sm10_25

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Problem 10.25. Show that the noise variance of the in-phase component nI(t) of the 
band-pass noise is the same as the band-pass noise n(t) variance; that is, for a band-pass 
noise bandwidth BN
2
0( )I Nn t N B⎡ ⎤ =⎣ ⎦E 
 
Solution
Recall the spectra of narrowband noise n(t) and its in-phase component shown in 
Figure 8.23. The variance of a random process 
( )In t
( ) (0) ( )xx t R X f df
∞
−∞= = ∫ , where X(f) is 
the power spectral density of x(t). Therefore, 
 
2
0
0
Var[ ( )] ( )
( )
2 2
2
2
N
n t n t
S f df
N B
N B
∞
−∞
⎡ ⎤= ⎣ ⎦
=
= ⋅ ⋅
= ⋅
∫
E
 
 
Where we have used the fact that for a bandpass signal BT = 2B, that is twice the 
lowpass bandwidth. Similarly, the variance of the in-phase noise is 
 
2
0
Var[ ( )] ( )
( )
2
I
I I
n
n t n t
S f df
N B
∞
−∞
⎡ ⎤= ⎣ ⎦
=
= ⋅
∫
E
 
 
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