6.1 Lab 5: filter specification

Filter specification

Please refer to Pictoralial filter specification for a pictorial description of how the various filter constraints correspond to the frequencyresponse of a low-pass filter. Note that the Pictoralial filter specification does not imply that the system response must be equiripple. The constraints on the system response, to be discussedshortly, are on the maximum allowable peak errors. However, in general, an equiripple solution to a given filterspecification will require the lowest filter order.

The full filter specification is given in filter specifications . The parameters $f_{\mathrm{p}}$ and $f_{\mathrm{s}}$ define the end of the pass-band and the beginning of the stop-band, respectively, on the frequency axis that hasbeen normalized to 1. The parameters ${}_{\mathrm{p}}$ and ${}_{\mathrm{s}}$ define the maximum allowable ripple, in Decibels, in the pass-band and the stop-band, respectively. The parameter $\forall f, \left|f\right|\le f_p\colon \left|({}_g)\right|\le 9$ defines the maximum allowable deviation from a group delay of 9 in the pass-band. The parameter ${}_{\mathrm{g}}$ is the group delay of the system and is defined by the following:

The parameter $({}_g)$ is defined as:

Thus, $\forall f, \left|f\right|\le f_p\colon \left|({}_g)\right|\le 9$ states that the group delay is not allowed to deviate more than 9 samples in the pass-band. A system thathas constant group delay will have a phase response that is generalized linear-phase. Thus, deviation from constant groupdelay is a measure of the deviation of the phase response from linear. The MATLAB command grpdelay computes the group delay of a system as a function of frequency given thefilter coefficients $a$ and $b$ .