# dsp.FIRRateConverter

Sample rate converter

## Description

The dsp.FIRRateConverter System object™ performs sampling rate conversion by a rational factor on a vector or matrix input. The FIR rate converter cascades an interpolator with a decimator. The rate converter (as shown in the schematic) conceptually consists of an upsampler, followed by a combined anti-imaging and anti-aliasing FIR filter, followed by a downsampler. The coefficients of the anti-imaging and anti-aliasing FIR filter can be specified through the Numerator property, or can be automatically designed by the object using the designMultirateFIR function. For an example, see Resample Signal Using FIR Rate Converter.

The upsampler increases the sample rate of the signal by a factor L and the downsampler reduces the sample rate of the signal by a factor M. Use upsampling and downsampling factors that are relatively prime or coprime. The resulting discrete-time signal has a sample rate that is L/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 perform sampling rate conversion:

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

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

## Creation

### Description

example

firrc = dsp.FIRRateConverter returns an FIR sample rate converter object with an interpolation factor of 3 and a decimation factor of 2. The object designs the FIR filter coefficients using the designMultirateFIR(3,2) function.

firrc = dsp.FIRRateConverter(L,M) returns an FIR sample rate converter with the integer-valued InterpolationFactor property set to L and the DecimationFactor property set to M. The object designs its filter coefficients based on the rate conversion factors that you specify while creating the object, using the designMultirateFIR(L,M) function. The designed filter corresponds to a lowpass filter with normalized cutoff frequency no greater than min(π/L,π/M) in radial frequency units.

firrc = dsp.FIRRateConverter(L,M,'Auto') returns an FIR sample rate converter where the NumeratorSource property is set to 'Auto'. In this mode, every time there is an update in the rate conversion factors, the object redesigns the filter using the design method specified in DesignMethod.

example

firrc = dsp.FIRRateConverter(L,M,num) returns an FIR sample rate converter with the Numerator property set to num.

firrc = dsp.FIRRateConverter(L,M,method) returns an FIR sample rate converter with the DesignMethod property set to method. When you pass the design method as an input, the NumeratorSource property is automatically set to 'Auto'.

firrc = dsp.FIRRateConverter(___,Name,Value) returns an FIR sample rate converter 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.

Example: firrc = dsp.FIRRateConverter('FullPrecisionOverride','false') enables the fixed-point data types to be controlled through the individual fixed-point property settings.

firrc = dsp.FIRRateConverter(L,M,'legacy') returns an FIR sample rate converter where the filter coefficients are designed using firpm(70,[0 0.28 0.32 1],[1 1 0 0]). The designed filter has a cutoff frequency of π/3 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.

Interpolation factor L, specified as a positive integer.

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

Decimation factor M, specified as a positive integer.

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.

• 'Auto' –– The numerator coefficients are designed automatically using the design method specified in DesignMethod.

Numerator coefficients of the FIR filter, specified as a row vector in powers of z-1.

The following equation defines the transfer function of the filter:

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

The vector b = [b(0), b(1), …, b(N)] represents the vector of filter coefficients.

Use a lowpass filter with normalized cutoff frequency no greater than min(1/L,1/M), where L is the interpolation factor and M is the decimation factor. The gain of the lowpass filter should equal L, the interpolation factor. Choose the length of the filter to be a multiple of the interpolation factor. To design such a filter, use the designMultirateFIR function. For an example, see Resample Signal Using FIR Rate Converter. All initial filter states are zero.

#### 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 the design method specified in DesignMethod. 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

Design method of the FIR filter coefficients, specified as one of the following:

• 'Kaiser' –– Kaiser method. Approximate combined anti-aliasing and anti-imaging lowpass filter using the designMultirateFIR function.

• 'ZOH' –– Zero order hold method. Hold the input sequence values.

• 'Linear' –– Linear interpolation method.

#### Dependencies

This property is visible only when you set the NumeratorSource property to 'Auto', or if you pass the 'auto' keyword as an input while creating the object.

### Fixed-Point Properties

Specify whether to use full precision rules. If you set FullPrecisionOverride to true, which is the default, the object computes all internal arithmetic and output data types using full precision rules. These rules provide the most accurate fixed-point numerics. It also turns off the display of other fixed-point properties because they do not apply individually. These rules guarantee that no quantization occurs within the object. Bits are added, as needed, to ensure that no roundoff or overflow occurs. If you set FullPrecisionOverride to false, fixed-point data types are controlled through individual fixed-point property settings. For more information, see Full Precision for Fixed-Point System Objects.

