ABSTRACT
Jean Baptiste Joseph Fourier (1768-1830), French mathematician, known also as an Egyptologist and administrator, was the first to present the series and transform that bear his name He was the first to propose that the periodic waveforms could be represented by a sum of sinusoids His major work, The Analytic Theory of Heat in 1822, stated the equations governing heat transfer in solids Although Fourier’s results were not enthusiastically received by the scientific world, his work provided the impetus for later work on trigonometric series and the theory of functions of a real variable
Fourier was born in France in 1768, the son of a tailor Joseph’s mother died when he was nine years old and his father died the following year Fourier first attended the local military school conducted by Benedictine monks He showed proficiency in mathematics and later became a teacher in mathematics at the same school In 1793,
he became involved in politics and joined the local Revolutionary Committee Due to his political involvement, he narrowly escaped death twice He died in his bed in Paris on May 16, 1830
This chapter is concerned with a means of analyzing systems with periodic excitations The notion of periodic functions was introduced in Chapter 1, where the importance of sinusoids (or cosinusoids) was emphasized In this chapter, we introduce Fourier series, a premier tool for analyzing periodic signals
Although the Fourier series has a long history involving many individuals, the Fourier series is named after Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist In 1822, Fourier was the first to suggest that any periodic function can be represented as a sum of sinusoids Such a representation provides a powerful tool for analyzing signals and systems System responses to periodic signals are of practical interest because these signals are common
Fourier analysis (Fourier series in this chapter and Fourier transform in the next chapter) plays a major role in system analysis for a number of reasons First, Fourier analysis leads to the frequency spectrum of a continuous-time signal The frequency spectrum displays the various sinusoidal components that make up the signal Engineers think of signals in terms of their frequency spectra and of systems in terms of their frequency response Second, Fourier analysis converts time-domain signals into frequency-domain representation that lends new insight into the nature and the properties of the signals and systems For many purposes, the frequencydomain representations are more convenient to analyze, synthesize, and process Third, in the frequency domain linear systems are described by linear algebraic equations that can be easily solved, in contrast to the time-domain representation, where they are described by linear differential equations It is for these reasons that Fourier analysis is used extensively today in science and engineering
This chapter begins with the trigonometric Fourier series and how to determine the coefficients of the series Then, we consider the complex exponential Fourier series We will discuss some properties of Fourier series We will cover circuit analysis, filtering, and spectrum analyzers as engineering applications of Fourier series We will finally see how we can use MATLAB® to plot line spectra
The Fourier series can be represented in three ways, the sine-cosine, amplitudephase, and complex exponential
A periodic signal is one that repeats itself every T s In other words, a continuous time signal x(t) satisfies
x t x t nT( ) ( )= + (41) where
n is an integer T is the fundamental period of x(t)
Any periodic function can be expressed as an infinite series consisting of sine or cosine functions Thus, x(t) can be expressed as
x t a a t b t a t b t
a t b
( ) cos sin cos sin cos
= + + + +
+ +
2 2
w w w w
w sin3 0w t + (42)
This can be written as
x t a
a n t b n tn n n
( ) ( cos sin )= + + =
w w
(43)
where
w p0
2= T
(44)
ω0 is known as the fundamental frequency (in rad/s) The coefficients as are called the Fourier cosine coefficients including a0, the constant term, which is in reality the 0th cosine term, and bs are called the Fourier sine coefficients The Fourier coefficient a0 in the earlier equation is the constant or dc component of x(t) The Fourier coefficients an and bn (for n ≠ 0) are the amplitudes of the sinusoids in the ac component of x(t)
The Fourier series of a periodic signal x(t) is a decomposition of x(t) into a dc component and an ac component that consists of an infinite series of harmonic sinusoids
