# Haar Basis in Signal processing

I want to help one of my friends who studies engineering. He has a homework at the signal processing course. I think I realize what I have to do, but since I don't have their course, I do not fully understand what I have to do. It goes like this:

$$H=\begin{pmatrix}1&-1&0&0&0&0&0&0 \\ 0&0&1&-1&0&0&0&0 \\ 0&0&0&0&1&-1&0&0\\ 0&0&0&0&0&0&1&-1 \\ 1&1&-1&-1&0&0&0&0\\ 0&0&0&0&1&1&-1&-1 \\ 1&1&1&1&-1&-1&-1&-1\\ 1&1&1&1&1&1&1&1 \end{pmatrix}$$

The exercise says to check that $H$ consists of independent line vectors (that's easy). Then it says to find the coordinates of $x=[1,1,...1]$ and $y=[1,-1,...,1,-1]$ in the Haar Basis defined by $H$.

Does this means to find the coordinates in the usual way, i.e. solve the equation $zH=x$?

(I do not need you to solve the system (I'll use Matlab for that), just say if I'm right or not. Thank you.)

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Maybe this is more suited to dsp.stackexchange.com ? – tdc Jan 11 '12 at 12:48

The Haar basis is the basis of the row vectors, not of the column vectors, so I presume you'd want to solve the equation $zH=x$ for row vectors instead. That also makes sense in that $x$ and $y$ are being specified as row vectors.
Isn't a normalization missing or is that omitted for practical/computational reasons? In any case, I think it would be worth pointing out that no matlab is needed, as up to multiplication with a diagonal matrix the inverse is $H^T$. – t.b. Oct 18 '11 at 19:50
@joriki: Sorry about the confusion. In the problem there are two vectors for which I must find the coordinates. Those are the systems I want to solve, namely $zH=x$ and $zH=y$. I edited the question, to correct my mistake. – Beni Bogosel Oct 18 '11 at 20:39