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Name: SOLUTION (Havlicek)
Section:
Laboratory Exercise 3
DISCRETE-TIME SIGNALS: FREQUENCY-DOMAIN
REPRESENTATIONS
3.1 DISCRETE-TIME FOURIER TRANSFORM
Project 3.1 DTFT Computation
A copy of Program P3_1 is given below:
% Program P3_
% Evaluation of the DTFT
clf;
% Compute the frequency samples of the DTFT
w = -4pi:8pi/511:4*pi;
num = [2 1];den = [1 -0.6];
h = freqz(num, den, w);
% Plot the DTFT
subplot(2,1,1)
plot(w/pi,real(h));grid
title('Real part of H(e^{j\omega})')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,1,2)
plot(w/pi,imag(h));grid
title('Imaginary part of H(e^{j\omega})')
xlabel('\omega /\pi');
ylabel('Amplitude');
pause
subplot(2,1,1)
plot(w/pi,abs(h));grid
title('Magnitude Spectrum |H(e^{j\omega})|')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,1,2)
plot(w/pi,angle(h));grid
title('Phase Spectrum arg[H(e^{j\omega})]')
xlabel('\omega /\pi');
ylabel('Phase in radians');
Answers:
Q3.1 The expression of the DTFT being evaluated in Program P3_1 is -
1
1
j
z
H e
z
−
ω
−
The function of the pause command is - to pause execution of a Matlab program. Without
arguments, pause waits for the user to type any key. With an argument, pause pauses
for a number of seconds specified by the argument.
Q3.2 The plots generated by running Program P3_1 are shown below:
Real part of H(e
j ω
ω
π
Amplitude
Imaginary part of H(e
j ω
ω
π
Amplitude
Magnitude Spectrum |H(e
j ω
ω
π
Amplitude
Phase Spectrum arg[H(e
jω
)]
ω
π
Phase in radians
The plots generated by running the modified Program P3_1 are shown below:
Real part of H(e
j ω
ω
π
Amplitude
Imaginary part of H(e
j ω
ω
π
Amplitude
Magnitude Spectrum |H(e
j ω
ω
π
Amplitude
Phase Spectrum arg[H(e
jω
)]
ω
π
Phase in radians
The DTFT is a periodic_ function of ω.
Its period is - 2π
The jump in the phase spectrum is caused by - a branch cut in the arctan function used by
angle in computing the phase. “angle” returns the principal branch of arctan.
The phase spectrum evaluated with the jump removed by the command unwrap is as given
below:
Phase Spectrum arg[H(e
j ω
)]
ω
π
Phase in radians
Q3.4 The required modifications to Program P3_1 to evaluate the given DTFT of Q3.4 are given below:
% Program P3_1D
% Evaluation of the DTFT
clf;
% Compute the frequency samples of the DTFT
w = -4pi:8pi/511:4*pi;
num = [1 3 5 7 9 11 13 15 17];
den = 1;
h = freqz(num, den, w);
% Plot the DTFT
subplot(2,1,1)
plot(w/pi,real(h));grid
title('Real part of H(e^{j\omega})')
xlabel('\omega /\pi');
ylabel('Amplitude');
subplot(2,1,2)
plot(w/pi,imag(h));grid
title('Imaginary part of H(e^{j\omega})')
xlabel('\omega /\pi');
ylabel('Amplitude');
pause
Magnitude Spectrum |H(e
j ω
ω
π
Amplitude
Phase Spectrum arg[H(e
j ω
)]
ω
π
Phase in radians
The DTFT is a periodic function of ω.
Its period is - 2π
The jump in the phase spectrum is caused by - “angle” returns the principal value of the
arc tangent.
