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THE DFT
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Shown in the figure below is a plot of h ( l - k ) R 4 ( n ) .
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[CHAP. 6
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Multiplying by P ( k ) and summing from k = 0 to k = 3 , we find that y ( l ) = 4. Repeating for n = 2 and n = 3, we have
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y ( n ) = h ( n ) @ .r(n) = 6 ( n )
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+ 46(n - I) + 26(n
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+ 26(n - 3 )
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+ 36(n - 5 )
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By comparison, the linear convolution of h ( n ) with x ( n ) is the following six-length sequence:
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h(n)* x(n) = S(n)
+ &(ti - 1) + 26(n
+ 2&n
Another way to perform circular convolution is to compute the DFTs of each sequence. multiply, and compute the inverse DFT.
EXAMPLE 6.4.2
Let us perform the N -point circular convolution of . r l ( n )and x 2 ( n ) where
Because the N-point DFTs of x,(n) and x 2 ( n ) are
X l ( k )= X2(k) =
xw ; ~
,,=O
k=O else
then Therefore, the N-point circular convolution of x l ( n ) with s z ( n ) is the inverse DFT of X ( k ) , which is
x(n) =
O ( n 5 N - l else
Circular Versus Linear Convolution
In general, circular convolution is not the same as linear convolution. However, there is a simple relationship between circular and linear convolution that illustrates what steps must be taken in order to ensure that they are the same. Specifically, let x ( n ) and h ( n ) be finite-length sequences and let y ( n ) be the linear convolution
y(n) =x(n)* h(n)
The N-point circular convolution of x ( n ) with h ( n ) is related to y ( n ) as follows:
CHAP. 61
THE D m
23 1
In other words, the circular convolution of two sequences is found by performing the linear convolution and
aliasing the result. R N ( n ) = Ofork
An important property that follows from Eq. (6.10) is that if y ( n ) is of length N or less, y ( n - k N ) # Oand h(n)@ x(n) = h(n)* x(n)
that is, circular convolution is equivalent to linear convolution. Thus, if h ( n ) and x ( n ) are finite-length sequences oflength NI and N 2 , respectively, y ( n ) = h ( n ) * x ( n ) is of length N I Nr - I, and the N -point circularconvolution is equivalent to linear convolution provided N 2 N I N 2 - 1 .
EXAMPLE 6.4.3 Let us find the four-point circular convolution of the sequences h(n) and x ( n ) in Example 6.4.1. Because the linear convolution is
y(n) = 6(n)
we may set up a table to evaluate the sum
+ S(n - I ) + 2S(n - 2) + 26(n - 3) + 3S(n - 5)
Note This is done by listing the values of the sequence y(n k N ) in a table and summing these values for n = 0, 1,2,3. that the only sequences that have nonzero values in the interval 0 5 n 5 3 are y(n) and y(n 4), and these are the only sequences that need be listed. Thus, we have
Summing the columns for 0 5 n 5 3, we have
which is the same as computed in Example 6.4.1.
6.5 SAMPLING THE DTFT
Let x ( n ) be a sequence with a DTFT X ( e J U ) ,and consider the finite-length sequence y ( n ) of length N whose DFT coefficients are obtained by sampling X ( e j U ) at wk = 2nk/N :
Because the DTFT is equal to the z-transform evaluated around the unit circle, these DFT coefficients may also be obtained by sampling X ( z ) at N equally spaced points around the unit circle at zk = exp(j2nklN):
These sampling points are illustrated in Fig. 6-3 for N = 8. To express the sequence values y ( n ) in terms of x ( n ) , we begin by finding the inverse DFT of Y (k):
Because the DFT coefficients Y ( k ) are samples of the DTFT of x ( n ) ,
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