Module-3 : Transmission Lecture-5 (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-5 Equalization K.U.Leuven/ESAT-SISTA

Prelude Comments on lectures being too fast/technical * I assume comments are representative for (+/-)whole group * Audience = always right, so some action needed…. To my own defense :-) * Want to give an impression/summary of what today’s transmission techniques are like (`box full of mathematics & signal processing’, see Lecture-1). Ex: GSM has channel identification (Lecture-6), Viterbi (Lecture-4),... * Try & tell the story about the maths, i.o. math. derivation. * Compare with textbooks, consult with colleagues working in transmission... Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Prelude Good news Bad news : * New start (I): Will summarize Lectures (1-2-)3-4. -only 6 formulas- * New start (II) : Starting point for Lectures 5-6 is 1 (simple) input-output model/formula (for Tx+channel+Rx). * Lectures 3-4-5-6 = basic dig.comms principles, from then on focus on specific systems, DMT (e.g. ADSL), CDMA (e.g. 3G mobile), ... Bad news : * Some formulas left (transmission without formulas = fraud) * Need your effort ! * Be specific about the further (math) problems you may have. Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Lecture-5 : Equalization Problem Statement : Optimal receiver structure consists of * Whitened Matched Filter (WMF) front-end (= matched filter + symbol-rate sampler + `pre-cursor equalizer’ filter) * Maximum Likelihood Sequence Estimator (MLSE), (instead of simple memory-less decision device) Problem: Complexity of Viterbi Algorithm (MLSE) Solution: Use equalization filter + memory-less decision device (instead of MLSE)... Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Lecture-5: Equalization - Overview Summary of Lectures (1-2-)3-4 Transmission of 1 symbol : Matched Filter (MF) front-end Transmission of a symbol sequence : Whitened Matched Filter (WMF) front-end & MLSE (Viterbi) Zero-forcing Equalization Linear filters Decision feedback equalizers MMSE Equalization Fractionally Spaced Equalizers Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Summary of Lectures (1-2-)3-4 Channel Model: Continuous-time channel =Linear filter channel + additive white Gaussian noise (AWGN) ... h(t) + ? ? n(t) AWGN transmitter channel receiver (to be defined) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Summary of Lectures (1-2-)3-4 Transmitter: * Constellations (linear modulation): n bits -> 1 symbol (PAM/QAM/PSK/..) * Transmit filter p(t) : r(t) transmit pulse s(t) n(t) p(t) + AWGN transmitter h(t) channel ? ... receiver (to be defined) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Summary of Lectures (1-2-)3-4 Transmitter: -> piecewise constant p(t) (`sample & hold’) gives s(t) with infinite bandwidth, so not the greatest choice for p(t).. -> p(t) usually chosen as a (perfect) low-pass filter (e.g. RRC) p(t) t Example transmit pulse s(t) p(t) transmitter t discrete-time symbol sequence continuous-time transmit signal Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Summary of Lectures (1-2-)3-4 Receiver: In Lecture-3, a receiver structure was postulated (front-end filter + symbol-rate sampler + memory-less decision device). For transmission of 1 symbol, it was found that the front-end filter should be `matched’ to the received pulse. 1/Ts p(t) h(t) + front-end filter transmit pulse n(t) AWGN transmitter channel receiver Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Summary of Lectures (1-2-)3-4 Receiver: In Lecture-4, optimal receiver design was based on a minimum distance criterion : Transmitted signal is Received signal p’(t)=p(t)*h(t)=transmitted pulse, filtered by channel Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Summary of Lectures (1-2-)3-4 Receiver: In Lecture-4, it was found that for transmission of 1 symbol, the receiver structure of Lecture 3 is indeed optimal ! p’(t)=p(t)*h(t) sample at t=0 1/Ts p(t) h(t) + p’(-t)* front-end filter transmit pulse n(t) AWGN transmitter channel receiver Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Summary of Lectures (1-2-)3-4 Receiver: For transmission of a symbol sequence, the optimal receiver structure is... 1/Ts p(t) h(t) + p’(-t)* front-end filter transmit pulse n(t) AWGN sample at t=k.Ts transmitter channel receiver Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Summary of Lectures (1-2-)3-4 Receiver: This receiver structure is remarkable, for it is based on symbol-rate sampling (=usually below Nyquist-rate sampling), which appears to be allowable if preceded by a matched-filter front-end. Criterion for decision device is too complicated. Need for a simpler criterion/procedure... Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Summary of Lectures (1-2-)3-4 Receiver: 1st simplification by insertion of an additional (magic) filter (after sampler). * Filter = `pre-cursor equalizer’ (see below) * Complete front-end = `Whitened matched filter’ p’(-t)* front-end filter 1/Ts receiver n(t) + AWGN transmit pulse p(t) transmitter h(t) channel 1/L*(1/z*) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Summary of Lectures (1-2-)3-4 Receiver: The additional filter is `magic’ in that it turns the complete transmitter-receiver chain into a simple input-output model: p(t) 1/Ts h(t) + p’(-t)* front-end filter transmit pulse 1/L*(1/z*) n(t) AWGN transmitter channel receiver Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Summary of Lectures (1-2-)3-4 Receiver: The additional filter is `magic’ in that it turns the complete transmitter-receiver chain into a simple input-output model: = additive white Gaussian noise means interference from future (`pre-cursor) symbols has been cancelled, hence only interference from past (`post-cursor’) symbols remains Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Summary of Lectures (1-2-)3-4 Receiver: Based on the input-output model one can compute the transmitted symbol sequence as A recursive procedure for this = Viterbi Algorithm Problem = complexity proportional to M^N ! (N=channel-length=number of non-zero taps in H(z) ) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Problem statement (revisited) Cheap alternative for MLSE/Viterbi ? Solution: equalization filter + memory-less decision device (`slicer’) Linear filters Non-linear filters (decision feedback) Complexity : linear in number filter taps Performance : with channel coding, approaches MLSE performance Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Preliminaries (I) Our starting point will be the input-output model for transmitter + channel + receiver whitened matched filter front-end Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Preliminaries (II) PS: z-transform is `shorthand notation’ for discrete-time signals… …and for input/output behavior of discrete-time systems H(z) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Preliminaries (III) PS: if a different receiver front-end is used (e.g. MF instead of WMF, or …), a similar model holds for which equalizers can be designed in a similar fashion... Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Preliminaries (IV) PS: properties/advantages of the WMF front end additive noise = white (colored in general model) H(z) does not have anti-causal taps pps: anti-causal taps originate, e.g., from transmit filter design (RRC, etc.). practical implementation based on causal filters + delays... H(z) `minimum-phase’ : =`stable’ zeroes, hence (causal) inverse exists & stable = energy of the impulse response maximally concentrated in the early samples Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Preliminaries (V) `Equalization’: compensate for channel distortion. Resulting signal fed into memory-less decision device. In this Lecture : - channel distortion model assumed to be known - no constraints on the complexity of the equalization filter (number of filter taps) Assumptions relaxed in Lecture 6 Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Zero-forcing & MMSE Equalizers 2 classes : Zero-forcing (ZF) equalizers eliminate inter-symbol-interference (ISI) at the slicer input Minimum mean-square error (MMSE) equalizers tradeoff between minimizing ISI and minimizing noise at the slicer input Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Zero-forcing Equalizers Zero-forcing Linear Equalizer (LE) : - equalization filter is inverse of H(z) - decision device (`slicer’) Problem : noise enhancement ( C(z).W(z) large) H(z) C(z) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Zero-forcing Equalizers Zero-forcing Linear Equalizer (LE) : - ps: under the constraint of zero-ISI at the slicer input, the LE with whitened matched filter front-end is optimal in that it minimizes the noise at the slicer input - pps: if a different front-end is used, H(z) may have unstable zeros (non-minimum-phase), hence may be `difficult’ to invert. Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Zero-forcing Equalizers Zero-forcing Non-linear Equalizer Decision Feedback Equalization (DFE) : - derivation based on `alternative’ inverse of H(z) : (ps: this is possible if H(z) has , which is another property of the WMF model) - now move slicer inside the feedback loop : H(z) 1-H(z) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Zero-forcing Equalizers moving slicer inside the feedback loop has… - beneficial effect on noise: noise is removed that would otherwise circulate back through the loop - beneficial effect on stability of the feedback loop: output of the slicer is always bounded, hence feedback loop always stable Performance intermediate between MLSE and linear equaliz. D(z) H(z) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Zero-forcing Equalizers Decision Feedback equalization (DFE) : - general DFE structure C(z): `pre-cursor’ equalizer (eliminates ISI from future symbols) D(z): `post-cursor’ equalizer (eliminates ISI from past symbols) H(z) C(z) D(z) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Zero-forcing Equalizers Decision Feedback equalization (DFE) : - Problem : Error propagation Decision errors at the output of the slicer cause a corrupted estimate of the postcursor ISI. Hence a single error causes a reduction of the noise margin for a number of future decisions. Results in increased bit-error rate. H(z) C(z) D(z) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Zero-forcing Equalizers `Figure of merit’ receiver with higher `figure of merit’ has lower error probability is `matched filter bound’ (transmission of 1 symbol) DFE-performance lower than MLSE-performance, as DFE relies on only the first channel impulse response sample (eliminating all other ‘s), while MLSE uses energy of all taps . DFE benefits from minimum-phase property (cfr. supra, p.20) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

