1

Designing Pilot-Symbol-Assisted Modulation (PSAM) Schemes for Time-

Varying Fading Channels

Tharaka Lamahewa

Research School of Information Sciences and Engineering

ANU College of Engineering and Computer Science

The Australian National University (ANU)

Collaborators: Dr. Parastoo Sadeghi (ANU), Prof. Rod Kennedy (ANU), Prof. Predrag Rapajic (University of Greenwich, UK), Mr. Sean Zhou (ANU),

Dr. Salman Durrani (ANU)

2

Outline

Introduction Background and Motivation

Contributions Optimum design of PSAM schemes in time-

varying fading channels Optimum design under spatially correlated

antenna arrays Conclusions

3

System Modely = g x` +n`

Fading Input Noise

Received Signal:

4

Adaptive Power Allocation (1)• Aim: To save the overall transmitter power by adapting the power

depending on the fading channel quality

5

Adaptive Power Allocation (2)• Early studies: Perfect (Genie aided) channel state information

(CSI) at Rx and possibly delayed CSI at Tx

6

Genie Aided Power Allocation Strategy• Solution: Waterfill in time, with water level 0 (Goldsmith and

Varaiya 1997)

Ed(½) =1½0

¡1½; ½ ½0

Ed(½) = 0; ½<½0

½= jgj2

7

Adaptive Power Allocation (3)

Noise

Variance:

• Later studies: A vicious genie provides imperfect CSI at Rx and (possibly delayed) imperfect CSI at Tx

• The channel is measured using a probe signal continuously

Good GenieBad Genie

g = g +~g

¾2~g( )

8

Power Allocation Strategy under Imperfect Channel State Information

• It is still a waterfilling strategy in time (with a more complex solution), with some water level 0 (Klein and Gallager 01, Yoo and Goldsmith 04 & 06)

½= jgj2

Ed(½) = f (½;½0;¾2~g); ½ ½0Ed(½) = 0; ½<½0

9

Our Model-Based Power Adaptation with Periodic Feedback

• There is no genie

• Channel estimate is based on pilot symbol transmission

• reduction in available resources

• possible tradeoffs.

• Channel estimates are fed back after some delay.

Pilot Symbols Data Symbols

Block Length = T

10

First Step

Let us consider the case where there is no feedback, for comparison purposes, and optimize pilot transmission parameters

11

System Model without Feedback (1)

Pilot symbols are transmitted periodically for channel estimation

12

System Model without Feedback (2)

State equation: g =®g ¡ 1+w`

First-order AR or Gauss-Markov channel model:

n` » NC (0;N0)

g » NC (0;¾2g)Observation equation:

fading

y = g x` +n`

inputnoise

small fast channellarge slow channel

13

Three Parameters for Optimization How often (pilot spacing)?

How strong (pilot symbol power)?

How many (pilot cluster size)? One1

P1 D2 D3 …... DT-1 DT

´ =1T;

1. M. Dong, L. Tong, and B. M. Sadler, “Optimal insertion of pilot symbols for transmissions over time-varying flat fading channels,” IEEE Trans. Signal Processing, vol. 52, no. 5, pp. 1403–1418, May 2004.

Ep =°´E;

Ed =1¡ °1¡ ´

E

14

Our Objective Function: A Capacity Lower Bound

A closed-form expression for the information capacity of the time-varying Gauss-Markov fading channel in the presence of imperfect channel estimation is still unavailable.

We utilize a lower bound to the channel capacity

15

Our Objective Function: A Capacity Lower Bound1

1. M. Medard, “The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel,” IEEE Trans. Inform. Theory, vol. 46, no. 3, pp. 933–946, May 2000.

Note the presence of signal power in the denominator

C(g) ¸ CLB (g) = log³1+

jgj2Ed¾2~gEd+N0

´

Channel estimation: g= g+~g

estimate error

y = g x` +~g x` +n`

useful part additional noise

effective instantaneous SNR

16

Average Capacity Lower Bound

• The lower bound is an average capacity per symbol and not per transmission block

• The lower bound, by itself, does not take into account models of channel time variations

• The lower bound, by itself, does not tell us how to optimize the two pilot parameters and

CLB =Egnlog³1+ jgj2Ed

¾2~gEd+N0

´o

expectations over channel estimate realizations

17

First Contribution (no Feedback Yet)

Invoke the autoregressive (AR) or Gauss-Markov channel model and Kalman filtering principles to maximize the capacity lower bound

Analyse optimum pilot scheme parameters pilot power γ and pilot spacing T

to maximise the capacity lower bound for a wide range of SNR and normalized fading rates

18

Estimation errorincreases

Kalman Filtering

P1 D2 D3 …... DT-1 DT

M(1) =¾2~g(1)

g1;~g1

Estimate (true filtering)

