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Equalize modulated signals using linear filtering

The `comm.LinearEqualizer`

System object™ uses a linear filter tap delay line with a weighted sum to equalize modulated
signals transmitted through a dispersive channel. The equalizer object adaptively adjusts tap
weights based on the selected algorithm. For more information, see Algorithms.

To equalize modulated signals using a linear filter:

Create the

`comm.LinearEqualizer`

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?.

creates a
linear equalizer System object to adaptively equalize a signal.`lineq`

= comm.LinearEqualizer

sets properties using one or more name-value pairs. For example,
`lineq`

= comm.LinearEqualizer(`Name`

,`Value`

)`comm.LinearEqualizer('Algorithm','RLS')`

configures the equalizer
object to update tap weights using the recursive least squares (RLS) algorithm. Enclose
each property name in quotes.

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.

`Algorithm`

— Adaptive algorithm`'LMS'`

(default) | `'RLS'`

| `'CMA'`

Adaptive algorithm used for equalization, specified as one of these values:

`'LMS'`

— Update the equalizer tap weights using the Least Mean Square (LMS) Algorithm.`'RLS'`

— Update the equalizer tap weights using the Recursive Least Square (RLS) Algorithm.`'CMA'`

— Update the equalizer tap weights using the Constant Modulus Algorithm (CMA).

**Data Types: **`char`

| `string`

`NumTaps`

— Number of equalizer taps`5`

(default) | positive integerNumber of equalizer taps, specified as a positive integer. The number of equalizer
taps must be greater than or equal to the value of the `InputSamplesPerSymbol`

property.

**Data Types: **`double`

`StepSize`

— Step size`0.01`

(default) | positive scalarStep size used by the adaptive algorithm, specified as a positive scalar. Increasing the step size reduces the equalizer convergence time but causes the equalizer output estimates to be less stable.

**Tip**

To determine the maximum step size allowed, use the `maxstep`

object function.

**Tunable: **Yes

To enable this property, set Algorithm
to `'LMS'`

or `'CMA'`

.

**Data Types: **`double`

`ForgettingFactor`

— Forgetting factor`0.99`

(default) | scalar in the range (0, 1]Forgetting factor used by the adaptive algorithm, specified as a scalar in the range (0, 1]. Decreasing the forgetting factor reduces the equalizer convergence time but causes the equalizer output estimates to be less stable.

**Tunable: **Yes

To enable this property, set Algorithm
to `'RLS'`

.

**Data Types: **`double`

`InitialInverseCorrelationMatrix`

— Initial inverse correlation matrix`0.1`

(default) | scalar | matrixInitial inverse correlation matrix, specified as a scalar or an
*N*_{Taps}-by-*N*_{Taps}
matrix. *N*_{Taps} is equal to the NumTaps
property value. If you specify `InitialInverseCorrelationMatrix`

as a
scalar, *a*, the equalizer sets the initial inverse correlation matrix
to *a* times the identity matrix:
*a*(`eye`

(*N*_{Taps})).

**Tunable: **Yes

To enable this property, set Algorithm
to `'RLS'`

.

**Data Types: **`double`

`Constellation`

— Signal constellation`pskmod(0:3,4,pi/4)`

(default) | vectorSignal constellation, specified as a vector. The default value is a QPSK
constellation generated using this code: `pskmod(0:3,4,pi/4)`

.

**Tunable: **Yes

**Data Types: **`double`

`ReferenceTap`

— Reference tap`3`

(default) | positive integerReference tap, specified as a positive integer less than or equal to the NumTaps property value. The equalizer uses the reference tap location to track the main energy of the channel.

**Data Types: **`double`

`InputDelay`

— Input signal delay`0`

(default) | nonnegative integerInput signal delay in samples relative to the reset time of the equalizer, specified
as a nonnegative integer. If the input signal is a vector of length greater than 1, then
the input delay is relative to the start of the input vector. If the input signal is a
scalar, then the input delay is relative to the first call of the System object and to the first call of the System object after calling the `release`

or `reset`

object function.

**Data Types: **`double`

`InputSamplesPerSymbol`

— Number of input samples per symbol`1`

(default) | positive integerNumber of input samples per symbol, specified as a positive integer. Setting this property to any number greater than one effectively creates a fractionally spaced equalizer. For more information, see Symbol Tap Spacing.

**Data Types: **`double`

`TrainingFlagInputPort`

— Enable training control input`false`

(default) | `true`

Enable training control input, specified as `false`

or
`true`

. Setting this property to `true`

enables the
equalizer training flag input tf.

**Tunable: **Yes

**Data Types: **`logical`

`AdaptAfterTraining`

— Update tap weights when not training`true`

(default) | `false`

Update tap weights when not training, specified as `true`

or
`false`

. If this property is set to `true`

, the
System object uses decision directed mode to update equalizer tap weights. If this
property is set to `false`

, the System object keeps the equalizer tap weights unchanged after training.

**Tunable: **Yes

**Data Types: **`logical`

`AdaptWeightsSource`

— Source of adapt tap weights request`'Property'`

(default) | `'Input port'`

Source of adapt tap weights request, specified as one of these values:

`'Property'`

— Specify this value to use the AdaptWeights property to control when the System object adapts tap weights.`'Input port'`

— Specify this value to use the`aw`

input to control when the System object adapts tap weights.

To enable this property, set Algorithm
to `'CMA'`

.

**Data Types: **`char`

| `string`

`AdaptWeights`

— Adapt tap weights`true`

(default) | `false`

Adapt tap weights, specified as `true`

or `false`

.
If this property is set to `true`

, the System object updates the equalizer tap weights. If this property is set to
`false`

, the System object keeps the equalizer tap weights unchanged.