MMSE Equalizers Zero-forcing equalizers: minimize noise at slicer input under zero-ISI constraint Generalize the criterion of optimality to allow for residual ISI at the slicer & reduce noise variance at the slicer =Minimum mean-square error equalizers Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

MMSE Equalizers MMSE Linear Equalizer (LE) : - combined minimization of ISI and noise leads to H(z) C(z) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

MMSE Equalizers - signal power spectrum (normalized) - noise power spectrum (white) - for zero noise power -> zero-forcing - (in the nominator) is a discrete-time matched filter, often `difficult’ to realize in practice (stable poles in H(z) introduce anticausal MF) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

MMSE Equalizers MMSE Decision Feedback Equalizer : MMSE-LE has correlated `slicer errors’ (=difference between slicer in- and output) MSE may be further reduced by incorporating a `whitening’ filter (prediction filter) E(z) for the slicer errors E(z)=1 -> linear equalizer Theory & formulas : see textbooks H(z) C(z)E(z) 1-E(z) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Fractionally Spaced Equalizers Motivation: All equalizers (up till now) based on (whitened) matched filter front-end, i.e. with symbol-rate sampling, preceded by an (analog) front-end filter matched to the received pulse p’(t)=p(t)*h(t). Symbol-rate sampling = below Nyquist-rate sampling (aliasing!). Hence matched filter is crucial for performance ! MF front-end requires analog filter, adapted to channel h(t), hence difficult to realize... A fortiori: what if channel h(t) is unknown ? Synchronization problem : correct sampling phase is crucial for performance ! Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Fractionally Spaced Equalizers Fractionally spaced equalizers are based on Nyquist-rate sampling, usually 2 x symbol-rate sampling (if excess bandwidth < 100%). Nyquist-rate sampling also provides sufficient statistics, hence provides appropriate front-end for optimal receivers. Sampler preceded by fixed (i.e. channel independent) analog anti-aliasing (e.g. ideal low-pass) front-end filter. `Matched filter’ is moved to digital domain (after sampler). Avoids synchronization problem associated with MF front-end. Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Fractionally Spaced Equalizers Input-output model for fractionally spaced equalization : `symbol rate’ samples : `intermediate’ samples : may be viewed as 1-input/2-outputs system Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Fractionally Spaced Equalizers Discrete-time matched filter + Equalizer (LE) : Fractionally spaced equalizer (LE) : 1/2Ts F(f) MF(z) 2 C(z) equalizer 1/2Ts F(f) C(z) 2 Fractionally spaced equalizer Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Fractionally Spaced Equalizers Fractionally spaced equalizer (DFE): Theory & formulas : see textbooks & Lecture 6 1/2Ts F(f) C(z) 2 D(z) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Conclusions Cheaper alternatives to MLSE, based on equalization filters + memoryless decision device (slicer) Symbol-rate equalizers : -LE versus DFE -zero-forcing versus MMSE -optimal with matched filter front-end, but several assumptions underlying this structure are often violated in practice Fractionally spaced equalizers (see also Lecture-6) Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

Assignment 3.1 Symbol-rate zero-forcing linear equalizer has i.e. a finite impulse response (`all-zeroes’) filter is turned into an infinite impulse response filter Investigate this statement for the case of fractionally spaced equalization, for a simple channel model and discover that there exist finite-impulse response inverses in this case. This represents a significant advantage in practice. Investigate the minimal filter length for the zero-forcing equalization filter. Module-3 Transmission Marc Moonen Lecture-5 Equalization K.U.Leuven-ESAT/SISTA