BE-IT <IT-05104>

Digital Signal Processing

Sample Question an Answer

Chapter – 31. Determine the Z-transform and sketch the ROC of the signals

(a) x (n) =(1/2)n u(n)(b) x (n) =3u(-n-1) (10-Marks)

2. State and prove ANY THREE properties of the Z-transform.(20-Marks)

3. Determine the Z-transform and sketch the ROC of the signal x(n)=[3(2)n -4(3)n ]u(n).

(10-Marks)4. Determine the system function and the unit sample response of the system describe by the difference equation. y(n)=1/2 y(n-1)+2x(n) (10-Marks)

5. Determine the inverse Z-transform of

X(Z)=

When

(a) ROC: |Z|>1(b) ROC:| Z|<0.5(c) ROC: 0.5<|Z|<1 (20-Marks)

6. Determine the causal signal x(n) having the Z-transform.

X(Z)= (20-Marks)

7. The well-known Fibonacci sequence of integer numbers is obtained by computing each term as the sum of the two previous ones. The first few terms of the sequence are

1,1,2,3,4,5,8. Determine a closed-form expression for the nth term of the Fibonacci sequence.

(20-Marks)

8. Determine the step response of the systemy(n)= ∞y(n-1) + x(n) -1<∞<1

when the initial condition is y(-1)=1 (20-Marks)

9. Determine the unit step response of the system describe by the difference equation y(n)=0.9y(n-1)-0.81y(n-2)+x(n)

under the following initial condition: y(-1)=y(-2)=1 (20-Marks)

10. Determine the transient and steady-state response of the system characterized by the Difference equation

y(n)=0.5y(n-1)+x(n) when the input signal is x(n)=10 cos(π n/4)u(n). The system initially at rest (i.e, it is relaxed) (20-Marks)11. Determine the inverse Z-transform of

H(Z)=

Specify the ROC of H(Z) and determine h(n) for the following conditions:(a) The system is stable.(b) The system is causal.(c) The system is purely anticausal. (20-Marks)

12. Compute the convolution of the following signals by means of Z-transform x1(n)=

x2(n)=(1/2)n u(n) (20-Marks)

13. Use the one side Z-transform to determine the zero input response, yzi (n), n in the following case.

y(n)-1.5y(n-1)+0.5y(n-2)=0,y(-1)=1,y(-2)=0 (10-Marks)

14. Compute the zero state response for the following pair of system and input signal.

h(n)=(2/5)n u(n),x(n)=u(n)-u(n-7) (10-Marks)

15. Determine the response of the system described by the difference equation

Y(n)=5/6 y(n-1)- 1/6 y(n-2) + x(n)

to the input signal x(n)= (n)-1/3 (n-1).

Take the initial condition y(-1)=1 and y(-2)=0 (20-Marks)

16. Determine the unit step response of the causal system described by the difference equation

y(n)=y(n-1) + x(n) (10-Marks)

17. By using schur-cohn stability test ,determine if the system having the function

H(Z)= is stable. (10Marks)

18. We want to design a causal discrete time LTI system with the property that if the input is x(n)=(1/2)n u(n)-1/4(1/2)n+1 u(n-1) then the output is y(n)=(1/3)n u(n).

(a) Determine the system function H(Z) and the impulse response of a system that satisfies the forgoing conditions.

(b) Find the difference equation that characterizes this system.(c) Determine a realization of the system that requires a minimum number possible

amount of money.(d) Determine if the system is stable. (20-Marks)

19. Determine the interconnection of the systems shown in Fig: where h(n)=an u(n),-1<a<1.

(a) Determine the impulse response of the overall system and determine if it is causal andstable.

(b) Determine a realization of the system using the minimum numbers of adders, multipliersand delay elements. (20-Marks)

20. Consider the system

H(Z)=

(a) Determine impulse response.(b) The zero state step response. (20-Marks)

21. Consider the system

H(Z)=

Determine the step response if y(-1)=1 and y(-2)=2 (20-Marks)

22. Determine the Fourier and series and the power density spectrum of the rectangular pulse signal illustrated in fig.

23. Determine the Fourier transform and the energy density spectrum of the signal defined as

x(t)=

(20-Marks)

24. Determine the spectrum of the signal.x(n)=cos w0n

when (a) w0= (a) w0= (20-Marks)

25. Determine the Fourier series coefficients and the power density spectrum of the signal show in fig:

(20-Marks)

X(w)=26. Determine the signal x(n) corresponding to the spectrum

X(w)=

(20-Marks)

27. Determine the Fourier transform and the energy density spectrum of the sequence

x(n)=

which is illustrated in fig.

