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Lagrange and Water Filling algorithm. Speaker : Kuan -Chou Lee Date : 2012/8/20. Lagrange and Water Filling Algorithm (1/4). Recall that the capacity of an ideal, band-limited, AWGN channel is - PowerPoint PPT Presentation
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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