Wireless Network Design: Optimization Models and Solution ...

16 Dinesh Rajan
of SNR is given in Figure 2.4 for different values of M. Clearly, we can see that
for a given SNR the error rate is higher for larger M. Also, for a given M, the error
probability decreases with increasing SNR. In the presence of multiple users and
other interference sources, the probability of error would depend on the signal to
interference ratio (SIR) instead of just the SNR. The error performance can be further
decreased at a given SNR using error control codes as discussed in Section 2.4.
The probability of bit error can be evaluated from the probability of symbol error
as follows. Typically, bits are assigned to symbols using a process known as gray
coding in which neighboring symbols differ in only 1 bit as shown in Figure 2.3.
With such a bit assignment process, a symbol error will cause only a single bit error
with high probability. Hence the probability of bit error Pbit−error ≈ 1
log2 (M) Pe.
The transmitted signal in the above case is actually a signal in baseband. Typically,
this signal is modulated up to a pass-band frequency, fc before transmission.
This pass band signal could be obtained as s(t)e− j2π fct which is however, a complex
signal. To make the transmitted signal real, we can add its complex conjugate
to obtain sp(t) = s(t)e− j2π fct + s(t)e j2π fct .
Fig. 2.3 Euclidean distance based detection and error calculation for various modulation schemes.
The figure also demonstrates the concept of gray coding, in which neighboring symbols only differ
in 1 bit.