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9.3 1D FIR Filter Design Methods

1D filter design functions.

EquirippleFilter returns linear-phase FIR filters with a frequency response that is optimal in the sense of minimizing the maximum approximation error [McC73b]. The approximation error is the difference between the desired and the actual frequency responses. The approximation error is spread evenly across the passbands and stopbands of the filter, resulting in a magnitude response that exhibits equal-sized ripples across the frequency domain (see Figure 9.1.1). The EquirippleFilter design is a Chebyshev approximation problem and uses the Remez exchange algorithm to find a solution.
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It is easy to verify the frequency response characteristics of the filter. This is accomplished by computing the discrete-time Fourier transform (DTFT) of filter h [Opp89]. The DTFT and the better-known DFT are related; the DFT is a sampled representation of the DTFT of a sequence. Here we evaluate the DTFT of h.
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Options of EquirippleFilter.

FrequencySamplingFilter is a simple linear-phase FIR filter design method that returns a filter with an optimum mean squared error fitted to a set of amplitude values spaced uniformly in the normalized frequency range . There are two possible sequences of sample positions:

Options for FrequencySamplingFilter.

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Tables have been published that, for a given filter specification, give the value of the best transition coefficient(s) [Rab70]. These coefficients minimize the maximum deviation of the frequency response of the synthesized filter from the frequency response of the desired filter. This is known as minimax optimization.

1D window sequences.

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Typically, the length of the filter is not known and must be estimated from the filter specifications. Formulas have been developed to assist the designer [ Her73, Kai74 ] in calculating the length given the width of the transition region and maximum tolerable peak ripple or stopband attenuation.

Filter design utility functions.

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