## Friday, June 29, 2012

### Morse code detection using modified Morlet Wavelet transformation

In my previous blog post I shared some experiments with wavelets using available online tools.  In order to build better understanding on how to apply Morlet wavelets in detecting Morse code in noisy signals I wrote a little test application using Octave.   The application creates noisy Morse code and does Continuous Wavelet Transform (CWT) using modified Morlet  wavelet as well as Short Term Fourier Transform (STFT).  The Octave application allows changing various parameters and visualizing the impact on the plotted graphs and images.  See discussion on modification to Morlet wavelet below.

The starting point was original noisy Morse code signal  that has -12.2 dB SNR as shown in the figure 1 below.  It is virtually impossible to detect the Morse signal buried in the noise. When listening the audio I can hear a faint sound in the noise but I have much difficulties recognizing any of the characters.

For reference, the signal-to-noise ratio calculation is done the following way:

SNR=20*log10(norm(morsecode)/norm(morsecode-noisey_morsecode));

 Figure 1.   Morse code with -12.2 dB SNR (signal-to-noise ratio)

To discover the Morse code signal I used Short Time Fourier Transformation (STFT)  with the following parameters ( x contains noisy audio,  sampling rate is Fs=4000 )

z = stft(x,1/Fs,4,128,1,512);

A very faint Morse signal is now visible at 600 Hz frequency as a horizontal pattern of "dits and "dahs" in figure 2. below.  It is very difficult to "read" from this figure 3 below what the message is.  Some "dits" and "dahs" are more visible but noise makes it difficult to detect the pattern or to decode the message.

 Figure 2.  Waterfall spectrogram of the Morse code with -12.2dB SNR.

Modified Morlet Wavelet power spectrum is shown on figure 3.  On Scale axis (vertical) you can see  at S = 8 as a horizontal pattern of lighter "dits" and "dahs".  Looking at the pattern you can almost see "dah-dit-dah-dit"  "dah dah dit dah"  "dah dit dit" "dit"  "dit dah" "dah dah dit" "dit dah dah dah dah" "dit dah dit dit"  "dit"   aka  "CQ DE AG1LE".  This represents the peak energy of the power spectrum  after Wavelet transformation. I am taking absolute value to show the envelope of the wavelet better  (see morlet.m file below, calculation is done by this line:    coeffs(k,:)=abs(fftshift(ifft(fft(w).*fft(sigin)))); )

 Figure 3.  Wavelet Power Spectrum of the Morse code with -12.2 dB SNR

I plotted Wavelet coefficient C(t,8) values  (corresponding Scale = 8 above) on  x-y graph below in figure 4.  Morse code "dits" and "dahs"  are quite visible as signal peaks above threshold value of 2000.  Note that these are  abs(C(t,8))  values and low pass filtered to show envelope better. See morletdemo.m below for details.

 Figure 4.  Modified Morlet coefficient C(8) values.

Original Morlet  wavelets (scales 1 to 16 shown below in figure 5.) have variable wavelet pulse length and frequency as follows:

for k=1:scale,
t=(-M/2:M/2-1);

% Calculate Morelet Wavelet w=e-(at^2)*cos(2*pi*fo*t)
const = 1/(sigma*K*sqrt(k)); % k impacts relative amplitude
e = exp(-((sigma*t/k).^2));    % k impacts pulse length (t/k)
phase = cos(2*pi*fo*t/k);    % k impacts frequency
w = const*e.*phase;
plot(w)
end

While experimenting with these wavelets it was quite difficult find the optimal wavelet to extract signal from noise.

 Figure 5.  Morlet wavelets 1...16
Looking at the impact of various parameters it became obvious that by modifying the wavelet to keep the duration constant improves the situation a lot.  I modified a single line and the corresponding wavelet graph is below in figure 6.

e = exp(-((sigma*t).^2));    % removed k to keep the wavelet duration constant.

Note that wavelet bandwidth sigma depends on Morse speed - I did several experiments and established the following relationship  sigma  = (1.2/speed)/w   where w is the number of the wavelet.

