digitalFilter

Digital filter

Description

Use a digitalFilter object to store digital filter coefficients and, if available, filter design specifications. You can use the object to analyze the filter response and to filter signals.

Use filter in the form dataOut = filter(d,dataIn) to filter a signal with a digitalFilter object d. You can also use the filtfilt and fftfilt functions with digitalFilter objects. For a complete list of functions that you can use with digitalFilter objects, see Object Functions.
Use Filter Analyzer to visualize a digitalFilter object.
Use Filter Designer to import and modify a digitalFilter object.

Creation

Create digitalFilter objects that contain coefficients representing Cascaded Transfer Functions (CTF). (since R2026a)

Note

To create a digitalFilter object with design specifications, use designfilt or the Filter Designer app.

Use the syntax d = designfilt(response,Name=Value) to design a digital filter d with response type response and design specifications listed as name-value arguments.
Use the syntax designfilt(d) or Filter Designer to edit an existing filter, d. The filter properties are otherwise read-only.

Syntax

d = digitalFilter(B,A)

d = digitalFilter(B,A,g)

d = digitalFilter(___,SampleRate=Fs)

Description

d = digitalFilter(B,A) creates a digital filter d with CTF numerator coefficients B and CTF denominator coefficients A. The filter uses normalized frequencies.

d = digitalFilter(B,A,g) specifies the scale values, g, across filter sections.

d = digitalFilter(___,SampleRate=Fs) sets a sample rate Fs to the filter for any of the previous syntaxes.

example

Input Arguments

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`B,A` — Cascaded transfer function (CTF) coefficients
scalars | vectors | matrices

Cascaded transfer function (CTF) coefficients, specified as scalars, vectors, or matrices. B and A list the numerator and denominator coefficients of the cascaded transfer function, respectively.

B must be of size L-by-(m + 1) and A must be of size L-by-(n + 1), where:

L represents the number of filter sections.
m represents the order of the filter numerators.
n represents the order of the filter denominators.

For more information about the cascaded transfer function format and coefficient matrices, see Specify Digital Filters in CTF Format.

Note

If any element of A(:,1) is not equal to 1, then digitalFilter normalizes the filter coefficients by A(:,1). In this case, A(:,1) must be nonzero.

Data Types: double
Complex Number Support: Yes

`g` — Scale values
`1` (default) | scalar | vector

Scale values, specified as a real-valued scalar or as a real-valued vector with L + 1 elements, where L is the number of CTF sections. The scale values represent the distribution of the filter gain across the sections of the cascaded filter representation.

The digitalFilter function applies a gain to the filter sections using the scaleFilterSections function depending on how you specify g:

Scalar — The function distributes the gain uniformly across all filter sections.
Vector — The function applies the first L gain values to the corresponding filter sections and distributes the last gain value uniformly across all filter sections.

Data Types: double

`Fs` — Sample rate of the digital filter
`"normalized"` | positive scalar

Sample rate of the digital filter, specified as one of these:

"normalized" — digitalFilter sets the digital filter d to use normalized frequency.
Positive scalar — digitalFilter sets the sample rate of the digital filter d to Fs Hz.

Data Types: char | string | single | double

Properties

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A digitalFilter object can have sets of properties that depend on how you create the object and whether the object represents a single-stage digital filter or a cascade of digital filters (digital filter cascade). All properties are read-only.

If the digitalFilter object represents a single-stage digital filter, including coefficients in CTF format, then the object has Coefficients properties.
If the digitalFilter object represents a digital filter cascade, then the object has Stages properties. (since R2026a)
If the digitalFilter object has design specifications, then the object has Specifications properties.

Coefficients

`Numerator` — Numerator coefficients
row vector | matrix

Numerator coefficients, returned as a row vector or as an L-by-(m + 1) matrix, where L is the number of CTF sections and m is the numerator order. See Return Digital Filters in CTF Format for more information.

