mohatta2015.pdf

THE MATH 11
Figure 1.10
4-ASK with Gray mapping.
A commonly used root-Nyquist pulse shape is root-raised cosine. Its autocorrelation
function is given by
Rp{t)
(sm(nt/T)\( cos(ßnt/T) \
~ \ nt/T ) \\-{2βφγ) ■
(L21)
where β is the rolloff. The RRC waveform and its autocorrelation function are
shown in Fig. 1.11 for a rolloff of 0.22 (22% excess bandwidth).
1.3.2 Channel
The transmitted signal passes through a communications channel on the way to
the receive antenna, of a particular device. We can model this aspect of the channel
as a linear filter and characterize this filter by its impulse response. The actual,
physical channel may consist of hundreds of paths on a continuum of path delays.
Fortunately, for an arbitrary channel, the channel response can be modeled as a
finite-impulse-response (FIR) filter, using a tap-spacing that meets the Nyquist
sampling criterion (sampling rate at least twice the bandwidth) for the transmitted
signal (typically between 1 and 2 samples per symbol period). The accuracy of this
model depends on how many tap delays are used.
Regulatory bodies typically limit the amount of bandwidth a wireless signal
is allowed to occupy. Thus, the channel is bandlimited. Theoretically, for root-
Nyquist pulse shaping, the radio bandwidth must be at least as large as the symbol
rate (baud rate) (the baseband equivalent bandwidth is half the baud rate, giving a
Nyquist sampling period of one symbol period). Conversely, for a given bandwidth,
the symbol rate with root-Nyquist pulse shaping is limited to the radio bandwidth
or twice the baseband bandwidth. This limit in symbol rate is sometimes referred
to as the Nyquist rate.
However, in most systems, a slightly larger bandwidth is used, giving rise to the
notion of excess bandwidth. When excess bandwidth is low, it is reasonable to