243
CHAPTER
13 Continuous Signal Processing
Continuous signal processing is a parallel field to DSP, and most of the techniques are nearly
identical. For example, both DSP and continuous signal processing are based on linearity,
decomposition, convolution and Fourier analysis. Since continuous signals cannot be directly
represented in digital computers, don't expect to find computer programs in this chapter.
Continuous signal processing is based on mathematics; signals are represented as equations, and
systems change one equation into another. Just as the digital computer is the primary tool used
in DSP, calculus is the primary tool used in continuous signal processing. These techniques have
been used for centuries, long before computers were developed.
The Delta Function
Continuous signals can be decomposed into scaled and shifted delta functions,
just as done with discrete signals. The difference is that the continuous delta
function is much more complicated and mathematically abstract than its discrete
counterpart. Instead of defining the continuous delta function by what it is, we
will define it by the characteristics it has.
A thought experiment will show how this works. Imagine an electronic circuit
composed of linear components, such as resistors, capacitors and inductors.
Connected to the input is a signal generator that produces various shapes of
short pulses. The output of the circuit is connected to an oscilloscope,
displaying the waveform produced by the circuit in response to each input
pulse. The question we want to answer is: how is the shape of the output
pulse related to the characteristics of the input pulse? To simplify the
investigation, we will only use input pulses that are much shorter than the
output. For instance, if the system responds in milliseconds, we might use input
pulses only a few microseconds in length.
After taking many measurement, we come to three conclusions: First, the
shape of the input pulse does not affect the shape of the output signal. This
The Scientist and Engineer's Guide to Digital Signal Processing 244
is illustrated in Fig. 13-1, where various shapes of short input pulses
produce exactly the same shape of output pulse. Second, the shape of the
output waveform is totally determined by the characteristics of the system,
i.e., the value and configuration of the resistors, capacitors and inductors.
Third, the amplitude of the output pulse is directly proportional to the area
of the input pulse. For example, the output will have the same amplitude
for inputs of: 1 volt for 1 microsecond, 10 volts for 0.1 microseconds,
1,000 volts for 1 nanosecond, etc. This relationship also allows for input
pulses with negative areas. For instance, imagine the combination of a 2
volt pulse lasting 2 microseconds being quickly followed by a -1 volt pulse
lasting 4 microseconds. The total area of the input signal is zero, resulting
in the output doing nothing.
Input signals that are brief enough to have these three properties are called
impulses. In other words, an impulse is any signal that is entirely zero
except for a short blip of arbitrary shape. For example, an impulse to a
microwave transmitter may have to be in the picosecond range because the
electronics responds in nanoseconds. In comparison, a volcano that erupts
for years may be a perfectly good impulse to geological changes that take
millennia.
Mathematicians don't like to be limited by any particular system, and
commonly use the term impulse to mean a signal that is short enough to be
an impulse to any possible system. That is, a signal that is infinitesimally
narrow. The continuous delta function is a normalized version of this type
of impulse. Specifically, the continuous delta function is mathematically
defined by three idealized characteristics: (1) the signal must be
infinitesimally brief, (2) the pulse must occur at time zero, and (3) the pulse
must have an area of one.
Since the delta function is defined to be infinitesimally narrow and have a fixed
area, the amplitude is implied to be infinite. Don't let this bother you; it is
completely unimportant. Since the amplitude is part of the shape of the
impulse, you will never encounter a problem where the amplitude makes any
difference, infinite or not. The delta function is a mathematical construct, not
a real world signal. Signals in the real world that act as delta functions will
always have a finite duration and amplitude.
Just as in the discrete case, the continuous delta function is given the
mathematical symbol: . Likewise, the output of a continuous system in *( )
response to a delta function is called the impulse response, and is often
denoted by: . Notice that parentheses, ( ), are used to denote continuous h ( )
signals, as compared to brackets, [ ], for discrete signals. This notation is
used in this book and elsewhere in DSP, but isn't universal. Impulses are
displayed in graphs as vertical arrows (see Fig. 13-1d), with the length of the
arrow indicating the area of the impulse.
