离散时间信号处理PPT_第八章 离散傅里叶变换

1 x n N
k e j 2 / N kn , (8.4) X
k 0

N 1
To obtain the sequence of Fourier series coefficients X k from the periodic sequence x n , we exploit the orthogonality of the set of complex exponential sequences. We obtain
8.0 Introduction
In Chapters 2 and 3 we discussed the representation of sequences and linear time-invariant systems in terms of the Fourier and z-transforms, respectively. For finite-duration sequences, it is possible to develop an alternative Fourier representation, referred to as the discrete Fourier transform (DFT). The DFT is itself a sequence rather than a function of a continuous variable, and it corresponds to samples, equally spaced in frequency, of the Fourier transform of the signal.
8.1 Representation of Periodic Sequences: The Discrete Fourier Series
Consider a sequence x n that is periodic with period N, so that x n x n rN for any integer values of n and r . As with continuous-time periodic signals, such a sequence can be represented by a Fourier series corresponding to a sum of harmonically related complex exponential sequences, i.e., these periodic complex exponentials are of the form j 2 / N kn (8.1)
0 X k nWNkn WN 1 N 1 n 0
In this case , X k is the same for all k. thus, substituting Eq.(8.15) leads to the representation
1 N 1 kn 1 N 1 j 2 / N kn x n n rN WN e N k 0 N k 0 r
Equations (8.9) and (8.4) together are an analysis-synthesis pair and will be referred to as the discrete Fourier series (DFS) representation of a periodic sequence. For convenience in notation, these equations are often written in terms of the complex quantity j 2 / N
1, n rN , r any int eger , x n n rN r 0, otherwise

Since x n n for 0 n N 1 , the DFS coefficients are found, using Eq.(8.11), to be
n e j 2 / N rn X r , (8.8) x
n 0
N 1
Thus, the Fourier series coefficients X k in Eq.(8.4) are obtained from x n by the relation
8 THE DISCRETE FOURIER TRANSFORM
8.0 Introduction 8.1 Representation of Periodic Sequences: The Discrete Fourier Series 8.2 Properties of the Discrete Fourier Series 8.4 Sampling the Fourier Transform 8.5 Fourier Representation of Finite-Duration Sequence: The Discrete Fourier Transform 8.6 Properties of the Discrete Fourier Transform 8.7 Linear Convolution Using the Discrete Fourier Transform
1 N 1 j 2 / N k r n 1, k r mN , m an integer ,(8.7) e N n 0 0, otherwise
When it is applied to the summation in brackets in Eq.(8.6), the results is
Substituting Y k into Eq.(8.12) gives
r
1 y n N
0 N k N kn WN 1 W k 0
N 1
In this case , y n 1 for all n. comparing this result with the results for x n and X k of Example 8.1, we see that Y k N x k and y n X n . In Section 8.2.3, we will show that this example is a special case of a more general duallity property.

Example 8.2 Duality in the Discrete Fourier Series
Here we let the discrete Fourier series coefficients be the periodic impulse train Y k N k rN
(8.2)
exponentials ek n in Eq.(8.1) are identical for values of k separated by N; i.e., e0 n eN n , e1 n eN 1 n and , in general.
To see this , note that the harmonically related complex
ek N n e
Where
j 2 / N k N n
e
j 2 / N kn j 2 n
e
e
j 2 / N kn
ek n
j 2 / N rn x n e n 0 N 1
1 N 1 j 2 / N k r n X k e ,(8.6) k 0 N n 0
N 1
The following identity expresses the orthogonality of the complex exponentials:
1 n x N
W X k N kn
In both of these equations, X k and x n are periodic sequences. We will sometimes find it convenient to use the notation
ek n e ek n rN
Where k is an integer, and the Fourier series representation then has the form

1 x n N
j 2 / N kn X k e k
WN e
With this notation, the DFS analysis-synthesis pair is expressed as follows: Analysis equation: N 1
W kn X k x n N
n 0
N 1 k 0
Synthesis equation:
1 N 1 j 2 / N rn j 2 / N k r n x n e X k e ,(8.5) n 0 n 0 N k 0
N 1
N 1
After interchanging the order of summation on the righthand side, we see that Eq.(8.5) becomes
j 2 / N k N n X k N x n e n 0 N 1
N 1 j 2 / N kn j 2 n x n e e X k n 0
For any integer k.
DFS x n X k
to signify the relationships of Eqs(8.11) and (8.12).
Example 8.1 Discrete Fourier Series of a Periodic Impulse Train
We consider the periodic impulse train
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k x n e j 2 / N kn , (8.9) X
n 0
N 1
Note that the sequence X k is periodic with period n i.e., X 0 X N , X 1 X N 1 and , more generally,
is an integer.
x n need contain only N of these complex exponentials,
and hence, it has the form
Thus, the Fourier series representation of a periodic sequence