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SOUTHERN METHODIST UNIVERSITY
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
INTRODUCTION TO SIGNAL PROCESSING
ECE3372
SPRING 2023
Homework #10 Alt.
Spectrum Analysis and Filter Design
for Speech Enhancement
Speech enhancement describes a set of signal processing techniques designed to improve the
quality of voice signals collected by audio sensors, typically microphones, in less-than-ideal situa-
tions. Speech enhancement is important for a number of real-world applications, including voice
calls, audio messaging, and automated speech recognition. In the simplest situation, a single mi-
crophone records an audio signal that, after analog-to-digital conversion, is represented by
x[n]
=
N
s[n]+[n],
i=1
where {ni[n]}, 1 ≤ i ≤ N are N noise signals that are corrupting the measurement of the desired
speech signal s[n]. The goal of speech enhancement is to digitally process the sampled signal x[n]
by one or more filters in cascade to attenuate the noises {ni[n]} and enhance the speech signal s[n].
The design of these filters is generally based on some form of analysis of the signal x[n].
In this assignment, you will use frequency analysis to determine the natures of the noises that
corrupt a recorded speech signal that you have been assigned. In this case, the noises are static in
frequency, which simplifies the required processing to a set of linear, time-invariant filters. From
this analysis, you are to design filters that attenuate the noises, thereby revealing the content of
the speech signal.
The tasks to complete are as follows:
1. Listen to the recorded signal that you have been assigned, stored as the variable x with
sample rate fs, using MATLAB's sound command. Keep the volume low to protect your
ears! Describe qualitatively what you hear. What noises appear to be present in the signal?
Can you tell there is a speech signal present? If so, can you understand what is being said?
If you can understand what is being said, attempt a written transcription of the recorded
message.
2. In MATLAB, type help pspectrum to learn what the command does. Describe briefly what
the function computes. How is this related to what we discussed about the FFT function
in class? Using the pspectrum command, plot the frequency spectrum in dB of your noisy
signal from DC to half the sample rate of the signal. Using the graph, carefully determine
the frequency values of the tonal noises.
1 3. Design individual second-order IIR notch filters to remove each of the tonal noises in the
signal, one for each tone. Apply the notch filters to the signal x one by one in a cascade
fashion using multiple calls to MATLAB's filter command. Then, listen to the output of
the combined system as designed. Have you removed the tones? Note that you may need to
adjust the frequency position of each of the notches by a few Hertz to adequately null out
each of the tonal noises in the recorded signal. Once you are satisfied with the sound of the
result, do the following:
a) Give the values of the coefficients of the notch filter(s) that you have designed, and find
the pair of 3dB cutoff frequencies for each of the notch filters, verifying that they are on
either side of the tonal noises.
b) Plot the frequency response of each of the notch filters on separate graphs using MAT-
LAB's freqz command.
c) Plot both the recorded signal (top) and the output of the system you've designed (bot-
tom) on the same figure in separate sub-graphs.
What do you hear in the processed signal at this point? Can you tell there is a speech signal
present? If so, can you understand what is being said? If you can understand what is being
said at this point, attempt a written transcription of the recorded message.
4. Now, perform a second simple frequency analysis of the output of the system to see what
frequency content remains in the signal. Plot the frequency spectrum in dB from DC to half
the sample rate of the processed signal, and carefully determine the frequency range of any
broadband noise that is present in the processed signal. What kind of filter is needed to
remove the broadband noise?
5. Using MATLAB's fir1 command. design a simple FIR filter to remove the broadband noise
that you identified in the previous step. Describe how you came to the choice of this filter
design. On a separate plot, plot the impulse response of the filter that you've designed, and
comment on its general structure and shape. Then, apply this filter to your processed signal
to produce a new processed signal that combines both the notch filtering and the FIR filter
that you've designed, and listen to the newly-processed output. What do you hear in the
processed signal at this point? Can you tell there is a speech signal present? If so, can you
understand what is being said? What else can you tell about the signal now (the type of
voice? the apparent gender of the talker? anything else?)? Transcribe the message. Do you
think this newly-processed signal could be recognized by a speech recognition engine after
processing?
Some (but not all) useful MATLAB commands: cos, filter, fft, freqz, sound, figure,
plot, subplot, xlabel, ylabel, title, legend.
Your writeup should include written responses, important numerical values, all requested plots,
and the MATLAB code used in your processing and the answers to the questions above. The due
date/time for this assignment is 9:00am, Tuesday, April 30, 2024.
2/n SOUTHERN METHODIST UNIVERSITY
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
INTRODUCTION TO SIGNAL PROCESSING
ECE3372
SPRING 2023
Name:
Homework #9 Alt.
