1. (Text problem 11.2) Consider the signal x(n) = a"u(n), |a| < 1. (a) Determine the spectrum X (w). (b) The signal x(n) is applied to a downsampler that subsamples by a factor of 2, with output y1(n). Recall that with y(n) as the result of downsampling x(n) by a factor of D, D-1 1 2nk Y (w) ΣΧ D k=0 Use this formula to determine the spectrum Y1(w). (c) Let y2(n) = x(2n). Find Y2(w) by taking the discrete-time Fourier transform Y2(w) = E Y2(n)e-jun n=-0 and show that it is the same as Y1(w).

Introductory Circuit Analysis (13th Edition)
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Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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1. (Text problem 11.2) Consider the signal x(n) = a"u(n), |a| < 1.
(a) Determine the spectrum X (w).
(b) The signal x(n) is applied to a downsampler that subsamples by a factor of 2,
with output y1(n). Recall that with y(n) as the result of downsampling x(n) by
a factor of D,
D-1
1
2nk
-
Y (w) =
Σχ
D
k=0
Use this formula to determine the spectrum Y1(w).
(c) Let y2(n) = x(2n). Find Y2(w) by taking the discrete-time Fourier transform
Y2(w) = y2(n)e¬jun
n=-0
and show that it is the same as Y1(w).
Transcribed Image Text:1. (Text problem 11.2) Consider the signal x(n) = a"u(n), |a| < 1. (a) Determine the spectrum X (w). (b) The signal x(n) is applied to a downsampler that subsamples by a factor of 2, with output y1(n). Recall that with y(n) as the result of downsampling x(n) by a factor of D, D-1 1 2nk - Y (w) = Σχ D k=0 Use this formula to determine the spectrum Y1(w). (c) Let y2(n) = x(2n). Find Y2(w) by taking the discrete-time Fourier transform Y2(w) = y2(n)e¬jun n=-0 and show that it is the same as Y1(w).
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