# How do wavelet levels depend on one another?

I am working with wavelets for my thesis, and I would like to know if there is an intuitive dependency between the levels of a DWT.

To make it clear: The DWT performs a convolution, and then the result is downsampled, right?

Like in this example of a convolution below.

Now when we downsample dyadically, we throw away every second value, right? Like this:

• Can we say that 2 adjacent values in the original are projected onto 1 value in the transform?

For example, the first value in the first level of the DWT would then be "responsible" for values 1 and 2 in the original. So that we would know where e.g. a pattern in a high DWT level actually comes from.

• Does this kind of projection exist?
• If it does not work, why not?
• Is the convolution/downsampling performed differently than I imagined?

I may be a little confused by your question (intuitive dependency?) but I think these two sources will help you wrap your head around the relationship between levels in a DWT:

I think one way to think of it is because a DWT can be inverted, you can take a signal and perform a DWT on it and then you can reverse that DWT to get the original signal. So there's a clear dependance between levels in a DWT. I don't expect this answer to be accepted but i hope it can point you in the right direction to what you're looking for.

What you're missing is that the wavelet transform is a lossless conversion. If you give it $$N$$ points as an input, you must get $$N$$ points as an output.
In the case of the wavelet transform, each step gives you a low-frequency result and a high-frequency result, each with $$\frac{N}{2}$$ points. If you keep both of them (and all your math is done with infinite precision), then you can completely reconstruct the original.