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Lecture5

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The document summarizes a lecture on equalization techniques for digital communications. It begins with the lecturer acknowledging feedback that previous lectures moved too quickly and were too technical. It then provides a high-level overview of equalization techniques, including: - Zero-forcing equalization using linear filters and decision feedback equalizers - MMSE equalization - Fractionally spaced equalizers It also summarizes the key concepts from previous lectures on digital transmission models and the optimal receiver structure involving a whitened matched filter front-end and maximum likelihood sequence estimation (MLSE). The goal of equalization techniques is to provide a lower complexity alternative to MLSE for the decision device while approaching similar performance with the use of channel coding

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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/ Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven/ESAT-SISTA 4/5/00 p. 1
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... Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 2
Prelude Good 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. Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 3
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)... Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 4
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 Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 5
Summary of Lectures (1-2-)3-4 Channel Model: ak (symbols) ˆ ak h(t) ? transmitter + n(t) AWGN channel ... ? receiver (to be defined) Continuous-time channel =Linear filter channel + additive white Gaussian noise (AWGN) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 6
Summary of Lectures (1-2-)3-4 Transmitter: r(t) s(t) ˆ ak ak . Es p(t) h(t) ... transmit pulse transmitter + n(t) AWGN channel ? receiver (to be defined) * Constellations (linear modulation): n bits -> 1 symbol a k (PAM/QAM/PSK/..) * Transmit filter p(t) : s(t ) Es . ak . p(t kTs ) k Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 7
Summary of Lectures (1-2-)3-4 p(t) Transmitter: t s(t) Example ak . Es t p(t) discrete-time symbol sequence transmit pulse continuous-time transmit signal 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) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 8
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. a0 . Es p(t) h(t) transmit pulse transmitter Postacademic Course on Telecommunications 1/Ts + n(t) AWGN channel Module-3 Transmission Lecture-5 Equalization front-end filter u0 ˆ a0 receiver Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 9
Summary of Lectures (1-2-)3-4 Receiver: In Lecture-4, optimal receiver design was based on a minimum distance criterion : min a0 ,a1 ,...,aK | r (t ) ˆ ˆ ˆ ˆ ak . p' (t kTs ) |2 dt Es . k • Transmitted signal is • Received signal s(t ) Es . ak . p(t kTs ) k r (t ) Es . ak . p' (t kTs ) n(t ) k • p’(t)=p(t)*h(t)=transmitted pulse, filtered by channel Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 10
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 ! min a0 u0 ˆ p’(t)=p(t)*h(t) sample at t=0 a0 . Es p(t) h(t) transmit pulse transmitter Postacademic Course on Telecommunications ˆ ( Es .g 0 ).a0 2 + n(t) AWGN channel Module-3 Transmission Lecture-5 Equalization 1/Ts p’(-t)* u0 front-end filter ˆ a0 receiver Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 11
Summary of Lectures (1-2-)3-4 • Receiver: For transmission of a symbol sequence, the optimal receiver structure is... K K Es . k 1 l 1 ak . Es p(t) h(t) transmit pulse transmitter Postacademic Course on Telecommunications + n(t) AWGN channel Module-3 Transmission Lecture-5 Equalization 2 ˆ* ak .uk uk min a0 ,...,aK ˆ ˆ ˆ* ˆ ak .g k l .al K ˆ ak k 1 1/Ts p’(-t)* front-end filter receiver sample at t=k.Ts Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 12
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... Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 13
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’ K min a0 ,...,aK ˆ ˆ K ym m 1 2 ˆ ak .hm k k 1 uk ak . Es p(t) transmit pulse transmitter Postacademic Course on Telecommunications h(t) + n(t) AWGN channel 1/Ts p’(-t)* front-end filter Module-3 Transmission Lecture-5 Equalization yk ˆ ak 1/L*(1/z*) receiver Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 14
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 inputoutput model: yk h0 .ak h1..ak h2 ..ak yk (h0 h1.z 1 h2 .z 2 h3 .z 3 ...).ak  1 2 h3 .ak 3 ... wk wk H (z) uk ak . Es p(t) transmit pulse h(t) 1/Ts p’(-t)* front-end n(t) filter ˆ ak + AWGN transmitter channel Postacademic Course on Telecommunications yk Module-3 Transmission Lecture-5 Equalization 1/L*(1/z*) receiver Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 15
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 inputoutput model: yk h0 .ak h1.ak 1 h2 .ak 2 h3 .ak 3 ... wk wk = additive white Gaussian noise means interference from future (`pre-cursor) symbols has been cancelled, hence only interference from past (`post-cursor’) symbols remains h1 h Postacademic Course on Telecommunications 2 ... 0 Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 16
Summary of Lectures (1-2-)3-4 Receiver: Based on the input-output model yk h0 .ak h1..ak h2 ..ak 1 2 h3 .ak 3 ... wk one can compute the transmitted symbol sequence as K min a0 ,...,aK ˆ ˆ K ym m 1 2 ˆ ak .hm k k 1 A recursive procedure for this = Viterbi Algorithm Problem = complexity proportional to M^N ! (N=channel-length=number of non-zero taps in H(z) ) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 17
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 Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 18
Preliminaries (I) • Our starting point will be the input-output model for transmitter + channel + receiver whitened matched filter front-end yk h0 .ak h1.ak 1 h2 .ak ak ak h0 h1 2 h3 .ak ak 1 h2 2 3 ... wk ak h3 wk Postacademic Course on Telecommunications 3 yk Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 19
Preliminaries (II) • PS: z-transform is `shorthand notation’ for discrete-time signals… A( z ) ai .z i a0 .z 0 a1.z 1 a2 . z 2 .... h0 .z 0 h1.z 1 h2 .z 2 .... i 0 H ( z) hi .z i i 0 …and for input/output behavior of discrete-time systems yk h0 .ak h1.ak 1 h2 .ak hence Y ( z ) H ( z ).A( z ) W ( z ) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization 2 h3 .ak A(z ) 3 ... wk Y (z ) H(z) W (z ) Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 20
Preliminaries (III) • PS: if a different receiver front-end is used (e.g. MF instead of WMF, or …), a similar model holds yk ~ ... h 2 .ak 2 ~ h 1.ak 1 ~ ~ h0 .ak h1.ak 1 ~ h2 .ak 2 ~ ... wk for which equalizers can be designed in a similar fashion... Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 21
Preliminaries (IV) PS: properties/advantages of the WMF front end • additive noise wk = white (colored in general model) • H(z) does not have anti-causal taps h 1 h 2 ... 0 pps: anti-causal taps originate, e.g., from transmit filter design (RRC, etc.). practical implementation based on causal filters + delays... • H(z) `minimum-phase’ : 1 =`stable’ zeroes, hence (causal) inverse H ( z ) exists & stable = energy of the impulse response maximally concentrated in the early samples Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 22
Preliminaries (V) yk h0 .ak h1.ak 1 h2 .ak 2 h3 .ak 3 ...     wk  ISI NOISE • `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 Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 23
Zero-forcing & MMSE Equalizers yk h0 .ak h1.ak 1 h2 .ak 2 h3 .ak 3 ...     wk  ISI NOISE 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 Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 24
Zero-forcing Equalizers Zero-forcing Linear Equalizer (LE) : - equalization filter is inverse of H(z) - decision device (`slicer’) C ( z) A(z ) H 1 ( z) ˆ A( z ) Y (z ) C(z) H(z) W (z ) • Problem : noise enhancement ( C(z).W(z) large) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 25