 Figure 6.   Morlet Wavelet

### CONCLUSIONS

With a small modification to Morlet wavelet the CWT works better than STFT in extracting the signal from the noise even at -12 dB SNR.  There are many similarities to Matched Filter method that I described in this blog post. Perhaps the main difference is the selected wavelet shape (Morlet) and the fact we use FFT to make the convolution very fast.

Further work could include the following tasks:
• program  the modified  Morlet Wavelet algorithm in C++
• implement this functionality in FLDIGI  CW decoder module
• test Wavelet based decoder with real life signals from HF bands

SOFTWARE
The Octave scripts are listed below.  The above results were created by running the following command on Octave:

%  noise level,  signal freq, sampling rate, morsespeed
morletdemo(2,600,4000,20)

The software prints these lines

text = CQ de AG1LE
file = CQ.wav
SNR =-12.226566
Fo =  1.2000
sigma =  0.0075000

and plots the figures.

File Morlet.m

% Project 2
% Time-Frequency Representations
% Andy Doran, modified by  AG1LE Mauri Niininen

% function coeffs = cwvt(sigin,scale,quiet,sigma,fo)
%
% sigin  = sampled input signal (Should be a row vector)
% scale  = number of real, positive scale you want (1:scales)
% quiet  = plot suppression
%          1 -> suppress all plots
%          2 -> suppress wavelet plots only
%          3 -> suppress scalogram only
% sigma =0.015625;     Morlet Wavelet bandwidth
% fo = 0.25;           Center frequency of Wavelet
% coeffs = scales-by-length(sigin) matrix returning CWT of sigin at
%          each scale from 1 to scale
%
% This function takes an input signal and computes the Continuous Wavelet
% Transform at different scales using a sampled Morlet Wavelet
%
% Morelet Wavelet w(t) = (1/sigma*K)*exp-((sigma*t)^2)*cos(2*pi*fo*t)

function coeffs = morlet(sigin,scale,quiet,sigma,fo)

K = 1;           % Not sure what this is, so set to 1

M = length(sigin);
coeffs = zeros(scale,M);

for k=1:scale,
t=(-M/2:M/2-1);

% Calculate Morelet Wavelet w=e-(at^2)*cos(2*pi*fo*t)
const = 1/(sigma*K*sqrt(k)); % k impacts relative amplitude
e = exp(-((sigma*t).^2));    % removed k to keep the wavelet duration constant
phase = cos(2*pi*fo*t/k);    % k impacts frequency
w = const*e.*phase;

% Plot wavelet in time domain and frequency domain
if ((quiet ~= 1) & (quiet ~= 2))
figure(3)
if (k == 1)  % Clear plot on initial run-through
clf
end
subplot(scale,2,(2*k)-1)
plot(w)
txt = ['Modified Morlet Wavelet at scale ', num2str(k)];
title(txt)
%figure(4)
%if (k == 1)  % Clear plot on initial run-through
%  clf
%end
subplot(scale,2,2*k)
plot(abs(fft(w)))
txt = ['Frequency Spectra of Morlet Wavelet at scale ', num2str(k)];
title(txt)
end
% Calculate CWT of sigin using circular convolution
%    coeffs(k,:)=ifft(fft(w).*fft(sigin));
coeffs(k,:)=abs(fftshift(ifft(fft(w).*fft(sigin))));
end

% Coeffs should be real anyway, this just accounts for numerical error
% in circular convolution that may give small imaginary parts
coeffs = real(coeffs);

% Plot scalogram and check against MATLAB's CWT routine
if ((quiet ~= 1) & (quiet ~= 3))
figure(1)
%clf
map = jet();
colormap(map);
imagesc(coeffs);
axis xy;
txt = ['abs|C(t,s)| for s = 1 to ' num2str(scale)];
title(txt)
ylabel('s')
xlabel('t')
figure(2);
plot(sigin);
title('original signal');
%figure(4);
%clf
%cwt(sigin,1:scale,'morl','plot');  % Call MATLAB's CWT routine
%title('CWT Output from MATLAB')
end

