HW6F13_sol

Notice that xn 64 4n1 un 1 which is a left sided

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Unformatted text preview: |X (ejω )| = 1 3 82 81 − 2 9 ∠X (ejω ) = −ω − arctan , · cos(ω ) sin(ω ) 9 − cos(ω ) (c). Notice that x(n) = −64 · 4−n−1 u(−n − 1) which is a left-sided sequence. Its z -transform is given by X (z ) = and its ROC is given by −64 · z 1−4·z 1 4 Since on every point of the unit circle the X (z ) diverges, its DTFT does not exist. 0 ≤ |z | < 338 Problem 13.11. Find the DTFTs of the following sequences: π 3n (a). x(n) = cos π 3n (b). x(n) = cos π 3n (c). x(n) = cos π 3n + 2j · sin (d). x(n) = cos π 3n · cos sin − 2π 3 . . π 6n π 6n . . In each case, plot the magnitude and phase of the DTFT. Solution. (a). π π n sin n 3 3 2π 1 sin n = DTFT 2 3 −jπ 2π 2π = δ ω− −δ ω+ 2 3 3 π 2π π 2π = δ ω− e−j π/2 + δ ω + 2 3 2 3 X (ejω ) = DTFT cos 3 ej π/2 2 1.5 2.5 1 0.5 |X(ejω)| ∠ X(e...
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