To better understand real world impulses, look into the night sky at a planet
and a star, for instance, Mars and Sirius. Both appear about the same
brightness and size to the unaided eye. The reason for this similarity is not
Chapter 13- Continuous Signal Processing 245
*
Linear
System
Linear
System
Linear
System
Linear
System
a.
b.
c
d.
(t)
FIGURE 13-1
The continuous delta function. If the input to a linear system is brief compared to the resulting
output, the shape of the output depends only on the characteristics of the system, and not the shape
of the input. Such short input signals are called impulses. Figures a,b & c illustrate example input
signals that are impulses for this particular system. The term delta function is used to describe a
normalized impulse, i.e., one that occurs at and has an area of one. The mathematical symbols t ' 0
for the delta function are shown in (d), a vertical arrow and . *(t)
obvious, since the viewing geometry is drastically different. Mars is about
6000 kilometers in diameter and 60 million kilometers from earth. In
comparison, Sirius is about 300 times larger and over one-million times
farther away. These dimensions should make Mars appear more than
three-thousand times larger than Sirius. How is it possible that they look
alike?
These objects look the same because they are small enough to be impulses to
the human visual system. The perceived shape is the impulse response of the
eye, not the actual image of the star or planet. This becomes obvious when the
two objects are viewed through a small telescope; Mars appears as a dim disk,
while Sirius still appears as a bright impulse. This is also the reason that stars
twinkle while planets do not. The image of a star is small enough that it can
be briefly blocked by particles or turbulence in the atmosphere, whereas the
larger image of the planet is much less affected.
The Scientist and Engineer's Guide to Digital Signal Processing 246
y(t ) ' m
%4
&4
x(J) h(t& J) dJ
EQUATION 13-1
The convolution integral. This equation
defines the meaning of: . y (t ) ' x (t )th(t )
Convolution
Just as with discrete signals, the convolution of continuous signals can be
viewed from the input signal, or the output signal. The input side
viewpoint is the best conceptual description of how convolution operates.
In comparison, the output side viewpoint describes the mathematics that
must be used. These descriptions are virtually identical to those presented
in Chapter 6 for discrete signals.
Figure 13-2 shows how convolution is viewed from the input side. An input
signal, , is passed through a system characterized by an impulse response, x (t)
, to produce an output signal, . This can be written in the familiar h (t) y (t)
mathematical equation, . The input signal is divided into y (t) ' x(t)th(t)
narrow columns, each short enough to act as an impulse to the system. In
other words, the input signal is decomposed into an infinite number of scaled
and shifted delta functions. Each of these impulses produces a scaled and
shifted version of the impulse response in the output signal. The final output
signal is then equal to the combined effect, i.e., the sum of all of the individual
responses.
For this scheme to work, the width of the columns must be much shorter
than the response of the system. Of course, mathematicians take this to the
extreme by making the input segments infinitesimally narrow, turning the
situation into a calculus problem. In this manner, the input viewpoint
describes how a single point (or narrow region) in the input signal affects
a larger portion of output signal.
In comparison, the output viewpoint examines how a single point in the output
signal is determined by the various values from the input signal. Just as with
discrete signals, each instantaneous value in the output signal is affected by a
section of the input signal, weighted by the impulse response flipped
left-for-right. In the discrete case, the signals are multiplied and summed. In
the continuous case, the signals are multiplied and integrated. In equation
form:
This equation is called the convolution integral, and is the twin of the
convolution sum (Eq. 6-1) used with discrete signals. Figure 13-3 shows how
this equation can be understood. The goal is to find an expression for
calculating the value of the output signal at an arbitrary time, t. The first
step is to change the independent variable used to move through the input
signal and the impulse response. That is, we replace t with J (a lower case
Chapter 13- Continuous Signal Processing 247
a
b
c
a
b
c
time (t) time(t)
x(t) y(t) Linear
System
FIGURE 13-2
Convolution viewed from the input side. The input signal, , is divided into narrow segments, x(t)
each acting as an impulse to the system. The output signal, , is the sum of the resulting scaled y (t)
and shifted impulse responses. This illustration shows how three points in the input signal contribute
to the output signal.
time (J) time(t)
x(J) y(t)
t t h(t-J)
?