File  stft.m

function y = STFT(x, sampling_rate, window, window_length, step_dist, padding)
%
%  y = STFT(x, sampling_rate, window, window_length, step_dist, padding)
%
%  STFT produces a TF image of "x".
%  The output is also stored in "y".
%
%  For "window", use one of the following inputs:
%  rectangular    = 1
%  Hamming        = 2
%  Hanning        = 3
%  Blackman-Tukey = 4
%
%  The time scale is associated with the center of the window,
%  if the window is of odd length.  Otherwise, the window_length/2
%  is used.  "Step_dist" determines the stepping distance between the number
%  of samples, and is arranged to maintain the proper time index
%  provided by "sampling_rate" in seconds.  "Padding" is the
%  total length of the windowed signal before the fft, which is
%  accomplished by zero padding.
%
%  Developed by Timothy D. Dorney
%               Rice University
%               April, 1999
%               tdorney@ieee.org
%
%  Coded using MATLAB 5.X.X.
%  See http://www.clear.rice.edu/elec631/Projects99/mit/index2.htm
%
% REVISION HISTORY
%
% VERSION 1.0.0 APR. 21, 1999 TIM DORNEY
%

if (nargin ~= 6)
disp('STFT requires 6 input arguments!')
return;
end
if ((window < 1) | (window > 4))
window = 1;
disp('The argument "window" must be between 1-4, inclusively.  Window set to 1!');
end
if ((step_dist < 1) | (round(step_dist) ~= step_dist))
step_dist = 1;
disp('The argument "step_dist" must be an integer greater than 0.  Step_dist set to 1!');
end
if (sampling_rate <= 0)
disp('The argument "sampling_rate" must be greater than 0.');
break;
end
if (padding < window_length)
disp('The argument "padding" must be non-negative.  Padding set to "window_length"!');
end

if (window == 1)
WIN = ones(1,window_length);
elseif (window == 2)
WIN = hamming(window_length)';
elseif (window == 3)
WIN = hanning(window_length)';
elseif (window == 4)
WIN = blackman(window_length)';
end

[m,n] = size(x);
if (m ~= 1)
X = x';
else
X = x;
end
[m,n] = size(X);
if (m ~= 1)
disp('X must be a vector, not a matrix!');
break;
end

LENX = length(X);
IMGX = ceil(LENX/step_dist);
IMGY = (padding/2) + 1;
else
end

y = zeros(IMGX,IMGY);

if (window_length/2 == round(window_length/2))
CENTER = window_length/2;
x_pad_st = window_length - CENTER - 1;
x_pad_fi = window_length - CENTER;
else
CENTER = (window_length+1)/2;
x_pad_st = window_length - CENTER;
x_pad_fi = window_length - CENTER;
end

iter = 0;
for kk = 1:step_dist:LENX
iter = iter + 1;
XX = X(kk:(kk + window_length - 1));
YY = XX .* WIN;
ZZ = abs(fft(YY, padding));
y(iter,:) = ZZ(1:IMGY);
end
figure(6);
freq = (1/sampling_rate)/2;
imagesc([0:(step_dist*sampling_rate):(sampling_rate*(LENX-1))], ...
[0:(freq/(IMGY-1)):freq],y');
xlabel('Time (seconds)');
ylabel('Frequency (Hz)');
axis('xy')

File  morletdemo.m

function morletdemo(noisy,freq,Fs,speed);

x = morse('CQ de AG1LE','CQ.wav',noisy,freq,Fs,speed);

w = 8;         %  peak will be at wavelet # w
Fo = freq / (Fs/w)     % tell wavelet transform where wavelet center frequency is
sigma  = (1.2/speed)/w  % wavelet bandwidth - impacts time resolution

c = morlet(x',16,2,sigma,Fo);         % do Morlet wavelet transform
y = filter(ones(1,20)/20,1,c(w,:));   % y = low pass filter C(t,w) wavelet
figure(4)
plot(y);                              % plot C(t,w) envelope

z = stft(x,1/Fs,4,128,1,512); % plot spectrogram of the signal using Short Term FFT
end;

