mohatta2015.pdf

THE MATH 37
The correlation in (2.27) can be interpreted as a convolution of r(r) with g(t) =
h*(—T) evaluated at t = 0. As convolution is referred to as filtering and filter
response g(t) is matched to the signal waveform, we also refer to formation of z as
matched filtering.
Observe that 2.26 also includes the term S(0)\Sj\ 2 . For M-PSK modulation, \Sj\ 2
is 1 for all j. Thus, when searching for the Sj to maximizes the LLF, this term
can be omitted. As for S(i), we will see this again in Chapter 6 when developing
maximum likelihood sequence detection.
2.3.2 Output SNR and error rate performance
Let's examine output SNR. of the matched filter. First, consider BPSK, for which
Sj is either +1 or — 1. From (2.26), we see that only the real part of z would be
used. Substituting (2.22) into (2.27) and taking the real part gives the following
model
z r \= V^ s + n, (2.29)
where n is AWGN with variance N 0 /2. We've replaced E s with E b (energy-per-bit)
to emphasize that a symbol represents one bit. From this model, it is straightforward
to compute the output SNR as
SNR 0 = -^- = 2E b /N n , (2.30)
where subscript o emphasizes that it is output SNR.
The decision variable z r has two PDFs, shown in Fig. 2.3, depending on the value
of s. It can be shown that if both possibilities are equi-likely, then the probability
of bit error P b is minimized using the decision rule:
6 = sign(z r ). (2.31)
This is equivalent to using a detection threshold of 0, such that S = +1 if z > 0.
Note that P b is also referred to as modem bit error rate (BER). We will use the
terms interchangeably.
Without loss of generality, consider the case s = +1. The probability of error is
then
P b = Pr{z r