Digital Filter/Characterization

Characterization

A digital filter is characterized by its transfer function, or equivalently, its difference equation. Mathematical analysis of the transfer function can describe how it will respond to any input. As such, designing a filter consists of developing specifications appropriate to the problem (for example, a second-order low pass filter with a specific cut-off frequency), and then producing a transfer function which meets the specifications.

The transfer function for a linear, time-invariant, digital filter can be expressed as a transfer function in the Z-domain; if it is causal, then it has the form:

$H(z)=\frac{B(z)}{A(z)}=\frac{b_0+b_1{z}^{-1}+b_2{z}^{-2}+\cdots +b_N{z}^{-N}}{1+a_1{z}^{-1}+a_2{z}^{-2}+\cdots +a_M{z}^{-M}}$

where the order of the filter is the greater of N or M. See Z-transform's LCCD equation for further discussion of this transfer function.

This is the form for a recursive filter, which typically leads to an infinite impulse response (IIR) behaviour, but if the denominator is made equal to unity, i.e. no feedback, then this becomes a finite impulse response (FIR) filter.