x(J)
Linear
System
FIGURE 13-3
Convolution viewed from the output side. Each value in the output signal is influenced by many
points from the input signal. In this figure, the output signal at time t is being calculated. The input
signal, , is weighted (multiplied) by the flipped and shifted impulse response, given by . x (J) h( t&J)
Integrating the weighted input signal produces the value of the output point, y (t)
Greek tau). This makes and become and , respectively. x (t) h(t) x (J) h(J)
This change of variable names is needed because t is already being used to
represent the point in the output signal being calculated. The next step is to
flip the impulse response left-for-right, turning it into . Shifting the h(&J)
flipped impulse response to the location t, results in the expression becoming
. The input signal is then weighted by the flipped and shifted impulse h(t&J)
response by multiplying the two, i.e., . The value of the output x (J) h(t&J)
signal is then found by integrating this weighted input signal from negative to
positive infinity, as described by Eq. 13-1.
If you have trouble understanding how this works, go back and review the same
concepts for discrete signals in Chapter 6. Figure 13-3 is just another way of
describing the convolution machine in Fig. 6-8. The only difference is that
integrals are being used instead of summations. Treat this as an extension of
what you already know, not something new.
An example will illustrate how continuous convolution is used in real world
problems and the mathematics required. Figure 13-4 shows a simple
continuous linear system: an electronic low-pass filter composed of a single
resistor and a single capacitor. As shown in the figure, an impulse entering this
system produces an output that quickly jumps to some value, and then
exponentially decays toward zero. In other words, the impulse response of
this simple electronic circuit is a one-sided exponential. Mathematically, the
The Scientist and Engineer's Guide to Digital Signal Processing 248
Time
-1 0 1 2 3
0
1
2
Time
-1 0 1 2 3
0
1
2
*(t) h(t)
R
C
Amplitude
Amplitude
FIGURE 13-4
Example of a continuous linear system. This electronic circuit is a low-pass filter composed of a single resistor
and capacitor. The impulse response of this system is a one-sided exponential.
h(t ) ' 0
h(t ) ' "e &" t
for t $ 0
for t < 0
x(t ) ' 1 for 0 # t # 1
x(t ) ' 0 otherwise
Time
-1 0 1 2 3
0
1
2
Time
-1 0 1 2 3
0
1
2
Time
-1 0 1 2 3
0
1
2
x(t) h(t) y(t)
FIGURE 13-5
Example of continuous convolution. This figure illustrates a square pulse entering an RC low-pass filter (Fig.
13-4). The square pulse is convolved with the system's impulse response to produce the output. Amplitude
Amplitude
Amplitude
impulse response of this system is broken into two sections, each represented
by an equation:
where (R is in ohms, C is in farads, and t is in seconds). Just as in " ' 1/RC
the discrete case, the continuous impulse response contains complete
information about the system, that is, how it will react to all possible signals.
To pursue this example further, Fig. 13-5 shows a square pulse entering the
system, mathematically expressed by:
Since both the input signal and the impulse response are completely known as
mathematical expressions, the output signal, , can be calculated by y (t)
evaluating the convolution integral of Eq. 13-1. This is complicated by
the fact that both signals are defined by regions rather than a single
Chapter 13- Continuous Signal Processing 249
c. Full overlap a. No overlap b. Partial overlap
J
0 1 0 1 0 1
J J
t t t
(t > 1) (0 # t # 1) (t < 0)
FIGURE 13-6
Calculating a convolution by segments. Since many continuous signals are defined by regions, the convolution
calculation must be performed region-by-region. In this example, calculation of the output signal is broken into
three sections: (a) no overlap, (b) partial overlap, and (c) total overlap, of the input signal and the shiftedflipped
impulse response.
