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

The first term in (2.14) is simply the total power in the process x(t) which is obtained by
integrating the power spectral density over all values of f, given by
E{|x(t)| 2 } =
∫ ∞
−∞
G(f)df. (2.15)
In order to evaluate the second term, define a differential increment in the process X(f)
as dX(f) = X(f + df) − X(f). It can be shown, using the same techniques to derive
equation (2.12), that the correlation of two differential increments is given by
⎧
⎪⎨ G(f)df, f = f ′
E{dX(f)d ∗ X(f ′ )} =
⎪⎩ 0, f ≠ f ′ .
(2.16)
A substitution of (2.16) into the the second term in (2.14) yields
∫ ∞ ∫ ∞
−∞
−∞
e i2π(f−f ′ )t E{dX(f)dX ∗ (f ′ )} =
∫ ∞
−∞
G(f)df. (2.17)
To facilitate the analysis of the third term, first compute E{x ∗ (t)dX(f)} as follows:
E{x ∗ (t)dX(f)}
= E{x ∗ (t)[X(f + df) − X(f)]}
=
=
=
=
∫ ∞
−∞
∫ ∞
−∞
∫ ∞
−∞
∫ ∞
−∞
e −i2π(f+df)t′ − e −i2πft′
−2πit ′ E{x ∗ (t)x(t ′ )}dt ′
e −i2π(f+df)t′ − e −i2πft′
−2πit ′
(∫
G(f ′ )e −i2πf ′ ∞
t
−∞
∫ ∞
−∞
G(f ′ )e −i2πf ′t Φ f+df,f (f ′ )df ′
G(f ′ )e i2πf ′ (t ′ −t) df ′ dt ′
e −i2π(f+df)t′ − e −i2πft′
−2πit ′ e i2πf ′ t ′ dt ′ )
df ′
= G(f)e −i2πft df. (2.18)
A substitution of (2.18) into the the third term in (2.14) yields
∫ ∞
−∞
e i2πft E{x ∗ (t)dX(f)} =
∫ ∞
−∞
G(f)df. (2.19)
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