Least-Squares Linear-Phase FIR Filter Design

Let the FIR filter length be samples, with even, and suppose we'll initially design it to be centered about the time origin. Then the frequency response is given on our frequency grid by

Enforcing even symmetry in the impulse response, i.e., , gives a zero phase FIR filter which we can later right-shift samples to make a causal, linear phase filter. In this case, the frequency response reduces to a sum of cosines:

or in matrix form:

(Note that Remez exchange algorithms are also based on this formulation internally.)

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Subsections

• Matrix Formulation: Optimal Design, Cont'd
• Least Squares Optimization
• Geometrical Interpretation of Least Squares
• Matlab Support for Least-Squares FIR Filter Design

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.