FIRFilters

FIR filters, or finite impulse response filters, are digital filters whose impulse response h[n] has finite duration. The output y[n] is the convolution of the input x[n] with h[n], expressed as y[n] = sum_{k=0}^{M} h[k] x[n−k], where M is the filter order. Because the impulse response is finite, FIR filters are inherently stable for bounded inputs, and they can be designed to realize a wide range of frequency responses.

A notable property of FIR filters is the possibility of linear phase. If the coefficients are symmetric (h[n] = h[M−n]) or anti-symmetric, the filter exhibits linear phase, meaning it preserves the waveform shape within the passband while applying the desired phase shift. The frequency response H(ω) is the discrete-time Fourier transform of h[n].

Design methods for FIR filters include the window method, the frequency sampling method, and optimal approaches. The window method starts from an ideal, infinite impulse response and truncates it with a window function such as Rectangular, Hamming, Hann, Blackman, or Kaiser, trading transition width against ripple. The Parks–McClellan algorithm (Remez exchange) designs equiripple FIRs that minimize the maximum error in a specified passband and stopband. Frequency sampling specifies a desired response at a set of frequencies and derives h[n] by interpolation. Least-squares methods optimize filter coefficients to best fit a target response in a defined sense.

In practice, longer filter orders give sharper transitions but higher computational cost. Symmetric coefficients can reduce multiplications by about half. FIR filtering can be implemented efficiently with direct convolution for small filters or FFT-based techniques for long filters.

Applications of FIR filters are widespread, including anti-aliasing and reconstruction filters in sampling systems, audio processing, communications, smoothing, and general digital signal processing tasks that require robust, stable filtering with controllable phase characteristics.