y(t ) ' 0 for t < 0
y(t ) ' m
t
0
1@ "e &"( t& J) dJ
y(t ) ' e &"t [ e "J ] /
t
0
y(t ) ' e &"t [ e "t & 1 ]
y(t ) ' 1 & e & "t
y(t ) ' m
4
&4
x(J) h(t& J) dJ (start with Eq. 13-1)
(plug in the signals)
for 0 # t # 1
(evaluate the integral)
(reduce)
mathematical expression. This is very common in continuous signal
processing. It is usually essential to draw a picture of how the two signals
shift over each other for various values of t. In this example, Fig. 13-6a
shows that the two signals do not overlap at all for . This means that t < 0
the product of the two signals is zero at all locations along the J axis, and
the resulting output signal is:
A second case is illustrated in (b), where t is between 0 and 1. Here the two
signals partially overlap, resulting in their product having nonzero values
between and . Since this is the only nonzero region, it is the only J' 0 J' t
section where the integral needs to be evaluated. This provides the output
signal for , given by: 0 # t #1
The Scientist and Engineer's Guide to Digital Signal Processing 250
y(t ) ' m
1
0
1@ "e &"( t& J) dJ
y(t ) ' e &"t [ e "J] /
1
0
y(t ) ' [e "& 1] e &"t
(plug into Eq. 13-1)
for t > 1
(evaluate the integral)
Figure (c) shows the calculation for the third section of the output signal, where
t > 1. Here the overlap occurs between and , making the J' 0 J' 1
calculation the same as for the second segment, except a change to the limits
of integration:
The waveform in each of these three segments should agree with your
knowledge of electronics: (1) The output signal must be zero until the input
signal becomes nonzero. That is, the first segment is given by for y (t ) ' 0
. (2) When the step occurs, the RC circuit exponentially increases to match t < 0
the input, according to the equation: . (3) When the input is y (t ) ' 1& e &" t
returned to zero, the output exponentially decays toward zero, given by the
equation: (where , the voltage on the capacitor just y (t ) ' ke &" t k ' e "& 1
before the discharge was started).
More intricate waveforms can be handled in the same way, although the
mathematical complexity can rapidly become unmanageable. When faced
with a nasty continuous convolution problem, you need to spend significant
time evaluating strategies for solving the problem. If you start blindly
evaluating integrals you are likely to end up with a mathematical mess. A
common strategy is to break one of the signals into simpler additive
components that can be individually convolved. Using the principles of
linearity, the resulting waveforms can be added to find the answer to the
original problem.
Figure 13-7 shows another strategy: modify one of the signals in some linear
way, perform the convolution, and then undo the original modification. In this
example the modification is the derivative, and it is undone by taking the
integral. The derivative of a unit amplitude square pulse is two impulses, the
first with an area of one, and the second with an area of negative one. To
understand this, think about the opposite process of taking the integral of the
two impulses. As you integrate past the first impulse, the integral rapidly
increases from zero to one, i.e., a step function. After passing the negative
impulse, the integral of the signal rapidly returns from one back to zero,
completing the square pulse.
Taking the derivative simplifies this problem because convolution is easy
when one of the signals is composed of impulses. Each of the two impulses
in contributes a scaled and shifted version of the impulse response to x )(t )
Chapter 13- Continuous Signal Processing 251
Time
-1 0 1 2 3
-2
-1
0
1
2
Time
-1 0 1 2 3
-2
-1
0
1
2
h(t) yN(t)
Time
-1 0 1 2 3
0
1
2
Time
-1 0 1 2 3
0
1
2
Time
-1 0 1 2 3
0
1
2
x(t) h(t) y(t)
I d/dt
Time
-1 0 1 2 3
-2
-1
0
1
2
xN(t)
Amplitude
Amplitude
Amplitude
FIGURE 13-7
A strategy for convolving signals. Convolution problems can often be simplified by clever use of the rules
governing linear systems. In this example, the convolution of two signals is simplified by taking the derivative
of one of them. After performing the convolution, the derivative is undone by taking the integral.
Amplitude
Amplitude
Amplitude
the derivative of the output signal, . That is, by inspection it is known y )(t)
that: . The output signal, , can then be found by y )(t ) ' h(t ) & h (t&1) y (t )
plugging in the exact equation for , and integrating the expression. h(t )
A slight nuisance in this procedure is that the DC value of the input signal is
lost when the derivative is taken. This can result in an error in the DC value
of the calculated output signal. The mathematics reflects this as the arbitrary
constant that can be added during the integration. There is no systematic way
of identifying this error, but it can usually be corrected by inspection of the
problem. For instance, there is no DC error in the example of Fig. 13-7. This
is known because the calculated output signal has the correct DC value when
t becomes very large. If an error is present in a particular problem, an
appropriate DC term is manually added to the output signal to complete the
calculation.
