## Description

1) Impulse responses of some LTI systems are given below. Let π₯[π] = 3

π be the input signal of these

systems. Determine those systems for which their output signals can be expressed as π¦[π] = πΆ 3

π where

C is a complex constant. Explain formally.

a) β[π] = (

1

2

)

π

π’[π] + 2

ππ’[π]

b) β[π] = (

1

2

)

π

π’[π] + 2

ππ’[βπ]

c) β[π] = 5

ππ’[βπ]

d) β[π] = 3

ππ’[π]

2) Find the impulse responses of the stable LTI systems having the following system functions. Which of

them are causal? Plot the pole-zero diagrams and show their ROCs.

a) π»(π§) =

2π§

β1+1

(1β

1

4

π§β1)(1+

2

3

π§β1)(1+

1

4

π§β1)

b) π»(π§) =

π§β4

(1β3π§β1)(1β5π§β1)

c) π»(π§) =

π§

β1

(1β

1

4

π§β1)(1β

1

3

π§β1)

d) π»(π§) =

1

(1β

1

2

π§β1)

3

3) The impulse response of a LTI system is

β[π] = πΏ[π] β β2πΏ[π β 1] + πΏ[π β 2].

a) Find the system function π»(π§). Plot the pole-zero diagram, indicate ALL poles and zeros, show

the ROC.

b) Does this system have a frequency response? Why? If yes, plot its magnitude and phase.

c) Find the output of this system to the following input signals

π₯1

[π] = sin (

π

4

π +

π

4

)

π₯2

[π] = sin (

π

4

π +

π

4

) π’[π]

π₯3

[π] = sin (

π

4

π +

π

4

) + sin (

3π

4

π)

d) Comment on the relationship between the frequency response and zero locations of π»(π§).

4) The system function of a LTI system is

π»(π§) =

1 β β2π§

β1 + π§

β2

1 β 2π§

β1

.

When the input is sin (

π

2

π), the output of this system is β

2

5

sin (

π

2

π + tanβ1

1

2

).

a) Find the impulse response of this system.

b) Is the system causal?

c) Find the difference equation for this system.

6) The z-transform, π(π§), of a right βsided sequence π₯[π] exists for π§ = 4π

ππ, 0 β€ π < 2π. Show that

π(π§) exists for = 4.1π

ππ, 0 β€ π < 2π , but not necessarily for π§ = 3.9π

ππ, 0 β€ π < 2π .

7) What are the ROCs of the z-transforms of the following sequences?

a) π₯[π] = πΏ[π + 3] + πΏ[π β 3]

b) π₯[π] = πΏ[π + 3]

c) π₯[π] = πΏ[π β 3]

8) Let π₯[π] = πΏ[π + 1] + (

1

2

)

π

π’[π]. Find the z-transforms of the following sequences. What are the

ROCs? State all poles and zeros.

a) π₯[π]

b) π₯[π β 5]

c) ππ₯[π]

d) cos (

π

2

π) π₯[π]

9) The pole-zero plot of the system function, π»(π§), of a stable LTI system is

shown. It is known that π»(1) = 1.

a. Show the ROC. Determine the impulse response β[π].

b. Let β1

[π] = β[βπ + 2]. Sketch the pole-zero plot for π»1

(π§) show

its ROC.

Re

Im

-1/2 1/2 3

1

10) The output of a stable LTI system is π¦[π] = πΏ[π + 1] + 2πΏ[π] + πΏ[π β 1] when its input is π₯[π] =

β2πΏ[π + 2] β 4πΏ[π + 1] + 4πΏ[π β 1] + 2πΏ[π β 2]. Find its impulse response β[π].

11) Problem 3.30 of textbook.

12) Problem 3.52 of textbook.

13) Problem 3.58 of textbook.

14) Let π₯[π] = πΏ[π] + 3πΏ[π β 1] + πΏ[π β 2].

a. Plot π₯[π] and its periodic extension, π₯Μ[π], for π = 3 and π = 5.

π₯Μπ[π] = β π₯[π β ππ]

β

π=ββ

= π₯ [((π))

π

]

b. Find the Discrete Fourier Series (DFS) coefficients, πΜ

3

[π], of π₯Μ3

[π]. Write the DFS

representation of π₯Μ3

[π].

c. Find the Discrete Fourier Series (DFS) coefficients, πΜ

5

[π], of π₯Μ5

[π]. Write the DFS

representation of π₯Μ5

[π].

d. Find the DTFT, π(π

ππ), of π₯[π]. Plot its magnitude and phase.

e. Verify that πΜ

3

[π] = π(π

ππ)|

π=π

2π

3

and πΜ

5

[π] = π(π

ππ)|

π=π

2π

5

, i.e., uniformly spaced

samples of DTFT of π₯[π]. Show these samples on the magnitude and phase plot of π(π

ππ).

f. Compute the sample values of π₯Μ3

[π] and π₯Μ5

[π] using their DFS representations and their

DFS coefficients you found in parts (b) and (c), respectively.

