Page 176 - Introduction to Statistical Pattern Recognition

158 Introduction to Statistical Pattern Recognition

E(F(k)] =mzWk' = { nm for k = 0
11-1
g-3 0 fork#O, (4.1 15)

n-l n-1
= (CR(U)W~")(C,W'"-"~)
II =o s =o

11-1
nzR(u)W'" for k = t
II =o (4.116)
0 for k f 4 ,

where R (-1.-s) = R (s--1.) for a real x(t) is used to derive the third line from the
second, and F*(t) is the complex conjugate of F(2). The variances and covari-
ances of the F(k)'s are

Var(F(k)) =E(F(k)F*(k)} - E(F(k)}E(F*(k)J

and