# What information could be extracted from this Discrete Wavelet Transform of a time series data?

I have decomposed this time series data in 5 levels using Discrete Wavelet Transform Daubechies 4.

I am trying to understand what kind of information one can extract from applying DWT to a time series.

I will not tell you where this data comes from, to see if you reach a conclusion that corresponds to the truth.

The original data has 512 samples and encompasses 256 seconds.

Decomposing the data in 5 levels gives me this:

The blue line is the low frequency coefficients and the red line is the high frequency coefficients

LEVEL 1

LEVEL 2

LEVEL 3

LEVEL 4

LEVEL 5

What kind of information can you gather from these graphs and their respective values? Can you see the signal frequency? What level contains the main frequency? What other information can you extract from these?

Please explain it in layman's terms, cause my math is little bit rusty.

Thanks.

Given that you have done the orthonormal DWT transform, Parseval's theorem applies and then most of the energy (integral over function value squared) will be in the 1st level high pass band, since the function values there squared are in range about $$[11^2,14^2]$$ and in the other bands $$[0^2,4^2]$$. So the dominant frequency should be roughly among the top 1/2 of Fourier frequencies.

Since the decomposition done is dyadic it halves the frequency domain for each level of decomposition we don't get any higher resolution than that for this frequency band.

As you can see we can put level 4 LP "on top of" level 4 HP to get same number of samples as level 3 HP, and then iterate. This is basically what the inverse transform of a dyadically subsampled DWT does.

1. Blowing up,
2. Inverse transforming
3. Adding together. Now we have LP of step 1 "lower" level.
4. Iterate from step 1 until we have inverse transformed all levels.
• Thanks. Just a doubt I have. Suppose I have a time series sampled at 30 Hz. I decompose it one time and in that level I am seeing coefficients that represent half the frequency, that is 15 Hz. So, if I understand correctly, the coefficients in that band will represent frequencies from 0 to 15 Hz, right? If half of the coefficients represent low frequencies and half represent high frequencies is each half representing 7.5 Hz? I mean from 0 to 7.5Hz for the low frequency and from 7.5 to 15 Hz for the other half? I cannot wrap my head around this. – SpaceDog Apr 6 at 0:10
• If I want to say in text what each set of coefficients represent, what would I say? These low frequencies coefficients represent __________________________ and these high frequencies coefficients represent _________________________ please fill the lines... 😃 – SpaceDog Apr 6 at 0:13
• another fast question 😃 can the main frequency be extracted by the coefficients? if the explanation is extensive, please just point me in the right direction.Thanks – SpaceDog Apr 6 at 0:21
• @SpaceDog Maybe we can make a new question about it. Today I will be busy running around town but maybe tomorrow. – mathreadler Apr 6 at 11:51
• Another question would be redundant, part of this question was not answered yet. Thanks. – SpaceDog Apr 6 at 14:20