File  morse.m

function code=morse(varargin)
% MORSE converts text to playable morse code in wav format
%
% SYNTAX
% morse(text)
% morse(text,file_name);
% morse(text,file_name,noise_multiplier);
% morse(text, file_name,noise_multiplier,code_frequency);
% morse(text, file_name,noise_multiplier,code_frequency,sample_rate);
% morse(text, file_name,noise_multiplier,code_frequency,sample_rate, code_speed_wpm, zero_fill_to_N);
% morse(text, file_name,noise_multiplier,code_frequency,sample_rate, code_speed_wpm, zero_fill_to_N, play_sound);
%
% Description:
%
%   If the wave file name is specified, then the funtion will output a wav
%   file with that file name.  If only text is specified, then the function
%   will only play the morse code wav file without saving it to a wav file.
%   If a noise multiplier is specified, zero mean addative white Gaussian
%   noise is added with 'amplitude' noise_multiplier.
%
% Examples:
%
%   morse('Hello');
%   morse('How are you doing my friend?','morsecode.wav');
%   morse('How are you doing my friend?','morsecode.wav', 0.01);
%   morse('How are you doing my friend?','morsecode.wav', 0.01, 440, ,20, Fs);
%   x = morse('How are you doing my friend?','morsecode.wav', 0.01, 440, 20, Fs, 2^20,1); %(to play the file, and make the length 2^20)
%
%   Version: 1.1 \$  \$Date: 08-Jul-2010
%   Modifications: Rob Frohne, KL7NA
%Defualt values
Fs=48000;
noise_multiplier = 0;
f_code = 375;
code_speed = 20;
text = varargin{1}
if nargin>=2
file = varargin{2}
end
if nargin>=3
noise_multiplier = varargin{3};
end
if nargin>=4
f_code = varargin{4};
end
if nargin>=5
Fs = varargin{5};
end
if nargin>=6
code_speed = varargin{6};
end
if nargin>=7
length_N = varargin{7};
end
if nargin>=8
playsound = varargin{8};
end
t=0:1/Fs:1.2/code_speed; %One dit of time at w wpm is 1.2/w.
t=t';
Dit = sin(2*pi*f_code*t);
ssp = zeros(size(Dit));
#Dah fixed by Zach Swena
t2=0:1/Fs:3*1.2/code_speed;  # one Dah of time is 3 times  dit time
t2=t2';
Dah = sin(2*pi*f_code*t2);
lsp = zeros(size(Dah));    # changed size argument to function of Dah
#Dah = [Dit;Dit;Dit];
#lsp = zeros(size([Dit;Dit;Dit]));
% Defining Characters & Numbers
A = [Dit;ssp;Dah];
B = [Dah;ssp;Dit;ssp;Dit;ssp;Dit];
C = [Dah;ssp;Dit;ssp;Dah;ssp;Dit];
D = [Dah;ssp;Dit;ssp;Dit];
E = [Dit];
F = [Dit;ssp;Dit;ssp;Dah;ssp;Dit];