A periodic function x(t) can be expanded as a Fourier series only if it fulfills the Dirichlet conditions given as
1 x(t) should be integrable over any period; that is,
| ( ) |x t dt t
< ¥ +
2 x(t) has only a finite number of maxima and minima over any period 3 x(t) has only a finite number of discontinuities over any period
The process of determining the Fourier coefficients a0, an, and bn is called Fourier analysis The coefficients can be determined as follows:
a T
x t dt T
1= ò ( )
(45)
a T
x t n t dtn T
= ò2 0 0
( )cos w
(46)
b T
x t n t dtn T
= ò2 0 0
( )sin w
(47)
The fact that sine and cosine functions are orthogonal over a period T leads to the following trigonometric integrals:
sin cosn t dt n t dt T T
w w0 0
0= =ò ò (48)
sin cosn t n t dt T
w w0 0
0 0ò = (49)
sin sin cos cos ,n t m t dt n t m t dt m n T T
w w w w0 0
00ò ò= = ¹ (410)
0 2
n t dt T n t dt T T
w wò ò= = (411)
These integrals will be used in finding the Fourier coefficients To find a0, integrate both sides of Equation 43 over one period
x t dt a dt a n t dt b n t dt T T
n( ) cos sin 0
é
ë ê ê
ù
û ú ú
+ =
w w T
a T
ë ê ê
ù
û ú ú
= + +
0 0 0
(412)
Using the identities in Equation 48, the two integrals involving the ac terms become zero Therefore,
x t dt a T T
( ) =ò 0 0
or
a T
x t dt T
1= ò ( ) (413)
To find an, multiply both sides of Equation 43 by cos mω0t and integrate over one period
x t m t dt a m t dt a n t m t dt T T T
n( )cos cos cos cosw w w w0 0
ë ê ê
ù
û ú ú
+ é
ë ê ê
ù
û ú ú
= + +
b n t m t dt
T a
0 2
sin cosw w
(414)
The integral containing a0 and bn becomes zero according to Equations 48 and 49, respectively The integral containing an will be zero except when m = n, in which case it is T/2 according to Equation 411 Thus,
x t m t dt a T m n T
n( )cos ,w0 0
2ò = = or
a T
x t n t dtn T
= ò2 0 0
( )cos w (415)
To find bn, multiply both sides of Equation 43 by sin mω0t and integrate over one period
x t m t dt a m t dt a n t m t dt T T
( )sin sin cos sinw w w w0 0
é
ë ê ê
ù
û ú ú
+ é
ë ê ê
ù
û ú ú
= + +
b n t m t dt
T b
0 0 2
sin sinw w
(416)
The integral containing a0 and an becomes zero according to Equations 48 and 410, respectively The integral containing bn will be zero except when m = n, in which case it is T/2 according to Equation 411 Thus,
x t n t dt b T m n T
n( )sin ,w0 0
2ò = = or
b T
x t n t dtn T
= ò2 0 0
( )sin w (417)
Since x(t) is periodic the integration of Equation 45 over one full period from to to to + T or from –T/2 to T/2 instead of 0 to T gives the same results
Equation 43 is the sine-cosine form of Fourier series An alternative representation is the amplitude-phase (or polar) form:
x t a A n tn n n
( ) cos( )= + + =
w f (418)
This combines the sine-cosine pair at frequency nω0 into a single sinusoid We can apply the trigonometric identity
cos( ) cos cos sin sinA B A B A B+ = - (419)
to the ac terms in Equation 418 to get
a A n t a A n t
A
+ + = +
+ -
å åcos( ) ( cos )cos
( sin )
w f f w
f sin n t n
(420)
When we equate the coefficients of the series expansions in Equations 43 and 420, we obtain
a A b An n n n n n= = -cos , sinf f (421)
A a b b a
= + = - -2 2 1, tanf (422)
These may also be related in a compact complex form as
A a jbn n n n< = -f (423)
The frequency components of the signal x(t) can be displayed in terms of the amplitude and phase spectra The frequency spectrum of x(t) is the combination of both the amplitude and phase spectra
Example 4.1
TABLE 4.1 For Example 4.1
Example 4.2
The sine and cosine functions can be represented in terms of complex exponentials It turns out that we can use complex exponentials to represent Fourier series In many respects, this makes for a simpler representation
The Fourier series representation of x(t) can be expressed in complex exponential form as
x t c en jn t
( ) = =-¥
where ω0 = 2π/T is the fundamental frequency the coefficients cn are given by
c T
x t e dtn jn t T
= -ò1 0 0
( ) w (425)
The exponential Fourier series of a periodic signal x(t) is a representation that is the sum of the complex exponentials at positive and negative harmonic frequencies
By substituting the following Euler’s identities in Equation 43
1 2
0 0= +[ ]- (426)
sin n t j e e
1 2 2
0 0 0 0= -[ ] = - -[ ]- - (427)
we get
x t a a e e jb e e
a
( ) = + +( ) - -( )
= +
1 2
( ) ( )a jb e a jb en n jn t n n jn t n
- + +éë ùû -
(428)
So the new coefficient cn will be
c a
c a jb
c
c c a jb
=
= - = <
= = +-
,
( ) ,
( )*
q (429)
so that x(t) becomes
x t c c e c en jn t
( ) = + +éë ùû-- =
The concise form is
x t c e c en jn t
n( ) ( )= = =-¥
The complex Fourier coefficients cn can be readily obtained as follows using Equations 46 and 47 for an, bn
c a T
x t dt T
1= = ò ( ) (432) For n = 1, 2, 3, …, we have
c a jb
T x t n t j n t dt
T x t en
= - = - =ò -( ) ( )(cos sin ) ( )2 1 1
0w w wò dt (433)
Similarly,
c c a jb
- = = + = ò* ( ) ( )2
The last expression is equivalent to stating that for n = −1, −2, −3, …
c T
x t e dtn jn t T
= -ò1 0 0
( ) w (434)
The three Equations 432, 433, 434 can be combined into one expression
c T
x t e dt nn jn t T
= = ± ± ±-ò1 0 1 2 30 0
( ) , , , ,w for … (435)
Therefore, the complex Fourier series is
x t c en jn t
( ) = =-¥
Equation 436 is also called complex frequency spectrum of x(t), which is formed by both the complex amplitude spectrum (even symmetry) and the phase spectrum (odd symmetry)
The three forms of Fourier series are now summarized as follows:
Sine-Cosine Form
x t a a n t b n tn n n
( ) ( cos sin )= + + =
w w (437)
Amplitude-Phase Form
x t a A n tn n n
( ) cos( )= + + =
w f (438)
Complex Exponential Form
x t c en jn t
( ) = =-¥
The coefficients of the three forms of Fourier series (sine-cosine form, amplitudephase form, and exponential form) are related by
A a jb cn n n n nÐ = - =f 2 (440)
or
c c
a b b a
n n= Ð = +
Ð - = Ð-| | tanq f 2 2
2 (441)
if only an > 0 Note that the phase θn of cn is equal to ϕn
Example 4.3
TABLE 4.2 For Example 4.3
Example 4.4
Some properties of Fourier series are discussed in this section These properties provide us a better understanding of the Fourier series
The Fourier series expansion for periodic signals x(t) and y(t) with the same period is given by
x t jn tn n
( ) exp[ ]= =-¥
åa w0 (442)
y t jn tn n
( ) exp[ ]= =-¥
åb w0 (443)
If z(t) is a linear combination of x(t) and y(t), then
z t k x t k y t( ) ( ) ( )= +1 2 (444)
where k1 and k2 are arbitrary constants Then we can write
z t k k jn t jn tn n n
( ) ( )exp[ ] exp[ ]= + = =-¥
å å1 2 0 0a b w g w (445)
This implies that the Fourier coefficients are
g a bn n nk k= +1 2 (446)
The linearity property is easily extended to a linear combination of an arbitrary number of signals with period T
When a time shift is applied to a periodic signal x(t), the period T of the signal is preserved The Fourier series coefficient γnof the resulting signal x(t − τ) may be expressed as
g t w w t l w l ln
T x t jn t dt j
T x jn d= - - = - -ò1 10 0
( )exp[ ] exp[ ] ( )exp[ ] T
n jn
ò = -a w texp[ ]0
(447)
where αnis the nth Fourier series coefficient of x(t) Here, we have changed variables by letting λ = t−τ That is, if
x t FS n( ) ¾ ®¾ a then
x t e FS
n jn( )- ¾ ®¾ -t a w t0 (448)
One consequence of this property is that, when a periodic signal is shifted in time, the magnitude of the Fourier coefficient remains unaltered, that is, |γn| = |αn|
Time reversal property, when applied to a continuous-time signal, results in a time reversal of the corresponding sequence of Fourier series coefficients To determine the Fourier series coefficients of y(t) = x(−t), consider
x t jn t Tn
( ) exp-= -é ëê
ù ûú
åa p (449)
By substituting n = −m, we get
y t x t jm t Tm
( ) ( ) exp= - = é ëê
ù ûú
åa p (450)
Thus, we conclude that
x t
x t
( )
( )
¾ ®¾
- ¾ ®¾ -
a
a (451)
An interesting consequence of time reversal is that when x(t) is even, that is, if x(−t) = x(t), then its Fourier series coefficients are also even, that is, α−n = αn Similarly, when x(t) is odd, that is, if x(−t) = −x(t), then its Fourier series coefficients are also odd, that is, α−n = −αn
Time scaling operation applies directly to each of the harmonic component of x(t) If
x(t) has the Fourier series representation x t jn tn n
( ) exp[ ]= =-¥
¥å a w0 , then the Fourier series representation of the time-scaled signal x(at) is
x at jn atn n
( ) exp[ ]= =-¥
åa w0 (452)
That is,
x at FS
n( ) ¾ ®¾ a (453)
The Fourier series coefficients have not changed, the Fourier series representation has changed because of the change in the fundamental frequency aω0
Three types of symmetries have been discussed-even, odd, and half-wave odd symmetry
A function x(t) is even if x(t) = x(−t), and odd if x(t) = −x(−t) Any function x(t) is a sum of an even function and an odd function, and this can be done in only one way
We know that any periodic function x(t), with period 2π, has a Fourier expansion of the form
x t a a n t b n tn n n
( ) cos( ) sin( )= + + =
w w (454)
If x(t) is even, then all the bns vanish and the Fourier series is simply
x t a a n tn n
( ) cos( )= + =
w (455)
If x(t) is odd, then all the a0, ans vanish and the Fourier series is simply
x t b n tn n
( ) sin( )= =
(456)
