Let the complex Fourier coefficients c, in Eq. (5.4) be expressed as in .NET framework

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Let the complex Fourier coefficients c, in Eq. (5.4) be expressed as
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FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
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[CHAP. 5
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A plot of lckl versus the angular frequency w is called the amplitude spectrum of the periodic signal x ( t ) , and a plot of 4, versus w is called the phase spectrum of x(t 1. Since the index k assumes only integers, the amplitude and phase spectra are not continuous curves but appear only at the discrete frequencies k o , , . They are therefore referred to as discrete frequency spectra or line spectra. For a real periodic signal x ( t ) we have c - , = c:. Thus,
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Hence, the amplitude spectrum is an even function of w , and the phase spectrum is an odd function of o for a real periodic signal.
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C . Power Content of a Periodic Signal:
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In Chap. 1 (Prob. 1.18) we introduced the average power of a periodic signal x ( t ) over any period as
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If x ( t ) is represented by the complex exponential Fourier series in Eq. ( 5 . 4 ) ,then it can be shown that (Prob. 5.14)
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Equation ( 5 . 2 1 ) is called Parserlal's identity (or Parse~lal'stheorem) for the Fourier series.
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5.3 THE FOURIER TRANSFORM
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From Fourier Series to Fourier Transform:
t ) be a nonperiodic signal of finite duration, that is,
x(t)=0
Itl> TI
Such a signal is shown in Fig. 5 - l ( a ) . Let x,,)(t) be a periodic signal formed by repeating x ( r ) with fundamental period T,, as shown i n " ~ i5~l.( b). If we let To --+ m, we have lim x T , l t ) = x ( t )
TO+=
The complex exponential Fourier series of xril(t)is given
where Since ~ , , ~) (= x ( r ) for t
It 1 < T,,/2
and also since x( t ) = 0 outside this interval, Eq. (5.24a)
CHAP. 51
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
Fig. 5-1 (a) Nonperiodic signal x(r);( b )periodic signal formed by periodic extension of x(r ).
can be rewritten as
Let us define X(w) as
Then from Eq. (5.246) the complex Fourier coefficients c, can be expressed as
Substituting Eq. (5.26) into Eq. (5.231, we have
As To -+ m, o, = 27r/T, becomes infinitesimal ( w , Eq. (5.27) becomes
0). Thus, let w,, = Aw. Then
FOURIER ANALYSIS O F T I M E SIGNALS AND SYSTEMS
[CHAP. 5
Therefore, x,Jt)
Aw-0
X(k Am) ejkA"'Aw ,
The sum on the right-hand side of Eq. (5.29) can be viewed as the area under the function X(w) ei"', as shown in Fig. 5-2. Therefore, we obtain
which is the Fourier representation of a nonperiodic x(t).
k Aw
Fig. 5-2
Graphical interpretation of Eq. (5.29).
B. Fourier Transform Pair: The function X ( o ) defined by Eq. (5.25) is called the Fourier transform of x(t), and Eq. (5.30) defines the inuerse Fourier transform of X(o). Symbolically they are denoted by
and we say that x(t) and X(w) form a Fourier transform pair denoted by
44-X(4
C. Fourier Spectra:
(5.33)
The Fourier transform X(w) of x(t) is, in general, complex, and it can be expressed as eJd(") X(o) =(X(o)( (5.34)
By analogy with the terminology used for the complex Fourier coefficients of a periodic signal x(t), the Fourier transform X(w) of a nonperiodic signal x(t) is the frequencydomain specification of x(t) and is referred to as the spectrum (or Fourier spectrum) of x ( t ). The quantity I X( w)( is called the magnitude spectrum of x(t), and $(w) is called the phase spectrum of x(t).
CHAP. 51
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
If x ( t ) is a real signal, then from Eq. (5.31) we get
Then it follows that
Ix(-o)l
= Ix(o)l
4(-4
-$(@)
(5.36b)
Hence, as in the case of periodic signals, the amplitude spectrum IX(o)( is an even function and the phase spectrum 4 ( o ) is an odd function of o .
D. Convergence of Fourier Transforms:
Just as in the case of periodic signals, the sufficient conditions for the convergence of X ( o ) are the following (again referred to as the Dirichlet conditions): 1. x ( l ) is absolutely integrable, that is,
2. x ( t ) has a finite number of maxima and minima within any finite interval. 3. x ( t ) has a finite number of discontinuities within any finite interval, and each of these discontinuities is finite.
Although the above Dirichlet conditions guarantee the existence of the Fourier transform for a signal, if impulse functions are permitted in the transform, signals which do not satisfy these conditions can have Fourier transforms (Prob. 5.23).
E. Connection between the Fourier Transform and the Laplace Transform:
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