%% 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. % % 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: 1.1.12.1 $ $Date: 2004/07/28 04:09:57 $ %% Code structure % 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: % % - a script that runs link % simulations for linear and DFE equalizers % % - a script that runs link simulations % for ideal and imperfect MLSE equalizers % % - a script that generates a binary phase % shift keying (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. % %% 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, 4th Ed., for more % details on the channel.) Set initial states of data and noise generators. % Set the Eb/No range. % System simulation parameters Fs = 1; % sampling frequency (notional) nBits = 2048; % number of BPSK symbols per vector maxErrs = 50; % target number of errors at each Eb/No maxBits = 1e8; % maximum number of symbols at each Eb/No % Modulated signal parameters M = 2; % order of modulation Rs = Fs; % symbol rate nSamp = Fs/Rs; % samples per symbol Rb = Rs * log2(M); % bit rate dataState = 999983; % initial state of data generator % Channel parameters chnl = [0.227 0.460 0.688 0.460 0.227]'; % channel impulse response chnlLen = length(chnl); % channel length, in samples EbNo = 0:14; % in dB BER = zeros(size(EbNo)); % initialize values noiseState = 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 parameters nWts = 31; % number of weights algType1 = 'rls'; % RLS algorithm for first data block at each Eb/No forgetFactor = 0.999999; % parameter of RLS algorithm algType2 = 'lms'; % LMS algorithm for remaining data blocks stepSize = 0.00001; % parameter of LMS algorithm % DFE parameters - use same update algorithms as linear equalizer nFwdWts = 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 estimation % 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 parameters tbLen = 30; % MLSE equalizer traceback length numStates = M^(chnlLen-1); % number of trellis states [mlseMetric, mlseStates, mlseInputs] = deal([]); const = pskmod(0:M-1, M); % signal constellation mlseType = 'ideal'; % perfect channel estimates at first mlseMode = 'cont'; % no MLSE resets % Channel estimation parameters chnlEst = chnl; % perfect estimation initially prefixLen = 2*chnlLen; % cyclic prefix length excessEst = 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 iterations dfeEq1 = 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 for a listing of the simulation code % for the adaptive equalizers. firstRun = true; % flag to ensure known initial states for noise and data eqType = '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 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 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 for a listing of the simulation code % for the MLSE equalizers. mlseType = 'imperfect'; eqber_mlse;