ECE 461
Fall 2006
Probability of Bit Error for M -ary Modulation
• Assuming that M = 2ν for some positive integer ν, we can map the symbols of any M -ary
signaling scheme to ν-bit vectors. To compare modulation schemes with different constellation
sizes, it is useful to plot the average bit error probability for M -ary modulation versus the bit
Eb
Es
SNR γb = N
= νN
.
0
0
• For M -ary orthogonal signaling, exploiting the perfect symmetry in the constellation, it is easy
to show that irrespective of how bits are assigned to symbols, we have
Pb =
2ν−1
2(log2 M −1)
Pe =
Pe
M −1
M −1
We can hence get Pb as a function of γb , based on expressions for Pe in terms of γs .
• For linear modulation, finding an exact expression for P b as a function of γb is difficult except
in special cases such as BPSK and QPSK. Furthermore, P b depends on how bits are assigned to
symbols (except of course in the special case of BPSK).
• Gray Coding. Assign bits to symbols so that nearest neighbors differ by exactly 1 bit. Gray
coding is possible for many linear modulation schemes (e.g., PSK, PAM), but not all (see HW 7).
• Nearest Neighbor Approximation (NNA) for P b . Let the symbol m be represented by the bit
vector bm = [b1,m · · · bν,m ]> , and define:
Ndmin (bm , i) = # NN’s of bm that differ from bm in the i-th bit position .
Then
and
s
P ({i-th bit position in error} | {m sent}) ≈ N dmin (bm , i) Q 
Pb,i

d2min (m) 
2N0

s
M
2 (m)
d
1 X
min
.
= P {i-th bit position in error} ≈
Ndmin (bm , i) Q 
M
2N0
m=1
Finally
Pb =
ν
1X
Pb,i .
ν
i=1
For Gray coded constellations, the NNA approximation for P b is at most equal to Pe /ν (why?).
c
V.V.
Veeravalli, 2006
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