Back to DSP thread. Lets talk about digital system equation. Every digital system can be described using this equation.

The output of digital filter generally consist of previous inputs and previous outputs.

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 ´(n). So assume that x(n)= ´(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·´(n)=b;

h(1)=a·h(0)+b·´(1)=ab;

…….

h(n)= ba^{n} 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.

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