Lagrange and Water Filling algorithm

1. Lagrange and Water Filling algorithm Speaker : Kuan-Chou Lee Date : 2012/8/20

2. 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.

3. 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 [1], Page. 716-717

4. Lagrange and Water Filling Algorithm(3/4) • Under the constraint on , the choice of that maximizes C may be determined by maximizing the Lagrangianfunction • 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

5. Lagrange and Water Filling Algorithm(4/4) • From the KKT conditions, . [2], Page. 716-717

6. On the Optimal Power Allocation forNonregenerative OFDM Relay Links I. –Hammerstromand A. –Wittneben, “On the optimal power allocation for nonregenerativeOFDM relay links,” in Proc. IEEE ICC, pp.4463 – 4468, Jun. 2006.

7. System Model (1/7) • Problem : Allocating the subcarrier power of the relayed signal to maximize the channel capacity. • Solution : Lagrange and Water Filling Algorithm Fig.1. Dual-hop relay communication system comprising source (S), relay (R) and destination (D) terminals.

8. System Model (2/7) • Transmitted signal : • Average transmission power for all subcarriers : • Received signal at the relay node : • Nonregenerative relay (variable-gain relaying scheme) :

9. System Model (3/7) • Received signal at the destination node : • Signal to noise power ratio (SNR)

10. 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

11. System Model (5/7) • Set up the Lagrangianfunction • The derivative of the Lagrangian with respect to • Setting to zero, we get

12. System Model (6/7) • From the KKT conditions • Another KKT condition is that • If , :

13. System Model (7/7) • If : • After some algebraic manipulations • where .

14. 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)

15. 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.