## Description

1) A discrete-time signal ๐ฅ[๐] is obtained by sampling a continuous-time sinusoidal signal

๐ฅ๐

(๐ก) = 4 sin(20000๐๐ก +

๐

13)

at a sampling rate of 3 kHz.

a) Describe the set of all other continuous-time sinusoidal signals (their frequencies) that yield ๐ฅ[๐] when

sampled at 3 kHz.

b) Describe the set of all other sampling frequencies that yield ๐ฅ[๐] from ๐ฅ๐

(๐ก).

2) Among the following sinusoidal signals, identify periodic ones and find their fundamental periods.

sin(1.74๐๐ + 3.1) , sin(1.74๐๐ + 3.1๐), cos (15.74๐๐ +

3๐

8

), cos(โ๐๐) , cos(๐โ๐๐), cos(๐โ2 ๐)

3) A linear sistem whose response to a shifted impulse, ๐ฟ[๐ โ ๐], is โ๐

[๐] = (๐ โ ๐)๐ข[๐ โ ๐]. Find the output, ๐ฆ[๐]

of this system for an arbitrary input, ๐ฅ[๐]. Is this system time-invariant? Explain clearly by using ๐ฆ[๐ โ ๐] and

๐ฅ[๐ โ ๐].

4) What does the following system do? Is it linear, time-invariant? Prove your answer.

๐ฆ[๐] =

{

๐ฅ [

๐

2

] ๐๐ ๐ ๐๐ ๐๐ฃ๐๐

๐ฅ [

๐ โ 1

2

] + ๐ฅ [

๐ + 1

2

]

2

๐๐ ๐ ๐๐ ๐๐๐

5) Are the following systems causal, stable? Justify/discuss/prove.

๐ฆ[๐] = 2

๐ฟ[๐+1] + ๐ฅ[๐ โ 3]

๐ฆ[๐] = {

๐ฆ[โ๐ฟ[๐ โ 1]] + ๐ฅ[๐ โ 3] ๐ > 0

2

๐๐ฅ[๐ โ 3] ๐ โค 0

6) Let ๐ฆ[๐] = ๐ฅ[๐] โ ๐ฅ[๐] where โ is the convolution operator. If the first and the last non-zero elements of ๐ฅ[๐] are

equal to ๐ฅ[โ6] = โ3 and ๐ฅ[24] = โ4, respectively, find the first and the last non-zero elements of ๐ฆ[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

๏ท

๏ธ

๏ถ

๏ง

๏จ

๏ฆ

๏ท ๏ญ

๏ธ

๏ถ

๏ง

๏จ

๏ฆ

๏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[๐].

b) Find the frequency response, ๐ป(๐

๐๐).

c) Plot the magnitude response, |๐ป(๐

๐๐)| and phase response, โก๐ป(๐

๐๐), in MATLAB using โfreqzโ command.

d) Let ๐ฅ[๐] = cos (

๐

3

๐) + sin (

๐

2

๐ +

๐

4

) be the system input. Find the output ๐ฆ[๐].

e) Prove that ๐ป(๐

๐๐) = ๐ปโ

(๐

๐(2๐โ๐)

). Does this equality hold for an arbitrary โ[๐]? Explain.

9) In the [โ๐, ๐) interval, the DTFT, ๐(๐

๐๐), of a real sequence ๐ฅ[๐] is nonzero only in [๐, ๐],๐ < 0 < ๐, ๐ โ ๐ =

๐

2

;

otherwise it is arbitrary.

a) Plot the magnitude and phase of a typical ๐(๐

๐๐) for โโ < ๐ < โ.

b) Express the DTFTs, ๐๐ถ(๐

๐๐) and ๐๐(๐

๐๐), respectively, of cos (

๐

5

๐)๐ฅ[๐] and sin (

๐

5

๐)๐ฅ[๐] in terms of

๐(๐

๐๐).

c) Assume that ๐(๐

๐๐) is real valued and still complies with the specifications above. Plot typical magnitudes

and phases for ๐(๐

๐๐), ๐๐ถ(๐

๐๐) and ๐๐(๐

๐๐).

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

๐ง[๐] = {

1 + sin (

๐

3

๐) + sin (

2๐

3

๐), ๐ = 0, 1,โฆ 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.