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While studying about Wavelet Transform, I'm confused as why this is called a [1 -1] Filter , Here is what i'm talking about.

In this hypothetical example the student does fairly well the first half of the term then neglects his or her studies for the last half. Thus the exam scores for the term were 80%, 80%, 80%, 80%, 0%, 0%, 0%, and 0%* We can tell the average of all the scores (40%) and when the scores “tanked” after the 4th exam just by looking. Knowing the answer in advance, however, is a good way to learn and to verify the wavelet transforms. Then we can use them with confidence on real-world data where we can’t simply “eyeball” the final values.

We will now walk through the CWT process step by step using the simplest of the wavelet filters on this example. We begin by comparing the humble Haar wavelet filter, [1 –1], with the data as shown

80 80 80 80 0 0 0 0

If We now keep subtracting each value from the next value we will eventually get,

[0, 0, 0, 80, 0, 0, 0]

Now If I want to stretch the Filter to three points i.e If the filter is stretched from [1 –1] to [1 0 –1].

How will I calculate these points as I did in case of two point Filter where i simply subtract each value from the next value ?

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1 Answer 1

up vote 3 down vote accepted

I have the same book as you. :-)

The [1 -1] filter is a a simple differentiator, also known as, the Haar mother wavelet. You need to understand convolution to understand what he is saying. On the first level, you convolve your signal with [1 -1]. Then on the next level, you convole your signal with [1 0 -1], etc. That is all the CWT is doing. (Look at convolution first). If you understand that well, you can understand CWT very easily.

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Thank you very much, P.S: this is the best Wavelet book i've ever found :) There is another book 'Wavelet Toolbox for Use with Matlab' by Michel Misiti,Yves Misiti,Georges Oppenheim,Jean-Michel Poggi and that is great too –  Sufiyan Ghori Feb 21 '13 at 18:11
    
could you tell me why we padded zero on both side of the original sequence (80 80 80 80 0 0 0 0) -> (0 80 80 80 80 0 0 0 0 0) when using haar wavelet ? @Mohammad –  Sufiyan Ghori Feb 21 '13 at 18:29
    
...This, again, has to do with the convolution. The author is showing you the convolution as a series of sliding dot products. I would highly recommend you read and understand convolution before doing the CWT or Wavelets in general. –  Mohammad Feb 21 '13 at 18:36

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