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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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