(20-Marks)

28. Determine the Fourier transform of the signalx(n)=a -1<a<1

(20-Marks)

29. Consider the full-wave rectified sinusoid in fig

(a) Determine its spectrum Xa(F).(b) Compute the power of the signal.(c) plot the power spectrum density.(d) Check the validity of parseval’s relation for this signals. (20-Marks)

30. Compute the Fourier transform of the following signals. (a) x(n)=u(n)-u(n-6) (b) x(n)=2n u(n-6)

(c) x(n)=cos( )[u(n)-u(n-6)] (20-Marks)

BE. IT <IT. 05104>Digital Signal Processing

Sample Questions and Answers

CHAPTER -3

1. Determine the Z-transform and sketch the ROC of the signal. (10 marks )(a) x(n)= (1/2)n u(n)(b) x(n)= 3u(-n-1)

Solution:(a) x(n) = (1/2)n u(n)

X(Z) = x(n) Z-n

= (1/2)n Z-n

= (1/2 Z-1)n

= ROC: | |< 1

ROC: |Z | > ½

(b) x(n) = 3u(-n-1)

X(Z) = x(n) Z-n

= 3u (-n-1) Z-n

= 3 (Z)l (where l=-n)

= 3

= ROC: |Z| <1

2. State and prove ANY THREE properties of the Z-transform.“Scaling in the z-Domain” (20-Marks)IF x(n) X(Z) ROC: r1 < |Z| < r2

Then an x(n) X(a-1Z) ROC: |a| r1 < |Z| < r2

For any constant a, real or complex.

Proof: By definition,

Z{ an x(n) } = an x(n)Z-n

= x(n) (a-1Z)-n

= X (a-1Z)

Since the ROC of X(Z) is r1 < |Z| < r2, the ROC of X(a-1Z) is r1 < |a-1Z| < r2 (or) |a| r1 < |Z| < |a| r2

“Time Reversal”

IF x(n) X(Z) ROC: r1 < |Z| < r2

Then x(-n) X(Z-1)ROC: < |Z| <

Proof: By definition,

Z {x(-n) } = x(-n) Z-n

= x(l) (Z-1)-l

= X(Z-1)where l=-n

The ROC of X(Z-1) is

r1 < |Z-1| < r2 or equivalently < |Z| <

“Differentiation in the Z-Domain”

IF x(n) X(Z)

Then nx(n) -Z

Proof: By differentiating both sides of Z transform equation

= x(n) (-n) Zn-1

= -Z-1 {n x(n)} Z-n

= -Z-1 Z {n x(n)}

= Z {n x(n) }

Both transforms have the same ROC:

3. Determine the Z-transform and sketch the ROC of the signal x(n)=[3(2)n – 4(3)n]

u(n).(10 marks)

Solution:x(n)=[3(2)n

– 4(3)n] u(n).X(Z) = Z {x(n)} = Z [ 3(2)n – 4(3)n ] u(n)] = Z { 3(2)n

u(n) } – Z { 4(3)n u(n) }

= -

ROC: |Z| > 2 ; ROC: |Z| > 3

X(Z) = - ; ROC: |Z| > 3

Fig: ROC of X(Z)

4. Determine the system function and the unit sample response of the system described by the difference equation.

y(n) = y (n-1) + 2x(n) (10 marks)

Solution:

y(n) = y (n-1) + 2x(n)

Y(Z) = Z-1 Y(Z) + 2 X(Z)

Y(Z) (1- Z-1) = 2 X(Z)

=

H(Z) =

h(n) = 2 ( )n u(n)

This is the unit sample response of the system.

