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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

[

]

()

[]

[]

[]

[]

[]

[][ ]

[]

N

S

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

=

=)(

2

1E

and the SNR is given by

1

10

SNR R

RR

−

=

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