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. Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 26
Zero-forcing Equalizers Zero-forcing Non-linear Equalizer Decision Feedback Equalization (DFE) : - derivation based on `alternative’ inverse of H(z) : A(z ) ˆ A( z ) Y (z ) H(z) W (z ) 1-H(z) (ps: this is possible if H(z) has h0 1 another property of the WMF model) , which is - now move slicer inside the feedback loop : Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 27
Zero-forcing Equalizers A(z ) Y (z ) ˆ A( z ) H(z) W (z ) D(z) D( z ) 1 H ( z ) 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. Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 28
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) A(z ) Y (z ) C(z) H(z) W (z ) Postacademic Course on Telecommunications ˆ A( z ) Module-3 Transmission Lecture-5 Equalization D(z) Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 29
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. A(z ) Y (z ) H(z) W (z ) Postacademic Course on Telecommunications ˆ A( z ) C(z) Module-3 Transmission Lecture-5 Equalization D(z) Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 30
Zero-forcing Equalizers `Figure of merit’ LE DFE MLSE MF • 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 h0 (eliminating all other hi ‘s), while MLSE uses energy of all taps hi . DFE benefits from minimum-phase property (cfr. supra, p.20) MF Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 31
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 Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 32
MMSE Equalizers MMSE Linear Equalizer (LE) : A(z ) ˆ A( z ) Y (z ) C(z) H(z) W (z ) - combined minimization of ISI and noise leads to 1 ) * z * 1 S A ( z ).H ( z ).H ( * ) SW ( z ) z S A ( z ).H * ( C ( z) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization 1 ) * z * 1 H ( z ).H ( * ) z H *( Marc Moonen K.U.Leuven-ESAT/SISTA 2 n 4/5/00 p. 33
MMSE Equalizers 1 ) * z * 1 S A ( z ).H ( z ).H ( * ) SW ( z ) z S A ( z ).H * ( C ( z) - 1 ) * z * 1 H ( z ).H ( * ) z H *( 2 W S A (z ) 1 signal power spectrum (normalized) 2 SW ( z ) noise power spectrum (white) W 1 for zero noise power -> zero-forcing C ( z ) H ( z ) * 1 H ( * ) (in the nominator) is a discrete-time matched filter, z often `difficult’ to realize in practice (stable poles in H(z) introduce anticausal MF) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 34
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 A(z ) Y (z ) ˆ A( z ) C(z)E(z) H(z) W (z ) 1-E(z) • E(z)=1 -> linear equalizer • Theory & formulas : see textbooks Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 35
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 ! Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 36
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. Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 37
Fractionally Spaced Equalizers • Input-output model for fractionally spaced equalization : `symbol rate’ samples : yk ~ ... h0 .ak ~ h1.ak 1 ~ h2 .ak 2 ~ ... wk `intermediate’ samples : yk 1/ 2 ~ ... h1/ 2 .ak ~ h3/ 2 .ak 1 ~ h5 / 2 .ak 2 ~ ... wk 1/ 2 • may be viewed as 1-input/2-outputs system Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 38
Fractionally Spaced Equalizers • Discrete-time matched filter + Equalizer (LE) : 1/2Ts r (t ) F(f) MF(z) 2 C(z) ˆ A( z ) equalizer • Fractionally spaced equalizer (LE) : 1/2Ts r (t ) F(f) C(z) 2 ˆ A( z ) Fractionally spaced equalizer Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 39
Fractionally Spaced Equalizers • Fractionally spaced equalizer (DFE): 1/2Ts r (t ) F(f) C(z) ˆ A( z ) 2 D(z) • Theory & formulas : see textbooks & Lecture 6 Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 40
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) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 41
Assignment 3.1 • Symbol-rate zero-forcing linear equalizer has H 1 ( z) C ( z) i.e. a finite impulse response (`all-zeroes’) filter H ( z) h0 h1.z 1 h2 .z 2 is turned into an infinite impulse response filter C ( z ) 1 /(h0 1 h1.z h2 .z 2 ) • Investigate this statement for the case of fractionally spaced equalization, for a simple channel model yk yk h0 .ak 1/ 2 h1.ak h1/ 2 .ak 1 h2 .ak h3 / 2 .ak 1 2 h5 / 2 .ak 2 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. Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 42

