# 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. To design an FIR anti-aliasing filter, use the designMultirateFIR function. For an example, see Reduce the Sample Rate of an Audio Signal.

The FIR filter filters the data in each channel of the input using a direct-form FIR filter. The downsampler that follows downsamples each channel of filtered data by discarding M–1 consecutive samples following each sample that is retained. 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 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, firdecim, which applies an FIR filter with a cutoff frequency of 0.4*pi radians/sample to the input and downsamples the filter output by factor of 2.

firdecim = dsp.FIRDecimator(decimFactor,num) returns an FIR decimator with the integer-valued DecimationFactor property set to decimFactor 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.

## 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 either:

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

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

Specify the numerator coefficients of the FIR filter as 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/DecimationFactor. To design an effective anti-aliasing filter, use the designMultirateFIR function. For an example, see Decimate a Sum of Sine Waves.

#### Dependencies

This property applies when NumeratorSource is set to 'Property'.

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: double
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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Note: This example runs only in R2016b or later. If you are using an earlier release, replace each call to the function with the equivalent step syntax. For example, myObject(x) becomes step(myObject,x).

This example shows how to decimate a sum of sine waves with angular frequencies of pi/4 and 2pi/3 radians/sample by a factor of two. To prevent aliasing, the FIR decimator uses an anti-aliasing lowpass filter that filters out the 2pi/3 radians/sample component before downsampling. To design such a filter, use the designMultirateFIR function.

x = cos(pi/4*(0:95)')+sin(2*pi/3*(0:95)');
M = 2;
num = designMultirateFIR(1,M);
firdecim = dsp.FIRDecimator(M,num);

Decimate the sinusoidal signal using this decimator.

y = firdecim(x);

Plot the original and resampled signals.

subplot(211);
stem(x(1:length(x)/2),'b','markerfacecolor',[0 0 1]);
title('Input Signal');
subplot(212);
stem(y,'b','markerfacecolor',[0 0 1]);
title('Output--Lowpass filtered and downsampled by 2');

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.

This example shows how to reduce the sampling rate of an audio signal by 1/2 and plays it.

'single');
num = designMultirateFIR(1,2);
firdecim = dsp.FIRDecimator(2,num);

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 counter clockwise 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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