why does wavelet approximation need to multiply sqrt(2)

I am learning wavelet transform, the image below is one of my examples. Notices that when down sampling the sample signal, the approximation gets multied by about sqrt(2). I know the scaling function has integral one. this property should keep the amplitude of the signal remain the same, i think. Do I need to divide sqrt(2) for each level?

This document https://www.mathworks.com/help/wavelet/ref/idwt.html, has exmple for deconstructing and reconstructing function with wavelet. But it doesn't multiply sqrt(2).

Answer 1

[Sudarsanan A K]

Hello Konoha,

I understand that you are interested in exploring the scaling and normalization of wavelet coefficients within the multi-level wavelet decomposition.

When you perform a multi-level wavelet decomposition in MATLAB:

• The signal is processed with two filters: a low-pass filter (scaling function) and a high-pass filter (wavelet function).
• The filtered signal is then downsampled by half, which is a key step in wavelet decomposition.

For the normalization and scaling of wavelet coefficients:

• The scaling function is normalized so that it integrates to one, helping to preserve the signal's energy.
• Downsampling can affect the energy of the signal if not compensated for, but this is taken care of in the wavelet transform.

In MATLAB's Wavelet Toolbox:

• Functions like "wavedec" for 1-D signals and "wavedec2" for 2-D signals, automatically handle normalization to ensure energy preservation.
• The filters used are designed to keep the energy consistent across decomposition levels.
• Users generally do not need to manually adjust coefficients for energy preservation.

Here is an example script that performs a two-level decomposition on a simple 1D signal using the Haar wavelet and checks the energy preservation:

% Define a simple 1D signal
signal = [1, 3, -2, 4, -1, 5, 3, -3];

% Calculate the energy of the original signal
original_energy = sum(signal.^2);

% Perform a two-level DWT using the Haar wavelet
[C, L] = wavedec(signal, 2, 'haar');

% Extract the approximation and detail coefficients from C
A2 = appcoef(C, L, 'haar', 2); % Level 2 Approximation Coefficients
D2 = detcoef(C, L, 2);         % Level 2 Detail Coefficients
D1 = detcoef(C, L, 1);         % Level 1 Detail Coefficients

% Calculate the energy of the wavelet coefficients
transformed_energy = sum(A2.^2) + sum(D2.^2) + sum(D1.^2);

% Display the results
disp(['Original Energy: ', num2str(original_energy)]);

Original Energy: 74

disp(['Transformed Energy: ', num2str(transformed_energy)]);

Transformed Energy: 74

To know more about the amplitude changes and ensure you are interpreting the results correctly, you can additionally refer to the MathWorks documentation:

https://mathworks.com/help/wavelet/ref/wavedec.html

I hope this helps!