Predict (no pilots)

g = ®¡ 1g1

M ( ) =¾2~g( ) =¾2g ¡ ®2(`¡ 1)(¾2g ¡ M (1))

19

Revisit the Capacity Lower Bound

P1 D2 D3 …... DT-1 DT

SNR decreases

g = ®¡ 1g1®<1

½inst( ) =jg j2Ed

¾2~g( )Ed+N0

y = g x` +~g x` +n`

signal noise

20

Revisit the Lower Bound (cnt’d)

P1 D2 D3 …... DT-1 DT

Capacity per symboldecreases

Instantaneous capacity per symbol:

Instantaneous capacity per block:

CLB (½1) =1T

TX

`=2

CLB ( ;½1)

CLB ( ;½inst) = log¡1+½inst( )

¢

21

LB versus Pilot Power and Spacing

Average

SNR = 10 dB

®= J 0¡2¼fD Ts

¢, fD Ts =0:1

22

Equal Power Comparison

Pilot spacing is optimized

23

Sensitivity of LB to Power Ratio

24

Sensitivity of LB to Pilot Spacing

25

Power Adaptation Based on Periodic Feedback

26

System Model with Feedback

Switching period: T

• This model is still idealistic in the sense that the feedback link is noiseless.

27

A Closer Look at the Transmitter ½ = jg j2

Ed(½) = f (½;½0;¾2~g` );

Ed(½) = 0;

Zero-delay power control: actual power allocation at time is performed based on the channel estimate g` = ®`¡ 1 g1

`

g` = ®`¡ 1 g1

28

Transmission Block Structure

P1 D2 D3 …... Dd …. DT-1 DT

Feedback

Transmitted

Fixed Power Strategy

Adaptive Power Strategy

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Power Distribution

P1 D2 D3 …... Dd …. DT-1 DT

Total Power: Adaptive Power per Symbol

Ed(½1; ) = f (½;¾2~g` ;¢¢¢)

TX

`=d+1

E ½1

nEd(½1; )

o= (1¡ °)

(T ¡ d)TT ¡ 1

E

Ed = (1¡ °)(d¡ 1) £ T

T ¡ 1E

Ep = °E £ T

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Capacity Lower Bound per Block

P1 D2 D3 …... Dd …. DT-1 DT

Fixed Power Capacity Adaptive Power Capacity

1T

TX

`=d+1

log

Ã

1+®2(`¡ 1)½1Ed(½1; )¾2~g`Ed(½1; ) +N0

!

1T

dX

`=2

log

Ã

1+®2(`¡ 1)½1Ed¾2~g`Ed+N0

!

+CL B =

No Information

0+

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Optimization Problem Maximize the total capacity per block subject

to the power constraint in the adaptive transmission mode

maxE(`;½1)¸ 0

1T

Z (dX

`=2

logµ1+

®2(`¡ 1)½1Ed¾2~g( )Ed +N0

¶

+TX

`=d+1

logµ1+

®2(`¡ 1)½1E( ;½1)¾2~g( )E( ;½1) +N0

¶)

f (½1)d½1;

subject toTX

`=d+1

ZE( ;½1)f (½1)d½1 = (1¡ °)

(T ¡ d)(T ¡ 1)

E £ T

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Optimum Solution

A generalized water-filling algorithm

E( ;½1) =

½¡ a

c+ b

º c; ½1 > º

®2(`¡ 1) ;0; otherwise,

a = N0(2M ( ) +®2(`¡ 1)½1);

b=qº®2(`¡ 1)½1(º®2(`¡ 1)½1 +4M ( )2 +4M ( )®2(`¡ 1)½1);

c= 2M ( )(M ( ) +®2(`¡ 1)½1);

º : threshold (or associated to thewater level)

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Slow Fading

0 2 4 6 8 10 12 14 150.5

1

1.5

2

2.5

3

3.5

SNR budget, , dBOpt

imum

Cap

acity

low

er b

ound

, C

LB,

bits

/ch

use f

DT

S = 0.01

w/o feedback, data & pilot equal power

0 2 4 6 8 10 12 14 150.5

1

1.5

2

2.5

3

3.5

SNR budget, , dBOpt

imum

Cap

acity

low

er b

ound

, C

LB,

bits

/ch

use f

DT

S = 0.01

w/o feedback, data & pilot equal power

w/o feedback

0 2 4 6 8 10 12 14 150.5

1

1.5

2

2.5

3

3.5

SNR budget, , dBOpt

imum

Cap

acity

low

er b

ound

, C

LB,

bits

/ch

use f

DT

S = 0.01

w/o feedback, data & pilot equal power

w/o feedback

delay-less feedback

0 2 4 6 8 10 12 14 150.5

1

1.5

2

2.5

3

3.5

SNR budget, , dBOpt

imum

Ca

paci

ty lo

wer

bou

nd,

CLB

, bi

ts/c

h us

e fD

TS

= 0.01

w/o feedback, data & pilot equal power

w/o feedback

delay-less feedback

delay = 8

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Messages For the slow fading, even after a feedback delay of 8,

we get good SNR improvement in low SNR In high SNR, even the delay-less feedback does not

offer much compared to the case, where pilot power and spacing are optimized

However, it continues to offer improvements at high SNR, compared the equal power allocation strategy.

