In digital communications, there is a phenomenon known as “InterSymbol Interference” (ISI) when transmitting data over a multipath channel. What does this mean in layman terms?
There is a high probability where a chunk of bits of information will interfere with each other through “wireless” transmission– distorting the signal.
Adaptive Equalization (Adaptive EQ)
The objective of this simulation is to investigate the performance of an adaptive equalizer for data transmission over a multipath channel that causes intersymbol interference (ISI).
The data generator module is used to create a sequence of complexvalued information symbol s[n]
. For this simulation, I will assume QPSK symbols. In other data will be drawn from the set {a+ja, a−ja,−a+ja,−a−ja}
, where a
represents the signal amplitude that is chosen according to a given signaltonoise ratio (SNR). Assuming that the noise has unit power, then SNR = 20log10(√2a)
To verify performance, here is the list of requirements:

A channel filter module will be used as an FIR filter with impulse response
c[n]
that simulates the channel distortion. 
A noise generator module will be used to generate additive noise that is present in any digital communication system. We assume unitpower, complex Gaussian noise.

The adaptive equalizer module is a length
M+1
FIR filterh[n]
whose coefficients are adjusted using either the LMS or the normalizedLMS algorithm. 
A decision device module takes the output of the equalizer and will quantize it to one of the four possible transmitted symbols in QPSK, based on whichever is closest.

A plot displaying the error
e[n]
as a function ofn
will be shown, averaged over theP
experiments.
The Code

To generate random complex data sequences from a given SNR value, I do the following sequence: generate the symbol table in
amplitude_to_qpskSet()
, next I generate random “complex binary” data by randomizing the real and complex components separately usingsign(randn())
ingenerate_QPSK_data()
, and finally I map the random complex data to the generated symbol table inqpsk_mod()
. The combination will ultimately give a complex unitvector in QPSK coordinates. 
I send the random complex data through the channel by using
filter(c,1, sn)
; This essentially convolutes the input with the channel. 
I add N amounts of noise using
xn = channel_out + (randn(size(channel_out))+ 1j*randn(size(channel_out)))/sqrt(2)
. 
I update my LMS and Normalized filter coefficients (of size
M+1
) usingh = h + ( mu * conj(e(n))*xn_shifts )
; andhn = hn + ( lambda * conj(en(n))*xn_shifts) ./ ( (xn_shifts)\' * (xn_shifts ) )

