Decoding Error-Correction Codes Utilizing Bit-Error Probability ...

It is first shown that X(f) is a process of orthogonal increments. Recall that a process
X(f) is said to have orthogonal increments if
E{|X(f 1 ) − X(f 2 )| 2 } < ∞ (2.6)
and, if whenever the frequencies f 1 , f 2 , f 3 , f 4 satisfy the inequality, f 1 < f 2 < f 3 < f 4 , the
correlation of the two finite increments, [X(f 1 ) − X(f 2 )] and [X(f 3 ) − X(f 4 )], satisfy the
relation
E{[X(f 1 ) − X(f 2 )][X(f 3 ) − X(f 4 )] ∗ } = 0, (2.7)
where “ ∗ ” denotes complex conjugation (Doob [5, pg. 99]). Since X(f) exists as a limit
in the mean-square sense, it is straightforward to show that equation (2.6) is satisfied. In
order to verify equation (2.7), first express the integral in (2.5) more simply by
X(f) =
∫ ∞
−∞
e −i2πft − 1
x(t)dt, (2.8)
−2πit
where it exists in the same sense that equation (2.5) exists - as a limit in the mean-square
sense. Consider a finite increment in X(f), given by
X(f 1 ) − X(f 2 ) =
∫ ∞
−∞
e −i2πf1t − e −i2πf 2t
x(t)dt. (2.9)
−2πit
Now define the real square pulse function by
⎧
⎪⎨ 1, f 1 < f < f 2 ,
Φ f1 ,f 2
(f)
⎪⎩ 0, otherwise.
The inverse Fourier transform of Φ f1 ,f 2
(f) is given by
∫ ∞
−∞
Φ f1 ,f 2
(f)e i2πft df = ei2πf2t − e i2πf 1t
, (2.10)
2πit
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