34)
We can write down the PSD of the linear filter output as
(1.35)
Now let us apply the above observations to the case when the linear time-invariant filter is excited with
noise N(t) that has a Gaussian distribution and whose spectral density is white. The PSD is expressed as
(1.36)
SN ( f )
No
2
SY( f ) ZH( f ) Z2 SX ( f )
SX( f ) 3
`
`
RX(t) ej2pft dt
Y(t) 3
`
`
h(t)X(t t) dt
RX (t) E5x (t t)x (t)6
RX (t1, t2) E5x (t1)x (t2)6
erfc(x)
2 2p 3
`
x
et2 dt 1 erf(x)
F(x) 1
1
2
erfc cx mx
s22 d
erf(x)
2 2p 3
x
0
et2dt
F(x)
1
2 a1 erf cx mx
s22 db
WIRELESS TOPICS 37
h(t) X(t) Y(t)
FIGURE 1.48 Filter convolution example.
and shown graphically in Fig. 1.49.
To obtain the autocorrelation function, we simply use the inverse Fourier transform to get
(1.37)
where is the Dirac delta function whose value equals 1 only when t0. The autocorrelation function
is graphically expressed in Fig. 1.50.
'(t)
RN (t)
No
2
'(t)
38 CHAPTER ONE
No/2
f
SN( f )
FIGURE 1.49 Power spectral density of white noise.
No/2
t
RN(?„)
0
FIGURE 1.50 Autocorrelation function of white noise.
What this tells us is that if we take two different samples of this noise signal, they are uncorrelated
with each other.
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