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Transmit Power and Bit Allocations for OFDM Systems in a Fading Channel

2002년 12월 26일

서울대학교 전기.컴퓨터 공학부 이동통신 연구실

장 지 호

(2)

Contents

• Introduction

• Transmit power allocation in a single user OFDM

– Conventional power allocation schemes – Frequency-time domain power allocation

• Transmit power allocation in a multiuser OFDM

– Subcarrier assignment for multiple users – Power allocation for subcarriers

• Practical algorithm for transmit power and bit allocations

– With integer number of bits constraint – Practical considerations

• Conclusions

(3)

• Orthogonal Frequency Division Multiplexing (OFDM)

– Potential solution for high data rate system in the future – Parallel transmission over a number of subcarriers

– Flat fading for each subcarrier – Robust to multipath fading

– Wireless LANs, Broadband Wireless Access, DAB, DVB, etc.

Introduction

Frequency

Time

Symbol Duration

Frequency

Time

Symbol Duration

Single carrier system OFDM system

(4)

• Transmit power and bit allocations for OFDM

– With the knowledge of the channel state information (CSI) at the transmitter

– Transmit power and number of bits for each subcarrier are adaptively allocated according to the CSI

– Can increase the data rate, can use the resources (power, time, frequency bandwidth) more efficiently

OFDM Transmitter

Slowly time-varying chanel

OFDM Receiver

Transmit power and bit allocations

Subchannels' gains estimator Error-free and no delay

feedback

Data in Data Out

(5)

• Capacity :

Single user OFDM systems

noise CH

IFFT / Parallel

to Serial

/ Guard Interval Insertion Serial

to Parallel user data

} {d

Decision / Parallel

to Serial

z1

zn

zN

user data ˆ} {d Guard

Interval Removal

/ Serial

to Parallel

/ FFT p1

pn

pN



 

 

N

n

n n

n N W N

i g i p N

i W g C

1 0

2

] [ ] 1 [

log ]})

[ ({

(6)

• Conventional power allocation schemes

– Equal power allocation

• When the CSI is not known at the transmitter

• Transmit power:

– Frequency domain power allocation

• Usually in a fixed channel, may not fully exploit the time- varying nature of the fading channel

• Transmit power:

• Examples:

N i P pn[ ] 0



 

 

 [ ]

] 1 [ ]

[ 0

i i g

N W i N

p

n FA

n

constraint:

0 1

] [i P p

N

n

n

]

FA[i

FA[i1]

(7)

• Proposed frequency-time domain power allocation

– Jointly optimized both in frequency and time domains – Exploitation of time variation of the fading channel

– Transmit power : – Examples :



 

 

 [ ]

] 1

[ 0

i g N

W i N

p

n FTA

n  constraint:

0 1 0 p [i]f (g )dg P

N

n

n n

n



FTA

(8)

• Capacity gains :

Capacity gain for ‘frequency-time domain power allocation’ is the upper bound of the capacity gain for ‘frequency domain power allocation’

Results

[%]

100

[%],

100

EQ EQ FTA

FTA EQ

EQ FA

FA C

C C

C C

C

0 2 4 6 8 10 12 14 16

0 5 10 15 20 25

FTA

FA, N=32

FA, N=16

FA, N=8

FA, N=4

FA, N=2

Capacity gain [%]

Average SNR [dB]

(9)

Multiuser OFDM systems

noise CH

IFFT / Parallel

to Serial

/ Guard Interval Insertion Kth user

data

1 ,

p1

1 ,

pk

1 ,

pK

p1,n

n

pk,

n

pK,

p1,N

N

pk,

N

pK,

Serial to Parallel kth user data

1st user data } {d1 } {dk

} {dK

Decision / Parallel

to Serial

1 ,

zk

n

zk,

N

zk,

kth user data ˆ} {dk Guard

Interval Removal

/ Serial

to Parallel

/ FFT

Generally formulated : multiple users can share a specific subcarrier simultaneously

(10)

• Total data rate





 

 

