y(n-k) is considered as previous outputs;

x(n-p) – previous inputs;

a_k and b­_p – coefficients;

This equation is convenient to define discrete system and extract various characteristics. Number N defines discrete filter tap (Tap - A FIR “tap” is simply a coefficient/delay pair).

Lets say we have digital filter made of one tap.

y(n)=a·y(n-1)+b·x(n)

First we calculate filter response function – h(n):

Response function is calculated from system reaction to discrete impulse δ(n). So assume that x(n)= δ(n) and y(n)=h(n). Initial conditions: y(-1)=h(-1)=0.

Then:

h(n)=0 when n<0;

h(0)=a·h(-1)+b·δ(n)=b;

h(1)=a·h(0)+b·δ(1)=ab;

………….

h(n)= baⁿ where n>0

Lets say we have a=0.7; b=1;

Then we get discrete system characteristics:

This equation in example describes IIR (Infinite Impulse Response) discrete systems, because there is a feedback element a·y(n-1).

There is another type of discrete filter – FIR (Finite Impulse Response)

Finite response filter equation is much simpler:

As you may noticed from equation – IIR filter have infinite number of response impulses while FIR filter have a finite number of impulses. IIR and FIR systems are different in their properties and realizations.