Q3.5 The required modifications to Program P3_1 to plot the phase in degrees are indicated below:
Only the last paragraph of the code needs to be changed to:
% plot phase in degrees
subplot(2,1,2)
plot(w/pi,180*angle(h)/pi);grid
title('Phase Spectrum arg[H(e^{j\omega})]')
xlabel('\omega /\pi');
ylabel('Phase in degrees');
Project 3.2 DTFT Properties
Answers:
Q3.6 The modified Program P3_2 created by adding appropriate comment statements, and adding
program statements for labeling the two axes of each plot being generated by the program is
given below:
% Program P3_2B
% Time-Shifting Properties of DTFT
clf;
w = -pi:2*pi/255:pi; % freqency vector for evaluating DTFT
D = 10; % Amount of time shift in samples
num = [1 2 3 4 5 6 7 8 9];
% h1 is the DTFT of original sequence
% h2 is the DTFT of the time shifted sequence
h1 = freqz(num, 1, w);
h2 = freqz([zeros(1,D) num], 1, w);
subplot(2,2,1)
% plot the DTFT magnitude of the original sequence
plot(w/pi,abs(h1));grid
title('Magnitude Spectrum of Original Sequence','FontSize',8)
xlabel('\omega /\pi');
ylabel('Amplitude');
% plot the DTFT magnitude of the shifted sequence
subplot(2,2,2)
plot(w/pi,abs(h2));grid
title('Magnitude Spectrum of Time-Shifted Sequence','FontSize',8)
xlabel('\omega /\pi');
ylabel('Amplitude');
% plot the DTFT phase of the original sequence
subplot(2,2,3)
plot(w/pi,angle(h1));grid
title('Phase Spectrum of Original Sequence','FontSize',8)
xlabel('\omega /\pi');
ylabel('Phase in radians');
% plot the DTFT phase of the shifted sequence
subplot(2,2,4)
plot(w/pi,angle(h2));grid
title('Phase Spectrum of Time-Shifted Sequence','FontSize',8)
xlabel('\omega /\pi');
ylabel('Phase in radians');
The parameter controlling the amount of time-shift is - D
Magnitude Spectrum of Original Sequence
ω
π
Amplitude
Magnitude Spectrum of Time-Shifted Sequence
ω
π
Amplitude
Phase Spectrum of Original Sequence
ω
π
Phase in radians
Phase Spectrum of Time-Shifted Sequence
ω
π
Phase in radians
From these plots we make the following observations: As before, of course, the time shift
has no effect on the magnitude spectrum. However, there is a very significant change
to the phase. As before, the time shift adds phase to the DTFT, making the slope of the
phase spectrum steeper. However, in this case with D=5 instead of D=10, the
(negative) increase in the slope is less than it was before.
Q3.9 Program P3_2 was run for the following values of the time-shift and for the following values of
length for the sequence –
- Length 4, time shift D=3.
- Length 16, time shift D=12.
The plots generated by running the modified program are given below:
Magnitude Spectrum of Original Sequence
ω
π
Amplitude
Magnitude Spectrum of Time-Shifted Sequence
ω
π
Amplitude
Phase Spectrum of Original Sequence
ω
π
Phase in radians
Phase Spectrum of Time-Shifted Sequence
ω
π
Phase in radians
Magnitude Spectrum of Original Sequence
ω
π
Amplitude
Magnitude Spectrum of Time-Shifted Sequence
ω
π
Amplitude
Phase Spectrum of Original Sequence
ω
π
Phase in radians
Phase Spectrum of Time-Shifted Sequence
ω
π
Phase in radians
From these plots we make the following observations: Increasing the length makes the
magnitude spectrum more narrow (i.e., makes the signal more “low pass”). It also
makes the phase steeper (i.e., the slope more negative). This is because, if we think of
the sequence as the impulse response of an LTI system, increasing the length adds more
Magnitude Spectrum of Original Sequence
ω
π
Amplitude
Magnitude Spectrum of Frequency-Shifted Sequence
ω
π
Amplitude
Phase Spectrum of Original Sequence
ω
π
Phase in radians
Phase Spectrum of Frequency-Shifted Sequence
ω
π
Phase in radians
From these plots we make the following observations: Both the magnitude and phase
spectra are shifted right by wo, which is given by 0.4π in this case. Note that the
frequency shifted signal was obtained by multiplying the original sequence pointwise
with a complex-valued exponential sequence. Thus, the frequency shifted sequence is
also complex-valued and it’s DTFT does not have conjugate symmetry.
