# dsp.FrequencyDomainFIRFilter

Filter input signal in frequency domain

## Description

The `dsp.FrequencyDomainFIRFilter`

System object™ implements frequency-domain, fast Fourier transform (FFT)-based filtering to
filter a streaming input signal. In the time domain, the filtering operation involves a
convolution between the input and the impulse response of the finite impulse response (FIR)
filter. In the frequency domain, the filtering operation involves the multiplication of the
Fourier transform of the input and the Fourier transform of the impulse response. The
frequency-domain filtering is efficient when the impulse response is very long. You can
specify the filter coefficients directly in the frequency domain by setting
`NumeratorDomain`

to `"Frequency"`

.

This object uses the overlap-save and overlap-add methods to perform the frequency-domain
filtering. For an example showing these two methods, see Frequency Domain Filtering Using Overlap-Add and Overlap-Save. For filters with a long
impulse response length, the latency inherent to these two methods can be significant. To
mitigate this latency, the `dsp.FrequencyDomainFIRFilter`

object partitions the
impulse response into shorter blocks and implements the overlap-save and overlap-add methods
on these shorter blocks. To partition the impulse response, set the
`PartitionForReducedLatency`

property to `true`

. For an
example, see Reduce Latency Through Partitioned Numerator. For more details on these
two methods and on reducing latency through impulse response partitioning, see Algorithms.

The `dsp.FrequencyDomainFIRFilter`

object can model multiple-input multiple-output (MIMO) systems by supporting multiple filters
in the time domain and the frequency domain. For an example, see Filter Input Signal Using 3-by-2 MIMO System. The object does not support
partitioning the impulse response when you specify multiple filters. You can also specify
multiple paths between each transmitter (input channel) and receiver (output channel) pair
using the `NumPaths`

property. For an example, see Filter Input Signal Through Multiple Propagation Paths. For more information on
modeling MIMO systems, see Modeling MIMO System with Multiple Propagation Paths.* (since R2023b)*

To filter the input signal in the frequency domain:

Create the

`dsp.FrequencyDomainFIRFilter`

object and set its properties.Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?

## Creation

### Syntax

### Description

creates a
frequency domain FIR filter System object that filters each channel of the input signal independently over time in the
frequency domain using the overlap-save or overlap-add method.`fdf`

= dsp.FrequencyDomainFIRFilter

creates a frequency domain FIR filter object with the `fdf`

= dsp.FrequencyDomainFIRFilter(num)`Numerator`

property set to `num`

.

**Example: **`dsp.FrequencyDomainFIRFilter(fir1(400,2*2000/8000))`

creates a frequency domain FIR filter System object with each specified property set to the specified value. You can use this
syntax with any previous input argument combinations.`fdf`

= dsp.FrequencyDomainFIRFilter(`Name=Value`

)

**Example: **`dsp.FrequencyDomainFIRFilter(Method="Overlap-add")`

## Properties

Unless otherwise indicated, properties are *nontunable*, which means you cannot change their
values after calling the object. Objects lock when you call them, and the
`release`

function unlocks them.

If a property is *tunable*, you can change its value at
any time.

For more information on changing property values, see System Design in MATLAB Using System Objects.

`Method`

— Frequency-domain filter method

`"Overlap-save"`

(default) | `"Overlap-add"`

Frequency-domain filter method, specified as either
`"Overlap-save"`

or `"Overlap-add"`

. For more
details on these two methods, see Algorithms.

`NumeratorDomain`

— Numerator domain

`"Time"`

(default) | `"Frequency"`

Domain of the filter coefficients, specified as one of the following:

`"Time"`

–– Specify the time-domain filter numerator in the`Numerator`

property.`"Frequency"`

–– Specify the filter's frequency response in the`FrequencyResponse`

property.

`Numerator`

— FIR filter coefficients

`fir1(100,0.3)`

(default) | row vector | matrix

FIR filter coefficients, specified as a row vector or a matrix.

When you set `PartitionForReducedLatency`

to
`true`

, `Numerator`

must be a row vector and
represents a single filter.

