mohatta2015.pdf
signal processing from power amplifier operation control point of view
signal processing from power amplifier operation control point of view
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THE MATH 9<br />
where u(t) = u r (t) + j«i(i) is the complex envelope of the radio signal. We can<br />
model the system at the complex envelope level, referred to as complex baseband,<br />
rather than having to include the carrier frequency term.<br />
We will assume the receiver radio extracts the complex envelope from the received<br />
signal. For example, the real part of the complex envelope can be obtained by<br />
multiplying by y/2 cos(2nf c t) and using a baseband filter that passes the signal.<br />
Mathematically,<br />
y r (t) = x(t)V2cos{2nf c t) = u r (t)2cos 2 (2TT/ C Í) - Ui(t)2sm(2nf c t)cos(2nf c t).<br />
(1.17)<br />
Using the fact that cos 2 (^4) = 0.5(1 + cos(2A)), we obtain<br />
y r (t) = u r (t) + u r {t) cos{2n2f c t) - Ui(t)2sm(2nf c t)cos{2nf c t) (1.18)<br />
A filter can be used to eliminate the second and third terms on the right-hand side<br />
(r.h.s.). Similarly, the imaginary part of the complex envelope can be obtained by<br />
multiplying by \/2sin(27r/ c i) and using a baseband filter that passes the signal.<br />
Notice that we have switched to a continuous time waveform u(t). Thus, when<br />
we send symbols one after another, we have to explain how we transition from<br />
one symbol to the next. We will see that each discrete symbol has a pulse shape<br />
associated with it, which explains how the symbol gets started and finishes up in<br />
time.<br />
1.3.1 Transmitter<br />
At the transmitter, modem symbols are transmitted sequentially as<br />
where<br />
oo<br />
x{t) = y r Ë~ s Σ s(m)p(t - mT), (1.19)<br />
m=—oo<br />
• E s is the average received energy per symbol,<br />
• s(m) is the complex (modem) symbol transmitted during symbol period m,<br />
and<br />
• p(t) is the symbol waveform or pulse shape (usually purely real).<br />
The symbols are normalized so that E{|s(m)| 2 } = 1, where E{·} denotes expected<br />
value. 2 The pulse shape is also normalized so that J_ \p(t)\ 2 dt = 1.<br />
In (119) we have assumed a continuous (infinite) stream of symbols. In practice,<br />
a block of N s symbols is usually transmitted as a packet. Usually N s is sufficiently<br />
large that the infinite model is reasonable for most symbols in the block. Theoretically,<br />
symbols on the edge of the block should be treated differently. However, in<br />
most cases, it is reasonable (and simpler) to treat all the symbols the same.<br />
In general, a symbol can be one of M possible values, drawn from the set S =<br />
{Sj\j = 1...M}. These M possible complex symbol values can have different<br />
2 In this case, expectation is taken over all possible symbol values.