Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 47

Denoting the impulse response by h(n), i.e. h ð n Þ¼ F f d ð n Þg; one obtains the
discrete convolution equation allowing to calculate the output signal as a function
of the input signal and the system's impulse response:

y ð n Þ¼ X

1

x ð k Þ h ð n k Þ ð 4 : 32 Þ

k ¼ 0

or symbolically

y ð n Þ¼ x ð n Þ h ð n Þ: ð 4 : 33 Þ

If both the input signal and the system follow the causal conditions, i.e.
x(n) = 0 and h(n) = 0 for n \ 0, then the discrete convolution can be written in
the form:

y ð n Þ¼ X

x ð k Þ h ð n k Þ¼ X

n n

x ð n k Þ h ð k Þ ð 4 : 34 Þ

k ¼ 0 k ¼ 0

Example 4.12 Applying the method of discrete convolution calculate step

response of the system with the impulse response h ð n Þ¼ exp ð n Þ:

Solution In accordance with ( 4.32 ) one can derive:

y ð n Þ¼ X

n

exp ð k Þ¼ 1 exp ð n þ 1 Þ
1 exp ð 1 Þ

:

k ¼ 0

Difference Equation
Another way of description of digital transferring of an input series into an output
one is the linear difference equation with constant coefficients:

y ð n Þ¼ X a k y ð n k Þþ X

M N 1

b k x ð n k Þ; ð 4 : 35 Þ

k ¼ 1 k ¼ 0

where a k ,b k are constant factors.
To solve Eq. 4.35 M initial conditions are needed. The solution, i.e. the function
of y(n), can be reached multifold. One of the ways is application of the Z trans-
form, the other is an iterative approach, that is easy when a few output samples are
sought only, or when digital means are applied.

Example 4.13 Let there be given a difference equation of the form:

y ð n Þ¼ x ð n Þþ ay ð n 1 Þ: