Alignment of Filters in `dwt` and `wavedec`

[tycohen]

I am trying to implement a matrix form of the DWT (for MCMC sampling reasons) and am comparing to pywt.dwt and wavedec (v 1.3.0) to confirm that I have constructed my matrix correctly. I am encountering a shift in the coefficients from dwt that I did not expect.

nsamp = 64
wavelet = "db4"
level = 1.
signal = np.zeros(nsamp)
signal[0] = 1.
cA_pywt, cD_pywt = pywt.dwt(signal, wavelet, mode="per")
# compute matrix W
# w_k = W @ signal
# cA_matrix = w_k[: nsamp // (2 ** (level + 1))]
# cD_matrix = w_k[nsamp // (2 ** (level + 1)) : nsamp // (2 ** level)]
# plot cA and cD

I would have thought that for a delta function, the peak would be at cA_pywt[0], not cA_pywt[1]. Are the filters not aligned so that at k=0, wavelet.dec_lo[0] is aligned with signal[0]? Also, during decimation, is every even or every odd sample selected?

[tycohen]

I've dug through the C code and done some tests to attempt to answer my own question. Please correct me if I'm wrong. Using mode="periodization", the signal is shifted so that the center of the filter is aligned with the first sample. So in my example above, for "db4", wavelet.dec_lo[0] is aligned with signal[4]. So I'll modify my question: Why is this the convention?

As for the decimation, with mode="periodization" if the filter is of even length then the even samples are kept and if the filter is of odd length the odd samples are kept.