Digital Filter Design

Digital Filter Design Methods	Output Response—Digital Filtering of Signals
Poles and Zeros of Digital Filters

The Wolfram Language provides a comprehensive set of methods for designing digital filters.

Digital Filter Design Methods

LeastSquaresFilterKernel	create an FIR filter using the least-squares method
FrequencySamplingFilterKernel	create an FIR filter using frequency sampling
EquirippleFilterKernel	create an FIR filter using the equiripple method
ToDiscreteTimeModel	create an IIR digital filter from an analog prototype

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 sample.

H(^{j ω}){

1,	\|ω\| ≤ ω_c
0,	otherwise

The impulse response is obtained by taking the inverse discrete-time Fourier transform of the filter specification:

Evaluating this expression for integer values of

from

will return a filter of length

Create a lowpass FIR filter of length 15 with a cutoff frequency of 1.2 radians per sample:

This method minimizes the mean-squared error between the ideal specification and the resulting FIR filter:

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:

Apply the window to the lowpass FIR filter h:

Different windows return different attenuations. Select a window type by the degree of the desired attenuation.

Compare attenuation levels of four selected windows

Frequency Sampling Method

Frequency sampling allows the creation of an FIR filter by specifying the filter's amplitude spectrum on the interval 0≤ ω≤ π at a finite number of uniformly spaced frequency locations.

Sample an ideal lowpass filter at

for

from 0 to 5:

Create an odd-length, even-symmetry lowpass filter:

Show the ideal lowpass filter, amplitude samples, and the resulting filter:

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:

Plot the magnitude response of an equiripple multiband filter:

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.

ToDiscreteTimeModel

discrete-time approximation of an analog filter

Create an IIR filter from an analog prototype.

Create an analog lowpass filter of Chebyshev type with a cutoff frequency at

radians per second:

Transform the analog filter to a digital filter using bilinear transformation:

Show the frequency response of the resulting IIR filter:

Poles and Zeros of Digital Filters

TransferFunctionPoles	extract poles of analog filters
TransferFunctionZeros	extract zeros of analog filters

Poles and zeros of analog filters.

Compute and show zeros of a lowpass filter:

Output Response—Digital Filtering of Signals

FIR Filters

ListConvolve	convolve an FIR filter with data
ImageConvolve	convolve an FIR filter with image
DiscreteConvolve	symbolic convolution of two signals

Apply FIR filters to signals.

Create a lowpass equiripple filter:

Import and filter an audio signal:

Lowpass filtering of a scanline of the red channel of an image:

Lowpass filtering of an image:

IIR Filters

RecurrenceFilter	apply an IIR filter to a signal
RecurrenceTable	iteratively solve the recurrence equation
OutputResponse	output response of a filter

Apply IIR filters to signals.

Define a transfer function of a digital Butterworth filter:

Compute the impulse response of the IIR filter:

Compute the impulse response by using the coefficients of the transfer function model:

Compute the impulse response using OutputResponse:

Compute the impulse response by solving the corresponding recurrence equation model of the filter using RecurrenceTable: