# dsp.FIRDecimator

Polyphase FIR decimator

## Description

The dsp.FIRDecimator System object™ resamples vector or matrix inputs along the first dimension. The FIR decimator (as shown in the schematic) conceptually consists of an anti-aliasing FIR filter followed by a downsampler.

The FIR filter filters the data in each channel of the input using a direct-form FIR filter. The FIR filter coefficients can be specified through the Numerator property, or can be automatically designed by the object using the designMultirateFIR function. The designMultirateFIR function designs an anti-aliasing FIR filter. The downsampler that follows the FIR filter downsamples each channel of filtered data by taking every M-th sample and discarding the M – 1 samples that follow. M is the value of the decimation factor that you specify. The resulting discrete-time signal has a sample rate that is 1/M times the original sample rate.

Note that the actual object algorithm implements a direct-form FIR polyphase structure, an efficient equivalent of the combined system depicted in the diagram. For more details, see Algorithms.

To resample vector or matrix inputs along the first dimension:

1. Create the dsp.FIRDecimator object and set its properties.

2. Call the object with arguments, as if it were a function.

Under specific conditions, this System object also supports SIMD code generation. For details, see Code Generation.

## Creation

### Description

example

firdecim = dsp.FIRDecimator returns an FIR decimator object with a decimation factor of 2. The object designs the FIR filter coefficients using the designMultirateFIR(1,2) function.

firdecim = dsp.FIRDecimator(M) returns an FIR decimator with the integer-valued DecimationFactor property set to M. The object designs its filter coefficients based on the decimation factor M that you specify while creating the object, using the designMultirateFIR(1,M) function. The designed filter corresponds to a lowpass with a cutoff at π/M in radial frequency units.

firdecim = dsp.FIRDecimator(M,'Auto') returns an FIR decimator with the NumeratorSource property set to 'Auto'. In this mode, every time there is an update in the decimation factor, the object redesigns the filter using designMultirateFIR(1,M).

firdecim = dsp.FIRDecimator(M,num) returns an FIR decimator with the DecimationFactor property set to M and the Numerator property set to num.

firdecim = dsp.FIRDecimator(___,Name,Value) returns an FIR decimator object with each specified property set to the specified value. Enclose each property name in quotes. You can use this syntax with any previous input argument combinations.

firdecim = dsp.FIRDecimator(M,'legacy') returns an FIR decimator where the filter coefficients are designed using fir1(35,0.4). The designed filter has a cutoff frequency of 0.4π radians/sample.

## Properties

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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.

Decimation factor M, specified as a positive integer. The FIR decimator reduces the sampling rate of the input by this factor. The number of input rows must be a multiple of the decimation factor.

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

FIR filter coefficient source, specified as one of the following:

• 'Property' –– The numerator coefficients are specified through the Numerator property.

• 'Input port' –– The numerator coefficients are specified as an input to the object algorithm.

• 'Auto' –– The numerator coefficients are designed automatically using the designMultirateFIR(1,M) function.

Numerator coefficients of the FIR filter, specified as a row vector in powers of z–1. The following equation defines the system function for a filter of length N+1:

$H\left(z\right)=\sum _{l=0}^{N}{b}_{l}{z}^{-l}$

The vector b = [b0, b1, …, bN] represents the vector of filter coefficients.

To prevent aliasing as a result of downsampling, the filter transfer function should have a normalized cutoff frequency no greater than 1/M. To design an effective anti-aliasing filter, use the designMultirateFIR function. For an example, see Decimate Sum of Sine Waves.

#### Dependencies

This property is visible only when you set NumeratorSource to 'Property'.

When NumeratorSource is set to 'Auto', the numerator coefficients are automatically redesigned using designMultirateFIR(1,M). To access the filter coefficients in the automatic design mode, type objName.Numerator in the MATLAB® command prompt.

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

Specify the implementation of the FIR filter as either Direct form or Direct form transposed.

### Fixed-Point Properties

Flag to use full-precision rules for fixed-point arithmetic, specified as one of the following:

• true –– The object computes all internal arithmetic and output data types using the full-precision rules. These rules provide the most accurate fixed-point numerics. In this mode, other fixed-point properties do not apply. No quantization occurs within the object. Bits are added, as needed, to ensure that no roundoff or overflow occurs.

• false –– Fixed-point data types are controlled through individual fixed-point property settings.

For more information, see Full Precision for Fixed-Point System Objects and Set System Object Fixed-Point Properties.

Rounding method for fixed-point operations. For more details, see rounding mode.

#### Dependencies

This property is not visible and has no effect on the numerical results when the following conditions are met:

• FullPrecisionOverride set to true.

• FullPrecisionOverride set to false, ProductDataType set to 'Full precision', AccumulatorDataType set to 'Full precision', and OutputDataType set to 'Same as accumulator'.

Under these conditions, the object operates in full precision mode.

Overflow action for fixed-point operations, specified as one of the following:

• 'Wrap' –– The object wraps the result of its fixed-point operations.

• 'Saturate' –– The object saturates the result of its fixed-point operations.

For more details on overflow actions, see overflow mode for fixed-point operations.

#### Dependencies

This property is not visible and has no effect on the numerical results when the following conditions are met:

• FullPrecisionOverride set to true.

• FullPrecisionOverride set to false, OutputDataType set to 'Same as accumulator', ProductDataType set to 'Full precision', and AccumulatorDataType set to 'Full precision'

Under these conditions, the object operates in full precision mode.

Data type of the FIR filter coefficients, specified as:

• Same word length as input –– The word length of the coefficients is the same as that of the input. The fraction length is computed to give the best possible precision.

• Custom –– The coefficients data type is specified as a custom numeric type through the CustomCoefficientsDataType property.

Word and fraction lengths of the coefficients data type, specified as an autosigned numerictype (Fixed-Point Designer) with a word length of 16 and a fraction length of 15.

#### Dependencies

This property applies when you set the CoefficientsDataType property to Custom.

Data type of the product output in this object, specified as one of the following:

• 'Full precision' –– The product output data type has full precision.

• 'Same as input' –– The object specifies the product output data type to be the same as that of the input data type.

• 'Custom' –– The product output data type is specified as a custom numeric type through the CustomProductDataType property.

For more information on the product output data type, see Multiplication Data Types.

#### Dependencies

This property applies when you set FullPrecisionOverride to false.

Word and fraction lengths of the product data type, specified as an autosigned numeric type with a word length of 32 and a fraction length of 30.

#### Dependencies

This property applies only when you set FullPrecisionOverride to false and ProductDataType to 'Custom'.

Data type of an accumulation operation in this object, specified as one of the following:

• 'Full precision' –– The accumulation operation has full precision.

• 'Same as product' –– The object specifies the accumulator data type to be the same as that of the product output data type.

• 'Same as input' –– The object specifies the accumulator data type to be the same as that of the input data type.

• 'Custom' –– The accumulator data type is specified as a custom numeric type through the CustomAccumulatorDataType property.

#### Dependencies

This property applies when you set FullPrecisionOverride to false.

Word and fraction lengths of the accumulator data type, specified as an autosigned numeric type with a word length of 32 and a fraction length of 30.

#### Dependencies

This property applies only when you set FullPrecisionOverride to false and AccumulatorDataType to 'Custom'.

Data type of the object output, specified as one of the following:

• 'Same as accumulator' –– The output data type is the same as that of the accumulator output data type.

• 'Same as input' –– The output data type is the same as that of the input data type.

• 'Same as product' –– The output data type is the same as that of the product output data type.

• 'Custom' –– The output data type is specified as a custom numeric type through the CustomOutputDataType property.

#### Dependencies

This property applies when you set FullPrecisionOverride to false.

Word and fraction lengths of the output data type, specified as an autosigned numeric type with a word length of 16 and a fraction length of 15.

#### Dependencies

This property applies only when you set FullPrecisionOverride to false and OutputDataType to 'Custom'.

## Usage

### Description

example

y = firdecim(x) outputs the filtered and downsampled values, y, of the input signal, x.

y = firdecim(x,num) uses the FIR filter, num, to decimate the input signal. This configuration is valid only when the 'NumeratorSource' property is set to 'Input port'.

### Input Arguments

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Data input, specified as a column vector or a matrix of size P-by-Q. The number of input rows, P, must be a multiple of the DecimationFactor property. The input columns represent the Q independent channels.

This object supports variable-size input and does not support complex unsigned fixed-point inputs.

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

FIR filter coefficients, specified as a row vector.

#### Dependencies

This input is accepted only when the 'NumeratorSource' property is set to 'Input port'.

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

### Output Arguments

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FIR decimated output, returned as a column vector or a matrix of size P/M-by-Q, where M is the decimation factor.

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

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 freqz Frequency response of discrete-time filter System object fvtool Visualize frequency response of DSP filters info Information about filter System object cost Estimate cost of implementing filter System object polyphase Polyphase decomposition of multirate filter generatehdl Generate HDL code for quantized DSP filter (requires Filter Design HDL Coder) impz Impulse response of discrete-time filter System object coeffs Returns the filter System object coefficients in a structure
 step Run System object algorithm release Release resources and allow changes to System object property values and input characteristics reset Reset internal states of System object

## Examples

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Decimate a sum of sine waves by a factor of 2 and by a factor of 4.

Note: If you are using R2016a or an earlier release, replace each call to the object with the equivalent step syntax. For example, obj(x) becomes step(obj,x).

