DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERINGINDIAN INSTITUTE OF TECHNOLOGY ROORKEE

EC-202: Signals & SystemsB.Tech (ECE,CSE) & IDD (ECW,CSI) IInd Year

Spring Semester 2013 End-Term Examination Maximum Marks :100Note: Answer all questions.

1. (a) (2)Using the generalized function definition of impulse, show that

d (at) =1|a|d (t).

(b) (3)Show that causality for a continuous-time system is equivalent to the following state-ment:For any time t0 and any input x(t) such that x(t) = 0 for t < t0, the correspondingoutput must also be zero for t < t0.

(c) (3)Consider a discrete-time system with input x[n] and output y[n] related by

y[n] =n+n0

Âk=n�n0

x[k]

where n0 is a finite positive integer.i. Is the system linear?

ii. Is the system time-invariant?Justify your answers.

(d) (4)Let

y(t) = u(t +1)�u(t�1)� t (u(t)�u(t�1))= �(1/2) · x(�3t +2).

Determine and sketch x(t).

(e) (3)Determine whether the following signals are periodic, and for those which are, findthe fundamental period.

i. x[n] = •k=�• (d [n�4k]�d [n�1�4k])

ii. x[n] = cos(n8 �p)

iii. x[n] = cos(p2 n)cos(p

4 n)

2. (a) (4)Suppose that x(t) = (u(t +1)�u(t�1)) and h(t) = x(t/a), where 0 < a 1.i. Determine and sketch y(t) = x(t)⇤h(t).

ii. If dy(t)/dt contains only three discontinuities, what is the value of a?.

(b) (4)An LTI system has step response given by s(t) = e�tu(t)� e�2tu(t). Determine theoutput of this system y(t) given an input x(t) = d (t�p)� cos(

p3)u(t).

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(c) (6)Let x[n] and y[n] be two real-valued discrete-time signals. The autocorrelation func-tions fxx[n] and fyy[n] of x[n] and y[n], respectively are defined as

fxx[n] =+•

Âm=�•

x[m]x[m�n] fyy[n] =+•

Âm=�•

y[m]y[m�n]

and the cross-correlation functions are given by

fxy[n] =+•

Âm=�•

x[m]y[m�n] fyx[n] =+•

Âm=�•

y[m]x[m�n]

i. Express fxx[n] and fxy[n] in terms of convolution. Is fxy = fyx?.ii. Let x[n] be the input to an LTI system with unit sample response h[n], and let the

corresponding output be y[n]. Show how fxy[n] and fyy[n] can be viewed as theoutput of LTI systems with fxx[n] as the input. (Do this by explicitly specifyingthe impulse response of each of the two systems in terms of h[n].)

(d) (6)Determine whether the LTI systems characterized by the following impulse responsesare invertible. If yes, find the impulse response of the inverse system.

i. h(t) = d (t)+d (t +2) ii. h(t) = e�2tu(t) iii. h[n] =�1

5

�nu[n]

3. (a) (5)Consider a causal continuous-time LTI system whose input x(t) and output y(t) arerelated by the following differential equation:

ddt

y(t)+4y(t) = x(t).

Find the Fourier series representation of the output y(t) for each of the followinginputs:

i. x(t) = cos(2pt)ii. x(t) = sin(4pt)+ cos(6pt +p/4).

(b) (4)Suppose we are given the following information about a periodic signal x(t) withperiod T = 2 and Fourier series coefficients ak.

i. x(t) is real and odd ii. ak = 0 for |k|> 1 iii. 12R 2

0 |x(t)|2dt = 1.

Specify two signals that satisfy these conditions.

(c) (6)Find the Fourier transform of the following signals

i. x(t) =⇢

e j10t , |t|< p0, |t|> p

ii. x(t) = ddt {(e

�3tu(t))⇤ (e�tu(t�2))}

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(d) (5)Let x(t) be a real signal with Fourier transform X( jw). Suppose we are given thefollowing facts:

i. |t|e�|t| FT ! Re{X( jw)} ii. x(t) = 0 for t 0.

Find x(t).

4. (a) (4)A signal x(t) = 3cos(6pt)+ cos(16pt)+2cos(20pt) is sampled at a rate 25% abovethe Nyquist rate. Sketch the spectrum of the sampled signal. How would you recon-struct x(t) from these samples?

(b) (8)i. Using the duality between the discrete-time Fourier transform synthesis equationand the continuous-time Fourier series analysis equation, show that:

x[n] =sin(Wn)

pnFT ! X(e jw) =

⇢1, |w|W0, W < w p .

ii. Determine the following sum:+•

Âk=�•

sin(pk/5)sin(pk/7)k2 .

(c) (4)Using the DTFT of discrete-time rectangular pulse and the properties of DTFT, deter-mine the DTFT of the signal

x[n] = 2d [n+2]+3d [n+1]+4d [n]+3d [n�1]+2d [n�2].

(d) (4)Consider a discrete-time LTI system with impulse response

h[n] =

8<

:

1, 0 n 2�1, �2 n�10, otherwise

.

Given that the input to the system is

x[n] =+•

Âk=�•

d [n�4k],

determine the Fourier series coefficients of the output y[n].

5. (a) (3)Consider a continuous-time LTI system for which the input x(t) and output y(t) arerelated by the differential equation

d2y(t)dt2 �

dy(t)dt�2y(t) = x(t).

i. Determine H(s) as a ratio of two polynomials in s. Sketch the pole-zero patternof H(s).

ii. Determine h(t) if the system is assumed to be stable.(b) (4)Let x(t) be the sampled signal specified as

x(t) =•

Ân=0

e�nT d (t�nT ), where T > 0.

i. Determine X(s), including its region of convergence.ii. Sketch the pole-zero plot of X(s).

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(c) (3)It is desired to design a lowpass Butterworth filter with the following specifications:

i. pass band (0 |w| 10 radians/sec)ii. pass band gain(0.9 |H( jw)| 1)

iii. stop band (|w|� 20 radians/sec)iv. stop band gain (|H( jw)| 0.10)

Write down the equations to determine the order and 3 dB cut-off frequency of theButterworth filter.

(d) (2)Sketch the poles of a 6th order Butterworth analog lowpass filter with a 3 dB cut-offfrequency of 1 kHz, in the s-plane.

(e) (3)Find the inverse z-transform of

X(z) =1

1,024

"1,024� z�10

1� 12z�1

#, |z|> 0.

(f) (6)Consider an LTI system with input x[n] and output y[n] for which

y[n�1]� (17/4)y[n]+ y[n+1] = x[n].

Using Cauchy residue theorem, determine the unit sample response (impulse response)of the system if it is assumed to be stable.

(g) (4)A discrete-time system has two poles at z = 0 and z = 0.5 and a zero at z = 2. Showusing the geometrical evaluation that the magnitude response of the system is unity.

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