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Lecture5

  • 1. 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/ Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven/ESAT-SISTA 4/5/00 p. 1
  • 2. 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... Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 2
  • 3. Prelude Good 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. Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 3
  • 4. 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)... Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 4
  • 5. 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 Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 5
  • 6. Summary of Lectures (1-2-)3-4 Channel Model: ak (symbols) ˆ ak h(t) ? transmitter + n(t) AWGN channel ... ? receiver (to be defined) Continuous-time channel =Linear filter channel + additive white Gaussian noise (AWGN) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 6
  • 7. Summary of Lectures (1-2-)3-4 Transmitter: r(t) s(t) ˆ ak ak . Es p(t) h(t) ... transmit pulse transmitter + n(t) AWGN channel ? receiver (to be defined) * Constellations (linear modulation): n bits -> 1 symbol a k (PAM/QAM/PSK/..) * Transmit filter p(t) : s(t ) Es . ak . p(t kTs ) k Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 7
  • 8. Summary of Lectures (1-2-)3-4 p(t) Transmitter: t s(t) Example ak . Es t p(t) discrete-time symbol sequence transmit pulse continuous-time transmit signal 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) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 8
  • 9. 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. a0 . Es p(t) h(t) transmit pulse transmitter Postacademic Course on Telecommunications 1/Ts + n(t) AWGN channel Module-3 Transmission Lecture-5 Equalization front-end filter u0 ˆ a0 receiver Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 9
  • 10. Summary of Lectures (1-2-)3-4 Receiver: In Lecture-4, optimal receiver design was based on a minimum distance criterion : min a0 ,a1 ,...,aK | r (t ) ˆ ˆ ˆ ˆ ak . p' (t kTs ) |2 dt Es . k • Transmitted signal is • Received signal s(t ) Es . ak . p(t kTs ) k r (t ) Es . ak . p' (t kTs ) n(t ) k • p’(t)=p(t)*h(t)=transmitted pulse, filtered by channel Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 10
  • 11. 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 ! min a0 u0 ˆ p’(t)=p(t)*h(t) sample at t=0 a0 . Es p(t) h(t) transmit pulse transmitter Postacademic Course on Telecommunications ˆ ( Es .g 0 ).a0 2 + n(t) AWGN channel Module-3 Transmission Lecture-5 Equalization 1/Ts p’(-t)* u0 front-end filter ˆ a0 receiver Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 11
  • 12. Summary of Lectures (1-2-)3-4 • Receiver: For transmission of a symbol sequence, the optimal receiver structure is... K K Es . k 1 l 1 ak . Es p(t) h(t) transmit pulse transmitter Postacademic Course on Telecommunications + n(t) AWGN channel Module-3 Transmission Lecture-5 Equalization 2 ˆ* ak .uk uk min a0 ,...,aK ˆ ˆ ˆ* ˆ ak .g k l .al K ˆ ak k 1 1/Ts p’(-t)* front-end filter receiver sample at t=k.Ts Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 12
  • 13. 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... Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 13
  • 14. 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’ K min a0 ,...,aK ˆ ˆ K ym m 1 2 ˆ ak .hm k k 1 uk ak . Es p(t) transmit pulse transmitter Postacademic Course on Telecommunications h(t) + n(t) AWGN channel 1/Ts p’(-t)* front-end filter Module-3 Transmission Lecture-5 Equalization yk ˆ ak 1/L*(1/z*) receiver Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 14
