Robust Transmit Power Control for Cognitive Radio Peyman Setoodeh & Simon Haykin

1. Robust Transmit Power Control for Cognitive RadioPeyman Setoodeh & Simon Haykin S-88.4221 Postgraduate Seminar on Signal Processing 1 P Furqan Ahmed

2. Outline • Introduction • Orthogonal Frequency Division Multiplexing • Cognitive Radio Environment • Iterative Waterfilling Algorithm • Stochastic versus Robust Optimization • IWFA as variational inequality problem • Transient behaviour analysis of CRNs • Sensitivity Analysis • Computer Experiments • Summary and Conclusions • Future work

3. Introduction • Cognitive Radio Network • Multiuser system • Users competing for limited resources • Dynamic wireless environment • Transmit Power Control • Robust Resource Allocation Algorithms • Equilibrium and transient behavior analysis • Some tools from control theory i.e. variational inequalities and projected dynamical systems

4. Introduction Difference between Cognitive Radio and conventional wireless communication? Secondary user perspective Cognitive Signal Processing Cycle

5. Introduction • Why OFDM for CRs? • Spectrum holes come and go • Flexible spectrum shaping • Divide PU bandwidth into subcarriers • Minimize interference • Main resources in CRNs • Bandwidth Allocation (Sub carrier assignment) • Power( can be manipulated by CR users)

6. Introduction • Two ways to build a CRN • Evolutionary: Build around existing infrastructure • Revolutionary: No base stations; joint channel assignment and power control required • CRN: Combination of discrete events and continuous dynamics • Discrete events: users coming and going • Continuous events: power evolves continuously

7. Orthogonal Frequency Division Multiplexing • Multicarrier modulation scheme • Wide-band signal is converted into a number of narrow band signals • Frequency selective fading channel is converted into flat fading channels • Advantages • Efficient spectrum utilization • First built in frequency domain and transformed into time • Interleaving over OFDM symbols provides robustness against losses due to flat fading and noise

8. Orthogonal Frequency Division Multiplexing • Advantages • Spectrum tails overlap each other, at center frequency of each subcarrier all other subcarriers are zero. • Takes care of ISI caused by multipath delay of wireless channels • Less ISI leads to less complexity of equalization at receiver • Frequency diversity, flexibility, robustness etc

9. Cognitive Radio Environment • Receiver performs radio scene analysis • Information is sent to Tx • Predictive model can be used to predict holes • Competitive multiuser environment • Finding global optimum is not feasible • Goal is to obtain suboptimal solution in a reasonable time • Game Theory • Nash Equilibrium: For CRNs Nash equilibrium solution is reasonably good even though it may not be best solution

10. Cognitive Radio Environment • For CRN framework • IWFA is a good candidate for finding Nash Equilibrium solution • Why IWFA? • Transmit power control problem is formulated as a game or distributed convex optimization problem • Implemented in decentralized manner • Algorithm has linear convergence • Each user acts greedily to optimize its own performance • Can be shown to converge by LCP and NCP formulations

11. Cognitive Radio Environment • Drawbacks • It is suboptimal • Defenseless against clever selfish users • Resource Allocation in CRs • Using IWFA framework • More robustness

12. Iterative Waterfilling Algorithm • Formulating transmit power control in IWFA framework • Assume m active Tx-Rx pairs in ROI and n subcarriers in OFDM potentially available for communication • Let PS be subset of subcarriers that are being used by primary users and cannot be assigned to CRs • Main goal is spectral efficiency • Utility function is the data rate

13. IWFA • IWFA lets user i solve the following optimization problem: Where pik is user i’s transmit power over subcarrier k. CAPk = maximum allowable interference at subcarrier k

14. IWFA • Noise plus interference experienced by other user i at subcarrier k because of transmission of other users is Can be measured at receiver Background noise power Normalized interference gain from transmitter j to receiver i at subcarrier k SNR Gap Channel gain

15. IWFA • Path loss • In IWFA each user assumes other users powers are fixed • User i assumes pkjis fixed for i not equal to j • Concave maximization or convex minimization problem • Fi or –Fi • Decentralized implementation • No need to know pkj; for all j • Interference measured by receiver • No scaling problem

16. IWFA • Power Updation schemes • IWF • Simultaneous IWF • Asynchronous IWF • Let pi denote action of i user; action of remaining m-1 users is denoted by p-i

17. IWFA • Iterative waterfilling • Simultaneous iterative waterfilling • Asynchronous iterative waterfilling • Asynchronous scheme is more realistic than others

18. Stochastic optimization versus Robust Optimization • CRN is a highly dynamic environment • Users are moving all the time; they can join or leave in stochastic manner • Interference plus noise term in objective function and constraint are both time varying • IWFA takes form of an optimization problem under uncertainity • Stochastic Optimization • Robust Optimization

19. Stochastic optimization versus Robust Optimization • Stochastic Optimization • Let noise plus interference be combination of two terms: a nominal term and a perturbation term Objective function for stochastic optimization In practice a little is known about distribution of random noise therefore its not a very good approach

