The null steering beamformer is used to cancel a plane wave arriving from a known direction and thus produces a null in the response pattern of the plane wave’s direction of arrival.A beam with unity response in the desired direction and nulls in interference directions may be formed by estimating beamformer weights shown in Figure 1 using suitable constraints.

Assume that S0 is the steering vector in the direction where unity response is required and that S1, …, Sk are k steering vectors associated with k directions where nulls are required. The desired weight vector is the solution of the following simultaneous equations:

	(3.1)
	(3.2)

Using matrix notation, this becomes

(3.3)

where A is a matrix with columns being the steering vectors associated with all directional sources including the look direction, that is,

(3.4)

and e1 is a vector of all zeros except the first element which is one, that is,

(3.5)

For k = L − 1, A is a square matrix. Assuming that the inverse of A exists, which requires that all steering vectors are linearly independent , the solution for the weight vector is given by

(3.6)

In case the steering vectors are not linearly independent, A is not invertible and its pseudo inverse can be used in its place.
It follows from (3.6) that due to the structure of the vector e1, the first row of the inverse of matrix A forms the weight vector. Thus, the weights selected as the first row of the inverse of matrix A have the desired properties of unity response in the look direction and nulls in the interference directions.
When the number of required nulls are less than L − 1, A is not a square matrix. A suitable estimate of weights may be produced using

(3.7)

Although the beam pattern produced by this beamformer has nulls in the interference directions, it is not designed to minimize the uncorrelated noise at the array output. It is possible to achieve this by selecting weights that minimize the mean output power subject to above constraints.

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In the next page optimal beamformer is discussed.

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