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Frequency-Domain FIR Filter

Filter input signal in the frequency domain

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  • Frequency-Domain FIR Filter block

Libraries:
DSP System Toolbox / Filtering / Filter Implementations

Description

The Frequency-Domain FIR Filter block 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 becomes more efficient than time-domain filtering as the impulse response grows longer. You can specify the filter coefficients directly in the frequency domain by setting Numerator domain to Frequency.

This block uses the overlap-save and overlap-add methods to perform the frequency-domain filtering. For filters with a long impulse response length, the latency inherent to these two methods can be significant. To mitigate this latency, the Frequency-Domain FIR Filter block 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, select the Partition numerator to reduce latency check box. For more details on these two methods and on reducing latency through impulse response partitioning, see Algorithms.

The Frequency-Domain FIR Filter block can model single-input multiple-output (SIMO) and multiple-input multiple-output (MIMO) systems by supporting multiple filters in the time domain and the frequency domain. For examples that show how to model such systems, see Filter Input Signal Using 1-by-2 SIMO System and Filter Input Signal Using 3-by-2 MIMO System. You can also specify multiple paths between each input channel and output channel pair using the Number of propagation paths parameter. For more information on modeling MIMO systems, see Modeling MIMO System with Multiple Propagation Paths. (since R2024a)

Examples

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This example uses:

Open Model

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

Open the overlapAddSave.slx model. The Lowpass FIR Filter Design block in the model designs the lowpass FIR filter with a filter order of 400 and a cutoff frequency of 0.5 in normalized frequency units. Use the Discrete FIR Filter block and the two Frequency-Domain FIR Filter blocks to filter the input signal using these coefficients.

The input signal in the model is a noisy sinusoidal signal with two tones, one at 100 Hz and the other at 3 kHz. The signal has a sample rate of 8000 Hz and a frame size of 256. The noise is a white Gaussian noise with a mean of 0 and a variance of 0.01.

Pass the input signal to the three filters. Account for the output latency of the frequency-domain FIR filters while passing the input to the time-domain FIR filter. View the filtered outputs in the time scope.

Run the model. The outputs of all the three filters match exactly.

This example uses:

Open Script

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

Open the reduceLatency.slx model. The Lowpass FIR Filter Design block in the model designs the lowpass FIR filter with a filter order of 4000 and a cutoff frequency of 0.5 in normalized frequency units. Use Frequency-Domain FIR Filter blocks to filter the input signal using these coefficients.

The input signal in the model is a noisy sinusoidal signal with two tones, one at 100 Hz and the other at 3 kHz. The signal has a sample rate of 8000 Hz and a frame size of 256. The noise is a white Gaussian noise with a mean of 0 and a variance of 0.01. Pass this data through the two frequency-domain filters.

Select the Partition numerator to reduce latency parameter in one of the Frequency-Domain FIR Filter blocks. Set the partition length to 256.

The latency of the frequency-domain FIR filter is given by $FFT length - numerator length + 1$. By default, the FFT length is equal to twice the numerator length. This makes the latency proportional to the impulse response length. For the frequency-domain FIR filter with no partition, the latency is 4002 samples, and for the frequency-domain FIR filter with partition, the latency is 256 samples.

Run the model. Compare the outputs of the two frequency-domain filters in the time scope. To compare the two outputs, delay the input to the partitioned frequency-domain FIR filter by 3746 samples (4002 $-$ 256). As you can see in the time scope, the outputs match exactly.

This example uses:

Open Model

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.

Open the specifyFrequencyResponse.slx model. The input is a noisy sinusoidal signal with two tones, one at 100 Hz and the other at 3 kHz. The noise is a white Gaussian noise with a mean of 0 and a variance of 0.01. Pass this signal to the Frequency-Domain FIR Filter block and the Discrete FIR Filter block.

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. Specify the time-domain impulse response in the Discrete FIR Filter block. Compute the FFT of this impulse response and specify this response as the frequency response of the frequency-domain FIR filter. In the Frequency-Domain FIR Filter block, set the time-domain numerator length to the number of elements in the time-domain impulse response.

Here is the MATLAB® code to design the filter.

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

Run the model and visualize the frequency-domain FIR filter output and the time-domain FIR filter output in the time scope. To compare the outputs, delay the time-domain FIR filter input by the latency of the frequency-domain FIR filter.

The outputs of both the filters match exactly.

Since R2024a

This example uses:

Open Model

Filter an input signal using a 1-by-2 SIMO system with two distinct paths between the input and each output. Partition the filters to reduce latency.

Open the filterSignalSIMOSystem.slx model. Design four lowpass FIR filters with different cutoff frequencies and concatenate the coefficients along the first dimension. The order for each filter is 4000 and the sampling rate is 8000 Hz. The SIMO system models one input channel, two output channels, and two paths between each input channel and output channel pair.

Pass the filter coefficients you design through the Num port of the Frequency-Domain FIR Filter block. Set the partition length of the filter to 256 and the number of propagation paths to 2. The block uses the Overlap-save method by default.

