Adaptive Equalizer Simulation (Part I)

This script simulates a communication link with PSK modulation, raised-cosine pulse shaping, multipath fading, and adaptive equalization.

It is the first of two parts: Part I (adapteqpt1.m) sets simulation parameters and creates channel and equalizer objects. Part II (adapteqpt2.m) performs a link simulation based on these settings, which are stored in the MATLAB® workspace. For information on the Part II script, enter demo('toolbox','comm') at the MATLAB prompt, and select "Adaptive Equalizer Simulation (Part II)".

Part I sets up three equalization scenarios, and calls Part II multiple times for each of these. Each call corresponds to a transmission block. The pulse shaping and multipath fading channel retain state information from one block to the next. For visualizing the impact of channel fading on adaptive equalizer convergence, the simulation resets the equalizer state every block.

To experiment with different simulation settings, you can edit the Part I script. For instance, you can set the ResetBeforeFiltering property of the equalizer object to 0, which will cause the equalizer to retain state from one block to the next.

Contents

Modulation and Transmission Block Transmit/Receive Filters Additive White Gaussian Noise Simulation 1: Frequency-Flat Fading Channel Simulation 2: Frequency-Selective Fading Channel and Linear Equalizer Simulation 3: Decision Feedback Equalizer (DFE)

Modulation and Transmission Block

Set parameters related to PSK modulation and the transmission block. The block comprises three contiguous parts: training sequence, payload, and tail sequence. All use the same PSK scheme. The training and tail sequences are used for equalization. We create a local random stream with known seed and state to be used by random number generators. Using this local stream ensures that the results generated by this demo will be repeatable.

Tsym = 1e-6; % Symbol period (s)bitsPerSymbol = 2; % Number of bits per PSK symbolM = 2.^bitsPerSymbol; % PSK alphabet size (number of modulation levels)nPayload = 400; % Number of payload symbolsnTrain = 100; % Number of training symbolsnTail = 20; % Number of tail symbolspskmodObj = modem.pskmod(M); % modulator objecthStream = RandStream('mt19937ar', 'Seed', 12345); % Local random number stream% Training sequence symbolsxTrainSym = randi(hStream, [0 M-1], 1, nTrain);% Modulated training sequencexTrain = modulate(pskmodObj, xTrainSym);

% Tail sequence symbolsxTailSym = randi(hStream, [0 M-1], 1, nTail);% Modulated tail sequencexTail = modulate(pskmodObj, xTailSym);

Transmit/Receive Filters

Create structures containing information about the transmit and receive filters (txFilt and rxFilt). Each filter has a square-root raised cosine frequency response, implemented with an FIR structure.

The transmit and receive filters incorporate upsampling and downsampling, respectively, and both use an efficient polyphase scheme (see Part II script for more information). These multirate filters retain state from one transmission block to the next, like the channel object (see "Simulation 1: Frequency-flat fading channel" below).

The peak value of the impulse response of the filter cascade is 1. The transmit filter uses a scale factor to ensure unit transmitted power.

To construct the pulse filter structures, this script uses an auxiliary function adapteq_buildfilter.m. An improved approach would be to use multirate filter objects from the Filter Design Toolbox™.

% Filter parametersnSymFilt = 8; % Number of symbol periods spanned by each filterosfFilt = 4; % Oversampling factor for filter (samples per symbol)rolloff = 0.25; % Rolloff factorTsamp = Tsym/osfFilt; % TX signal sample period (s)cutoffFreq = 1/(2*Tsym); % Cutoff frequency (half Nyquist bandwidth)orderFilt = nSymFilt*osfFilt; % Filter order (number of taps - 1)

% Filter responses and structuressqrtrcCoeff = firrcos(orderFilt, cutoffFreq, rolloff, 1/Tsamp, ... 'rolloff', 'sqrt');txFilt = adapteq_buildfilter(osfFilt*sqrtrcCoeff, osfFilt, 1);rxFilt = adapteq_buildfilter(sqrtrcCoeff, 1, osfFilt);

Additive White Gaussian Noise

Set signal-to-noise ratio parameter for additive white Gaussian noise.

EsNodB = 20; % Ratio of symbol energy to noise power spectral density (dB)snrdB = EsNodB - 10*log10(osfFilt); % Signal-to-noise ratio per sample (dB)

Simulation 1: Frequency-Flat Fading Channel

Begin with single-path, frequency-flat fading. For this channel, the receiver uses a simple 1-tap LMS (least mean square) equalizer, which implements automatic gain and phase control.

The Part II script (adapteqpt2.m) runs multiple times. Each run corresponds to a transmission block. The equalizer resets its state and weight every transmission block. (To retain state from one block to the next, you can set the

ResetBeforeFiltering property of the equalizer object to 0.)

Before the first run, the Part II script displays the initial properties of the channel and equalizer objects. For each run, a MATLAB figure shows signal processing visualizations. The red circles in the signal constellation plots correspond to symbol errors. In the "Weights" plot, blue and magenta lines correspond to real and imaginary parts, respectively. (The HTML version of this demo shows the last state of the visualizations.)

simName = 'Frequency-flat fading'; % Used to label figure window.

% Multipath channelfd = 30; % Maximum Doppler shift (Hz)chan = rayleighchan(Tsamp, fd); % Create channel object.chan.ResetBeforeFiltering = 0; % Allow state retention across blocks.