**Tunable: **Yes

To enable this property, set AdaptWeightsSource to `'Property'`

and set AdaptAfterTraining to `true`

.

**Data Types: **`logical`

`InitialWeightsSource`

— Source for initial tap weights`'Auto'`

(default) | `'Property'`

Source for initial tap weights, specified as

`'Auto'`

— Initialize the tap weights to the algorithm-specific default values, as described in the InitialWeights property.`'Property'`

— Initialize the tap weights using the InitialWeights property value.

**Data Types: **`char`

| `string`

`InitialWeights`

— Initial tap weights`0`

or `[0;0;1;0;0]`

(default) | scalar | column vectorInitial tap weights used by the adaptive algorithm, specified as a scalar or vector.
The default is `0`

when the Algorithm
property is set to `'LMS'`

or `'RLS'`

. The default is
`[0;0;1;0;0]`

when the Algorithm
property is set to `'CMA'`

.

If you specify `InitialWeights`

as a vector, the vector length
must be equal to the NumTaps
property value. If you specify `InitialWeights`

as a scalar, the
equalizer uses scalar expansion to create a vector of length NumTaps with
all values set to `InitialWeights`

.

**Tunable: **Yes

To enable this property, set InitialWeightsSource to `'Property'`

.

**Data Types: **`double`

`WeightUpdatePeriod`

— Tap weight update period`1`

(default) | positive integerTap weight update period in symbols, specified as a positive integer. The equalizer updates the tap weights after processing this number of symbols.

**Data Types: **`double`

also specifies training flag `y`

= lineq(`x`

,`tsym`

,`tf`

)`tf`

. The System object starts training when `tf`

changes from
`false`

to `true`

(at the rising edge). The
System object trains until all symbols in `tsym`

are processed. The
input `tsym`

is ignored when `tf`

is
`false`

. To enable this syntax, set the Algorithm
property to `'LMS'`

or `'RLS'`

and TrainingFlagInputPort property to `true`

.

also specifies adapts weights flag `y`

= lineq(`x`

,`aw`

)`aw`

. The System object adapts the equalizer tap weights when `aw`

is
`true`

. If `aw`

is `false`

, the
System object keeps the weights unchanged. To enable this syntax, set the Algorithm
property to `'CMA'`

and AdaptWeightsSource property to `'Input port'`

.

`x`

— Input signalcolumn vector

Input signal, specified as a column vector. The input signal vector length must be equal to an integer multiple of the InputSamplesPerSymbol property value. For more information, see Symbol Tap Spacing.

**Data Types: **`double`

**Complex Number Support: **Yes

`tsym`

— Training symbolscolumn vector

Training symbols, specified as a column vector of length less than or equal to the
length of input `x`

. The input `tsym`

is ignored
when `tf`

is `false`

.

To enable this argument, set the Algorithm
property to `'LMS'`

or `'RLS'`

.

**Data Types: **`double`

**Complex Number Support: **Yes

`tf`

— Training flag`true`

| `false`

Training flag, specified as `true`

or `false`

.
The System object starts training when `tf`

changes from
`false`

to `true`

(at the rising edge). The
System object trains until all symbols in `tsym`

are processed. The
input `tsym`

is ignored when `tf`

is
`false`

.

To enable this argument, set the Algorithm
property to `'LMS'`

or `'RLS'`

and TrainingFlagInputPort property to `true`

.

**Data Types: **`logical`

`aw`

— Adapt weights flag`true`

| `false`

Adapt weights flag, specified as `true`

or
`false`

. If `aw`

is `true`

, the
System object adapts weights. If `aw`

is `false`

,
the System object keeps the weights unchanged.

To enable this argument, set the Algorithm
property to `'CMA'`

and AdaptWeightsSource property to `'Input port'`

.

**Data Types: **`logical`

`y`

— Equalized symbolscolumn vector

Equalized symbols, returned as a column vector that has the same length as input
signal `x`

.

`err`

— Error signalcolumn vector

Error signal, returned as a column vector that has the same length as input signal
`x`

.

`weights`

— Tap weightscolumn vector

Tap weights, returned as a column vector that has NumTaps
elements. `weights`

contains the tap weights from the last tap
weight update.

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)

`comm.LinearEqualizer`

`isLocked` | Determine if System object is in use |

`clone` | Create duplicate System object |

`info` | Characteristic information about the equalizer object |

`maxstep` | Maximum step size for LMS equalizer convergence |

`mmseweights` | Linear equalizer MMSE tap weights |

Create a BPSK modulator and an equalizer System object™, specifying a linear LMS equalizer having eight taps and a step size of 0.03.

bpsk = comm.BPSKModulator; eqlms = comm.LinearEqualizer('Algorithm','LMS','NumTaps',8,'StepSize',0.03);

Change the reference tap index of the equalizer.

eqlms.ReferenceTap = 4;

Build a set of test data. Receive the data by convolving the signal.

x = bpsk(randi([0 1],1000,1)); rxsig = conv(x,[1 0.8 0.3]);

Use `maxstep`

to find the maximum permitted step size.

mxStep = maxstep(eqlms,rxsig)

mxStep = 0.1384

Equalize the received signal. Use the first 200 symbols as the training sequence.

y = eqlms(rxsig,x(1:200));

Apply linear equalization using the least mean squares (LMS) algorithm to recover QPSK symbols passed through a multipath AWGN channel.