35

Fast Fading

0 5 10 150.2

0.4

0.6

0.8

1

1.2

SNR budget, , dBOpt

imum

Cap

acity

low

er b

ound

, C

LB,

bits

/ch

use f

DT

S = 0.1

Blue: Delay 3

Pink: Without Feedback

Delay-less Feedback

36

Messages Idealistically (no feedback delay), we would

get a good SNR improvement at low SNR However, with a modest delay of d = 3, the

performance rapidly drops No point in power adaptation for such a fast

fading rate Note that, we have forced the transmitter that,

in spite of feedback delay, perform power adaptation for at least one symbol

37

Optimal Block Length, Slow Fading

0 5 10 15 2010

15

20

25

30

35

40

SNR budget, , dB

Opt

imim

Pilo

t sp

acin

g, T

*fDT

S = 0.01

w/o feedbackwith feedback, delay d = 0with feedback, delay d = 8

38

Message In the slow fading regime, the optimal block

length without feedback is large enough to accommodate power adaptation

39

Optimal Block Length, Fast Fading

0 5 10 15 20

3

4

5

6

7

8

9

SNR budget, , dB

Opt

imim

Pilo

t sp

acin

g, T

*fD

TS

= 0.1

w/o feedbackwith feedback, delay d = 0with feedback, delay d = 3with feedback, delay d = 8

40

Message In the fast fading regime, the optimal block

length without feedback is so small that it cannot accommodate any power adaptation

Note that the optimal T* is less than half of what would have in a bandlimited fading channel according to the Nyquist rate (fDT = 0.1 Tmin=10)

41

Optimal Pilot Power Allocation

0 5 10 15 200

0.2

0.4

0.6

0.8

1O

ptim

im P

ilot

pow

er r

atio

, *

SNR budget, , dB

with feedback, fD

Ts = 0.01

w/o feedback, fD

Ts = 0.01

with feedback, fD

Ts = 0.1

w/o feedback, fD

Ts = 0.1

42

Message In the slow fading, the pilot power ratio in the

without feedback scheme, is a reasonable choice for the feedback scheme too.

43

Multiple Antennas at the Receiver

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A Capacity Lower Bound: SIMO

Instantaneous capacity per symbol:

CLB ;` = E h `

nlog2

¯¯I +½d(½dM ss;`+I )

¡ 1h`hy`

¯¯¯o

½d : data symbol SNR;

h` : channel estimate;

M ss;` : steady-state channel covariancematrix

45

LB vs SNR: Spatially i.i.d. Channels

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

SNR budget, , dB

CLB

in b

its p

er

tra

nsm

issi

on

fD

Ts = 0.01, N

r=8

fD

Ts = 0.01, N

r=4

fD

Ts = 0.11, N

r=8

fD

Ts = 0.11, N

r=4

Solid lines: using opt. parameters corresponding to SIMO boundDashed lines: using opt. parameters corresponding to SISO bound

46

LB vs SNR: Spatially Correlated Channels

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

SNR budget, , dB

CLB

in b

its p

er

tra

nsm

issi

on

fD

Ts = 0.01, N

r=8

fD

Ts = 0.01, N

r=4

fD

Ts = 0.11, N

r=8

fD

Ts = 0.11, N

r=4

Solid lines: using opt. parameters corresponding to SIMO boundDashed lines: using opt. parameters corresponding to SISO bound

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Conclusions We considered the achievable information rates in

autoregressive fading channels with and without feedback under pilot-assisted channel estimation

Under realistic delays, feedback can provide some gain at low SNR and in moderately slow fading channels (fDTs = 0.01) and even at high SNR (compared to equal power allocation case)

In the relatively fast-fading case (fDTs = 0.1), use equal power allocation and just optimize the pilot spacing

By optimally designing the training parameters for SISO systems, the same parameters can be used to achieve near optimum capacity in both spatially i.i.d. and correlated SIMO systems.

48

Current and Future Research What is the price of providing feedback in terms of

information rate loss in the reverse channel? How much power should we allocate for feedback transmission? Requires proper modelling and formulation.

So far, we have considered SISO and SIMO, extending the results to the MIMO

Model-based comparison of PSAM and superimposed training in terms of achievable information rates