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EE322 Homework Problems for Signal Processing, Assignments of Electrical and Electronics Engineering

Wright State University (WSU) - DaytonElectrical and Electronics Engineering

A series of homework problems related to signal processing, including finding fundamental frequencies, fourier series representation, sampling frequencies, aliasing errors, and dfs coefficients. Students are required to use matlab for calculations and plotting.

Typology: Assignments

Pre 2010

Uploaded on 08/18/2009

koofers-user-24m
koofers-user-24m 🇺🇸

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EE322 First Homework Problem
Summer 2007
Instructor: Kefu Xue, Ph.D.
1. Given an analog signal xa(t)=2sin(250πt +π
3)+cos(550πt −π
3)+2
(a) Find the fundamental frequency ω0and Fourier series representation and its co-
efficients Xkusing Fourier series recognition method.
(b) Select a sampling frequency which is 100 Hz higher than the Nyquist Frequency
(twice of the highest frequency component) of the signal xa(t)(in Hz).
(c) Is there an aliasing error in this sampling process? Why?
(d) Find the expression for the sampled sequence x(n).
(e) Is the sequence x(n)periodic? If yes, find the minimum period (in number of
samples) of the sequence. If not, explain why.
(f) Find DFS coefficients using MatlabTM.MatchingtheMatlab
TM result with your
hand calculated result.
2. For the same analog signal in the previous problem (1), select the sampling frequency
which is 4 times of the Nyquist frequency (4 times over sampling) for the expression
of x(n).
(a) What is the minimum period (in number of samples) of the sequence x(n)at this
time?
(b) Find DFS coefficients using MatlabTM.MatchingtheMatlab
TM result with your
hand calculated result.
3. For the same analog signal in problem 1, if we take 3 samples in each period of the
highest frequency component in xa(t)to form the expression of x(n), what is the
sampling frequency? Is there an aliasing error in this sampling process? Why? What
is the period of x(n)in this case?
4. Compare an analog signal wa(t) = 2 cos(250πt)+cos(550πt)+1 with the signal in
problem 1.
(a) Find the Fourier series representation and its coefficients Xkof the new signal
using Fourier series recognition method. What are the differences in their coeffi-
cients?
(b) In MatlabTM,plot3period(3N)ofw(n)and x(n)sampled with the 4 times over
sampling frequency. What is the time duration (in second) of the plotted signals?
(c) Plot each frequency components of w(n)and x(n)in separate plots with the same
length (3N). From observation, comments on the following questions:
i. How many periods of analog signals wa(t)and xa(t)in 3Nsamples of w(n)
and x(n)? Do your analysis to see if it matches your calculation.
pf3

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Download EE322 Homework Problems for Signal Processing and more Assignments Electrical and Electronics Engineering in PDF only on Docsity!

EE322 First Homework Problem

Summer 2007

Instructor: Kefu Xue, Ph.D.

  1. Given an analog signal xa(t) = 2 sin(250πt + π 3 ) + cos(550πt − π 3 ) + 2

(a) Find the fundamental frequency ω 0 and Fourier series representation and its co- efficients Xk using Fourier series recognition method. (b) Select a sampling frequency which is 100 Hz higher than the Nyquist Frequency (twice of the highest frequency component) of the signal xa(t) (in Hz). (c) Is there an aliasing error in this sampling process? Why? (d) Find the expression for the sampled sequence x(n). (e) Is the sequence x(n) periodic? If yes, find the minimum period (in number of samples) of the sequence. If not, explain why. (f) Find DFS coefficients using MatlabT M^. Matching the MatlabT M^ result with your hand calculated result.

  1. For the same analog signal in the previous problem (1), select the sampling frequency which is 4 times of the Nyquist frequency (4 times over sampling) for the expression of x(n).

(a) What is the minimum period (in number of samples) of the sequence x(n) at this time? (b) Find DFS coefficients using MatlabT M^. Matching the MatlabT M^ result with your hand calculated result.

  1. For the same analog signal in problem 1, if we take 3 samples in each period of the highest frequency component in xa(t) to form the expression of x(n), what is the sampling frequency? Is there an aliasing error in this sampling process? Why? What is the period of x(n) in this case?
  2. Compare an analog signal wa(t) = 2 cos(250πt) + cos(550πt) + 1 with the signal in problem 1.

(a) Find the Fourier series representation and its coefficients Xk of the new signal using Fourier series recognition method. What are the differences in their coeffi- cients? (b) In MatlabT M^ , plot 3 period ( 3 N) of w(n) and x(n) sampled with the 4 times over sampling frequency. What is the time duration (in second) of the plotted signals? (c) Plot each frequency components of w(n) and x(n) in separate plots with the same length ( 3 N). From observation, comments on the following questions: i. How many periods of analog signals wa(t) and xa(t) in 3 N samples of w(n) and x(n)? Do your analysis to see if it matches your calculation.

Summer 2007 page 2

ii. How many periods of each frequency components do you see in the plots? Show your analysis to match your observation. iii. Comment on the wave forms of signals wa(t) and xa(t) and their relationship with their signal components (time/phase delay and amplitude).

  1. Given the FS coefficients: C 0 = 0. 5 , C 1 = 3e−j^ , C 3 = −j, and C 5 = ej^

π (^6) of a real-valued periodic signal, xa(t), with a period equal to 2 · 10 −^3 seconds, find the expression of the signal.

  1. A middle C piano note (262 Hz) will be sent by a AM traffic radio station (550 KHz) in the morning 4am as the test signal. The signal expression is shown as follows.

[2 + sin(262 · 2 πt − 1)] cos(550 · 103 · 2 πt)

Find and sketch the line spectrum of the radio station signal. What is the Nyquist Frequency of the signal?

  1. Consider the following analog signal :

x(t) = sin(120πt) +

sin(360πt) 3

sin(600πt) 5

sin(840πt) 7

sin(1080πt) 9 which is a square-wave with signal components up to 9th harmonic and a fundamental frequency of 120 π radian/second.

(a) Select the sampling frequency that is twice the Nyquist frequency. The sampling duration should include 4 periods of the square-wave. Generate all of the harmonic components and add them together to form the composite signal. Generate hard copies of the plots for the waveforms with up to 5th, 7th and 9th harmonic contents (3 plots), respectively. Observe the changes in the waveform on the screen as each harmonic content is being added into the signal. (b) Change the sampling frequency to 750 Hz and sample 4 periods of the signal using this new sampling frequency. Generate hard copies of the plots for the waveforms with up to 5th, 7th and 9th harmonic contents (3 plots), respectively. (c) Compare and contrast these six plots and discuss your observation and conclusions in a few sentences. (d) Just for fun, you can include different delays in the harmonic components and then add them together to see the changes in signal waveforms. This is so called harmonic distortion.

  1. Text book problems: 8.2-4, 8.2-6, 8.3-4, 8.4-1 and 8.4-3(a,b,c).

Tips:

  • Use a text editor to put all the MatlabT M^ commands in one file (yourfile.m) referred to as m-file and run the m-file in MatlabT M^. It will save your time.