Initialize simulation variables.

```
M = 4; % QPSK
numSymbols = 10000;
numTrainingSymbols = 1000;
chtaps = [1 0.5*exp(1i*pi/6) 0.1*exp(-1i*pi/8)];
```

Generate QPSK-modulated symbols. Apply multipath channel filtering and AWGN impairments to the symbols.

```
data = randi([0 M-1],numSymbols,1);
tx = pskmod(data,M,pi/4);
rx = awgn(filter(chtaps,1,tx),25,'measured');
```

Create a linear equalizer System object and display the default configuration. Adjust the reference tap to `1`

. Check the maximum permitted step size. Equalize the impaired symbols.

eq = comm.LinearEqualizer

eq = comm.LinearEqualizer with properties: Algorithm: 'LMS' NumTaps: 5 StepSize: 0.0100 Constellation: [1x4 double] ReferenceTap: 3 InputDelay: 0 InputSamplesPerSymbol: 1 TrainingFlagInputPort: false AdaptAfterTraining: true InitialWeightsSource: 'Auto' WeightUpdatePeriod: 1

eq.ReferenceTap = 1; mxStep = maxstep(eq,rx)

mxStep = 0.3154

[y,err,weights] = eq(rx,tx(1:numTrainingSymbols));

Plot the constellation of the impaired and equalized symbols.

```
constell = comm.ConstellationDiagram('NumInputPorts',2);
constell(rx,y)
```

Plot the equalizer error signal and compute the error vector magnitude (EVM) of the equalized symbols.

plot(abs(err)) grid on; xlabel('Symbols'); ylabel('|e|');title('Equalizer Error Signal')

errevm = comm.EVM; evm = errevm(tx,y)

evm = 11.7710

Plot the equalizer tap weights.

subplot(3,1,1); stem(real(weights)); ylabel('real(weights)'); xlabel('Tap'); grid on; axis([0 6 -0.5 1]) title('Equalizer Tap Weights') subplot(3,1,2); stem(imag(weights)); ylabel('imag(weights)'); xlabel('Tap'); grid on; axis([0 6 -0.5 1]) subplot(3,1,3); stem(abs(weights)); ylabel('abs(weights)'); xlabel('Tap'); grid on; axis([0 6 -0.5 1])

Demonstrate linear equalization by using the least mean squares (LMS) algorithm to recover QPSK symbols passed through an AWGN channel. Apply different equalizer training schemes and show the symbol error magnitude.

**System Setup**

Simulate a QPSK-modulated system subject to AWGN. Transmit packets composed of 200 training symbols and 1800 random data symbols. Configure a linear LMS equalizer to recover the packet data.

M = 4; numTrainSymbols = 200; numDataSymbols = 1800; SNR = 20; trainingSymbols = pskmod(randi([0 M-1],numTrainSymbols,1),M,pi/4); numPkts = 10; lineq = comm.LinearEqualizer('Algorithm','LMS', ... 'NumTaps',5,'ReferenceTap',3,'StepSize',0.01);

**Train the Equalizer at the Beginning of Each Packet With Reset**

Use prepended training symbols when processing each packet. After processing each packet, reset the equalizer. This reset forces the equalizer to train the taps with no previous knowledge. Equalizer error signal plots for the first, second, and last packet show higher symbol errors at the start of each packet.

jj = 1; figure for ii = 1:numPkts b = randi([0 M-1],numDataSymbols,1); dataSym = pskmod(b,M,pi/4); packet = [trainingSymbols;dataSym]; rx = awgn(packet,SNR); [~,err] = lineq(rx,trainingSymbols); reset(lineq) if (ii ==1 || ii == 2 ||ii == numPkts) subplot(3,1,jj) plot(abs(err)) title(['Packet # ',num2str(ii)]) xlabel('Symbols') ylabel('Error Magnitude') axis([0,length(packet),0,1]) grid on; jj = jj+1; end end

**Train the Equalizer at the Beginning of Each Packet Without Reset**

Process each packet using prepended training symbols. Do not reset the equalizer after each packet is processed. By not resetting after each packet, the equalizer retains tap weights from training prior packets. Equalizer error signal plots for the first, second, and last packet show that after the initial training on the first packet, subsequent packets have fewer symbol errors at the start of each packet.

release(lineq) jj = 1; figure for ii = 1:numPkts b = randi([0 M-1],numDataSymbols,1); dataSym = pskmod(b,M,pi/4); packet = [trainingSymbols;dataSym]; channel = 1; rx = awgn(packet*channel,SNR); [~,err] = lineq(rx,trainingSymbols); if (ii ==1 || ii == 2 ||ii == numPkts) subplot(3,1,jj) plot(abs(err)) title(['Packet # ',num2str(ii)]) xlabel('Symbols') ylabel('Error Magnitude') axis([0,length(packet),0,1]) grid on; jj = jj+1; end end

**Train the Equalizer Periodically**

Systems with signals subject to time-varying channels require periodic equalizer training to maintain lock on the channel variations. Specify a system that has 200 symbols of training for every 1800 data symbols. Between training, the equalizer does not update tap weights. The equalizer processes 200 symbols per packet.

```
Rs = 1e6;
fd = 20;
spp = 200; % Symbols per packet
b = randi([0 M-1],numDataSymbols,1);
dataSym = pskmod(b,M,pi/4);
packet = [trainingSymbols; dataSym];
stream = repmat(packet,10,1);
tx = (0:length(stream)-1)'/Rs;
channel = exp(1i*2*pi*fd*tx);
rx = awgn(stream.*channel,SNR);
```

Set the `AdaptAfterTraining`

property to `false`

to stop the equalizer tap weight updates after the training phase.

release(lineq) lineq.AdaptAfterTraining = false

lineq = comm.LinearEqualizer with properties: Algorithm: 'LMS' NumTaps: 5 StepSize: 0.0100 Constellation: [1x4 double] ReferenceTap: 3 InputDelay: 0 InputSamplesPerSymbol: 1 TrainingFlagInputPort: false AdaptAfterTraining: false InitialWeightsSource: 'Auto' WeightUpdatePeriod: 1

Equalize the impaired data. Plot the angular error from the channel, the equalizer error signal, and signal constellation. As the channel varies, the equalizer output does not remove the channel effects. The output constellation rotates out of sync, resulting in bit errors.

