OPTIMUM POWER ALLOCATION FOR PARALLEL GAUSSIAN CHANNELS WITH ARBITRARY INPUTS DISTRIBUTION

1. OPTIMUM POWER ALLOCATION FOR PARALLEL GAUSSIAN CHANNELS WITH ARBITRARY INPUTS DISTRIBUTION BY: ORAI COKER

2. Outline • Introduction • Problem formulation • Optimum Power Allocation policy • Waterfilling Policy-Gaussian Inputs • Graphical Interpretations: MMSE Power Charts • Graphical Interpretations: Mercury Waterfilling • High power and low Power Regime • Application: DSL • Conclusion

3. Introduction • For independent parallel Gaussian –noise channels, the mutual information is maximized under an average power Constraint by independent Gaussian inputs. -This power is allocated according to the water filling policy. • In practice, however discrete signaling constellations ( m-PSK, m-QAM, etc.) are the input signals instead of the ideal Gaussian signals. • This project formulates the power allocation policy that maximizes the mutual information over parallel channels with these non-ideal input distributions Graphical Interpretation: Mercury/water filling.

4. Problem Formulation The Input-Output relationship on the th channel with noise zero-mean unit variance, independent of the noise on the other channels. Power constraint

5. Problem Formulation (contd.) Introducing for convenience…….. with power allocation constrained by Defining the input-output mutual Information on the channel as: The problem posed is: What is the Power Allocation that maximizes the mutual information with given input distributions while satisfying the power constraint:

6. Optimum Power Allocation Policy Introducing two very important theorems that are very essential for the determination of this Power Allocation Policy are: Theorem 1: For any distribution of the inputs satisfies: Theorem 2 : The Power Allocation With such that:

7. Optimum Power Allocation Policy (contd.) From this two theorems the Power to be allocated to each channel can then explicitly be represented as : ……1 Where can be calculated from ……2 as sum of allocated power must be 1; Hence the power allocation boils down to a two-step process: • Solve for in (2 ) • Use to identify using (1 )

8. Waterfilling Policy-Gaussian Inputs which reduces Theorem 2 to the Waterfilling policy Where plays the role of the water level.

9. Waterfilling Policy-Gaussian Inputs

10. Graphical Interpretations: MMSE Power Charts

11. Graphical Interpretations: MMSE Power Charts (contd)

12. Graphical Interpretations: MMSE Power Charts (contd.)

13. Graphical Interpretations: Mercury Waterfilling Let us define, for any arbitrary input distribution: such that, for a Gaussian input, for all . The function enables the interpretation of our power allocation policy, which we refer to as Mercury/Waterfilling.

14. Graphical Interpretations: Mercury Waterfilling (contd) • For each of the channels, set up a unit-base vessel solid up to a height • Choose . Pour mercury onto each of the vessels until its height reaches .

15. Graphical Interpretations: Mercury Waterfilling (contd) • Water fill, keeping identical upper level of water in all vessels, • until the water level reaches

16. Graphical Interpretations: Mercury Waterfilling (contd) • The water height over the mercury on the th vessel • gives .

17. High power and low Power Regime In Low power Regime: The Waterfilling policy is identical to the optimum Mercury/Waterfilling policy. For Proper complex input distributions ..its the same High Power Regime: In stark contrast with waterfilling: Given equal constellations, the stronger a channel, the less power it is allocated.

18. Application: DSL • The smaller the constellation, the higher the excess power at all values of P . • Regardless of the constellation, the excess power vanishes for small P • For large P, high excess powers

19. Conclusion • The relationship between the nonlinear MMSE an the mutual information is used in formulating the power allocation policy for independent Gaussian-noise channels driven by arbitrary (not necessarily identical) input constellations. • Mercury/Waterfilling, is a geometric representation of the optimum policy that generalizes the classical solution for ideal Gaussian inputs. • As long as the input constellations are proper complex they are almost as good as Gaussian inputs in the low-power region and thus, Waterfilling is almost optimum for proper complex inputs in that region. • In the moderate-to-high power regions, the principle that stronger channels should be allocated more power, may not hold with constellations of practical interest. In fact, in some circumstances, the Mercury/Waterfilling policy mandates exactly the opposite.

20. Questions?