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CHAP. 6 1
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THE DFT
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Periodic Convolution
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1f h(n) and i ( n ) are periodic with a period N with DFS coefficients &(k) and g ( k ) , respectively, the sequence with DFS coefficients F(k)= 4 (k)f (k)
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is formed by periodically convolving h(n) with R(n) as follows:
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Notationally, the periodic convolution of two sequences is written as
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The only difference between periodic and linear convolution is that, with periodic convolution, the sum is only evaluated over a single period, whereas with linear convolution the sum is taken over all values of k.
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EXAMPLE 6.2.2
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Let us periodically convolve the two sequences pictured below that have a period N = 6.
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The periodic convolution of two sequences may be performed graphically, analytically, or using the DFS. In this problem, we will use the graphical approach. We begin by plotting P(n - k ) versus k . This sequence, for n = 0. is illustrated below.
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The value of J(0)is then Found by summing the product i ( k ) f ( - k ) from X- = 0 to k = 5. The result is j ( 0 ) = 1 . Next, f ( - k ) is shifted to the right by one and multiplied by h ( k ) . Because the only two nonzero values of P(1 - k ) are at k = 4.5. the product h"(k)f(l - k ) is equal to zero, and J ( I ) = 0 . This process is continued until we have one period of J ( n ) . The result is illustrated below.
T H E DFT
[CHAP. 6
6.3 DISCRETE FOURIER TRANSFORM
The DFT is an important decomposition for sequences that are finite in length. Whereas the DTFT is a mapping from a sequence to a function of a continuous variable, w ,
the DFT is a mapping from a sequence, x ( n ) , to another sequence, X ( k ) ,
The DFT may be easily developed from the discrete Fourier series representation for periodic sequences. Let
x ( n ) be a finite-length sequence of length N that is equal to zero outside the interval [0, N - I]. A periodic sequence i ( n ) may be formed from x ( n ) as follows:
This periodic extension may be expressed as follows:
i ( n ) = x ( n mod N) r ~ ( ( n ) ) ~
where ( n mod N ) and ((n))N are taken to mean "n modulo N ." That is to say, if n is written in the form n = kN + I where 0 5 I < N ,
( n mod N) = ( ( n ) ) = I ~
For example, (( 13))8 = 5 and ( ( - 6 ) ) x = 2. A periodic sequence may be expanded using the DFS as in Eq. ( 6 . 1 ) . Because x ( n ) = 2 ( n ) for n = 0, 1, . . . , N - 1 , x ( n ) may similarly be expanded as follows:
Because the DFS coefficients are periodic, if we let X ( k ) be one period of i ( k ) and replace%(k) in the sum with X ( k ) , then we have
The sequence X ( k ) is called the N-point DFT of x ( n ) . These coefficients are related to x ( n ) as follows:
Equations ( 6 . 4 ) and ( 6 . 5 ) form a DFT pair, and we write
This expansion is valid for complex-valued as well as real-valued sequences. Comparing the definition of the DFT of x ( n ) to the DTFT, it follows that the DFT coefficients are samples of the DTFT:
CHAP. 61
THE DFT
Alternatively, the DFT coefficients correspond to N samples of X ( z ) that are taken at N equally spaced points around the unit circle:
X ( k ) = X(z)lz=exp(j nklNJ 6.4 DFT PROPERTIES
In this section, we list some of the properties of the DFT. Because each sequence is assumed to be finite in length, some care must be exercised in manipulating DFTs.
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