B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

566 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
The integral appearing on the right-hand side of Eq. (10.121) is computed and plotted in
Fig. 10.28 (P e M vs. Eb/N). This plot shows an interesting behavior for the case of M = oo.
By properly taking the limit of P e M in Eq. (10.121) as M --+ oo, it can be shown that 5
lim P e M = { o l
M-+oo
Eb/N < loge 2
Eb/N loge 2
Because the signal power S; = EhR b , where Rh is the bit rate, it follows that for error-free
communication,
Eb 1
N - e
- > log 2 = -- or
1.44
Si 1
-- >--
NR b
- 1.44
(10.122)
Hence,
This shows that M-ary orthogonal signaling can transmit error-free data at a rate of up to
1.44 Si/N bit/s as M --+ oo (see Fig. 10.28).
Bit Error Rate (BER) of Orthogonal Signaling
For PAM and MPSK, we have shown that, by applying the Gray code, P b = P e M / 1og2 M.
This result is not valid for MFSK because the errors that predominate in PAM and MPSK, are
those in which a symbol is mistaken for its immediate neighbor. We can use the Gray code to
assign the adjacent symbols codes that differ in just one digit. In MFSK, on the other hand, a
symbol is equally likely to be mistaken for any of the remaining M - I symbols. Hence, P(E),
the probability of mistaking one particular M-ary symbol for another, is equally likely,
P(E) =
P e M
M - I
=
PeM
2 k - I
If an M-ary symbol differs by 1 bit from N1 number of symbols, and differs by 2 bits from N2
number of symbols, and so on, then N E , the average number of bits in error in reception of an
M-ary symbol, is
k
N E
=
L nN nP(E)
n=l
k
L
n=l
P e M
- nNn --
2
k - 1
= n(k)
2 k - I L.., n
n=l
This is an average number of bits in error in a sequence of k bits (one M-ary symbol).
Consequently, the BER, P b , is this figure divided by k,
k » I