Two Mark Questions with Answers of Linear Time Invariant-Discrete Time Systems

Two Mark Questions with Answers of Linear Time Invariant-Discrete Time Systems

1. Determine the system function of the discrete time system described by the difference equation.

y(n) = 0.5 y(n-1) + x(n)

Solution :

y(n) = 0.5 y(n-1) + x(n)

Taking z transform on both sides.

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2. Define system function of the discrete time system. May-02.

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3. Determine the convolution of the signals x(n) = {2, -1, 3, 2} and h(n) = {1, -1, 1, 1} May-12

Convolution by multiplication method is given below.

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4. Write the general difference equation relating input and output of a system. May-03

The generalized difference equation is given as

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Here

y(n-k) ⇒ Previous output

x(n-k) ⇒ Previous input.

5. If u(n) is the impulse response of the system, what is its step response May-98

Here h(n) = u(n) and input x(n) = u(n)

Hence output y(n) = h(n) * x(n) = u(n) * u(n).

6. Define system function.

The system function is the z transform of the unit sample response of the system.

H(z) = z{h(n)} = Y(z)/X(z)

System function is the ratio of z transform of the output to z transform of input.

7. Write the difference equation for non recursive system.

The difference equation of non recursive system is given as

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Thus output depends upon past and present inputs only.

 

8. find the step response of the system if the impulse response h(n) = δ(n-2)-δ(n-1) May-11

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9. Find the frequency response of a linear shift invariant system whose input and output satisfy the difference equation.

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Taking z transform of above equation

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Frequency response is obtained by calculating H (z) on unit circle so we have to

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10. What is the linear convolution of the two signals {2, 3, 4} and {1, -2, 1} Dec-04

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11. If x(n) and y(n) are discrete variable functions, what is its convolution sum. Dec-13

Convolution sum of x(n) and y(n) are given as

Convolution sum = eeeeeeeeeeeee

12. Consider a system whose impulse response is h(t) = e-|t|. Is this system causal (or) non causal?

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Since h(t) ≠ 0 for t < 0, the system is non causal.

13. Check whether the system with system function eeeeeeeeeeeee with Roc |z| < 0.5 is causal or a stable?

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Poles of this system are located at z = 0.5 and z = 2

All the poles should be located inside the unit circle in for stable and causal system. But here all the poles are not located inside the unit circle. So this system is not causal and stable.

 

14. Using z transform, check whether the following system is stable. Eeeeeeeeeeeeeeee June-14

Eeeeeeeeeeeeeeee

Poles of this system are located at z = 1/2 and z = 3

All the poles should be located inside the unit circle for stable. But in this case pole z = 3 is not located inside the unit circle. So the given system is not stable system.

15. Is the discrete time system descrited by the difference equation y(n) = x(-n) is causal ? June-13

y(n) = x(-n)

Put n = -1 in given equation

When n = -1, the output depends upon future i/p

Hence the given system is not causal system.

16. Determine the transfer function of the system described by y(n) = a y(n-1) + x(n) Dec -05

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

Taking z transform on both sides of equation.

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17. What are the basic building blocks to realize any structure? May-08

Basic building blocks to realize any structure is shown in figure.

For continuous time system:

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For discrete time system:

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18. Obtain the convolution of

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19. Convolve the following two sequences: x(n) = {1, 1, 1, 1} and h(n) = {2, 2} Dec-12

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Hence x(n) * h(n) = {2, 4, 4, 4, 2}

20. How z transform is related to Fourier transform May-02

Fourier transform is basically z transform evaluated on the unit circle.

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21) Find the system response x(n) = u(n) & h(n) = δ(n) + δ(n-1) [Nov '11]

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22) Define convolution sum with its equation.

eeeeeeeeeeee The convolution sum (or) the Linear convolution gives the output or response of a DT system which is the convolution of the input sequence impulse response sequence.

23) Check whether the system with eeeeeeeeee With ROC |z| < 1/2 is causal & stable

(i) H(z) = eeeeeeeeeeeeeee Taking inv z-transform, one of the terms of h(n) = 2n u(n)

Since h(n) ≠ 0 for n < 0, it is a Non-Causal system.

(ii) One of the poles of H(z) is outside the unit circle. Hence, it is an Unstable system.

 

24) Give the impulse response of a linear time invariant system as h(n) = sin (πn), check whether the system is stable or not. [Nov '14]

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25) In terms ROC, state the condition for an LTI-DR system to be causal & stable. [Nov '14]

For LTI-DT system to be causal, h(n) = 0 for n < 0

For an LTI-DT system to be stable, eeeeeeeeee

26) Write the nth order difference Equation.

eeeeeeeeeeeee, where, 'N' is called Order of the system.

27) Distingush between Recursive & Non-recursive systems. [Nov '15]

A Recursive system is one in which the output is dependent not only on its present inputs, but also on past outputs.

A Non-Recursive system is one in which the output is dependent only on its present inputs & not on past outputs.

28) Convolve the following signals, x(n) = {1, 1, 3} & h(n) = {1, −4, 1}. [Nov '15]

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29) List the 3 steps to obtain convolution.

(i) Folding h(k) ; h(-k)

(ii) Shifting h(k) ; h(n0-k)

(iii) Multiplication ; x(k) h(n0-k)

(iv) Summation ; eeeeeeeeeee

30) What is state transition matrix? [Nov '15]

The state - Transition matrix is used to find the solution to a general state-space representation of a linear system.

State variable equation:

Q(n+1) = A Q (n) + B x (n) : State Equation

y(n) = C Q (n) + D x x (n) : Output Equation