EE430 Homework 1 solution

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1) A discrete-time signal ๐‘ฅ[๐‘›] is obtained by sampling a continuous-time sinusoidal signal
๐‘ฅ๐‘
(๐‘ก) = 4 sin(20000๐œ‹๐‘ก +
๐œ‹
13)
at a sampling rate of 3 kHz.
a) Describe the set of all other continuous-time sinusoidal signals (their frequencies) that yield ๐‘ฅ[๐‘›] when
sampled at 3 kHz.
b) Describe the set of all other sampling frequencies that yield ๐‘ฅ[๐‘›] from ๐‘ฅ๐‘
(๐‘ก).
2) Among the following sinusoidal signals, identify periodic ones and find their fundamental periods.
sin(1.74๐œ‹๐‘› + 3.1) , sin(1.74๐œ‹๐‘› + 3.1๐œ‹), cos (15.74๐œ‹๐‘› +
3๐œ‹
8
), cos(โˆš๐œ‹๐‘›) , cos(๐œ‹โˆš๐œ‹๐‘›), cos(๐œ‹โˆš2 ๐‘›)
3) A linear sistem whose response to a shifted impulse, ๐›ฟ[๐‘› โˆ’ ๐‘˜], is โ„Ž๐‘˜
[๐‘›] = (๐‘› โˆ’ ๐‘˜)๐‘ข[๐‘› โˆ’ ๐‘˜]. Find the output, ๐‘ฆ[๐‘›]
of this system for an arbitrary input, ๐‘ฅ[๐‘›]. Is this system time-invariant? Explain clearly by using ๐‘ฆ[๐‘› โˆ’ ๐‘š] and
๐‘ฅ[๐‘› โˆ’ ๐‘š].
4) What does the following system do? Is it linear, time-invariant? Prove your answer.
๐‘ฆ[๐‘›] =
{

 

