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Generalized Window Method

Reiterating and expanding on points made in §4.6.3, 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 (§4.5), 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 graphic equalizer, cubic splines [286] could be used to connect the desired band gains.5.12

  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 (see §4.8 below).

  3. Perform the inverse Fourier transform 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 nonparametrically, 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. As we'll discuss in Chapter 8, it is necessary to precisely limit the FIR filter length to avoid time-aliasing in an FFT-convolution implementation.

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Minimum-Phase Filter Design
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Hilbert Transform Design Example