When you set
`PartitionForReducedLatency`

to `false`

,
`Numerator`

can be a row vector or a matrix of size
*F*-by-*NumLen*, where *F* is the
number of filters and *NumLen* is the filter length. If
*F* is greater than 1, then its value must be a multiple of the
product of the number of input channels (columns) *T* and the number of
paths *P* you specify through the `NumPaths`

property. The multiplication factor determines the number of receivers (output channels)
*R* and equals *F*/(*T* ×
*P*).* (since R2023b)*

The coefficient values can change during simulation but the size of the numerator must remain constant.

**Tunable: **Yes

#### Dependencies

To enable this property, set `NumeratorDomain`

to
`"Time"`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

**Complex Number Support: **Yes

`FrequencyResponse`

— Filter frequency response

`fft(fir1(100,0.3),202)`

(default) | row vector | matrix

Frequency response of the filter, specified as a row vector or a matrix.

When you set `PartitionForReducedLatency`

to
`true`

, `FrequencyResponse`

must be a
2*PL*-by-*N* matrix and represents a single filter,
where *PL* is the partition size and *N* is the number
of partitions.

When you set
`PartitionForReducedLatency`

to `false`

,
`FrequencyResponse`

can be a row vector or a matrix of size
*F*-by-*FFTLength*, where *F* is
the number of filters. If *F* is greater than 1, then its value must be
a multiple of the product of the number of input channels (columns) *T*
and the number of paths *P* you specify through the
`NumPaths`

property. The multiplication factor determines the
number of receivers (output channels) *R* and equals
*F*/(*T* × *P*).* (since R2023b)*

**Tunable: **Yes

#### Dependencies

To enable this property, set `NumeratorDomain`

to
`"Frequency"`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

**Complex Number Support: **Yes

`NumeratorLength`

— Time-domain numerator length

`101`

(default) | positive integer

Time-domain numerator length, specified as a positive integer.

#### Dependencies

To enable this property, set `NumeratorDomain`

to
`"Frequency"`

and `PartitionForReducedLatency`

to `false`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`FilterIsReal`

— Flag to specify if filter is real

`true`

(default) | `false`

Flag to specify if the filter coefficients are all real, specified as
`true`

or `false`

.

#### Dependencies

To enable this property, set `NumeratorDomain`

to
`"Frequency"`

.

`PartitionForReducedLatency`

— Flag to partition numerator to reduce latency

`false`

(default) | `true`

Flag to partition numerator to reduce latency, specified as:

`false`

–– The filter uses the traditional overlap-save or overlap-add method. The latency in this case is`FFTLength`

– length(`Numerator`

) + 1.`true`

–– In this mode, the object partitions the numerator into segments of length specified by the`PartitionLength`

property. The filter performs overlap-save or overlap-add on each partition, and combines the partial results to form the overall output. The latency is now reduced to the partition length.

`FFTLength`

— FFT length

`[]`

(default) | positive integer

FFT length, specified as a positive integer. The default value of this property,
`[]`

, indicates that the FFT length is equal to twice the numerator
length. The FFT length must be greater than or equal to the numerator length.

#### Dependencies

To enable this property, set `NumeratorDomain`

property to
`"Time"`

and `PartitionForReducedLatency`

property to `false`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`PartitionLength`

— Numerator partition length

`32`

(default) | positive integer

Numerator partition length *PL*, specified as a positive integer
less than or equal to the length of the numerator.

#### Dependencies

To enable this property, set the `NumeratorDomain`

property to
`"Time"`

and `PartitionForReducedLatency`

property to `true`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`Latency`

— Filter latency

`102`

(default) | positive integer

This property is read-only.

Filter latency in samples, returned as an integer greater than 0. When
`PartitionForReducedLatency`

is:

`false`

–– The latency is equal to`FFTLength`

– length(`Numerator`

) + 1.`true`

–– The latency is equal to the partition length.

**Data Types: **`uint32`

`NumPaths`

— Number of propagation paths between each transmitter and receiver pair

`1`

(default) | positive integer

*Since R2023b*

Number of propagation paths *P* between each transmitter (input
channel) and receiver (output channel) pair, specified as a positive integer. Each path
is represented by a unique filter. When you specify the number of paths to be greater
than `1`

, the filter models multipath propagation.

