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USN 06EC44

Fourth Semester B.E. Degree Examination, December 2010

Signals and SystemsTime: 3 hrs. Max. Marks: 100

1

Note: Answer any FIVE full questions, selecting

at least TWO questions from each part.PART-A

a. Distinguish between: i) Periodic and non-periodic signals and ii)

random signals.b. A signal x(t) is as shown in figure Ql (b). Find its even and odd parts.

't-l~J- "0

--~----2 -\ I'" 1- 3 \::-"'7

Detenninistic and

(04Marks)

(06Marks)

Fig. Ql (b)

c. Two signals x(t) and get)are as shown in figure Ql (c). Express the signals x(t) in tenns of

~. ~~~

<c,q;~9(!.

~~J--l-- I 'L-

D I 'J- d r:--7

2

Fig. Ql (c) - (i) Fig. Ql (c) - (ii)

d. A system is described by yen)=(n + l)x(n). Test the system for (i) memory less

(ii) Causality (iii) Linearity (iv) Timeinvarianceand (v) Stability. (04Marks)

a. An LTI system has impulse response hen) =[U(n) - U(n - 4)]. Find the output of the

systemif the input x(n) =[U(n + 10)- 2U(n + 5)+ U(n - 6)]. Sketch the output. (08Marks)

b. Showthat an aribitrarysignalx(n) can be expressedas a sumofweightedandtimeshifted00

impulses, x(n) = L x(k)8(n - k).K=-oo

(04Marks)

c. An LTI system is described by an impulse response h(t) =[U(t-1) - U(t - 2)]. Find the

outputof the system if the input x(t) is as shown in figure Q2 (c). Sketchthe output.(08Marks)

-it+- < ~

Fig. Q2 (c)3 a. Two LTI systems with impulse responses hl(n) and h2(n) are connected in cascade. Derive

the expression for the impulse response if the two systems are replaced by a single system.. (04Marks)

b. An LTI system has its impulse response, h(n) = 4-n U(2 - n). Detennine whether the system

is memory less, stable and causal. (04Marks)

c. A systemis described by a differential equation,

d2y~) +3dyet) +2y( t) = x( t) + dx(t). Detennine its forced response if the inputdt dt dt

x(t) =[cost + sin t]U(t) (06Marks)

10f2

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d. Draw the direct form I and direct form II:)mplementations for the following difference

equation,yen)+!y(n -I)-y(n - 3)=3x(n -1) +2x(n - 2). (06Marks)2

a. State and,prove.convolution property of continuous time Fourier series.

b. Find the DTFS co-efficients of the signal shown in figure.Q4 (b),-J~t~. ~\'1

.

t.

~.

.' -

~) ,I. 1

3

4

'It ""-"

06EC44

(06Marks)

(08Marks)

Fig. Q4 (b)

c. Determine the time domain signal x(t), whose Fourier co-efficient are,

3 -JX 3 JX

X(K)= -e 4, K = -1 ; X(K) =-e 4, K = 1 ; X(K)= 0 otherwise. (06Marks)2 2PART-B

a. State and prove the following propertiesof Fourier transform:

i) Frequency shifting property ii) Time differentiationproperty (06Marks)

b. Show that the Fourier transform of a train of impulses of unit height separated by T sees is

also a train of impulses of height roo= 27t separated by roo= 27t sec. (08 Marks)T T

c. Find the DTFT of following signals and draw its magnitude spectrum:

i) x(n) = anU(n); lal< 1 ii) x(n) = 8(n) unit impulse. -

a. The system produces an output yet)=e-tu(t) for an inputof x(t) =e-2tu(t). Determine the

frequency response and impulse response of the system. (08Marks)

b. State and prove samplingtheorem for low pass signals. (07Marks)c. Find the Nyquist rate for the following signals:

i) x(t) = 25eJ500m ii) [x(t)} = [1+ 0.1 sin(2007tt)] cos(20007tt) (05Marks)

a. State and prove time reversal and differentiation in Z-domain properties of Z-transforms.(06Marks)

b. Find the Z-transforms of following sequences including R.O.C.: i) x(n) =_p_)nu(-n-l)

\2

ii) x(n) = alnl lal < 1

5

6

7

c. The Z-transfonn of sequence x(n) is given by, X(z) =. Z(Z2- 4z +5)(z - 3)(z - 2)(z -1)

Find x(n) for the following ROC's:

i) 2<lzl<3 ii) Izl>3 iii) Izi< 1.

8

(06Marks)

(06Marks)

(08Marks)

a. Solve the following linear constant co-efficient difference equation using z-transform

method: yen) _iy(n -1) +~y(n - 2) =(~)

n

.

u(n) with initial conditions, y(-1) = 4, y(-2)= 10.2 2 4

(10Marks)

b. A causal system has input x(n) and output yen). Find the impulse response of the system if,113

x(n) = 8(n) + 48(n-l) -g8(n - 2); yen) =8(n) - 48(n -1)

Find th~ output of the system if the input isGru(n)....* * * *

20f2

(10Marks)