This method also works for signals that can be reduced to impulses by taking
the derivative multiple times. In the jargon of the field, these signals are called
piecewise polynomials. After the convolution, the initial operation of multiple
derivatives is undone by taking multiple integrals. The only catch is that the
lost DC value must be found at each stage by finding the correct constant of
integration.
The Scientist and Engineer's Guide to Digital Signal Processing 252
x (t ) ' 1
B m
%4
0
Re X(T) cos(Tt ) & Im X(T) sin(Tt ) dT
EQUATION 13-2
The Fourier transform synthesis equation. In this equation, is the time x(t)
domain signal being synthesized, and & are the real and ReX(T) Im X(T)
imaginary parts of the frequency spectrum, respectively.
Before starting a difficult continuous convolution problem, there is another
approach that you should consider. Ask yourself the question: Is a
mathematical expression really needed for the output signal, or is a graph of
the waveform sufficient? If a graph is adequate, you may be better off to
handle the problem with discrete techniques. That is, approximate the
continuous signals by samples that can be directly convolved by a computer
program. While not as mathematically pure, it can be much easier.
The Fourier Transform
The Fourier Transform for continuous signals is divided into two categories,
one for signals that are periodic, and one for signals that are aperiodic.
Periodic signals use a version of the Fourier Transform called the Fourier
Series, and are discussed in the next section. The Fourier Transform used with
aperiodic signals is simply called the Fourier Transform. This chapter
describes these Fourier techniques using only real mathematics, just as the last
several chapters have done for discrete signals. The more powerful use of
complex mathematics will be reserved for Chapter 31.
Figure 13-8 shows an example of a continuous aperiodic signal and its
frequency spectrum. The time domain signal extends from negative infinity to
positive infinity, while each of the frequency domain signals extends from zero
to positive infinity. This frequency spectrum is shown in rectangular form
(real and imaginary parts); however, the polar form (magnitude and phase) is
also used with continuous signals. Just as in the discrete case, the synthesis
equation describes a recipe for constructing the time domain signal using the
data in the frequency domain. In mathematical form:
In words, the time domain signal is formed by adding (with the use of an
integral) an infinite number of scaled sine and cosine waves. The real part
of the frequency domain consists of the scaling factors for the cosine waves,
while the imaginary part consists of the scaling factors for the sine waves. Just
as with discrete signals, the synthesis equation is usually written with
negative sine waves. Although the negative sign has no significance in this
discussion, it is necessary to make the notation compatible with the complex
mathematics described in Chapter 29. The key point to remember is that
some authors put this negative sign in the equation, while others do not.
Also notice that frequency is represented by the symbol, T, a lower case
Chapter 13- Continuous Signal Processing 253
FIGURE 13-8
Example of the Fourier Transform. The time domain signal, , extends from negative to positive infinity. x(t )
The frequency domain is composed of a real part, , and an imaginary part, , each extending from ReX(T) Im X(T)
zero to positive infinity. The frequency axis in this illustration is labeled in cycles per second (hertz). To
convert to natural frequency, multiply the numbers on the frequency axis by 2B.
Frequency (hertz)
0 20 40 60 80 100 120 140
0
20
40
60
80
100
b. Re X(T)
Frequency (hertz)
0 20 40 60 80 100 120 140
0
20
40
60
80
100
c. Im X(T)
Time (milliseconds)
-50 -40 -30 -20 -10 0 10 20 30 40 50
-8
-4
0
4
8
a. x(t)
Time Domain Frequency Domain
Amplitude Amplitude
Amplitude
ReX(T) ' m
%4
&4
x(t ) cos (Tt ) dt EQUATION 13-3
The Fourier transform analysis equations. In
this equation, & are the real ReX(T) Im X(T)
and imaginary parts of the frequency
spectrum, respectively, and is the time x(t)
domain signal being analyzed. ImX(T) ' & m
%4
&4
x(t ) sin (Tt ) dt
Greek omega. As you recall, this notation is called the natural frequency,
and has the units of radians per second. That is, , where f is the T' 2Bf
frequency in cycles per second (hertz). The natural frequency notation is
favored by mathematicians and others doing signal processing by solving
equations, because there are usually fewer symbols to write.