15)

a. Find the 3-point and 5-point DFTs (π3

[π] and π5

[π]) of π₯[π] given in Question-14.

b. What is the relationship between π3

[π] and πΜ

3

[π], and π5

[π] and πΜ

5

[π]?

c. How would you find π₯[π] using its 3-point and 5-point DFTs?

d. Find π3

[π] and π5

[π] using MATLAB.

16) Let π¦[π] = πΏ[π β 2] + 3πΏ[π β 3] + πΏ[π β 4] and π§[π] = 3πΏ[π] + πΏ[π β 1] + πΏ[π β 4]

a. Plot π¦[π] and π§[π].

b. Relate π¦[π] and π§[π] to π₯[π] of Question-14

c. Find the 5-point DFTs, π5

[π] and π5

[π], of π¦[π] and π§[π]. Do they have 3-point DFTs? Why?

17) Let πΜ [π] = π(π

ππ)|

π=π

2π

2

, i.e., two (uniformly spaced) samples from each period of π(π

ππ), DTFT

of π₯[π] in Question-14.

a. Find the periodic sequence π€Μ[π] whose DFS coefficients are πΜ [π].

b. Find the relationship between π€Μ[π] and π₯[π].

18) (Generalization of the result in Question-17) Let π₯[π] be an arbitrary sequence with a DTFT

π(π

ππ)and

πΜ [π] = π(π

ππ)|

π=π

2π

π

i.e., π (uniformly spaced) samples from each period of π(π

ππ). Also let π€Μ[π] be the periodic

sequence whose DFS coefficients are πΜ [π].

a) Show that

π€Μ[π] = β π₯[π β ππ]

β

π=ββ

b) Assuming that π₯[π] has finite length π. Comment on the cases π β₯ π and π < π.

c) Verification using MATLAB. You may use the following code to verify for different values of

π and π.

clear all

close all

N = 10;

n = 0:(N-1);

x = 1:N;

M = 3;

% M = 5;

% M = 7;

% M = 10;

% M = 15;

W_M = exp(-j*2*pi/M);

F = W_M .^ n ;

for k = 0:(M-1)

DFT_matrix(k+1,:) = F.^k;

end

Z = DFT_matrix * x’;

z = ifft(Z)

19) Let π₯[π] = 3πΏ[π] β 2πΏ[π β 1] + πΏ[π β 2] + πΏ[π β 3] β 2πΏ[π β 4] β πΏ[π β 5].

You do not need to compute any DFTs in parts (a)-(c)!

a) Let π3

[π] = π(π

ππ)|π=π

2π

3

, π = 0,1,2. Find the sequence π€3

[π] whose 3-point DFT is π3

[π].

b) Let π5

[π] = π(π

ππ)|π=π

2π

5

, π = 0,1,2,3,4. Find the sequence π€5

[π] whose 5-point DFT is

π5

[π].

c) Let π8

[π] = π(π

ππ)|π=π

2π

8

, π = 0,1,2,3,4,5,6,7. Find the sequence π€8

[π] whose 8-point DFT is

π8

[π].

Let β[π] = 2πΏ[π] β 1πΏ[π β 1] be the impulse response of a LTI system.

You do not need to compute any DFTs in parts (d)-(f)!

d) Let π»3

[π] be the 3-point DFT of β[π]. Find the sequence π¦3

[π] whose 3-point DFT is π3

[π]π»3

[π].

e) Let π»5

[π] be the 5-point DFT of β[π]. Find the sequence π¦5

[π] whose 5-point DFT is π5

[π]π»5

[π].

f) Let π»8

[π] be the 8-point DFT of β[π]. Find the sequence π¦8

[π] whose 8-point DFT is

π8

[π]π»8

[π].

g) Describe the relationships between the sequence π¦[π] = π₯[π] β β[π] and the sequences

π¦3

[π], π¦5

[π], π¦8

[π].

20) Let π₯[π] be a sequence of length π; π is even. Let π[π] be its π-point DFT.

a) Show that π[π] can be written as

π[π] = πΈ [((π))π

2

] + π

βππ2π

π π [((π))π

2

] π = 0,1, β¦ , π β 1

where πΈ[π] and π[π] are the π

2

-point DFTs of π[π] = π₯[2π] and π[π] = π₯[2π + 1], respectively.

b) Assume that π₯[π] is real.

i. Count the number of real multiplications and real additions in the direct computation of π[π].

ii. Count the number of real multiplications and real additions in the computation of [π]

according to the right hand side of the above expression.

iii. Compare the numbers of arithmetic operations in these two cases.