20. Stochastic optimization versus Robust Optimization • Robust Optimization • Based on worst case analysis • Each user tries to maximizes its utility function whereas other users and environment are trying to minimize that users utility • Suboptimality is traded in favor of robustness • Similar studies done for DSL • Conservative approach • Perturbation term in objective function makes it non-convex

21. Stochastic optimization versus Robust Optimization • Robust Optimization

22. IWFA as a variational inequality problem • Variational Inequality provides a tool for • formulating a variety of equilibrium problems • Analysing problems in terms of existence and uniqueness of solutions • It contains as special cases well known mathematical problems like nonlinear equations, optimization problems and complementarity problems • Providing us with algorithms with accompanying convergence analysis for computational purpose

23. IWFA as a variational inequality problem • Optimization Problems & Variational Inequality

24. IWFA as a variational inequality problem Geometric Interpretation of Variational Inequality problem x x-x* -F(x*) Normal Cone x* F(x*) Feasible set K

25. IWFA as a variational inequality problem • Nash equilibrium game can be formulated as variational inequality [53][54] etc. Our optimization problem is: P* is the Nash Equilibrium of game if and only if it is the solution of variational inequality problem given by VI(K,F)

26. IWFA as a variational inequality problem And K and F are given by:

27. IWFA as a variational inequality problem • IWFA as variational inequality for CRs • Based on [42], a similar work for DSL • The Lagrangian of optimization problem for users i can be written as:

28. IWFA as a variational inequality problem KKT conditions are given by: This KKT condition is a mixed linear complementarity system (proof given in paper) and therefore it corresponds to affine variational inequality problem!

29. IWFA as a variational inequality problem • Affine Variational Inequality system defined by:

30. Transient behavior analysis of CRNs • Behavior of network can change drastically over time • Equilibrium allocation of resources • Transient behavior of network • For equilibrium allocation of resources • IWFA as a variational inequality • Transient behavior of network • Projected Dynamic systems theory • Using PDS we can associate a differential equation with given Variational Inequality

31. Transient behavior analysis of CRNs • Projected Dynamic Systems • To study equilibrium problem in a dynamic framework • Associate an ODE to the VI • Stationary points of ODE coincide with solutions of VI • To incorporate the constraints use a projection operator • Resulting dynamic model enables us to study transient behavior of network

32. Transient behavior analysis of CRNs Dynamic Problems Static Problems Unconstrained World Dynamical Systems System of nonlinear equations Constrained World Variational Inequality Problems Projected Dynamic Systems

33. Transient behavior analysis of CRNs Variational Inequality Projected Dynamic System

34. Transient behavior analysis of CRNs • Projection operator Projection y of x on set K

35. Transient behavior analysis of CRNs Solution trajectory Constraint set Classical dynamic system Projected dynamic system

36. Transient behavior analysis of CRNs Trajectory of a projected dynamic system that evolves both in interior and on the boundary of constraint set Constraint set

37. Sensitivity Analysis of Equilibrium Solutions • Stability • Ability to maintain equilibrium against external perturbations • Condition for stability • If each user’s receiver has the proper distance from its own transmitter which is short compared to its distance from other active transmitters in the network, then it can be guaranteed that the network will reach a stable unique equilibrium.

38. Computer Experiments • Robust IWFA versus Classic IWFA • Spectrum hole disappears users increase power increasing interference • Addition of more users also increase interference • First scenario • m=5 nodes, n=2 subcarriers • At fourth time step 2 new users join network • Comparison of powers and data rates

39. Computer Experiments Robust IWFA versus Classic IWFA

40. Computer Experiments • Second scenario • m=5 nodes and n=4 available subcarriers • At fourth time step, two users join • At eighth time step third subcarrier is no longer available (i.e. a spectrum hole disappears)

41. Computer Experiments

42. Computer Experiments

43. Computer Experiments • Projected Dynamic Systems Scenario • 3 users and 3 subcarriers • All three subcarriers are idle • 3 D space: p1 p2 p3 • Second subcarrier is not available • 2 D space: p1 p3 plane • Third subcarrier is not available • 1 D space: p1 line • Third subcarrier becomes available again • 2 D space: p1 p3 plane • Second carrier becomes available again • 3 D space: p1 p2 p3

44. Computer Experiments

45. Computer Experiments • Sensitivity Analysis • System is perturbed and equilibrium points are calculated again • Perturbed system converges to original system

46. Computer Experiments Stability results for different subcarriers

47. Computer Experiments

48. Summary and Conclusions • CRN is a dynamic environment with a stochastic behavior • Fast convergence required to use resources before they disappear • Decentralized algorithms are better • Modified IWFA works for CRNs • A dynamic model for evolution of state using PDS • State trajectory enters higher and lower dimensions depending on holes

49. Future Work • Present work is about resource allocation in competitive CRNs • Cooperative scenarios can be considered to study gains acheivable by coordination

50. Questions