pressure Time Time Figure 3.1 Plots of recorded pressure amplitudes versus time for two sounds. THE LOGIC OF FOURIER ANALYSIS Luckily, there is a technique called Fourier analysis that can solve these prob
lems. The technique is based on the fact that any continuous waveform (or
continuous graph line) can be broken down into a set of pure sinusoidal
waves of infinite duration whose frequencies, amplitudes, and relative phases
are easily quantified and compared. This list of measured components is
mathematically equivalent to a complete description of the waveform. We can
even synthesize the original sound by generating each of the component sinu
soidal waves and adding them together with the proper phase relationships.
Not only can any complex waveform be broken down and reconstituted by
separating and recombining the appropriate set of sinusoidal waves, but also
the pressure at any time t can be predicted exactly by summing the values of
each of the component waves at that time t. The pressure PU) of a complex
waveform at time t thus equals an infinite sum of cosine (or sine) waves, each
of which has a specific amplitude, frequency, and relative phase. Alge
braically, this is written as P(t) = P0 + 219,, cos(27rfnt + on)
n=1 where Po is the mean pressure around which the waveform oscillates, 13,, is the
pressure amplitude of the nth cosine wave in the set, f" is the frequency of the
nth cosine wave, and (15,, is the relative phase of the nth component. (Note that
we could have used sine instead of cosine waves in this formula; since sine
and cosine waves differ only by phase, using sine waves would shift the val
ues of 45"). In principle, it will take an infinite number of components to com
pletely describe any given complex waveform. In practice, animals’ ears and
scientists’ instruments cannot measure an infinite number of components, and
hence they focus on a finite number of the louder components. Usually this;
approximation is sufficient to characterize and compare sounds. The mean
value of the waveform (P0) is normally the ambient pressure. Where it is
higher or lower, we say that the waveform has a nonzero DC component
(from the electrical analogue of direct current). Faurier Analysis 43 Suppose we have decomposed a complex waveform into its Fourier co—
sine components. How do we summarize the necessary information to de
scribe the sound? There are many ways to graph and tabulate the Fourier in—
formation. One method is shown in Figure 3.2. The Fourier components are
arranged by their frequencies along the xaxes of two graphs. On one graph,
the amplitude of each component is indicated by the height of a bar over the
frequency of that component. This is called a frequency spectrum (or if aver—
age intensities that equal the squares of amplitudes are used, a power spec
trum). On the second graph, the size and direction of each bar relative to the
dotted line represents the phase of that Fourier component. This is called a
phase spectrum. Since it is the relative alignment of the component waves
that determines the overall waveform of the sum, one of the components is
often chosen as the reference and its phase is arbitrarily set to zero. The phase
spectrum then indicates the phase of each other component relative to this
reference. If one component peaks a bit later than the reference component,
then its relative phase is positive; if it peaks earlier, then its phase is negative
with respect to the reference component. These two graphs combined sum—
marize all the data required to describe or reconstitute the original signal.
They are in fact mathematically equivalent to it. We shall introduce other
methods for summarizing Fourier information in later sections. In this chapter, we shall mainly focus on the Fourier analysis of sounds.
You should not forget, however, that Fourier analysis can be used to charac
terize any continuous pattern. Most of our attention centers on graphs of
sound pressure versus time recorded at a given location. We could also graph
the distribution of sound pressures versus distance from the signaler, and this
too could be characterized by a Fourier decomposition into pure sine or co
sine waves. In later chapters, we shall use frequency decomposition to charac
terize visual colors and the electrical discharges of fish. Ecologists use Fourier Frequency spectrum Phase spectrum +21: Relative phase
0 Amplitude l " —27n
Frequency of component Frequency of component Figure 3.2 Fourier description of a sound. A complete description of a sound is
provided by indicating the amplitudes of all Fourier components in a frequency
spectrum and the relative phases of all components in a phase spectrum. 44 Chapter 3 analysis to characterize the spatial pattern of ecological variables, such as
grass height along a transect, or by doing Fourier analysis in two dimensionS,
they can characterize the degree of patchiness in different habitats. Clearly the
method hasgreat utility in a number of fields. FOURIER ANALYSIS OF PERIODIC SIGNALS When we View a record of a sound as a graph of pressure versus time, we are
said to be examining the sound in the time domain. The time domain wave—
form is a complete description of the sound at the recording location. Like all
signals, this waveform can be broken down into its Fourier parts. Thus a set of
tables or graphs listing all the frequencies, amplitudes, and relative phases of
the Fourier components of the sound is also a complete description of it. Such
a description is said to be given in the frequency domain. Since either descrip—
tion is complete, it doesn’t matter in principle which we use to characterize a
