Review of basic ideas from digital communication - Signal Processing and Optimization for Transceiver Systems - page 27

Im

Re

Figure 2.14 . The 2-bit QAM constellation, also known as a QPSK constellation.

An important property of Gray codes is that, for reasonably high SNR, the
symbol error rate can be related to the bit error rate in a simple manner. Thus,
assume the SNR at the input of the detector is large enough, so that, when there
is a symbol error, the estimated symbol is an adjacent symbol (rather than a
symbol that is far away). In this case, only one bit is in error. Thus, in a symbol
stream with N symbols, if there are K symbol errors then there are K bit errors
as well. The symbol error rate (SER) is K/N , but since N symbols have Nb
bits, the bit error rate (BER) is K/Nb. Thus, for a Gray coded system with b
bits,

BER = SER
b

. (2 . 30)

From Eq. (2.25) we know that the symbol error probabilities for the PAM and
QAM systems (for fixed SNR) are related approximately by

P e,QAM ( b ) ≈ 2 P e,P AM ( b/ 2) . (2 . 31)

Using this, a simple relation between the bit error rates can be derived. Thus,
dividing both sides of the preceding equation by b we get

P e,QAM ( b )
b

2 P e,P AM ( b/ 2)
b

= P e,P AM ( b/ 2)
b/ 2

≈ ,

which shows that the BER for b-bit QAM is identical to BER for b/ 2 bit PAM:

BER QAM ( b ) ≈ BER PAM ( b/ 2) . (2 . 32)

Thus, for a given error rate, the QAM system can transmit twice as many bits.
However, QAM also requires twice as much bandwidth compared to PAM, as
we shall see in Sec. 2.4.3.