Page 343 - Fundamentals of Radar Signal Processing

T
T
respectively, h t and h w. The power in the signal component is therefore
H * T
T
*
H
h t t h, and in the interference component it is h w w h. Because the
interference power is a random variable (RV), its expected value is used to get
T
meaningful results. The expected value of w*w is the interference covariance
matrix S I

(5.6)

It follows also that , and . With this definition the SIR
becomes

(5.7)

As in Chap. 4, the filter h that maximizes Eq. (5.7) is found using the
Schwarz inequality, which in a form suitable for vector-matrix manipulations is

(5.8)

where and equality occurs if and only if p = kq for some scalar

constant k. To apply Eq. (5.8), first note that the matrix SI will be positive
definite so that it can be factored into the form S = A A for some matrix A; that
H
I
H –1
is, A is the “square root” of S in some sense. Define p = Ah and q = (A ) t*.
I
H
* T
H
2
This choice is contrived so that p q = h t* and therefore |p q| = h t t h,
H
H
which is the numerator of Eq. (5.7). The Schwarz inequality then gives

(5.9)

Rearranging Eq. (5.9) to isolate the SIR of Eq. (5.7) shows that

(5.10)

with equality only when p = kq. The optimal weight vector therefore satisfies

H -1 *
Ah = k(A ) t , or, with k = 1
opt