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* denotes the complex conjugate.
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F. Time Reversal:
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FOURIER ANALYSIS O F DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6
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G. Time Scaling:
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In Sec. 5.4D the scaling property of a continuous-time Fourier transform is expressed as [Eq. (5.5211
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However, in the discrete-time case, x[an] is not a sequence if a is not an integer. On the other hand, if a is an integer, say a = 2, then x[2n] consists of only the even samples of x[n]. Thus, time scaling in discrete time takes on a form somewhat different from Eq. (6.47). Let m be a positive integer and define the sequence x[n/m] = x [ k ] Then we have if n = km, k = integer ifn#km
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Equation (6.49) is the discrete-time counterpart of Eq. (6.47). It states again the inverse relationship between time and frequency. That is, as the signal spreads in time (m > I), its Fourier transform is compressed (Prob. 6.22). Note that X(rnR) is periodic with period 27r/m since X ( R ) is periodic with period 27r.
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H. Duality:
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In Sec. 5.4F the duality property of a continuous-time Fourier transform is expressed as [Eq. (5.5411
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There is no discrete-time counterpart of this property. However, there is a duality between the discrete-time Fourier transform and the continuous-time Fourier series. Let
From Eqs. (6.27) and (6.41)
Since fl is a continuous variable, letting R = t and n
-k in Eq. (6.51), we have
CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
Since X ( t ) is periodic with period To = 27r and the fundamental frequency oo= 27r/T0 = 1 , Eq. (6.53) indicates that the Fourier series coefficients of ~ ( twill be x [ - k ] . This duality ) relationship is denoted by
~ ( t )c , = x [ - k ] B
where FS denotes the Fourier series and c, are its Fourier coefficients.
(6.54)
I. Differentiation in Frequency:
J. Differencing:
The sequence x [ n ] - x [ n - 11 is called the firsf difference sequence. Equation (6.56) is easily obtained from the linearity property (6.42) and the time-shifting property (6.43).
K. Accumulation:
Note that accumulation is the discrete-time counterpart of integration. The impulse term on the right-hand side of Eq. (6.57) reflects the dc or average value that can result from the accumulation.
L. Convolution:
As in the case of the z-transform, this convolution property plays an important role in the study of discrete-time LTI systems.
M. Multiplication:
FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6
where @ denotes the periodic convolution defined by [Eq. (2.70)]
The multiplication property (6.59) is the dual property of Eq. (6.58).
N. Additional Properties:
If x[n] is real, let
where x,[n] and xo[n] are the even and odd components of x[n], respectively. Let
, x [ n ] t X ( n ) = A ( R ) +jB(R)
IX(R)leJe(n)
(6.61)
Then
Equation (6.62) is the necessary and sufficient condition for x[n] to be real. From Eqs. (6.62) and (6.61) we have A( - R ) = A ( R ) Ix(-fl)I= Ix(R)I B(-R) +a)
-B(R) -9(~)
(6.64a) (6.646)
From Eqs. ( 6 . 6 3 ~ (6.636), and (6.64~) see that if x[n] is real and even, then X(R) is )~ we real and even, while if x[n] is real and odd, X(R) is imaginary and odd.
0. Parseval's Relations:
Equation (6.66 ) is known as Parseual's identity (or Parseual's theorem) for the discrete-time Fourier transform. Table 6-1 contains a summary of the properties of the Fourier transform presented in this section. Some common sequences and their Fourier transforms are given in Table 6-2.
CHAP. 6 FOURIER ANALYSIS O F DISCRETE-TIME SIGNALS AND SYSTEMS 1
Table 6-1. Properties of the Fourier Transform
Property
Sequence
Fourier transform
Periodicity Linearity Time shifting Frequency shifting Conjugat ion Time reversal Time scaling Frequency differentiation First difference Accumulation
Convolution Multiplication Real sequence Even component
Odd component
Parseval's relations
FOURIER ANALYSIS OF DISCRETE-TIME SlGNALS AND SYSTEMS
[CHAP. 6
Table 6-2. Common Fourier Transform Pairs
sin Wn ,o<w<sr 7 n 7
THE FREQUENCY RESPONSE OF DISCRETE-TIME LTI SYSTEMS Frequency Response:
In Sec. 2.6 we showed that the output y [ n ] of a discrete-time LTI system equals the convolution of the input x [ n ] with the impulse response h [ n ] ; that is,
Applying the convolution property (6.581, obtain we
CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS
where Y(R), X(R), and H(R) are the Fourier transforms of y [ n ] , x [ n ] , and h [ n ] , respectively. From Eq. ( 6.68) we have
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