Which part of the Bookkeeping vector do I select in order to reconstruct the signal using 3rd, 4th and 5th level coefficients with waverec?

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Eashan Gandhi 2021-4-3

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回答： Abhimenyu 2024-4-4

I have a signal that I have decomposed using wavedec, which outputs the wavelet coefficients, C and the bookkeeping vector, L. I want to reconstruct the signal using the 3rd, 4th, and 5th level coefficients. Which part of the Bookkeeping vector should be selected in order to reconstruct the signal using waverec?

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Answer 1

Abhimenyu 2024-4-4

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Hi Eashan,

From the information shared, I inferred that you are trying to reconstruct the signal using the 3rd, 4th, and 5th level "wavelet coefficients". The "bookkeeping vector", L provides the lengths of the various levels of decomposition stored in the wavelet coefficients", C. The structure of C is such that it begins with the approximation coefficients at the final level of decomposition, followed by the detail coefficients for each level in descending order.

Given a decomposition up to level N, the structure of C and L is as follows:

C = [A_N, D_N, D_{N-1}, ..., D_1]
L = [length(A_N), length(D_N), length(D_{N-1}), ..., length(D_1), original_signal_length]

To reconstruct the signal using just the 3rd, 4th, and 5th level coefficients, the "wavelet coefficients", C can be modified to include only the coefficients needed to reconstruct by setting others to zero. The specific parts of the "bookkeeping vector" are not required.

Assuming 5-level decomposition, here is an example MATLAB R2024A code that can reconstruct the signal using the 3rd, 4th, and 5th level coefficients:

Creating an example signal for reference:

% Create a sample signal
t = linspace(0, 1, 512); % Time vector
signal = sin(2 * pi * 5 * t) + 0.5 * sin(2 * pi * 10 * t); % Sample signal

Decomposition using the MATLAB "wavedec" function and 'db1' wavelet:

% Wavelet decomposition
wname = 'db1'; % Daubechies wavelet with 1 vanishing moment
[coeffs, L] = wavedec(signal, 5, wname);

Zeroing out the coefficients that are not required for reconstruction. Here 1st and 2nd level coefficients are reduced to zero:

% Calculate the cumulative sum of lengths in L to find indices
cumL = cumsum(L);
% Zero out detail coefficients for levels 1 and 2 in coeffs
% For level 1
coeffs(cumL(end-2)+1:cumL(end-1)) = 0;
% For level 2
coeffs(cumL(end-3)+1:cumL(end-2)) = 0;

Reconstruct the signal using MATLAB "waverec" function:

% Reconstruct the signal using the modified coeffs and original L
reconstructed_signal = waverec(coeffs, L, wname);

Plotting the results:

% Plot the original and reconstructed signal

figure;

plot(t, signal, 'b', 'DisplayName', 'Original Signal');

hold on;

plot(t, reconstructed_signal, 'r--', 'DisplayName', 'Reconstructed Signal');

legend show;

xlabel('Time');

ylabel('Amplitude');

title('Signal Reconstruction Using Levels 3, 4, and 5');

grid on;

To understand more about the MATLAB's "wavedec", "cumsum", "waverec" functions, please follow the below mentioned MATLAB R2024A documentation links respectively:

Regards,

Abhimenyu