% Adaptive equalizernWeights = 1; % Single weightstepSize = 0.1; % Step size for LMS algorithmalg = lms(stepSize); % Adaptive algorithm objecteqObj = lineareq(nWeights, alg, pskmodObj.Constellation); % Equalizer object

% Link simulationnBlocks = 50; % Number of transmission blocks in simulationfor block = 1:nBlocks, adapteqpt2; end % Run Part II script in loop.chan =

ChannelType: 'Rayleigh' InputSamplePeriod: 2.5000e-007 DopplerSpectrum: [1x1 doppler.jakes] MaxDopplerShift: 30 PathDelays: 0 AvgPathGaindB: 0 NormalizePathGains: 1 StoreHistory: 0 StorePathGains: 0 PathGains: 1.3607 + 0.9826i ChannelFilterDelay: 0 ResetBeforeFiltering: 0 NumSamplesProcessed: 0

eqObj =

EqType: 'Linear Equalizer' AlgType: 'LMS' nWeights: 1 nSampPerSym: 1 RefTap: 1 SigConst: [1x4 double] StepSize: 0.1000 LeakageFactor: 1 Weights: 0 WeightInputs: 0 ResetBeforeFiltering: 1

NumSamplesProcessed: 0

Simulation 2: Frequency-Selective Fading Channel and Linear Equalizer

Simulate a three-path fading channel (frequency-selective fading). The receiver uses an 8-tap linear RLS (recursive least squares) equalizer with symbol-spaced taps. The simulation uses the channel object from Simulation 1, but with modified properties.

simName = 'Linear equalization of frequency-selective fading channel';

% Multipath channelchan.PathDelays = [0 0.9 1.5]*Tsym; % Path delay vector (s)chan.AvgPathGaindB = [0 -3 -6]; % Average path gain vector (dB)

% Adaptive equalizernWeights = 8;forgetFactor = 0.99; % RLS algorithm forgetting factoralg = rls(forgetFactor); % RLS algorithm objecteqObj = lineareq(nWeights, alg, pskmodObj.Constellation); % Equalizer objecteqObj.RefTap = 3; % Reference tap

% Link simulation. Store BER values.for block = 1:nBlocks, adapteqpt2; BERvect(block)=BER; endavgBER2 = mean(BERvect) % Average BER over transmission blockschan =

ChannelType: 'Rayleigh' InputSamplePeriod: 2.5000e-007 DopplerSpectrum: [1x1 doppler.jakes] MaxDopplerShift: 30 PathDelays: [0 9.0000e-007 1.5000e-006] AvgPathGaindB: [0 -3 -6] NormalizePathGains: 1 StoreHistory: 0 StorePathGains: 0 PathGains: [1x3 double]

ChannelFilterDelay: 4 ResetBeforeFiltering: 0 NumSamplesProcessed: 0

eqObj =

EqType: 'Linear Equalizer' AlgType: 'RLS' nWeights: 8 nSampPerSym: 1 RefTap: 3 SigConst: [1x4 double] ForgetFactor: 0.9900 InvCorrInit: 0.1000 InvCorrMatrix: [8x8 double] Weights: [0 0 0 0 0 0 0 0] WeightInputs: [0 0 0 0 0 0 0 0] ResetBeforeFiltering: 1 NumSamplesProcessed: 0

avgBER2 =

1.0000e-004

Simulation 3: Decision Feedback Equalizer (DFE)

The receiver uses a DFE with a six-tap fractionally spaced forward filter (two samples per symbol) and two feedback weights. The DFE uses the same RLS algorithm as in Simulation 2. The receive filter structure is reconstructed to account for the increased number of samples per symbol. This simulation uses the same channel object as in Simulation 2.

simName = 'Decision feedback equalizer (DFE)';

% Receive filter

nSamp = 2; % Two samples per symbol at equalizer inputrxFilt = adapteq_buildfilter(sqrtrcCoeff, 1, osfFilt/nSamp);

% Adaptive equalizernFwdWeights = 6; % Number of feedforward equalizer weightsnFbkWeights = 2; % Number of feedback filter weightseqObj = dfe(nFwdWeights, nFbkWeights, alg, pskmodObj.Constellation, nSamp);eqObj.RefTap = 3; % Reference tap

for block = 1:nBlocks, adapteqpt2; BERvect(block)=BER; endavgBER3 = mean(BERvect)

block = 1; % Reset variable (in case Part II is run independently).chan =

ChannelType: 'Rayleigh' InputSamplePeriod: 2.5000e-007 DopplerSpectrum: [1x1 doppler.jakes] MaxDopplerShift: 30 PathDelays: [0 9.0000e-007 1.5000e-006] AvgPathGaindB: [0 -3 -6] NormalizePathGains: 1 StoreHistory: 0 StorePathGains: 0 PathGains: [1x3 double] ChannelFilterDelay: 4 ResetBeforeFiltering: 0 NumSamplesProcessed: 105450

eqObj =

EqType: 'Decision Feedback Equalizer' AlgType: 'RLS' nWeights: [6 2] nSampPerSym: 2 RefTap: 3 SigConst: [1x4 double] ForgetFactor: 0.9900 InvCorrInit: 0.1000 InvCorrMatrix: [8x8 double] Weights: [0 0 0 0 0 0 0 0] WeightInputs: [0 0 0 0 0 0 0 0] ResetBeforeFiltering: 1 NumSamplesProcessed: 0

avgBER3 =

0.0010

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