The decisions are commented on the code. In sum, the minimum distance between the filter output and a constellation point (QPSK value) is calculated and is then mapped to a constellation point using an index. This decision is made after the training sequence has gone (when
n >T
). 
The resulting LMS filter experiment is done P times and is then plotted using
stem()
to see the channel coefficients and filter coefficients, andsemilogy()
to see the LMS and Normalized LMS Learning Curve.
Adaptive Equalization Code
function adapt_equal( c, SNR, mu, lambda, M, N, T, P )
average_J= zeros(N,1);
average_Jn= zeros(N,1);
e = zeros(1,N);
en = zeros(1,N);
J = zeros(1,N);
Jn = zeros(1,N);
xn_shifts = zeros([N 1]);
f_out = zeros(N);
f_out_n = zeros(N);
%generate qpsk symbols
sn = generate_QPSK_data(SNR, N);
finite_sn = amplitude_to_qpskSet(SNR);
for p = 1:P
%go through channel
% channel_out=conv(c,sn);
channel_out = filter(c,1,sn);
%add noise
xn = channel_out + (randn(size(channel_out))+ 1j*randn(size(channel_out)))/sqrt(2);
%initialize filter size
h=zeros(M+1,1);
hn = zeros(M+1,1);
%per sample
for n=1:N
xn_shifts = [xn(n) ; xn_shifts(1:M)];
f_out(n) = h' * xn_shifts;
f_out_n(n) = hn' * xn_shifts;
%decision block
%LMS
error_decision = f_out(n)  finite_sn;
[~,decided_index] = min(abs(error_decision));
s_hat = finite_sn(decided_index);
%normalized LMS
error_decision_n = f_out_n(n)  finite_sn;
[~,decided_index_n] = min(abs(error_decision_n));
s_hat_n = finite_sn(decided_index_n);
%if training
if n < T
e(n) = sn(n)  f_out(n);
en(n) = sn(n)  f_out_n(n);
else
e(n) = s_hat  f_out(n);
en(n) = s_hat_n  f_out_n(n);
end
%update coeff
h = h + ( mu * conj(e(n))*xn_shifts );
hn = hn + ( lambda * conj(en(n))*xn_shifts) ./ ( (xn_shifts)' * (xn_shifts ) );
J(n)=abs(e(n));
Jn(n) = abs(en(n));
average_J(n)=average_J(n)+J(n);
average_Jn(n)=average_Jn(n)+Jn(n);
end
average_J=average_J/P;
average_Jn=average_Jn/P;
subplot(3,3,1)
cplot(f_out)
title('Filter Output')
subplot(3,3,2)
stem(h)
% axis([2 2 1 1])
title('Adaptive Filter Impulse Response Coefficients (Real Component)')
xlabel('n')
subplot(3,3,3)
stem(c)
% axis([2 2 1 1])
title('Channel Impulse Response Coefficients')
xlabel('n')
subplot(3,3,4)
cplot(xn)
axis([35,35,35,35])
title('Channel with Noise')
subplot(3,3,5)
cplot(e)
title('Error (complex)')
subplot(3,3,6)
cplot(channel_out)
axis([35,35,35,35])
title('All Data through channel')
subplot(3,3,7)
cplot(sn)
title('Data Input (QPSK)')
subplot(3,3,[8 9])
drawnow
semilogy(average_J)
hold on
semilogy(average_Jn)
title('Learning curve abs(e(n))')
xlabel('time step n')
legend('LMS', 'Normalized LMS')
hold off
end
end
Data Generation (BPSK) Code
function sn = generate_QPSK_data(sig_noise,numOfData)
for n = 1:numOfData
lookupTable = amplitude_to_qpskSet(sig_noise); %QPSK lookup table using SNR
%generate random digital data
complex_binary = complex(sign(randn(numOfData,1)0.5),sign(randn(numOfData,1)0.5));
%map data to the QPSK lookup table
sn(n,1) = qpsk_mod(complex_binary, lookupTable);
end
%end data generator module!!!!
end
function signal = amplitude_to_qpskSet(SNR)
%returns column vector
%assume noise has unit power
a = ( 10 ^ (SNR/20) ) / sqrt(2);
signal = [complex(a,a), complex(a,a), complex(a,a), complex(a,a)];
end
function output = qpsk_mod(data, qpsk)
for k = 1:size(data)
if angle(data(k)) == angle(complex(1,1))
output = qpsk(1); %first quad
elseif angle(data(k)) == angle(complex(1,1))
output = qpsk(2); %fourth quad
elseif angle(data(k)) == angle(complex(1,1))
output = qpsk(3); %second quad
elseif angle(data(k)) == angle(complex(1,1))
output = qpsk(4); %third quad
else
output = 0;
end
end
output = output.';
end
SPlane Plotting
function cplot(v)
drawnow
% pause(0.00001)
plot(real(v(1:end)),imag(v(1:end)),'x')
axis([18,18,18,18])
end
RESULTS & CONCLUSION
Test Cases
To simulate adaptive equalization, I ran three test cases: a base case, a different channel (complex impulses), and different step sizes. The figures below show the last snapshot of the simulation in their respective order.
Looking at the three figures, what seems to be most effective is step size. One can see the learning curve decrease per time step. Though not shown here, each experiment snapshot demonstrated that the filter output was attempting to cluster at the Data Input constellations.
To conclude, this experiment simulated the adaptive equalizer environment, and with the correct step size, one can recover distorted data by canceling channel and noise effects using LMS.