Adaptive Interference Nulling and Direction of Arrival Estimation in GPS Dual-Polarized Antenna Receiver - Adaptive Signal Processing in Wireless Communications - page 413

where E [.] denotes statistical expectation. In practice, r is replaced by its time-average
estimates, ˆ , and can be expressed as

T

1

∑

ˆ

H

R = x () k x ( . k (14 .15)

T

k = 1

Denote the 2 N -by-1 complex beamformer weight vector for the N dual-polarized anten-
nas as

T

w = [ wwww … ww ] . (14 .16)

1 HVHV 1 2 2 NH NV

The corresponding antenna array output y ( k ) at the dual-polarized antenna array is
given by

H

yk () = wx (). k (14 .17)

Equations (14.11)-(14.17) and the analyses hereafter are also valid and applicable to
the European, Russian, and Chinese satellite navigation systems [6].

14.4 Single versus Multiple Antijam Beamformers

The minimum variance distortionless response (MVDR) technique is an efficient tool
to mitigate interference without compromising the desired signals. The array weights
are computed to form unit gains toward the DOAs of the GPS signals, and to place deep
nulls toward the directions of the interferers. This is achieved by minimizing the output
power subject to unit-gain constraints,

H H

min wrw subjectto C wf = , (14 .18)

w

where the constraint matrix C represents the GPS steering matrix a D (θ). The optimal
weights for the above constrained minimization problem can be obtained as [40]

= −

1 H − − 1 1

wrCC rC f
opt

( ) , (14 .19)

where f is a D -by-1 vector of unit values, f = [1 1 … 1] T . Applying the optimal weights
to the received data vector x ( k ) yields the interference beamformer output

H

yk () = wx (). k (14.20)

opt

It is noted that the power inversion method is a special case of equation (14.19), where
the constraint matrix is a vector of zero values except the first element, which has a unit
value. Also, f is a scalar equal to 1.