Page 175 - Introduction to Statistical Pattern Recognition

4 Parametric Classifiers 157

let us define the discrete Fourier transform of the time-sampled values of a ran-
dom process, x(O), . . . ,x(n -l), as

I1 - I
F(k) = xx(L)Wk' (k = 0,. . . ,n-1) , (4.109)
i=o
where W is

2n
-JT
W=e (4.1 10)
and W satisfies

n-I n for = 0
k4 0 for 5#0. (4.111)

Then, the inverse Fourier transform becomes

111-1
~(5) = -xF(k)W-k' (t = 0,. . . ,n-1) . (4.1 12)
n k4

In a stationary process, the first and second order moments of x(k) must
satisfy

R (k-4) = E (x(k)x(t)] , (4.1 14)

where m and R (.) are called the mean and autocorrelation function of the pro-
cess. We assume that the process is real. Note that m is independent of k, and
R(.) depends only on the difference between k and 2 and is independent of k
itself.
Using (4.113) and (4.114), the expected values and second order
moments of the F(k)'s can be computed as follows.