G = [Dah;ssp;Dah;ssp;Dit];
H = [Dit;ssp;Dit;ssp;Dit;ssp;Dit];
I = [Dit;ssp;Dit];
J = [Dit;ssp;Dah;ssp;Dah;ssp;Dah];
K = [Dah;ssp;Dit;ssp;Dah];
L = [Dit;ssp;Dah;ssp;Dit;ssp;Dit];
M = [Dah;ssp;Dah];
N = [Dah;ssp;Dit];
O = [Dah;ssp;Dah;ssp;Dah];
P = [Dit;ssp;Dah;ssp;Dah;ssp;Dit];
Q = [Dah;ssp;Dah;ssp;Dit;ssp;Dah];
R = [Dit;ssp;Dah;ssp;Dit];
S = [Dit;ssp;Dit;ssp;Dit];
T = [Dah];
U = [Dit;ssp;Dit;ssp;Dah];
V = [Dit;ssp;Dit;ssp;Dit;ssp;Dah];
W = [Dit;ssp;Dah;ssp;Dah];
X = [Dah;ssp;Dit;ssp;Dit;ssp;Dah];
Y = [Dah;ssp;Dit;ssp;Dah;ssp;Dah];
Z = [Dah;ssp;Dah;ssp;Dit;ssp;Dit];
period = [Dit;ssp;Dah;ssp;Dit;ssp;Dah;ssp;Dit;ssp;Dah];
comma = [Dah;ssp;Dah;ssp;Dit;ssp;Dit;ssp;Dah;ssp;Dah];
question = [Dit;ssp;Dit;ssp;Dah;ssp;Dah;ssp;Dit;ssp;Dit];
slash_ = [Dah;ssp;Dit;ssp;Dit;ssp;Dah;ssp;Dit];
n1 = [Dit;ssp;Dah;ssp;Dah;ssp;Dah;ssp;Dah];
n2 = [Dit;ssp;Dit;ssp;Dah;ssp;Dah;ssp;Dah];
n3 = [Dit;ssp;Dit;ssp;Dit;ssp;Dah;ssp;Dah];
n4 = [Dit;ssp;Dit;ssp;Dit;ssp;Dit;ssp;Dah];
n5 = [Dit;ssp;Dit;ssp;Dit;ssp;Dit;ssp;Dit];
n6 = [Dah;ssp;Dit;ssp;Dit;ssp;Dit;ssp;Dit];
n7 = [Dah;ssp;Dah;ssp;Dit;ssp;Dit;ssp;Dit];
n8 = [Dah;ssp;Dah;ssp;Dah;ssp;Dit;ssp;Dit];
n9 = [Dah;ssp;Dah;ssp;Dah;ssp;Dah;ssp;Dit];
n0 = [Dah;ssp;Dah;ssp;Dah;ssp;Dah;ssp;Dah];
text = upper(text);
vars ={'period','comma','question','slash_'};
morsecode=[];
for i=1:length(text)
if isvarname(text(i))
morsecode = [morsecode;eval(text(i))];
elseif ismember(text(i),'.,?/')
x = findstr(text(i),'.,?/');
morsecode = [morsecode;eval(vars{x})];
elseif ~isempty(str2num(text(i)))
morsecode = [morsecode;eval(['n' text(i)])];
elseif text(i)==' '
morsecode = [morsecode;ssp;ssp;ssp;ssp];
end
morsecode = [morsecode;lsp];
end
if exist('length_N','var')
append_length = length_N - length(morsecode);
if (append_length < 0)
printf("Length %d isn't large enough for your message; it must be > %d.\n",length_N,length(morsecode));
return;
else
morsecode = [morsecode; zeros(append_length,1)];
end
end
noisey_morsecode = morsecode + noise_multiplier*randn(size(morsecode));
SNR=20*log10(norm(morsecode)/norm(morsecode-noisey_morsecode));
printf('SNR =%f\n',SNR);
if exist('file','var')
wavwrite(noisey_morsecode,Fs,16,file);
if exist('playsound')
system(['aplay ',file]);
end
else
soundsc(noisey_morsecode,Fs);
% wavplay(morsecode);
end
code = noisey_morsecode;
endfunction