`Denominator` — Denominator coefficients
`1` (default) | row vector | matrix

Denominator coefficients, returned as 1, a row vector, or an L-by-(n + 1) matrix, where L is the number of CTF sections and n is the denominator order. See Return Digital Filters in CTF Format for more information.

Stages

`Stage1,Stage2,...,StageN` — Filter stages
`digitalFilter` objects

Since R2026a

Filter stages, returned as digitalFilter objects. Each digitalFilter object represents a stage of a digital filter cascade. A stage can consist of one or more substages.

Specifications

The Specification properties returned in a digitalFilter object depend on the filter response and design specified when creating the object using designfilt. All properties are read-only.

`FrequencyResponse` — Frequency response type
`'lowpass'` | `'highpass'` | `'bandpass'` | `'bandstop'` | `'differentiator'` | ... | ...

Filter response type, returned as one of the values listed in the table.

Value	Description	Requires DSP System Toolbox™ License
`'lowpass'`	Lowpass filter
`'highpass'`	Highpass filter
`'bandpass'`	Bandpass filter
`'bandstop'`	Bandstop filter
`'differentiator'`	FIR differentiator filter
`'hilbert'`	FIR Hilbert transformer filter
`'arbmag'`	Filter of arbitrary-magnitude response
`'arbmagnphase'`	Filter of arbitrary magnitude and phase responses	✓
`'arbgrpdelay'`	IIR filter of arbitrary group delay	✓
`'fracdelay'`	FIR filter of fractional delay	✓
`'isinclp'`	Inverse sinc lowpass FIR filter	✓
`'isinchp'`	Inverse sinc highpass FIR filter	✓
`'notch'`	Multi-notch IIR filter	✓
`'peak'`	Multi-peak IIR filter	✓
`'complexlowpass'`	Complex FIR lowpass filter
`'complexhighpass'`	Complex FIR highpass filter

`ImpulseResponse` — Impulse response type
`'fir'` | `'iir'`

Impulse response type, returned as 'fir' or 'iir'.

`NormalizedFrequency` — Frequency normalization status
`1` (`true`) (default) | `0` (`false`)

Since R2026a

Frequency normalization status, returned as one of these:

1 (true) — The digitalFilter object works with normalized frequencies.
0 (false) — The digitalFilter object uses a sample rate in Hz, which is returned in the SampleRate property.

`SampleRate` — Sample rate
2 (default) | positive scalar

Sample rate, returned as a positive scalar expressed in hertz.

This property is visible only if NormalizedFrequency is false. (since R2026a)

A digitalFilter object can have more properties that describe its filter design. For more information about filter design properties, see Frequency Constraints, Filter Order, Magnitude Constraints, and Design Method.

Object Functions

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Filtering

`fftfilt`	FFT-based FIR filtering using overlap-add method
`filter`	1-D digital filter
`filtfilt`	Zero-phase digital filtering
`bandpass`	Bandpass-filter signals
`bandstop`	Bandstop-filter signals
`highpass`	Highpass-filter signals
`lowpass`	Lowpass-filter signals

Filter Responses

`freqz`	Frequency response of digital filter
`grpdelay`	Average filter delay (group delay)
`impz`	Impulse response of digital filter
`impzlength`	Impulse response length
`phasedelay`	Phase delay of digital filter
`phasez`	Phase response of digital filter
`stepz`	Step response of digital filter
`zerophase`	Zero-phase response of digital filter
`zplane`	Zero-pole plot for discrete-time systems

Filter Exploration

`filtord`	Filter order
`firtype`	Type of linear phase FIR filter
`getNumStages`	Get number of stages in digital filter cascade
`info`	Information about digital filter
`isallpass`	Determine whether filter is allpass
`isdouble`	Determine if digital filter coefficients are double precision
`isfir`	Determine if digital filter has finite impulse response
`islinphase`	Determine whether filter has linear phase
`ismaxphase`	Determine whether filter is maximum phase
`isminphase`	Determine whether filter is minimum phase
`issingle`	Determine if digital filter coefficients are single precision
`isstable`	Determine whether filter is stable

