Cryptography Reference
In-Depth Information
If all the s j ( t ) signals are transmitted with the same probability ( π j =1 /M ) ,
the term C j reduces to
E j / 2 . In addition, if all the s j ( t ) signals have in
addition the same energy E j = E (the case of phase or frequency shift keying
modulation), then the MAP criterion is simplified and becomes:
T
T
s j ( t ) if
r ( t ) s j ( t ) dt >
r ( t ) s n ( t ) dt
n
= jn =1 , 2 ,
···
,M
(2.61)
0
0
2.2.2 Performance of the coherent receiver
Amplitude shift keying with M states
For an M-ASK modulation, the s j ( t ) signals are of the form:
s j ( t )= A j h ( t )cos(2 πf 0 t + ϕ 0 )
with:
A j =(2 j
1
M ) Aj =1 , 2 ,
···
,M
They define a space of dimension N =1 and thus observation R at the output
of the demodulator reduces to a component r .
T
r =
r ( t ) ν ( t ) dt
0
with ν ( t )= T cos(2 πf 0 t + ϕ 0 ) .
For a continuous data stream, the estimation of the symbols a i is done by
integrating the product r ( t ) ν ( t ) on each time interval [ iT, ( i +1) T [ .Ifamatched
filter is used rather than an integrator, the sampling at the output of the filter
is realized at time ( i +1) T .
On the interval [0 ,T [ , assuming iid information data d i , all the amplitude
states have the same probability and decision rule (2.59) leads to:
1
2 s j >rs n
1
2 s n
A j if rs j
n
= j
(2.62)
where A j represents the amplitude estimated with:
A j cos(2 πf 0 t + ϕ 0 ) ν ( t ) dt = A j T
2
T
1
T
s j =
if f 0 >>
(2.63)
0
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