In digital communications, there is a phenomenon known as “Inter-Symbol 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 inter-symbol interference (ISI).

The data generator module is used to create a sequence of complex-valued 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 signal-to-noise ratio (SNR). Assuming that the noise has unit power, then `SNR = 20log10(√2a)`

To verify performance, here is the list of requirements:

1. A channel filter module will be used as an FIR filter with impulse response `c[n]` that simulates the channel distortion.

2. A noise generator module will be used to generate additive noise that is present in any digital communication system. We assume unit-power, complex Gaussian noise.

3. The adaptive equalizer module is a length `M+1` FIR filter `h[n]` whose coefficients are adjusted using either the LMS or the normalized-LMS algorithm.

4. 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.

5. A plot displaying the error `e[n]` as a function of `n` will be shown, averaged over the `P` experiments.

## The Code

1. 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 using `sign(randn())` in `generate_QPSK_data()`, and finally I map the random complex data to the generated symbol table in `qpsk_mod()`. The combination will ultimately give a complex unit-vector in QPSK coordinates.

2. I send the random complex data through the channel by using `filter(c,1, sn)`; This essentially convolutes the input with the channel.

3. I add N amounts of noise using `xn = channel_out + (randn(size(channel_out))+ 1j*randn(size(channel_out)))/sqrt(2)`.

4. I update my LMS and Normalized filter coefficients (of size `M+1`) using `h = h + ( mu * conj(e(n))*xn_shifts )`; and `hn = hn + ( lambda * conj(en(n))*xn_shifts) ./ ( (xn_shifts)\' * (xn_shifts ) )`

5. 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`).

6. The resulting LMS filter experiment is done P times and is then plotted using `stem()` to see the channel coefficients and filter coefficients, and `semilogy()` 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
``````

### S-Plane 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. Figure 1: Assignment Given Paramenters Figure 2: Change Channel Figure 3: Change Step Sizes

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.