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The output of the optimal PIC contains residual interference power and uncorrelated noise power. This section presents and analyzes the optimal PIC using an interference beamformer that eliminates all interference in the output while simultaneously reducing the contribution of uncorrelated noise in the output. For this case, let the interference beam be formed with the beamforming weights

Note that the above expression is similar to the expression for the weights of constrained optimal beamformer except that in this case the beam is constrained in the direction of the interference rather than the look direction. Thus, it can easily be verified that the interference beam formed with these weights has unity response in the interference direction and has a reduced response in the signal direction. The response of the interference beam in the signal direction depends on the signal source power and uncorrelated noise power. It can be shown that this choice of beamforming weights minimizes the sum of signal power and uncorrelated noise power in the interference channel output.
A substitution for V and U in (6.48) from (6.50) and (6.77), respectively, leads to the following expression for wˆ 1, the weight of the optimal PIC using the improved interference beamformer (IIB):

(6.78)

It follows from (6.78) that the weight, which minimizes the output power of PIC using the IIB is independent of the signal, the interference, and the uncorrelated noise powers. This weight depends only on the array geometry and relative directions of the two sources. The expressions for the signal power and the noise power at the output of the optimal PIC using the IIB are, respectively, given by

	(6.79)
and
	(6.80)

One observes from expressions (6.79) and (6.80) that the output signal power and the output noise power of the optimal PIC using the IIB are independent of the interference power. Thus, the optimal PIC using the IIB has completely suppressed the interference. Furthermore, the output signal power and output noise power depend on σn^2/LPS (ratio of uncorrelated noise power to signal power at signal beam output). The output signal power increases as σn^2/LPS decreases, and approaches the input signal power in the limit. Thus, in the presence of a strong signal source, the signal suppression by the optimal PIC using the IIB is negligible. The signal suppression becomes further reduced as the number of elements in the array is increased.

The total noise power at the output of the optimal PIC using the IIB is equal to the uncorrelated noise power at the output of the signal beam when ρ = 1. To investigate the effect of σn^2/LPS on the output noise power when ρ < 1, you can rewrite the quantity in the braces on the right side of (6.80) in the following form:

(6.81)

Since ρ < 1, it follows from (2.6.81) that the second term on the RHS is negative if ρ < (σn^2/LPS)^2. Thus, under this condition the quantity in the braces on the right side of (6.80) is less than unity and, hence, the uncorrelated noise power at the output of the PIC is less than the uncorrelated noise power at the output of the signal beam. Thus, the optimal PIC using the IIB reduces the uncorrelated noise when ρ < (σn^2/LPS)^2. On the other hand, when ρ > (σn^2/LPS)^2, the quantity in the braces on the right side of (6.80) is more than unity and the optimal PIC using the IIB increases the uncorrelated noise power. Note that at the output of the optimal PIC using the IIB, total noise consists of uncorrelated noise only: it increases as σn^2/LPS decreases and in the limit approaches σn^2/LPS.
Now the output SNR of the optimal PIC using the IIB is examined. Let this ratio be denoted by SNR(wI). It follows then from (6.79) and (6.80) that

(6.82)

Thus, the output SNR of the optimal PIC using the IIB is proportional to the input signal to uncorrelated noise ratio, the number of elements in the array, and the parameter ρ.

PIC with Conventional Interference Beamformer is discussed here.

PIC with Orthogonal Interference Beamformer is discussed here.

Back to PIC.

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