Signal Processing Toolbox

Linear Phase Filters

Except for cremez, all of the FIR filter design functions design linear phase filters only. The filter coefficients, or "taps," of such filters obey either an even or odd symmetry relation. Depending on this symmetry, and on whether the order n of the filter is even or odd, a linear phase filter (stored in length n+1 vector b) has certain inherent restrictions on its frequency response.

Linear Phase Filter Type	Filter Order	Symmetry of Coefficients	Response H(f), f = 0	Response H(f), f = 1 (Nyquist)
Type I	Even	even:	No restriction	No restriction
Type II	Odd		No restriction	H(1) = 0
Type III	Even	odd:	H(0) = 0	H(1) = 0
Type IV	Odd		H(0) = 0	No restriction

The phase delay and group delay of linear phase FIR filters are equal and constant over the frequency band. For an order n linear phase FIR filter, the group delay is n/2, and the filtered signal is simply delayed by n/2 time steps (and the magnitude of its Fourier transform is scaled by the filter's magnitude response). This property preserves the wave shape of signals in the passband; that is, there is no phase distortion.

The functions fir1, fir2, firls, remez, fircls, fircls1, and firrcos all design type I and II linear phase FIR filters by default. Both firls and remez design type III and IV linear phase FIR filters given a 'hilbert' or 'differentiator' flag. cremez can design any type of linear phase filter, and nonlinear phase filters as well.

Note    Because the frequency response of a type II filter is zero at the Nyquist frequency ("high" frequency), fir1 does not design type II highpass and bandstop filters. For odd-valued n in these cases, fir1 adds 1 to the order and returns a type I filter.

FIR Filter Design

Windowing Method