The analysis equations for continuous signals follow the same strategy as the
discrete case: correlation with sine and cosine waves. The equations are:
The Scientist and Engineer's Guide to Digital Signal Processing 254
h(t ) ' 0
h(t ) ' "e &" t for t $ 0
for t < 0
Re H(T) ' "2
"2 % T2
Re H(T) ' "e &" t
"2 % T2
[&" cos(Tt ) % Tsin(Tt ) ] /
%4
0
Re H(T) ' m
%4
0
"e &" t cos(Tt ) dt
Re H(T) ' m
%4
&4
h(t ) cos(Tt ) dt (start with Eq. 13-3)
(plug in the signal)
(evaluate)
Im H(T) '
&T"
"2 % T2
As an example of using the analysis equations, we will find the frequency
response of the RC low-pass filter. This is done by taking the Fourier
transform of its impulse response, previously shown in Fig. 13-4, and
described by:
The frequency response is found by plugging the impulse response into the
analysis equations. First, the real part:
Using this same approach, the imaginary part of the frequency response is
calculated to be:
Just as with discrete signals, the rectangular representation of the frequency
domain is great for mathematical manipulation, but difficult for human
understanding. The situation can be remedied by converting into polar
notation with the standard relations: Mag H(T) ' [Re H(T)2 % Im H(T)2]½
and . Working through the algebra Phase H(T) ' arctan [Re H(T) / Im H(T)]
Chapter 13- Continuous Signal Processing 255
Mag H(T) ' "
[ "2 % T2 ]1/2
Phase H(T) ' arctan & T
"
Frequency (hertz)
0 1000 2000 3000 4000 5000 6000
-1.6
-1.2
-0.8
-0.4
0.0
0.4
0.8
1.2
1.6
b. Phase
FIGURE 13-9
Frequency response of an RC low-pass filter. These curves were derived by calculating the Fourier
transform of the impulse response, and then converting to polar form.
Frequency (hertz)
0 1000 2000 3000 4000 5000 6000
0.0
0.2
0.4
0.6
0.8
1.0
1.2
a. Magnitude
Phase (radians)
Amplitude
provides the frequency response of the RC low-pass filter as magnitude and
phase (i.e., polar form):
Figure 13-9 shows graphs of these curves for a cutoff frequency of 1000 hertz
(i.e., ). "' 2B1000
The Fourier Series
This brings us to the last member of the Fourier transform family: the Fourier
series. The time domain signal used in the Fourier series is periodic and
continuous. Figure 13-10 shows several examples of continuous waveforms
that repeat themselves from negative to positive infinity. Chapter 11 showed
that periodic signals have a frequency spectrum consisting of harmonics. For
instance, if the time domain repeats at 1000 hertz (a period of 1 millisecond),
the frequency spectrum will contain a first harmonic at 1000 hertz, a second
harmonic at 2000 hertz, a third harmonic at 3000 hertz, and so forth. The first
harmonic, i.e., the frequency that the time domain repeats itself, is also called
the fundamental frequency. This means that the frequency spectrum can be
viewed in two ways: (1) the frequency spectrum is continuous, but zero at all
frequencies except the harmonics, or (2) the frequency spectrum is discrete,
and only defined at the harmonic frequencies. In other words, the frequencies
between the harmonics can be thought of as having a value of zero, or simply
The Scientist and Engineer's Guide to Digital Signal Processing 256
x(t ) ' a0
% j4
n'1
an cos (2Bf t n) & j4
n'1
bn sin (2Bf t n)
EQUATION 13-4
The Fourier series synthesis equation. Any periodic signal, , can x(t)
be reconstructed from sine and cosine waves with frequencies that are
multiples of the fundamental, f. The an and bn coefficients hold the
amplitudes of the cosine and sine waves, respectively.
a0
' 1
T m
T /2
&T /2
x (t ) dt an
' 2
T m
T /2
&T /2
x(t ) cos
2Btn
T
dt
bn
'
&2
T m
T /2
&T /2
x(t ) sin
2Btn
T
dt
EQUATION 13-5
Fourier series analysis equations. In these equations, is x(t)
the time domain signal being decomposed, is the DC a0
component, & hold the amplitudes of the cosine and an bn
sine waves, respectively, and T is the period of the signal,
i.e., the reciprocal of the fundamental frequency.
not existing. The important point is that they do not contribute to forming the
time domain signal.