K

k N

n

n k n

k TOTAL

i N

i W g

R

1 1

, 2

,

] 1 [

log ]})

[

({ 

• Constraint

0

1 1

, [i] P p

K

k N

n

n

k



5 1

5 ln

] 1 [

log ] [

1 2

] 5 [

. 1 5exp 1

, 2

,

] [ ,

,

. BER) ( i i b BER i

n k n

k

i b

n k

n k









• SINR

N W N i

g i p

i g i i K p

k j j

n k n

j

n k n k n

k

0 ,

1

, ,

, ,

,

] [ ] [

] [ ] ] [

[

(11)

• Derivation of the solution

– Two-step approach

– (1) Subcarrier assignment for users

• Theorem: a subcarrier should be assigned to only one

user who has the best channel gain for that subcarrier

1 2 3 4 5 6 7 8

subchannel index, n 474

.

0 0.286 2.014 0.943 3.583 1.551 0.102 1.044 887

.

0 0.043 0.480 0.182 0.744 1.373 0.289 3.650 586

.

0 2.578 1.226 1.538 1.513 0.024 0.479 0.124 021

.

0 0.781 0.616 0.014 2.052 0.092 0.115 0.856

user index, k

1

2

3

4

1 2 3 4 5 6 7 8

subchannel index, n 887

.

0 2.578 2.014 1.538 3.583 1.551 0.479 3.650

can be treated as a single user system ]

, [i gk n

(12)

• Total data rate

– constraint :

 

 

 

N

n

n n

n TOTAL

N W N

i G i p N

i W G R

1 0

2

1 ] [ ] 1 [

log ]})

[ ({

0 1

]

[

i P p

N

n

n

[ ], [ ], , [ ]

max ]

[ i g

1,

i g

2,

i g

,

i

G

n

n n

K n

different from the single user case

(13)

– (2) Power allocation for subcarriers

same as single user case except for the channel power gain

• Equal power allocation

• Frequency domain power allocation

• Frequency-time domain power allocation

N i P pn[ ] 0



 

 

 

] [ ] 1

[ ]

[ 0

i i G

N W i N

p

n FA

n  constraint:

0 1

] [i P p

N

n

n



 

 

 

] [ ] 1

[ 0

i G N

W i N

p

n FTA

n

constraint:

0 1 0 p [i]f (G )dG P

N

n

n n

n



(14)

data rates for the proposed methods increase as the number of users

increases 

multiuser diversity effects

data rates for the proposed methods are greater than AWGN capacity when K>22

1 2 4 8 16 32 64 128 256

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

22

SNR=10dB, N=256 AWGN capacity proposed-FA proposed-EQ FDMA-WF FDMA-EQ R TOTAL / W

Number of Users (K)

6 8 10 12 14 16 18 20 22 24

0 1 2 3 4 5 6 7 8

K=16, N=256

AWGN capacity proposed-FA proposed-EQ FDMA-WF FDMA-EQ

R TOTAL / W

Average SNR [dB]

Results

(15)

• Probability density functions

 





exp( )

) exp(

1 )

(

) exp(

) (

1 ,

,

n K

n n

n k n

k

G G

K G

f

g g

f

• Multiuser diversity

0 1 2 3 4 5 6 7 8 9 10

0.0 0.2 0.4 0.6 0.8 1.0

K=64 K=16K=32

K=4K=8 K=2 K=1

PDF, f(G n)

Gn

50 100 150 200 250

0 1 2 3 4 5 6 7

gain for user 3

gain for user 2 gain for user 1

Channel power gain (g k,n)

Frequency (n)

maximum gain for each subcarrier

(16)

[%]

100

[%],

100

EQ EQ FTA

FTA EQ

EQ FA

FA R

R R

R R

R

• Data rate gain:

as K and SNR increase, data rate gain decreases

0 2 4 6 8 10 12 14 16

0 5 10 15 20 25

Data rate gain [%]

Average SNR [dB]

K=2 FTA FA, N=32 FA, N=16 FA, N=8 FA, N=4 FA, N=2

0 2 4 6 8 10 12 14 16

0 1 2 3 4 5 6

Data rate gain [%]