sound. In practice, we usually have to examine both the time and frequency
domain images of a sound to completely understand its structure. For this rea
son, it is very useful to know the basic rules by which common types of sound
waveforms are translated into the frequency domain and vice versa. Most of
this chapter is devoted to summarizing the more common of these rules. The fundamental rule of Fourier analysis can be stated in the following
way: the more the time domain image of a signal deviates from that of a sin
gle sinusoidal wave of infinite duration, the greater the number of other sinu
soidal waves that must be added to the Fourier set and the larger must be the
amplitudes of these additional components. Signals can deviate from the case
of a single sinusoidal wave of infinite duration either in waveform or in re—
peatability. In the first case, signals have a basic pattern that repeats indefi~
nitely (as a sine wave does), but the shape of the repeating unit is not sinu
soidal. Any signal (including a sinusoidal wave) that repeats the same basic
pattern indefinitely is called periodic. All periodic signals last forever. How~
ever, they may differ in the pattern of the repeating unit. As the shape of the
repeating unit becomes more and more deviant from a sinusoidal one, the
more Fourier components we can expect to see in the signal’s frequency do—
main graphs and the greater the amplitudes these additional components will
exhibit. The consequences of varying the patterns of periodic waveforms on
the corresponding frequency domain representations of signals will be treated,
first in the next sections. The second type of deviation away from an infinitely long single sinu
soidal wave is reduced repeatability. Consider a signal which consists of 5 cy~
cles of a pure sine wave with no pressure oscillations before or after. This sig
nal is not periodic because it does not last forever. As signals become more
and more aperiodic, the basic rule of Fourier analysis predicts that we shall
have to add more and more additional sinusoidal components, and shift more
and more energy into these components, to provide a representative fre—
quency domain description of the sounds. We shall treat the Fourier analysis
of aperiodic signals after we have discussed periodic waveforms. Fourier Analysis 45 Sound pressure
or voltage Figure 3.3 Time domain
Time image of a pure sine wave. Fourier Analysis of a Simple Sine Wave Let us begin with a pure sine wave of infinite duration. Its time domain image
would appear as in Figure 3.3. If there is no DC component in this sound, the
dotted line will reﬂect ambient air pressure and the rise above and below this
line will represent the increases or decreases in local air pressure as the sound
disturbance passes our microphone. On most time domain instruments, early
events are plotted on the left side of the graph and later events on the right. To
measure the frequency of this sine wave on a time domain plot, we simply
measure the period. Since period and frequency are reciprocals, a measured pe—
riod of t seconds gives a frequency in Hz of f = 1 / i. If we want to measure the
amplitude, we record the peak or peakto—peak voltage from the plot. Voltage
can be converted into pressure if we know the sensitivity of our microphone. Suppose we now perform Fourier analysis on this signal. The frequency
spectrum would appear as in Figure 3.4. Notice that only a single vertical bar
is shown since only one pure sine wave is present and there is no DC compo
nent (which if present would appear as a second bar at a frequency of zero).
This single component should have a frequency equal to the reciprocal of the
period that we measured in the time domain image, and the amplitude of the
component should equal that taken off of the time domain plot. We could also
plot the phase spectrum of this signal, but since there are no other compo
nents, relative phase has no real meaning in this case. Types of Periodic Signals There are several ways in which this single sine wave of infinite duration
might be modified and still retain its periodicity. First, we could hold its shape
and frequency fixed, but let its amplitude vary in some periodic way. The Amplitude Figure 3.4 Frequency spec
Frequency trum of single pure sine wave. simplest case is to vary the amplitude sinusoidally. This is called sinusoidal
amplitude modulation. Alternatively, we could hold its amplitude and shape
fixed, but let its frequency vary in a periodic manner. Again, the simplest case
would vary the frequency sinusoidally. This is called sinusoidal frequency
modulation. A third modification would be to hold the amplitude and fre
quency constant, but change the repeating shape of the signal from a sine
wave to something else. We shall call such a signal a nonsinusoidal periodic
wave. It is useful to learn the frequency domain expectations for each of these
three simple cases because nearly any other periodic signal can be considered
as an additive combination of them. After you understand each one, we shall
show you how to break down more complicated signals into pieces and apply
the rules for one or more of the three basic cases to each piece. Sinusoidal Amplitude Modulation Suppose we take our infinitely long pure sine wave of frequency and modu
late its amplitude sinusoidally. This means that the frequency of the wave
would remain unchanged, but the amplitude of successive cycles would go up
and down in a sinusoidal way. We can determine whether the modulation was
sinusoidal by looking at the envelope of the time domain picture (Figure 3.5).
