Wireless Communication Elec 534 Set IV October 23, 2007

1. Wireless CommunicationElec 534Set IVOctober 23, 2007 Behnaam Aazhang

2. Reading for Set 4 • Tse and Viswanath • Chapters 7,8 • Appendices B.6,B.7 • Goldsmith • Chapters 10

3. Outline • Channel model • Basics of multiuser systems • Basics of information theory • Information capacity of single antenna single user channels • AWGN channels • Ergodic fast fading channels • Slow fading channels • Outage probability • Outage capacity

4. Outline • Communication with additional dimensions • Multiple input multiple output (MIMO) • Achievable rates • Diversity multiplexing tradeoff • Transmission techniques • User cooperation • Achievable rates • Transmission techniques

5. Dimension • Signals for communication • Time period T • Bandwidth W • 2WT natural real dimensions • Achievable rate per real dimension

6. Communication with Additional Dimensions: An Example • Adding the Q channel • BPSK to QPSK • Modulated both real and imaginary signal dimensions • Double the data rate • Same bit error probability

7. Communication with Additional Dimensions • Larger signal dimension--larger capacity • Linear relation • Other degrees of freedom (beyond signaling) • Spatial • Cooperation • Metric to measure impact on • Rate (multiplexing) • Reliability (diversity) • Same metric for • Feedback • Opportunistic access

8. Multiplexing Gain • Additional dimension used to gain in rate • Unit benchmark: capacity of single link AWGN • Definition of multiplexing gain

9. Diversity Gain • Dimension used to improve reliability • Unit benchmark: single link Rayleigh fading channel • Definition of diversity gain

10. Multiple Antennas • Improve fading and increase data rate • Additional degrees of freedom • virtual/physical channels • tradeoff between diversity and multiplexing Transmitter Receiver

11. Multiple Antennas • The model where Tc is the coherence time Transmitter Receiver

12. Basic Assumption • The additive noise is Gaussian • The average power constraint

13. Matrices • A channel matrix • Trace of a square matrix

14. Matrices • The Frobenius norm • Rank of a matrix = number of linearly independent rows or column • Full rank if

15. Matrices • A square matrix is invertible if there is a matrix • The determinant—a measure of how noninvertible a matrix is! • A square invertible matrix U is unitary if

16. Matrices • Vector X is rotated and scaled by a matrix A • A vector X is called the eigenvector of the matrix and lambda is the eigenvalue if • Then with unitary and diagonal matrices

17. Matrices • The columns of unitary matrix U are eigenvectors of A • Determinant is the product of all eigenvalues • The diagonal matrix

18. Matrices • If H is a non square matrix then • Unitary U with columns as the left singular vectors and unitary V matrix with columns as the right singular vectors • The diagonal matrix

19. Matrices • The singular values of H are square root of eigenvalues of square H

20. MIMO Channels • There are channels • Independent if • Sufficient separation compared to carrier wavelength • Rich scattering • At transmitter • At receiver • The number of singular vectors of the channel • The singular vectors are the additional (spatial) degrees of freedom

21. Channel State Information • More critical than SISO • CSI at transmitter and received • CSI at receiver • No CSI • Forward training • Feedback or reverse training

22. Fixed MIMO Channel • A vector/matrix extension of SISO results • Very large coherence time

23. Exercise • Show that if X is a complex random vector with covariance matrix Q its differential entropy is largest if it was Gaussian

24. Solution • Consider a vector Y with the covariance as X

25. Solution • Since X and Y have the same covariance Q then

26. Fixed Channel • The achievable rate with • Differential entropy maximizer is acomplex Gaussian random vector with some covariance matrix Q

27. Fixed Channel • Finding optimum input covariance • Singular value decomposition of H • The equivalent channel

28. Parallel Channels • At most parallel channels • Power distribution across parallel channels

29. Parallel Channels • A few useful notes

30. Parallel Channels • A note

31. Fixed Channel • Diagonal entries found via water filling • Achievable rate with power

32. Example • Consider a 2x3 channel • The mutual information is maximized at

33. Example • Consider a 3x3 channel • Mutual information is maximized by

34. Ergodic MIMO Channels • A new realization on each channel use • No CSI • CSIR • CSITR?

35. Fast Fading MIMO with CSIR • Entries of H are independent and each complex Gaussian with zero mean • If V and U are unitary then distribution of H is the same as UHV* • The rate

36. MIMO with CSIR • The achievable rate since the differential entropy maximizer is a complex Gaussian random vector with some covariance matrix Q

37. Fast Fading and CSIR • Finally, with • The scalar power constraint • The capacity achieving signal is circularly symmetric complex Gaussian (0,Q)

38. MIMO CSIR • Since Q is non-negative definite Q=UDU* • Focus on non-negative definite diagonal Q • Further, optimum

39. Rayleigh Fading MIMO • CSIR achievable rate • Complex Gaussian distribution on H • The square matrix W=HH* • Wishart distribution • Non negative definite • Distribution of eigenvalues

40. Ergodic / Fast Fading • The channel coherence time is • The channel known at the receiver • The capacity achieving signal b must be circularly symmetric complex Gaussian

41. Slow Fading MIMO • A channel realization is valid for the duration of the code (or transmission) • There is a non zero probability that the channel can not sustain any rate • Shannon capacity is zero

42. Slow Fading Channel • If the coherence time Tc is the block length • The outage probability with CSIR only with and

43. Slow Fading • Since • Diagonal Q is optimum • Conjecture: optimum Q is

44. Example • Slow fading SIMO, • Then and • Scalar

45. Example • Slow fading MISO, • The optimum • The outage

46. Diversity and Multiplexing for MIMO • The capacity increase with SNR • The multiplexing gain

47. Diversity versus Multiplexing • The error measure decreases with SNR increase • The diversity gain • Tradeoff between diversity and multiplexing • Simple in single link/antenna fading channels

48. Coding for Fading Channels • Coding provides temporal diversity or • Degrees of freedom • Redundancy • No increase in data rate

49. M versus D (0,MRMT) Diversity Gain (min(MR,MT),0) Multiplexing Gain