Average SNR [dB]

K=8 FTA FA, N=32 FA, N=16 FA, N=8 FA, N=4 FA, N=2

(17)

Practical algorithm for transmit power and bit allocations

• Practical algorithm

– Integer number of bits for each subcarrier – Practical modulation / demodulation

• Data rate maximization

– Constraints : total transmit power and bit error rate – Single user case is considered

• Existing algorithms

– Hughes-Hartogs : greedy search method, optimal – Chow : equal power allocation for subcarriers

– Leke : water-filling power allocation for subcarriers – Krongold : table lookup method

(18)

• Hughes-Hartogs’ algorithm

– Greedy search method – One additional bit is

allocated to the subchannel that requires the least

incremental power at each step

– Optimal performance – High complexity

n

n

p

p

ˆ, ˆ

ˆ 1 ˆ *

*

n

n

b

b

ˆ } {ˆ

min

* arg

n

n p

p

n

• Chow’s algorithm

– Total transmit power is equally distributed over all subcarriers

– Low complexity

N P p

n

 

 

 

log2 1 2

n n

n

g b p

] ˆ [

n n

round b b

n 5

. 1

) 5 ln(

1 52 . 1 5exp 1

2

BER g p BER

n n n

b n

n

(19)

50 100 150 200 250 0.0

0.5 1.0 1.5 2.0

pn,eq

n

• Examples:

pn

bn

n

n

50 100 150 200 250

0 1 2 3 4

gn

n

gn

Equal power

50 100 150 200 250

0 1 2 3 4 5

bn,eq

n

50 100 150 200 250

0 1 2 3 4 5

b^ n,eq

n

50 100 150 200 250

0.0 0.5 1.0 1.5 2.0 2.5 3.0

p^ n,eq

n

bits 502

(20)

• Leke’s algorithm

– (1) water-filling power allocation (2) flat power

allocation

– High complexity

N





NON

n n

ON g

P

N 1

2 ~

1 1

Y START

NON

n*

1 1~ 1

1 0

*

*

ON ON

ON n ON n

N N

g N N

p

N NON

order descending in

} {

~}

{gn sortgn

?

~ 1gn*

END

n

n g

p 21~ ,NON

, , n for 12

ON

n PN

p ,NON

, , n for 12

END Water-filling Flat power END

 

2 1

ˆ ˆ ,

, 1

log

2 ˆ 2 2

bn

n n

n n

n n n

p g

b round b

g b p

? ˆ 0

1

P p

N

n n

Y

adjust

N

,N , , n

for 12

START

/ P P

Initialize:

0

1 P P

pn

is calculated from (1) water-filling method or (2) flat power allocation with total power constraint P

(21)

50 100 150 200 250 0

1 2 3 4 5

b^ n,wf

n

50 100 150 200 250

0.0 0.5 1.0 1.5 2.0

pn,wf

n

• Examples:

pn

bn

n

n

50 100 150 200 250

0 1 2 3 4

gn

n

gn

Water-filling

50 100 150 200 250

0 1 2 3 4 5

bn,wf

n

50 100 150 200 250

0.0 0.5 1.0 1.5 2.0 2.5 3.0

p^ n,wf

n

bits 519

(22)

• Leke’s algorithm

P p

N

n

n

1

: (1) constraint

n

n g

p 1

: solution filling

water

2

 

2 1

ˆ

] ˆ [

1 log

1

2 ˆ 2 2 2

bn

n n

n n

n n n

n n

p g

b round b

g b p

p g

? ˆ

: (2) constraint

0 1

P p

N

n

n

adjust P

5 1

5 ln

1 52 . 1 5exp 1

. BER) (

BER bn

n

initialize: P P0

(23)

• Basic idea for the proposed algorithm

– relationship between water-filling level ( ) and transmit power

 

 