The original sine wave that is being modulated is called the carrier and here
has frequency f; the sine wave that is imposed on the amplitude of the carrier
is called the modulating wave. The frequency of the modulating wave can be
computed by measuring the amount of time it takes to go from one maximum
of the modulating wave to the next maximum. If t is the period of this modu
lating waveform, then the frequency of the modulating wave w = 1/ 1‘ Hz.
When we amplitude modulate a carrier signal, the requisite sine waves
that have to be added in the frequency domain appear as side bands around
the carrier. For a sinusoidally amplitude—modulated carrier with no DC com~
ponent, there will be two side bands, one on each side of the carrier (Figure
3.6). One will have frequency f ~ to and the other will have frequency f + to.
Both should have amplitudes less than that of the carrier. However, the
greater the amplitude of the modulating waveform, the larger the size of the _____~.____. Sinusoidal
/,—\ /,~_.\ /m0dulat1ngwave /,__\
\\
\ / \ \ /
x \ x ” Hl.. \‘HHH ""lH ‘ HUN!“ 1"" \ Carrier Sound pressure or voltage Time Figure 3.5 Time domain image of a sinusoidally amplitudemodulated sine wave. Fourier Analysis 47 Sinusoidal AM % .é‘ TL E Figure 3.6 Frequency spectrum of
sinusoidally amplitude~modulated
sine wave. Here f is the frequency of f ‘ w f f + w the sine wave carrier, and w is the
Frequency frequency of the modulating wave. two side bands. We have not shown the phase spectrum for this signal, but
you should realize that only one set of relative phase relationships for these
three Fourier components is likely to generate the original waveform. A com—
bination of these same three components at the amplitudes shown in Figure
3.6 that used any other set of relative phases might produce a summed wave—
form quite unlike sinusoidal amplitude modulation. You may have heard of
amplitude modulation by its abbreviation, AM, since this is one of the ways
in which sounds are encoded in radio waves. Sinusoidal Frequency Modulation Suppose we return to our original pure sine wave of frequency f, infinite du—
ration, and no DC component. This time, let’s hold the amplitude constant,
but modulate the frequency. This means that the period of successive waves
will change. Let us consider the simplest case in which the frequency varies
above and below f in a sinusoidal fashion. The time domain plot would ap
pear as in Figure 3.7. Period of modulating wave cu t: 1/ w — a
'6
>
H
0
5‘3
51.“ .____.....  .._.___
5
H
EL
“5
o e—i
m \ Lon estc cle Shortest c cle
S Y Y
length, T1 = l /fmin length, T2 = 1/ fmax
Time Figure 3.7 Time domain image of a sinusoidally frequencymodulated sine wave. W 48 Chapter 3 index. We can again compute how long it takes to go through one complete mod—
ulation cycle by examining the time domain plot. Let us denote this period by t
and its reciprocal, the modulation frequency, by w. The frequency spectrum of a
sinusoidally frequency—modulated sine wave again has a band at the carrier f
and side bands around However, unlike sinusoidal amplitude modulation,
frequency modulation (FM) generally produces multiple pairs of side bands
around These side bands occur as before at f — w and f + w, but also at f — 2a;
and f + 2w, f — 3w and f + 3w, etc. The amount of the total signal energy that gets
diverted away from the carrier and into the side bands depends on the modu
lation index. This is the ratio between the range of frequencies exhibited by the
modulated carrier and the modulating frequency w. To estimate the modulation
index for a frequency modulated signal, compute w as usual. Then compute the
time it takes for the longest cycle in the time domain plot and take its reciprocal.
