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Hani Mehrpouyan1,
Department of Electrical and Computer Engineering,
California State University, Bakersfield
Lecture 20 (Error Probability)February 20th, 2013
1 Some of the lectures notes here reproduced are taken from
course textbooks: Digital Communications: Fundamentals and
Applications B. Sklar. Communication Systems Engineering, J. G.
Proakis and MSalehi, and Lecture Notes for Digital Communication,
Queens University, Canada, S. Yousefi.
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Outline Error Rate for M-PSK
Differential Encoding and Differential PSK (DPSK):
Error Probability for Coherent PSK
Error Probability for DPSK
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Error Rate for M-PSK
For binary case (M = 2), we have already obtained both bit
andsymbol error rates (BPAM):
To extend the results for the M-ary case, we will consider
twogeneral methods of detection (demodulation) each with ownlevel
of complexity and performance:
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Error Rate for M-PSK Coherent or Phase-Coherent detection: This
method of detection
requires that the received signal r(t) and the
correlatingwaveforms in the demodulator ({n(t)}
Nn=1) be perfectly
synchronized with each other in time/carrier phase.
Non-coherent detection: otherwise.
In practice, due to propagation delays in the medium as well
asnon-idealities in the local oscillators in Tx and Rx, the phase
ofcos(2fct) (carrier phase) generated in the Rx might not belocked
to that of r(t) (i.e., phase of Tx oscillator).
There are a number of techniques to cause the carrier to
bephase-locked to the received signal.
One remedy is to use a Phase-Locked Loop (PLL). A PLL will
lockthe phase of the Rx to that of Tx. We do not discuss the
PLLtechniques/circuits in this course.
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Error Rate for M-PSK
Other alternative to the use of PLL circuits:transmit a replica
of the carrier signal with theinformation signal, e.g., in the AM
case (we add astrong carrier replica to the DSB-SC signal
toconstruct an AM signal).
This carrier component is referred to as the pilotsignal. The
pilot signal can be filtered from thereceived signal and be used
for coherentdemodulation.
What is the price to pay here?
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Differential Encoding and DifferentialPSK (DPSK):
Differential encoding is another method to combat ambiguitiesin
the phase of the received signal.
This is a modulation scheme with memory.
In PSK: we adopt a phase for a given block of information
(bits).
In DPSK: we alter the carrier phase with respect to the
previous(thus, differential) signaling interval (block of
information).
To show this very simple concept, let us compare QPSK
andDQPSK:
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DPSK
In QPSK
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DPSK In DQPSK:
Di-bit 00 from the source shift the phase by 0o
Di-bit 01 from the source shift the phase by 90o
Di-bit 11 from the source shift the phase by 180o
Di-bit 10 from the source shift the phase by 270o
Assume we have the binary sequence 1 1 0 1 0 1 1 0 0 0
totransmit.
As the schemes are both quaternary, we need to parse the
dateinto di-bits.
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DPSK
The transmitted signal s(t) is obtained by varying the phase of
acarrier according to the mappings discussed for QPSK and DQPSK
That is, theinformation isreally stored in thephase difference
oftwo successivesignaling intervals.
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DPSK Implication: system will not be sensitive to phase
ambiguities as
long they are stationary.
Consider a phase jitter of degrees stationary over a
fewsuccessive intervals. Then, if the perceived phases are:
Through differential detection:
which is the correct phase for the 2nd signaling interval.
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Error Probability for Coherent PSK
For the coherent case, all thearguments used in theintroduction
of demodulationand detection are valid:
The demodulator is perfectlyin synchronization with thereceived
signal and theobservation vector r has theGaussian PDF
presentedbefore.
The detector will implement
an appropriate detectionusing r (say MED, forequiprobable set in
AWGN).
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Error Probability for Coherent PSK For an equiprobable case, MED
involves simple Voronoi regions.
Considering the perpendicular bisectors of lines joining
theneighboring signal points in an M-PSK signal set, the
decisionregions turn out to be wedges with tips or apices at the
origin.
D1 is the decision region for s1(t): a wedge with a 2/M rad
angle.
Then, using the geometry of the constellation and
Voronoiregions, the error rate
will simplify to:
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Error Probability for DPSK
Coherent PSK naturally performs better than the simpler
DPSKvariant.
Comparison of BPSK and DBPSK: we do not discuss thederivations
of error probability for the differential PSK here. Forthe binary
case, the error rate is given by:
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