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Thus, the sequences q k [ n ]are distinct only over a range of No successive values of k.
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CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
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B. Discrete Fourier Series Representation:
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The discrete Fourier series representation of a periodic sequence x[n] with fundamental period No is given by
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where c, are the Fourier coefficients and are given by (Prob. 6.2)
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Because of Eq. (6.5) [or Eq. (6.6)], Eqs. (6.7) and (6.8) can be rewritten as
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where C , denotes that the summation is on k as k varies over a range of No successive integers. Setting k = 0 in Eq. (6.101, we have
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which indicates that co equals the average value of x[n] over a period. The Fourier coefficients c, are often referred to as the spectral coefficients of x[n].
C. Convergence of Discrete Fourier Series:
Since the discrete Fourier series is a finite series, in contrast to the continuous-time case, there are no convergence issues with discrete Fourier series.
D. Properties of Discrete Fourier Series:
I. Periodicity of Fourier Coeficients:
From Eqs. ( 6 . 5 ) and (6.7) [or (6.911, we see that
C,+N,
= Ck
which indicates that the Fourier series coefficients c, are periodic with fundamental period No.
2. Duality:
From Eq. (6.12) we see that the Fourier coefficients c, form a periodic sequence with fundamental period No. Thus, writing c, as c[k], Eq. (6.10) can be rewritten as
FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6
Let n
- m in Eq. (6.13). Then
Letting k
and m = k in the above expression, we get
Comparing Eq. (6.14) with Eq. (6.91, we see that (l/N,,)x[-k] of c[n]. If we adopt the notation x [ n ]B c k= c [ k ]
are the Fourier coefficients (6.15)
to denote the discrete Fourier series pair, then by Eq. (6.14) we have DFS 1 ~ [ nc--) ] -x[-k] No Equation (6.16) is known as the duality property of the discrete Fourier series.
3. Other Properties:
When x[n] is real, then from Eq. (6.8) or [Eq. (6.10)] and Eq. (6.12) it follows that * (6.17) C P k =CN,,-k= ck where
denotes the complex conjugate.
Even and Odd Sequences: When x[n] is real, let x[nl =xe[nl + ~ o [ n l where xe[n] and xo[n] are the even and odd components of x[n], respectively. Let x[n] S c k Then xe[n] Re[ck] (6.18~) (6.186) xo[n] 2%j Im[ck]
Thus, we see that if x[n] is real and even, then its Fourier coefficients are real, while if x[n] is real and odd, its Fourier coefficients are imaginary.
E. Parseval's Theorem:
If x[n] is represented by the discrete Fourier series in Eq. (6.9), then it can be shown that (Prob. 6.10)
Equation (6.19) is called Parseval's identity (or Parseual's theorem) for the discrete Fourier series.
CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
6 3 THE FOURIER TRANSFORM
A. From Discrete Fourier Series to Fourier Transform: Let x [ n ] be a nonperiodic sequence of finite duration. That is, for some positive integer N , ,
Such a sequence is shown in Fig. 6-l(a). Let x,Jn] be a periodic sequence formed by repeating x [ n ] with fundamental period No as shown in Fig. 6-l(b). If we let No - m, we , have
No+-
lim x N o [ n ]= x [ n ]
The discrete Fourier series of x N o [ n ] given by is
where
Fig. 6-1 ( a ) Nonperiodic finite sequence x [ n ] ; 6 ) periodic sequence formed by periodic extension of ( xhl.
FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
[CHAP. 6
Since xN,,[n] x [ n ] for In1 I and also since x [ n ] = 0 outside this interval, Eq. (6.22~) = N, can be rewritten as
Let us define X(R) as
Then, from Eq. (6.22b) the Fourier coefficients c , can be expressed as
Substituting Eq. (6.24) into Eq. (6.21), we have
From Eq. (6.231, X(R) is periodic with period 27r and so is eJRn.Thus, the product X(R)e*'" will also be periodic with period 27r. As shown in Fig. 6-2, each term in the 1" summation in Eq. (6.25) represents the area of a rectangle of height ~ ( k R , ) e ' ~ ~ 1and width R,. As No + m, 0, = 27r/N0 becomes infinitesimal (R, + 0) and Eq. (6.25) passes to an integral. Furthermore, since the summation in Eq. (6.25) is over N,, consecutive intervals of width 0, = 27r/N,,, the total interval of integration will always have a width 27r. Thus, as NO+ a: and in view of Eq. (6.20), Eq. (6.25) becomes
1 x [ n ] = - ~ ( 0ejRnd R ) (6.26) 27r 2 v Since X(R)e 'On is periodic with period 27r, the interval of integration in Eq. (6.26) can be taken as any interval of length 27r.
Fig. 6-2 Graphical interpretation of Eq. (6.25).
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