This gives a value of the lowest frequency, We next compute the time taken
for the shortest cycle visible in the time domain plot. Its reciprocal is the highest
frequency, fmax. The range of frequency is then fmax — fmm, and the modulation
index is (fmax  fmgnl/w. The greater the modulation index, the more energy in
the side bands of the frequency spectrum. These can be so large that they are in
fact larger than the energy in the carrier. A frequency spectrum for a sinu—
soidally frequency—modulated sine wave with a low modulation index is
shown in Figure 3.8; an example with a high modulation index is shown in
Figure 3.9. A thoughtful reader might have noticed that an FM signal with a suffi—
ciently low modulation index may have only one pair of side bands visible
and thus generate a spectrum identical to that of an AM signal with the same f
and to. In both cases one will see a carrier band at f and side bands at frequen
cies f — w and f + w. The amplitudes of carrier and side bands might even be
the same for the two graphs. How can this be when we know that the time
domain waveforms are quite different? The answer is that the two signals dif
fer markedly in the relative phases of the three Fourier components. If we add
together the three Fourier components with one set of phase relations, the re~
sult will be a pure AM signal. If we use another set of phase relations, the re—
sulting waveform will be a pure FM signal. For all other sets of phase rela—
tions, the waveform will be a mixture of FM and AM. Amplitude Figure 3.8 Frequency spectrum of a
sinusoidally frequencymodulated
sine wave with low modulation f—2w f—w f f+w fIZw
‘ Frequency Fourier Analysis 49 Amplitude Figure 3.9 Frequency spectrum of a
sinusoidally frequencymodulated 1“ 3w f“ M f“ w f f + w f + 2w f + 37“ sine wave with high modulation
Frequency index. Periodic Nonsinusoidal Signals Now consider a signal which is not a sine wave but instead exhibits a repeat—
ing pattern with some nonsinusoidal shape. Examples of such signals are
shown in Figure 3.10. Any signal that is periodic but is neither a sine wave
nor a modulated sine wave has a frequency spectrum called a harmonic se
ries. A harmonic series is a set of component frequencies that are all integer
multiples of some common frequency known as the fundamental of the se—
ries. Figure 3.11 shows a typical frequency spectrum for a nonsinusoidal peri—
odic signal. If the fundamental is w, then the other component frequencies 2w,
3w, 4w, etc. are called harmonics. Note that the differences in frequency between adjacent harmonics are
equal to each other and to the fundamental w. The time domain image of such
a signal will repeat the nonsinuoidal pattern to times per second. This fre
quency can be measured directly off of the time domain waveform by count
ing how many repeats occur in a fixed interval of time, or by measuring the
duration of one complete repeat (the period of the waveform) and taking its
reciprocal. Either method should result in a frequency that is equal to the dif
ference between the frequencies of the corresponding Fourier components.
Note that the amplitudes of higher harmonics tend to be smaller than those of Sawtooth wave Square wave Triangular wave Figure 3.10 Time domain images of various periodic nonsinusoidal signals. we 50 Chapter 3 Amplitude Figure 3.11 Frequency spectrum of a periodic nonsinusoidal signal. Such
a spectrum is called a harmonic series
built upon a fundamental frequency w. w 2w 3w 4w 5w 6m 7w
Frequency lower harmonics. Dirichlet’s rule states that for periodic signals with few
major discontinuities in their waveforms, the energy in higher harmonics of
the corresponding frequency spectrum will tend to fall off exponentially with
the frequency of the harmonic. Periodic nonsinusoidal waves that differ in shape but not in repeat rate w
will all generate harmonic series built upon a component spacing of w Hz.
However, the different waveforms will vary in how the energy is distributed
among the successive harmonics and in the phase spectra. The most conspic—
uous differences between the spectra of periodic nonsinusoidal signals in
volve the presence or absence of specific harmonics. In the simplest case, the
frequency spectrum of a nonsinusoidal periodic signal may exhibit or lack a
band at a frequency of zero. As we have seen earlier, the presence or lack of
such a band depends upon whether the time domain signal does or does not
contain a nonzero DC component (Figure 3.12). The remainder of the spectrum depends in part on whether the signal is
halfwave symmetric or halfwave asymmetric. A half—wave symmetric sig—
nal is one whose repeating waveform can be divided into two equal halves such that the second half is identical to the first half flipped upside down (B) '93 Sound pressure or voltage Time ‘ Time Figure 3.12 Presence or absence of a DC component in a square wave train. In both
examples, square pulses oceur at a rate of 10/ sec. In (A) the average deviation around
the ambient pressure (dotted line) is zero and no DC component is present. The fre
quency spectrum will not include a line at a frequency of zero. (B) All deviations from
ambient are greater than zero, producing a positive DC component. In this case, the
frequency spectrum would include a line at a frequency of zero. 3*?” Fourier Analysis 51 (A) (B) Sound pressure or voltage (D) Sound pressure or voltage @ Time Time Figure 3.13 Half—wave symmetry of periodic signals with single maxima and
minima. In each figure, the dotted horizontal line demarcates the average amplitude
for the entire signal. (A) and (B) show a symmetrical sawtooth, and (C) and (D) show
an asymmetrical one. To demonstrate symmetry in either case, draw a small box
around a single period, as in A and C. Divide this period in half with a dark vertical
line. Then reﬂect the second half of this period to the opposite side of the dotted line.