Q3.12 Program P3_3 was run for the following value of the frequency-shift – wo = -0.5π.
The plots generated by running the modified program are given below:
Magnitude Spectrum of Original Sequence
ω
π
Amplitude
Magnitude Spectrum of Frequency-Shifted Sequence
ω
π
Amplitude
Phase Spectrum of Original Sequence
ω
π
Phase in radians
Phase Spectrum of Frequency-Shifted Sequence
ω
π
Phase in radians
From these plots we make the following observations: In this case, the magnitude and
phase spectra are shifted left by π/2 rad. As before, the frequency shifted signal is
complex-valued, so the frequency shift causes a loss of the conjugate symmetry that
was present in the spectrum of the original signal. NOTE: you should keep in mind
that these spectra are all 2π-periodic; we are only displaying the fundamental period.
Q3.13 Program P3_3 was run for the following values of the frequency-shift and for the following
values of length for the sequence –
- Length 4, frequency shift wo = π.
- Length 16, frequency shift wo = -0.3π.
The plots generated by running the modified program are given below:
the impulse responses of LTI systems, then a shorter length implies a smaller delay
between the input and output of the system, which corresponds to a phase spectrum
with a negative slope that is less steep. Likewise, a longer length implies a longer
delay between the input and output, which corresponds to a phase spectrum with a
negative slope that is steeper. In both cases shown here, the effect of the frequency
shift is to translate both the magnitude and phase spectrum by wo radians. Whereas the
original sequences are real-valued and hence have conjugate symmetric spectra, the
frequency shifted sequences are complex-valued and do not exhibit any inherent
spectral symmetry.
Q3.14 The modified Program P3_4 created by adding appropriate comment statements, and adding
program statements for labeling the two axes of each plot being generated by the program is
given below:
% Program P3_4B
% Convolution Property of DTFT
clf;
w = -pi:2*pi/255:pi; % freqency vector for evaluating DTFT
x1 = [1 3 5 7 9 11 13 15 17]; % first sequence
x2 = [1 -2 3 -2 1]; % second sequence
y = conv(x1,x2); % time domain convolution of x1 and x
h1 = freqz(x1, 1, w); % DTFT of sequence x
h2 = freqz(x2, 1, w); % DTFT of sequence x
% hp is the pointwise product of the two DTFT's
hp = h1.*h2;
% h3 is the DTFT of the time domain convolution;
% it should be the same as hp
h3 = freqz(y,1,w);
% plot the magnitude of the product of the two original spectra
subplot(2,2,1)
plot(w/pi,abs(hp));grid
title('Product of Magnitude Spectra','FontSize',8)
xlabel('\omega /\pi');
ylabel('Amplitude');
% plot the magnitude spectrum of the time domain convolution
subplot(2,2,2)
plot(w/pi,abs(h3));grid
title('Magnitude Spectrum of Convolved Sequence','FontSize',8)
xlabel('\omega /\pi');
ylabel('Amplitude');
% plot the phase of the product of the two original spectra
subplot(2,2,3)
plot(w/pi,angle(hp));grid
title('Sum of Phase Spectra','FontSize',8)
xlabel('\omega /\pi');
ylabel('Phase in radians');
% plot the phase spectrum of the time domain convolution
subplot(2,2,4)
plot(w/pi,angle(h3));grid
title('Phase Spectrum of Convolved Sequence','FontSize',8)
xlabel('\omega /\pi');
ylabel('Phase in radians');
Q3.15 The plots generated by running the modified program are given below:
Product of Magnitude Spectra
ω
π
Amplitude
Magnitude Spectrum of Convolved Sequence
ω
π
Amplitude
Sum of Phase Spectra
ω
π
Phase in radians
Phase Spectrum of Convolved Sequence
ω
π
Phase in radians
From these plots we make the following observations: The DTFT magnitude and phase
spectra obtained by performing pointwise multiplication of the two DTFT’s of the
original sequences are identical to those obtained by performing time domain
convolution of the two original sequences; this verifies the convolution property of the
DTFT.