Filter Conversion

`double`	Cast coefficients of digital filter to double precision
`single`	Cast coefficients of digital filter to single precision
`cascade`	Cascade of filter system objects
`ctf`	Convert digital filter to coefficients in cascaded transfer function format
`ss`	Convert digital filter to state-space representation
`tf`	Convert digital filter to transfer function
`zpk`	Convert digital filter to zero-pole-gain representation
`filt2block`	Generate Simulink filter block
`setSampleRate`	Set sample rate of digital filter
`toMultirate`	Create multirate filter System object from digital FIR filter object

Examples

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Lowpass IIR Filter

Open Live Script

Design a lowpass IIR filter with order 8, passband frequency 35 kHz, and passband ripple 0.2 dB. Specify a sample rate of 200 kHz. Visualize the frequency response of the filter.

lpFilt = designfilt("lowpassiir",FilterOrder=8, ...
         PassbandFrequency=35e3,PassbandRipple=0.2, ...
         SampleRate=200e3);
freqz(lpFilt)

$Figure contains 2 axes objects. Axes object 1 with title Phase, xlabel Normalized Frequency (\times\pi rad/sample), ylabel Phase (degrees) contains an object of type line. Axes object 2 with title Magnitude, xlabel Normalized Frequency (\times\pi rad/sample), ylabel Magnitude (dB) contains an object of type line.$

Use the filter you designed to filter a 1000-sample random signal.

dataIn = randn(1000,1);
dataOut = filter(lpFilt,dataIn);

Output the filter coefficients, expressed as cascaded transfer functions.

ctfNum = lpFilt.Numerator

ctfNum = 4×3

    0.2666    0.5333    0.2666
    0.1943    0.3886    0.1943
    0.1012    0.2023    0.1012
    0.0318    0.0636    0.0318

ctfDen = lpFilt.Denominator

ctfDen = 4×3

    1.0000   -0.8346    0.9073
    1.0000   -0.9586    0.7403
    1.0000   -1.1912    0.5983
    1.0000   -1.3810    0.5090

Bandpass IIR Filters in CTF Format

Since R2026a

Open Live Script

Design bandpass IIR digital filters and filter a sinusoidal signal with harmonics.

Design a 10th-order bandpass IIR elliptic filter with passband edge frequencies of [0.3, 0.5] radians/sample, passband ripple of 0.1 dB, and stopband attenuation of 50 dB. Obtain the filter coefficients in CTF format.

N = 10;
Rp = 0.1;
Rs = 50;
Wp = [0.3 0.5];
[B,A] = ellip(N/2,Rp,Rs,Wp,"bandpass","ctf");

Create a digital filter object from the filter coefficients. Set the sample rate to 1000 Hz.

Fs = 1000;
dCoeff = digitalFilter(B,A,SampleRate=Fs);

Create a digital filter object from the filter specifications.

dSpecs = designfilt("bandpassiir",SampleRate=Fs, ...
    FilterOrder=N,DesignMethod="ellip", ...
    PassbandFrequency1=Wp(1)*Fs/2,PassbandFrequency2=Wp(2)*Fs/2, ...
    StopbandAttenuation1=Rs,PassbandRipple=Rp,StopbandAttenuation2=Rs);

Generate a sinusoidal signal with an amplitude of 1 V and a frequency of 60 Hz, and add third-, fifth-, and seventh-order harmonics. The harmonic amplitudes are 1 V, 0.2 V, and 0.1 V, respectively. Sample the signal for 1 second.

rng("default")
a = [1 1 0.2 0.1];
f = 60*[1 3 5 7];
t = (0:1/Fs:1)';
x = cos(2*pi*f.*t)*a' + 0.1*randn(size(t));

Filter the signal using the digitalFilter objects created from specifications and from CTF coefficients. Compare the input and output signals.

ySpecs = filtfilt(dSpecs,x);
yCoeff = filtfilt(dCoeff,x);

pspectrum([x ySpecs yCoeff],Fs)
title("Power Spectrum of Input and Output Signals")
legend(["Input" "Output (dSpecs filter)" "Output (dCoeff filter)"])
set(gca().Children(1),LineStyle="--")
ylim([-70 0])

Figure contains an axes object. The axes object with title Power Spectrum of Input and Output Signals, xlabel Frequency (Hz), ylabel Power Spectrum (dB) contains 3 objects of type line. These objects represent Input, Output (dSpecs filter), Output (dCoeff filter).