A periodic function possesses half-wave symmetry if it satisfies the constraints
x t x t
T( ) = - -æ è ç
ö ø ÷2
(457)
If a function is shifted one half period and inverted, it looks identical to the original, then it is called a half-wave symmetry A half-wave odd symmetry can be even, odd or neither
a a b n
a T
x t n t dt n
0 0 0
= = =
,
( )cos ,
and for even
for oddw
b T
x t n t dt nn
= ò4 0 0
( )sin ,w for odd
(458)
This shows that the Fourier series of a half-wave symmetric function contains only odd harmonics (Figure 410)
Parseval’s theorem states that if x(t) is a periodic function with period T, then the average power P of the signal is defined by
P T
x t dt T
= ò1 2 0
( ) (459)
Again let x(t) be an arbitrary periodic signal with period T and consider the Fourier series of x(t) given by Equation 424 By Parseval’s theorem, the average power P of the signal x(t) is given by
P cn n
= =-¥
Parseval’s theorem states that the average power of the signal x(t) over one period equals the sum of the squared magnitudes of all the complex Fourier coefficients
To prove Parseval’s theorem, assume x(t) has a complex Fourier series of the usual form:
x t c e T
c T
x t e dt
x t
( ) ,
( )
( )
= = æ è ç
ö ø ÷
=
w p0
= = =
= =
x t x t x t c e c x t e dt
P T
x t dt T
c
( ) ( ) ( ) ( )
( )
1 12
x t e dt
T c x t e dt
c c
P c
( )
( )
=
= ×
= =-¥
(461)
Parseval’s theorem can also be written in terms of the Fourier coefficients an, bn of the trigonometric Fourier series The engineering interpretation of this theorem is
as follows Suppose x(t) denotes an electrical signal (current or voltage), then from elementary circuit theory x2(t) is the instantaneous power (in a 1 ohm resistor) so that
P T
x t dt T
= ò1 2 0
( ) (462)
is the energy dissipated in the resistor during one period In terms of the trigonometric Fourier series, Equation 461 becomes
P a a bo n n n
= + +( ) =
1 2
(463)
In terms of the amplitude-phase Fourier series, we have
P a Ao n n
= + =
1 2
(464)
Table 43 summarizes the power associated with Fourier series coefficients
Example 4.5
TABLE 4.3 Power Associated with Fourier Series Coefficients
Example 4.6
Example 4.7
The Fourier series representation of x(t) can be expressed in complex exponential form as
x t c en jn t
( ) = =-¥
It is an infinite summation and to truncate x(t) to finite partial sum of xN(t) at n = N, we can write Equation 465 as follows:
x t c eN n jn t
( ) = =- å w0 (466)
We expect the signal approximation xN(t) to converge to x(t) as N → ∞ The behavior of the partial sum in the vicinity of discontinuity exhibits ripples and that the peak amplitude of these ripples does not seem to decrease with increasing N Gibbs showed that for the discontinuity of unit height, the partial sum exhibits an overshoot of 9% of the height of the discontinuity, no matter how large N becomes Thus, as N increases, the ripples in the partial sum become compressed toward the discontinuity, but for any finite value of N, the peak amplitude of the ripples remains constant This is known as Gibbs phenomenon The Gibbs phenomenon (also known ringing artifacts) is named after the American physicist J Willard Gibbs
Gibbs discovered that the truncated Fourier series approximation xN(t) of a signal x(t) with discontinuous will in general exhibit high-frequency ripples and overshoot x(t) near the discontinuities If such an approximation is used in practice, a large enough value of N should be chosen so as to guarantee that the total energy in this ripple is insignificant These are shown in the Figure 415 for various values of N We know that the energy in the approximation error vanishes and that the Fourier series representation of a discontinuous signal such as the square wave converges
Fourier analysis finds engineering applications in circuit analysis, filtering, amplitude modulation, rectification, harmonic distortion, and signal transmission through a linear system We will only consider three simple applications: circuit analysis, spectrum analyzers, and filters
Fourier analysis enables us to find the steady response of a circuit to a periodic excitation The first step involves finding the Fourier series expansion of the excitation This Fourier series representation may be regarded as a series-connected sinusoidal sources with each source having its own amplitude and frequency The second step is finding the response to each term in the Fourier series by the phasor techniques Finally, following the principle of superposition, we add all the individual responses
Example 4.8
Answer: v t n
n t no n
