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Problems 201
Clearly, ˜x 3(n) isdifferent from ˜x 2(n) ofProblem P5.3 even though both of them are
periodic with period 80.
1. Compute the DFS ˜X 3(k) of˜x 3(n), and plot samples of its magnitude and angle versus k.
2. What effect does the periodicity doubling have on the DFS?
3. Generalize this result to M-fold periodicity. In particular, show that if
⎡
⎤
then
⎢
⎥
˜x M (n) = ⎣˜x 1(n), ˜x 1(n),...,˜x 1(n) ⎦
} {{ }
M times PERIODIC
˜X M (Mk)=M ˜X 1(k), k=0, 1,...,N − 1
˜X M (k) = 0, k ≠0,M,...,MN
P5.6 Let X(e jω )bethe DTFT of a finite-length sequence
1. Let
x(n) =
{ n +1, 0 ≤ n ≤ 49;
100 − n, 50 ≤ n ≤ 99;
0, otherwise.
10-point
y 1(n) = IDFS [ X(e j0 ),X(e j2π/10 ),X(e j4π/10 ),...,X(e j18π/10 ) ]
Determine y 1(n) using the frequency sampling theorem. Verify your answer using
MATLAB.
2. Let
200-point
y 2(n) = IDFS [ X(e j0 ),X(e j2π/200 ),X(e j4π/200 ),...,X(e j398π/200 ) ]
Determine y 2(n) using the frequency sampling theorem. Verify your answer using
MATLAB.
3. Comment on your results in parts (a) and (b).
P5.7 Let ˜x(n) beaperiodic sequence with period N and let
ỹ(n) △ =˜x(−n) =˜x(N − n)
that is, ỹ(n) isaperiodically folded version of ˜x(n). Let ˜X(k) and Ỹ (k) bethe DFS
sequences.
1. Show that
Ỹ (k) = ˜X(−k) = ˜X(N − k)
that is, Ỹ (k) isalso a periodically folded version of ˜X(k).
2. Let ˜x(n) ={2, 4, 6, 1, 3, 5} PERIODIC with N =6.
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