通信原理(英文版)8[40页]

T
0
n 2 (t )dt
2

k dimensional combined probability density function of the received voltage r(t) = s(t) + n(t) When the symbol 0 is transmitted
f 0 (r )
f ( n)
where n f 0 H
2 n

2 n

k
exp n0

0
n 2 (t )dt
f (n) f k (n1 , n2 , , nk ) f (n1 ) f (n2 ) f (nk )

Note that f (n) is not a function of time. n is a k dimensional quantity, which can be regarded as a point on the k dimensional space. f (n) is decided only by the noise energy


The meaning of “optimum” － minimum error probability Decision rule for optimum receiving Received vector r may be regarded as a point in a k dimensional space. The k dimensional space can be partitioned into two regions:A0 and A1
f (ni ) ni2 exp 2 2 2 n n 1

where n － standard deviation of the noise n2 － variance of the noise
1

k dimensional combined probability density function of noise sample voltage
Chapter 8 Optimum Receiving of Digital Signal
8.1 Statistical Characteristics of Digital Signal
Assume the highest transmission frequency of the communication system is fH , and the received voltage is expressed by its sample. One dimensional probability density of the noise sample voltage If the voltage is sampled at the sampling rate 2fH in a symbol interval; then k samples are obtained: n1, n2, …, ni, …, nk , and each sample is a random variable with normal distribution. Its one dimensional probability density can be written as
f1 ( r )

1 2 n

k
1 exp n0
0 r (t ) s1 (t ) dt
T 2

where s1 (t) － signal waveform of the transmitted symbol 1
3
8.2 Optimum Receiving Criterion of Digital Signal
f k (n1 , n2 , , nk ) f (n1 ) f (n2 ) f (nk )


1 2 n

k
1 exp 2 2 n
ቤተ መጻሕፍቲ ባይዱ
ni2 i 1
k
Average power of the received noise in a symbol duration T: k k 1 k 2 1 1 T 2 1 2 ni 2 f T ni or T 0 n (t )dt 2 f T ni2 k i 1 i 1 H i 1 H Substituting the above equation into the uppermost equation, we obtain 1 T 1

1 2 n

k
1 exp n0
0 r (t ) s0 (t ) dt
T 2

where r (t) － sum of the signal and noise voltages s0 (t) － signal waveform of the transmitted symbol 0 When the symbol 1 is transmitted
A0 A
0
A1
A1

Decision rule: If the received vector r falls into the region A0, then the decision is symbol o transmitted; If the received vector r falls into the region A1, then the decision is symbol o transmitted;
4

Total symbol error probability:
Pe P(1) P( A0 / 1) P(0) P( A1 / 0)
where
P( A0 / 1) f1 (r )dr － conditional probability of
A0
the vector r fallen in region A0 when 1 is transmitted