压缩感知 压缩采样 稀疏采样

% SCALDMO demo script on the scalogram. % % SCALDMO is a demo script that illustrates the capabilities % of the scalogram function using the Morlet wavelet for % the signal analysis. % % SCALDMO may be accessed from the WTDEMO command interactive % menu, choosing the 'Scalogram' demo section. %-------------------------------------------------------- % Copyright (C) 1994, 1995, 1996, by Universidad de Vigo % % % Uvi_Wave is free software; you can redistribute it and/or modify it % under the terms of the GNU General Public License as published by the % Free Software Foundation; either version 2, or (at your option) any % later version. % % Uvi_Wave is distributed in the hope that it will be useful, but WITHOUT % ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or % FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License % for more details. % % You should have received a copy of the GNU General Public License % along with Uvi_Wave; see the file COPYING. If not, write to the Free % Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA. % % Author: Santiago Gonzalez Sanchez % Nuria Gonzalez Prelcic % e-mail: Uvi_Wave@tsc.uvigo.es %-------------------------------------------------------- echo on; %-------------------------------------------------------- % Let f(t) be a continous signal and F(a,b) its continous wavelet % transform. The representation of the modulus of the transform % (or its phase, if it is a complex signal) in the (b,a) open % half-plane is called a scalogram. The shift b is arbitrary and % the scale a is strictly positive (a>0). % The most used prototype function for signal analysis is the % Morlet wavelet (a windowed complex exponential). The "morletw" % function calculates this complex wavelet at any scale and % sampled at the specified rate. Figure 1 shows its real and % imaginary part at scale 1 and a sampling rate of 400 Hz, and % taking the second as time unit. [w,t]=morletw(400,1); figure(1) subplot(211); plot(t,real(w)); title('Real part') subplot(212); plot(t,imag(w)); title('Imaginary part') % Press any key to continue ... pause % -------------------------------------------------------- echo off clear w echo on % The "scalog" function returns the scalogram of a finite length % signal sampled at a certain rate. It uses the Morlet wavelet % as the analyzing function. We have to set the minimum and % maximum scale of the wavelet transform and the step between % them. Now, we will see some example signals and their corresponding % scalograms. % In figure 1 you can see the first example signal: a rectangular % pulse which has been sampled at 8 kHz. x=[zeros(1,362) ones(1,300) zeros(1,362)]; subplot(111); plot(x) % Now we are going to compute its scalogram from scale 1 to 20, % with step 0.3: figure(2) % Please, wait ... [sc,t,s]=scalog(x,8e3,1,20,0.3); % We can display the modulus of the transform with a 2-D plot, using % "show". The holes appearing in the scalogram are due to destructive % interference between the analyzing wavelets. colormap('pink') show(abs(sc),1,t,s); xlabel('Time'); ylabel('Scale') % Press any key to continue ... pause %-------------------------------------------------------- % To obtain a 3-D plot we can use the "srf" function: figure(3) colormap('pink') srf(abs(sc),5,5,t,s); % On the scalogram of a signal you can clearly detect abrupt changes % in it or in its derivatives. This is due to the good time localization % of the wavelet transform at low scales. % You can observe this in the preceding example, where the start and the % end of the pulse are very sharp transitions. These changes lead to local % maxima in the modulus of the WT. % % You can see this clearly in figure 3. % When the scale grows, the analyzing wavelet dilates and then the % scalogram focuses on long term details (the pulse itself). % Press any key to continue ... pause %-------------------------------------------------------- echo off clear s t sc echo on % You can set the time unit for Morlet wavelet at scale 1 or use the % default value, which is approximated according to the input sampling % rate. The effect of taking a shorter time unit is equivalent to decrease % the values of the employed scales. On the other hand, a longer time % unit corresponds to increase the scale (the prototype wavelet dilates). % For our example signal, let us set the time unit to 0.5 ms (half the % default value of 1 ms): % Please, wait ... colormap('pink') [sc,t,s]=scalog(x,8e3,1,20,0.3,0.5e-3); show(abs(sc),1,t,s) xlabel('Time'); ylabel('Scale') % As you can observe in the figure, the effect is similar to divide by 2 the % scale values of the specified range (1 to 20). % Press any key to continue ... pause %-------------------------------------------------------- echo off clear s t sc echo on % Now, setting the time unit to the double of the default value. % Please, wait ... [sc,t,s]=scalog(x,8e3,1,20,0.3,2e-3); show(abs(sc),1,t,s) xlabel('Time'); ylabel('Scale') % It is equivalent to use higher scales in the analysis. % Press any key to continue ... pause %-------------------------------------------------------- echo off close(2);close(3) clear s t sc echo on % Now, we are going to calculate the scalogram for a finite signal % composed by two tones starting and finishing at different instants. % For example, let us take 80 and 240 Hz tones, sampled at 8 kHz: x=[cos((2*pi*80/8000)*(0:723)) zeros(1,300)]+... [zeros(1,400) cos((2*pi*240/8000)*(0:623))]; plot(x) % The duration of the signal is 128 ms. The tone of lowest frequency % finishes 37.5 ms before the end of the signal and the other one % begins 50 ms later Figure 1 depicts this signal. % Press any key to continue ... pause %-------------------------------------------------------- % We analyze the signal using the "scalog" function, with the scale ranging % from 1 to 14 and a step of 0.1. % Please, wait ... figure(2); colormap('pink') [sc,t,s]=scalog(x,8e3,1,14,0.1); show(abs(sc),1,t,s) xlabel('Time'); ylabel('Scale') % Figure 2 shows its scalogram, where you can notice the presence of two % parallel strips of constant scale, corresponding to each one of the tones. % Moreover, you can observe their points of start and end. % Press any key to continue ... pause %-------------------------------------------------------- echo off clear s t sc echo on % In order to achieve a high temporal resolution, we calculate the transform % at lower scales: figure(3); colormap('pink') [sc,t,s]=scalog(x,8e3,0.3,0.75,0.03); show(abs(sc),10,t,s) xlabel('Time'); ylabel('Scale') % Figure 3 shows these 'zoomed' analysis of the signal. % Press any key to continue ... pause %-------------------------------------------------------- echo off clear s t sc close(2); close(3) echo on % Now, let us consider a a 'chirp' signal, as depicted in the figure. % The frequency grows linearly from 120 to 240 Hz, and the sampling % rate is 8000 Hz again. w=(2*pi*120/8000)*(1+(0:1023)/512); x=cos(w.*(0:1023)); plot(x) % We calculate its scalogram for the scales from 1 to 10, with a step 0.1 % between them. You can see it in figure 2. % Please, wait ... figure(2); colormap('pink') [sc,t,s]=scalog(x,8e3,1,1