Page 330 - Satellite Communications, Fourth Edition

310 Chapter Ten

With QPSK, the signal can be represented by the formulas given in
Table 10.1, which may be written generally as

n
e(t) 22 cosa t b (10.27)
0
4
Quadrupling this, followed by some trigonometric simplification,
results in

4
e (t) 3 2 cos 2a 0 t n b 1 cos 4a 0 t n b (10.28)
2 4 2 4
The last term on the right-hand side is selected by the bandpass filter
and is

1 n 1
cos 4a 0 t b cos(4 0 t n ) (10.29)
2 4 2

This is seen to consist of the fourth harmonic of the carrier, including
a constant-phase term that can be ignored. The fourth harmonic is
selected by the bandpass filter, and the operation of the circuit proceeds
in a similar manner to that for the BPSK signal.
Frequency multiplication can be avoided by use of a method known
as the Costas loop. Details of this, along with an analysis of the effects
of noise on the squaring loop and the Costas loop methods, will be
found in Gagliardi (1991). Other methods are also described in detail
in Franks (1980).

10.8 Bit Timing Recovery
Accurate bit timing is needed at the receiver in order to be able to sample
the received waveform at the optimal points. In the most common
arrangements, the clocking signal is recovered from the demodulated
waveform, these being known as self-clocking or self-synchronizing systems.
Where the waveform has a high density of zero crossings, a zero-crossing
detector can be used to recover the clocking signal. In practice, the received
waveform is often badly distorted by the frequency response of the trans-
mission link and by noise, and the design of the bit timing recovery circuit
is quite complicated. In most instances, the spectrum of the received wave-
form will not contain a discrete component at the clock frequency. However,
it can be shown that a periodic component at the clocking frequency is
present in the squared waveform for digital signals (unless the received
pulses are exactly rectangular, in which case squaring simply produces a
dc level for a binary waveform). A commonly used baseband scheme is