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20 www.rfdesign.com January 2002
B
inary phase shift keying (BPSK), in terms of
noise immunity per unit bandwidth, is one of the
most efficient binary data modulation techniques.
Yet, communications systems designers often
neglect this option because the design of a BPSK
demodulator is not as mathematically simple or
straightforward as frequency shift keying (FSK).
The prospect of having to apply thorough engineer-
ing rigor to the design of a BPSK demodulator can
be daunting. However, it is unlikely that any such
circuit will perform as well as it could if it were
implemented without fully understanding and
parameterizing its behavior.
Designing and implementing a Costas-loop carrier
recovery circuit and demodulator can be done simply
and inexpensively using only basic components.
BPSK Background
Simple BPSK modulation is the process of shift-
ing a carrier’s phase by 180° for one data symbol
while not shifting it for the other — known as
‘antipodal’ phase shift modulation. The mathemati-
cal equation for this process is:
(1)
where DATA
N
is restricted to ±1 and N is advanced
at a much lower rate than the frequency of the car-
rier (the cosine function). Shifting the phase of a
carrier (a sinusoid) by 180° is the same mathemati-
cal process as reversing the magnitude of a carrier
for one symbol and not the other. With identical
results, the following amplitude modulation process
can be substituted, interchangeably:
(2)
Modulation theory
The modulation techniques in Equation 1 and
Equation 2 are referred to as BPSK and double
side-band, suppressed carrier-amplitude modula-
tion (DSBSC-AM), respectively, and when the
phase shift is restricted to 180° between opposing
symbols, there is no difference.
As with the DSBSC-AM, the resultant BPSK RF
spectra are simply the baseband spectra mirrored by
the carrier frequency (see Figures 1, 2). The upper
sideband (the half of the BPSK spectra that exists
above the carrier) is identical to that of the modulat-
ing signal, except shifted up to where the carrier fre-
quency was the DC point in the spectra of the original
signal. The lower sideband (similarly, the part of the
modulated signal that exists below the carrier) con-
tains identical information to the upper sideband,
except its spectra is a mirror image of the carrier.
A mathematically simple demodulation scheme
multiplies the incoming RF signal by a coherent car-
rier (a carrier that is identical in frequency and
phase to the carrier that originally modulated the
BPSK signal). This is an application of the following
trigonometric identity:
(3)
where the product of two cosine functions is the sum
and the difference of the inner term of each. When
two cosine functions representing periodic time-
domain waveforms are multiplied together, the result
is two new cosines; the sum of the two frequencies
and the difference. Therefore, when the BPSK signal
is multiplied by a cosine function identical to the one
that modulated it, the original modulating data, plus
the same BPSK signal at twice the carrier frequency,
are produced. This is mathematically represented by:
(4)
BPSK t f t DATA t
ft ft
NcN
cc
(
)
•
(
)
=
(
)
•
(
)
•
(
)
cos
cos cos
2
22
π
ππ
cos cos cos cosab ab ab
(
)
•
(
)
=+
(
)
+−
(
)
1
2
BPSK t DATA t f tNNc
(
)
=
(
)
•
(
)
cos 2π
BPSK t f t DATA tNcN
(
)
=+
(
)
•
cos 2
2
π
π
Figure 1. Amplitude spectra of a typical binary data signal.
Practical
Costas loop
design
By Jeff Feigin
Designing a simple and
inexpensive BPSK Costas loop
carrier recovery circuit.
signal processing
aduo75
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