## Blocking Matrices

**Proof that the Blocking Matrices do Indeed Block Signals from Specified Directions.**

For some/certain signal processing tasks we wish to generate beam patterns with nulls in specified directions (or equivalently; filters with narrow notches at specified frequencies). Typically these can be used to null out stationary interference sources, directions to multipath images of a desired source, …

We assume that we have a number of sensors ( )which may be array elements, subarrays or formed beams and we wish to null signals from directions , and , and responses for a source in direction (being a column vector of complex signals one element for each sensor). In a narrowband system we may assume there are simply complex numbers representing the sensor amplitude and phase response. We let denote the normalised versions of these signals.

It is obvious (when pointed out anyway) that the matrix:

is a blocking matrix for directions (where is the matrix with for its columns). The space of possible sensor outputs constitutes a (complex) vector space of dimension and the subspace is the null space of and behaves like the identity transformation on the orthogonal complement of the null space.

To see that this is a blocking matrix for we need just consider:

so:

Now the columns of are the vectors and so these are zeros vectors as expected.

To see that behaves like the identity on the orthogonal complement consider in the orthogonal complement of the space spanned by the columns of . Then since each component of this is the dot product of a column of and and so zero. Hence:

(I hope that is all clear, WordPress LaTeX seems to have started deleting all the \ characters from the LaTeX strings again so there may still be some omissions from when I have tried to restore them)

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