15.6 Power Spectral Density
539
15.6 Power Spectral Density
For a stationary ergodic process, the temporal mean square is
F
2
(t)
= lim
T →∞
1
T
T /2
−T /2
F(t)F(t + τ )dt
(15.26)
Writing ω = 2π f , Eqs. (3.124) and (3.125) are written as
F(t) =
∞
−∞
F(t)e
i2π f t d f
(15.27)
and
F( f ) =
∞
−∞
F(t)e
−i2π f t dt
(15.28)
Substituting one F(t) of the integral of Eq. (15.26) by Eq. (15.27), we get
F
2
(t)
= lim
T →∞
1
T
T /2
−T /2
F(t)
∞
−∞
F( f )e
i2π f t d f dt
and
= lim
T →∞
∞
−∞
F( f )
1
T
⎡
⎢
⎣
T /2
−T /2
F(t)e
i2π f t dt
⎤
⎥
⎦d f
(15.29)
A comparison of the integral within the parenthesis of Eqs. (15.29)–(15.28) reveals
that the integral is the complex conjugate of F( f ). We shall indicate it as F ∗ ( f ).
Therefore, Eq. (15.29) becomes
F
2
(t)
= lim
T →∞
∞
−∞
1
T
F( f )F ∗ ( f )d f
= lim
T →∞
∞
−∞
1
T
|F( f )|
2 d f
or