Filter Design Toolbox 4.6

Equalization in Digital Communications

This demonstration illustrates the application of adaptive filters to channel equalization in digital communications.

Author(s): Scott C. Douglas

Contents

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

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

The received signal x[n] is generated by the transmitted 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

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])