For an example, see Filter Input Signal Through Multiple Propagation Paths.

#### Dependencies

To enable this property, the number of filters *F* must be
greater than 1 and you must set `PartitionForReducedLatency`

to
`false`

.

**Data Types: **`single`

| `double`

`SumFilteredOutputs`

— Option to sum filtered output contributions from all input channels

`true`

(default) | `false`

*Since R2023b*

Option to sum filtered output contributions from all input channels, specified as:

`true`

–– The object adds the filtered output from each input channel for a receiver to generate an*L*-by-*R*output matrix, where*L*is the input frame size (number of input rows) and*R*is the number of receivers (output channels).*R*equals*F*/(*T*x*P*), where*F*is the number of filters,*T*is the number of transmitters (input channels), and*P*is the value that you specify in the`NumPaths`

property.`false`

–– The object does not sum filtered output contributions from all input channels. The output is an*L*-by-*R*-by-*T*array.

For more information on how the object computes the output based on the value of the
`SumFilteredOutputs`

property, see Modeling MIMO System with Multiple Propagation Paths.

#### Dependencies

To enable this property, the number of filters *F* must be
greater than 1 and you must set `PartitionForReducedLatency`

to
`false`

.

**Data Types: **`logical`

## Usage

### Syntax

### Description

### Input Arguments

`input`

— Data input

matrix

Data input, specified as a matrix of size
*L*-by-*T*.

This object supports variable-size input signals. That is, you can change the input frame size (number of rows) even after calling the algorithm. However, the number of channels (columns) must remain constant.

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

### Output Arguments

`output`

— Filtered output

matrix | 3-*D* array

Filtered output, returned as a matrix or a 3-*D* array.

When you set
`SumFilteredOutputs`

to:

`true`

–– The object adds the filtered output from each input channel for a receiver to generate an*L*-by-*R*output matrix, where*L*is the input frame size (number of input rows) and*R*is the number of receivers (output channels).*R*equals*F*/(*T*x*P*), where*F*is the number of filters,*T*is the number of transmitters (input channels), and*P*is the value that you specify in the`NumPaths`

property.`false`

–– The object does not sum filtered output contributions from all input channels. The output is an*L*-by-*R*-by-*T*array.`output(:,j,k)`

refers to the output from the*k*^{th}input channel for the*j*^{th}receiver. For example,`output(:,3,2)`

indicates output from the second transmitter for the third receiver.

* (since R2023b)*

For more information on how the object computes the output, see Algorithms.

The output has the same data type and complexity as the input signal.

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

## Object Functions

To use an object function, specify the
System object as the first input argument. For
example, to release system resources of a System object named `obj`

, use
this syntax:

release(obj)

## Examples

### Frequency Domain Filtering Using Overlap-Add and Overlap-Save

Filter input signal using overlap-add and overlap-save methods, and compare the outputs to the output of a time-domain FIR filter.

**Initialization**

Design the FIR lowpass filter coefficients using the `designLowpassFIR`

function. The sampling frequency is 8 kHz. Specify a filter order of 400 and a cutoff frequency of 2 kHz. The impulse response has a length of 401.

order = 400; Fs = 8000; Fcutoff = 2000; imp = designLowpassFIR(FilterOrder=order,CutoffFrequency=2*Fcutoff/Fs);

Create two `dsp.FrequencyDomainFIRFilter`

objects and a `dsp.FIRFilter`

object. Set the numerator of all the three filters to `imp`

. Delay the FIR output by the latency of the frequency-domain filter.

fdfOA = dsp.FrequencyDomainFIRFilter(imp,Method="Overlap-add"); fdfOS = dsp.FrequencyDomainFIRFilter(imp,Method="Overlap-save"); fir = dsp.FIRFilter(Numerator=imp); dly = dsp.Delay(Length=fdfOA.Latency);

Create two `dsp.SineWave`

objects. The sine waves have a sample rate of 8000 Hz, frame size of 256, and frequencies of 100 Hz and 3 kHz, respectively. Create a `timescope`

object to view the filtered outputs.

