ParksMcClellanAlgorithmus

Parks–McClellan algorithm, also known as the Remez exchange algorithm for FIR filter design, is a method for designing finite impulse response filters with an optimal equiripple magnitude response. Introduced in 1972 by Lawrence Parks and James McClellan at Bell Labs, the algorithm solves the Chebyshev (minimax) approximation problem in the frequency domain. Given a desired frequency response defined over multiple bands, filter length N, and a weight function for error in each band, the algorithm iteratively selects a set of extremal frequencies where the error is largest and solves a linear system to determine the filter coefficients and the ripple level. This process uses the Remez exchange principle and guarantees an optimum solution under the alternation theorem: the error curve exhibits at least N+2 alternations in the design bands.

The result is an FIR filter whose magnitude response approximates the desired response with an equiripple behavior, i.e., equal ripple in the passband and stopbands according to the weights. The method is widely used in digital signal processing because it provides exact control over the error distribution and yields linear-phase filters when symmetric coefficients are used.

Applications include audio equalization, communication channel shaping, anti-aliasing filters, and other FIR designs. The Parks–McClellan algorithm is implemented in many software environments, notably in MATLAB as firpm or remez and in Python's SciPy as signal.remez.