Uncorrelated Noise Only

For a special case of the noise environment when no direction interference is present, the noise-only array correlation matrix is given by

(4.36)
Substituting the matrix in (4.16), a simple calculation yields  
(4.37)
Thus, the weights of the optimal processor in the absence of errors are the same as those of the conventional processor, implying that the conventional processor is the optimal processor for this case. Thus, in the absence of directional interferences the conventional processor yields the maximum output SNR and the array gain. The output SNR α and the array gain Gˆ of the optimal processor for this case are, respectively, given by
(4.38)

and

 

(4.39)
These quantities are independent of array geometry and depend only on the number of elements in the array.

 

One Directional Interference

Consider the case of a noise environment consisting of a directional interference of power PI and uncorrelated noise of power σn^2 on each element of the array. Let SI denote the steering vector in the direction of interference. For this case, the noise-only array correlation matrix is given by

(4.40)

Using the Matrix Inversion Lemma, this yields

 

(4.41)

The substitution for RN^–1 , rearrangement, and algebraic manipulation leads to the following expression for the output SNR:

(4.42)
The array gain is given by  
(4.43)
where  
(4.44)
   
is a scalar quantity and depends on the direction of the interference relative to the signal source and the array geometry, as discussed previously. It follows from (2.1) and (4.44) after rearrangement that
(4.45)

Thus, this parameter is characterized by the weights of the conventional processor. As this parameter characterizes the performance of the optimal processor, it implies that the performance of the optimal processor in terms of its interference cancelation capability depends to a certain extent on the response of the conventional processor to interference.An interesting special case is when the interference is much stronger compared to background noise, PI>>σn^2. For this case, these expressions may be approximated as

(4.46)
and  
(4.47)

When interference is away from the main lobe of the conventional processor ρ ≈ 1, it follows that the output SNR of the optimal processor in the presence of a strong interference is the same as that of the conventional processor in the absence of interference. This implies that the processor has almost completely canceled the interference, yielding a very large array gain.

The performance of the processor in terms of its output SNR and the array gain is not affected by the look direction constraint, as it only scales the weights. Therefore, the treatment presented above is valid for the unconstrained processor.
For the optimal beamformer to operate as described above and to maximize the SNR by canceling interferences, the number of interferences must be less than or equal to L – 2, as an array with L elements has L – 1 degrees of freedom and one has been utilized by the constraint in the look direction. This may not be true in a mobile communications environment due to the existence of multipath arrivals, and the array beamformer may not be able to achieve the maximization of the output SNR by suppressing every interference. However, the beamformer does not have to suppress interferences to a great extent and cause a vast increase in the output SNR to improve the performance of a mobile radio system. An increase of a few decibels in the output SNR can make possible a large increase in the system’s channel capacity.

Output signal to noise ratio and array gain of optimal beamformer is discussed here.

Back to optimal beamformer.

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