Guru Gobind Singh Indraprastha Vishwavidyalaya 2007-3rd Sem B.E Electronics & Communication Engineering SIGNAL & SYSTEM -- Question Paper
END TERM exam
THIRD SEMESTER[B.TECH] - DECEMBER-2007
SIGNAL & SYSTEM
(Please write your Exam Roll No.) Exam Roll No.
Tij mu Semester |B.Tccn|- Decf.mukk-2007
Paper Code:ETEC203 (Batch-2004-2006) Subjcct: Signal and Sysiams Paper ID:28203 ___
Time : 3 Hours Maximum Marks : 75
Note: Attempt all questions. Question no. 1 is compulsory. Internal choice is indicated._|
Q. 1 (a) If x,(l) and x2{t) :.jre periodic signals with time periods T, and T2 respectively, then under what conditions is the sum x(t) = Xi(t) + x2(t) periodic and what is the fundamental time period of x(t) if it is periodic. (3)
$ (b) The impulse response of a disc.ete time LT1 system is given by h[n] = an u[n].
Compute the unit step response o.: the system. (3)
(c) What are the dicrete-time Fourier series coefficients of the sequence x[n] = sin 271
L
> n. (3)
9
(d) Show that Jsinc(x)dx = I
and Jsinc (.<) dx = 1 (4)
(e) If x[n] is add, theii show that its z-transfomn x[z) has a zero at z = 1. (3)
(f) Le! x(s) and R be the Laplace transform and ROC of x(t). What wili be the Laplace transform and RCC of y(t) = -t x(t). (3)
(g) A continuous time LTI system which is also stable and causa! is described by the
differential equation. -+5y(t) = 24x(t), where y(t) is the output when the input dt
is x(l). What is th-3 final value s (eo) of the step response s(t) of this system. (3)
(h) Lei x(t) be a low pass signal wiiii band wiuti WHz. Compute the NyquiGi rste for
(i) x(t) + X(t-1) (ii) x2(t) (3)
Q.2 (a) The output y [n] and input x [n] ot a system are related as y[n] = 2k' x[k + I] (6.5)
(i) Find out whether the system is linear, time invariant, stable and causal.
(ii) Find the impulse respon.se of the system.
(b) Compute the um!l step response of an LT1 system whose impulse response is given
be"t0 = ic-' li0- <6)
OR
(c) The output y(i) and input x d) of a continuous lime system are related as
1 r*T-
>V) - J_r :X(zUix (6.5)
(i) Is this system linear, lim* invariant, causal and stable?
(ii) Find and sketch the impulse response of the system.
(d) Lsl x|_n] du h discrete tirrvj signal, and let y,[n] = x[2n] and y2[n] =
Jx[n/2] n is L-. cn.
(6J
Determine whether the following statements are true or false. Give reasons.
(i) If x[n] is periodic, ynj is also periodic.
(ii) If y2(n) is Deriodic, x[n) is also periodic.
(iii) If ynj is periodic, x[nj is also periodic.
Q.3 (a) Let p(t) be a signal such that pf!) = 0 for |t| > T/4. Define (he periodic signal x (t) =
a;
p(t - nT). Determine the Fourier series coefficient of x(t) in terms of the Fourier
n -
t-2-] -QQ J
(b) For an aperiodic discrete time sicnal x[n], show that, |x[n] |Z = ~ J]x(jw)] dw ,
R-OD 2H
where x(w) is ihe r.T transform o; x [n], (6)
OR
(c) A signal x[nj is periodic with time period N, and discrete time Fourier coefficients a*
satisfy the following: - (6.5)
(i) x [n] is red (ii) a, = 1 :
N-l - N-l
(iii) Vlx [n] | = 3N (iv) V x(n) - N find x [n]
n-Q j\-Q
(d) Let the Fourier 'ransform of n(t) by x(w). Find out the Fourier transform of the following signals in terms of x(w). (6) (i) yi(t) = Re [>(()] (ii) y(t) = x(i-t) + x (-1-t)
(iii) y?{t) = even part of { x{t) - c:;; wot)
Cl.4 (a) Let x,(w) and x2(wi bt the Fourier transforms of x,(t) and xz(t) respectively.
X!(W) = 0 |W|>W;
xz{w) = 0 |w| a w2
Compute the Nyquist sampling rale of y (t) = x,(t).x2(t) and p(t) = x,(t) x2(t) ~>convolulion. (6-5)
exp(-t) t > 0
(b) The impulse response of a LT1 system is defined by h(t) = < 1/2 t = 0.
0 t<0
Determine Ihe pilose delay and group delay of the system. Interpret the results (6)
OR
(c) Determine wheih;' each of the following statements is true or false. (6.5)
(i) The signal x(t) = u(t +T0) - u (t - T0) can undergo impulse train sampling without aiiasing if the sampling period T < 2 T0.
(ii) The signal x(t) with Fourier transform. X(jw) = u(w - w0) - u(w-w0) can undergo impulse train sampling without aliasing if the sampling period. T <
(iii) The signal x(t) with Fourier transform x (jw) = u(w) - u (w - w0) can undergo impulse train sampling without aliasing provided that the sampling period T <
2ti/wo.
(d) Which of the following systems is a linear phase system. Justify your answers. (6)
(i) H(w) = -- (ii) H(w) = 1
1 -f Jw (1 + jw)2
(iii) H(w) =-------
(1 + jw) (2 + jw)
Q.5 (a) The following information about a discrete time signal n(n) is given: - (6-5)
(i) x [n] is real and right sided. (ii) x [z] has exactly two poles.
1 ~
(iii) x [z] has tv/c zeroes at the orig-n. (iv) x [z] has a pole ai z = e 3
(v) \[1] = 8/3
Determine x(z) and specify it's ROC.
(b) Let Laplace transform of x(t) be x-s). What are the properties of the TOC of x(s) is
x(s) is a rational function of's. (6)
OR
(c) Determine the forced and natural response of an LT1 system described by the following differentia) equation and initial conditions. (6.5)
-f y(iHlOy(t) = !Ox(t) y(0 ) = 1 dt
y(t) is o/p and x(i) is input = u(t).
(d) A signal x[n] Is evL-n and has a rational transfer function (6)
(i) What constraints must the poles of such a signal satisfy?
'-(17/4)2''
(ii) If x{z; = r--T77T7---. Oetermine the ROC and find x[n).
II
|
Attachment: |
| Earning: Approval pending. |