2 1

ˆ

] ˆ [

) ( log

2 ˆ 2

 

bn

n n

n n

n n

p g

b round b

g b

? ˆ

: constraint

0 1

P p

N

n

n

adjust 

2 2 2

1 log

1

n n n

n n

g b p

p g

initialize 

(24)

gn

2

) 2 (

2

) 2 (

p1

1 2 3 4 5 6 7 8 subchannel

index, n

) 1 (

2

) 1 (

p1

) 0

1 (

2

p

) 0

2 (

2

p

) 2 (

p4 p5(2) p6(2)

) 2 (

p7 )

1 (

p3

) 0

2 (

3

p p8(2) 0

) 1 (

p4 p5(1) p6(1) p7(1)

) 0

1 (

8

p

) 2 ( )

1 ( )

2 ( )

1 ( )

2 ( )

1 ( ) 2 ( )

1 ( )

2 ( )

1

( , then , , ˆ ˆ , and ˆ ˆ

if

pnpn bnbn bnbn pnpn

• Effects of on the transmit power and number of bits

(25)

• Proposed algorithm

– Based on water-filling approach

– Only the water-filling level ( ) is adjusted – Low complexity

? ˆ 0

1

P p

N

n

n

START

END

  

 

2 1

ˆ

, 1

, ˆ 0 ˆ ,

, log

2 ˆ 2

bn

n n

ON ON

n

n n

n n

p g

N N

then b

if

b round b

g b

Initialize:

N N

g P

N

ON

N

n n





1 , 1

1 2

0

Y

N

n n ON

p

N P 1

2 0 ˆ

1 1

N

,N , , n

for 12

(26)

• Complexity comparisons

Loading algorithms Order of Operations

Hughes-Hartogs (optimal)

Chow (equal power allocation) Leke (water-filling, flat power) Krongold (lookup table)

Proposed

) ˆ log

(B N 2 N

O HH   ) (Iter N

O

) 2

log

(N 2 N Iter N

O    

)

(N M Iter N M

O    

) (Iter N

O

• : total number of loaded bits when the Hughes-Hartogs’ algorithm is used

• Iter : required number of iterations

• M : maximum number of bits mapped for a symbol in the constellation

HH

(27)

0 2 4 6 8 10 12 14 16 18 20 0

10 20 30 40 50 60

Number of Iterations

Average SNR [dB]

=0.4 =0.5 =0.6 =0.7 =0.8 =0.9

0 2 4 6 8 10 12 14 16 18 20

0 1 2 3 4 5 6 7

Data Rate Decrease [%]

Average SNR [dB]

=0.4 =0.5 =0.6 =0.7 =0.8 =0.9

Results

[%]

ˆ 100

ˆ ˆ

, 10

, 256

arg 3

HH HH et

t B

B Decrease B

Rate Data

BER N

• Effects of step size,

(28)

0 2 4 6 8 10 12 14 16 18 20 0

1 2 3 4 5 6 7 8

Data Rate Decrease [%]

Average SNR [dB]

Equal power Leke's flat power Krongold

Proposed, =0.7 Leke's water-filling

Complexity of the proposed is less than the Leke’s algorithm

(e.g.) at SNR=4 dB, O(8N) for the proposed and O(24N) for the Leke’s algorithm

Data Rate Decrease for the proposed is less than 1% when SNR>4dB

0 2 4 6 8 10 12 14 16 18 20

0 10 20 30 40 50 60 200400 600800 10001200

Number of Iterations

Average SNR [dB]

Optimal (Hughes-Hartogs) Equal power

Leke's flat power Krongold

Proposed, =0.7 Leke's water-filling

[%]

ˆ 100

ˆ ˆ

, 10

, 256

arg 3

HH HH et

t B

B Decrease B

Rate Data

BER N

(29)

지적 사항

• Practical considerations for transmit power and bit allocations

– Transmit power level constraint (transmit spectrum mask) – Effects of imperfect channel information

• 학위 논문

– Motivations 및 기존 연구 survey 보강 – 참고문헌 보강

– Conclusions 보강 – 논문 쪽 수

(30)

• Transmit power level constraint

– Transmit power level (transmit spectrum mask) is constrained by the regulations