The input contains two sinusoidal signals each with a frame length of 256. The input noise is white Gaussian with a mean of 0 and a variance of 0.01. The first sinusoidal signal has a frequency of 500 Hz and the second sinusoidal signal has a frequency of 1000 Hz.

Run the model. Pass the signal through the designed filter. View the spectrum of the input and the filtered output in the spectrum analyzer.

Since R2024a

This example uses:

Open Model

Filter an input signal using a 3-by-2 MIMO system with three input channels, two output channels, and one path between each input channel and output channel pair.

Open the filterSignalMIMOSystem.slx model. Design six lowpass FIR filters with different cutoff frequencies. The order for each filter is 400 and the sampling rate is 8000 Hz.

Pass the filter coefficients you design through the Num port of the Frequency-Domain FIR Filter block. The block uses the Overlap-save method by default. Select the Sum filtered output contributions from all input channels parameter.

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. The input noise is white Gaussian with a mean of 0 and a variance of 0.01.

Run the model. Pass the signal through the designed filter. View the spectrum of the input and the filtered output in the spectrum analyzer.

Extended Examples

Overlap-Add/Save

Overlap-Add/Save

Filter a sinusoid with the Overlap-Add and Overlap-Save FFT methods using the Frequency-Domain FIR filter block.

Ports

Input

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Data input, specified as a vector or a matrix of size L-by-T. This block 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 (number of columns) must remain constant.

This port is unnamed until you select the Specify coefficients from input port parameter.

Data Types: single | double
Complex Number Support: Yes

Specify the time-domain FIR filter coefficients as a row vector (single filter) or a matrix (multiple filters) (since R2024a) 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 Number of propagation paths parameter. The multiplication factor determines the number of output channels R and equals F/(T × P).

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

Dependencies

To enable this port, set Numerator domain to Time and select the Specify coefficients from input port parameter.

Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32
Complex Number Support: Yes

Specify the frequency-domain FIR filter coefficients as a row vector, matrix, or a 3-D array (since R2024a).

When you clear the Partition numerator to reduce latency parameter, the coefficients input at this port can be a row vector or a matrix of size F-by-FFTLength. The FFT length is equal to the length of the vector input. If F is greater than 1 (multiple filters) (since R2024a), 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 Number of propagation paths parameter. The multiplication factor determines the number of output channels R and equals F/(T × P).

When you select the Partition numerator to reduce latency parameter, Frequency response must be a matrix of size 2PL-by-N to represent a single filter or a 3-D array (since R2024a) of size 2PL-by-N-by-F to represent multiple filters, where PL is the partition size, N is the number of partitions, and F is the number of filters.

Dependencies

To enable this port, set Numerator domain to Frequency and select the Specify frequency response from input port parameter.

Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32
Complex Number Support: Yes

Output Arguments

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Filtered output, returned as a vector, matrix or a 3D array (since R2024a).

When the number of filters F is greater than 1 and you select the Sum filtered output contributions from all input channels parameter, the block adds the filtered output from each input channel 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 output channels. R equals F/(T x P), where F is the number of filters, T is the number of input channels, and P is the value that you specify in the Number of propagation paths parameter. (since R2024a)

When you clear the Sum filtered output contributions from all input channels parameter, the block 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 kth input channel for the jth output channel. For example, output(:,3,2) indicates output on the third output channel from the second input channel. (since R2024a)

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.

This port is unnamed until you select the Output filter latency parameter and click Apply.

Data Types: single | double
Complex Number Support: Yes

Filter latency, returned as a scalar. This latency is inherent to overlap-add and overlap-save methods and does not include the group delay of the filter. This port appears only when you select the Output filter latency check box.

This port is unnamed until you select the Output filter latency check box and click Apply.

Data Types: uint32

Parameters

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Specify the filtering method in the frequency domain as Overlap-save or Overlap-add. For more details on these two methods, see Algorithms.

Specify the domain of the filter coefficients as one of the following:

  • Time –– Specify the time-domain filter coefficients in the Filter coefficients parameter or through the Num input port.

  • Frequency –– Specify the filter's frequency response in the Frequency response parameter or through the NumFFT input port.

When you select this check box, the time-domain FIR filter coefficients are input through the port, Num. When you clear this check box, the coefficients are specified in the block dialog box through the Filter coefficients parameter.

To view the filter response, clear this check box, specify the coefficients in the block dialog box, and click on the View Filter Response button.

Dependencies

To enable this parameter, set Numerator domain to Time.

Specify the time-domain FIR filter coefficients as a row vector (single filter) or a matrix (multiple filters) (since R2024a) 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 Number of propagation paths parameter. The multiplication factor determines the number of output channels R and equals F/(T × P).

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

Tunable: Yes

Dependencies

To enable this parameter, set Numerator domain to Time and clear the Specify coefficients from input port parameter.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
Complex Number Support: Yes

When you select this check box, the FIR filter coefficients in the frequency domain are input through the port, NumFFT. When you clear this check box, the coefficients are specified in the block dialog box through the Frequency response parameter.

