firpm

firpm is a MATLAB function used to design finite impulse response FIR filters by applying the Parks–McClellan (Remez exchange) algorithm. The function seeks a linear-phase FIR filter whose frequency response best approximates a user-specified target response in the minimax sense, producing equiripple behavior in the realized error across defined frequency bands. This makes firpm a popular tool for creating low-pass, high-pass, band-pass, and band-stop filters with well-controlled passband and stopband ripples.

The design problem solved by firpm is framed in terms of an order, a frequency specification, and a desired amplitude response. The user provides a filter order n and a frequency vector f that spans the normalized range from 0 to 1 (0 corresponds to 0 Hz and 1 corresponds to the Nyquist frequency). A is the corresponding desired amplitude vector at those frequencies, and w optionally assigns weights to emphasize certain bands more than others. The algorithm iteratively adjusts the impulse response coefficients to minimize the maximum error between the actual and desired responses across the bands. There are also helper utilities such as firpmord to estimate a suitable order from ripple and bandwidth requirements.

Output of firpm is the filter coefficient vector b, suitable for use in convolution with signals. The method is known for producing equiripple error in the passband and stopband, and for yielding robust designs with relatively few taps compared to other optimization approaches. Firpm is part of the MATLAB Signal Processing Toolbox and is widely used in digital filter design and analysis.