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Filters - filter design and filtering

DSP.jl differentiates between [filter coefficients](@ref coefficient-objects) and [stateful filters](@ref stateful-filter-objects). Filter coefficient objects specify the response of the filter in one of several standard forms. Stateful filter objects carry the state of the filter together with filter coefficients in an implementable form (PolynomialRatio, Biquad, or SecondOrderSections). When invoked on a filter coefficient object, filt does not preserve state.

[Filter coefficient objects](@id coefficient-objects)

DSP.jl supports common filter representations. Filter coefficients can be converted from one type to another using convert.

ZeroPoleGain
PolynomialRatio
Biquad
SecondOrderSections

These filter coefficient objects support the following arithmetic operations: inversion (inv), multiplication (*) for series connection, and integral power (^) for repeated multiplication with itself. For example:

julia> H = PolynomialRatio([1.0], [1.0, 0.3])
PolynomialRatio{:z, Float64}(LaurentPolynomial(1.0), LaurentPolynomial(0.3*z⁻¹ + 1.0))

julia> inv(H)
PolynomialRatio{:z, Float64}(LaurentPolynomial(0.3*z⁻¹ + 1.0), LaurentPolynomial(1.0))

julia> H * H
PolynomialRatio{:z, Float64}(LaurentPolynomial(1.0), LaurentPolynomial(0.09*z⁻² + 0.6*z⁻¹ + 1.0))

julia> H^2
PolynomialRatio{:z, Float64}(LaurentPolynomial(1.0), LaurentPolynomial(0.09*z⁻² + 0.6*z⁻¹ + 1.0))

julia> H^-2
PolynomialRatio{:z, Float64}(LaurentPolynomial(0.09*z⁻² + 0.6*z⁻¹ + 1.0), LaurentPolynomial(1.0))

[Stateful filter objects](@id stateful-filter-objects)

DF2TFilter

DSP.jl's FIRFilter type maintains state between calls to filt, allowing you to filter a signal of indefinite length in RAM-friendly chunks. FIRFilter contains nothing more that the state of the filter, and a FIRKernel. There are five different kinds of FIRKernel for single rate, up-sampling, down-sampling, rational resampling, and arbitrary sample-rate conversion. You need not specify the type of kernel. The FIRFilter constructor selects the correct kernel based on input parameters.

FIRFilter

Filter application

filt
filt!
filtfilt
fftfilt
fftfilt!
tdfilt
tdfilt!
resample

Filter design

Most analog and digital filters are constructed by composing [response types](@ref response-types), which determine the frequency response of the filter, with [design methods](@ref design-methods), which determine how the filter is constructed.

The response type is Lowpass, Highpass, Bandpass or Bandstop and includes the edges of the bands.

The design method is Butterworth, Chebyshev1, Chebyshev2, Elliptic, or FIRWindow, and includes any necessary parameters for the method that affect the shape of the response, such as filter order, ripple, and attenuation.

[Filter order estimation methods](@ref order-est-methods) are available in buttord, cheb1ord, cheb2ord, and ellipord if the corner frequencies for different IIR filter types are known. remezord can be used for an initial FIR filter order estimate.

analogfilter
digitalfilter

For some filters, the design method is more general or inherently implies a response type; these [direct design methods](@ref direct-design-methods) include remez which designs equiripple FIR filters of all types, and iirnotch which designs a 2nd order "biquad" IIR notch filter.

For a more general application of creating a digital filter from s-domain representation of an analog filter, one can use bilinear:

bilinear

[Filter response types](@id response-types)

Lowpass
Highpass
Bandpass
Bandstop

The interpretation of the frequencies Wn, Wn1 and Wn2 depends on wether an analog or a digital filter is designed.

  1. If an analog filter is designed using analogfilter, the frequencies are interpreted as analog frequencies in radians/second.
  2. If a digital filter is designed using digitalfilter and the sampling frequency fs is specified, the frequencies of the filter response type are normalized to fs. This requires that the sampling frequency and the filter response type use the same frequency unit (Hz, radians/second, ...). If fs is not specified, the frequencies of the filter response type are interpreted as normalized frequencies in half-cycles/sample.

[Filter design methods](@id design-methods)

IIR filter design methods

Butterworth
Chebyshev1
Chebyshev2
Elliptic

[Filter order estimation methods](@id order-est-methods)

IIR filter order estimation methods

buttord
cheb1ord
cheb2ord
ellipord

FIR filter order estimation methods

remezord

FIR filter design methods

FIRWindow

[Direct filter design methods](@id direct-design-methods)

remez
iirnotch

Filter response

freqresp
phaseresp
grpdelay
impresp
stepresp

Miscellaneous

coefb
coefa

Examples

Construct a 4th order elliptic lowpass filter with normalized cutoff frequency 0.2, 0.5 dB of passband ripple, and 30 dB attentuation in the stopband and extract the coefficients of the numerator and denominator of the transfer function:

responsetype = Lowpass(0.2)
designmethod = Elliptic(4, 0.5, 30)
tf = convert(PolynomialRatio, digitalfilter(responsetype, designmethod))
numerator_coefs = coefb(tf)
denominator_coefs = coefa(tf)

Filter the data in x, sampled at 1000 Hz, with a 4th order Butterworth bandpass filter between 10 and 40 Hz:

responsetype = Bandpass(10, 40)
designmethod = Butterworth(4)
filt(digitalfilter(responsetype, designmethod; fs=1000), x)

Filter the data in x, sampled at 50 Hz, with a 64 tap Hanning window FIR lowpass filter at 5 Hz:

responsetype = Lowpass(5)
designmethod = FIRWindow(hanning(64))
filt(digitalfilter(responsetype, designmethod; fs=50), x)

Estimate a Lowpass Elliptic filter order with a normalized passband cutoff frequency of 0.2, a stopband cutoff frequency of 0.4, 3 dB of passband ripple, and 40 dB attenuation in the stopband:

(N, ωn) = ellipord(0.2, 0.4, 3, 40)