# 20100216

Experiment with combining signals to increase SNR.

I will try to add the signals together by computing their phase differences programmatically, shifting them, and adding them together.

First, some setup

```foo = csvread('sig11_tag150.2_J20062.csv');
Fs = 1000000;  % Sampling Frequency

% Construct a 200KHz BPF
N      = 10;      % Order
Fpass1 = 198000;  % First Passband Frequency
Fpass2 = 202000;  % Second Passband Frequency
Apass  = 1;       % Passband Ripple (dB)
h  = fdesign.bandpass('N,Fp1,Fp2,Ap', N, Fpass1, Fpass2, Apass, Fs);
Hd = design(h, 'cheby1');

samples = 670000:710000;

% Filter the four channels
out1 = filter(Hd, foo(samples,2));
out2 = filter(Hd, foo(samples,3));
out3 = filter(Hd, foo(samples,4));
out4 = filter(Hd, foo(samples,5));
```

Now compute the cross-correlation, based on the first signal

```N = xcorr(out1, out2);
x1 = fliplr(N(1:length(out1))');
N = xcorr(out1, out3);
x2 = fliplr(N(1:length(out1))');
N = xcorr(out1, out4);
x3 = fliplr(N(1:length(out1))');
```

Now work out the required shift

Four band-pass filtered signals from sig11_tag150.2_J20062.csv showing samples 685000-685100, which occurred during the CW transmission. Original signals and phase-corrected signals are shown.
```[a, shift2] = max(x1(1:30));
[a, shift3] = max(x2(1:30));
[a, shift4] = max(x3(1:30));
```

Then shift those signals

```outs1 = out1 % First signal is not shifted, but give it a consistent name.
outs2 = vertcat(out2(shift2:length(out2)), zeros(shift2-1,1));
outs3 = vertcat(out3(shift3:length(out3)), zeros(shift3-1,1));
outs4 = vertcat(out4(shift4:length(out4)), zeros(shift4-1,1));

origsig = [out1(15000:15100), out2(15000:15100), out3(15000:15100), out4(15000:15100)];
newsig = [out1(15000:15100), outs2(15000:15100), outs3(15000:15100), outs4(15000:15100)];

% Compare these two plots to check that they're now coherent

figure
subplot(2,1,1)
plot(origsig)
axis tight
title('Original signals')
subplot(2,1,2)
plot(newsig)
title('Shifted signals')
axis tight

% Now examine the addition of all of those

newsig = out1+outs2+outs3+outs4;

```

Now we'll look at the SNR of the individual signals and the combined one

FFT plot of band-pass filtered signal 1 (out1) and coherently summation of four BP filtered signals showing dB vs frequency. Samples 670000:710000 (40ms around CW transmission) from sig11_tag150.2_J20062.csv.
```
N = length(out1);
k=-(N-1)/2:(N-1)/2;
T=N/Fs;
freq=k/T;

figure
subplot(2,1,1)
bleh=fft(outs1);
plot(freq, abs(bleh));
Pyy = bleh .* conj(bleh);
Pyy = 10 * log10(Pyy);
plot (freq, Pyy)

subplot(2,1,2)
bleh=fft(newsig);
plot(freq, abs(bleh));
Pyy = bleh .* conj(bleh);
Pyy = 10 * log10(Pyy);
plot (freq, Pyy)

```

Well.. it looks like the signal has gone to a maximum of 22dB (note, not sure this scale makes sense), up from 0 (or from 15.1 dB in out3) and the noise is also about 10dB lower, so an SNR improvement of around 30 dB!

How about if the shift is applied to the unfiltered signal?

```wide1 = foo(samples,2);
wide2 = foo(samples,3);
wide3 = foo(samples,4);
wide4 = foo(samples,5);

wide2 = vertcat(wide2(shift2:length(wide2)), zeros(shift2-1,1));
wide3 = vertcat(wide3(shift3:length(wide3)), zeros(shift3-1,1));
wide4 = vertcat(wide4(shift4:length(wide4)), zeros(shift4-1,1));

widesig = wide1 + wide2 + wide3 + wide4;

figure
subplot(2,1,1)
bleh=fft(wide3);
plot(freq, abs(bleh));
Pyy = bleh .* conj(bleh);
Pyy = 10 * log10(Pyy);
plot (freq, Pyy)
title('Wide3')

subplot(2,1,2)
bleh=fft(widesig);
plot(freq, abs(bleh));
Pyy = bleh .* conj(bleh);
Pyy = 10 * log10(Pyy);
plot (freq, Pyy)
title('Combined wide')

```
FFT plot of unfiltered signal 3 (out3, which had the highest SNR of the four samples) and coherently summation of four unfiltered signals showing dB vs frequency. Samples 670000:710000 (40ms around CW transmission) from sig11_tag150.2_J20062.csv.

This results in a signal increase from 9.719 dB (wide1) to 23.73 dB and noise increase from -16 dB to -9dB. The best signal received was 16.21dB on wide3, so this is still not bad (though I'm not sure it would ever make sense to combine unfiltered signals)

Out of interest, how does this approach compare to naively mashing uncorrected signals together?

```dodgysig =  foo(samples,2) + foo(samples,3) + foo(samples,4) + foo(samples,5);
bleh=fft(outdodgy);
plot(freq, abs(bleh));
Pyy = bleh .* conj(bleh);
Pyy = 10 * log10(Pyy);
plot (freq, Pyy)
title('Unshifted combined signals')
```

Gives a maximum signal of 17.32 dB and noise of -9dB.

So just combining them is slightly better than just using the best signal alone, and 6dB worse than fiddling around with all of this. Or thereabouts. It might be interesting to combine them all and then filter them:

```outdodgy = filter(Hd, dodgysig);
```

Max signal of 16.2 dB, and much less noise (< -60dB).

In summary:

• Filtering is obviously good for reducing noise.
• Coherently combining them works well, but this approach assumes that you know in advance that the signal is present

Could next work on samples with multiple blips and try to pull them out by examining an fft output for maxima over a detection threshold.