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Unformatted text preview: 3, −1, 5}, w[n] = {0, 7, 0, 5}. a) Determine if a sequence that satisﬁes x y = w can be found. If so, ﬁnd y [n]. If not,
prove it does not exist.
b) Given that W [k ] = DFT4 {w[n]} = Im{G[k ]}, ﬁnd g [3] − g [1].
2
c) Find q [n] whose DFT is given by: Q[k ] = W [(k − 2) mod 4] W4 k . 2 Problem №3 (15%)
A DFT engineer would like to use windowing and zero padding to analyze the DFT of a
sampled signal.
a) Given that the signal was sampled using a sampling frequency of 128 Hz, and that the
engineer intends to use a rectangular window and apply the DFT of length N = 64 to
the signal, what is the minimal distance (in Hz) between two distinguishable frequency
components of the signal?
b) Which of the following two is the proper procedure:
• Zeropad, multiply by a window, apply FFT;
• Multiply by a window, zeropad, apply DFT. Problem №4 (25%)
In STFT analysis, the DTFT is applied to windowed versions of a sequence x[n], namely
X [n, λ) = DTFT{x[m + n] w[m]} = ∞
x[n + m]w[m]e−λm . m=−∞ Alternatively, one can apply the DTFT to the autocorrelation sequence a[n] of the windowed
signal x[m + n] w[m] which is deﬁned as:
a[m, n] = ∞
x[n + l] w[l] x[n + m + l] w[m + l]. l=−∞ Find the DTFT A[n, λ) of a[m, n] in terms of X [n, λ). (Assume real sequence and window.) Problem №5 (10%)
ω

✻H (e ) 1 ❈
❈ ❈ ❈
❈ ❈ ❈
❈ ❈ ❈
❈❈ ω π
3 ✲ Considering the above response of an FIR generalized linear phase (GLP) ﬁlter, which
of the following statements is correct:
a) The ﬁlter is Type I;
b) The ﬁlter is Type II;
c) The ﬁlter is Type III;
d) The ﬁlter type cannot be determined from the graph.
Explain your answer. 4...
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This document was uploaded on 02/28/2014 for the course ECE 413 at University of Waterloo, Waterloo.
 Fall '13
 OlegMichailovich
 Electrical Engineering, Digital Signal Processing, Signal Processing

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