Q3.16 Program P3_4 was run for the following two different sets of sequences of varying lengths –
- Length of x1 = 8; x1[n] = ( )
1 2
n
for 0 ≤ n ≤ 7;
Length of x2 = 4; x2[n]= [0.25 0.25 0.25 0.25]
- Length of x1 = 16; x1[n] = ( )
3 4
n
− for 0 ≤ n ≤ 15;
Length of x2 = 8; x2[n]= [1 3 5 7 9 11 13 15]
The plots generated by running the modified program are given below:
are identical to the magnitude and phase spectra obtained from the DTFT of the time
domain convolution of the two original sequences.
Q3.17 The modified Program P3_5 created by adding appropriate comment statements, and adding
program statements for labeling the two axes of each plot being generated by the program is
given below:
% Program P3_5B
% Modulation Property of DTFT
clf;
w = -pi:2*pi/255:pi; % freqency vector for evaluating DTFT
x1 = [1 3 5 7 9 11 13 15 17]; % first sequence
x2 = [1 -1 1 -1 1 -1 1 -1 1]; % second sequence
% y is the time domain pointwise product of x1 and x
y = x1.*x2;
h1 = freqz(x1, 1, w); % DTFT of sequence x
h2 = freqz(x2, 1, w); % DTFT of sequence x
h3 = freqz(y,1,w); % DTFT of sequence y
% plot the magnitude spectrum of x
subplot(3,1,1)
plot(w/pi,abs(h1));grid
title('Magnitude Spectrum of First Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
% plot the magnitude spectrum of x
subplot(3,1,2)
plot(w/pi,abs(h2));grid
title('Magnitude Spectrum of Second Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
% plot the magnitude spectrum of y
% it should be 1/2pi times the convolution of the DTFT's
% of the two original sequences.
subplot(3,1,3)
plot(w/pi,abs(h3));grid
title('Magnitude Spectrum of Product Sequence')
xlabel('\omega /\pi');
ylabel('Amplitude');
Q3.18 The plots generated by running the modified program are given below:
Magnitude Spectrum of First Sequence
ω
π
Amplitude
Magnitude Spectrum of Second Sequence
ω
π
Amplitude
Magnitude Spectrum of Product Sequence
ω
π
Amplitude
From these plots we make the following observations: The DTFT of the product sequence
y is 1/2π times the convolution of the DTFT’s of the two sequences x1 and x2, as
expected. The low-pass mainlobe of the DTFT of x1 combines with the high-pass
mainlobe of the DTFT of x2 to produce a high-pass mainlobe centered at ±π in the
magnitude spectrum of the product signal. The low-pass mainlobe of the DTFT of x
combines with the low-pass sidelobes of the DTFT of x2 to produce a low-pass smooth
region of relatively lower gain centered at DC in the magnitude spectrum of the product
signal.
Q3.19 Program P3_5 was run for the following two different sets of sequences of varying lengths –
- Length of x1 = 8; x1[n] = ( )
1 2
n
for 0 ≤ n ≤ 7;
Length of x2 = 8; x2[n]= [1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8]
- Length of x1 = 16; x1[n] = ( )
3 4
n
− for 0 ≤ n ≤ 15;
Length of x2 = 16; x2[n]= [1 3 5 7 9 11 13 15 0 0 0 0 0 0 0 0]
The plots generated by running the modified program are given below:
