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Module-3 : Transmission Lecture-6 (4/5/00)

Learn about adaptive equalizers for channels with unknown or time-varying characteristics, including training sequence-based equalizer design and channel identification.

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Module-3 : Transmission Lecture-6 (4/5/00)

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  1. Module-3 : TransmissionLecture-6 (4/5/00) Marc Moonen Dept. E.E./ESAT, K.U.Leuven marc.moonen@esat.kuleuven.ac.be www.esat.kuleuven.ac.be/sista/~moonen/ Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven/ESAT-SISTA

  2. Lecture 6 : Adaptive Equalization Problem Statement : • Equalizers of Lecture-5 assume perfect knowledge of channel distortion (impulse response h(t)) and possibly also noise characteristics (variance/color) • What if channel is unknown or time-varying (e.g. mobile communications)... ? • Channel model identification and/or (direct) equalizer design based on training sequences (and/or decision directed operation) Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  3. Lecture 6 : Adaptive Equalization -Overview • Equalizers design subject to complexity constraint (=finite number of filter taps) • Training sequence based direct equalizer design • Training sequence based channel identification • Recursive/adaptive algorithms LMS (1965), RLS, Fast RLS • Blind Equalization • Postscript:Adaptive filters in digital communications Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  4. Complexity Constrained Equalizer Design • In Lecture-5, equalizer design ignores complexity issues (filter lengths,..) • If (=practical approach) the number of equalizer filter coefficients (`taps’) is fixed, then what would be an optimal equalizer ? • MMSE criterion based approach zero-forcing criterion generally not compatible with complexity constraint. (+ noise enhancement) Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  5. C(z) H(z) Complexity Constrained Equalizer Design Example : linear equalizer design • complexity constraint : (3 taps) • MMSE-LE equalizer is such that the slicer input is as close as possible (in expected value, E{.}) to transmitted symbol : Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  6. Complexity Constrained Equalizer Design Solution is given by Wiener Filter Theory: ….ignore formula! Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  7. Complexity Constrained Equalizer Design • Formula allows to compute MMSE equalizer from channel coefficients, noise variance, etc. • Similar formulas for DFE, fractionally spaced equalizers,… • Conclusion : Necessary theory available • Wiener Filter theory = basis for adaptive filter theory, see below. • Here: immediately move on to training sequence based equalizer design, which may be viewed as a`deterministic version’ of the above (with true symbol/sample values instead of expected values). Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  8. Training sequence based equalizer design • If the channel is unknown and/or time-varying, a fixed sequence of symbols (`training sequence’) may be transmitted for channel `probing’. • example : GSM -> 26 training bits in each burst of 148 bits (=17% `overhead’) • In the receiver, based on the knowledge of the training sequence, the channel model is identified and/or an equalizer is designed accordingly. Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  9. H(z) Training sequence based equalizer design • Assume simple channel model (linear filter +AWGN) • Assume transmitted training sequence is • Received samples are • Optimal (`least squares’) linear equalizer compareto page 5 ! Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  10. Training sequence based equalizer design • In matrix notation this is .… ...remember matrix algebra? (`overdetermined set of equations’) Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  11. Training sequence based equalizer design • `Least Squares’ (LS) solution is .… compareto page 6 ! Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  12. Training sequence based equalizer design • PS: possibly incorporate `delay optimization’ : check delay within a range, and then pick one that gives smallest error norm Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  13. Training sequence based equalizer design • Similar least squares problem for fractionally spaced eq..… ...optimal solution Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  14. Training sequence based equalizer design • Similar least squares problem for DFE..… …optimal solution Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  15. Training sequence based channel identification • Alternatively, the training sequence may be used to estimate a channel model, from which then an optimal equalizer (see Lecture-5) is computed (or by means of which an MLSE receiver is designed (ex: GSM)) Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  16. H(z) Training sequence based channel identification • Assume simple channel model (linear filter +AWGN) • Assume transmitted training sequence is • Received samples are • Optimal (`least squares’) channel model is compareto page 9 ! Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  17. Training sequence based channel identification • In matrix notation this is .… …optimal solution Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  18. Training sequence based channel identification • Conclusion : MMSE-optimal equalizer design (LE, DFE, FS) or channel identification may be reduced to solving an overdetermined set of linear equations A.x=b in the least squares sense where the optimal solution is always given as • `Fast algorithms’ available (e.g. Levinson, Schur), that exploit matrix structure (`constant along diagonals’) • In practice, sometimes iterative procedures (e.g. steepest descent) are used to find the optimal solution (a la LMS, see below). Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  19. Recursive/Adaptive Algorithms • Up till now we considered `batch processing’ : at the end of the training sequence, the complete batch of data is processed… • Is it possible to process data on a `per-sample’ basis, i.e. process samples as they come in? • Answer=Yes : `Adaptive Filters’ References : S. Haykin, `Adaptive filter theory’, Prentice-Hall 1996. M. Moonen & I. Proudler : `Introduction to adaptive filtering’, free copy @ www-address. Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  20. Recursive/Adaptive Algorithms • Starting point is the (common) least squares problem (=overdetermined set of equations) • Whenever new samples come in, a new row (=equation) is added to the underlying set of equations, and so the optimal solution vector x may be re-computed • Most adaptive filtering algorithms have the following form Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  21. Recursive/Adaptive Algorithms Example : Least-Mean-Squares (LMS) Algorithm (Widrow 1965) (=channel identification example of p.17) • is step-size parameter, controls adaptation speed. If too large -> divergence. Need for proper tuning ! Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  22. Recursive/Adaptive Algorithms Example : Least-Mean-Squares (LMS) Algorithm • LMS is a `stochastic gradient algorithm’, i.e. steepest descent algorithm for the least squares problem, with instantaneous estimates of the gradient. • LMS (and variants) are by far the most popular algorithms in practical systems. Reason = simple (to understand & to implement) • Complexity = O(N), where N is the number of filter taps (=dimension of x). • Disadvantage : often (too) slow convergence (e.g. 1000 training symbols) Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  23. Recursive/Adaptive Algorithms Example : `Normalized’ LMS • Normalize step-size parameter, i.e. use • For guaranteed convergence : hence simpler tuning Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  24. Recursive/Adaptive Algorithms Recursive Least Squares (RLS) algorithms : • Also of the form but now exact update for the solution vector (unlike LMS) • Fast convergence (unlike LMS) • Complexity is O(N^2), where N is the number of filter coefficients (dimension of x), which is often too much for practical systems • Formulas : see textbooks Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  25. Recursive/Adaptive Algorithms `Fast’ Recursive Least Squares (RLS) algorithms: • Reduce complexity of RLS algorithm by exploiting special properties (structure) of the involved matrices (cfr. supra: `constant along the diagonals’) • Convergence = RLS convergence ! • Complexity is O(N), where N is the number of filter coefficients (dimension of x), which approaches LMS-complexity. • Great algorithms, but hardly used in practice :-( • Formulas : see textbooks Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  26. Blind Equalization • Problem Statement : channel identification or equalizer initialization based on channel outputs only, i.e. without having to transmit a training sequence ?? • LMS-type algorithms : (constant modulus, Godard,…) simple but slow convergence (>1000 training symbols) Reference : S. Haykin (ed.), `Blind deconvolution’, Prentice-Hall 1994 • Algorithms based on higher-order statistics • Algorithms based on `2nd-order’ statistics or deterministic properties : fast, but mostly complex Reference : vast recent literature (IEEE Tr. SP,...) Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  27. Postscript Adaptive Filters in Digital Communications Adaptive filters are used in dig.comms. systems for -equalization (cfr. supra) -channel identification (cfr supra) -echo cancellation -interference suppression -etc.. Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  28. Postscript Basis for adaptive filter theory is Wiener filter theory • Prototype Wiener filtering scheme : Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  29. Postscript Prototype adaptive filtering scheme : • 2 operations: filtering + adaptation Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  30. Postscript Adaptive filters for channel identification : Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  31. Postscript Adaptive filters for channel identification : • line echo cancellation in a telephone network Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  32. Postscript Adaptive filters for channel identification : • echo cancellation in full-duplex modems Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  33. Postscript Adaptive filters for channel identification : • acoustic echo cancellation for conferencing Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  34. Postscript Adaptive filters for channel identification : • hands-free telephony Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  35. Postscript Adaptive filters for channel equalization : Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  36. Postscript Adaptive filters for channel equalization : decision-directed operation Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  37. Postscript Adaptive filters for channel equalization and interference cancellation (see also Lecture-10) Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  38. Conclusions • Training sequence based channel identification and/or equalization : Least squares optimization criterion provides common framework/solution procedure for LE, DFE, fractionally spaced equalization,.. • Recursive/adaptive implementation -simple & cheap (but slow) : LMS -fast (but sometimes too expensive) : RLS, Fast RLS Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

  39. Assignment 3.2 • Return to the zero-forcing fractionally spaced equalizer of assignment 3.1. • Run to your favorite computer & simulation program (e.g. Matlab, Simulink,…) & simulate a transmitter/channel/receiver system as follows: • Transmitter : random 2-PAM training symbols +1,-1 • Channel : choose (random) values for the hi’s in the model. No additive noise. • Receiver : NLMS-based adaptive zero-forcing equalizer. Select appropriate filter length (see Assignment 3.1). • Experiment with the step-size parameter, and observe convergence behavior. • Experiment with shorter and longer equalizer filter lengths. Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

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