Wireless Communication Elec 534 Set III October 11, 2007

1. Wireless CommunicationElec 534Set IIIOctober 11, 2007 Behnaam Aazhang

2. Reading for Set 3 • Tse and Viswanath • Chapters 5.4,6 • Appendices B.8,B.9 • Goldsmith • Chapters 4

3. Model • A simple discrete time model where h is a complex Gaussian distributed fading coefficient • Channel distribution information (CDI) at transmitter and receiver • Channel state information at receiver (and CDI) • Channel state information at transmitter and receiver (and CDI)

4. Channel Distribution Information (CDI) • Achievable rate • Finding the maximizer is non trivial • For Rayleigh independent channel coefficients • Maximizing input is discrete with finite number of mass points • Mass at zero

5. Channel Distribution Information (CDI) • Achievable rate computed numerically • Maximizing input distribution computed numerically • Not much to discuss—little analytical results

6. CSI Model • State of the channel S (a function of h ) • Known to the receiver as V • Known to the transmitter as U

7. CSIR • Channel state as a part of channel output since fading (or more precisely CSIR) is independent of the channel input r b v n

8. CSIR • Proof

9. Ergodic • The achievable rate when CSIR but no CSI at transmitter

10. Ergodic Capacity • The model • Perfect channel state information at receiver

11. Ergodic • The achievable rate is not a variable in time • If channel gain changes instantaneously the rate does not change • The rate is achieved over a long long codebook across different realizations of the channel • Long long decoding delay

12. Fading • Fading does not improve Ergodic capacity • The key to the proof is Jensen’s inequality

13. Example • A flat fading (frequency nonselective) with independent identically distributed channel gain as

14. Example • CSIR no CSIT • Other system assumptions

15. Example • The three possible signal to noise ratios

16. Example • Ergodic capacity

17. Example • Average SNR • The capacity of AWGN channel with the average SNR

18. CSITR r b v u n

19. CSITR • The mutual information • Capacity when there is CSI at transmitter and receiver • The original definition is not applicable • Define fading channel capacity

20. CSITR • A result for multi-state channel due to Wolfowitz capacity for each state • Applied to CSITR

21. Ergodic Capacity • Channel state information at transmitter and receiver • Power adjusted with constraint

22. Achievable Rate with CSITR • Constraint optimization • Solving via differentiation

23. Power Control • The solution is • Temporal water filling • Variable rate and variable power • Different size code books • Multiplexing encoders and decoders

24. Power Control Minimum Channel Quality Threshold Allocated Power Better Channels

25. Water Filling Solution

26. Capacity with CSITR • The maximized rate • The threshold not a function of average power limit

27. Power Control for Fading Channels

28. Example • A flat fading (frequency nonselective) with independent identically distributed channel gain as

29. Example • CSITR • Other system assumptions

30. Example • The three possible signal to noise ratios

31. Example • Calculate the threshold • If the weakest channel is not used a consistent threshold emerges

32. Example • Ergodic capacity

33. Example • Average SNR • The capacity of AWGN channel with the average SNR

34. Probability of Outage • Achieving ergodic channel capacity • Codewords much be longer than coherence time • Slow fading channels have long coherence times • Ergodic capacity more relevant in fast fading cases

35. Outage • A burst with signal to noise ratio • Probability of outage • Capacity with outage • Information sent over a burst • Limited decoding delay • Nonzero probability of decoding error

36. Outage • The minimum required channel gain depends on the target rate. • When instantaneous mutual information is less than target rate depends on the channel realization

37. Outage • Probability of outage (CSIR) • Fading channel (CSIR)

38. CSI at Transmitter and Receiver • Use CSITR to meet a target rate • Channel inversion • Minimize outage • Truncated channel inversion

39. Outage • Probability of outage with CSITR • Fading channel with CSITR

40. Power Control • Outage minimization • The solution for CSITR • Truncation with channel inversion

41. Power Control Minimum Channel Quality Threshold Rayleigh PDF Allocated Power Better Channels

42. Minimum Channel Quality Threshold Power Control Realization

43. Outage Capacity • Target probability of outage • Fixed power • The outage capacity

44. Frame Error Rate • An appropriate performance metric • In many examples, probability of outage is a lower bound to Frame Error Rate

45. Frequency Selective • Recall the input output relationship

46. Capacity for Frequency Selective Channels • Consider a time invariant channel • CSI is available at transmitter and receiver • Block frequency selective fading • An equivalent parallel channel model

47. CSITR: Frequency Selective • The sum of rates • The power distribution

48. Power Control • The power distribution threshold • Spectral water filling • Variable rate and variable power across channels • Different size code books • Multiplexing encoders and decoders

49. Achievable Rate • The rate

50. Frequency Selective Fading • Continuous transfer function • Power distribution across spectrum