sm9_21

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Problem 9.21. Suppose that the receiver bandpass-filter magnitude response ( )BPH f 
has symmetry about cf± and noise bandwidth BT. From the properties of narrowband 
noise described in Section 8.11, what is the spectral density SN(f) of the in-phase and 
quadrature components of the narrowband noise n(t) at the output of the filter? Show that 
the autocorrelation of n(t) is 
 
( ) ( ) cos(2 )N cR fτ ρ τ π τ= 
 
where [ ]1( ) ( )NS fρ τ −= F ; justify the approximation ( ) 1ρ τ ≈ for TB/1<τ . 
 
Solution 
Let the noise spectral density of the bandpass process be SH(f) then 
 
20( ) ( )
2H BP
NS f H f= 
From Section 8.11, the power spectral densities of the in-phase and quadrature 
components are given by 
 
( ) ( ),
( )
0, otherwise
H c H c T
N
S f f S f f f B
S f
⎧ − + + ≤⎪= ⎨⎪⎩
/ 2
. 
 
Since the spectrum SH(f) is symmetric about fc, , the spectral density of the in-phase and 
quadrature components is 
 
2
0( )( )
0 ot
BP c T
N
H f f N f BS f
⎧ − <⎪= ⎨⎪⎩
/ 2
herwise
c
 (1) 
 
Note that if |HBP(f)| is symmetric about fc then |HBP(f-fc)| will be symmetric about 0. 
Consequently, the power spectral densities of the in-phase and quadrature components 
are symmetric about the origin. This implies that the corresponding autocorrelation 
functions are real valued (since they are related by the inverse Fourier transform). In 
Problem 9.20, we shown that if the autocorrelation function of the in-phase component is 
real valued then autocorrelation of n(t) is ( ) ( ) cos(2 )
I IN n n
R R fτ τ π τ= . If we denote 
 
[ ] 21 1( ) ( ) ( ) ( )
I In n N O BP c
R S f N H fρ τ τ − − f⎡ ⎤= = = −⎣ ⎦F F 
 
 
 
Continued on next slide 
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Problem 9.21 continued 
 
then the autocorrelation of the bandpass noise is 
 ( )tfR cN πτρτ 2cos)()( = 
 
For 1/ TBτ � (there is a typo in the text), we have 
 
( )
( )
0
( ) ( ) exp 2
( ) cos 2
N
N
S f j f df
S f f df
ρ τ π τ
π τ
∞
−∞
∞
= −
=
∫
∫
 
 
due to the real even-symmetric nature of SN(f). If the signal has noise bandwidth BT then 
 
 
( )
( )
0
0
0
( ) ( ) cos 2
( )cos 0
( )
a constant
T
T
T
B
N
B
N
B
N
S f f df
S f df
S f df
ρ τ π τ≈
≈
=
=
∫
∫
∫
 
 
where the second line follows from the assumption that 1/ TBτ � . With suitable scaling 
the constant can be set to one. 
 
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only 
to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that 
permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.