sm9_1

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Problem 9.1 In practice, we often cannot measure the signal by itself but must measure 
the signal plus noise. Explain how the SNR would be calculated in this case. 
 
Solution 
Let r(t) = s(t) + n(t) be the received signal plus noise. Assuming the signal is independent 
of the noise, we have that the received power is 
 [ ]
( )[ ][ ] [ ] [ ][ ] [ ] [ ] [ ]
NS
tntntsts
tntntsts
tnts
trR
++=
++=
++=
+=
=
0
)()()(2)(
)()()(2)(
)()(
)(
22
22
2
2
0
EEEE
EEE
E
E
 
 
where S is the signal power and N is the average noise power. We then measure the noise 
alone 
 [ ]
N
tnR
=
= )(21 E 
 
and the SNR is given by 
 
1
10SNR
R
RR −= 
 
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