More About

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Digital Filter Cascades

A digital filter can be represented as a single stage or as a cascade of multiple stages. Accordingly, a digitalFilter object can represent either a single-stage digital filter or a cascade of digital filters (digital filter cascade).

Assume these filters: a bandstop FIR filter d1, bandpass IIR filter d2, and a Hilbert transformer FIR filter d3. The following block diagram represents the digital filter cascade d, which has three stages: d1, d2, and d3.

To create a digital filter cascade, use the cascade function. For example, the following code creates a digital filter cascade with stages d1, d2, and d3 from their corresponding digitalFilter objects.

d = cascade(d1,d2,d3);

You can cascade filters in multiple hierarchy levels. That way, you can get a multi-level digital filter cascade. In a multi-level digital filter cascade, each stage can have multiple substages. For example, the bandpass IIR filter d2 can be represented as a cascade of a lowpass IIR filter d21 and a highpass IIR filter d22. The following code creates a two-level cascaded filter with stages d1, d2, and d3, where d2 is a digital filter cascade with stages d21 and d22. All the inputs are digitalFilter objects.

d2 = cascade(d21,d22);
d = cascade(d1,d2,d3);

Cascaded Transfer Functions

Partitioning an IIR digital filter into cascaded sections improves its numerical stability and reduces its susceptibility to coefficient quantization errors. The cascaded form of a transfer function H(z) in terms of the L transfer functions H₁(z), H₂(z), …, H_L(z) is

$H (z) = \prod_{l = 1}^{L} H_{l} (z) = H_{1} (z) \times H_{2} (z) \times \dots \times H_{L} (z) .$

Specify Digital Filters in CTF Format

You can specify digital filters in the CTF format for analysis, visualization, and signal filtering. Specify a filter by listing its coefficients B and A. You can also include the filter scaling gain across sections by specifying a scalar or vector g.

Filter Coefficients

When you specify the coefficients as L-row matrices,

$B = [\begin{matrix} b_{11} & b_{12} & \dots & b_{1, m + 1} \\ b_{21} & b_{22} & \dots & b_{2, m + 1} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ b_{L 1} & b_{L 2} & \dots & b_{L, m + 1} \end{matrix}], A = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1, n + 1} \\ a_{21} & a_{22} & \dots & a_{2, n + 1} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{L 1} & a_{L 2} & \dots & a_{L, n + 1} \end{matrix}],$

it is assumed that you have specified the filter as a sequence of L cascaded transfer functions, such that the full transfer function of the filter is

$H (z) = \frac{b_{11} + b_{12} z^{- 1} + \dots + b_{1, m + 1} z^{- m}}{a_{11} + a_{12} z^{- 1} + \dots + a_{1, n + 1} z^{- n}} \times \frac{b_{21} + b_{22} z^{- 1} + \dots + b_{2, m + 1} z^{- m}}{a_{21} + a_{22} z^{- 1} + \dots + a_{2, n + 1} z^{- n}} \times \dots \times \frac{b_{L 1} + b_{L 2} z^{- 1} + \dots + b_{L, m + 1} z^{- m}}{a_{L 1} + a_{L 2} z^{- 1} + \dots + a_{L, n + 1} z^{- n}},$

where m ≥ 0 is the numerator order of the filter and n ≥ 0 is the denominator order.

