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Performance of Communication Systems Corrupted by Noise Chap. 7
The probability of symbol error (also called the word error rate, WER) is not easily
related to the BER. However, bounds on the relationship between the BER and the WER are
[Couch, 1993]
1
K P(E) … P e … (M>2)
M-1 P(E)
where P e is the BER, P(E) is the WER, and M = 2 K. When errors are made under the usual
operating conditions of low error rate (say P e 6 10 -3 ), the error symbol selected is usually the
“nearest neighbor” to the correct symbol on the signal constellation. This results in a BER
near the lower bound. For this case, the BER is almost equal to the lowest bound if the bit-tosymbol
mapping is a Gray code (see Table 3–1) since there is only a one bit change (error) for
the nearest neighbor symbol. For example, if M = 128 (K = 7), then
0.143 P(E) … P e … 0.504 P(E)
Under the usual operating conditions of low BER, say P e 6 10 -3 , the BER would be near the
lower bound, so P e ≈ 0.143 P(E) for M = 128.
Synchronization
As we have seen, three levels of synchronization are needed in digital communication
systems:
1. Bit synchronization.
2. Frame, or word, synchronization.
3. Carrier synchronization.
Bit and frame synchronizations were discussed in Chapter 3.
Carrier synchronization is required in receivers that use coherent detection. If the spectrum
of the digital signal has a discrete line at the carrier frequency, such as in equally likely
OOK signaling, a PLL can be used to recover the carrier reference from the received signal.
This was described in Chapter 4 and shown in Fig. 4–24. In BPSK signaling, there is no discrete
carrier term, but the spectrum is symmetrical about the carrier frequency. Consequently,
a Costas loop or a squaring loop may be used for carrier sync recovery. These loops were
illustrated in Figs. 5–3 and P5–60. As indicated in Sec. 5–4, these loops may lock up with a
180° phase error, which must be resolved to ensure that the recovered data will not be complemented.
For QPSK signals, the carrier reference may be obtained by a more generalized
Costas loop or, equivalently, by a fourth-power loop [Spilker, 1977]. These loops have a fourphase
ambiguity of 0, ;90, or 180°, which must also be resolved to obtain correctly demodulated
data. These facts illustrate once again why noncoherent reception techniques (which can
be used for OOK, FSK, and DPSK) are so popular.
Bit synchronizers are needed at the receiver to provide the bit sync signal for clocking
the sample-and-hold circuit and, if used, the matched-filter circuit. The bit synchronizer was
illustrated in Fig. 3–20.
All the BER formulas that we have obtained assume that noise-free bit sync and carrier
sync (for coherent detection) are available at the receiver. Of course, if these sync signals are