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1. Workshop on Random Matrix Theory and Wireless Communications Bridging the Gaps:Free Probability and Channel Capacity • Antonia Tulino • Università degli Studi di Napoli • Chautauqua Park, Boulder, Colorado,July 17, 2008 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAA

2. Linear Vector Channel noise=AWGN+interference N-dimensional output K-dimensional input (NK)channel matrix • Variety of communication problems by simply reinterpreting K, N, and H • Fading • Wideband • Multiuser • Multiantenna

3. Role of the Singular Values Mutual Information: Ergodic case: Non-Ergodic case:

4. Role of the Singular Values Minumum Mean-Square Error (MMSE) :

5. Independent and Identically distributed entries • Separable Correlation Model • UIU-Model with independent arbitrary distrbuted entries • with which is uniformly distributed over the manifold of complex matrices such that H-Model

6. Gaussian Erasure Channels Flat Fading & Deterministic ISI: Random erasure mechanisms: • link congestion/failure (networks) • cellular system with unreliable wired infrastructure • impulse noise (DSL) • faulty transducers (sensor networks)

7. i.i.d. with uniform distribution in [0, 1]d, d-Fold Vandermonde Matrix • Sensor networks • Multiantenna multiuser communications • Detection of distributed Targets

8. an i.i.d sequence Flat Fading & Deterministic ISI:

9. Formulation = asymptotically circulant matrix (stationary input (with PSD ) = asymptotically circulant matrix Grenander-Szego theorem Eigenvalues ofS

10. the water levelis chosen so that: where is the waterfilling input power spectral density given by: Deterministic ISI & |Ai|=1 Key Tool: Grenander-Szego theorem on the distribution of the eigenvalues of large Toeplitz matrices

11. Deterministic ISI & Flat Fading Key Question: The distribution of the eigenvalues of a large-dimensional random matrix: • S = asymptotically circulant matrix • A = random diagonal fading matrix

12. Ai =ei ={0,1} Key Question: The distribution of the eigenvalues of a large-dimensional random matrix: • S = asymptotically circulant matrix • E = random 0-1 diagonal matrix

13. with X a nonnegative random variable whose distribution is while g is a nonnegative real number. RANDOM MATRIX THEORY:- & Shannon-Transform The - and Shannon-transform of an nonnegative definite random matrix , with asymptotic ESD A. M. Tulino and S. Verdú“Random Matrices and Wireless Communications,”Foundations and Trends in Communications and Information Theory, vol. 1, no. 1, June 2004.

14. Theorem: The Shannon transform and -transforms are related through: where is defined by the fixed-point equation RANDOM MATRIX THEORY:Shannon-Transform • Abe a nonnegative definite random matrix.

15. is monotonically increasing with g • which is the solution to the equation • is monotonically decreasing with y Property of

16. Theorem: -Transform Theorem: The -transform of is where is the solution to:

17. Theorem: Shannon-Transform Theorem: The Shannon-transform of is where a and n are the solutions to:

19. Stationary Gaussian inputs with power spectral Flat Fading & Deterministic ISI: Theorem: The mutual information is: with

20. Stationary Gaussian inputs with power spectral Flat Fading & Deterministic ISI: Theorem: The mutual information is: with

21. Stationary Gaussian inputs with power spectral Flat Fading & Deterministic ISI: Theorem: The mutual information is: with

22. Special Case: No Fading  

23. Special Case: Memoryless Channels  

24. Stationary Gaussian inputs with power spectral Special case: Gaussian Erasure Channels Theorem: The mutual information is: with

25. Stationary Gaussian inputs with power spectral with : Flat Fading & Deterministic ISI: Let Theorem: The mutual information is:

26. • ─ n = 200 Example n=200

27. • n = 1000 ─ Example n=1000

28. with so that: Input Optimization Theorem: • be an random matrix • such that i-th column of A. M. Tulino, A. Lozano and S. Verdú“Capacity-Achieving Input Covariance for Single-user Multi-Antenna Channels”,IEEE Trans. on Wireless Communications 2006

29. Input Optimization Theorem: The capacity-achieving input power spectral density is: where and is chosen so that

30. the waterfilling solutionfor g the fading-free water level for g Input Optimization Corollary: Effect of fading on the capacity-achieving input power spectral density = SNR penalty with k< 1regulates amount of water admitted on each frequency tailoring the waterfilling for no-fading to fading channels.

31. Theorem: -Transform Theorem: The -transform of is where is the solution to:

32. Proof: Key Ingredient • We can replace S by it circulant asymptotic equivalent counterpart, =FLF† • Let Q = EF, denote by qi the ith column of Q, and let

33. Proof: Matrix inversion lemma:

34. Proof:

35. Proof: Lemma:

36. Asymptotics • Low-power ( ) • High-power ( )

37. Asymptotics: High-SNR At large SNR we can closely approximate it linearly need and S0 where High-SNR dB offset High-SNR slope

38. Asymptotics: High-SNR Theorem: Let , and the generalized bandwidth,

39. Asymptotics • Sporadic Erasure (e!0) • Sporadic Non-Erasure (e!1)

40. Memoryless noisy erasure channel High SNR where is the water level of the PSD that achieves Low SNR Asymptotics: Sporadic Erasures (e0) Theorem: For any output power spectral densityand Theorem: For sporadic erasures:

41. Theorem: Optimizingover with with the maximum channel gain Asymptotics: Sporadic Non-Erasures (e1) Theorem:

42. S(f) =1 Bounds: Theorem: The mutual information rate is lower bounded by: Equality

43. Bounds: Theorem: The mutual information rate is upper bounded by:

44. d-Fold Vandermonde Matrix Diagonal matrix (either random or deterministic) with supported compact measure Diagonal matrix (either random or deterministic) with supported compact measure

45. d-Fold Vandermonde Matrix Theorem: The -transform of is The Shannon-Transformr is

46. d-Fold Vandermonde Matrix Theorem: The p-moment of is:

47. Summary • Asymptotic distribution of A S A ---new result at the intersection of the asymptotic eigenvalue distribution of Toeplitz matrices and of random matrices --- • The mutual information of a Channel with ISI and Fading. • Optimality of waterfilling in the  presence of fading known at the receiver. • Easily computable asymptotic expressions in various regimes (low and high SNR) • New result for d-fold Vandermond matrices and on their product with diagonal matrices