mohatta2015.pdf

THE MATH 41
Thus, in the complex plane, we want to find the possible symbol value such that
ASj is closest, in Euclidean distance, to z. In general, we will denote the operation
in (2.50) as
s = detect(z, A), (2.51)
where
detect(«, A) = arg min \z - ASj\ 2 . (2.52)
Thus, the detect function has two inputs: the decision variable z and the amplitude
reference A.
For BPSK, S = { + 1, -1}, and (2.50) simplifies to
s = sign(Re{yTz}). (2.53)
Multiplication by A* can be omitted if A is purely real and positive. For QPSK,
the I bit can be detected using (2.53) and the Q bit is detected using
For Gray-coded QAM, s can be expressed as
s = sign(Im{.4*z}). (2.54)
s = s¡+js Q , (2.55)
where s¡ and SQ are \/M-ary ASK symbols. These can be detected separately
using the real and imaginary parts of A*z. Using the complex-variable definition
of output SINR. gives
\h(t)\ 2 dt. (2.56)
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With QAM and multiple bits per ASK symbol, ML symbol detection uses thresholds
that are different than minimum-BER bit detection [Sim05]. However, the
differences are only significant at low SNR, where QAM is typically not used.
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2.3.5 Partial MF
In practical receiver designs, it is convenient to work with a set of digital signal
samples, rather than the continuous-valued, continuous-time waveform r(i). Such
digital samples can be easily stored in a digital memory devices and processed using
digital signal processing devices. While it is often not necessary to model the effects
of digitizing the sample values, it is important to consider how often the received
signal is sampled and what filtering occurs prior to sampling.
In the remaining chapters, we will focus primarily on equalization designs operating
on the sampled output of a front-end filter matched to the pulse shape (TDM)
or chip pulse shape (CDM, OFDM). Such a front-end filter is often used in practice
for a number of reasons.
1. It reduces the bandwidth of the signal, reducing the sampling rate needed to
meet the Nyquist criterion.
2. It suppresses signals in adjacent frequency bands, sometimes called blocking
signals, reducing the number of bits per sample needed.