( ) cos( tan )= +
- -
å 69 32 2 1
1 p p p/
As we have observed, the Fourier series provides the spectrum of a signal By providing the spectrum of a signal x(t), the Fourier series helps us identify which frequencies are playing an important role in the shape of the output and which ones are not For example, audible sounds have significant components in the frequency range of 20 Hz-15 kHz, while microwaves range from 24 to 300 GHz
We regard a periodic signal as band-limited if its amplitude spectrum contains only a finite number of coefficients An or cn For a band-limited signal, the Fourier series becomes (truncated)
x t c e a A n tn jn t n N
( ) cos( )= = + + =- = å åw w f0 0 0
(467)
From this, we see that we need only 2N + 1 terms (namely, a0, A1, A2,…, AN, ϕ1, ϕ2, …, ϕN) to completely specify x(t) if ω0 is known This leads to the sampling theorem
The sampling theorem states that a band-limited periodic signal whose Fourier series contains N harmonics is uniquely specified by its values at 2N + 1 instants in one period
A spectrum analyzer is an instrument that displays the frequency components (spectra lines) of a signal Spectrum analyzers are commercially available in various sizes and shapes Figure 418 illustrates a typical one An oscilloscope shows the signal in the time domain, while the spectrum analyzer shows the signal in the frequency domain Spectrum analyzer is perhaps the most useful instrument for circuit analysis It is widely used to measure the frequency response, noise and spurious signal analysis, and more
Filters are frequency-selective devices used in electronics and communications systems Here, we investigate how to design filters to select the fundamental component
(or any desired harmonic) of the input signal and reject other harmonics This filtering process cannot be accomplished without the Fourier series expansion of the input signal
A filter is a device designed to pass signals with desired frequencies and block or attenuate others
As shown in Figure 419, there are four types of filters:
1 A lowpass filter (LPF) passes low frequencies and rejects high frequencies 2 A highpass filter (HPF) passes high frequencies and rejects low frequencies 3 A bandpass filter (BPF) passes frequencies within a frequency band and
blocks or attenuates frequencies outside the band 4 A bandstop filter (BSF) passes frequencies outside the frequency band and
stops or attenuates frequencies within the band
For the purpose of illustration, we will consider two cases: a LPF and BPF A typical LPF is shown in Figure 420 The output of a LPF depends on the
input signal vi(t) (with fundamental frequency ω0), the transfer function H(ω) of the filter, and the corner or half-power frequency ωc = 1/RC The LPF characteristically passes the dc and low-frequency components, while blocking or attenuating the high-frequency components If we make ωc sufficiently large (ωc ≫ ω0, eg, making C small), a large number of the harmonics can be passed On the other hand, if we make ω sufficiently small (ωc ≪ ω0), we can block out all the ac components and pass only dc
A typical BPF is shown in Figure 421 The output of a BPF depends on the input signal vi(t) (with fundamental frequency ω0), the transfer function H(ω) of the filter, its bandwidth B, and its center frequency ωc The filter passes all the harmonics of the input signal within a band of frequencies (ω1 < ω.< ω2) centered around ωc and rejects other harmonics
Example 4.9
Although MATLAB does not help in finding the Fourier series of a signal, it can help in plotting line spectra and examining the convergence of a Fourier series The following examples show us how
Example 4.10
Example 4.11
1 A periodic function x(t) repeats itself every period T, that is, x(t + T) = x(t) 2 A periodic function x(t) can expressed in terms of sinusoids using Fourier
series:
x t a a n t b n tn n n
( ) ( cos sin )= + + =
w w
where ω0 = 2π/T is the fundamental frequency The Fourier series decomposes the signal into the dc component a0 and ac components
There are three basic forms of Fourier series representation: the sine-cosine form, the amplitude-phase form, and the exponential form
3 The Fourier coefficients of sine-cosine form are determined as
a T
x t dt
a T
x t n t dt
b T
x t n t dt
=
=
=
( )
( )cos
( )sin
w
w
4 The amplitude-phase form of Fourier series is
x t a A n tn n n
( ) cos( )= + + =
w f
where
A a b b
= + = - -2 2 1, tanf
5 The exponential (or complex) form of Fourier series is
x t c en jn t
( ) = =-¥
c T
x t e dtn jn t T
= -ò1 0 0
( ) w
and w p0 2= /T .