Specify the rounding method as one of | Ceiling | Convergent | Floor | Nearest | Round | Simplest | Zero |.

#### Dependencies

This property applies only if the object is not in full precision mode.

Specify the overflow action as one of | Wrap | Saturate |.

#### Dependencies

This property applies only if the object is not in full precision mode.

Specify the filter coefficient fixed-point data type as one of | Same word length as input | Custom |.

Specify the filter coefficient fixed-point type as a numerictype (Fixed-Point Designer) object with a Signedness of Auto.

#### Dependencies

This property applies only when the CoefficientsDataType property is Custom.

Specify the product fixed-point data type as one of | Full precision | Same as input | Custom |.

Specify the product fixed-point type as a scaled numerictype (Fixed-Point Designer) object with a Signedness of Auto.

#### Dependencies

This property applies only when the ProductDataType property is Custom.

Specify the accumulator fixed-point data type as one of | Full precision | Same as product | Same as input | Custom |.

Specify the accumulator fixed-point type as a scaled numerictype (Fixed-Point Designer) object with a Signedness of Auto.

#### Dependencies

This property applies only when the AccumulatorDataType property is Custom.

Specify the output fixed-point data type as one of | Same as accumulator | Same as product | Same as input | Custom |.

Specify the output fixed-point type as a scaled numerictype (Fixed-Point Designer) object with a Signedness of Auto.

#### Dependencies

This property applies only when the OutputDataType property is Custom.

## Usage

### Description

example

y = firrc(x) resamples the input x and returns the resampled signal y.

### Input Arguments

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Data input, specified as a column vector or a matrix. The number of input rows P must be a multiple of the decimation factor M. A P-by-Q matrix input is treated as Q independent channels.

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

### Output Arguments

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Resampled output, returned as a column vector or a matrix. The number of rows in the output signal is given by PL/M, where P is the number of input rows, L is the interpolation factor, and 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 coeffs Returns the filter System object coefficients in a structure polyphase Polyphase decomposition of multirate filter generatehdl Generate HDL code for quantized DSP filter (requires Filter Design HDL Coder)
 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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Resample a 100 Hz sine wave signal by a factor of 3/2.

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).

Create a dsp.SineWave object which generates a sinusoidal signal with 54 samples per frame, contains a tone at 100 Hz, and has a sample rate of 1250 Hz.

sine = dsp.SineWave(1,100,'SampleRate',1250,'SamplesPerFrame',54);

% Grab a frame
x = sine();

% Calculate input time vector
tx = (0:length(x)-1)/sine.SampleRate;

Design Default Filter

Create a dsp.FIRRateConverter object. The object uses a combined anti-imaging and anti-aliasing FIR filter. By default, this filter is designed using the designMultirateFIR function. The function designs the filter based on the rate conversion factors that you specify, and stores the coefficients in the Numerator property. For an interpolation factor of 3 and a decimation factor of 2, the object designs the coefficients using designMultirateFIR(3,2).

firrc = dsp.FIRRateConverter(3,2);

Resample by a Factor of 3/2

Resample the signal by a factor of 3/2.

y = firrc(x);

Plot the original and resampled signals. In order to plot the two signals on the same plot, you must account for the output delay of the FIR rate converter and the scaling introduced by the filter. Use the outputDelay function to compute the delay value introduced by the rate converter. Shift the output by this delay value.

Visualize the input and the resampled signals. The input and output values coincide every 3 output samples, and every other input sample, owing to the conversion rate of 3/2.

FsIn = sine.SampleRate;
[delay,FsOut] = outputDelay(firrc,FsIn=FsIn);
% Calculate output times for vector y in input units
ty = (0:length(y)-1)/FsOut-delay;

stem(tx,x,'filled',MarkerSize=4)
hold on
stem(ty,y)
hold off
xlim([0.0 0.0145])
ylim([-1.5 1.5])
legend('Original input','Resampled')

Resample by a Factor of 5/3 in Automatic Filter Design Mode

Now change the interpolation factor to 5 and the decimation factor to 3. In order for the filter design to be updated automatically based on the new rate conversion factors, set the NumeratorSource property to 'Auto'. Alternately, you can pass the keyword 'auto' as an input while creating the object. The object then operates in the automatic filter design mode. Every time there is a change in the rate conversion factors, the object updates the filter design accordingly.

release(firrc)
firrc.NumeratorSource = 'Auto';
firrc.InterpolationFactor = 5;
firrc.DecimationFactor = 3
firrc =
dsp.FIRRateConverter with properties:

InterpolationFactor: 5
DecimationFactor: 3
NumeratorSource: 'Auto'
DesignMethod: 'Kaiser'

Show all properties

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

Resample the signal with the updated rate conversion values.

yAuto = firrc(x);

Plot the original and resampled signals. Recalculate the output delay and the output sample rate since the rate conversion factors have changed. Note that the input and output coincide every 3 input samples, and every 5 output samples, owing to the 5/3 conversion factor.