To view the filter response, clear this check box, specify the frequency response in the block dialog box, and click on the View Filter Response button.

Dependencies

To enable this parameter, set Numerator domain to Frequency.

Specify the frequency response of the filter as a row vector, matrix, or a 3-D array (since R2024a).

When you clear the Partition numerator to reduce latency parameter, Frequency response can be a row vector or a matrix of size F-by-FFTLength. The FFT length is equal to the length of the Frequency response vector. If F is greater than 1 (multiple filters) (since R2024a), 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 Number of propagation paths parameter. The multiplication factor determines the number of output channels R and equals F/(T × P).

When you select the Partition numerator to reduce latency parameter, Frequency response must be a matrix of size 2PL-by-N to represent a single filter or a 3-D array (since R2024a) of size 2PL-by-N-by-F to represent multiple filters, where PL is the partition size, N is the number of partitions, and F is the number of filters.

Tunable: Yes

Dependencies

To enable this parameter, set Numerator domain to Frequency and clear the Specify frequency response from input port parameter.

Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32
Complex Number Support: Yes

Since R2024a

Specify the number of propagation paths P between each input channel and output channel pair 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. The algorithm sums up the filtered output contributions from all paths between each input channel and output channel pair.

For an example, see Filter Input Signal Using 1-by-2 SIMO System.

Dependencies

To enable this parameter, the number of filters F must be greater than 1.

Data Types: single | double

Since R2024a

Select this check box to sum filtered output contributions from all input channels. When you select this check box, the block adds the filtered output from each input channel 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 output channels. R equals F/(T x P), where F is the number of filters, T is the number of input channels, and P is the value that you specify through the Number of propagation paths parameter.

When you clear this parameter, the block 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 block computes the output based on the value of this parameter, see Modeling MIMO System with Multiple Propagation Paths.

Dependencies

To enable this property, the number of filters F must be greater than 1.

Select this parameter to partition the numerator to reduce latency. When you select this parameter, the block partitions the numerator into segments of length specified by the Numerator partition length parameter. The filter performs overlap-save or overlap-add on each partition, and combines the partial results to form the overall output. This option reduces latency to the partition length.

When you clear this parameter, the filter uses the traditional overlap-save or overlap-add method. The latency in this case is FFT length – NumLen + 1, where NumLen is the filter length. See the Filter coefficients parameter for more information on the filter length.

Specify the partition length of the numerator PL as a positive integer less than or equal to the length of the numerator.

Dependencies

To enable this parameter, set Numerator domain to Time and select the Partition numerator to reduce latency parameter.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Specify the time-domain numerator length as a positive integer.

Dependencies

To enable this parameter, set Numerator domain to Frequency and clear the Partition numerator to reduce latency check box.

Data Types: single | double | int8 | int16 | int32 | uint8 | uint16 | uint32

When you select this check box, the FFT length equals twice the numerator length. When you clear this check box, you specify the FFT length through the FFT length parameter.

Dependencies

To enable this parameter, set Numerator domain to Time and clear the Partition numerator to reduce latency parameter.

The FFT length you specify must be greater than or equal to the length of the numerator you specify in the Filter coefficients parameter.

Dependencies

To enable this parameter, set Numerator domain to Time, clear the Partition numerator to reduce latency and the Inherit FFT length from numerator length parameters.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Select this parameter to specify that the filter coefficients are all real.

Dependencies

To enable this parameter, set Numerator domain to Frequency.

When you select this check box and click Apply, the block outputs the filter latency through the latency port.

Clicking this button opens the filter visualizer and displays the magnitude response of the frequency domain FIR filter. The response is based on the parameters you select in the block dialog box. To update the response while the visualizer is running, modify the parameters in the dialog box and click Apply.

You can configure the plot settings and the frequency response measurements from the interface of the visualizer.

On the Scope tab, you can enable the legend, specify to plot both magnitude and phase responses of the filter, modify the plot settings, generate a script to re-create the plot, and even save or share the settings. On the Measurements tab, you can enable data cursors, and display the peak values of the filter response.

For more details on the filter visualizer interface and its tools, see Configure Filter Visualizer.

Dependencies

To enable this button, clear the Specify coefficients from input port and the Specify frequency response from input port parameters.

Specify the type of simulation to run. You can set this parameter to:

  • Interpreted execution –– Simulate model using the MATLAB® interpreter. This option shortens startup time.

  • Code generation –– Simulate model using generated C code. The first time you run a simulation, Simulink® generates C code for the block. The C code is reused for subsequent simulations as long as the model does not change. This option requires additional startup time but provides faster subsequent simulations.

Block Characteristics

Data Types

double | single

Direct Feedthrough

no

Multidimensional Signals

no

Variable-Size Signals

yes

Zero-Crossing Detection

no

Algorithms

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

References

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

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2017b

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See Also

Objects

Blocks