optimizing psk for correlated data

Optimizing PSK for Correlated Data

Blake BorgesonRice UniversityClemson SURE Project

Advised by Dr. Carl BaumClemson University

Basic Road Map

Background Ideas Correlated data transmission Phase Shift Keying (PSK)

Altering the receiver Altering the transmitter Conclusions, directions

Basic Road Map

Background Ideas Correlated data transmission Phase Shift Keying (PSK)

Altering the receiver Altering the transmitter Conclusions, directions

Correlated Data--Introduction

Goal: transmit, receive correlated data Markov state machine: models real data

Yields desired correlation values, e.g.,

qp

qpRxx

)1()1(

Correlated Data—Example

Analysis in MATLAB:

p=0.03, q=0.59

“Mr. PSK”

Phase Shift Keying (PSK)

M-ary PSK:

Optimum receiver correlates with sine and cosine:

)2

2cos()( mEMtcfAts

xT

yT

Decision

algorithm 321ˆˆˆ bbb

Received bits

s(t)+n(t)

T

dt0

)(

T

dt0

)(

)2sin(1

tfc

s

)2cos(1

tfc

s

PSK Representation

Traditional transmitter: evenly spaced points on the circle

Traditional receiver: corresponding equal pie wedges

Basic Road Map

Background Ideas Correlated data transmission Phase Shift Keying (PSK)

Altering the receiver Altering the transmitter Conclusions, directions

Altering the Receiver: MAP

MAP, maximum a posteriori probability: choose sm to maximize probability that sm was transmitted, given received r, i.e.,

Other gains: take into account previous bit, next bit, or both

)|(.maxargˆ rsPs ms

mm

p = q = 0.001

Gains from Altering Receiver

10-6

10-5

10-4

10-3

10-2

10-1

0 5 10 15

8-PSK Receivers, p=.01, q=.01

Traditional rcvr

Pe: p

rob.

of b

it er

ror

Eb/N

0 (dB)

Traditional receiver never gains

Gains from Altering Receiver

10-6

10-5

10-4

10-3

10-2

10-1

0 5 10 15

8-PSK Receivers, p=.01, q=.01

Traditional rcvr

MAP receiver

Pe: p

rob.

of b

it er

ror

Eb/N

0 (dB)

MAP algorithm:

prior probabilities

Gains from Altering Receiver

10-6

10-5

10-4

10-3

10-2

10-1

0 5 10 15

8-PSK Receivers, p=.01, q=.01

Traditional rcvr

MAP receiver

MAP, Prev. bit

Pe: p

rob.

of b

it er

ror

Eb/N

0 (dB)

Algorithm:

prior probabilities plus guess of preceding (previous) bit

Gains from Altering Receiver

10-6

10-5

10-4

10-3

10-2

10-1

0 5 10 15

8-PSK Receivers, p=.01, q=.01

Traditional rcvr

MAP receiver

MAP, Prev. bit

MAP, Next bit

Pe: p

rob.

of b

it er

ror

Eb/N

0 (dB)

Algorithm:

prior probabilities plus guess of following (next) bit

Gains from Altering Receiver

10-6

10-5

10-4

10-3

10-2

10-1

0 5 10 15

8-PSK Receivers, p=.01, q=.01

Trad. rcvrMAP receiverMAP, Prev. bitMAP, Next bitMAP, Both prev, next

Pe: p

rob.

of b

it er

ror

Eb/N

0 (dB)

Algorithm:

prior probabilities plus guesses of both preceding and following bits

Putting Gains into Perspective

10-6

10-5

10-4

10-3

10-2

10-1

0 5 10 15

8-PSK MAP, Both prev. and next bits

p=q=0.5p=q=0.1p=q=0.01p=q=0.001p=q=0.0001

Pe:

prob. o

f bit

err

or

Eb/N

0 (dB)

All decision algorithms: higher correlation more gain

Even playing field: set p, q for comparison

Basic Road Map

Background Ideas Correlated data transmission Phase Shift Keying (PSK)

Altering the receiver Altering the transmitter Conclusions, directions

Altering the Transmitter

Idea: equation gives angle for each symbol Requirements

Use prior probabilities For all , limit is traditional receiver

Resulting formula:

)(i

)(i

8

1i

)ln(

)ln(

48

1

ii

The Altered Transmitter

Resulting transmission points: shifted

Here:

beta = .000001

p=0.01, q=0.5

000

001

011

010

110

111

101

100

The Altered Transmitter

Resulting transmission points: shifted

Here: beta = .1

p=0.01, q=0.5000

100101

111

110

010

011001

Gains from Altering Transmitter

Moderate correlation values moderate gains for MAP

10-6

10-5

10-4

10-3

10-2

10-1

0 5 10 15

8-PSK Receivers, p=.01, q=.5"Morgan's drawing"

Trad, no MAP

MAP receiver

MAP using prev, next

Pe: p

rob.

of b

it er

ror

Eb/N

0 (dB)

Gains from Altering Transmitter

10-6

10-5

10-4

10-3

10-2

10-1

0 5 10 15

8-PSK Receivers, p=.01, q=.5"Morgan's drawing"

Trad, no MAP

MAP receiver

MAP using prev, next

Altered, beta=.000001

Pe: p

rob.

of b

it er

ror

Eb/N

0 (dB)

Moderate correlation values moderate gains for MAP

~.5-1dB gain over best MAP at reasonable Pe values

Conclusions

A successful alternative Correlated data, PSK transmission Source coding impractical

Future directions Simplified algorithms Bandwidth tradeoffs

References

Proakis and Salehi. Communications Systems Engineering. Prentice Hall, 2002.

Komo, John J. Random Signal Analysis in Engineering Systems. The Academic Press, 1987.

Hogg and Tanis. Probability and Statistical Inference. Prentice Hall, 2001.