数字通信课件8

6
5-2-3 Probability of error for M-ary biorthogonal signals
❏ M-ary bi-orthogonal signals r [ n1 , n2 , , nM ], if s1 ( t ) is transmitted.
C '( r , s M / 2 ) [ n1 , n2 , , nM / 2 ][0, 0, ..., ] nM / 2 rM / 2
To minimize error probability under ML detection and equal prior for M-ary equalpower bi-orthogonal signals.
Hence , d ML ( r ) arg max M / 2 m M / 2 C '( r , sm ) m0 Since C '( r , sm ) C '( r , s m ) , u arg max1 m M / 2 C '( r , sm )
The decision is correct for transmitting s1 when r1 0 and r1 max 2 m M / 2 rm
P ( correct s1 ) P (r : r1 0and r1 max 2 m M / 2 rm s1 )
SNR

N0
, Pe , BFSK
d 12 SNR , d12 2 . 2N 0


– The larger the SNR, the smaller the probability of error. – The larger the distance among signals, 4 the smaller the probability of error.
1 ' 1 M M 1 (10 log10 ')dB 10 log dB (10 log10 )dB M M 1 save 10 log dB.( 3dB for M 2) M
Since the inter-symbol distances (as well as their relative positions) remain the same, the POE is the same as M-ary orthogonal signals.
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s1, s2 , ..., sM
is called the simplex signals.
5-2-4 Probability of error for simplex signals
Transmitted energy of simplex signals is reduced.
3
Review——Probability of error for binary
modulation
Binary antipodal signals (BPAM)
For equal prior, 0.
Pe , BFSK 1 1 - N 2 0 1 - 2 N 0 N 0
Digital Communications
Chapter 5: Optimum Receivers for the Additive White Gaussian Noise Channel (5-2-3~5-2-10)
1
Review——Viterbi algorithm
Compute two metrics for the two signal paths entering a node at each stage of the trellis search Remove the one with larger Euclidean distance The survivor path for each node is then extended to the next state. The number of paths searched reduced by a factor of two at each stage
2 k b )2 / 2
dx
The larger the M is, the better the system performance.
5
5-2-3 Probability of error for M-ary biorthogonal signals
❏ Summary for the POE study Binary bi-orthogonal performs better than binary orthogonal in bit-POE. (Require twice bit-SNR to achieve the same bit-POE) M-ary orthogonal performs better with larger M. There is a Shannon-Limit for minimum bit-SNR for M-ary orthogonal to achieves arbitrarily small bit-POE. ❏ How about M-ary bi-orthogonal? How good can it performs over M-ary orthogonal?

0
( x )dx P (( r2 , ..., rM ) : x r2 and ...and x rM / 2 r1 x , s1 ) Pr1 s 1
9
5-2-3 Probability of error for M-ary biorthogonal signals
2
1 2( x 2 )
M / 2 1
1 x2 / 2 e dx 2
where / N 0.
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5-2-3 Probability of error for M-ary bi-orthogonal signals 101
5
symbol-POE BPSK < symbol-POE QPSK – Symbol-POE comparison does not really tell the winner in performance. The Shannon-limit remains the same.
sm [ am1, am 2 , ..., amK ] For m=1,…M, its center is
1 s [ M 1 am 1 , M m 1
M
1 a m 2 , ..., M m 1
M
a
m 1
M
mK
].
Define new channel symbols as
sm s sm
C '( r , s2 ) [ n1 , n2 , , nM / 2 ][0, , ..., 0] n2 r2
C '( r , s1 ) [ n1 , n2 , , nM / 2 ][ , 0, ..., 0] ( n1 ) r1
C '( r , s1 ) [ n1 , n2 , , nM / 2 ][ , 0, ..., 0] ( n1 ) r1
C '( r , s2 ) [ n1 , n2 , , nM / 2 ][0, , ..., 0] n2 r2
7 C '( r , sM / 2 ) [ n1 , n2 , , nM / 2 ][0, 0, ..., ] nM / 2 rM /2
5-2-3 Probability of error for M-ary biorthogonal signals
Review——Probability of error for binary
modulation
M-ary equal-power orthogonal signals
2 k 1 Peb , MFSK k Pe , MFSK 2 1 1 2 k 1 ( x M -1 1 x e k 2 2 1
10 2 5
2
2
10 3 5
10 4 5
2
2
10 5 5
M Capacity limit (-1.6dB)
10
6
2
SNR per bit, b (dB)
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பைடு நூலகம்
5-2-4 Probability of error for simplex signals
❏ Simplex signals from orthogonal signals Given the vector representations of (orthogonal and equal-power) channel symbols
2
Review——Viterbi ML-decision
Optimal Viterbi ML-decision maker : to wait until there is only one possibility for the transmitted symbol. (The decision delay could be as large as the transmitted codeword length.) Suboptimal Viterbi ML-decision maker : Set a limit to the decision delay. = If there are more than one survivor paths (which results in more than one possible decoding results) remain for time period limited, just select the one with smaller metric, and forcefully remove the others.