Wireless Communication Elec 534 Set I September 9, 2007

1. Wireless CommunicationElec 534Set ISeptember 9, 2007 Behnaam Aazhang

2. The Course • Light homework • Team project • Individual paper presentations • Mid October • Team project presentations • Early December

3. Multiuser Network • Multiple nodes with information

4. Outline • Transmission over simple channels • Information theoretic approach • Fundamental limits • Approaching capacity • Fading channel models • Multipath • Rayleigh • Rician

5. Outline • Transmission over fading channels • Information theoretic approach • Fundamental limits • Approaching achievable rates • Communication with “additional” dimensions • Multiple input multiple (MIMO) • Achievable rates • Transmission techniques • User cooperation • Achievable rates • Transmission techniques

6. Outline • Wireless network • Cellular radios • Multiple access • Achievable rate region • Multiuser detection • Random access

7. Why Information Theory? • Information is modeled as random • Information is quantified • Transmission of information • Model driven • Reliability measured • Rate is established

8. Information • Entropy • Higher entropy (more random) higher information content • Random variable • Discrete • Continuous

9. Communication • Information transmission • Mutual information Channel Useful Information Maximum useful information Noise; useless information

10. Wireless Interference • Information transmission Channel Useful Information Maximum useful information Randomness due to channel Noise; useless information

11. Multiuser Network • Multiple nodes with information

12. References • C.E. Shannon, W. Weaver, A Mathematical Theory Communication, 1949. • T.M. Cover and J. Thomas, Elements of Information Theory, 1991. • R. Gallager, Information Theory and Reliable Communication, 1968. • J. Proakis, Digital Communication, 4th edition • D. Tse and P. Viswanath, Fundamentals of Wireless Communication, 2005. • A. Goldsmith “Wireless Communication” Cambridge University Press 2005

13. References • E. Biglieri, J. Proakis, S. Shamai, Fading Channels: Information Theoretic and Communications, IEEE IT Trans.,1999. • A. Goldsmith, P. Varaiya, Capacity of Fading Channels with Channel Side Information, IEEE IT Trans. 1997. • I. Telatar, Capacity of Multi-antenna Gaussian Channels, European Trans. Telecomm, 1999. • A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity, Part I. Systemdescription,” IEEE Trans. Commun.,Nov. 2003. • ——, “User cooperation diversity. Part II. Implementation aspects andperformance analysis,” IEEE Trans. Commun.,Nov. 2003. • J. N. Laneman, D. N. C. Tse, and G. W. Wornell,“Cooperative diversityin wireless networks: Efficientprotocols and outage behavior,” IEEETrans. Inform. Theory, Dec. 2004. • M.A. Khojastepour, A. Sabharwal, and B. Aazhang, “On capacity of Gaussian ‘cheap’ relay channel,” GLOBECOM, Dec.2003.

14. Reading for Set 1 • Tse and Viswanath • Chapters 5.1-5.3, 3.1 • Appendices A, B.1-B.5 • Goldsmith • Chapters 1, 4.1,5 • Appendices A, B, C

15. Single Link AWGN Channel • Model where r(t) is the baseband received signal, b(t) is the information bearing signal, and n(t) is noise. • The signal b(t) is assumed to be band-limited to W. • The time period is assumed to be T. • The dimension of signal is N=2WT

16. Signal Dimensions • A signal with bandwidth W sampled at the Nyquist rate. • W complex (independent) samples per second. • Each complex sample is one dimension or degree of freedom. • Signal of duration T and bandwidth W has 2WT real degrees of freedom and can be represented 2WT real dimensions

17. Signals in Time Domain • Sampled at Nyquist rate • Example: three independent samples per second means three degrees of freedom Voltage 1 second time 1/W

18. Signal in Frequency Domain • Bandwidth W at carrier frequency fc Power Carrier frequency fc frequency W

19. Baseband Signal in Frequency Domain • Passband signal down converted • Bandwidth W Power frequency W

20. Sampling • The baseband signal sampled at rate W Where • Sinc function is an example of expansion basis

21. Model • There are N orthonormal basis functions to represent the information signal space. • For example, • The discrete time version

22. Noise • Assumed to be a Gaussian process • Zero mean • Wide sense stationary • Flat power spectral density with height • Passed through a filter with BW of W • Samples at the rate W are Gaussian • Samples are independent

23. Noise • Projection of noise • Projections, nionto orthonormal bases fi(t) are • zero mean • Gaussian • Variance

24. Noise • The samples of noise are Gaussian and independent • The received signal given the information samples are also Gaussian

25. Model • The discrete time formulation can come from sampling the received signal at the Nyquist rate of W • The final model • The discrete time model could have come from projection or simple sampling

26. Statistical Model • Key part of the model • The discrete time received signals are independent since noise is assumed white

27. Entropy • Differential entropy • Differential conditional entropy with

28. Example • A Gaussian random variable with mean and variance • The differential entropy is • If complex then it is • Among all random variables with fixed variance Gaussian has the largest differential entropy

29. Proof • Consider two zero mean random variables X and Y with the same variance • Assume X is Gaussian Variance of X

30. Proof • Kullback-Leibler distance Due to Gibbs inequality!

31. Gibbs’ Inequality • The KL distance is nonnegative

32. Capacity • Formally defined by Shannon as where the mutual information with

33. Capacity • Maximum reliable rate of information through the channel with this model. • In our model

34. Mutual Information • Information flow Channel Useful Information Maximum useful information Noise; useless information

35. Capacity • In this model the maximum is achieved when information vector has mutually independent and Gaussian distributed elements.

36. AWGN Channel Capacity • The average power of information signal • The noise variance

37. AWGN Capacity • The original Shannon formula per unit time • An alternate with energy per bit

38. Achievable Rate and Converse • Construct codebook with • N-dimensional space • Law of large numbers • Sphere packing

39. Sphere Packing • Number of spheres (ratio of volumes) • Non overlapping • As N grows the probability of codeword error vanishes • Higher rates not possible without overlap

40. Achievable Rate and Converse • Construct codebook with bits in N channel use

41. Achieving Capacity • The information vector should be mutually independent with Gaussian distribution • The dimension N should be large • Complexity • Source has information to transmit • Full buffer • Channel is available • No contention for access • Point to point

42. Achieving Capacity • Accurate model • Statistical • Noise • Deterministic • Linear channel • Signal model at the receiver • Timing • Synchronization

43. Approaching Capacity • High SNR: • Coded modulation with large constellation size • Large constellation with binary codes • Low SNR: • Binary modulation • Turbo coding • LDPC coding

44. Constellations and Coding