[y,err] = lineq(rx,trainingSymbols); figure subplot(2,1,1) plot(tx, unwrap(angle(channel))) xlabel('Time (sec)') ylabel('Channel Angle (rad)') title('Angular Error Over Time') subplot(2,1,2) plot(abs(err)) xlabel('Symbols') ylabel('Error Magnitude') grid on title('Time-Varying Channel Without Retraining')

scatterplot(y)

Set the `TrainingInputPort`

property to `true`

to configure the equalizer to retrain the taps when signaled by the `trainFlag`

input. The equalizer trains only when `trainFlag`

is `true`

. After every 2000 symbols, the equalizer retrains the taps and keeps lock on variations of the channel. Plot the angular error from the channel, equalizer error signal, and signal constellation. As the channel varies, the equalizer output removes the channel effects. The output constellation does not rotate out of sync and bit errors are reduced.

release(lineq) lineq.TrainingFlagInputPort = true; symbolCnt = 0; numPackets = length(rx)/spp; trainFlag = true; trainingPeriod = 2000; eVec = zeros(size(rx)); yVec = zeros(size(rx)); for p=1:numPackets [yVec((p-1)*spp+1:p*spp,1),eVec((p-1)*spp+1:p*spp,1)] = ... lineq(rx((p-1)*spp+1:p*spp,1),trainingSymbols,trainFlag); symbolCnt = symbolCnt + spp; if symbolCnt >= trainingPeriod trainFlag = true; symbolCnt = 0; else trainFlag = false; end end figure subplot(2,1,1) plot(tx, unwrap(angle(channel))) xlabel('t (sec)') ylabel('Channel Angle (rad)') title('Angular Error Over Time') subplot(2,1,2) plot(abs(eVec)) xlabel('Symbols') ylabel('Error Magnitude') grid on title('Time-Varying Channel With Retraining')

scatterplot(yVec)

Simulate a system with delay between the transmitted symbols and received samples. Typical systems have transmitter and receiver filters that result in a delay. This delay must be accounted for to synchronize the system. In this example, the system delay is introduced without transmit and receive filters. Linear equalization, using the least mean squares (LMS) algorithm, recovers QPSK symbols.

Initialize simulation variables.

```
M = 4; % QPSK
numSymbols = 10000;
numTrainingSymbols = 1000;
mpChan = [1 0.5*exp(1i*pi/6) 0.1*exp(-1i*pi/8)];
systemDelay = dsp.Delay(20);
snr = 24;
```

Generate QPSK-modulated symbols. Apply multipath channel filtering, a system delay, and AWGN to the transmitted symbols.

data = randi([0 M-1],numSymbols,1); tx = pskmod(data,M,pi/4); % OQPSK delayedSym = systemDelay(filter(mpChan,1,tx)); rx = awgn(delayedSym,snr,'measured');

Create equalizer and EVM System objects. The equalizer System object specifies a linear equalizer that uses the LMS algorithm.

lineq = comm.LinearEqualizer('Algorithm','LMS', ... 'NumTaps',9,'ReferenceTap',5); evm = comm.EVM('ReferenceSignalSource', ... 'Estimated from reference constellation');

**Equalize Without Adjusting Input Delay**

Equalize the received symbols.

[y1,err1,wts1] = lineq(rx,tx(1:numTrainingSymbols,1));

Find the delay between the received symbols and the transmitted symbols by using the `finddelay`

function.

rxDelay = finddelay(tx,rx)

rxDelay = 20

Display the equalizer information. The latency value indicates the delay introduced by the equalizer. Calculate the total delay as the sum of `rxDelay`

and the equalizer latency.

eqInfo = info(lineq)

`eqInfo = `*struct with fields:*
Latency: 4

totalDelay = rxDelay + eqInfo.Latency;

Until the equalizer output converges, the symbol error rate is high. Plot the error output, `err1`

, to determine when the equalized output converges.

plot(abs(err1)) xlabel('Symbols') ylabel('Error Magnitude') title('Equalizer Error Signal')

The plot shows excessive errors beyond the 1000 symbols training period. When demodulating symbols and computing symbol errors, to account for the unconverged output and the system delay between the equalizer output and transmitted symbols, skip the first 2000 symbols.

dataRec1 = pskdemod(y1(2000+totalDelay:end),M,pi/4); symErrWithDelay = symerr(data(2000:end-totalDelay),dataRec1)

symErrWithDelay = 5999

evmWithDelay = evm(y1)

evmWithDelay = 29.5795

The error rate and EVM are high because the receive delay was not accounted for in the equalizer System object.

**Adjust Input Delay in Equalizer**

Equalize the received data by using the delay value to set the `InputDelay`

property. Because `InputDelay`

is a nontunable property, you must release the `lineq`

System object to reconfigure the `InputDelay`

property. Equalize the received symbols.

release(lineq) lineq.InputDelay = rxDelay

lineq = comm.LinearEqualizer with properties: Algorithm: 'LMS' NumTaps: 9 StepSize: 0.0100 Constellation: [1x4 double] ReferenceTap: 5 InputDelay: 20 InputSamplesPerSymbol: 1 TrainingFlagInputPort: false AdaptAfterTraining: true InitialWeightsSource: 'Auto' WeightUpdatePeriod: 1

[y2,err2,wts2] = lineq(rx,tx(1:numTrainingSymbols,1));

Plot the tap weights and equalized error magnitude. A stem plot shows the equalizer tap weights before and after the system delay is removed. A 2-D line plot shows the slower equalizer convergence for the delayed signal as compared to the signal with the delay removed.