Start with a cosine wave that has an angular frequency of $\frac{\pi }{4}$ radians/sample.

x = cos(pi/4*(0:95)');

Design Default Filter

Create a dsp.FIRDecimator object. To prevent aliasing, the object uses an anti-aliasing lowpass filter before downsampling. By default, the anti-aliasing lowpass filter is designed using the designMultirateFIR function. The function designs the filter based on the decimation factor that you specify, and stores the coefficients in the Numerator property. For a decimation factor of 2, the object designs the coefficients using designMultirateFIR(1,2).

firdecim = dsp.FIRDecimator(2)
firdecim =
dsp.FIRDecimator with properties:

DecimationFactor: 2
NumeratorSource: 'Property'
Numerator: [0 -1.0054e-04 0 3.8704e-04 0 -0.0010 0 0.0022 0 ... ]
Structure: 'Direct form'

Show all properties

Visualize the filter response using fvtool. The designed filter meets the ideal filter constraints that are marked in red. The cutoff frequency is approximately half the spectrum.

fvtool(firdecim)

Decimate by 2

Decimate the cosine signal by a factor of 2.

y = firdecim(x);

Plot the original and the decimated signals. In order to plot the two signals on the same plot, obtain the time affine transformation parameters. Use these parameters to compute the output sample times. The ${\mathit{n}}^{\mathrm{th}}$ output sample time equals $\mathrm{scale}×\mathit{n}+\mathrm{delay}$.

scale = 1/firdecim.DecimationFactor;
delay = length(firdecim.Numerator)/(2*firdecim.DecimationFactor);

nx = (0:length(x)-1);
ty = ((0:length(y)-1)-delay)/scale;

After a short transition, the output converges to a cosine of frequency $\frac{\pi }{2}$ as expected, which is twice the frequency of the input signal $\frac{\pi }{4}$. Due to the decimation factor of 2, the output samples coincide with every other input sample.

stem(ty,y,'filled',MarkerSize=4); hold on;
stem(nx,x); hold off;
xlim([-10,22])
ylim([-2.5 2.5])
legend('Decimated by 2 (y)','Input signal (x)');

Add a High Frequency Component to Input and Decimate

Add another frequency component to the input signal, a sine with an angular frequency of $\frac{2\pi }{3}$ radians/sample. Since $\omega =\text{\hspace{0.17em}}\frac{2\pi }{3}$ is above the FIR lowpass cutoff, $\frac{\pi }{2}$, the frequency $\frac{2\pi }{3}$ radians/sample is filtered out from the signal.

xhigh = x + 0.2*sin(2*pi/3*(0:95)');
release(firdecim)
yhigh = firdecim(xhigh);

Plot the input signal, decimated signal, and the output of the low frequency component. The decimated signal yhigh has the high frequency component filtered out. yhigh is almost identiical to the output of the low frequency component y.

stem(ty,yhigh,'filled',MarkerSize=4); hold on;
stem(nx,xhigh);
stem(ty,y,':m',MarkerSize=7);
hold off;
xlim([-10,22])
ylim([-2.5 2.5])
legend('Decimated by 2 (yhigh)',...
'Input signal with the high tone added (xhigh)',...
'Decimated by 2 - low tone only (y)');

Decimate by 4 in Automatic Filter Design Mode

Now decimate by a factor of 4. In order for the filter design to be updated automatically based on the new decimation factor, set the NumeratorSource property to 'Auto'. Alternately, you can pass 'auto' as the keyword while creating the object. The object then operates in the automatic filter design mode. Every time there is a change in the decimation factor, the object updates the filter design.

release(firdecim)
firdecim.NumeratorSource = 'Auto';
firdecim.DecimationFactor = 4
firdecim =
dsp.FIRDecimator with properties:

DecimationFactor: 4
NumeratorSource: 'Auto'
Structure: 'Direct form'

Show all properties

To access the filter coefficients in the automatic mode, type firdecim.Numerator in the MATLAB command prompt.

The designed filter occupies a narrower passband that is approximately a quarter of the spectrum.

fvtool(firdecim)

Decimate the cosine signal by a factor of 4. After a short transition, the output converges to a cosine of frequency $\pi$ as expected, which is four times the lower frequency component of the input signal $\frac{\pi }{4}$. This time, the amplitude of the output is half the amplitude of the input since the gain of the FIR at $\omega =\frac{\pi }{4}$ is exactly $\frac{1}{2}$. The high frequency component $\frac{2\pi }{3}$ diminishes by the lowpass FIR whose cutoff frequency is $\frac{\pi }{4}$.

yAuto = firdecim(xhigh);

Plot the input signal with the high frequency component added, low frequency component scaled by 1/2, and the decimated signal. Recalculate the time affine transformation parameters since the decimation factor has changed.

scale = 1/firdecim.DecimationFactor;
delay = length(firdecim.Numerator)/(2*firdecim.DecimationFactor);
tyAuto = ((0:length(yAuto)-1)-delay)/scale;

stem(tyAuto,yAuto,'filled',MarkerSize=4); hold on;
stem(nx,xhigh);
stem(nx,x/2,'m:',MarkerSize=7); hold off;

xlim([-20,36])
ylim([-2.5 2.5])
legend('Decimated by 4 (yAuto)',...
'Input signal with the high frequency component added (xhigh)',...
'Low tone input scaled by 1/2');

Reduce the sample rate of an audio signal by a factor of 2 and play the decimated signal using the audioDeviceWriter object.