– Proposed algorithm needs to be modified

• Effects of imperfect channel information

– Due to the feedback delay in time varying fading channels – Accurate channel estimation is assumed

– Doppler frequency and feedback delay need to be considered

Practical Considerations

(31)

• Proposed algorithm needs to be modified

(1) Transmit power level constraint

 





 

 

 

 

 

 min [log2( )] , log2 1 2 ˆn roundgn floor pmaskgn b



 

 

N P

pmask

0

10log10

SM

• Spectrum Margin

Data rate for H.-H.

algorithm decreases as SM decreases

0 2 4 6 8 10 12 14 16 18 20

101 102 103

Data rate for Hughes-Hartogs' algorithm

H.-H., SM=Infinte H.-H., SM=6dB H.-H., SM=5dB H.-H., SM=4dB H.-H., SM=3dB H.-H., SM=0dB

Data Rate (B^ HH)

Average SNR [dB]

(32)

As SM decreases, the difference of Data Rate Decrease between proposed

algorithm and Chow’s algorithm decreases

[%]

ˆ 100

ˆ ˆ

, 10

, 256

arg 3

HH HH et

t B

B Decrease B

Rate Data

BER N

0 2 4 6 8 10 12 14 16 18 20

0 1 2 3 4 5 6 7 8

Data Rate Decrease [%]

Average SNR [dB]

Chow, SM=Infinite Chow, SM=6dB Chow, SM=5dB Chow, SM=4dB Chow, SM=3dB Chow, SM=0dB

0 2 4 6 8 10 12 14 16 18 20

0 1 2 3 4 5 6 7 8

Data Rate Decrease [%]

Average SNR [dB]

Proposed, SM=Infinite Proposed, SM=6dB Proposed, SM=5dB Proposed, SM=4dB Proposed, SM=3dB Proposed, SM=0dB

(33)

(2) Effects of imperfect channel information













2

1 1

) cos(

2 ,

2

1 1

, 10

, max

,

log 10

) ( )

(

) (

L

l U

u

T D f j u l

L

l U

u

u l actual

u l u f

e D

c

c dB SNR

dB SNR

dB SNR

L

l

l l iT s kT k

i h

1

] [

] [ ]

,

[

 

U

u

u l u D

u l

i iT c j f iT

1

, max

,

, exp (2 cos( ) )

]

[

 







 

 

L

l m

l

l T

f m T j

f m S iT f

i H

1

2 exp ]

[ )

,

( 2

1 1

) cos(

2 ,

2

1 1

,

, max



,



L

l U

u

T D f j u l

L

l U

u

u l

actual D f u lu

e c

c SNR

SNR

• SNR mismatch error is a function of

(Doppler frequency x feedback delay)

(34)

actual

SNR SNR

actual

SNR SNR

Over-loaded:

higher data rate

worse BER performance Under-loaded:

lower data rate

better BER performance

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

0 1 2 3 4 5 6

PDF

SNR [dB]

fD,maxDfT=0.001 fD,maxDfT=0.005 fD,maxDfT=0.01

(35)

0 1 2 3 4 5 6 10-3

10-2 10-1 100

1-CDF

SNR [dB]

fD,maxDfT=0.001 fD,maxD

fT=0.005 fD,maxD

fT=0.01

• at 1% outage: 0.2dB SNR loss for 1.0dB SNR loss for 2.0dB SNR loss for

 SNR margin is required in loading

001 .

max 0

, D T

fD f

005 .

max 0

, D T

fD f

01 .

max 0

, D T

fD f

(36)

• Examples

Parameters for High Speed Downlink Packet Access (HSDPA)

Carrier frequency 2 GHz

Speed 0.3 km/hr 0.03 km/hr

Doppler frequency 0.55 Hz 0.055 Hz

Frame size 2 msec

Parameters for IEEE 802.11a Wireless LAN

Carrier frequency 5 GHz

Speed 0.3 km/hr 0.03 km/hr

Doppler frequency 1.39 Hz 0.139 Hz

Frame size 628 usec (54Mbps data rate, maximum length)

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