subplot(2,1,1) stem([real(wts1),real(wts2)]) xlabel('Taps') ylabel('Tap Weight Real') legend('rxDelayed','rxDelayRemoved') grid on subplot(2,1,2) stem([imag(wts1),imag(wts2)]) xlabel('Taps') ylabel('Tap Weight Imaginary') legend('rxDelayed','rxDelayRemoved') grid on

figure plot([abs(err1),abs(err2)]) xlabel('Symbols') ylabel('Error Magnitude') legend('rxDelayed','rxDelayRemoved') grid on

Plot error output of the equalized signals, `rxDelayed`

and `rxDelayRemoved`

. For the signal that has the delay removed, the equalizer converges during the 1000 symbol training period. When demodulating symbols and computing symbol errors, to account for the unconverged output and the system delay between the equalizer output and transmitted symbols, skip the first 500 symbols. Reconfiguring the equalizer to account for the system delay enables better equalization of the signal, and reduces symbol errors and the EVM.

eqInfo = info(lineq)

`eqInfo = `*struct with fields:*
Latency: 4

totalDelay = rxDelay + eqInfo.Latency; dataRec2 = pskdemod(y2(500+totalDelay:end),M,pi/4); symErrDelayRemoved = symerr(data(500:end-totalDelay),dataRec2)

symErrDelayRemoved = 0

evmDelayRemoved = evm(y2(500+totalDelay:end))

evmDelayRemoved = 9.4435

Recover QPSK symbols with a linear equalizer by using the constant modulus algorithm (CMA) and EVM-based taps training. When using blind equalizer algorithms, such as CMA, train the equalizer taps by using the `AdaptWeights`

property to start and stop training. Helper functions are used to generate plots and apply phase correction.

Initialize system variables.

rng(123456); M = 4; % QPSK numSymbols = 100; numPackets = 5000; raylChan = comm.RayleighChannel('PathDelays',[0 1], ... 'AveragePathGains',[0 -12],'MaximumDopplerShift',1e-5); SNR = 50; adaptWeights = true;

Create the equalizer and EVM System objects. The equalizer System object specifies a linear equalizer by using the CMA adaptive algorithm. Call the helper function to initialize figure plots.

lineq = comm.LinearEqualizer('Algorithm','CMA', ... 'NumTaps',5,'ReferenceTap',3, ... 'StepSize',0.03,'AdaptWeightsSource','Input port')

lineq = comm.LinearEqualizer with properties: Algorithm: 'CMA' NumTaps: 5 StepSize: 0.0300 Constellation: [1x4 double] ReferenceTap: 3 InputSamplesPerSymbol: 1 AdaptWeightsSource: 'Input port' InitialWeightsSource: 'Auto' WeightUpdatePeriod: 1

info(lineq)

`ans = `*struct with fields:*
Latency: 2

evm = comm.EVM('ReferenceSignalSource', ... 'Estimated from reference constellation'); [errPlot,evmPlot,scatSym,adaptState] = initFigures(numPackets,lineq);

**Equalization Loop**

To implement the equalization loop:

Generate PSK data packets.

Apply Rayleigh fading and AWGN to the transmission data.

Apply equalization to the received data and phase correction to the equalizer output.

Estimate the EVM and toggle the

`adaptWeights`

flag to`true`

or`false`

based on the EVM level.Update the figure plots.

for p=1:numPackets data = randi([0 M-1],numSymbols,1); tx = pskmod(data,M,pi/4); rx = awgn(raylChan(tx),SNR); rxDelay = finddelay(rx,tx); [y,err,wts] = lineq(rx,adaptWeights); y = phaseCorrection(y); evmEst = evm(y); adaptWeights = (evmEst > 20); updateFigures(errPlot,evmPlot,scatSym,adaptState, ... wts,y(end),evmEst,adaptWeights,p,numPackets) end

rxDelay

rxDelay = 0

The figure plots show that, as the EVM varies, the equalizer toggles in and out of decision-directed weight adaptation mode.

**Helper Functions**

This helper function initializes figures that show a quad plot of simulation results.

function [errPlot,evmPlot,scatter,adaptState] = initFigures(numPkts,lineq) yVec = nan(numPkts,1); evmVec = nan(numPkts,1); wVec = zeros(lineq.NumTaps,1); adaptVec = nan(numPkts,1); figure subplot(2,2,1) evmPlot = stem(wVec); grid on; axis([1 lineq.NumTaps 0 1.8]) xlabel('Taps'); ylabel('|Weights|'); title('Tap Weight Magnitude') subplot(2,2,2) scatter = plot(yVec, '.'); axis square; axis([-1.2 1.2 -1.2 1.2]); grid on xlabel('In-phase'); ylabel('Quadrature'); title('Scatter Plot'); subplot(2,2,3) adaptState = plot(adaptVec); grid on; axis([0 numPkts -0.2 1.2]) ylabel('Training'); xlabel('Symbols'); title('Adapt Weights Signal') subplot(2,2,4) errPlot = plot(evmVec); grid on; axis([1 numPkts 0 100]) xlabel('Symbols'); ylabel('EVM (%)'); title('EVM') end

This helper function updates figures.

function updateFigures(errPlot,evmPlot,scatSym, ... adaptState,w,y,evmEst,adaptWts,p,numFrames) persistent yVec evmVec adaptVec if p == 1 yVec = nan(numFrames,1); evmVec = nan(numFrames,1); adaptVec = nan(numFrames,1); end yVec(p) = y; evmVec(p) = evmEst; adaptVec(p) = adaptWts; errPlot.YData = abs(evmVec); evmPlot.YData = abs(w); scatSym.XData = real(yVec); scatSym.YData = imag(yVec); adaptState.YData = adaptVec; drawnow limitrate end

This helper function applies phase correction.

function y = phaseCorrection(y) a = angle(y((real(y) > 0) & (imag(y) > 0))); a(a < 0.1) = a(a < 0.1) + pi/2; theta = mean(a) - pi/4; y = y * exp(-1i*theta); end

Recover QPSK symbols in fading environments with a linear equalizer, using the least mean squares (LMS) algorithm. Use the `reset`

object function to equalize independent packets. Use helper functions to generate plots. This example also shows symbol-based processing and frame-based processing.