Note: If you are using R2016a or an earlier release, replace each call to the object with the equivalent step syntax. For example, obj(x) becomes step(obj,x).

Note: The audioDeviceWriter System object™ is not supported in MATLAB Online.

Create a dsp.AudioFileReader object. The default audio file read by the object has a sample rate of 22050 Hz.

'single');

Create a dsp.FIRDecimator object and specify the decimation factor to be 2. The object designs the filter using designMultirateFIR(1,2) and stores the coefficients in the Numerator property of the object.

firdecim = dsp.FIRDecimator(2)
firdecim =
dsp.FIRDecimator with properties:

DecimationFactor: 2
NumeratorSource: 'Property'
Numerator: [0 -1.0054e-04 0 3.8704e-04 0 -0.0010 0 0.0022 0 ... ]
Structure: 'Direct form'

Show all properties

Create an audioDeviceWriter object. Specify the sample rate to be 22050/2.

audioDeviceWriter with properties:

Device: 'Default'
SampleRate: 11025

Show all properties

Read the audio signal using the file reader object, decimate the signal by a factor of 2, and play the decimated signal.

while ~isDone(afr)
frame = afr();
y = firdecim(frame);
end

release(afr);
pause(0.5);

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## Algorithms

The FIR decimation filter is implemented efficiently using a polyphase structure. For more details on polyphase filters, see Polyphase Subfilters.

To derive the polyphase structure, start with the transfer function of the FIR filter:

$H\left(z\right)={b}_{0}+{b}_{1}{z}^{-1}+...+{b}_{N}{z}^{-N}$

N+1 is the length of the FIR filter.

You can rearrange this equation as follows:

$H\left(z\right)=\begin{array}{c}\left({b}_{0}+{b}_{M}{z}^{-M}+{b}_{2M}{z}^{-2M}+..+{b}_{N-M+1}{z}^{-\left(N-M+1\right)}\right)+\\ {z}^{-1}\left({b}_{1}+{b}_{M+1}{z}^{-M}+{b}_{2M+1}{z}^{-2M}+..+{b}_{N-M+2}{z}^{-\left(N-M+1\right)}\right)+\\ \begin{array}{c}⋮\\ {z}^{-\left(M-1\right)}\left({b}_{M-1}+{b}_{2M-1}{z}^{-M}+{b}_{3M-1}{z}^{-2M}+..+{b}_{N}{z}^{-\left(N-M+1\right)}\right)\end{array}\end{array}$

M is the number of polyphase components, and its value equals the decimation factor that you specify.

You can write this equation as:

$H\left(z\right)={E}_{0}\left({z}^{M}\right)+{z}^{-1}{E}_{1}\left({z}^{M}\right)+...+{z}^{-\left(M-1\right)}{E}_{M-1}\left({z}^{M}\right)$

E0(zM), E1(zM), ..., EM-1(zM) are the polyphase components of the FIR filter H(z).

Conceptually, the FIR decimation filter contains a lowpass FIR filter followed by a downsampler.

Replace H(z) with its polyphase representation.

Here is the multirate noble identity for decimation.

Applying the noble identity for decimation moves the downsampling operation to before the filtering operation. This move enables you to filter the signal at a lower rate.

You can replace the delays and the decimation factor at the input with a commutator switch. The switch starts on the first branch 0 and moves in the counterclockwise direction as shown in this diagram. The accumulator at the output receives the processed input samples from each branch of the polyphase structure and accumulates these processed samples until the switch goes to branch 0. When the switch goes to branch 0, the accumulator outputs the accumulated value.

When the first input sample is delivered, the switch feeds this input to the branch 0 and the decimator computes the first output value. As more input samples come in, the switch moves in the counter clockwise direction through branches M−1, M−2, and all the way up to branch 0, delivering one sample at a time to each branch. When the switch comes to branch 0, the decimator outputs the next set of output values. This process continues as data keeps coming in. Every time the switch comes to the branch 0, the decimator outputs y[m]. The decimator effectively outputs one sample for every M samples it receives. Hence the sample rate at the output of the FIR decimation filter is fs/M.

## Extended Capabilities

### Topics

Introduced in R2012a

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