# Discrete Wavelet Transform (DWT) Filter Bank

I have some stumbling block in my thesis writing.

Do we use the same filter pair while implementing DWT filter bank with downsampling of the filter output, or filters do change also from level to level? Though on the Wiki page and in the book "Biosignal and Medical Image Processing, 3rd edition" by J. L. Semmlow and B. Griffel there are same labelling of filters at each decomposition level on the filter bank schemes:  , the description in Wikipedia states: "The filter output of the low-pass filter g in the diagram above is then subsampled by 2 and further processed by passing it again through a NEW low- pass filter g and a high- pass filter h with half the cut-off frequency of the previous one.."(c)

This resulted in big confusion for me. Logically, as I have been thinking before, for DWT we "travel" in frequency domain by signal downsampling and leaving the filters untouched (while e.g. in Stationary Wavelet Transform it is achieved through upsampling the filters themselves without signal modification). The word "new" in Wiki frustrated me a lot.

Please, help me to resolve this issue.

With respect,

-Andrey.

• The same set of filter coefficients are used at every level of decomposition. Aug 1, 2019 at 14:30
• Are you satisfied with some answer? Apr 24, 2022 at 9:10

In the DWT scheme, whether it is the classical $$2$$-band or the $$M$$-band wavelet setting, the very same analysis filter bank (lowpass/highpass + subsampling) is used at each level. Under this condition, one can derive the cascade algorithm that provides the spectrum of the scaling function: $$\Phi(\omega)= \prod_{k=1}^\infty \frac {1} {\sqrt 2} H\left( \frac {\omega} {2^k}\right) \Phi^{(\infty)}(0)$$ where iterated half-cut-off frequencies are apparent.