Digital Filter Design
Mathematica provides a comprehensive set of methods for designing digital filters.
Digital Filter Design Methods
Digital filter design methods.
Least-Squares Method
This method obtains a finite impulse response (FIR) from a given prototype filter specification in the frequency domain by means of the inverse discrete-time Fourier transform.
Define a zero-phase ideal lowpass filter with a cutoff frequency of 1.2 radians per second.
The impulse response is obtained by taking the inverse discrete-time Fourier transform of the filter specification.
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Evaluating this expression for integer values of 
from 
to 
will return a filter of length 
.
Create a lowpass FIR filter of length 15 with a cutoff frequency of 1.2 radians per second.
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This method minimizes the mean-squared error between the ideal specification and the resulting FIR filter.
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The least-squares method is also known as the window-based method. The mean-squared error can be diminished by applying a smoothing window to the FIR returned by LeastSquaresFilterKernel.
Create a Hann window of length 15.
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Apply the window to the lowpass FIR filter
h.
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Different windows return different attenuations. Select a window type by the degree of the desired attenuation.
Compare attenuation levels of four selected windows
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Frequency Sampling Method
Frequency sampling allows the creation of an FIR filter by specifying the filter's amplitude spectrum on the interval 
at a finite number of uniformly spaced frequency locations.
Sample an ideal lowpass filter at
for
from 0 to 5.
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Create an odd-length, even-symmetry lowpass filter.
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Show the ideal lowpass filter, amplitude samples, and the resulting filter.
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Equiripple Method
The Parks-McClellan-Rabiner (1979) algorithm is one of the most popular methods of designing linear-phase FIR filters. It minimizes the maximum deviation from the desired ideal frequency response and results in filters with equal ripples in each of the bands of the filter.
Create a multiband filter of length 55.
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Plot the magnitude response of an equiripple multiband filter.
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Create a Digital Filter from an Analog Prototype
A popular method of creating digital filters is to transform analog prototypes to their digital equivalents using the bilinear transformation. The result is an infinite impulse response (IIR) filter, represented as a TransferFunctionModel.
Create an IIR filter from an analog prototype.
Create an analog lowpass filter of Chebyshev type with a cutoff frequency at
radians per second.
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Transform the analog filter to a digital filter using bilinear transformation.
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Show the frequency response of the resulting IIR filter.
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Poles and Zeros of Digital Filters
Poles and zeros of analog filters.
Compute and show zeros of a lowpass filter.
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Output Response—Digital Filtering of Signals
FIR Filters
Apply FIR filters to signals.
Create a lowpass equiripple filter.
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Import and filter an audio signal.
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Lowpass filtering of a scanline of the red channel of an image.
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Lowpass filtering of an image.
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IIR Filters
Apply FIR filters to signals.
Define a transfer function of a digital Butterworth filter.
Compute the impulse response of the IIR filter.
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Compute the impulse response by using the coefficients of the transfer function model.
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Compute the impulse response by solving the corresponding recurrence equation model of the filter using
RecurrenceTable.
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Lowpass filtering of a scanline of the red channel of an image.
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Lowpass filtering of an image.
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