๐‘ฅ [
๐‘›
2
] ๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘’๐‘ฃ๐‘’๐‘›
๐‘ฅ [
๐‘› โˆ’ 1
2
] + ๐‘ฅ [
๐‘› + 1
2
]
2
๐‘–๐‘“ ๐‘› ๐‘–๐‘  ๐‘œ๐‘‘๐‘‘
5) Are the following systems causal, stable? Justify/discuss/prove.
๐‘ฆ[๐‘›] = 2
๐›ฟ[๐‘›+1] + ๐‘ฅ[๐‘› โˆ’ 3]
๐‘ฆ[๐‘›] = {
๐‘ฆ[โˆ’๐›ฟ[๐‘› โˆ’ 1]] + ๐‘ฅ[๐‘› โˆ’ 3] ๐‘› > 0
2
๐‘›๐‘ฅ[๐‘› โˆ’ 3] ๐‘› โ‰ค 0
6) Let ๐‘ฆ[๐‘›] = ๐‘ฅ[๐‘›] โˆ— ๐‘ฅ[๐‘›] where โˆ— is the convolution operator. If the first and the last non-zero elements of ๐‘ฅ[๐‘›] are
equal to ๐‘ฅ[โˆ’6] = โˆ’3 and ๐‘ฅ[24] = โˆ’4, respectively, find the first and the last non-zero elements of ๐‘ฆ[๐‘›].
7) Calculate the value for output, y[n], as
n ๏‚ฎ ๏‚ฅ
for the LTI system, whose input is x[n]= u[n] and its impulse
response is equal to
๏› ๏ [ ]
3
1
[ ] 2
2
1
3
1
h n u n u n
n n๏€ญ
๏ƒท
๏ƒธ
๏ƒถ
๏ƒง
๏ƒจ
๏ƒฆ
๏ƒท ๏€ญ
๏ƒธ
๏ƒถ
๏ƒง
๏ƒจ
๏ƒฆ
๏€ฝ
8) Consider the following difference equation of a causal LTI system,
๏› ๏ ๏› 1๏ ๏› ๏ ๏› 1๏ ๏› 2๏
2
1
y n ๏€ญ y n ๏€ญ ๏€ฝ x n ๏€ญ x n ๏€ญ ๏€ซ x n ๏€ญ
a) Find the impulse response, โ„Ž[๐‘›].
b) Find the frequency response, ๐ป(๐‘’
๐‘—๐œ”).
c) Plot the magnitude response, |๐ป(๐‘’
๐‘—๐œ”)| and phase response, โˆก๐ป(๐‘’
๐‘—๐œ”), in MATLAB using โ€œfreqzโ€ command.
d) Let ๐‘ฅ[๐‘›] = cos (
๐œ‹
3
๐‘›) + sin (
๐œ‹
2
๐‘› +
๐œ‹
4
) be the system input. Find the output ๐‘ฆ[๐‘›].
e) Prove that ๐ป(๐‘’
๐‘—๐œ”) = ๐ปโˆ—
(๐‘’
๐‘—(2๐œ‹โˆ’๐œ”)
). Does this equality hold for an arbitrary โ„Ž[๐‘›]? Explain.
9) In the [โˆ’๐œ‹, ๐œ‹) interval, the DTFT, ๐‘‹(๐‘’
๐‘—๐œ”), of a real sequence ๐‘ฅ[๐‘›] is nonzero only in [๐‘Ž, ๐‘],๐‘Ž < 0 < ๐‘, ๐‘ โˆ’ ๐‘Ž =
๐œ‹
2
;
otherwise it is arbitrary.
a) Plot the magnitude and phase of a typical ๐‘‹(๐‘’
๐‘—๐œ”) for โˆ’โˆž < ๐œ” < โˆž.
b) Express the DTFTs, ๐‘‹๐ถ(๐‘’
๐‘—๐œ”) and ๐‘‹๐‘†(๐‘’
๐‘—๐œ”), respectively, of cos (
๐œ‹
5
๐‘›)๐‘ฅ[๐‘›] and sin (
๐œ‹
5
๐‘›)๐‘ฅ[๐‘›] in terms of
๐‘‹(๐‘’
๐‘—๐œ”).
c) Assume that ๐‘‹(๐‘’
๐‘—๐œ”) is real valued and still complies with the specifications above. Plot typical magnitudes
and phases for ๐‘‹(๐‘’
๐‘—๐œ”), ๐‘‹๐ถ(๐‘’
๐‘—๐œ”) and ๐‘‹๐‘†(๐‘’
๐‘—๐œ”).
10) Textbook question 2.55
11) Textbook question 2.60
12) Textbook question 2.65
The following is a computer problem.
13) MATLAB PART
a) Create a MATLAB function named myconv in a new m-file. This function will compute and return the
convolution of two finite length sequences using multiplications and delay operations.
b) Generate the sequences x[n]=[ 1 2 3 4 5 4 3 2 1] and h[n]=[1 1 1].

Convolve these sequences using myconv. Compute the convolution output and give the graph of it (using
stem command). Verify the result using conv command. Plot the result on the same graph you plot
myconv output.
c) Read the description of freqz command of MATLAB. Using this command, plot the 1024-point sampled
magnitude and phase response of the filter h[n]. Comment on its magnitude and phase chracteristics. Based
on your comment and the convolved sequence in part b), explain the reason of smoothing effect of this filter.
d) Now, consider the filter h2[n]=[1 -2 1]. Convolve x[n] with h2[n] and plot the result using stem command.
e) Using freqz command, plot the 1024-sampled magnitude and phase response of
the filter h2[n]. Comment on its magnitude and phase chracteristics. Based on your comment and the
convolved sequence in part d), explain the effect of this filter.
f) Generate the following sequence in MATLAB
n=0 n=0
n=0
๐‘ง[๐‘›] = {
1 + sin (
๐œ‹
3
๐‘›) + sin (
2๐œ‹
3
๐‘›), ๐‘› = 0, 1,โ€ฆ 59
0, otherwise .
Plot z[n]. Using freqz command, plot the 1024-point sampled magnitude and phase response of it. Comment
on the results.
g) Convolve the sequence z[n] with the filter h[n]. Denote the output by y[n]. Plot y[n]. Using freqz command,
plot the 1024-point sampled magnitude and phase response of it. Comment on the results.