TABLE OF CONTENTS The design techniques described in the previous section are based on converting an analog filter into a digital filter. It is also possible to perform the design directly in the time domain without any reference to an analog filter. This section describes several methods for designing a digital filter directly.
Let h d( n) be the unit sample response of an ideal filter that is to be approximated by a causal filter that has a unit sample response, h( n), and a rational system function,
Because H( z) has p + q + 1 free parameters, it is generally possible to find values for the coefficients a( k) and a( k) so that h( n) = h d( n) for n = 0, 1, , p + q. The procedure that is used to find these coefficients is to write H( z) = B( z)/ A( z) as follows,
and note that, in the time domain, the left-hand side corresponds to a convolution
note that b( n) is a finite-length sequence that is equal to zero for n < 0 and n > q). Setting h( n) = h d( n) for n = 0, 1, , p + q
