ece iv signals & systems [10ec44] assignment

Signals and Systems 10EC44

Dept of ECE/SJBIT Page 1

Assignment Questions

UNIT 1: INTRODUCTION

1. What are even and Odd signals 2. Find the even and odd components of the following signals

a. tcostsintsintcos)t(x

b. 432 9531)( ttttx

c. t10tcos)t1()t(x 33

3. What are periodic and A periodic signals. Explain for both continuous and discrete cases. 4. Determine whether the following signals are periodic. If they are periodic find the fundamental

period.

a. 2))t2((cos)t(x

b. )n2cos()n(x

c. n2cos)n(x

5. Define energy and power of a signal for both continuous and discrete case. 6. Which of the following are energy signals and power signals and find the power or energy of the

signal identified.

a.

otherwise0

2t1,t2

1t0,t

)t(x

b.

otherwise0

10n5,n10

5n0,n

)n(x

c.

0

5.0t5.0tcos5)t(x

d.

otherwise0

4n4,nsin)n(x

7. The raised cosine pulse is shown in fig.1 and is defined by

0

1w

1)t(cos2

1

)t(x

Determine the total energy of the signal x(t)

8. Explain with examples various operations that can be performed on dependent and independent



variables of a continuous time function and discrete time function.

9. A sinusoidal signal x(t)=3 cos )6/t200( is passed through a square law device defined by the

input output relation y(t)=x2(t).

10. A rectangular pulse x(t) is defined by

otherwise0

Tt0A)t(x

This pulse is applied to an integrator whose output is defined by

1

0

d)(x)t(y

find the total energy of the output.

11. A Trapezoidal signal as shown in fig. 2 is time scaled producing output y(t)=x(t). Sketch y(t) for a=5, a=0.2.

12. Problem no.1.14 (a,e,f,g) page 6.3, signals and systems by Simon Haykin. 13. A continuous-time signal x ( t ) is shown in Fig. 1-17. Sketch and label each of the following

signals.

( a ) x(t - 2); ( b ) x(2t); ( c ) x(t/2); (dl x ( - t )

14. A discrete-time signal x [ n ] is shown in Fig. 1-19. Sketch and label each of the following

signals.

(a) x[n 2] (b) x[2n] ( c ) x [ - n ] ( d ) x [ - n + 2]

15. Determine the even and odd components of the following signals:



16. Let x(t) be an arbitrary signal with even and odd parts denoted by xe(t) and xo(t), respectively. Show that

17. Let x[n] be an arbitrary sequence with even and odd parts denoted by xe[n] and xo[n] respectively.

Show that

18. Determine whether or not each of the following signals is periodic. If a signal is periodic, determine its fundamental period.

19. Show that if x[n] is periodic with period N, then

20. Evaluate the following integrals:

21. Consider a continuous-time system with the input-output relation



Determine whether this system is ( a ) linear, ( b ) time-invariant, ( c ) causal.

22. Consider a continuous-time system with the input-output relation

Determine whether this system is ( a ) linear, ( b ) time-invariant.

23. Give an example of a linear time-varying system such that with a periodic input the corresponding output is not periodic.

24. A system is called invertible if we can determine its input signal x uniquely by observing its output signal y. This is illustrated in Fig. 1-43. Determine if each of the following systems is invertible. If

the system is invertible, give the inverse



UNITS 2&3: TIME-DOMAIN REPRESENTATIONS FOR LTI SYSTEMS 1 & 2

1. Show that if x(n) is input of a linear time invariant system having impulse response h(n), then the output of the system due to x(n) is

k

)kn(h)k(x)n(y

2. Use the definition of convolution sum to prove the following properties 1. x(n) * [h(n)+g(n)]=x(n)*h(n)+x(n)*g(n) (Distributive Property) 2. x(n) * [h(n)*g(n)]=x(n)*h(n) *g(n) (Associative Property) 3. x(n) * h(n) =h(n) * x(n) (Commutative Property)

3. Prove that absolute summability of the impulse response is a necessary condition for stability of a discrete time system.