21) Let β[π] = 2πΏ[π] β πΏ[π β 1] + πΏ[π β 2] be the impulse response of a LTI system and

π₯[π] = [1 2 3 4 β 1 β 2 β 3 β 4 1 2 3 4] 0 β€ π < 11

be an input to this system. The output π¦[π] will be found by using the overlap-add method. Take

the length, πΏ, of the input segments as πΏ = 4.

a) How many point DFTs will be used in this computation?

b) How many input segments are there? Write all of them.

c) Find the response of the system to the individual input segments (use MATLAB; find DFTs,

multiply them and then take inverse DFT).

d) Obtain the whole output sequence by using the responses to input segments.

22) Overlap-save method will be used for the setting in Question-16.

Take the length, πΏ, of the input segments as πΏ = 6. Use 7-point DFTs. Answer parts (b)-(d).

23) π₯[π] is a finite length sequence defined to be zero for π < 0 and π > π β 1. DTFT of π₯[π] is π(π

ππ).

a) π1

[π] is obtained by taking πΏ , (π

2

< πΏ < π), uniform samples of π(π

ππ) in 0 β€ π < 2. π¦1

[π] is

the result of πΏ-point IDFT of π1

[π]. Find the indices of samples of π¦1

[π] such that π¦1

[π] = π₯[π].

b) Let π₯[π] = [1 2 β 1 β 2 3], 0 β€ π β€ 4 and π2

[π] = π

βπ

4π

5 π[π] where π[π] is the 5-point DFT

of π₯[π]. Find the 5-point IDFT of π2

[π].

c) Two finite length sequences π₯1

[π] = [1 β 1 β 2 3 2 1] , 0 β€ π β€ 5, and π₯2[π] = [2 1 3 4 β

1 2] , 0 β€ π β€ 5 are given. Find the 6-point circular convolution of these sequences.

d) Let π₯[π] = [3 β 1 4 1 2 β 1 β 3], 0 β€ π β€ 6. Find the periodic even part of this sequence,

π₯ππ[π]. Also find the 7-point DFT of π₯ππ[π].

24) Let π₯[π] = [0 1 3 β 2 2 4], 0 β€ π β€ 5, and π[π] be its 10-point DFT.

Answer the following questions by using the DFT properties.

a) Find the sequence whose 10-point DFT is π

2

[π].

b) Find the sequence whose 10-point DFT is π

ππππ[π].

c) Find the sequence whose 5-point DFT is π[π] = π[2π], π = 0,1, β¦ ,4.

25) Let π£[π] be a real sequence of length 2π with a 2π-point DFT π[π]. π1

[π] and π2

[π] are obtained

from the even and odd indexed sequences of π£[π] as,

π1

[π] = π£[2π], π2

[π] = π£[2π + 1], 0 β€ π β€ π β 1

with π-point DFTβs πΊ1

[π] and πΊ2

[π] respectively. Define a new complex sequence

π₯[π] = π1

[π] + π π2

[π], 0 β€ π β€ π β 1 .

a) Find π

β

[( (βπ))

π

], in terms of πΊ1

[π] and πΊ2

[π].

b) Determine πΊ1

[π] and πΊ2

[π], in terms of π[π].

c) Determine the 2π-point DFT π[π] in terms of π-point DFTs πΊ1

[π] and πΊ1

[π].

26) π₯[π] is a finite length sequence, nonzero only for 0 β€ π β€ 5 and β[π] is a sequence nonzero only

for 0 β€ π β€ 4. Let π6

[π] = π6

[π]π»6

[π] where π6

[π] and π»6

[π] are the 6-point DFTs of π₯[π] and β[π],

respectively. 6-point IDFT of π6

[π], π¦6

[π], is

π¦6

[π] = [β4 β 6 β 5 1 14 0 ] 0 β€ π β€ 7.

Similarly, π¦8

[π] is the 8-point IDFT of π8

[π] = π8

[π]π»8

[π] where π8

[π] and π»8

[π] are the 8-point DFTs

of π₯[π] and β[π], respectively. π¦8[π] is

π¦8

[π] = [1 7 0 β 3 β 1 14 0 β 7 β 11] 0 β€ π β€ 7

Find the sample values of the sequence, π¦[π] = π₯[π] β β[π], which is the linear convolution of π₯[π] and

β[π].

27) The Fourier series coefficients of a periodic sequence π₯Μ[π] are

πΜ[π] = [6 4 β 2 4] for π = 0,1,2,3

a) Find π₯Μ[π].

b) Find the sequence π₯[π] whose whose DFT is π[π] = {

πΜ[π] π = 0,1,2,3

0 π. π€.

c) Find the sequence π¦[π] whose DFT is

π[π] = {

πΜ[π β 2] π = 0,1,2,3

0 π. π€.

d) Find the sequence π€[π] whose DFT is

π[π] = {

π [((π + 2))

4

] π = 0,1,2,3

0 π. π€.

