remez

Remez, short for the Remez exchange algorithm, is an iterative method for computing a minimax (Chebyshev) approximation to a desired frequency response using a finite impulse response (FIR) filter. In digital signal processing, it is most commonly associated with designing linear-phase FIR filters via the Parks–McClellan approach, and is frequently referred to by the function name remez in software toolboxes.

The problem is to find FIR coefficients h[n] that make the filter’s frequency response H(ω) approximate a target D(ω) over specified frequency bands (passband, stopband, and transition bands), while minimizing the maximum weighted error E(ω) = W(ω)[D(ω) − H(ω)]. W(ω) assigns relative importance to different bands. For a fixed-length, linear-phase FIR filter, the optimization reduces to a finite set of linear equations and a constraint that the error reaches equal magnitude at a set of extremal frequencies.

The algorithm proceeds iteratively. An initial set of extremal frequencies is chosen, and a linear system is solved to obtain the filter coefficients and the error magnitude. The actual error is evaluated across the bands; if the set of extremal points is not optimal, some of them are replaced by new frequencies where the error is largest, enforcing alternating signs. This exchange continues until convergence, typically when the extremal set stabilizes or changes fall below a tolerance.

The result is an equiripple error pattern in the specified bands, with a minimax-optimal overall deviation. The Parks–McClellan implementation of the method supports various filter lengths, symmetry types (Type I–IV), and user-defined passband/stopband specifications. It is widely implemented in DSP toolboxes, including MATLAB/Octave (remez) and SciPy.