Transmit Power and Bit Allocations for OFDM Systems in a Fading Channel
2002년 12월 26일
서울대학교 전기.컴퓨터 공학부 이동통신 연구실
장 지 호
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
• 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
• 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
• 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 ]})
[ ({
• 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]
• 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
• 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]
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• 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
, ,
, ,
,
] [ ] [
] [ ] ] [
[
• 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
• Total data rate
–
–
– constraint :
Nn
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 ndifferent from the single user case
– (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
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>221 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
• 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
[%]
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 decreases0 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
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
• 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, ˆˆ 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 nn
g b p
] ˆ [
n n
round b b
pˆ
n 5. 1
) 5 ln(
1 52 . 1 5exp 1
2
BER g p BER
n n n
b n
n
50 100 150 200 250 0.0
0.5 1.0 1.5 2.0
pn,eq
n
• Examples:
pn
bn
bˆn
pˆn
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
• 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
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
bˆn
pˆn
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
• Leke’s algorithm
P p
N
n
n
1
: (1) constraint
n
n g
p 1
: solution filling
water
2
2 1ˆ
] 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
• Basic idea for the proposed algorithm
– relationship between water-filling level ( ) and transmit power
2 1ˆ
] 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
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
pn pn bn bn bn bn pn pn• Effects of on the transmit power and number of bits
• 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
• 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
BˆHH
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,
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
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
• 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
• Proposed algorithm needs to be modified
(1) Transmit power level constraint
min [log2( )] , log2 1 2 ˆn round gn 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]
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
(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)
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
SNR [dB]
fD,maxDfT=0.001 fD,maxDfT=0.005 fD,maxDfT=0.01
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
• 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)