1. A Multipath Channel Estimation Algorithm using the Kalman Filter. Rupul Safaya

2. 2 Organization zIntroduction zTheoretical Background zChannel Estimation Algorithm zConclusions zFuture Work

3. 3 Introduction

4. 4 Definitions: zChannel: In its most General sense can describe everything from the source to the sink of the radio signal. Including the physical medium. zIn this work “Channel” refers to the physical medium. zChannel Model: Is a mathematical representation of the transfer characteristics of the physical medium. zChannel models are formulated by observing the characteristics of the received signal. zThe one that best explains the received signal behavior is used to model the channel. zChannel Estimation: The process of characterizing the effect of the physical medium on the input sequence.

5. 5 General Channel Estimation Procedure

6. 6 zAim of any channel estimation procedure: zMinimize some sort of criteria, e.g. MSE. zUtilize as little computational resources as possible allowing easier implementation. zA channel estimate is only a mathematical estimation of what is truly happening in nature. zWhy Channel Estimation? zAllows the receiver to approximate the effect of the channel on the signal. zThe channel estimate is essential for removing inter symbol interference, noise rejection techniques etc. zAlso used in diversity combining, ML detection, angle of arrival estimation etc.

7. 7 Training Sequences vs. Blind Methods Training Sequence methods: zSequences known to the receiver are embedded into the frame and sent over the channel. zEasily applied to any communications system. zMost popular method used today. zNot too computationally intense. zHas a major drawback: It is wasteful of the information bandwidth. Blind Methods: z No Training sequences required z Uses certain underlying mathematical properties of the data being sent. z Excellent for applications where bandwidth is scarce. z Has the drawback of being extremely computationally intensive z Thus hard to implement on real time systems. zThere Are two Basic types of Channel Estimation Methods:

8. 8 Algorithm Overview zConsider a radio communications system using training sequences to do channel estimation. zThis thesis presents a method of improving on the training sequence based estimate without anymore bandwidth wastage. zJakes Model: Under certain assumptions we can adopt the Jakes model for the channel. yThis allows us to have a second estimate independent of the data based (training sequence) estimate. zThe Kalman estimation algorithm uses these two independent estimates of the channel to produce a LMMSE estimate. zPerformance improvement: As a result of using the Jakes model in conjunction with the data based estimates there is a significant gain in the channel estimate.

9. 9 Theoretical Background

10. 10 Signal Multipath

11. 11 Multipath zSignal multipath occurs when the transmitted signal arrives at the receiver via multiple propagation paths. zEach path can have a separate phase, attenuation, delay and doppler shift associated with it. zDue to signal multipath the received signal has certain undesirable properties like Signal Fading, Inter-Symbol-Interference, distortion etc. zTwo types of Multipath: yDiscrete: When the signal arrives at the receiver from a limited number of paths. yDiffuse: The received signal is better modeled as being received from a very large number of scatterers.

12. 12 Diffuse Multipath

13. 13 Tap Delayed Line Channel Model

14. 14 Tap-Delay Line Model

15. 15 Tap gain functions

16. 16 Model Parameters

17. 17 Jakes Model

18. 18

19. 19 Jakes Spectrum

20. 20

21. 21 The Channel Estimation Algorithm

22. 22 Introduction zAim: To improve on the data-only estimate. zJakes model: We have adopted the Jakes model for the radio channel. zTap-gains as auto-regressive processes: The Jakes power spectrum is used to represent the tap-gains as AR processes. zState-Space Representation: We have two independent estimates of the process from the data-based estimate and the Jakes model. yThese are used to formulate a State-Space representation for the tap- gain processes. yAn appropriate Kalman filter is derived from the state-space representation. zDerivation: The algorithm is developed first for a Gauss-Markov Channel and then for the Jakes Multipath channel

23. 23 AR representation of the Tap-gains zGeneral form: Any stationary random process can be represented as an infinite tap AR process. zThe current value is a weighted sum of previous values and the plant noise.

24. 24

25. 25 Data based estimator

26. 26

27. 27 Tracking a Gauss-Markov Channel

28. 28 zA Gauss-Markov tap-gain process has an exponential autocorrelation.

29. 29 Kalman Filter Derivation

30. 30 Kalman filter equations

31. 31 Simulation Parameters

32. 32 Results: Channel Estimation for a single ray Gauss-Markov channel

33. 33 MSE for the estimator:

34. 34 Tracking a single Jakes Tap-Gain Process

35. 35 Single ray Jakes Channel zConsider a single ray line of sight radio channel. zMore Complex Channel: The underlying channel model is no longer a single tap AR process. zAR representation: The tap-gain process with the Jakes spectrum is a stationary process. We can represent it as an AR process. yParameters: We derive the co-efficients for the process from the closed form expression of the Jakes channel-shaping filter. zState-Space representation: Using the AR model and the data based estimator, a state-space representation is derived. zKalman tracking filter: Similar to the Gauss-Markov case, a Kalman filter to track the process is derived from the State-Space representation.

36. 36 AR representation of the Jakes Process

37. 37

38. 38 Model Validation

39. 39 AR process length comparison

40. 40

41. 41 Kalman filter Derivation

42. 42

43. 43

44. 44 Simulation Parameters

45. 45

46. 46 Results

47. 47 Error covariance:

48. 48

49. 49 Multipath Channel Estimation

50. 50 Multipath Jakes Channel zConsider a multipath radio channel. zAssume the Jakes model on each path. zAR representation: For the Multipath case, a modification of the single ray AR system model is presented. zState-Space representation: Using the AR model and the data based estimator, a state-space representation is derived. zKalman tracking filter:Once again a vector Kalman filter is used to track the tap-gain functions.

51. 51 System model

52. 52

53. 53 Observation Model

54. 54

55. 55 Simulation parameters

56. 56

57. 57 Results

58. 58

59. 59

60. 60 Error Covariance

61. 61

62. 62 Conclusions zDeveloped a Kalman filter based channel estimation algorithm for the Multipath radio channel. zSignificant gain in performance over a training sequence based estimator. zThis improvement is obtained without wasting any more bandwidth. zAlso allows us to predict the channel state without having to wait for data.

63. 63 Future Work zUse Of Multiple Sampling Rates: yInstead of waiting for the data to arrive at the end of every frame we can run the Kalman filter at a higher rate than the frame rate. yIn the absence of a data based estimate perform the time-update portion of the algorithm and do a measurement update when data is received. yAllows estimates to be available as required. zDifferent process models on each path: yIn case the process model varies with path, we can still use the Kalman filter but with some modifications to the system matrix. zCorrelated paths: yFor correlated paths the Kalman filter needs to be modified.