**Setup**

Initialize system variables, create an equalizer System object, and initialize the plot figures.

M = 4; % QPSK numSym = 1000; numTrainingSym = 100; numPackets = 5; numTaps = 9; ttlNumSym = numSym + numTrainingSym; raylChan = comm.RayleighChannel('PathDelays',[0 1], ... 'AveragePathGains',[0 -9], ... 'MaximumDopplerShift',0, ... 'PathGainsOutputPort',true); SNR = 35; rxVec = zeros(ttlNumSym,numPackets); txVec = zeros(ttlNumSym,numPackets); yVec = zeros(ttlNumSym,1); eVec = zeros(ttlNumSym,1); lineq1 = comm.LinearEqualizer('Algorithm','LMS', ... 'NumTaps',numTaps,'ReferenceTap',5, ... 'StepSize',0.01,'TrainingFlagInputPort',true); [errPlot,wStem,hStem,scatPlot] = initFigures(ttlNumSym,lineq1, ... raylChan.AveragePathGains);

**Symbol-Based Processing**

For symbol-based processing, provide one symbol at the input of the equalizer. Reset the equalizer state and channel after processing each packet.

for p = 1:numPackets trainingFlag = true; for q=1:ttlNumSym data = randi([0 M-1],1,1); tx = pskmod(data,M,pi/4); [xc,pg] = raylChan(tx); rx = awgn(xc,25); [y,err,wts] = lineq1(rx,tx,trainingFlag);

Disable training after processing `numTrainingSym`

training symbols.

if q == numTrainingSym trainingFlag = false; end updateFigures(errPlot,wStem,hStem,scatPlot,err,wts,y,pg,q,ttlNumSym); txVec(q,p) = tx; rxVec(q,p) = rx; end

After processing each packet, reset the channel System object to get a new realization of channel taps and the equalizer System object to restore the default set of tap weights.

```
reset(raylChan)
reset(lineq1)
end
```

**Packet-Based Processing**

For packet-based processing, provide one packet at the input of the equalizer. Each packet contains `ttlNumSym`

symbols. Because the training duration is less than the packet length, you do not need to specify the start-training input.

yVecPkt = zeros(ttlNumSym,numPackets); errVecPkt = zeros(ttlNumSym,numPackets); wgtVecPkt = zeros(numTaps,numPackets); lineq2 = comm.LinearEqualizer('Algorithm','LMS', ... 'NumTaps',9,'ReferenceTap',6,'StepSize',0.01); for p = 1:numPackets [yVecPkt(:,p),errVecPkt(:,p),wgtVecPkt(:,p)] = ... lineq2(rxVec(:,p),txVec(1:numTrainingSym,p)); for q=1:ttlNumSym updateFigures(errPlot,wStem,hStem,scatPlot, ... errVecPkt(q,p),wgtVecPkt(:,p),yVecPkt(q,p),pg,q,ttlNumSym); end

After processing each packet, reset the channel System object to get a new realization of channel taps and the equalizer System object to restore the default set of tap weights.

```
reset(raylChan)
reset(lineq2)
end
```

**Helper Functions**

The helper function initializes the figures.

function [errPlot,wStem,hStem,scatPlot] = initFigures(ttlNumSym,lineq,pg) yVec = nan(ttlNumSym,1); eVec = nan(ttlNumSym,1); wVec = zeros(lineq.NumTaps,1); figure; subplot(2,2,1); wStem = stem(wVec); axis([1 lineq.NumTaps 0 1.8]); grid on xlabel('Taps'); ylabel('|Weights|'); title('Tap Weight Magnitude') subplot(2,2,2); hStem = stem([0 abs(pg) 0]); grid on; xlabel('Taps'); ylabel('|Path Gain|'); title('Channel Path Gain Magnitude') subplot(2,2,3); errPlot = plot(eVec); axis([1 ttlNumSym 0 1.2]); grid on xlabel('Symbols'); ylabel('|Error Magnitude|'); title('Error Magnitude') subplot(2,2,4); scatPlot = plot(yVec,'.'); axis square; axis([-1.2 1.2 -1.2 1.2]); grid on; xlabel('In-phase'); ylabel('Quadrature'); title(sprintf('Scatter Plot')); end

This helper function updates the figures.

function updateFigures(errPlot,wStem,hStem,scatPlot, ... err,wts,y,pg,p,ttlNumSym) persistent yVec eVec if p == 1 yVec = nan(ttlNumSym,1); eVec = nan(ttlNumSym,1); end yVec(p) = y; eVec(p) = abs(err); errPlot.YData = abs(eVec); wStem.YData = abs(wts); hStem.YData = [0 abs(pg) 0]; scatPlot.XData = real(yVec); scatPlot.YData = imag(yVec); drawnow limitrate end

Use the linear equalizer in nonadaptive mode. Use the `mmseweights`

object function to calculate the minimum mean squared error (MMSE) solution and use the weights returned as the set of tap weights for the linear equalizer.

