# What is the “fourth” Haar basis function of the 4×4 Haar basis?

I know the three Haar basis functions

//mother wavelet:
[1 1 -1 -1]
//dilation:
[1 -1 0 0]
//translation:
[0 0 1 -1]


However, the Haar basis consists of four basis elements: The mother wavelet, the translation, the dilation and another, fourth element, which I don't understand. According to this, this fourth element is

[1 1 1 1]


But how would this one look like, and how is it obtained (dilation/translation, whatever combination)? This confuses me.

Btw: is the Haar basis defined as

[ 1  1  1  1
1  1 -1 -1
1 -1  0  0
0  0  1 -1]


or as the transpose of this?

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If I understand the notation right, it looks like the [1 1 1 1] element is just the DC coefficient, i.e. the mean of the input signal.

By symmetry, the Haar wavelet integrates to 0 over its domain. Thus, if you didn't include the DC coefficient, adding a constant bias to the input signal would not change its Haar transform at all.

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This has been very helpful, thanks! Say, I want to apply H_4 (as given above) to a signal x, do I use the matrix I gave above or its transpose? Just to be clear of the definition. –  ptikobj Aug 21 '11 at 14:19
@ptikobj: That surely depends on whether you're using row or column vectors to represent your data. (If you're using column vectors, I assume you should use the matrix as given. If you're using row vectors, use the transpose.) –  Ilmari Karonen Aug 21 '11 at 18:05