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Random Processes and Spectral Analysis Chap. 6
The two processes are said to be uncorrelated if
for all t. They are said to be orthogonal if
for all t.
Let y(t) be the output process of a linear system and x(t) the input process, where
y(t) = x(t) * h(t). Then
and
R xy (t) = [x(t)] [y(t)]
R xy (t) = 0
R y (t) = h(-t) * h(t) * R x (t)
y (f) = ƒH(f) 2 x (f)
where H(f) = F[h(t)].
The equivalent bandwidth of a linear system is defined by
B =
1
2
ƒ H(f 0 ) ƒ L0
q
ƒ H(f)ƒ 2 df
where H( f ) is the transfer function of the system and f 0 is usually taken to be the frequency
where |H( f )| is a maximum. Similarly, the equivalent bandwidth of a random process x(t) is
B =
1
x (f 0 ) L0
q
x (f) df =
If the input to a linear system is a Gaussian process, the output is another Gaussian
process.
A real stationary bandpass random process can be represented by
v(t) = Re{g(t)e j(v ct+u c ) }
where the complex envelope g(t) is related to the quadrature processes x(t) and y(t). Numerous
properties of these random processes can be obtained and are listed in Sec. 6–7. For example,
x(t) and y(t) are independent Gaussian processes when the PSD of v(t) is symmetrical about
f = f c , f 7 0, and v(t) is Gaussian. Properties for SSB bandpass processes are also obtained.
The matched filter is a linear filter that maximizes the instantaneous output signal
power to the average output noise power for a given input signal waveshape. For the case of
white noise, the impulse response of the matched filter is
h(t) = Cs(t 0 - t)
R x (0)
2 x (f 0 )
where s(t) is the known signal waveshape, C is a real constant, and t 0 is the time that the output
signal power is a maximum. The matched filter can be realized in many forms, such as the
integrate-and-dump, the correlator, and the transversal filter configurations.