Lagrange and Water Filling algorithm

Lagrange and Water Filling algorithm

Speaker : Kuan-Chou LeeDate : 2012/8/20

Graduate Institute of Communication Engineering, NTU

Lagrange and Water Filling Algorithm(1/4)

Recall that the capacity of an ideal, band-limited, AWGN channel is

C is capacity in bits/s, W is the channel bandwidth, Pav is the average transmitted power, N0 is noise variance. In a multicarrier system, with Δf sufficiently small, the subchannel has capacity

H(f) is the frequency response of a nonideal, band-limited channel with a bandwidth W. noise variance is Φnn(f). is the transmitted power in Δf.

pp. 2

2

0

log 1 avPC WWN

2

2log 1 i i

i

nn i

fP f H fC f

f f

Graduate Institute of Communication Engineering, NTU

Lagrange and Water Filling Algorithm(2/4)

Hence, the total capacity of the channel is

In the limit as , we obtain the capacity of the overall channel in bits/s. The object of the problem is maximizing the capacity can be formulate as:

subject to

pp. 3

0f

2

21 1

log 1 .N N

i i

ii i nn i

P f H fC C f

f

2

2max , where log 1W

nn

P f H fC C df

f

,

0.

totalWP f df P

P f

[1], Page. 716-717

SDhS D

Graduate Institute of Communication Engineering, NTU

Lagrange and Water Filling Algorithm(3/4) Under the constraint on , the choice of that maximizes C

may be determined by maximizing the Lagrangian function

where λf and are the Lagrange multiplier, which is chosen to satisfy the constraint. By using the calculus if variations to perform the maximization, we find that the optimum distribution of transmitted signal power is the solution to the equation

pp. 4

P f P f

2

2log 1 ,f totalWnn

P f H fvP f P f df vP

f

2

2

2

0,ln 2

1ˆ ˆˆ ˆ, where ln 2 and ln 2

nnf

nn nn

f f f

nn

H f fv

f f P f H f

v v vf H f P f

Graduate Institute of Communication Engineering, NTU

Lagrange and Water Filling Algorithm(4/4) From the KKT conditions, .

pp. 5

[2], Page. 716-717

ˆ 0f P f

2

2 2

1ˆ ˆ0 ,

1 , ( ).ˆ

f

nn

nn nn

vf H f P f

f fP f K f W

v H f H f

2

nn f

H f

K

WIRELESS Communication LAB

Graduate Institute of Communication Engineering, NTU

On the Optimal Power Allocation forNonregenerative OFDM Relay Links

I. –Hammerstrom and A. –Wittneben, “On the optimal power allocation for nonregenerative OFDM relay links,” in Proc. IEEE ICC, pp.4463 – 4468, Jun. 2006.

Graduate Institute of Communication Engineering, NTU

System Model (1/7)

Problem : Allocating the subcarrier power of the relayed signal to maximize the channel capacity.

Solution : Lagrange and Water Filling Algorithm

pp. 7

R RDhSRhS D

N-1...10

Fig.1. Dual-hop relay communication system comprising source (S), relay (R) and destination (D) terminals.

Graduate Institute of Communication Engineering, NTU

System Model (2/7)

Transmitted signal :

Average transmission power for all subcarriers :

Received signal at the relay node :

Nonregenerative relay (variable-gain relaying scheme) :

pp. 8

[ ] [ ]-1

0=

1 2exp , 0,1, , 1,N

i

j nix i X i n NNNp

=

æ ö÷ç = -÷ç ÷çè øå L

[ ]2E 1X ié ùºê úë û

[ ] [ ] [ ] [ ]SR R , 0,1, , 1,R i H i X i W i i N= + = -L

[ ] [ ]i R ia

[ ]( ) [ ] [ ] [ ] [ ]( )2 2 2 2R R SR RE .i P i R i P i H ia sé ù= = +ê úë û

Graduate Institute of Communication Engineering, NTU

System Model (3/7)

Received signal at the destination node :

Signal to noise power ratio (SNR)

pp. 9

[ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]( ) [ ][ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

RD D

RD SR R D

RD SR RD R D

,

, 0,1, , 1,

.

D i H i i R i W i

H i i H i X i W i W i i N

H i H i i X i H i i W i W i

aa

a a

= += + + = -= + +

L

[ ] [ ] [ ]

[ ] [ ] [ ]( )[ ][ ]

2 2R RD SR

2 2RR D

2 2 22 2 RSR R DR RD R2 2 2 2R D R D

.1

i ii

i i

P i H i H iP i b a

a P i bH iP i H i

s srs ss

s s s s

= = + +++

Graduate Institute of Communication Engineering, NTU

System Model (4/7)

The total capacity of the channel is

, the object of the problem is maximizing the capacity can be formulate as:

subject to

pp. 10

R2 2

1 1 1 R

1 1 1log 1 log 1 .2 2 2 1

N N Ni i

i ii i i i i

P i b aC C f f

a P i b

R2

1 R

1max , where log 1 ,2 1

Ni i

i i i

P i b aC C

a P i b

R

1

R

,

0, for =1, , .

N

totali

P i P

P i i N

1f

Graduate Institute of Communication Engineering, NTU

System Model (5/7)

Set up the Lagrangian function

The derivative of the Lagrangian with respect to

Setting to zero, we get

pp. 11

RR 2 R R

1 R

R R R R1 1

, , log 1 ,1

where and .

Ni i T T

totali i i

N NT T

ii i

P i b aL v v P

a P i b

P i P i

P λ λ P 1 P

λ P 1 P

RP i

R

R R R

, , 1 ,ln 2 1 1

i ii

i i i

L v a b vP i a P i b P i b

P λ

R R

ln 2 ln 2 ,1 1

i ii

i i i

a bva P i b P i b

Graduate Institute of Communication Engineering, NTU

System Model (6/7)

From the KKT conditions

Another KKT condition is that

If , :

pp. 12

R R

.1 1

i i

i i i

a bva P i b P i b

ln 2 0, ln 2 .i i v v

R 0.i P i

R

R R

0.1 1

i i

i i i

a bv P ia P i b P i b

1i i

i

a bva

R 0P i R 0P i

R R

0.1 1

i i

i i i

a bva P i b P i b

Graduate Institute of Communication Engineering, NTU

System Model (7/7)

If :

After some algebraic manipulations

where .

pp. 13

1i i

i

a bva

R 0P i R 0P i

[ ] { }max 0,x x+ =

[ ]R

41 1 1 1 ,2

i i

i i

a bP ib a v

+

*é ùæ ö÷çê ú÷= + - -ç ÷ê úç ÷¢çè øë û

Graduate Institute of Communication Engineering, NTU

Conclusion

The objective function (Maximize Capacity? Minimize total Power or bit error rate?) Constraint (Power, Resource) Lagrange function (Derivation)

Solve the optimization problem (i.e., Obtain the power allocation among the subcarrier)

pp. 14

Graduate Institute of Communication Engineering, NTU

Reference[1] J. G. Proakis, Digital Communications, 4rd ed. New York: McGraw-Hill, 2001.[2] I. –Hammerstrom and A.-Wittneben, “On the optimal power allocation for nonregenerative OFDM relay links,” IEEE ICC , pp.4463-4468, Jun. 2006.

pp. 15