Recreating the `dwt` function from Matlab

Question

I am trying to implement DWT (Discrete Wavelet Transform) on a given signal, without using the dwt function in Matlab. My approach was to get the synthesis filter coefficients using,

[LoD, HiD, LoR, HiR] = wfilters('db4')

and then get the approximation and detail coefficients by convolving my input signal with the filters and then downsampling by 2

a = downsample(conv(x,LoD),2);
d = downsample(conv(x,HiD),2);

But the output for this doesn't match the output I get after using the Matlab function.

[a,d] = dwt(x,LoD,HiD)

Do I need to pad my signal for it to match the output? I have tried symmetric padding but it didn't help much.

Answer 1

MATLAB uses, by default, a symmetric padding of the signal.
So the procedure is something like:

1. Pad the signal with symmetric extension.
2. Apply convolution with valid shape.
3. Downsample the result by factor 2.

The downsampling is basically something like: vA(firstIdx:2:lastIdx);.
Where firstIdx = 2; and lastIdx = length(vX) + length(vF) - 1;.

Where vX is the input signal and vF is the filter used.

This is an example how to replicate:

clear();

% Parameters
numSamples  = 27;
wavType     = 'db4';

% Data
vA = randn(numSamples, 1);
[vLoD, vHiD, vLoR, vHiR] = wfilters(wavType); %<! Filters
numCoeff = length(vLoD); %<! All filters same length

firstIdx = 2;
lastIdx  = numSamples + numCoeff - 1;

% Padding
vB = padarray(vA, numCoeff - 1, 'symmetric', 'both');

% LoD
vY = conv(vB, vLoD, 'valid');
vYLoD = vY(firstIdx:2:lastIdx);

% HiD
vY = conv(vB, vHiD, 'valid');
vYHiD = vY(firstIdx:2:lastIdx);

% Reference
[vYLoDRef, vYHiDRef] = dwt(vA, wavType);

max(abs(vYLoD - vYLoDRef))
max(abs(vYHiD - vYHiDRef))

It will generate practically zero difference.