If you specify both B and A as vectors, it is assumed that the underlying system is a one-section IIR filter (L = 1), with B representing the numerator of the transfer function and A representing its denominator.
If B is scalar, it is assumed that the filter is a cascade of all-pole IIR filters with each section having an overall system gain equal to B.
If A is scalar, it is assumed that the filter is a cascade of FIR filters with each section having an overall system gain equal to 1/A.

Note

To convert second-order section matrices to cascaded transfer functions, use the sos2ctf function.
To convert a zero-pole-gain filter representation to cascaded transfer functions, use the zp2ctf function.

Coefficients and Gain

If you have an overall scaling gain or multiple scaling gains factored out from the coefficient values, you can specify the coefficients and gain as a cell array of the form {B,A,g}. Scaling filter sections is especially important when you work with fixed-point arithmetic to ensure that the output of each filter section has similar amplitude levels, which helps avoid inaccuracies in the filter response due to limited numeric precision.

The gain can be a scalar overall gain or a vector of section gains.

If the gain is scalar, the value applies uniformly to all the cascade filter sections.
If the gain is a vector, it must have one more element than the number of filter sections L in the cascade. Each of the first L scale values applies to the corresponding filter section, and the last value applies uniformly to all the cascade filter sections.

If you specify the coefficient matrices and gain vector as

it is assumed that the transfer function of the filter system is

$H (z) = g_{S} (g_{1} \frac{b_{11} + b_{12} z^{- 1} + \dots + b_{1, m + 1} z^{- m}}{a_{11} + a_{12} z^{- 1} + \dots + a_{1, n + 1} z^{- n}} \times g_{2} \frac{b_{21} + b_{22} z^{- 1} + \dots + b_{2, m + 1} z^{- m}}{a_{21} + a_{22} z^{- 1} + \dots + a_{2, n + 1} z^{- n}} \times \dots \times g_{L} \frac{b_{L 1} + b_{L 2} z^{- 1} + \dots + b_{L, m + 1} z^{- m}}{a_{L 1} + a_{L 2} z^{- 1} + \dots + a_{L, n + 1} z^{- n}}) .$

Return Digital Filters in CTF Format

Get the filter coefficients by specifying to return B = d.Numerator and A = d.Denominator from a digitalFilter object d. That way, you can have digital filters in CTF format for analysis, visualization and signal filtering.

Filter Coefficients

When you specify to return the numerator and denominator coefficients in the CTF format, the L-row matrices B and A are returned as

$B = [\begin{matrix} b_{11} & b_{12} & \dots & b_{1, m + 1} \\ b_{21} & b_{22} & \dots & b_{2, m + 1} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ b_{L 1} & b_{L 2} & \dots & b_{L, m + 1} \end{matrix}], A = [\begin{matrix} 1 & a_{12} & \dots & a_{1, n + 1} \\ 1 & a_{22} & \dots & a_{2, n + 1} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & a_{L 2} & \dots & a_{L, n + 1} \end{matrix}],$

such that the full transfer function of the filter is

$H (z) = \frac{b_{11} + b_{12} z^{- 1} + \dots + b_{1, m + 1} z^{- m}}{1 + a_{12} z^{- 1} + \dots + a_{1, n + 1} z^{- n}} \times \frac{b_{21} + b_{22} z^{- 1} + \dots + b_{2, m + 1} z^{- m}}{1 + a_{22} z^{- 1} + \dots + a_{2, n + 1} z^{- n}} \times \dots \times \frac{b_{L 1} + b_{L 2} z^{- 1} + \dots + b_{L, m + 1} z^{- m}}{1 + a_{L 2} z^{- 1} + \dots + a_{L, n + 1} z^{- n}},$

where m ≥ 0 is the numerator order of the filter and n ≥ 0 is the denominator order.

References

[1] Lyons, Richard G. Understanding Digital Signal Processing. Upper Saddle River, NJ: Prentice Hall, 2004.

Version History

Introduced in R2014a

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R2026a: Create `digitalFilter` object using CTF format

You can create digital filters represented as coefficients of cascaded transfer functions.