The Fourier series synthesis equation creates a continuous periodic signal
with a fundamental frequency, f, by adding scaled cosine and sine waves
with frequencies: f, 2f, 3f, 4f, etc. The amplitudes of the cosine waves are
held in the variables: etc., while the amplitudes of the sine a1, a2, a3, a4,
waves are held in: and so on. In other words, the "a" and "b" b1, b2, b3, b4,
coefficients are the real and imaginary parts of the frequency spectrum,
respectively. In addition, the coefficient is used to hold the DC value of a0
the time domain waveform. This can be viewed as the amplitude of a cosine
wave with zero frequency (a constant value). Sometimes is grouped with a0
the other "a" coefficients, but it is often handled separately because it
requires special calculations. There is no coefficient since a sine wave b0
of zero frequency has a constant value of zero, and would be quite useless.
The synthesis equation is written:
The corresponding analysis equations for the Fourier series are usually
written in terms of the period of the waveform, denoted by T, rather than the
fundamental frequency, f (where ). Since the time domain signal is f ' 1/T
periodic, the sine and cosine wave correlation only needs to be evaluated over
a single period, i.e., to , 0 to T, -T to 0, etc. Selecting different &T/2 T/2
limits makes the mathematics different, but the final answer is always the same.
The Fourier series analysis equations are:
Chapter 13- Continuous Signal Processing 257
0 f 2f 3f 4f 5f 6f
A
0
0 f 2f 3f 4f 5f 6f
A
0
0 f 2f 3f 4f 5f 6f
A
0
0 f 2f 3f 4f 5f 6f
A
0
0 f 2f 3f 4f 5f 6f
A
0
0 f 2f 3f 4f 5f 6f
A
0
A
a. Pulse
b. Square
c. Triangle
d. Sawtooth
e. Rectified
Time Domain Frequency Domain
A
A
A
A
A
k
T
d = k/T
t = 0
t = 0
t = 0
t = 0
t = 0
t = 0
f. Cosine wave
(all even harmonics are zero)
a0
' 0
an
' 2A
nB
sin nB
2
bn
' 0
( in this example) d ' 0.27
a0
' A d
bn
' 0
an
' 2A
nB
sin (nBd )
(all even harmonics are zero)
a0
' 0
an
' 4A
(nB)2
bn
' 0
a0
' 0
an
' 0
bn
' A
nB
a0
' 2A/B
bn
' 0
an
'
&4A
B(4n 2&1)
(all other coefficients are zero)
a1
' A
FIGURE 13-10
Examples of the Fourier series. Six common time domain waveforms are shown, along with the equations to
calculate their "a" and "b" coefficients.
The Scientist and Engineer's Guide to Digital Signal Processing 258
-T 0 T 2T 3T -2T -3T
A
0
-T/2 T/2
-k/2 k/2
Time
FIGURE 13-11
Example of calculating a Fourier series. This is a pulse train with a duty cycle of d = k/T. The
Fourier series coefficients are calculated by correlating the waveform with cosine and sine waves
over any full period. In this example, the period from -T/2 to T/2 is used.
Amplitude
x(t ) ' 0
x(t ) ' A for -k/2 # t # k/2
otherwise
a0
' 1
T m
T/2
&T/2
x (t ) dt (start with Eq. 13-5)
a0
' 1
T m
k/2
&k/2
A dt
a0
' A k
T
a0
' A d
(plug in the signal)
(evaluate the integral)
(substitute: d = k/T)
Figure 13-11 shows an example of calculating a Fourier series using these
equations. The time domain signal being analyzed is a pulse train, a square
wave with unequal high and low durations. Over a single period from &T/2
to , the waveform is given by: T/2
The duty cycle of the waveform (the fraction of time that the pulse is "high")
is thus given by . The Fourier series coefficients can be found by d ' k/T
evaluating Eq. 13-5. First, we will find the DC component, : a0
This result should make intuitive sense; the DC component is simply the
average value of the signal. A similar analysis provides the "a" coefficients:
Chapter 13- Continuous Signal Processing 259
an
' 2
T m
T/2
&T/2
x(t ) cos
2Btn
T
dt
an
' 2
T m
k /2
&k /2
A cos
2Btn
T
dt
an
' 2A
T
T
2Bn
sin 2Btn
T /
k/2
&k/2
an
' 2A
nB
sin(Bnd )
(start with Eq. 13-4)
(plug in the signal)
(evaluate the integral)
(reduce)
The "b" coefficients are calculated in this same way; however, they all turn out
to be zero. In other words, this waveform can be constructed using only cosine
waves, with no sine waves being needed.
The "a" and "b" coefficients will change if the time domain waveform is
shifted left or right. For instance, the "b" coefficients in this example will be
zero only if one of the pulses is centered on . Think about it this way. t ' 0
If the waveform is even (i.e., symmetrical around ), it will be composed t ' 0
solely of even sinusoids, that is, cosine waves. This makes all of the "b"
coefficients equal to zero. If the waveform if odd (i.e., symmetrical but
opposite in sign around ), it will be composed of odd sinusoids, i.e., sine t ' 0
waves. This results in the "a" coefficients being zero. If the coefficients are
converted to polar notation (say, Mn and 2n coefficients), a shift in the time
domain leaves the magnitude unchanged, but adds a linear component to the
phase.
To complete this example, imagine a pulse train existing in an electronic
circuit, with a frequency of 1 kHz, an amplitude of one volt, and a duty cycle
of 0.2. The table in Fig. 13-12 provides the amplitude of each harmonic
contained in this waveform. Figure 13-12 also shows the synthesis of the
waveform using only the first fourteen of these harmonics. Even with this
number of harmonics, the reconstruction is not very good. In mathematical
jargon, the Fourier series converges very slowly. This is just another way of
saying that sharp edges in the time domain waveform results in very high
frequencies in the spectrum. Lastly, be sure and notice the overshoot at the
sharp edges, i.e., the Gibbs effect discussed in Chapter 11.
An important application of the Fourier series is electronic frequency
multiplication. Suppose you want to construct a very stable sine wave
oscillator at 150 MHz. This might be needed, for example, in a radio
The Scientist and Engineer's Guide to Digital Signal Processing 260
Time (milliseconds)
0 1 2 3 4
-0.5
0.0
0.5
1.0
1.5 frequency amplitude
(volts)
DC 0.20000
1 kHz 0.37420
2 kHz 0.30273
3 kHz 0.20182
4 kHz 0.09355
5 kHz 0.00000
6 kHz -0.06237
7 kHz -0.08649
8 kHz -0.07568
9 kHz -0.04158
10 kHz 0.00000
11 kHz 0.03402
12 kHz 0.05046
!
123 kHz 0.00492
124 kHz 0.00302
125 kHz 0.00000
126 kHz -0.00297
!
803 kHz 0.00075
804 kHz 0.00046
805 kHz 0.00000
806 kHz -0.00046
FIGURE 13-12
Example of Fourier series synthesis. The waveform
being constructed is a pulse train at 1 kHz, an
amplitude of one volt, and a duty cycle of 0.2 (as
illustrated in Fig. 13-11). This table shows the
amplitude of the harmonics, while the graph shows
the reconstructed waveform using only the first
fourteen harmonics.
Amplitude (volts)
transmitter operating at this frequency. High stability calls for the circuit to
be crystal controlled. That is, the frequency of the oscillator is determined by
a resonating quartz crystal that is a part of the circuit. The problem is, quartz
crystals only work to about 10 MHz. The solution is to build a crystal
controlled oscillator operating somewhere between 1 and 10 MHz, and then
multiply the frequency to whatever you need. This is accomplished by
distorting the sine wave, such as by clipping the peaks with a diode, or running
the waveform through a squaring circuit. The harmonics in the distorted
waveform are then isolated with band-pass filters. This allows the frequency
to be doubled, tripled, or multiplied by even higher integers numbers. The
most common technique is to use sequential stages of doublers and triplers to
generate the required frequency multiplication, rather than just a single stage.
The Fourier series is important to this type of design because it describes the
amplitude of the multiplied signal, depending on the type of distortion and
harmonic selected.