Trying to understand WAVELET TRANSFORM Frequncy-Time diagram

Question

I am reading

The Wavelet Tutorial

Part III MULTIRESOLUTION ANALYSIS & THE CONTINUOUS WAVELET TRANSFORM

by Robi Polikar

The author explains about the following fig and says;

Note that boxes have a certain non-zero area, which implies that the value of a particular point in the time-frequency plane cannot be known. All the points in the time-frequency plane that falls into a box is represented by one value of the WT.

I am unable to interpret this fig and also that why value of a particular point in the time-frequency plane cannot be known. I can always intersect two lines one passing through a particular value of t and another through f. The intersecting point will give me exact location of frequency at particular time (and V.V). I am not able to understand what thsi diagram is all about and what is its significance?

Answer 1

I believe by the WT, you are talking about the discrete wavelet transform, DWT.

This can be thought of as a subsampling of the continuous wavelet transform, CWT. In the case of the DWT, we pick frequencies of the form $2^{j-1}$ for ($j=1,2,\dots$) and then pick times seperated by multiples of $2^j$.

You can see this in the diagram: As frequency increases, boxes double in height and half in width.

I believe what the author means by the statement

Note that boxes have a certain non-zero area, which implies that the value of a particular
point in the time-frequency plane cannot be known.

is that the DWT, is an 'averaging' of the CWT within the boxes i.e. the boxes represent only one value and that is the average of the the values within it from the CWT. Hence as the boxes have non-zero area, it is impossible to know any of the exact values making up this average, so it is impossible to know the exact value of any point in the time-frequency plane.