Initialize simulation variables.

```
M = 4; % QPSK
numSymbols = 10000;
numTrainingSymbols = 1000;
chtaps = [1 0.5*exp(1i*pi/6) 0.1*exp(-1i*pi/8)];
EbN0 = 20;
```

Generate QPSK modulated symbols. Apply delayed multipath channel filtering and AWGN impairments to the symbols.

```
data = randi([0 M-1], numSymbols, 1);
tx = pskmod(data, M, pi/4);
rx = awgn(filter(chtaps,1,tx),25,'measured');
```

Create a linear equalizer System object configured to use CMA algorithm, set the `AdaptWeights`

property to `false`

, and the `InitialWeightsSource`

property to `Property`

. Calculate the MMSE weights. Set the initial tap weights to the calculated MMSE weights. Equalize the impaired symbols.

eq = comm.LinearEqualizer('Algorithm','CMA','AdaptWeights',false,'InitialWeightsSource','Property')

eq = comm.LinearEqualizer with properties: Algorithm: 'CMA' NumTaps: 5 StepSize: 0.0100 Constellation: [1x4 double] InputSamplesPerSymbol: 1 AdaptWeightsSource: 'Property' AdaptWeights: false InitialWeightsSource: 'Property' InitialWeights: [5x1 double] WeightUpdatePeriod: 1

wgts = mmseweights(eq,chtaps,EbN0)

`wgts = `*5×1 complex*
0.0005 - 0.0068i
0.0103 + 0.0117i
0.9694 - 0.0019i
-0.3987 + 0.2186i
0.0389 - 0.1756i

eq.InitialWeights = wgts; [y,err,weights] = eq(rx);

Plot constellation of impaired and equalized symbols.

```
constell = comm.ConstellationDiagram('NumInputPorts',2);
constell(rx,y)
```

Plot the equalizer error signal and compute the error vector magnitude of the equalized symbols.

plot(abs(err)) grid on; xlabel('Symbols'); ylabel('|e|')

errevm = comm.EVM; evm = errevm(tx,y)

evm = 140.6177

Plot equalizer tap weights.

subplot(3,1,1); stem(real(weights)); ylabel('real(weights)'); xlabel('Tap'); grid on; axis([1 8 -0.5 1]) line([eq.NumTaps+0.5 eq.NumTaps+0.5], [-0.5 1], 'Color', 'r', 'LineWidth', 1) title('Equalizer Tap Weights') subplot(3,1,2); stem(imag(weights)); ylabel('imag(weights)'); xlabel('Tap'); grid on; axis([1 8 -0.5 1]) line([eq.NumTaps+0.5 eq.NumTaps+0.5], [-0.5 1], 'Color', 'r', 'LineWidth', 1) subplot(3,1,3); stem(abs(weights)); ylabel('abs(weights)'); xlabel('Tap'); grid on; axis([1 8 -0.5 1]) line([eq.NumTaps+0.5 eq.NumTaps+0.5], [-0.5 1], 'Color', 'r', 'LineWidth', 1)

Demonstrate linear equalization by using the least mean squares (LMS) algorithm to recover QPSK symbols passed through an AWGN channel. Process the signal sample-by-sample.

**System Setup**

Simulate a QPSK-modulated system subject to AWGN. Transmit packets composed of 200 training symbols and 1800 random data symbols. Configure a linear LMS equalizer to recover the packet data.

M = 4; numTrainSymbols = 200; numDataSymbols = 1800; SNR = 20; trainingSymbols = pskmod(randi([0 M-1],numTrainSymbols,1),M,pi/4); numPkts = 10; lineq = comm.LinearEqualizer('Algorithm','LMS', ... 'NumTaps',5,'ReferenceTap',3,'StepSize',0.01,'TrainingFlagInputPort',true);

**Main Loop**

Use prepended training symbols when processing each packet. After processing each packet, reset the equalizer. This reset forces the equalizer to train the taps with no previous knowledge. Equalize the received signal sample-by-sample. For each packet, use the first 200 symbols for training.

subPlotCnt = 1; figure for ii = 1:numPkts b = randi([0 M-1],numDataSymbols,1); dataSym = pskmod(b,M,pi/4); packet = [trainingSymbols;dataSym]; rx = awgn(packet,SNR); y = zeros(numDataSymbols+numTrainSymbols,1); err = zeros(numDataSymbols+numTrainSymbols,1); for jj = 1:numDataSymbols+numTrainSymbols if jj <= numTrainSymbols [y(jj),err(jj)] = lineq(rx(jj),trainingSymbols(jj),true); else [y(jj),err(jj)] = lineq(rx(jj),1i,false); end end reset(lineq) if (ii ==1 || ii == 2 ||ii == numPkts) subplot(3,1,subPlotCnt) plot(abs(err)) title(['Packet # ',num2str(ii)]) xlabel('Symbols') ylabel('Error Magnitude') axis([0,length(packet),0,1]) grid on; subPlotCnt = subPlotCnt+1; end end

Demonstrate linear equalization by using the least mean squares (LMS) algorithm to recover QPSK symbols passed through an AWGN channel. Process a packet that has training symbols at the beginning in multiple passes. Compare results to equalization processing the full packet in a single pass.

**System Setup**

Simulate a QPSK-modulated system subject to AWGN. Transmit packets composed of 200 training symbols and 1800 random data symbols. Configure a linear LMS equalizer to recover the packet data.

M = 4; numTrainSymbols = 200; numDataSymbols = 1800; SNR = 20; trainingSymbols = pskmod(randi([0 M-1],numTrainSymbols,1),M,pi/4); b = randi([0 M-1],numDataSymbols,1); dataSym = pskmod(b,M,pi/4); packet = [trainingSymbols;dataSym]; rx = awgn(packet,SNR); lineq = comm.LinearEqualizer('Algorithm','LMS', ... 'NumTaps',5,'ReferenceTap',3,'StepSize',0.01);

**Process Packet in One Pass**

Use prepended training symbols when processing each packet. After processing each packet, reset the equalizer. This reset forces the equalizer to train the taps with no previous knowledge. Equalize the received signal sample-by-sample. For each packet, use the first 200 symbols for training.

subPlotCnt = 1; figure [y1,err1] = lineq(rx,trainingSymbols); reset(lineq) plot(abs(err1)) title("Single Pass Processing") xlabel('Symbols') ylabel('Error Magnitude') axis([0,length(packet),0,1]) grid on;

**Process Packet in Multiple Passes**

Use prepended training symbols when processing each packet. After processing each packet, reset the equalizer. This reset forces the equalizer to train the taps with no previous knowledge. Equalize the received signal sample-by-sample. For each packet, use the first 200 symbols for training.

lineq = comm.LinearEqualizer('Algorithm','LMS', ... 'NumTaps',5,'ReferenceTap',3,'StepSize',0.01,'TrainingFlagInputPort',true); frameLen = 100; numFrames = (numDataSymbols+numTrainSymbols) / frameLen; figure y2 = zeros(numDataSymbols+numTrainSymbols,1); err2 = zeros(numDataSymbols+numTrainSymbols,1); idx = 1:frameLen; symbolCnt = 0; for jj = 1:numFrames if symbolCnt < numTrainSymbols [y2(idx),err2(idx)] = lineq(rx(idx),trainingSymbols(idx),true); else [y2(idx),err2(idx)] = lineq(rx(idx),1i*ones(frameLen,1),false); end idx = idx + frameLen; symbolCnt = symbolCnt + frameLen; end reset(lineq) plot(abs(err2)) title("Multipass Processing") xlabel('Symbols') ylabel('Error Magnitude') axis([0,length(packet),0,1]) grid on;

Results from equalization using single pass and multipass approaches match.

outputsEqual = isequal(y1,y2)

`outputsEqual = `*logical*
1

errorsEqual = isequal(err1,err2)

`errorsEqual = `*logical*
1

You can configure the equalizer to operate as a symbol-spaced equalizer or as a fractional symbol-spaced equalizer.

To operate the equalizer at a symbol-spaced rate, specify the number of samples per symbol as

`1`

. Symbol-rate equalizers have taps spaced at the symbol duration. Symbol-rate equalizers are sensitive to timing phase.To operate the equalizer at a fractional symbol-spaced rate, specify the number of input samples per symbol as an integer greater than

`1`

and provide an input signal oversampled at that sampling rate. Fractional symbol-spaced equalizers have taps spaced at an integer fraction of the input symbol duration. Fractional symbol-spaced equalizers are not sensitive to timing phase.

Linear equalizers can remove intersymbol interference (ISI) when the frequency response of a channel has no null. If a null exists in the frequency response of a channel, linear equalizers tend to enhance the noise. In this case, use decision feedback equalizers to avoid enhancing the noise.

A linear equalizer consists of a tapped delay line that stores samples from the input signal. Once per symbol period, the equalizer outputs a weighted sum of the values in the delay line and updates the weights to prepare for the next symbol period.

Linear equalizers can be symbol-spaced or fractional symbol-spaced.

For a symbol-spaced equalizer, the number of samples per symbol,

*K*, is 1. The output sample rate equals the input sample rate.For a fractional symbol-spaced equalizer, the number of samples per symbol,

*K*, is an integer greater than 1. Typically,*K*is 4 for fractionally spaced equalizers. The output sample rate is 1/*T*and the input sample rate is*K/T*, where*T*is the symbol period. Tap-weight updating occurs at the output rate.

This schematic shows a linear equalizer with *L* weights, a symbol period of
*T*, and *K* samples per symbol. If
*K* is 1, the result is a symbol-spaced linear equalizer instead of a
fractional symbol-spaced linear equalizer.

In each symbol period, the equalizer receives *K* input samples at the
tapped delay line. The equalizer then outputs a weighted sum of the values in the tapped
delay line and updates the weights to prepare for the next symbol period.

For more information, see Equalization.

For the LMS algorithm, in the previous schematic, *w* is a vector of all
weights *w _{i}*, and

*w*_{new} =
*w*_{current} +
(*StepSize*)
*u**e**.

The step size used by the adaptive algorithm is specified as a positive scalar. Increasing the
step size reduces the equalizer convergence time but causes the equalized output signal to
be less stable. To determine the maximum step size allowed when using the LMS adaptive
algorithm, use the `maxstep`

object
function. The * operator denotes the complex conjugate and the error calculation *e* = *d* -
*y*.

For the RLS algorithm, in the previous schematic, *w* is the vector of all
weights *w _{i}*, and

$$K=\frac{Pu}{(ForgettingFactor)+{u}^{H}Pu}.$$

The forgetting factor used by the adaptive algorithm is specified as a scalar in the range (0,
1]. Decreasing the forgetting factor reduces the equalizer convergence time but causes the
equalized output signal to be less stable. *H* denotes the Hermitian
transpose. Based on the current inverse correlation matrix, the new inverse correlation
matrix is

$${P}_{\text{new}}=\frac{(1-K{u}^{H}){P}_{\text{current}}}{ForgettingFactor}.$$

Based on the current set of weights, the RLS algorithm creates the new set of weights as

*w*_{new} =
*w*_{current}+*K***e*.

The * operator denotes the complex conjugate and the error calculation *e* = *d* - *y*.

For the CMA adaptive algorithm, in the previous schematic, *w* is the vector
of all weights *w _{i}*, and

*w*_{new} =
*w*_{current} +
(*StepSize*)
*u***e*.

The step size used by the adaptive algorithm is specified as a positive scalar. Increasing the
step size reduces the equalizer convergence time but causes the equalized output signal to
be less stable. To determine the maximum step size allowed by the CMA adaptive algorithm,
use the `maxstep`

object
function. The * operator denotes the complex conjugate and the error calculation
*e* = *y*(*R* -
|*y*|^{2}), where *R* is a constant related to the signal
constellation.

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