28) Let π₯[0] = π, π₯[1] = π, π₯[2] = π, π₯[3] = π and π₯[π] = 0 for π < 0 and π > 3. Let

π¦[π] = π(π

ππ)|

π=π

2π

3

π = 0,1,2

Find the 3-point DFT of π¦[π] in terms of {π, π, π, π}

29) Let π΄π[π] denote π-point DFT of a sequence π[π].

Let π[π] β πΌπ·πΉπ{π4

[π]π»4[π]} and π [π] β πΌπ·πΉπ{π5

[π]π»5[π]}

π[π] = {

[1 3 0 β 2] π = 0,1,2,3

0 π. π€.

π [π] = {

[6 0 β 1 β 2 β 1] π = 0,1,2,3,4

0 π. π€.

Find the output of the system with impulse response β[π] when π₯[π] is its input.

30) π₯[π] and β[π] are 4-point sequences. Their linear convolution is π¦[π], i.e. π¦[π] = π₯[π] β β[π].

π¦[π] is given as π¦[π] = [β2 1 5 1 β 2 β 2 1] for π = 0,1, β¦ 6.

Let π§[π] = π₯ [((π β 2))

4

].

Find the 4-point circular convolution of π§[π] and β[π].

31) Let π₯[π] be a π-point real sequence (π₯[π] = 0, π β {0,1, β¦ , π β 1}).

a) What is the periodic conjugate symmetric part, π₯ππ[π], (periodic even part in this case) of π₯[π]?

Write its definition.

Let π₯[π] = πΏ[π] β πΏ[π β 1] + 2πΏ[π β 2] for the following parts.

b) Find and plot π₯ππ[π].

c) Express the DFT, πΜ[π], of π₯ππ[π] in terms of π[π], the DFT of π₯[π]. Also,

i. Find πΜ[π] of your π₯[π] in part (a).

ii. Plot πΜ[π].

d) Plot the periodic conjugate anti-symmetric part, πππ[π], of π[π].

e) Find the sequence, π₯ΜΏ[π] whose DFT is πππ[π].

32) Let π₯[π] = 3πΏ[π] + 2πΏ[π β 1] + πΏ[π β 2] + 2πΏ[π β 3]

a) Plot π₯[π].

b) Find and plot the DFT, π[π], (magnitude and phase) of π₯[π].

c) Find the sequence, π¦[π], whose DFT is π₯[π] = 3πΏ[π] + 2πΏ[π β 1] + πΏ[π β 2] + 2πΏ[π β 3].

d) You are given a software that computes π-point DFTs of complex sequences. You are not

allowed to modify it. How can you use it to compute inverse-DFTs. Clearly describe any preand/or post-processing needed.

33) Which of the following sequencesβ DFTs can be expressed as π΄[π]π

ππΌπ where π΄[π] is real or

imaginary, and πΌ is a constant. For those that can be expressed as such find π΄[π] and πΌ. (DFT sizes are

equal to the lengths of the sequences.)

a)

b)

3

2

-3

-2

1 2 3 π

4

2

1 2 3 π

c)

34)

a) Determine the polynomial result of (1+3z-1

-4z-2

)( -1+2z-1

-3z-2 +z-3+7z-4

) using βconvβ command in

MATLAB.

b) Let us consider

π(π§) =

1β3π§

β1+4π§

β2

1βπ§β1+π§β2βπ§β3

Using βresiduezβ command, determine the inverse z transform of π(π§).

c) Let us consider

π(π§) =

1β0.2π§

β1β1.2π§

β2

1β0.9π§β1+0.81π§β2

.

Using βzplaneβ command, plot the pole-zero plot of π(π§). Plot also the magnitude and phase

characteristics of it using βfreqzβ command. Comment on the relationship between the frequency

response and zero & pole locations of π(π§).

35)

a) Write a MATLAB function named βmydftβ which computes the N-point DFT of a given sequence.

Find the 9-point DFT of x[n]=[ 1 2 3 4 5 6 7] using this function. Plot the result.

b) Find the 9-point DFT of x[n] in part a) using βfftβ function. Verify your function comparing the

results of part a) and part b).

c) Consider π₯[π] = 2cos (

π

13

π β

π

8

) + 5 sin (

π

5

π +

π

2

) .

Obtain 130-point DFT of π₯[π] for π = 0, β¦ ,129. Plot the magnitude and phase of DFT and

explain your observations. At which frequency bins the peaks of the magnitude of DFT occur?

What are the magnitudes at these frequency bins? What are the phases of DFT at these

frequency bins? What are the magnitudes at the other frequency bins? Comment on your

observations.

4

2

1 2 3 4 5 6 π