  • 15. 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 inputoutput model: yk h0 .ak h1..ak h2 ..ak yk (h0 h1.z 1 h2 .z 2 h3 .z 3 ...).ak  1 2 h3 .ak 3 ... wk wk H (z) uk ak . Es p(t) transmit pulse h(t) 1/Ts p’(-t)* front-end n(t) filter ˆ ak + AWGN transmitter channel Postacademic Course on Telecommunications yk Module-3 Transmission Lecture-5 Equalization 1/L*(1/z*) receiver Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 15
  • 16. 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 inputoutput model: yk h0 .ak h1.ak 1 h2 .ak 2 h3 .ak 3 ... wk wk = additive white Gaussian noise means interference from future (`pre-cursor) symbols has been cancelled, hence only interference from past (`post-cursor’) symbols remains h1 h Postacademic Course on Telecommunications 2 ... 0 Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 16
  • 17. Summary of Lectures (1-2-)3-4 Receiver: Based on the input-output model yk h0 .ak h1..ak h2 ..ak 1 2 h3 .ak 3 ... wk one can compute the transmitted symbol sequence as K min a0 ,...,aK ˆ ˆ K ym m 1 2 ˆ ak .hm k k 1 A recursive procedure for this = Viterbi Algorithm Problem = complexity proportional to M^N ! (N=channel-length=number of non-zero taps in H(z) ) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 17
  • 18. 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 Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 18
  • 19. Preliminaries (I) • Our starting point will be the input-output model for transmitter + channel + receiver whitened matched filter front-end yk h0 .ak h1.ak 1 h2 .ak ak ak h0 h1 2 h3 .ak ak 1 h2 2 3 ... wk ak h3 wk Postacademic Course on Telecommunications 3 yk Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 19
  • 20. Preliminaries (II) • PS: z-transform is `shorthand notation’ for discrete-time signals… A( z ) ai .z i a0 .z 0 a1.z 1 a2 . z 2 .... h0 .z 0 h1.z 1 h2 .z 2 .... i 0 H ( z) hi .z i i 0 …and for input/output behavior of discrete-time systems yk h0 .ak h1.ak 1 h2 .ak hence Y ( z ) H ( z ).A( z ) W ( z ) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization 2 h3 .ak A(z ) 3 ... wk Y (z ) H(z) W (z ) Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 20
  • 21. Preliminaries (III) • PS: if a different receiver front-end is used (e.g. MF instead of WMF, or …), a similar model holds yk ~ ... h 2 .ak 2 ~ h 1.ak 1 ~ ~ h0 .ak h1.ak 1 ~ h2 .ak 2 ~ ... wk for which equalizers can be designed in a similar fashion... Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 21
  • 22. Preliminaries (IV) PS: properties/advantages of the WMF front end • additive noise wk = white (colored in general model) • H(z) does not have anti-causal taps h 1 h 2 ... 0 pps: anti-causal taps originate, e.g., from transmit filter design (RRC, etc.). practical implementation based on causal filters + delays... • H(z) `minimum-phase’ : 1 =`stable’ zeroes, hence (causal) inverse H ( z ) exists & stable = energy of the impulse response maximally concentrated in the early samples Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 22
  • 23. Preliminaries (V) yk h0 .ak h1.ak 1 h2 .ak 2 h3 .ak 3 ...     wk  ISI NOISE • `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 Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 23
  • 24. Zero-forcing & MMSE Equalizers yk h0 .ak h1.ak 1 h2 .ak 2 h3 .ak 3 ...     wk  ISI NOISE 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 Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 24
  • 25. Zero-forcing Equalizers Zero-forcing Linear Equalizer (LE) : - equalization filter is inverse of H(z) - decision device (`slicer’) C ( z) A(z ) H 1 ( z) ˆ A( z ) Y (z ) C(z) H(z) W (z ) • Problem : noise enhancement ( C(z).W(z) large) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 25
  • 26. 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. Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 26
  • 27. Zero-forcing Equalizers Zero-forcing Non-linear Equalizer Decision Feedback Equalization (DFE) : - derivation based on `alternative’ inverse of H(z) : A(z ) ˆ A( z ) Y (z ) H(z) W (z ) 1-H(z) (ps: this is possible if H(z) has h0 1 another property of the WMF model) , which is - now move slicer inside the feedback loop : Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 27
