%% BER Performance of Several Equalizer Types% This script shows the BER performance of several types of equalizers in a% static channel with a null in the passband.  The script constructs and% implements a linear equalizer object and a decision feedback equalizer (DFE)% object.  It also initializes and invokes a maximum likelihood sequence% estimation (MLSE) equalizer.  The MLSE equalizer is first invoked with perfect% channel knowledge, then with a straightforward but imperfect channel% estimation technique.%% As the simulation progresses, it updates a BER plot for comparative analysis% between the equalization methods.  It also shows the signal spectra of the% linearly equalized and DFE equalized signals.  It also shows the relative% burstiness of the errors, indicating that at low BERs, both the MLSE algorithm% and the DFE algorithm suffer from error bursts.  In particular, the DFE error% performance is burstier with detected bits fed back than with correct bits fed% back.  Finally, during the "imperfect" MLSE portion of the simulation, it% shows and dynamically updates the estimated channel response.%% This script relies on several other scripts and functions to perform link% simulations over a range of Eb/No values.  These files are as follows:%% <eqber_adaptive.html eqber_adaptive> - a script that runs link% simulations for linear and DFE equalizers%% <eqber_mlse.html eqber_mlse> - a script that runs link simulations% for ideal and imperfect MLSE equalizers%% <eqber_siggen.html eqber_siggen>   - a script that generates a BPSK% signal with no pulse shaping, then processes it through the channel and adds% noise%% eqber_graphics - a function that generates and updates plots showing the% performance of the linear, DFE, and MLSE equalizers.  Type "edit% eqber_graphics" at the MATLAB command line to view this file.%% The scripts eqber_adaptive and eqber_mlse illustrate how to use adaptive and% MLSE equalizers across multiple blocks of data such that state information is% retained between data blocks.%% To experiment with this demo, you can change such parameters as the channel% impulse response, the number of equalizer tap weights, the recursive least% squares (RLS) forgetting factor, the least mean square (LMS) step size, the% MLSE traceback length, the error in estimated channel length, and the maximum% number of errors collected at each Eb/No value.%   Copyright 1996-2004 The MathWorks, Inc.%   $Revision: 1.1.4.1 $  $Date: 2004/06/30 23:03:12 $%% Signal and channel parameters% Set parameters related to the signal and channel.  Use BPSK without any pulse% shaping, and a 5-tap real-valued symmetric channel impulse response.  (See% section 10.2.3 of Digital Communications by J. Proakis for more details on the% channel.)  Set initial states of data and noise generators.  Set the Eb/No% range.% System simulation parametersFs      = 1;      % sampling frequency (notional)nBits   = 2048;   % number of BPSK symbols per vectormaxErrs = 50;     % target number of errors at each Eb/NomaxBits = 1e8;    % maximum number of symbols at each Eb/No% Modulated signal parametersM          = 2;            % order of modulationRs         = Fs;           % symbol ratenSamp      = Fs/Rs;        % samples per symbolRb         = Rs * log2(M); % bit ratedataState  = 999983;       % initial state of data generator% Channel parameterschnl       = [0.227 0.460 0.688 0.460 0.227]';  % channel impulse responsechnlLen    = length(chnl);      % channel length, in samplesEbNo       = 0:14;              % in dBBER        = zeros(size(EbNo)); % initialize valuesnoiseState = 999917;            % initial state of noise generator%% Adaptive equalizer parameters% Set parameter values for the linear and DFE equalizers.  Use a 31-tap linear% equalizer, and a DFE with 15 feedforward and feedback taps.  Use the recursive% least squares (RLS) algorithm for the first block of data to ensure rapid tap% convergence.  Use the least mean square (LMS) algorithm thereafter to ensure% rapid execution speed.% Linear equalizer parametersnWts         = 31;       % number of weightsalgType1     = 'rls';    % RLS algorithm for first data block at each Eb/NoforgetFactor = 0.999999; % parameter of RLS algorithmalgType2     = 'lms';    % LMS algorithm for remaining data blocksstepSize     = 0.00001;  % parameter of LMS algorithm% DFE parameters - use same update algorithms as linear equalizernFwdWts      = 15;       % number of feedforward weights nFbkWts      = 15;       % number of feedback weights%% MLSE equalizer and channel estimation parameters, and initial visualization% Set the parameters of the MLSE equalizer.  Use a traceback length of six times% the length of the channel impulse response.  Initialize the equalizer states.% Set the equalization mode to "continuous", to enable seamless equalization% over multiple blocks of data.  Use a cyclic prefix in the channel esimation% technique, and set the length of the prefix.  