# Continuous wavelet transform subsampling at lower frequencies?

I am not completely confident in my understanding of wavelet theory, but since I am currently in the process of creating a CWT scalogram tool, I have to ask:

Is there a theorem allowing us to subsample at low frequencies and obtain correct scale/frequency results?

• I don't understand your question. I suggest you give more detail. – Dave Kielpinski Aug 14 '17 at 22:27

The core of continuous wavelet transform

$$C_s(\tau,a) = \int s(t) \psi^*_{\tau,a}(t) dt$$

discretization is that (this is only one of the numerous possibilities) one can find constants $$a_0$$ and $$b_0$$, whose product $$a_0 b_0 ($$C_\psi$$ depending on the wavelet), such that the following sampling pattern

$$c_{j,k} = C_s(k b_0 a_0^j,a_0^j), \quad (j,k)\in \mathbb{Z}$$

keeps all the information contained in $$C_s(\tau,a)$$. The subsampling at low frequencies is bigger, i.e. one need less samples since large wavelets smooth the signal.

The varying sampling space from low (top) to high (bottom) is illustrated below: