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Fourier Analysis of Continuous-Time Signals and Systems
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5.1 INTRODUCTION
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In previous chapters we introduced the Laplace transform and the z-transform to convert time-domain signals into the complex s-domain and z-domain representations that are, for many purposes, more convenient to analyze and process. In addition, greater insights into the nature and properties of many signals and systems are provided by these transformations. In this chapter and the following one, we shall introduce other transformations known as Fourier series and Fourier transform which convert time-domain signals into frequency-domain (or spectral) representations. In addition to providing spectral representations of signals, Fourier analysis is also essential for describing certain types of systems and their properties in the frequency domain. In this chapter we shall introduce Fourier analysis in the context of continuous-time signals and systems.
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FOURIER SERIES REPRESENTATION OF PERIODIC SIGNALS Periodic Signals:
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In Chap. 1 we defined a continuous-time signal x ( t ) to be periodic if there is a positive nonzero value of T for which x(t + T )= x ( t ) all t (5.1) The fundamental period To of x ( t ) is the smallest positive value of T for which Eq. (5.1) is satisfied, and l / T o =fo is referred to as the fundamental frequency. Two basic examples of periodic signals are the real sinusoidal signal
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and the complex exponential signal X ( t )= e~%' where oo= 2n-/To = 2n-fo is called the fundamental angular frequency. B. Complex Exponential Fourier Series Representation: The complex exponential Fourier series representation of a periodic signal x ( t ) with fundamental period To is given by (5.3)
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FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
[CHAP. 5
where c , are known as the complex Fourier coefficients and are given by
where 1 denotes the integral over any one period and 0 to To or - T o / 2 to T 0 / 2 is " , commonly used for the integration. Setting k = 0 in Eq. ( 5 3 , we have
which indicates that co equals the average value of x ( t ) over a period. When x ( t ) is real, then from Eq. ( 5 . 5 ) it follows that
C - , =C,
(5.7)
where the asterisk indicates the complex conjugate.
C. Trigonometric Fourier Series:
The trigonometric Fourier series representation of a periodic signal x ( t ) with fundamental period T,, is given by
x(t)= -
(a,cos ko,t
+ b , sin k w o t )
w0 = -
(5.8)
where a , and b, are the Fourier coefficients given by
x ( t ) c o s kw,tdt
(5.9a)
The coefficients a , and b, and the complex Fourier coefficients c , are related by (Prob. 5.3)
From Eq. ( 5 . 1 0 ) we obtain
+ ( a , - jb,)
= 2 Re[c,]
c - , = : ( a k + jb,) b,
=0 =
(5.11)
When x ( t ) is real, then a , and b, are real and by Eq. (5.10) we have
a, Even and Odd Signals:
- 2 Im[c,]
(5.12)
If a periodic signal x ( t ) is even, then b, cosine terms:
x(t) = a0
and its Fourier series ( 5 . 8 ) contains only
a , cos kwot
o0= -
CHAP. 51
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
If x ( t ) is odd, then a,
and its Fourier series contains only sine terms: m 2T T x(t) = b, sin kwot w0= k= 1 To
D. Harmonic Form Fourier Series:
Another form of the Fourier series representation of a real periodic signal x ( t ) with fundamental period To is
Equation (5.15) can be derived from Eq. (5.8) and is known as the harmonic form Fourier series of x(t). The term Co is known as the d c component, and the term C, cos(kwot - 0,) is referred to as the kth harmonic component of x(t). The first harmonic component -~ C, C O S ( ~ , 8,) is commonly called the fundamental component because it has the same fundamental period as x(t). The coefficients C, and the angles 8, are called the harmonic amplitudes and phase angles, respectively, and they are related to the Fourier coefficients a, and b, by
For a real periodic signal ~ ( r ) the Fourier series in terms of complex exponentials as , given in Eq. (5.4) is mathematically equivalent to either of the two forms in Eqs. (5.8) and (5.15). Although the latter two are common forms for Fourier series, the complex form in Eq. (5.4) is more general and usually more convenient, and we will use that form almost exclusively.
E. Convergence of Fourier Series:
It is known that a periodic signal x(t) has a Fourier series representation if it satisfies the following Dirichlet conditions:
x ( t ) is absolutely integrable over any period, that is,
2. x(t) has a finite number of maxima and minima within any finite interval of t . 3. x(t) has a finite number of discontinuities within any finite interval of t, and each of these discontinuities is finite. Note that the Dirichlet conditions are sufficient but not necessary conditions for the Fourier series representation (Prob. 5.8).
F. Amplitude and Phase Spectra of a Periodic Signal:
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