frameLen = 256; sin_100Hz = dsp.SineWave(Frequency=100,SampleRate=Fs,... SamplesPerFrame=frameLen); sin_3KHz = dsp.SineWave(Frequency=3e3,SampleRate=Fs,... SamplesPerFrame=frameLen); ts = timescope(TimeSpanSource="property",TimeSpan=5*frameLen/Fs,... YLimits=[-1.1 1.1],... ShowLegend=true,... SampleRate=Fs,... ChannelNames={"Overlap-add","Overlap-save","Direct-form FIR"});

**Streaming**

Stream 1e4 frames of noisy input data. Pass this data through the frequency domain filters and the time-domain FIR filter. View the filtered outputs in the time scope.

numFrames = 1e4; for idx = 1:numFrames x = sin_100Hz() + sin_3KHz() + 0.01*randn(frameLen,1); yOA = fdfOA(x); yOS = fdfOS(x); yFIR = fir(dly(x)); ts([yOA,yOS,yFIR]); end

The outputs of all the three filters match exactly.

### Reduce Latency Through Partitioned Numerator

Partition the impulse response length of a frequency domain FIR filter. Compare the outputs of the partitioned filter and the original filter.

Design the FIR lowpass filter coefficients using the `designLowpassFIR`

function. The sampling frequency is 8 kHz. Specify a filter order of 4000 and a cutoff frequency of 2 kHz. The impulse response is of length 4001.

order = 4000; Fs = 8000; Fcutoff = 2000; imp = designLowpassFIR(FilterOrder=order,CutoffFrequency=2*Fcutoff/Fs);

Create a `dsp.FrequencyDomainFIRFilter`

with coefficients set to the `imp`

vector. The latency of this filter is given by , which is equal to 4002. By default, FFT length is equal to twice the numerator length. This makes the latency proportional to the impulse response length.

fdfOS = dsp.FrequencyDomainFIRFilter(imp,Method="Overlap-save"); fprintf('Frequency domain filter latency is %d samples\n',fdfOS.Latency);

Frequency domain filter latency is 4002 samples

Partition the impulse response to blocks of length 256. The latency after partitioning is proportional to the block length.

fdfRL = dsp.FrequencyDomainFIRFilter(imp,Method="Overlap-save",... PartitionForReducedLatency=true,... PartitionLength=256); fprintf('Frequency domain filter latency is %d samples\n',fdfRL.Latency);

Frequency domain filter latency is 256 samples

Compare the outputs of the two frequency domain filters. The latency of `fdfOS`

is 4002, and the latency of `fdfRL`

is 256. To compare the two outputs, delay the input to `fdfRL`

by 4002 - 256 samples.

dly = dsp.Delay(Length=(fdfOS.Latency-fdfRL.Latency));

Create two `dsp.SineWave`

objects. The sine waves have a sample rate of 8000 Hz, frame size of 256, and frequencies of 100 Hz and 3 kHz, respectively. Create a `timescope`

object to view the filtered outputs.

frameLen = 256; sin_100Hz = dsp.SineWave(Frequency=100,SampleRate=Fs,... SamplesPerFrame=frameLen); sin_3KHz = dsp.SineWave(Frequency=3e3,SampleRate=Fs,... SamplesPerFrame=frameLen); ts = timescope(TimeSpanSource="property",TimeSpan=5*frameLen/Fs,... YLimits=[-1.1 1.1],... ShowLegend=true,... SampleRate=Fs,... ChannelNames={'Overlap-save With Partition','Overlap-save Without Partition'});

Stream 1e4 frames of noisy input data. Pass this data through the two frequency domain filters. View the filtered outputs in the time scope. The outputs match exactly.

numFrames = 1e4; for idx = 1:numFrames x = sin_100Hz() + sin_3KHz() + .1 * randn(frameLen,1); yRL = fdfRL(dly(x)); yOS = fdfOS(x); ts([yRL,yOS]); end

### Specify Frequency Response of the Frequency-Domain FIR Filter

Specify the numerator coefficients of the frequency-domain FIR filter in the frequency domain. Filter the input signal using the overlap-add method. Compare the frequency-domain FIR filter output to the corresponding time-domain FIR filter output.

