adaptequaldemo.m

自适应滤波器设计是现在滤波器设计的重要一方面

%% Equalization in Digital Communications 
% This demonstration illustrates the application of adaptive filters to
% channel equalization in digital communications.
%
% Author(s): Scott C. Douglas
% Copyright 1999-2005 The MathWorks, Inc.


%% Introduction
% Channel equalization is a simple way of mitigating the detrimental
% effects caused by a frequency-selective and/or dispersive communication
% link between sender and receiver.  For this demonstration, all signals
% are assumed to have a digital baseband representation.  During the
% training phase of channel equalization, a digital signal s[n] that is
% known to both the transmitter and receiver is sent by the transmitter to
% the receiver. The received signal x[n] contains two signals:
%        the signal s[n] filtered by the channel impulse response, and
%        an unknown broadband noise signal v[n].
% The goal is to filter x[n] to remove the inter-symbol interference
% (ISI) caused by the dispersive channel and to minimize the effect of
% the additive noise v[n].  Ideally, the output signal would closely 
% follow a delayed version of the transmitted signal s[n].  

%% Transmitted Input Signal s[n]
%
% A digital signal carries information through its discrete structure.
% There are several common baseband signaling methods.  We shall use
% a 16-QAM complex-valued symbol set, in which the input signal takes
% one of sixteen different values given by all possible combinations
% of {-3, -1, 1, 3} + j*{-3, -1, 1, 3}, where j = sqrt(-1).  Let's 
% generate a sequence of 5000 such symbols, where each one is equiprobable.

ntr = 5000;
j = sqrt(-1);
s = sign(randn(1,ntr)).*(2+sign(randn(1,ntr)))+...
    j*sign(randn(1,ntr)).*(2+sign(randn(1,ntr)));
plot(s,'o');
axis([-4 4 -4 4]); 
axis('square');
xlabel('Re\{s(n)\}');
ylabel('Im\{s(n)\}');
title('Input signal constellation');

%% Transmission Channel
%
% The transmission channel is defined by the channel impulse response 
% and the noise characteristics.  We shall choose a particular 
% channel that exhibits both frequency selectivity and dispersion.
% The noise variance is chosen so that the received signal-to-noise
% ratio is 30 dB.

b = exp(j*pi/5)*[0.2 0.7 0.9];
a = [1 -0.7 0.4];
% Transmission channel filter
channel = dfilt.df2t(b,a);
% Impulse response
hFV = fvtool(channel,'Analysis','impulse');
legend(hFV, 'Transmission channel');
set(hFV, 'Color', [1 1 1])
%%

% Frequency response
set(hFV, 'Analysis', 'freq')
%%

%% Received Signal x[n]
%
% The received signal x[n] is the signal s[n] filtered by
% the channel impulse response with additive noise v[n].
% We shall assume a complex Gaussian noise signal for
% the additive noise.

sig = sqrt(1/16*(4*18+8*10+4*2))/sqrt(1000)*norm(impz(channel));
v = sig*(randn(1,ntr) + j*randn(1,ntr))/sqrt(2);
x = filter(channel,s) + v;
plot(x,'.');
xlabel('Re\{x[n]\}');
ylabel('Im\{x[n]\}');
axis([-40 40 -40 40]);
axis('square');
title('Received signal x[n]');
set(gcf, 'Color', [1 1 1])

%%  Training Signal
%  
% The training signal is a shifted version of the
% original transmitted signal s[n].  This signal would
% be known to both the transmitter and receiver.  

d = [zeros(1,10) s(1:ntr-10)];

%% Trained Equalization
% 
% To obtain the fastest convergence, we shall use the
% conventional version of a recursive least-squares
% estimator.  Only the first 2000 samples are used
% for training.  The output signal constellation shows
% clusters of values centered on the sixteen different
% symbol values--an indication that equalization has
% been achieved. 

P0 = 100*eye(20);
lam = 0.99;
h = adaptfilt.rls(20,lam,P0);
ntrain = 1:2000;
[y,e] = filter(h,x(ntrain),d(ntrain));
plot(y(1001:2000),'.');
xlabel('Re\{y[n]\}');
ylabel('Im\{y[n]\}');
axis([-5 5 -5 5]);
axis('square');
title('Equalized signal y[n]');
set(gcf, 'Color', [1 1 1])

%% Training Error e[n]
%
% Plotting the squared magnitude of the error signal 
% e[n], we see that convergence with the RLS algorithm
% is fast. It occurs in about 60 samples with the
% equalizer settings chosen.

semilogy(ntrain,abs(e).^2);
xlabel('Number of iterations');
ylabel('|e[n]|^2')
title('Squared magnitude of the training errors');
set(gcf, 'Color', [1 1 1])

%% Decision-Directed Adaptation
%
% Once the equalizer has converged, we can use 
% decision-directed adaptation to continue adaptation
% during periods where no training data are available.
% In such cases, the desired signal d[n] is replaced
% by a quantized version of the output signal y[n] that
% is nearest to a valid symbol in the transmitted signal.
% We can use the RLS adaptive algorithm to implement
% this decision-directed algorithm in a sample-by-sample
% mode.

e = [e(1:2000) zeros(1,3000)];
h.PersistentMemory = true;
for n=2001:5000
    yhat = h.Coefficients*[x(n);h.States];
    ydd = round((yhat+1+j)/2)*2-1-j;
    if (abs(real(ydd))>3)
        ydd = 3*sign(real(ydd)) + imag(ydd);
    end
    if (abs(imag(ydd))>3)
        ydd = real(ydd) + 3*sign(imag(ydd));
    end
    e(n) = d(n) - yhat;
    [yhat,edd] = filter(h,x(n),ydd);
end

%% Comparison with Trained Adaptation
%
% If the symbol decisions are correct, then decision-
% directed adaptation produces identical performance
% to trained adaptation.  We can compare the error
% sequence from the combined training/decision-directed
% adaptive equalizer with one that uses training data
% over the whole received signal.  A sudden jump in
% the difference in the error signals indicates an
% incorrect symbol decision was used in the decision-
% directed algorithm.  So long as these errors are
% infrequent enough, the effects of these errors decay
% away, and the decision-directed equalizer's performance 
% remains similar to that of the trained equalizer.

reset(h);
[ytrain,etrain] = filter(h,x,d);
n = 1:5000;
semilogy(n,abs(e),n,abs(etrain),n,abs(e-etrain));
xlabel('number of iterations');
ylabel('|e[n]|^2');
title('Trained and trained/decision-directed equalizers');
legend('Trained/Decision-Directed','Trained','Difference');
set(gcf, 'Color', [1 1 1])


displayEndOfDemoMessage(mfilename)