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juriguen

Member

Registered: 2009-07-05

Posts: 59

Singular Value Decomposition of noisy data: null space

Hi all

I am trying to understand the following problem a bit further, but I lack algebra background. Say that I have a noisy sequence of measurements (a vector) of length N (say it is a column vector, Nx1), that can be written as:

where

is an ideal vector of samples, and

is additive white Gaussian noise. So, the components

are iid, of zero mean and have variance

.

Now, we filter the noisy samples using:

where

is a matrix of size (PxN). The rows of the matrix are orthogonal, but not orthonormal, and so the filtered noise

becomes coloured Gaussian noise.

Moreover, suppose that now I form a Toeplitz matrix using the filtered measurements

:

The matrix can of any size ((P-M+1)x(M+1)), not necessarily square. And it is also not a condition that P>2M, even though the example I wrote is like that.

The idea is that, now, I have the following:

It is easy to prove that matrix

is diagonal, but not a multiple of the identity matrix. In fact, assume that the matrix

is such that the diagonal values of

are symmetric and decreasing from the ends to the central value.

Now, consider that doing SVD on

yields

non-zero entries as singular values of

. This means that the null-space of

is spanned by the columns of

related to the (M-K) zero entries in

.

But, does anyone know how the SVD of

would relate to that of

? If the noise was not coloured, it is straightforward to see that the null-eigenspace is preserved. But I don't know what happens with coloured noise.

Thanks in advance
Jose

Last edited by juriguen (2010-11-11 23:53:29)

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