**Initialization**

Design the FIR lowpass filter coefficients using the `designLowpassFIR`

function. The sampling frequency is 8 kHz, and the cutoff frequency of the filter is 2 kHz. The time-domain impulse response has a length of 401. Compute the FFT of this impulse response and specify this response as the frequency response of the frequency-domain FIR filter. Set the time-domain numerator length, specified by the `NumeratorLength`

property, to the number of elements in the time-domain impulse response.

order = 400; Fs = 8000; Fcutoff = 2000; imp = designLowpassFIR(FilterOrder=order,CutoffFrequency=2*Fcutoff/Fs); H = fft(imp,2*numel(imp)); oa = dsp.FrequencyDomainFIRFilter(NumeratorDomain="Frequency",... FrequencyResponse=H,... NumeratorLength=numel(imp),... Method='Overlap-add'); fprintf('Frequency domain filter latency is %d samples\n',oa.Latency);

Frequency domain filter latency is 402 samples

Create a `dsp.FIRFilter`

System object and specify the numerator as the time-domain coefficients computed using the `designLowpassFIR`

function, `imp`

. Delay the FIR output by the latency of the frequency-domain FIR filter.

fir = dsp.FIRFilter(Numerator=imp); dly = dsp.Delay(Length=oa.Latency);

Create two `dsp.SineWave`

objects. The sine waves generated have a sample rate of 8000 Hz, frame size of 256, and frequencies of 100 Hz and 3 kHz, respectively. Create a `timescope`

object to view the filtered outputs.

frameLen = 256; sin_100Hz = dsp.SineWave(Frequency=100,SampleRate=Fs,... SamplesPerFrame=frameLen); sin_3KHz = dsp.SineWave(Frequency=3e3,SampleRate=Fs,... SamplesPerFrame=frameLen); ts = timescope(YLimits=[-1.1 1.1],... TimeSpanSource="property",TimeSpan=5*frameLen/Fs,... ShowLegend=true,... SampleRate=Fs,... ChannelNames={'Overlap-add','Direct-form FIR'});

**Streaming**

Stream 1e4 frames of noisy input data. Pass this data through the frequency-domain FIR filter and the time-domain FIR filter. View the filtered outputs in the time scope. The outputs of both the filters match exactly.

numFrames = 1e4; for idx = 1:numFrames x = sin_100Hz() + sin_3KHz() + 0.01 * randn(frameLen,1); y1 = oa(x); y2 = fir(dly(x)); ts([y1,y2]); end

### Filter Input Signal Using 3-by-2 MIMO System

*Since R2023b*

Design six lowpass FIR filters with varying cutoff frequencies. The filter order for each filter is 400 and the sampling rate is 8000 Hz. The system models three transmitters (input channels), two receivers (output channels), and one path between each transmitter-receiver pair.

order = 400; Fs = 8000; num = [% Transmitter 1, Receiver 1 designLowpassFIR(FilterOrder=order,CutoffFrequency=0.1);... % Transmitter 1, Receiver 2 designLowpassFIR(FilterOrder=order,CutoffFrequency=0.15);... % Transmitter 2, Receiver 1 designLowpassFIR(FilterOrder=order,CutoffFrequency=0.2);... % Transmitter 2, Receiver 2 designLowpassFIR(FilterOrder=order,CutoffFrequency=0.25);... % Transmitter 3, Receiver 1 designLowpassFIR(FilterOrder=order,CutoffFrequency=0.3);... % Transmitter 3, Receiver 2 designLowpassFIR(FilterOrder=order,CutoffFrequency=0.35)];

Initialize the `dsp.FrequencyDomainFIRFilter`

object with the array of filters. Specify the filtering method as `"Overlap-save"`

. The `SumFilteredOutputs`

property is `true`

by default.

filt = dsp.FrequencyDomainFIRFilter(Numerator=num,... Method='Overlap-save')

filt = dsp.FrequencyDomainFIRFilter with properties: Method: 'Overlap-save' NumeratorDomain: 'Time' Numerator: [6x401 double] NumPaths: 1 SumFilteredOutputs: true FFTLength: [] Latency: 402

