sm9_22

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Problem 9.22. Assume that, in the DSB-SC demodulator of Fig. 9.6, there is a phase 
error φ in the synchronized oscillator such that its output is cos(2 )cf tπ φ+ . Find an 
expression for the coherent detector output and show that the post-detection SNR is 
reduced by the factor 2cos φ . 
 
Solution 
The signal at the input to the coherent detector of Fig. 9.6 is x(t) where 
 
( ) ( ) ( ) cos(2 ) ( )sin(2 )
( ) cos(2 ) ( )cos(2 ) ( )sin(2 )
I c Q c
c c I c Q
x t s t n t f t n t f t
cA m t f t n t f t n t f t
π π
π π
= + −
= + − π 
 
The output of mixer2 in Fig. 9.6 is 
 
[ ]
[ ] [ ]
( ) ( ) cos(2 )
( ) ( ) cos(2 )cos(2 ) ( )sin(2 )cos(2 )
1 1 1 1( ) ( ) cos ( )sin ( ) ( ) cos(4 ) ( )sin(4 )
2 2 2 2
c
c I c c Q c c
c I Q c I c Q c
v t x t f t
A m t n t f t f t n t f t f t
A m t n t n t A m t n t f t n t f t
π φ
π π φ π π φ
φ φ π φ
= +
= + + − +
= + + + + + − π φ+
 
With the higher frequency components will be eliminated by the low pass filter, the 
received message at the output of the low-pass filter is 
 
1 1 1( ) ( ) cos ( ) cos ( )sin
2 2 2c I Q
y t A m t n t n tφ φ φ= + + 
 
To compute the post-detection SNR we note that the average output message power in 
this last expression is 
 
2 21 cos
4 c
A P φ 
 
and the average output noise power is 
 
2 2
0 0
1 1 12 cos 2 sin 2
4 4 4
N W N W N Wφ φ⋅ + ⋅ = ⋅ 0 
 
where [ ] [ ] WNtntn QI 022 )()( == EE . Consequently, the post-detection SNR is 
 
2 2 2 2
0 0
1/ 4 cos cosSNR
1/ 4 2 2
c cA P A P
N W N W
φ φ= =⋅ 
 
Compared with (9.23), the above post-detection SNR is reduced by a factor of 2cos φ . 
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