R2026a: Use digital filter cascades

A digitalFilter object can represent a single-stage digital filter or a digital filter cascade.

The NormalizedFrequency property indicates if the digitalFilter object uses normalized frequency instead of a sample rate. If NormalizedFrequency is 0 (false), then the object shows the SampleRate property.
For a digital filter cascade, the Stage1, Stage2, …, StageN properties of a digitalFilter cascade object represent the stages of the filter.

These new object functions enables you to create, analyze, and modify a digital filter cascade.

cascade — Cascade two or more digital filter objects.
ctf — Return filter coefficients in cascaded-transfer-function (CTF) format. If the filter is a cascade of digital filters, it returns the coefficients of all the stages in CTF format.
getNumStages — Get the number of stages of a digital filter or digital filter cascade.
setSampleRate —Set the sample rate of a digital filter or digital filter cascade.

R2024b: Use `Numerator` and `Denominator` properties in `digitalFilter` object

Obtain the numerator and denominator coefficients of digital filters designed with Cascaded Transfer Functions. Starting in R2024b, the Numerator and Denominator properties replace the Coefficients property.

For a digitalFilter object filt, you must update your code to get the filter coefficients B (numerator) and A (denominator).

Impulse Response Type	Original Code in R2024a or Earlier	Updated Code in R2024b
IIR	B = filt.Coefficients(:,1:3); A = filt.Coefficients(:,4:6);	B = filt.Numerator; A = filt.Denominator;
FIR	B = filt.Coefficients;	B = filt.Numerator;

digitalFilter

Description

Creation

Syntax

Description

Input Arguments

B,A — Cascaded transfer function (CTF) coefficients scalars | vectors | matrices

g — Scale values 1 (default) | scalar | vector

Fs — Sample rate of the digital filter "normalized" | positive scalar

Properties

Coefficients

Numerator — Numerator coefficients row vector | matrix

Denominator — Denominator coefficients 1 (default) | row vector | matrix

Stages

Stage1,Stage2,...,StageN — Filter stages digitalFilter objects

Specifications

FrequencyResponse — Frequency response type 'lowpass' | 'highpass' | 'bandpass' | 'bandstop' | 'differentiator' | ... | ...

ImpulseResponse — Impulse response type 'fir' | 'iir'

NormalizedFrequency — Frequency normalization status 1 (true) (default) | 0 (false)

SampleRate — Sample rate 2 (default) | positive scalar

Object Functions

Filtering

Filter Responses

Filter Exploration

Filter Conversion

Examples

Lowpass IIR Filter

Bandpass IIR Filters in CTF Format

More About

Digital Filter Cascades

Cascaded Transfer Functions

Specify Digital Filters in CTF Format

Return Digital Filters in CTF Format

References

Version History

R2026a: Create digitalFilter object using CTF format

R2026a: Use digital filter cascades

R2024b: Use Numerator and Denominator properties in digitalFilter object

See Also

Live Editor Tasks

Apps

Objects

Functions

`B,A` — Cascaded transfer function (CTF) coefficients
scalars | vectors | matrices

`g` — Scale values
`1` (default) | scalar | vector

`Fs` — Sample rate of the digital filter
`"normalized"` | positive scalar

`Numerator` — Numerator coefficients
row vector | matrix

`Denominator` — Denominator coefficients
`1` (default) | row vector | matrix

`Stage1,Stage2,...,StageN` — Filter stages
`digitalFilter` objects

`FrequencyResponse` — Frequency response type
`'lowpass'` | `'highpass'` | `'bandpass'` | `'bandstop'` | `'differentiator'` | ... | ...

`ImpulseResponse` — Impulse response type
`'fir'` | `'iir'`

`NormalizedFrequency` — Frequency normalization status
`1` (`true`) (default) | `0` (`false`)

`SampleRate` — Sample rate
2 (default) | positive scalar

R2026a: Create `digitalFilter` object using CTF format

R2024b: Use `Numerator` and `Denominator` properties in `digitalFilter` object