The DTFT and Discrete-Time Systems
Assigned: Friday, March 22, 2024
Due: Friday, April 19, 2024
9.1. A linear, time-invariant system has a frequency response over ≤ π as
202
H()
= rect
πT
Find the output y[n] of the system if the input x[n] is
a) x[n] = sinc(πn/2)
1 b) x[n] = sinc(n)
c) x[n] = sinc²(πn/2)
21 9.2. Lathi, Problem 9.1-1, p. 899. [shown below]
9.1-1
Find the discrete-time Fourier series (DTFS)
and sketch their spectra |Dr] and [Dr for
0≤r≤No-1 for the following periodic signal:
x[n] 4 cos 2.4лп+2 sin 3.2лn
=
3 9.3. Lathi, Problem 9.1-5, p. 899. [shown below]
9.1-5 Find the discrete-time Fourier series and the
corresponding amplitude and phase spectra for
the x[n] shown in Fig. P9.1-5.
x[n]
3
-9
-6
-3
0
3
6
9
12 n
Figure P9.1-5
4/n SOUTHERN METHODIST UNIVERSITY
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
INTRODUCTION TO SIGNAL PROCESSING
ECE3372
SPRING 2023
Name:
Homework #8 Alt.
Discrete-Time Fourier Transform
Assigned: Friday, March 22, 2024
Due: Friday, April 12, 2024
8.1. First, plot the following signals over the range −6 ≤ n ≤ 6. Then, from the definition of the
DTFT in Eq. (9.19), find the DTFT X() for the following signals. Simplify the expressions
for X) using trigonometric relations.
a)
x[n]
=
1
1,
−3≤ n ≤0
2,
1≤ n ≤3
0,
otherwise b)
x[n]
=
J n n ≤ 3
{ b
otherwise
21 x[n]
=
{ ?
2n n≤3
otherwise
3 8.2. First, plot the following discrete-time spectra over the range –π ≤ N ≤ π. Then, from the
definition of the inverse DTFT in Eq. (9.18), find x[n] for the following signals. Simplify the
expressions for x[n] using trigonometric relations.
[Ω] < 1
a)
X(N)
=
2,
1 ≤ N ≤ 2
0,
2≤ N ≤π
[For your plot, choose No = π/2. For the solution of x[n], give a general expression for any
Ωρ.]
4 b) For any o in the range 0 < No < π,
X(N)
=
ΩΣ ΤΩΙ ΣΩΟ
{3²
ΩΟΣ ΙΩΙ ΣΠ
[For your plot, choose o =π/2. For the solution of x[n], give a general expression for
any No.]
5/n SOUTHERN METHODIST UNIVERSITY
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
INTRODUCTION TO SIGNAL PROCESSING
ECE3372
SPRING 2023
Name:
Homework #7 Alt.
Sampling
Assigned: Friday, March 22, 2024
Due: Friday, April 5, 2024
7.1. A continuous time signal x(t) has the continuous-time Fourier transform or spectrum given
by
X (w)
=
5, |w| < 25л rad/sec
0,
otherwise.
a) Sketch X(w) over the range −100π <w < 100”.
b) Suppose x(t) is ideally sampled by the unit impulse train with sampling period T
sec to produce x(t). Sketch X (w) over the range −100π <w < 100л. Does aliasing
occur? If so, what frequencies in the original signal are aliased?
= 0.02
1 c) Suppose x(t) is ideally sampled by the unit impulse train with sampling period T = 0.04
sec to produce x(t). Sketch X (w) over the range -100π <w < 100л. Does aliasing
occur? If so, what frequencies in the original signal are aliased?
d) Suppose x(t) is ideally sampled by the unit impulse train with sampling period T = 0.05
sec to produce x(t). Sketch X (w) over the range -100π <w < 100л. Does aliasing
occur? If so, what frequencies in the original signal are aliased?
21 7.2.
=
20
a) A continuous time signal x(t) = 10 sinc² (10πt) + cos(50πt) is sampled at a rate of fs
Hz. Find a mathematical expression for the spectrum of the sampled signal. Can x(t)
be reconstructed from the sampled signal via lowpass filtering? Explain.
=
50 Hz. Find
b) The same continuous time signal in part a) is sampled at a rate of fs
a mathematical expression for the spectrum of the sampled signal. Can x(t) be recon-
structed from the sampled signal via lowpass filtering? Explain.
3 c) A continuous time signal x(t) = 10 sinc²(10πt)+sin(50πt) is sampled at a rate of fs = 50
Hz. Can x(t) be reconstructed from its samples if the signal is sampled at a rate of
fs = 50 Hz? Use spectral representations to explain why or why not.
d) Repeat part c) for a sampling rate of fs:
=
51 Hz.
4 7.3. A continuous-time signal x(t) has the spectrum
w
X (w)
=
10πT
0,
"
20πw < 30π
otherwise.
a) Find the Nyquist rate for this signal based on the highest frequency contained in the
signal. Sketch the spectrum of the ideally-sampled signal x(t) if sampling is done at
the Nyquist rate. (NOTE: The spectrum has a slightly different shape for negative and
positive frequencies!)
=
b) A colleague of yours claims that, because the signal is bandlimited to 5 Hz, it can be
reconstructed if it is sampled at a rate of ƒs 10 Hz. Sketch the spectrum of the
ideally-sampled signal (t) for this sampling rate. Can x(t) be reconstructed from the
samples taken at this rate? If so, explain how reconstruction can be performed. If not,
explain why not.
5