In B, it is clear that the reﬂected second half is identical to the first half. This signal is
halfwave symmetric. In D, the reﬂected half is not identical to the first half (it is a
mirror image of it) and this wave is thus not half—wave symmetric. (Figure 3.13). A similar operation performed on a half—wave asymmetric sig—
nal will fail to produce a match. Once you determine which type of signal you
have, the frequency spectra of half—wave symmetric periodic signals are im—
mediate: all of these signals have frequency spectra in which only the odd
harmonics are present (Figure 3.14A). The spectra of half—wave asymmetric
signals are a bit more complicated, but usually have all harmonics present
(Figure 3.14B). The pattern of variation in the amplitudes of the harmonics of a half
wave asymmetric signal depends upon the number and spacing of successive
maxima and minima in the signal’s time domain waveform. Signals that have
only a single maximum and minimum per repeat period always have all har—
monics present, and these tend to decrease smoothly in amplitude according
to Dirichlet’s rule. Signals with more than one maximum and/ or minimum
per period have somewhat more complicated spectral rules. The trick is to ex—
amine the time domain waveform and to find the shortest interval between
either two successive maxima or two successive minima in a single period. i
g
l 52 Chapter 3 (A) H (B) H H 0.) OJ
E a
8 ‘d
14 54
(L. (L
Time Time
 1
: I
a) m I I
p ":1
a 1 3
04 
i“ a I
5 10 5 10
Frequency (kHz) Frequency (kHz) Figure 3.14 Frequency spectra of symmetric Versus asymmetric periodic signals.
(A) Waveform (top) and frequency spectrum (bottom) of a halfwave symmetric saw—
tooth ane with a period of 5 msec and thus a fundamental frequency of 200 Hz.
Note that only odd harmonics are present. (B) Waveform and frequency spectrum of the same signal as in A but with positive peaks clipped to generate a halfwave
asymmetric signal. Because the signal is asymmetric, all harmonics of the 200 Hz.
fundamental are present. In the clipped portion, there are two successive maxima
per period separated by 0.8 msec (= 1/1200 Hz). They generate a pattern of lobes
and nodes across harmonics, with the Width of a lobe equal to 1200 Hz (marked by
vertical dashed lines). Suppose this shortest time is I see and its reciprocal is the frequency 2 Hz. As
usual, we denote the overall period of the signal by t and the corresponding
repetition rate of the waveform by to. In general, all harmonics of the funda—
mental w will be present in the corresponding frequency spectrum. However,
the closer a harmonic of w is to a harmonic of z, the lower will be its ampli
tude. Harmonics of to that fall closest to a harmonic of 2 will be maximally re—
duced and are called nodes. The adjacent clusters of more intense harmonics
are called lobes. Nodes and lobes may not be visible among the very first har
monics of w, but they usually dominate the spectrum at higher frequencies
(Figure 3.14). Thus the frequency spectra of periodic signals with multiple maxima or
minima consist of harmonic series based on the frequency of the waveform
repeat but with adjacent harmonics grouped into lobes and separated by
nodes. The width of lobes is inversely related to the lag between adjacent
maxima or minima. When the lag between successive maxima or minima is
equal to onehalf the period of the waveform, then nodes will fall on the even Fourier Analysis 53 harmonics of the waveform repeat frequency w. This pattern will cause the re
sulting spectrum to resemble that of a half—wave symmetric wave with the
same period. Compound Signals Now we are ready to look at combinations of these three basic modifications of
a sine wave. Because our ears and many of the instruments used to analyze
sounds usually do not store or use phase spectra, we shall focus mainly on the
frequency spectra. This does not mean that phase is unimportant: clearly it is
very important if your analysis or the response of the receiving animal depends
on the time domain waveform. As we shall see below, there is also a limit to the
abilities of our instruments and any animal’s ears to analyze signals in the fre—
quency domain. Where this limit is reached, one has no choice but to examine
the time domain image, including its embedded phase information. However,
for the moment, we shall ignore the phase spectra of compound signals. The simplest compound signals include (1) nonsinusoidal modulation of
a simple sine carrier, (2) sinusoidal modulation of a nonsinusoidal carrier, and
(3) nonsinusoidal modulation of a nonsinusoidal carrier. The first step in each
case is to characterize the frequency spectrum of the carrier: What are you
starting with? If the carrier is a pure sine wave, then the carrier frequency
spectrum is a single line. If the carrier is a nonsinusoidal periodic signal, then
it can be broken down into its component harmonics. Then we can treat each
compdnent in the compound carrier as if it were the only carrier. The second
step is to examine the modulating waveform and break it down into its com
ponent sine waves. If modulation is sinusoidal, then you only have a single
modulating frequency to worry about. If the modulating waveform is nonsi—
nusoidal, then each of its component harmonic frequencies can be treated. as a
separate sinusoidal modulator of the carrier. The overall frequency spectrum
of the compound signal will be the sum of the results of modulating each car—
rier component by each modulating wave component according to the simple
rules. More explicitly, for every sinusoidal component in the modulating
waveform, we add side bands according to the appropriate rules for AM or
FM around each sinusoidal component of the carrier. As an example, consider the case of a pure sine wave that is pulsed
(turned on and off). Some frogs and insects makes sounds pulsed this way. The
modulating waveform is thus periodic but nonsinusoidal. Suppose that the
duration of each pulse is equal to the interval between pulses. This means that
the modulating waveform will be half—wave symmetric and in fact, is here a
square wave. Suppose the carrier is sinusoidal. The overall waveform is shown
in Figure 3.15. What does the frequency spectrum of this signal look like? We
ought to be able to predict its appearance using the simple rules we have
learned. The carrier will appear in the spectrum as a single line at frequency The modulating waveform, however, has a frequency spectrum that is a har—
monic series with only odd components present because it is a half—wave sym—
metric square wave. The rules for AM are to add one pair of side bands around
the carrier for each sine component in the modulating waveform. That means 54 Chapter 3 Sound pressure or voltage (A) Amplitude Amplitude T tzl/w q ‘Carrier Time Figure 3.15 Time domain image of a pulsed sine wave. The carrier here is a
pure sine wave of frequency The modulating waveform is a square wave with
frequency w. we expect bands at the frequency of the carrier f, and at the side bands f + w
and f ~ to. But this is not all. We also expect side bands at f + 3w and f — 3w, at + 5w and f — 5w, etc. For each sine component in the modulating waveform, we
must add two side bands around the carrier frequency in the spectrum. Be~
cause of Dirichlet’s rule, the energy in these side bands ought to fall off for
higher multiples of w. The frequency spectrum of the pulsed sine wave is
shOWn in Figure 3.16A. A special case of compound signals occurs when a carrier signal has a.
nonzero DC component. In addition to the component frequencies in the car
rier, there is thus an additional component at a frequency of zero (the DC (B) Amplitude f—w f f+w Frequency f+3w f+5w w f—w f f+zu Frequency Figure 3.16 Spectra of some com
pound signals. (A) Frequency spec—
trum of a pure sine wave at frequency
fpulsed at to times / sec. (B) Frequency
spectrum of a sine wave of frequency
f with a nonzero DC component sinu
soidally modulated at rate w. (C)
Frequency spectrum of a single sinu—
soid of frequency f with a nonzero DC component amplitude modulated
by periodic nonsinusoidal waveform
with repeat rate w. 3w f— 4w f— 2w f
Frequency f+2w f+4w Fourier Analysis 55 component). Because all carrier components receive side bands when modu
lated, the DC component will also receive side bands (but just for frequencies
greater than zero). Thus if a simple sine wave of frequency f has a nonzero DC
component and is sinusoidally modulated at a rate w, we shall see frequency
bands at w (= O + w), f — 10, f, and f + to (Figure 3.16B). If the modulating wave
form is periodic with repetition rate to but not sinuosidal, then we shall find a
series of bands at integral multiples of w as well as bands around the carrier
frequencies (Figure 3.16C). To see if you understand the additivity of these rules, try to predict the
frequency spectrum of the reverse situation—a carrier that is a square wave
but that is amplitude modulated sinusoidally. Frequency modulation gener~
ates much more complicated spectra than amplitude modulation since it usu
ally generates more than one pair of side bands per carrier component. HOW—
ever, the logic is no more complicated than for amplitude modulation. You
may Wish to try an example. Finally, try amplitude modulating a square wave
carrier with a square wave modulating waveform. Being able to move freely
between the time domain and frequency domain is a skill that is essential for
accurate analyses of animal sounds. Box 3.1 shows how the use of both do—
mains is often necessary for such work. FOURIER ANALYSIS OF APERIODIC SIGNALS All of the signals analyzed so far have been periodic. The theory presumes
that these signals last forever. However, many natural signals are given only a
few times, or change form in each successive rendition. Hence they may re
peat only rarely, if at all. This means that we have to add even more sine
waves in the frequency domain representations of the signals. The more ape—
riodic the signal, the more frequencies we have to add. Consider the follow~
ing example. In Figures 3.15 and 3.16A, we examined the frequency spectrum
of an infinite train of pulses containing a carrier frequency f. We assumed in
that case that the pulse duration equaled the interval between pulses. This as
sumption caused the modulating waveform to be half—wave symmetric and
the only side bands were based on odd harmonics of the modulating wave
form repeat rate to. Let us now consider a more general case in which pulse
durations do not necessarily equal inter—pulse intervals. The modulating
waveform will no longer be halfwave symmetric, and thus all harmonics of
the pulse rate to will appear in the spectrum as side bands around A piece of
such a waveform is shown in the top of Figure 3.17. The frequency domain picture of the top waveform in Figure 3.17 is a se—
ries of side bands around the carrier, f, with generally decreasing amplitudes.
