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barcode generator for ssrs THE DFT Evaluating X(k) at k = 0,we have the desired result in Software
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(a) The 2Npoint sequence y,(n) is formed from x(n) by padding with zeros. Therefore, Yl(k) corresponds to 2N equally spaced samples of X(z) around the unit circle: (b) The sequence y2(n) is formed by adding x(n  N) to x(n) (i.e., a delayed version of x(n)). If X(z) is the ~ ztransform of x(n), the ztransform of x(n  N) is z  X(Z). Therefore, CHAP. 61
and Y2(k) is
THE DFT
(c) The third sequence is formed by upsampling by a factor of 2 (i.e., by stretching x(n) in time by a factor of 2 and inserting a zero between each sample). The ztransform of y3(n) is Therefore, the 2Npoint DFT is
Thus, the DFT coefficients Y3(k)correspond to two periods of the coefficients X(k). Sampling the DTFT
Let h ( n ) be a finitelength sequence of length N with h ( n ) = 0 for n < 0 and n Fourier transform of h ( n ) is sampled at 3N equally spaced points: > N. The discretetime
Find the sequence g ( n ) that is the inverse DFT of the 3N samples H ( k ) = H ( e J w k ) . Because h(n) is a finitelength sequence of length N, it may be recovered from its Npoint DFT, which corresponds to N equally spaced samples of H (ei"). A sequence of length N may also be considered to be a sequence of length 3N, with the last 2N samples having a value of zero. The inverse DFT of the 3N equally spaced samples of H(eJU) corresponds to this 3Npoint sequence. Thus, g(n) = n=0, I....,N  1 else
Consider the finitelength sequence
4n) = [ I , 1, 1, 1 , 1, 11 and let X ( z ) be its ztransform. If we sample X ( z ) at zk = e x p ( j % k ) fork = 0, 1 , 2 , 3 , we obtain a set of DFT coefficients X ( k ) . Find the sequence, y ( n ) , that has a fourpoint DFT equal to these samples. Sampling X(z) at four equally spaced points around the unit circle produces an aliased version of x(n): Using the tabular method to evaluate this sum, noting that x(n) and x(n +4) are the only sequences that have nonzero values for 0 5 n 5 3, we have Therefore.
THE D m
[CHAP. 6
Consider a finitelength sequence x ( n ) that is zero outside the interval 10. N  11. Suppose that we form a new sequence i ( n ) as follows: where M < N. Find the Mpoint DFT of the sequence i(n) O l n c M otherwise expressing the answer in terms of the D T F T of x(n). This problem is most easily solved if we take advantage of what we know about the DFT. Recall that the Mpoint , DFT corresponds to M samples of the DTFT, X ( e J U )at wi = 2 n k l M for k = 0, 1, . . ., M  1. In addition, if these samples are used for the DFT coefficients of a sequence y ( n ) of length M, y ( n ) is related t o x ( n ) as follows:

