Generalized Window Method

Often we need a filter with a frequency response that is not analytically known. An example is a graphic equalizer in which a user may manipulate sliders in a graphical user interface to control the gain in each of several frequency bands. From the foregoing, the following procedure, based in spirit on the window method (§E.4), can yield good results:

1. Synthesize the desired frequency response as the smoothest possible interpolation of the desired frequency-response points. For example, in a graphical equalizer, cubic splines could be used to connect the desired band gains.^E.2
2. If the desired frequency response is real (as in simple band gains), either plan for a zero-phase filter in the end, or synthesize a desired phase, such as linear phase or minimum phase [247].
3. Perform the inverse Fourier transform (FFT) of the (sampled) desired frequency response to obtain the desired impulse response.
4. Plot an overlay of the desired impulse response and the window to be applied, ensuring that the great majority of the signal energy in the desired impulse response lies under the window to be used.
5. Multiply by the window.
6. Take an FFT (now with zero padding introduced by the window).
7. Plot an overlay of the original desired response and the response retained after time-domain windowing, and verify that the specifications are within an acceptable range.

In summary, FIR filters can be designed non-parametrically, directly in the frequency domain, followed by a final smoothing (windowing in the time domain) which guarantees that the FIR length will be precisely limited. It is necessary to precisely limit the FIR filter length to avoid time-aliasing in an FFT convolution implementation.

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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.