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One sided Z transform is very efficient tool for the solution of difference equations with nonzero initial condition. System function of LSI system can be obtained from its difference equation.

**SOLUTION OF DIFFERENTIAL EQUATION**

One sided
Z transform is very efficient tool for the solution of difference equations with
nonzero initial condition. System function of LSI system can be obtained from
its difference equation.

1. Difference equations are used to find out the
relation between input and output sequences. It is also used to relate system
function H(z) and Z transform.

2. The
transfer function H(ω) can be
obtained from system function H(z) by putting z=e^{j}^{ω}.

Magnitude
and phase response plot can be obtained by putting various values of ω.

Tutorial
problems:

**Q) **A difference equation of the
system is given below

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

Determine

a) System
function

b) Pole
zero plot

c) Unit
sample response

**Q) **A difference equation of the
system is given below

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

a) System
Function

b) Pole
zero plot

c)
Response of system to the input x(n) = nu(n)

d) Is the
system stable? Comment on the result.

**Q) **A difference equation of the
system is given below

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

Determine
a) System function

b) Pole
zero plot

c) Unit
sample response

d)
Transfer function

e)
Magnitude and phase plot

**Q) **A difference equation of the
system is given below

a. Y(n)=
0.5 y(n-1) + x(n) + x(n-1)

b. Y(n)=
x(n) + 3x(n-1) + 3x(n-2) + x(n-3)

a) System
Function

b) Pole
zero plot

c) Unit
sample response

d) Find
values of y(n) for n=0,1,2,3,4,5 for x(n)= δ(n) for no initial condition.

**Q) **Solve second order difference
equation

2x(n-2) –
3x(n-1) + x(n) = 3^{n-2} with x(-2)=-4/9 and x(-1)=-1/3.

**Q) **Solve second order difference
equation

x(n+2) +
3x(n+1) + 2x(n) with x(0)=0 and x(1)=1.

**Q) **Find the response of the system
by using Z transform

x(n+2) -
5x(n+1) + 6x(n)= u(n) with x(0)=0 and x(1)=1.

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