Optimal Beam Space Processor

Let an M – 1 dimensional vector w denote the adjustable weights of the auxiliary beams. It follows from Figure 9 that the output η(t) of the interference beam is given by

(6.7)

The output y(t) of the overall beam space processor is obtained by subtracting the interference beam output from the main beam, and thus is given by

(6.8)

The mean output power P(w) of the processor for a given weight vector w is given by

	(6.9)
where P0 is the mean power of the main beam given by
	(6.10)
Rqq is the correlation matrix of auxiliary beams defined as
	(6.11)

and Z denotes the correlation between the output of auxiliary beams and the main beam. It is defined as

	(6.12)
A substitution for q(t) and ψ(t) in (6.11) and (6.12) yields
	(6.13)
	(6.14)
Substituting for P0, Rqq and Z in (6.9), the expression for P(w) becomes
	(6.15)

Note that P(w) is a quadratic function of w and has a unique minimum. Let w denote weights that minimize P(w). Thus, it follows that

	(6.16)
Substituting (6.15) in (6.16) yields
	(6.17)

As B has rank M – 1, (B)H*R*B is of full rank and its inverse exists. Thus, (6.17) yields

(6.18)

Substituting for w = wˆ from (6.18) in (6.15), you obtain the following expression for the mean output power of the optimal processor:

(6.19)

Expressions for the mean output signal power may be obtained by replacing the array correlation matrix R by the signal only array correlation matrix RS in (6.15), yielding

	(6.20)
Since
	(6.21)
and
	(6.22)
it follows from (6.20) that
	(6.23)

Thus, when the blocking matrix B is selected such that (B)H*S0 = 0, there are no signal flows through the interference beam and the output signal power is present only in the main beam. When the main beam is taken as the conventional beam, that is,

(6.24)

the mean output signal power of the beam space processor becomes

(6.25)

Note that the signal power is independent of w.
Similarly, an expression for the mean output noise power may be obtained by replacing the array correlation matrix R by the noise-only array correlation matrix RN in (6.15), yielding

(6.26)

Substituting for w = wˆ from (6.18) in (6.26), you obtain the following expression for the mean output noise power of the optimal processor:

(6.27)

The output SNR of the optimal beam space processor then becomes

(6.28)

These expressions cannot be simplified further without considering specific cases. Later , a special case of beam space processor is considered where only one auxiliary beam is considered in the presence of one interference source to understand the behavior of beam space processors. The results are then compared with an element space processor. In the next section, a beam space processor referred to as the generalized side-lobe canceler (GSC) is considered. The main difference between the general beam space processor considered in this section and the GSC is that the GSC uses presteering delays.

In the next page Generalized Side-Lobe Canceler is discussed.

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