5. Determine the inverse Z-transform of

X(Z) = (20 marks)

when (a) ROC: |Z| > 1 (b) ROC: |Z| < 0.5

(c) ROC: 0.5 < |Z| < 1Solution:

X(Z) =

X(Z) =

X(Z) =

= =

A1 = (Z-1)

A1 =

=

A1 = 2

A2 = (Z-0.5)

A2 =

=

A2 = -1

=

X(Z) =

X(Z) =

(a) When the ROC is |Z| > 1 , the signal x(n) is causal arc terms are causal terms. x(n) = 2(1)n u(n) –(0.5)n u(n)

= (2-0.5n) u(n) (b) When the ROC is |Z| < 0.5 , the signal x(n) is anticausal.

Thus both terms result in anticausal components.x(n) = [-2 + (0.5)n ] u(-n-1)

(c) When the ROC is 0.5 < |Z| < 1 , the signal x(n) is two sided.Thus one of term is a causal part and the other is anticausal part. Thus,

x(n) = -2 (1)n u(-n-1) –(0.5)n u(n).

6. Determine the causal signal x(n) having the Z-transform

X(Z) = (20 marks)

Solution:

X(Z) =

X(Z) =

X(Z) has simple pole at p1 = -1 and a double pole at p2 = p3 = 1.Thus partial fraction exp

= =

A1 = (Z+1)

A1 =

A1 =

A1 =

A3 =

A3 =

A3 =

A2 = Z=1

A2 =

A2 =

A2 =

X(Z) =

=

x(n) =

x(n) =

7. The well-known Fibonancci sequence of integer numbers is obtained by computing each term as the sum of the two previous one. The first few terms of the sequence are 1,1,2,3,5,8,13,21,34,….Determine a closed –form expression for the nth term of the Fibonancci sequence. (20 marks)

Solution:y(n) = y(n-1) + y(n-2) initial condition n=0

eqn: (A)

,

8 . Determine the step response of the system , ,when the initial condition is y(-1)=1. (20-marks)

Soln: ;

;

9 . Determine the unit step response of the system described b difference equation.

under the following initial conditions:(20 marks)

Solution:

=

=

= =

B = A”

10. Determine the transient and steady state response of the system characterized by the difference equation.

y(n)=0.5 y(n-1) + x(n) when the input signal x(n) = 10 cos ( ) u(n)

The system is initially at rest (i.e it is relaxed). (20 marks)

Solution:y(n) = 0.5 y(n-1) + x(n)Y(Z) = 0.5 y(Z)Z-1 + X(Z)Y(Z) (1-0.5Z-1) = X(Z)

H(Z) =

One pole at Z=0.5

x(n) =10 cos ( ) u(n)

Y(Z) = H(Z) * (Z)

Y(Z) =

=

=

=

=

=

C = B = 6.79

Y(Z) =

y(n) = -1.9(0.5)n u(n) +2*6.79(1)n cos ( - 28.7) u(n)

The natural or transient rep: is ynr (n) = -1.9(0.5)n u(n)

The forced or steady state rep: is

yfr(n) = 13.58 cos ( - 28.7) u(n)

11 . Determine the inverse Z-transform of

.

Specify the ROC of H(Z) and determine h(n) for the following condition (a)The system is stable (b)The system is causal (c)The system is purely anticausal. (20 marks)

Solution:

The system has poles at and

(a) The system is stable : < < 3.

(b) The system is causal : < < 3.

(c) The system is anticausal : < < 3.

, system is unstable.

12. Compute the convolution of the following signals by means of Z-transform. (20-Marks)

Solution

=

=

=

=

=

13. Use the one-side Z-transform to determine the zero input response ,y in the following case.

; y(-1)=1,y(-2)=0 (10-Marks)

(where no input signal)

14. Compute the zero state response for the following pair of system and input signal.(10-Marks)

=

=

=

=

=

=

let

15 . Determine the response of the system described by the difference equation .

to the input signal

.

Take the initial condition: and . (20-marks)

Solution:,

16. Determine the unit step response of the causal system described by the difference Equation.

x(n)=u(n) X(z)=

=

=

Double pole at Z=1 The inverse Z transform of Y(z) is

y(n)=(n+1)u(n)