4. Compute the convolution y(t)= x(t)*h(t) of the following pair of signals:

5. Compute the convolution sum y[n] =x[n]* h[n] of the following pairs of sequences:

6. Show that if y (t) =x(t)* h(t), then

7. Let y[n] = x[n]* h[n]. Then show that

8. Show that

for an arbitrary starting point no.

9. The step response s ( t ) of a continuous-time LTI system is given by

Find the impulse response h(r) of the system.



10. The system shown in Fig. 2-31 is formed by connection two systems in parallel. The impulse responses of the systems are given by

(a) Find the impulse response h(t) of the overall system.

(b) Is the overall system stable?

11. Consider an integrator whose input x(t) and output y ( t ) are related by Hhgk

(a) Find the impulse response h(t) of the integrator.

( b) Is the integrator stable?

12. Consider a discrete-time LTI system with impulse response h[n] given by

Is this system memory less?

13. The impulse response of a discrete-time LTI system is given by

Let y[n] be the output of the system with the input. Find y[1] and y[4].

14. Consider a discrete-time LTI system with impulse response h[n] given by

( a ) Is the system causal?

( b ) Is the system stable?

15. Let y(t)= x(t)*h(t). Then show that



16. The input x ( t ) and the impulse response h ( t ) of a continuous time LTI system are given by

(a) Compute the output y (t) by Eq. (2.6).

(b) Compute the output y (t) by Eq. (2.10).

17. Compute the output y(t) for a continuous-time LTI system whose impulse response h (t) and the input x (t) are given by

18. Show that

19. Evaluate y (t) = x (t) * h(t), where x (t) and h (t) are shown in Fig. 2-6 (a) by analytical technique,

and (b) by a graphical method.



20. Consider a continuous-time LTI system described by

(a) Find and sketch the impulse response h(t) of the system.

(b) Is this system causal?

21. Let y (t) be the output of a continuous-time LTI system with input x(t) . Find the output of the system if the input is x

l(t) , where x

l (t) is the first derivative of x(t) .

22. Verify the BIBO stability condition for continuous-time LTI systems. 23. Consider a stable continuous-time LTI system with impulse response h ( t ) that is real and even. Show

that cos wt and sin wt are eigenfunctions of this system with the same real eigenvalue.

24. The continuous-time system shown in Fig. 2-19 consists of two integrators and two scalar multipliers. Write a differential equation that relates the output y(t) and theinput x( t ).



UNITS 4 & 5: FOURIER REPRESENTATION FOR SIGNALS 1 & 2

1. A continuous time periodic signal x(t) is real valued and has a fundamental period T=8, the non-

zero Fourier series coefficients for x(t) are j4aa,2aa 3311

Express x(t) in form

0kkkk )1w(cosA)t(x

2. For a continuous time signal (periodic)

)t3/5(sin4)t3/2cos(2)t(X determine the fundamental frequency 0w and Fourier

series coefficients ka such that

k

jkwk 0ea)t(x

3. Consider the following three continuous time periodic signals whose Fourier series representation are as follows:

100

0k

jkk

1 t50

2e.

2

1)1(x

100

100k

t50

2jk

2 e.)k(cos)t(x

100

100k

t50

2jk

3 e.2

ksinj)t(x

Using Fourier Series properties

i. Which of the three signals is / are real valued ii. Which of the three signals is / are even

4. State and prove the properties of continuous time Fourier Series 5. Obtain an expression to express a continuous time periodic signal in Fourier series representation

in (i) Trigonometric form (ii) Exponential form

6. State and prove parsevels relation for continuous time periodic signals 7. Obtain an expression for Fourier series representation of a discrete time periodic signals in



complex exponential form.