## Sunday, June 24, 2012

### Ultimate Morse Code Decoder?

Over the last few weeks I started reading articles and papers about wavelet transformation to figure out how to build an ultimate Morse code decoder. Wavelets have been used since 1980's in digital signal processing and wavelet transforms are now being adopted for a vast number of applications, often replacing the conventional  Fourier transformation.

One particular application is for smoothing/denoising data based on wavelet coefficient thresholding, also called wavelet shrinkage. By adaptively thresholding the wavelet coefficients that correspond to undesired frequency components smoothing and/or denoising operations can be performed.

### WAVELETS FOR DETECTING NOISY SIGNALS

I stumbled on this paper from Aly, Omar and Eldherbeni few days ago. The paper describes an algorithm for extracting and localizing an RF radar pulse from a noisy background. The algorithm combines two powerful tools: the wavelet packet analysis and higher-order-statistics (HOS). The use of the proposed technique makes detection and localization of RF radar pulses possible in very low signal-to-noise ratio conditions.

The proposed algorithm is able to detect and well localize RF radar pulses without a prior knowledge of the pulse parameters (e.g., its frequency and duration). The proposed  algorithm has been tested for SNR down to −24 dB and proved to work successfully. See figures 9 and 7 below on the results they achieved.

In my previous blog post  I used matched filter to extract Morse code pulses from noisy signals. Matched filter can be implemented  either in time or frequency domain. However,  one problem with narrow matched filter is "ringing" artifacts  - see figure 8  in the blog post. This creates uncertainty and jitter in pulse width.  According to above paper  one of the advantages of  using wavelets is possibility to obtain adapted tiling in the time-frequency plane, which is automatically generated based on the signal observation. This leads to improved time & frequency resolution compared to traditional short-time windowed Fast Fourier Transformation (SFFT) based methods.

### DIFFERENT WAVELET TRANSFORMATIONS

I found this website that provides online tools to visualize various wavelet types. Since the discovery of wavelets many years ago there has been active development and many different wavelets have been created for various purposes. This site provide very nice online visualization tool.

I created  a small noisy Morse code file using  morse.m  Octave function by Rob Frohne, KL7NA.  The above website allows only 2000 samples so I copied numbers to the above site (select "Your data" section).
Figures 1..3  below show a small noisy audio section with a "dit" tone  and start of "dah" tone and corresponding wavelet transformation (Morlet, Gaussian & Paul wavelet types) underneath.  There is also a global wavelet showing the variance by period on the right side.  The wavelet transformation values are color coded - red color is showing the highest energy at Period (scale) 8 clearly identifying where the tone starts and where it ends.  Different wavelets have slightly different properties as is quite visible looking at the figures below.

 Figure 1.  Morlet Wavelet - noisy Morse code

 Figure 2.  Gaussian Wavelet - noisy Morse code

 Figure 3.  Paul Wavelet - noisy Morse code

### MORLET WAVELET TRANSFORMATIONS

As the results from previous section demonstrate Morlet type wavelets seem to provide particularly good delineation of Morse code tones in noisy signals.   I did three more visualizations with  various level of noise included in the signals.  The first one (MORSE_SNR0) below has no noise and the signal is visible as clear red color  line on wavelet power spectrum image (period = 8). It is very easy to see where the signal ends and next one starts (see the gap between samples 1450 ...1900 ).

The following visualization (MORSE_SNR1) has approximately - 9 dB  SNR  and quite a lot of noise. The red line is still visible in the wavelet power spectrum image (period = 8). However,  noise spikes make it a bit more difficult to determine signal end and start timing. There is also other red noise signals at periods 32, 64 and 256. As these are on different wavelet periods they can be filtered out.

The last visualization (Morse_SNR2)  has  approximately  - 12 dB SNR  and it is quite difficult to see where the tone ends and next one starts. Looking from original signal it is almost impossible to tell where the tone is.  Global wavelet shows a peak at period 8  where most of the red dots are also aligned. There are many more red areas showing noise energy peaks.

### WAVELETS FOR PULSE TRAIN

I  did another experiment with Morse code pulse train.  This is the envelope of the noisy audio signal after detection and filtering. Pulse train has some sharp edges that should correspond to high frequency components as well as longer stable plateaus corresponding low frequency components.

I used  Haar and Gaussian wavelets in this experiment. Figure 4 and 5 below show the signal and corresponding wavelet transformation.  As expected the high frequency components are visible where the signal edges are.

Figure 4. Haar Wavelet - noisy Morse code pulse train

Notice the red high energy components between 600 - 750 ms in scale 128..512 range. This represents noise that is visible also on the time domain signal in figure 5.  In a wavelet filter implementation these  coefficients could be set to zero  to denoise the signal.

 Figure 5. Gaussian Wavelet - noisy Morse code pulse train

### CONCLUSIONS

Wavelet transformation  is a powerful signal processing tool to manipulate signals. Based on literature the wavelet transform gives better localization in the time-frequency domain than the discrete windowed Fourier transform.  Wavelets enable also flexible manipulation of the signals to remove noise, find signals buried under noise  etc.

For a real time CW decoding software like in FLDIGI  wavelet transformation could open a whole new performance level dealing with noisy signals.  More experimentation is definitely needed to unlock this potential.

73
Mauri AG1LE