[delay,FsOut] = outputDelay(firrc,FsIn=FsIn);
% Calculate output times for vector yAuto in input units
tyAuto = (0:length(yAuto)-1)/FsOut-delay;

stem(tx,x,'filled',MarkerSize=4)
hold on
stem(tyAuto,yAuto,'r')
hold off
xlim([0.0 0.015])
ylim([-1.5 1.5])
legend('Original input','Resampled')

Specify Signal Interpolation Model

In the automatic design mode, you can also specify the underlying D/A signal interpolation model through the DesignMethod property.

Set DesignMethod to 'linear' and change the interpolation factor to 11.

release(firrc)
firrc.DesignMethod = 'linear';
firrc.InterpolationFactor = 11;

Resample the signal using the linear interpolation model.

yLinear = firrc(x);

Plot the original and resampled signals. The output samples lie on a piecewise-linear curve. Note that the input and output coincide every three input samples and every 11 output samples as expected from the ratio 11/3.

[delay,FsOut] = outputDelay(firrc,FsIn=FsIn);
% Calculate output times for vector yLinear in input units
tyLinear = (0:length(yLinear)-1)/FsOut-delay;

stem(tx,x,'filled',MarkerSize=4)
hold on
stem(tyLinear,yLinear)
plot(tyLinear,yLinear,Color=[1 0 0 0.3])
hold off
xlim([0.0 0.009])
ylim([-1.5 1.5])
legend('Original input','Resampled')

Resample an audio signal from 48 kHz to 44 kHz and play the resampled 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 object reads an audio file that has a sample rate of 48 kHz.

L = 11;
M = 12;
'OutputDataType','single', ...
'SamplesPerFrame',4*M);

Create a dsp.FIRRateConverter object with an interpolation factor of L = 11 and a decimation factor of M = 12 (the co-prime representation of the ratio $44\mathit{k}/48\mathit{k}$), a reasonable approximation of the standard 44.1 kHz rate. The object designs the filter using designMultirateFIR(11,12) and stores the coefficients in the Numerator property of the object.

firrc = dsp.FIRRateConverter(L,M)
firrc =
dsp.FIRRateConverter with properties:

InterpolationFactor: 11
DecimationFactor: 12
NumeratorSource: 'Property'
Numerator: [0 2.3076e-05 5.4790e-05 9.3620e-05 1.3665e-04 ... ]

Show all properties

Create an audioDeviceWriter object. Specify the sample rate to be 44100 Hz.

Read the audio file, convert the sample rate of the audio signal, and play the resampled audio.

while ~isDone(afr)
audio1 = afr();
audio2 = firrc(audio1);
end
release(afr)

## Algorithms

The FIR rate converter is implemented efficiently using a polyphase structure.

To derive the polyphase structure, start with the transfer function of the FIR filter: This FIR filter is a combined anti-imaging and anti-aliasing 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:

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

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

You can write this equation as:

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

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

Conceptually, the FIR rate converter contains an upsampler, followed by a combined anti-imaging, anti-aliasing FIR filter H(z), which is followed by a downsampler.

Replace H(z) with its polyphase representation.

Here is the multirate noble identity for interpolation.

Applying the noble identity for interpolation moves the upsampling operation to after the filtering operation. This move enables you to filter the signal at a lower rate.

You can replace the upsampling operator, delay block, and the adder with a commutator switch. To account for the downsampler that follows, the switch moves in steps of size M. The switch receives the first sample from branch 0 and moves in the counter clockwise direction, each time skipping M−1 branches.

As an example, consider a rate converter with L set to 5 and M set to 3. The polyphase components are E0(z), E1(z), E2(z), E3(z), and E4(z). The switch starts on the first branch 0, skips branches 1 and 2, receives the next sample from branch 3, then skips branches 4 and 0, receives the next sample from branch 2, and so on. The sequence of branches from which the switch receives the data sample is [0, 3, 1, 4, 2, 0, 3, 1, ….].

The rate converter implements the L/M conversion by first applying the interpolation factor L to the incoming data, and using the commutator switch at the end to receive only 1 in M samples, effectively accounting for the dowsampling factor M. Hence, the sample rate at the output of the FIR rate converter is Lfs/M.

## Version History

Introduced in R2012a