1. Module-3 : Transmission Lecture-5 (4/5/00)
Marc Moonen
Dept. E.E./ESAT, K.U.Leuven
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven/ESAT-SISTA

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...
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

3. 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 = 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

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)...
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

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
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

6. 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

7. 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

8. 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

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.
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

10. 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

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 !
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

12. 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

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...
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

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’
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

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 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

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 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

17. 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

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
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

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.
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

27. 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

28. 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

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)
H(z)
C(z)
D(z)
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

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.
H(z)
C(z)
D(z)
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

31. 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

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
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

33. 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

34. 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

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
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

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 !
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

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.
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

38. 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

39. 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

40. 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

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)
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven-ESAT/SISTA

42. 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