Example Problems Based on z Transform Analysis of Discrete Time Systems

5.5.4 Example Problems Based on z Transform Analysis of Discrete Time Systems

(1) Compute the response of the system.  y(n) = 0.7 y(n-1) - 0.12 y(n-2) + x(n-1) + x(n-2) to the input x(n) = n u(n). Is the system stable?

Solution:

Calculation of system function H(z).

Given difference equation is

y(n) = 0.7 y(n−1) - 0.12 y(n-2) + x(n−1) + x(n-2)

Taking z transform of above equation.

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Stability of the system.

H(z) can be expressed in factored form is given as

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Poles: p₁ = 0.4 and p₂ = 0.3

Zeros: z = -1

Both the poles p₁ and p₂ are inside the unit circle |z|= 1. Hence this system is stable.

Response of the system :

We should calculate response y(n) of the system for the input x(n) = n u(n).

To calculate response y(n), first we should find Y(z)

Y(z) = H(z) . X(z) ............(2)

Calculation of X(z):

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After substituting A, B, C, D values y(z)/z becomes.

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Taking inverse z transform using standard relations.

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(2) Determine the system function and unit sample response of the system described by the difference equation.

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Solution:

Given difference equation

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Taking z transform of above equation

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Putting y(-1) = 0 in above equation

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Transfer function :

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Unit sample response:

By taking inverse z transform of H(z), we can get the value of h(n)

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(3) Find the output sequence y(n) of the system described by the equation y(n) = 0.7 y(n-1) - 0.1 y(n-2) + 2x(n) - x(n-2) for the input sequence x(n) = u(n)

Solution:

Given difference equation

(1-1) (5,0-3) (co-)

y(n) = 0.7 y(n-1) - 0.1 y(n-2) + 2x(n) - x(n-2)

Taking z transform of above equation.

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But x(n) = u(n)

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Putting X(z) values in (1)

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Calculation of y(n):

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Substituting A, B, C values in (2)

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Taking inverse z transform of above equation.

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(4) Solve the difference equation using z transform method x(n-2) - 9 x(n-1) + 18 x(n) = 0 with initial conditions x(-1) = 1. (May 03)

Solution:

The given difference equation is

x(n-2) - 9 x(n-1) + 18 x(n) = 0.

Taking unilateral z transform.

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By putting the initial conditions

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Calculation of x(n):

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Taking inverse z transform

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(5) Given H(z) = eeeeeeeeeeee Roc |z| > 0.4. Find the impulse response of the system. Dec-14 – 8 marks

Solution:

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Taking inverse z transform

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(6) Find the step response of the system y(n) + 1/3 y(n−1) = x(n) Dec 14-8 Marks

Solution:

y(n) + 1/3 y(n−1) = x(n)

We should calculate y (n) for the step input x(n) = u(n)

 

Take z transform of given equation

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Substitute X(z) values in (1)

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Now substitute A and B values

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Taking inverse z transform to get y(n)

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(7) Determine the range of K for which the system defined by the difference equation y(n) - 2k y(n-1) + k2 y(n-2) = x(n) is stable. Dec 11-Marks 4

Solution :

Condition for stable system: All the poles of system function (or) transfer function H(z) must lie in the unit circle.

y(n) - 2k y(n-1) + k2 y(n-2) = x(n)

Taking z transform of given equation

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There are multiple poles at z = k. For the system to be stable poles must lie in the unit circle.

Hence |k| < 1 for the system to be stable.

 

(8) Find the impulse response of the causal system y(n) - 5 y(n-1) = x(n) + x(n−1) with all initial conditions zero. Dec-2000 -8 Marks

Solution:

The given equation is

y(n) − 5 y(n-1) = x(n) + x(n−1)

Taking z transform of above equation

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Transfer function

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Divide (2) by z

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After substituting the values of A and B we get

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Taking inverse z transform using standard relations

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Hint :

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(9) A causal discrete time LTI system is described by

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(i) Determine the system function H(z)

(ii) Find the impulse response h(z) of the system.

(iii) Step response of the system.

 

Solution:

(i) System function H(z)

Given difference equation is

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Taking z transform of above equation

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(ii) Impulse response h(n)

By taking inverse z transform of H(z), we can find. h(n)

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Taking inverse z transform

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(iii) Step response:

Step response means we should calculate y(n) for the i/p x(n) = u(n).

We know that

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Taking z transform of above equation

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Taking inverse z transform using standard relations.

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(10) Find the output of the system whose input-output is related by y(n) = 7 y(n-1) - 12 y(n-2) + 2 x(n) - x(n-2) for the input x(n) = u(n) Dec-02, 16 Marks

Solution:

The given equation is

y(n) = 7 y(n-1) - 12 y(n-2) + 2 x(n) - x(n-2)

Taking z transform of above equation

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But i/p x(n) = u(n)

z transform of u(n) = eeeeeeeeeeeee

Substitute X(z) values in (1)

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After substituting A, B, C Y(z)/z becomes.

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Taking inverse z transform using standard relations

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