The input contains two sinusoidal signals each with a frame length of 256. The first sinusoidal signal contains tones at 100 Hz, 200 Hz, and at 300 Hz. The second sinusoidal signal contains tones at 2 kHz, 2.5 kHz, and at 3 kHz.

frameLen = 256; sin_Hz = dsp.SineWave(Frequency=[100 200 300],SampleRate=Fs,... SamplesPerFrame=frameLen); sin_KHz = dsp.SineWave(Frequency=[2e3 2.5e3 3e3],SampleRate=Fs,... SamplesPerFrame=frameLen);

Initialize a `spectrumAnalyzer`

object to view the spectrum of the input and the filtered output.

specScope = spectrumAnalyzer(SampleRate=Fs,PlotAsTwoSidedSpectrum=false, ... ChannelNames={'Transmitter 1','Transmitter 2','Transmitter 3', ... 'Receiver 1','Receiver 2'},ShowLegend=true);

Stream in 1e4 frames of the noisy input sinusoidal signal. The input noise is white Gaussian with a mean of 0 and a variance of 0.01. Pass the signal through the designed filter. Visualize the spectrum of the input and output signals in spectrum analyzer.

for idx = 1:1e4 x = sin_Hz()+sin_KHz()+0.01*randn(frameLen,3); y = filt(x); specScope([x,y]); end

### Filter Input Signal Through Multiple Propagation Paths

*Since R2023b*

Filter an input signal through three distinct paths between the input and the output by specifying three filters to model the frequency responses across each path.

Design multiple lowpass FIR filters with varying fractional delays. The filter order for each filter is 400 and the sampling rate is 8000 Hz.

order = 400; Fs = 8000; num = [designFracDelayFIR(0.1,order); % Path 1 designFracDelayFIR(0.2,order); % Path 2 designFracDelayFIR(0.3,order); % Path 3 ];

Initialize the `dsp.FrequencyDomainFIRFilter`

object with the array of filters. Set the filtering method to `"Overlap-add"`

and the number of propagation paths to 3. The object models a single-input single-output (SISO) system. The `SumFilteredOutputs`

property is `true`

by default.

filt = dsp.FrequencyDomainFIRFilter(Numerator=num, ... Method='overlap-add',NumPaths=3)

filt = dsp.FrequencyDomainFIRFilter with properties: Method: 'Overlap-add' NumeratorDomain: 'Time' Numerator: [3x400 double] NumPaths: 3 SumFilteredOutputs: true FFTLength: [] Latency: 401

The input contains two sinusoidal signals each with a frame length of 1024. The first sinusoidal signal contains a tone at 200 Hz. The second sinusoidal signal contains a tone at 4 kHz.

frameLen = 1024; sin_200Hz = dsp.SineWave(Frequency=200,SampleRate=Fs, ... SamplesPerFrame=frameLen); sin_4KHz = dsp.SineWave(Frequency=4e3,SampleRate=Fs, ... SamplesPerFrame=frameLen);

Initialize a `spectrumAnalyzer`

object to view the spectrum of the input and the filtered output.

specScope = spectrumAnalyzer(SampleRate=Fs,PlotAsTwoSidedSpectrum=false, ... ChannelNames={'Transmitter','Receiver'},ShowLegend=true);

Stream in 1e3 frames of the noisy input sinusoidal signal. The input noise is white Gaussian with a mean of 0 and a variance of 0.01. Pass the signal through the designed filter. Visualize the spectrum of the input and output signals in spectrum analyzer.

for idx = 1:1e3 x = sin_200Hz()+sin_4KHz()+0.01*randn(frameLen,1); y = filt(x); specScope([x,y]) end

## Algorithms

Overlap-save and overlap-add are the two frequency-domain FFT-based filtering methods this algorithm uses.

### Overlap-Save

The overlap-save method is implemented using the following approach:

The input stream is partitioned into overlapping blocks of size *FFTLen*, with an overlap factor of *NumLen* – 1 samples. *FFTLen* is the FFT length and *NumLen* is the length of the FIR filter numerator. The FFT of each block of input samples is computed and multiplied with the length-*FFTLen* FFT of the FIR numerator. The inverse fast Fourier transform (IFFT) of the result is performed, and the last *FFTLen* – *NumLen* + 1 samples are saved. The remaining samples are dropped.

