26) p(x)
1
s22p
e
(xmx)2
2s2
Var5g6 E5g26 [E5g6]2
s2g
3
`
`
(g mg)2 p(g) dg
Var5g6 E5[g E5g6]26
E5g y6 E5g6 E5y6 E5Kg6 K # E5g6
E5X6 mX 3
`
`
xP(x) dx
F(x) P(X x) 3
x
`
p(v) dv
3
`
`
psx) dx 1
psx)
d
dx
F(x) (` x `)
F(`) 1
F(`) 0 0 F(x) 1
F(x) PsX x) (` x `)
36 CHAPTER ONE
The CDF is defined as
(1.27)
where erf(x) is the error function and is defined as
(1.28)
The CDF can also be defined in terms of the complementary error function given below:
(1.29)
where the erfc(x) function is defined below along with its relationship to the earlier defined error
function.
(1.30)
1.4.2 Autocorrelation and Power Spectral Density (PSD)
Let us define the autocorrelation of a random process X(t) to be a function of two time instances, t1 and t2.
(1.31)
which can also be expressed, assuming t2 t1.
(1.32)
Let us consider the effects of a linear filter on a random
process. In Fig. 1.48, the random variable X(t) is input to the
filter with impulse response h(t).
The filter output is easily written in the time domain by convolving
the impulse response of the filter with the input signal.
(1.33)
Using the frequency domain, we can simply write down the power spectral density (PSD) of the input
as (units are Watts/Hz)
(1.
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