Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 48

Solve the equation for the input signal that is equal unity for n C 0 and is equal
zero otherwise. Assume initial condition y(-1) = 0.

Solution

n\0 ; y ð n Þ¼ 0

n ¼ 0 ; y ð 0 Þ¼ x ð 0 Þþ ay ð 1 Þ¼ 1

n ¼ 1 ; y ð 1 Þ¼ x ð 1 Þþ ay ð 0 Þ¼ 1 þ a

n ¼ 2 ; y ð 2 Þ¼ x ð 2 Þþ ay ð 1 Þ¼ 1 þ a ð 1 þ a Þ

and so on.
Iteratively one can calculate any required number of the signal y(n) samples.
Sometimes it is also possible to reach the solution in general form.

4.8.2 Discrete System Description in Frequency Domain

The alternative way of description of a linear discrete system is its frequency
response. This is a supplementary possibility, since sometimes one may
need responses in both time and frequency domains. On the other hand, one of
those descriptions is enough, since one can get the other one using Fourier
transform.
The frequency response of a linear system can be obtained using different
methods of description in time domain, especially the difference equations and the
impulse response (discrete convolution). In the first case from difference equation
one can get the system transfer function first, while in the second case the impulse
response is used directly.

Transfer Function Versus Difference Equation
Calculating Z transform of both sides of Eq. 4.35 and assuming zero initial con-
ditions yields:

Y ð z Þ¼ X a k z k Y ð z Þþ X

M N 1

b k z k X ð z Þ ð 4 : 36 Þ

k ¼ 1 k ¼ 0

and rearranging it the discrete transfer function is reached, in the form:

X ð z Þ ¼ H ð z Þ¼ P N 1

Y ð z Þ

k ¼ 0 b k z k

1 P k ¼ 1 a k z k ;

ð 4 : 37 Þ

which is directly related to the difference equation given before. This transfer
function can be used to find the system response to any input signal. This is done
by multiplying the system discrete transfer function by given Z transform of the
input signal, and calculating inverse transform of that product.