The latency of overlap-save is *FFTLen* – *NumLen* + 1. The first *FFTLen* – *NumLen* + 1 samples are equal to zero. The filtered value of the first input sample appears as the *FFTLen* – *NumLen* + 2 output sample.

Note that the FFT length must be larger than the numerator length, and is typically set to a value much greater than *NumLen*.

### Overlap-Add

The overlap-add method is implemented using the following approach:

The input stream is partitioned into blocks of length *FFLen* – *NumLen* + 1, with no overlap between consecutive blocks. Similar to overlap-save, the FFT of the block is computed, and multiplied by the FFT of the FIR numerator. The IFFT of the result is then computed. The first *NumLen* + 1 samples are modified by adding the values of the last *NumLen* + 1 samples from the previous computed IFFT.

The latency of overlap-add is *FFTLen* – *NumLen* + 1. The first *FFTLen* – *NumLen* + 1 samples are equal to zero. The filtered value of the first input sample appears as the *FFTLen* – *NumLen* + 2 output sample.

### Reduce Latency Through Impulse Response Partitioning

With an FFT length that is twice the length of the FIR numerator, the latency roughly equals the length of the FIR numerator. If the impulse response is very long, the latency becomes significantly large. However, frequency domain FIR filtering is still faster than the time-domain filtering. To mitigate the latency and make the frequency domain filtering even more efficient, the algorithm partitions the impulse response into multiple short blocks and performs overlap-save or overlap-add on each block. The results of the different blocks are then combined to obtain the final output. The latency of this approach is of the order of the block length, rather than the entire impulse response length. This reduced latency comes at the cost of additional computation. For more details, see [1].

### Modeling MIMO System with Multiple Propagation Paths

These diagrams present a generalized representation of a frequency-domain FIR filter using multiple filters to process signals between multiple transmitters and receivers. This diagram models multiple paths between each transmitter-receiver pair.

`SumFilteredOutputs`

is
`true`

The algorithm passes the first input channel through the first set of
*R* × *P* filters within which
((*i* − 1) × *P* + 1) to (*i* ×
*P*) filters compute the output for the *i*th
receiver. The algorithm passes the second input channel through the next set of
*R* × *P* filters, and the sequence continues
until the last input channel passes through the filter *F*.

The algorithm adds the filtered output contributions from all input channels before sending them to the respective receiver.

The input signal is of size

*L*-by-*T*. Each input channel corresponds to a signal sent by the specific transmitter.*L*is the input frame size (number of input rows)*T*is the number of transmitters (input channels)*F*is the number of filters. This value is the number of rows in the filter coefficients matrix.Filter coefficients is a matrix of size

*F*-by-*numLen*. Each row corresponds to a filter of length*numLen*.*P*is the number of propagation paths between each transmitter and receiver*R*is the number of receivers (output channels) and equals*F*/(*T*x*P*)Output signal is of size

*L*-by-*R*. Each receiver receives a specific output channel of size*L*-by-1.

`SumFilteredOutputs`

is
`false`

The algorithm concatenates the filtered outputs and does not sum across input
channels. The output is an
*L*-by-*R*-by-*T* array that you
can further process before sending it to the receiver. For example, if you add elements
in the *L*-by-*R*-by-*T* array across
the third dimension, the resultant *L*-by-*R* array is
same as the output signal you get when you set `SumFilteredOutputs`

to `true`

.

## References

[1] Stockham, T. G., Jr. "High Speed
Convolution and Correlation." *Proceedings of the 1966 Spring Joint Computer
Conference, AFIPS,* Vol 28, 1966, pp. 229–233.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

See System Objects in MATLAB Code Generation (MATLAB Coder).

## Version History

**Introduced in R2017b**

### R2023b: Support for multiple filters

You can model MIMO systems by specifying multiple filters in the time and frequency domains. You can also specify multiple paths between each input and output pair.

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