# 2 c find q n whose dft is given by qk w k 2 mod

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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: • Zero-pad, multiply by a window, apply FFT; • Multiply by a window, zero-pad, 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 auto-correlation 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.

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