4/5/00 p. 1 Postacademic Course on Telecommunications Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven/ESAT-SISTA Module-3 : Transmission Lecture-6 (4/5/00) Marc Moonen Dept. E.E./ESAT, K.U.Leuven

Postacademic Course on Telecommunications 4/5/00 p. 2 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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)

Postacademic Course on Telecommunications 4/5/00 p. 3 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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

Postacademic Course on Telecommunications 4/5/00 p. 4 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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)

Postacademic Course on Telecommunications 4/5/00 p. 5 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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 : H(z) C(z)

Postacademic Course on Telecommunications 4/5/00 p. 6 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Complexity Constrained Equalizer Design Solution is given by Wiener Filter Theory: ….ignore formula!

Postacademic Course on Telecommunications 4/5/00 p. 7 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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).

Postacademic Course on Telecommunications 4/5/00 p. 8 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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.

Postacademic Course on Telecommunications 4/5/00 p. 9 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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 H(z) compare to page 5 !

Postacademic Course on Telecommunications 4/5/00 p. 10 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Training sequence based equalizer design In matrix notation this is.…...remember matrix algebra? (`overdetermined set of equations’)

Postacademic Course on Telecommunications 4/5/00 p. 11 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Training sequence based equalizer design `Least Squares’ (LS) solution is.… compare to page 6 !

Postacademic Course on Telecommunications 4/5/00 p. 12 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Training sequence based equalizer design PS: possibly incorporate `delay optimization’ : check delay within a range, and then pick one that gives smallest error norm

Postacademic Course on Telecommunications 4/5/00 p. 13 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Training sequence based equalizer design Similar least squares problem for fractionally spaced eq..…...optimal solution

Postacademic Course on Telecommunications 4/5/00 p. 14 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Training sequence based equalizer design Similar least squares problem for DFE..… …optimal solution

Postacademic Course on Telecommunications 4/5/00 p. 15 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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))

Postacademic Course on Telecommunications 4/5/00 p. 16 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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 H(z) compare to page 9 !

Postacademic Course on Telecommunications 4/5/00 p. 17 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Training sequence based channel identification In matrix notation this is.… …optimal solution

Postacademic Course on Telecommunications 4/5/00 p. 18 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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).

Postacademic Course on Telecommunications 4/5/00 p. 19 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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 M. Moonen & I. Proudler : `Introduction to adaptive filtering’, free www-address.

Postacademic Course on Telecommunications 4/5/00 p. 20 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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

Postacademic Course on Telecommunications 4/5/00 p. 21 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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 !

Postacademic Course on Telecommunications 4/5/00 p. 22 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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 training symbols)

Postacademic Course on Telecommunications 4/5/00 p. 23 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Recursive/Adaptive Algorithms Example : `Normalized’ LMS Normalize step-size parameter, i.e. use For guaranteed convergence : hence simpler tuning

Postacademic Course on Telecommunications 4/5/00 p. 24 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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

Postacademic Course on Telecommunications 4/5/00 p. 25 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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

Postacademic Course on Telecommunications 4/5/00 p. 26 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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,...)

Postacademic Course on Telecommunications 4/5/00 p. 27 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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..

Postacademic Course on Telecommunications 4/5/00 p. 28 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Postscript Basis for adaptive filter theory is Wiener filter theory Prototype Wiener filtering scheme :

Postacademic Course on Telecommunications 4/5/00 p. 29 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Postscript Prototype adaptive filtering scheme : 2 operations: filtering + adaptation

Postacademic Course on Telecommunications 4/5/00 p. 30 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Postscript Adaptive filters for channel identification :

Postacademic Course on Telecommunications 4/5/00 p. 31 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Postscript Adaptive filters for channel identification : line echo cancellation in a telephone network

Postacademic Course on Telecommunications 4/5/00 p. 32 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Postscript Adaptive filters for channel identification : echo cancellation in full-duplex modems

Postacademic Course on Telecommunications 4/5/00 p. 33 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Postscript Adaptive filters for channel identification : acoustic echo cancellation for conferencing

Postacademic Course on Telecommunications 4/5/00 p. 34 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Postscript Adaptive filters for channel identification : hands-free telephony

Postacademic Course on Telecommunications 4/5/00 p. 35 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Postscript Adaptive filters for channel equalization :

Postacademic Course on Telecommunications 4/5/00 p. 36 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Postscript Adaptive filters for channel equalization : decision-directed operation

Postacademic Course on Telecommunications 4/5/00 p. 37 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA Postscript Adaptive filters for channel equalization and interference cancellation (see also Lecture-10)

Postacademic Course on Telecommunications 4/5/00 p. 38 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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

Postacademic Course on Telecommunications 4/5/00 p. 39 Module-3 Transmission Marc Moonen Lecture-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA 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.