These side bands will be separated by a frequency interval w (equal to the
pulse repeat rate). Because the modulating waveform is half—wave asymmet—
ric and has several successive maxima or minima (here the points of onset
and offset of the pulse), successive side bands will be even smaller in ampli—
tude when they have frequencies similar to harmonics of z (the reciprocal of
the pulse duration), and a bit larger in amplitude for frequencies intermediate mum many 56 Chapter 3 Pulse train of carrier f Pressure l T=1/z l \f=carrier
l____._______________——
Time 3 Single pulse of carrier f
% ... ...
5 ————JW\/\.——————
14
(L _‘~______>
r = 1/ z
W
Time Figure 3.17 Periodic versus aperiodic pulse trains of a pure sine wave. between the harmonics of 2 (nodes and lobes). The corresponding spectrum is
shown in Figure 3.18A. Now suppose we keep the carrier frequency and pulse duration fixed, but
increase the interval between pulses. This change will increase the repeat period (A) (B) pig (C) (D) Amplitude Frequency ( f) Figure 3.18 Effects on frequency spectrum of increasing repeat period of pulse.
(A) through (D) represent signals with increasing intervals between repeats. fis the carrier frequency, w is the frequency of the pulse repeat, and z is the reciprocal of the
pulse duration. D shows the spectrum of a single finite duration pulse. Fourier Analysis 57 1‘, decrease the pulse rate to, and thus bring adjacent side bands closer to the car
rier and to each other. Figure 3.18B shows the spectrum for an infinite train of
pulses with a lower pulse rate than that shown for Figure 3.18A, Note that the
nodes and lobes appear in the same locations because pulse duration, and thus
the internode frequency difference 2, has not changed. What does change is the
spacing of the many side bands around the carrier. Figure 3.18 C shows the spec
trum for an infinite pulse train but with an even longer interval between pulses,
and thus an even smaller value of it). Finally, in the limit we make the interval
between pulses infinitely long. The resulting waveform is shown in the bottom
of Figure 3.17: there is a single pulse that never repeats. Because t is now infi
nitely long, the value of w is infinitely small, and the side bands are thus infi—
nitely close together. The thin frequency component lines of continuously re—
peating signals are now replaced by continuous smears or bands of frequencies. The previous signal has a substantial duration. It is a new type of signal
only because it occurs without repetition. Now, consider what happens if we
reduce the duration of that single pulse to an infinitely short period of time.
The shorter the pulse duration, the larger 2, and thus the wider the lobes in—
cluding the main lobe around the carrier. In the limit (Figure 3.19A), we end
up with a single pulse of infinitely small duration and infinite amplitude (oth
erwise the pulse would have no energy being infinitely short in duration). Be—
cause its duration is infinitely short, the main lobe in its spectrum is infinitely
wide and fills the entire spectrum (Figure 3.193). All frequencies in this spec—
trum have the same energy. This signal is as deviant from a single pure sine
wave as is possible, and given our general rule for Fourier analysis, it is not surprising that we need every possible frequency at the same amplitude to
synthesize it. This imaginary pulse is called a delta function or delta pulse.
There is no such pulse in reality, but one can sometimes trick a machine or
even an ear into thinking it was fed such a signal. Note that another kind of signal, white noise, also has a frequency spec— 4,
trum consisting of all frequencies at approximately the same energies. White
noise has a time domain aneform that is entirely random, and the amplitude
at any time t has no significant correlation with the value at subsequent times.