  • 28. Zero-forcing Equalizers A(z ) Y (z ) ˆ A( z ) H(z) W (z ) D(z) D( z ) 1 H ( z ) 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. Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 28
  • 29. 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) A(z ) Y (z ) C(z) H(z) W (z ) Postacademic Course on Telecommunications ˆ A( z ) Module-3 Transmission Lecture-5 Equalization D(z) Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 29
  • 30. 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. A(z ) Y (z ) H(z) W (z ) Postacademic Course on Telecommunications ˆ A( z ) C(z) Module-3 Transmission Lecture-5 Equalization D(z) Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 30
  • 31. Zero-forcing Equalizers `Figure of merit’ LE DFE MLSE MF • 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 h0 (eliminating all other hi ‘s), while MLSE uses energy of all taps hi . DFE benefits from minimum-phase property (cfr. supra, p.20) MF Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 31
  • 32. 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 Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 32
  • 33. MMSE Equalizers MMSE Linear Equalizer (LE) : A(z ) ˆ A( z ) Y (z ) C(z) H(z) W (z ) - combined minimization of ISI and noise leads to 1 ) * z * 1 S A ( z ).H ( z ).H ( * ) SW ( z ) z S A ( z ).H * ( C ( z) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization 1 ) * z * 1 H ( z ).H ( * ) z H *( Marc Moonen K.U.Leuven-ESAT/SISTA 2 n 4/5/00 p. 33
  • 34. MMSE Equalizers 1 ) * z * 1 S A ( z ).H ( z ).H ( * ) SW ( z ) z S A ( z ).H * ( C ( z) - 1 ) * z * 1 H ( z ).H ( * ) z H *( 2 W S A (z ) 1 signal power spectrum (normalized) 2 SW ( z ) noise power spectrum (white) W 1 for zero noise power -> zero-forcing C ( z ) H ( z ) * 1 H ( * ) (in the nominator) is a discrete-time matched filter, z often `difficult’ to realize in practice (stable poles in H(z) introduce anticausal MF) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 34
  • 35. 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 A(z ) Y (z ) ˆ A( z ) C(z)E(z) H(z) W (z ) 1-E(z) • E(z)=1 -> linear equalizer • Theory & formulas : see textbooks Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 35
  • 36. 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 ! Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 36
  • 37. 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. Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 37
  • 38. Fractionally Spaced Equalizers • Input-output model for fractionally spaced equalization : `symbol rate’ samples : yk ~ ... h0 .ak ~ h1.ak 1 ~ h2 .ak 2 ~ ... wk `intermediate’ samples : yk 1/ 2 ~ ... h1/ 2 .ak ~ h3/ 2 .ak 1 ~ h5 / 2 .ak 2 ~ ... wk 1/ 2 • may be viewed as 1-input/2-outputs system Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 38
  • 39. Fractionally Spaced Equalizers • Discrete-time matched filter + Equalizer (LE) : 1/2Ts r (t ) F(f) MF(z) 2 C(z) ˆ A( z ) equalizer • Fractionally spaced equalizer (LE) : 1/2Ts r (t ) F(f) C(z) 2 ˆ A( z ) Fractionally spaced equalizer Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 39
  • 40. Fractionally Spaced Equalizers • Fractionally spaced equalizer (DFE): 1/2Ts r (t ) F(f) C(z) ˆ A( z ) 2 D(z) • Theory & formulas : see textbooks & Lecture 6 Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 40
  • 41. 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) Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 41
  • 42. Assignment 3.1 • Symbol-rate zero-forcing linear equalizer has H 1 ( z) C ( z) i.e. a finite impulse response (`all-zeroes’) filter H ( z) h0 h1.z 1 h2 .z 2 is turned into an infinite impulse response filter C ( z ) 1 /(h0 1 h1.z h2 .z 2 ) • Investigate this statement for the case of fractionally spaced equalization, for a simple channel model yk yk h0 .ak 1/ 2 h1.ak h1/ 2 .ak 1 h2 .ak h3 / 2 .ak 1 2 h5 / 2 .ak 2 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. Postacademic Course on Telecommunications Module-3 Transmission Lecture-5 Equalization Marc Moonen K.U.Leuven-ESAT/SISTA 4/5/00 p. 42