8. What is Gibbs phenomenon? State and prove the properties of discrete time Fourier series. 9. Obtain Fourier transform of

i. 0afor)t(u.e)t(x )t(a

ii. 0afor,e)t(x ta

iii. |||)t()t(x

10. Consider a rectangular pulse

1

1

T|t|,0

T|t|,1)t(x

Obtain Fourier transform of x(t)

11. Obtain the analysis and synthesis equation of Fourier Transform of n periodic signal.

12. Obtain x(t) if x(jw)

w|w|for0

w|w|for1

13. Obtain an expression for Fourier transform of a periodic signal 14. State and prove properties of continuos time Fourier transform. 15. A Discrete time periodic signal x(n) is real value and has fundamental period N=5. The non-zero

discrete time Fourier series coefficients. For x(n) are 3/j

444/j

220 e2aaeaa,1A

Express x(n) in the form

1kkkk0 )nw(sinAA)n(x



UNIT 6: APPLICATIONS OF FOURIER REPRESENTATIONS

1. The output of a system in response to an input )t(u.e)t(x t2 is )t(u.e)t(y t . Find the

frequency response and impulse response of this system.

2. Find the frequency response and impulse response of a system described by the differential equation

)t(x)t(xdt

d2)t(y2)t(y

dt

d3)t(y

dt

d

2

2

3. Find the frequency response and impulse response of a discrete time system described by the difference equation y(n-2)+5y(n-1)+6y(n)=8x(n-1)+18x(n)

4. Find the Fourier transform of impulse train given by

n

)nTt()t(p

T is fundamental period

5. Obtain an expression that relates discrete time Fourier transform and Discrete time Fourier series.

6. Find both DTFS & DTFT for the signal. )n2/(sin4)3/n8/3(cos2)n(x

7. Show that x(n) * h(n) )e(H)e(x jnjn where x(n) are the input and impulse response of a LTI

systems and )e(x jn and )e(H jn are their respective DFTF

8. If )n(u)2/j()n(h n is impulse response of a LTI system, then obtain the response of this

system to the input )3/n(cos3)n(x

9. Consider x(n)=cos )n16/7( )n16/9(cos using modulation property. Evaluate the effect of

computing the DFTF using only 2M+1 values of x(n) where |n| M

10. Consider x(t) =cos t . Assume that this signal is sampled at internals T=1/4, T=1 & T=1/2. Find the Fourier Transform of the sampled data for all the three cases.

11. Determine Fourier Transform pair associated with the signal whose DTFT is

jn

jn

e1

1)(x



UNITS 7&8: Z-Transforms 1&2

1. State and prove the properties of unilateral Z transform & ROC 2. Find Z transform of

)n(u)2/1()1n(u)n(X n

3. Determine Z transform, ROC pole and zero location of x(t) for

i. )n(u)3/1()n(u)2/1()n(x nn

ii. )n(ue)n(xn0

j

4. Find Z transform of X(n)={n(-1/2)

n x(n)} * (1/4)

-n u(n)

X(n)=an cos (0n)

5. Find inverse Z transform of

1z21

2

1z2

11

1)z(x

Assume a. signal is casual b. Signal has DTFT

4Z2Z2

4Z4Z10Z)z(x

2

23

with ROC |z| 1

6. Find the inverse Z transform of

)ze9.01()ze9.01(

Z1)z(H

14/j14/j

1

7. A system is described by the difference equation

)2n(x8/1)1n(x4/1)n(x)2n(y4

1)1yn)n(Y

Find the Transfer function of the Inverse system

Does a stable and causal Inverse system exists

8. Sketch the magnitude response for the system having transfer functions.



9. Find the z-transform of the following x[n]:

10. Given

(a) State all the possible regions of convergence.

(b) For which ROC is X (z) the z-transform of a causal sequence?

11. Show the following properties for the z-transform.

12. Derive the following transform pairs:

13. Find the z-transforms of the following x[n]:

14. Using the relation

find the z-transform of the following x[n]:



15. Using the z-transform

16. Find the inverse z-transform of X(z)= ea/z , z > 0 17. Using the method of long division, find the inverse z-transform of the following X ( z ) :

18. Consider the system shown in Fig. 4-9. Find the system function H ( z ) and its impulse response

h[n]

19. Consider a discrete-time LTI system whose system function H(z) is given by

(a) Find the step response s[n].

(b) Find the output y[n] to the input x[n] = nu[n].

20. Consider a causal discrete-time system whose output y[n] and input x[n] are related by

(a) Find its system function H(z).

(b) Find its impulse response h[n].

ece iv signals & systems [10ec44] assignment

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