Clearly, this is a very different time domain waveform from that of a delta e A. «7.95 aw (A) (B) Amplitude Pressure Time Frequency Figure 3.19 Time domain image (A) and frequency spectrum (B) of a single
instantaneous pulse. Amplitude MANY ANIMALS PRODUCE SOUNDS BY AMPLITUDE OR FREQUENCY MODULATION of a car—
rier signal. Other animals (and humans) produce signals that have periodic but nonsi
nusoidal waveforms. In both cases, the time domain waveforms of the resulting sig
nals will be periodic, and we can measure the amount of time (the period) required for
one complete cycle of each repeating pattern. The reciprocal of this period is the fre
quency w of the repeating pattern. If we examine the corresponding frequency spectra,
both cases will exhibit numerous bands evenly spaced along the frequency axis, and
the frequency difference between adjacent bands will equal the time domain measure
of w. If we didn’t know which animal made which sound, how could we tell which
sounds were the results of modulation and which were harmonic series? Harmonic series Harmonic series
FM signal (before filtering) (after filtering) pwq
PM w w m w w m H. E t—l—i w w o. w or t—l H a H 2
Frequency Frequency Frequency A first guess would be to see whether the spectral component with the lowest fre
quency was equal to equal to w or not. If it was, it is likely that the sound is a harmonic:
series. In addition, we would expect successive harmonics to show decreasing ampli
tudes because of Dirichlet conditions. However, these are not very powerful tests. What
if a sound that is generated as a harmonic series suffers the loss of its fundamental and
attenuation of its first few harmonics due to filtering during sound production. or trans
mission in the medium? The frequency spectrum of this filtered signal may now look
very much like that of a modulated signal, and the frequency of the lowest component
will not be equal to w. How is a Wise sound analyst to deal with these problems? The best approach is as follows. First, note that the time domain periodicity of a
harmonic series waveform is very robust to filtering. As long as at least two of the
original components of a harmonic series remain, (and these need not be adjacent
bands), the time domain image of a ﬁltered series will still show a periodicity equal to
the original w. This means that you should continue to use your time domain image to
determine w. You then determine the frequencies of each of the signal components, 1"",
visible in your frequency spectrum. If the original signal was a harmonic series, the
quotient of f, and w for each component should equal an integer. If these quotients are. Fourier Analysis 59 not integers, within the limits of your ability to measure to and the fi, then it is more
likely that your periodic signal was generated by modulation. Consider a situation in which sound generation or transmission filters out alterna
tive harmonics in a harmonic series. In this case, the value of to measured in the time
domain will be much lower than the difference measured between adjacent bands in
the corresponding frequency domain image. When there is a mismatch between time
domain and frequency domain estimates of w, always use the time domain measure.
You could still use the test outlined above to determine Whether the original signal
was a harmonic series or not. Some animals, such as birds, can produce two independent sine waves at the same
time. We saw in Chapter 2 that the combination of any two randomly selected sine
waves yields a time domain waveform with a periodic variation in amplitude. This
waxing and waning of the overall amplitude is called beating. If the two sine waves
being added have frequencies of f1 and f2 with f1 > f2, then the frequency of beating
w = f1 f2. Are beats a special case of harmonics? In general, the answer is no because
it will be rare that the quotients f1 /w and fz/w both equal integers for two randomly
selected frequencies f1 and f2 and a beating frequency m. For example, the combination
of two sine waves at frequencies 519 and 530 will produce beating at a w of 11 Hz.
However, neither 519/11 nor 531/11 equal integers. Thus the two frequencies are not
harmonically related. Where two combined frequencies are not harmonically related,
their combined waveform will not be truly periodic. If you look carefully at the sec
tions of time domain waveform separated by beat maxima, you will see that succes—
sive sections are usually not identical. As a rule, you will have to wait for f1 cycles of
the first frequency (or the equivalent, f2 cycles of the second), before the beating wave—
form really repeats itself. This corresponds to a repeat frequency of 1, which is periodic
in only a trivial sense. If it is rare to find two combined frequencies that produce truly periodic wave—
forms, imagine how much rarer it is to find a combination of three randomly selected
frequencies that is periodic. As with pairs of random frequencies, the maximum repeat
frequency is likely to be 1. From this point of View, it should be clear to you that peri—
odic signals, including both harmonic and modulation signals, are a very small subset
of the large number of possible waveforms generated by adding together randomly
chosen sine waves and that most such waveforms will not be periodic. function. Although the two types of signals show some similarities in the fre
quency domain, they also show major differences. Whereas the amplitudes of
all frequencies in the delta function frequency spectrum are always the same,
the amplitudes of different frequencies in white noise vary randomly, and it is
only the average for each frequency that is constant. The relative amplitudes of
each frequency in a very short segment of white noise would not be equal.
Similarly, the relative phases in a delta function are precisely defined and
g; fixed; the relative phases of different frequencies of white noise vary randomly. ...
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 Spring '09
 NIEH
 Frequency, Sine wave

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