mlpt

Create a signal with nonuniform sampling and verify good reconstruction when performing the mlpt and imlpt.

Create and plot a sine wave with non-uniform sampling.

timeVector = 0:0.01:1;
sineWave = sin(2*pi*timeVector)';

samplesToErase = randi(100,100,1);
sineWave(samplesToErase) = [];
timeVector(samplesToErase) = [];

figure(1)
plot(timeVector,sineWave,'o')
hold on

Perform the multiscale local 1-D polynomial transform (mlpt) on the signal. Visualize the coefficients.

[coefs,T,coefsPerLevel,scalingMoments] = mlpt(sineWave,timeVector);

figure(2)
stem(coefs)
title('Wavelet Coefficients')

Perform the inverse multiscale local 1-D polynomial transform (imlpt) on the coefficients. Visualize the reconstructed signal.

y = imlpt(coefs,T,coefsPerLevel,scalingMoments);

figure(1)
plot(T,y,'*')
legend('Original Signal','Reconstructed Signal')
hold off

Look at the total error to verify good reconstruction.

reconstructionError = sum(abs(y-sineWave))

reconstructionError = 
3.0396e-15

Specify nondefault dual moments by using the mlpt function. Compare the results of analysis and synthesis using the default and nondefault dual moments.

Create an input signal and visualize it.

T = (1:16)';
x = T.^2;
plot(x)
hold on

Perform the forward and inverse transform for the input signal using the default and nondefault dual moments.

[w2,t2,nj2,scalingmoments2] = mlpt(x,T);
y2 = imlpt(w2,t2,nj2,scalingmoments2);

[w3,t3,nj3,scalingmoments3] = mlpt(x,T,DualMoments=3);
y3 = imlpt(w3,t3,nj3,scalingmoments3,DualMoments=3);

Plot the reconstructed signal and verify perfect reconstruction using both the default and nondefault dual moments.

plot(y2,'o')
plot(y3,'*')
legend('Original Signal', ...
       'DualMoments = 3', ...
       'DualMoments = 2 (Default)');

fprintf('\nMean Reconstruction Error:\n');

Mean Reconstruction Error:

fprintf('  - Nondefault dual moments: %0.2f\n',mean(abs(y3-x)));

  - Nondefault dual moments: 0.00

fprintf('  - Default dual moments: %0.2f\n\n',mean(abs(y2-x)));

  - Default dual moments: 0.00

hold off

Resolution levels are the number of cascaded local polynomial smoothing operations. The details at each resolution level are obtained by predicting one half the samples based on a local polynomial interpolation of the other half. The difference between the predicted and actual values are the details at each resolution level. The scaling coefficients at each coarser resolution level are smoother versions of the higher resolution scaling coefficients. Only the final-level scaling coefficients are retained.

Increasing the number of resolution levels enables you to analyze narrowband coefficients for a computational and memory cost.

Create a dual-tone input signal, x, that contains high and low frequencies.

fs = 1000;
t = (0:1/fs:10)';
x = sin(499*pi.*t) + sin(2*pi.*t);

Use mlpt to obtain coefficients for minimum and maximum resolution levels. Print the computation time.

tic
[w1,~,nj1,m1] = mlpt(x,t,1);
computationTime1 = toc;
fprintf('Level one computation time: %0.2f\n',computationTime1)

Level one computation time: 1.80

tic
[w13,~,nj13,m13] = mlpt(x,t,13);
computationTime13 = toc;
fprintf('Level thirteen computation time: %0.2f\n',computationTime13)

Level thirteen computation time: 2.65

If your time instants are not known or specified, you can calculate the MLPT using default time instants.

Load a data signal corrupted with NaNs and with unknown time instants. Calculate the MLPT without specifying time instants. The resulting implied time instants is a vector of valid indices of the corrupted signal.

load('CorruptedData.mat');

[w,t,nj,scalingMoments] = mlpt(yCorrupt);

Calculate the inverse MLPT and visualize the results. Reinsert NaNs to visualize gaps in the signal.

z = imlpt(w,t,nj,scalingMoments);

zToPlot = NaN(numel(yCorrupt),1);
zToPlot(t) = z;

plot(yCorrupt,'k','LineWidth',2.5)
hold on
plot(zToPlot,'c','LineWidth',1)
hold off
legend('Original Signal','Reconstructed Signal')
xlabel('Time Instants')