Assume that the estimated length% of the channel impulse response is one sample longer than the actual length.% MLSE equalizer parameterstbLen      = 30;                 % MLSE equalizer traceback lengthnumStates  = M^(chnlLen-1);      % number of trellis states[mlseMetric, mlseStates, mlseInputs] = deal([]);const      = pskmod(0:M-1, M);   % signal constellationmlseType   = 'ideal';            % perfect channel estimates at firstmlseMode   = 'cont';             % no MLSE resets% Channel estimation parameterschnlEst = chnl;         % perfect estimation initiallyprefixLen = 2*chnlLen;  % cyclic prefix lengthexcessEst = 1;          % length of estimated channel impulse response                        % beyond the true length% Initialize the graphics for the simulation.  Plot the unequalized channel% frequency response, and the BER of an ideal BPSK system.idealBER = berawgn(EbNo, 'psk', M, 'nondiff');[hBER, hLegend, legendString, hLinSpec, hDfeSpec, hErrs, ...    hText1, hText2, hFit, hEstPlot] = eqber_graphics('init', chnl, EbNo, ...                                               idealBER, nBits);%% Construct RLS and LMS linear and DFE equalizer objects.% The RLS update algorithm is used to initially set the weights, and the LMS% algorithm is used thereafter for speed purposes.alg1 = eval([algType1 '(' num2str(forgetFactor) ')']);linEq1 = lineareq(nWts, alg1);alg2 = eval([algType2 '(' num2str(stepSize) ')']);linEq2 = lineareq(nWts, alg2);[linEq1.RefTap, linEq2.RefTap] = ...    deal(round(nWts/2));    % Set reference tap to center tap[linEq1.ResetBeforeFiltering, linEq2.ResetBeforeFiltering] = ...    deal(0);                % Maintain continuity between iterationsdfeEq1 = dfe(nFwdWts, nFbkWts, alg1);dfeEq2 = dfe(nFwdWts, nFbkWts, alg2);[dfeEq1.RefTap, dfeEq2.RefTap] = ...    deal(round(nFwdWts/2)); % Set reference tap to center forward tap[dfeEq1.ResetBeforeFiltering, dfeEq2.ResetBeforeFiltering] = ...    deal(0);                % Maintain continuity between iterations%% Linear equalizer% Run the linear equalizer, and plot the equalized signal spectrum, the BER, and% the burst error performance for each data block.  Note that as the Eb/No% increases, the linearly equalized signal spectrum has a progressively deeper% null.  This highlights the fact that a linear equalizer must have many more% taps to adequately equalize a channel with a deep null.  Note also that the% errors occur with small inter-error intervals, which is to be expected at such% a high error rate.%% See <eqber_adaptive.html eqber_adaptive> for a listing of the simulation code% for the adaptive equalizers.firstRun = true;  % flag to ensure known initial states for noise and dataeqType = 'linear';eqber_adaptive;%% Decision feedback equalizer% Run the DFE, and plot the equalized signal spectrum, the BER, and the burst% error performance for each data block.  Note that the DFE is much better able% to mitigate the channel null than the linear equalizer, as shown in the% spectral plot and the BER plot.  The plotted BER points at a given Eb/No value% are updated every data block, so they move up or down depending on the number% of errors collected in that block.  Note also that the DFE errors are somewhat% bursty, due to the error propagation caused by feeding back detected bits% instead of correct bits. The burst error plot shows that as the BER decreases,% a significant number of errors occurs with an inter-error arrival of five bits% or less.  (If the DFE equalizer were run in training mode at all times, the% errors would be far less bursty.)  %% For every data block, the plot also indicates the average inter-error interval% if those errors were randomly occurring.%% See <eqber_adaptive.html eqber_adaptive> for a listing of the simulation code% for the adaptive equalizers.eqType = 'dfe';eqber_adaptive; %% Ideal MLSE equalizer, with perfect channel knowledge% Run the MLSE equalizer with a perfect channel estimate, and plot the BER and% the burst error performance for each data block.  Note that the errors occur% in an extremely bursty fashion.  Observe, particularly at low BERs, that the% overwhelming percentage of errors occur with an inter-error interval of one or% two bits.%% See <eqber_mlse.html eqber_mlse> for a listing of the simulation code% for the MLSE equalizers.eqType = 'mlse';mlseType = 'ideal';eqber_mlse;%% MLSE equalizer with an imperfect channel estimate% Run the MLSE equalizer with an imperfect channel estimate, and plot the BER% and the burst error performance for each data block.  These results align% fairly closely with the ideal MLSE results.  (The channel estimation algorithm% is highly dependent on the data, such that an FFT of a transmitted data block% has no nulls.)  Note how the estimated channel plots compare with the actual% channel spectrum plot.%% See <eqber_mlse.html eqber_mlse> for a listing of the simulation code% for the MLSE equalizers.mlseType = 'imperfect';eqber_mlse;