Description
Problem 1: Problem 4.3 in R1 (Proakis 4th Edition)
Problem 2: Problem 4.6 (Part (b) only) in R1
Problem 3: Problem 4.7 (Part (a) only) in R1
Problem 4: Problem 4.9 (Part (d) only) in R1
Problem 5: Problem 4.10 (Parts (c) and (d) only) in R1
Problem 6: Problem 7.18 in R1
Problem 7: Problem 7.23 (Part (h) only, assume N odd) in R1
Problem 8:
Assume that an N-point DFT, performed on an N-sample x(n) sequence, results in a DFT frequency sample spacing of 100 Hz. What would be the DFT frequency domain sample spacing in Hz if the N-sample sequence x(n) is zero-padded with 4N zero-valued samples and we perform DFT on that extended time sequence?
Problem 9:
A system for discrete-time spectral analysis of a continuous-time signal is shown below:
xa(t)→ideal A/Dx(n)→w(n)xw(n)→64-Point FFTXw(k)Fs=1000 Hz
where w(n) is a rectangular window:
w(n)={164,for 0≤n≤630,otherwise
We have obtained the 64-point FFT, Xw(k), the magnitude of which is shown below with the vertical axis in dB scale.
The associated continuous-time input signal, xa(t), could be one or more of the following signals. Identify which one(s) of the signals below could have produced this FFT and clearly explain your reasoning.
Hint: Do not try to analyze the signals one at a time. Instead, first look at all the signals and, from what you know about DFT’s, try to divide and conquer!
Candidate signals:
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□ xa1(t)=10cos(550πt)
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□ xa2(t)=1000cos(550πt)
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□ xa3(t)=10ej550πt
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□ xa4(t)=1000ej550πt
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□ xa5(t)=10cos(531.25πt)
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□ xa6(t)=1000cos(531.25πt)
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□ xa7(t)=1000ej531.25πt
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□ xa8(t)=1000ej562.5πt
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□ xa9(t)=1000cos(562.5πt)
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□ xa10(t)=1000ej2562.5πt
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□ xa11(t)=1000cos(2550πt)
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□ xa12(t)=1000ej2550πt
MATLAB Exercises:
P5.10
Plot the DFT magnitude and angle of each of the following sequences using the DFT as a computation tool. Make an educated guess about the length N so that your plots are meaningful.
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x(n)=[cos(0.5πn)+jsin(0.5πn)][u(n)−u(n−51)]
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x(n)={1,2,3,4,3,2,1}
Note: For DFT calculation in MATLAB for this problem, you can either use the built-in fft function, or the following DFT function:
function [Xk] = dft(xn, N) % Computes Discrete Fourier Transform % [Xk] = dft(xn, N) % Xk = DFT coeff. array over 0 <= k <= N-1 % xn = N-point finite-duration sequence % N = Length of DFT n = [0:1:N-1]; % row vector for n k = [0:1:N-1]; % row vector for k WN = exp(-j*2*pi/N); % Wn factor nk = n' * k; % creates a N by N matrix of nk values WNnk = WN .^ nk; % DFT matrix Xk = xn * WNnk; % row vector for DFT coefficients
P5.38
An analog signal xa(t)=2sin(4πt)+5cos(8πt) is sampled at t=0.01n for n=0,1,…,N−1 to obtain an N-point sequence x(n). An N-point DFT is used to obtain an estimate of the magnitude spectrum of xa(t).
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From the following values of N, choose the one that will provide the accurate estimate of the spectrum of xa(t). Plot the real and imaginary parts of the DFT spectrum X(k).
(a) N=40
(b) N=50
(c) N=60 -
From the following values of N, choose the one that will provide the least amount of leakage in the spectrum of xa(t). Plot the real and imaginary parts of the DFT spectrum X(k).
(a) N=90
(b) N=95
(c) N=99