LeastSquaresFilterKernel

LeastSquaresFilterKernel[{{ω1,,ωk-1},{a1,,ak}},n]
creates a k-band finite impulse response (FIR) filter kernel of length n designed using a least squares method, given the specified frequencies ωi and amplitudes ai.

LeastSquaresFilterKernel[{"type",spec},n]
uses the full filter specification {"type",spec}.

Details and OptionsDetails and Options

  • LeastSquaresFilterKernel returns a numeric list of length n of the impulse response coefficients of an FIR filter that has the minimum mean-squared error.
  • The impulse response of the filter is computed using the inverse discrete-time Fourier transform.
  • In LeastSquaresFilterKernel[{"type",spec},n], filter specification can be any of the following:
  • {"Lowpass",ωc}lowpass filter with cutoff frequency ωc
    {"Highpass",ωc}highpass filter with cutoff frequency ωc
    {"Bandpass",{ωc1,ωc2}}bandpass filter with pass band from ωc1 to ωc2
    {"Bandpass",{{ω,q}}}bandpass filter with center frequency ω and quality factor q
    {"Bandstop",{ωc1,ωc2}}bandstop filter with stop band from ωc1 to ωc2
    {"Bandstop",{{ω,q}}}bandstop filter with center frequency ω and quality factor q
    {"Multiband",{ω1,,ωk-1},{a1,,ak}}multiband filter specification with k bands
    {"Differentiator",ωc}differentiator filter with cutoff frequency ωc
    {"Hilbert",ωc}Hilbert filter with cutoff frequency ωc
  • If "type" is omitted, "Multiband" is assumed.
  • Frequencies should be given in an ascending order such that 0<ω1<ω2<<ωk-1<π.
  • Amplitude value a1 corresponds to the frequency band 0 to ω1, and amplitude ak corresponds to the frequency band ωk-1 to π.
  • Amplitude values should be non-negative. Typically, values ai=0 specify a stopband, and values ai=1 specify a passband.
  • The quality factor q is defined as , with being the center frequency of a bandpass or bandstop filter. Higher values of q give narrower filters.
  • The kernel ker, returned by LeastSquaresFilterKernel, can be used in ListConvolve[ker,data] to apply the filter to data.
  • LeastSquaresFilterKernel takes a WorkingPrecision option, which specifies the precision to use in internal computations. The default setting is WorkingPrecision->MachinePrecision.

ExamplesExamplesopen allclose all

Basic Examples  (2)Basic Examples  (2)

A lowpass FIR kernel:

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Magnitude plot of the filter and its ideal lowpass prototype:

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A Bode plot of the filter:

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A multiband FIR kernel:

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Magnitude plot of the filter and its "brickwall" specification:

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Introduced in 2012
(9.0)
| Updated in 2015
(10.3)
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