sm9_20
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sm9_20


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Problem 9.20. Assume that the narrowband noise n(t) is Gaussian and its power spectral 
density SN(f) is symmetric about the midband frequency fc. Show that the in-phase and 
quadrature components of n(t) are statistically independent. 
 
Solution 
The narrowband noise n(t) can be expressed as: 
 
2
( ) ( ) cos(2 ) ( )sin(2 )
Re ( ) c
I c Q
j f t
n t n t f t n t f t
z t e \u3c0
c\u3c0 \u3c0= \u2212
\u23a1 \u23a4= \u23a3 \u23a6
, 
 
where nI(t) and nQ(t) are in-phase and quadrature components of n(t), respectively. The 
term z(t) is called the complex envelope of n(t). The noise n(t) has the power spectral 
density SN(f) that may be represented as shown below 
 
 
 
We shall denote ( )nnR \u3c4 , ( )I In nR \u3c4 and ( )Q Qn nR \u3c4 as autocorrelation functions of n(t), nI(t) 
and nQ(t), respectively. Then 
 [ ]
{ }
( ) ( ) ( )
( ) cos(2 ) ( )sin(2 ) ( ) cos(2 ( )) ( )sin(2 ( ))
1 1( ) ( ) cos(2 ) ( ) ( ) cos(2 (2 ))
2 2
1 ( ) ( )
2
I I Q Q I I Q Q
Q I I Q
nn
I c Q c I c Q c
n n n n c n n n n c
n n n n
R n t n t
n t f t n t f t n t f t n t f t
R R f R R f t
R R
\u3c4 \u3c4
\u3c0 \u3c0 \u3c4 \u3c0 \u3c4 \u3c4 \u3c0
\u3c4 \u3c4 \u3c0 \u3c4 \u3c4 \u3c4 \u3c0 \u3c4
\u3c4 \u3c4
= +
\u23a1 \u23a4 \u23a1= \u2212 \u22c5 + + \u2212 +\u23a3 \u23a6 \u23a3
\u23a1 \u23a4 \u23a1 \u23a4= + + \u2212 +\u23a3 \u23a6 \u23a3 \u23a6
\u23a1 \u23a4\u2212 \u2212\u23a3 \u23a6
E
E \u3c4 \u23a4+ \u23a6
1sin(2 ) ( ) ( ) sin(2 (2 ))
2 Q I I Qc n n n n c
f R R f t\u3c0 \u3c4 \u3c4 \u3c4 \u3c0 \u3c4\u23a1 \u23a4\u2212 + +\u23a3 \u23a6
 
Since n(t) is stationary, the right-hand side of the above equation must be independent of 
t, this implies 
( ) ( )
I I Q Qn n n n
R R\u3c4 \u3c4= (1) 
( ) ( )
I Q Q In n n n
R R\u3c4 \u3c4= \u2212 (2) 
 
 
Continued on next slide 
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to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that 
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Problem 9.20 continued 
Substituting the above two equations into the expression for Rnn(\u3c4), we have 
 
( ) ( )cos(2 ) ( )sin(2 )
I I Q Inn n n c n n c
R R f R f\u3c4 \u3c4 \u3c0 \u3c4 \u3c4 \u3c0= \u2212 \u3c4 (3) 
 
The autocorrelation function of the complex envelope ( ) ( ) ( )I Qz t n t jn t= + is 
 
 
*( ) ( ) ( )
2 ( ) 2 (
I I Q I
zz
n n n n
R E z t z t
R j R
\u3c4 \u3c4
)\u3c4 \u3c4
\u23a1 \u23a4= +\u23a3 \u23a6
= + (4) 
 
From the bandpass to low-pass transformation of Section 3.8, the spectrum of the 
complex envelope z is given bye 
 
( )
( )
0 oth
N c
Z
S f f f f
S f
\u23a7 + > \u2212\u23aa= \u23a8\u23aa\u23a9 erwise
c 
 
Since SN(f) is symmetric about fc, SZ(f) is symmetric about f = 0. Consequently, the 
inverse Fourier transform of SZ(f) = Rzz(\u3c4) must be real. Since ( )zzR \u3c4 is real valued, based 
on Eq. (4), we have 
 
( ) 0
Q In n
R \u3c4 = , 
 
which means the in-phase and quadrature components of n(t) are uncorrelated. Since the 
in-phase and quadrature components are also Gaussian, this implies that they are also 
statistically independent. 
 
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.