# Solution of Difference Equation using ZT

Updated on 2017/08/06 22:43

## Solution of Difference equation using ZT

The response of any discrete time system can be decomposed as:

**Total Response = zero state response + zero input response**

### Expected Questions

- Calculate impulse response of the system: take input x (n) as impulse sequence.
- Calculate step response of the system: take input x (n) as unit step sequence.
- Calculate impulse response of the system: i.e. calculate h(n).
- Calculate system / transfer function i.e. H(z). H(z)=Y(z)/X(z).
- Calculate zero state and input response of the system.

### Calculate impulse response of the system

**Questions**.

Compute the impulse response of the system given by difference equation:

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

**Solution:**

As we need to calculate the impulse response of the system that means input is impulse function.

Take ZT on both sides,

Z[y(n)]=Y(z) ,

According to time shifting property,

Convert it into positive powers of z, multiply numerator and denominator by Z^2,

Take IZT using partial fraction expansion method,

Write equation in partial fraction expansion form,

= 8.33

= 35

= -44.3

Taking IZT,

## References

- Notes by Prof. Priyanka Bhosale
- WikiNote Foundation