Post on 14-Nov-2014
description
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven/ESAT-SISTA
4/5/00p. 1
Postacademic Course on Telecommunications
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 Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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 Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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 Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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 Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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 Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 6
Summary of Lectures (1-2-)3-4
Channel Model:
Continuous-time channel
=Linear filter channel + additive white Gaussian noise (AWGN)
(symbols) kaka
n(t)
+
AWGN
transmitter receiver (to be defined)
h(t)
channel
...
??
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 7
Summary of Lectures (1-2-)3-4
Transmitter:
* Constellations (linear modulation): n bits -> 1 symbol (PAM/QAM/PSK/..)
* Transmit filter p(t) :
receiver (to be defined)
...
sk Ea .
r(t)
ka
transmit
pulse
s(t)
n(t)
p(t) +
AWGNtransmitter
h(t)
channel?
k
sks kTtpaEts )(..)(
ka
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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)
sk Ea .
transmit
pulse
s(t)
p(t)
transmitter
discrete-timesymbol sequence
continuous-timetransmit signal
tp(t)
t
Example
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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.
0afront-end
filter
1/Ts
receiver
n(t)
+
AWGN
sEa .0
transmit
pulse
p(t)
transmitter
h(t)
channel
0u
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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
k
sksaaa dtkTtpaEtrK
2ˆ,...,ˆ,ˆ |)('.ˆ.)(|min
10
k
sks kTtpaEts )(..)(
)()('..)( tnkTtpaEtrk
sks
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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 !
2
000ˆ ˆ)..(min0
agEu sa
0ap’(-t)*
front-end
filter
1/Ts
receiver
n(t)
+
AWGN
sEa .0
transmit
pulse
p(t)
transmitter
h(t)
channel
sample at t=0p’(t)=p(t)*h(t)
0u
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 12
Summary of Lectures (1-2-)3-4
• Receiver: For transmission of a symbol sequence, the optimal receiver structure is...
ka
p’(-t)*
front-end
filter
1/Ts
receiver
n(t)
+
AWGN
sk Ea .
transmit
pulse
p(t)
transmitter
h(t)
channelsample at t=k.Ts
ku
K
kkkllk
K
k
K
lksaa uaagaE
K1
*
1 1
*ˆ,...,ˆ .ˆ2ˆ..ˆ.min
0
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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 Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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’
ka
p’(-t)*front-end
filter
1/Ts
receiver
n(t)
+
AWGN
sk Ea .
transmit
pulse
p(t)
transmitter
h(t)
channel
ku
1/L*(1/z*)
ky
2
1 1ˆ,...,ˆ .ˆmin
0
K
m
K
kkmkmaa hay
K
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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:
kk
zH
k
kkkkkk
wazhzhzhhy
wahahahahy
....)...(
.........
)(
33
22
110
3322110
ka
p’(-t)*front-end
filter
1/Ts
receiver
n(t)
+
AWGN
sk Ea .
transmit
pulse
p(t)
transmitter
h(t)
channel
ku
1/L*(1/z*)
ky
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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
kkkkkk wahahahahy ....... 3322110
kw
0...21 hh
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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) )
kkkkkk wahahahahy ......... 3322110
2
1 1ˆ,...,ˆ .ˆmin
0
K
m
K
kkmkmaa hay
K
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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 Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 19
Preliminaries (I)
• Our starting point will be the input-output model for transmitter + channel + receiver whitened matched filter front-end
kkkkkk wahahahahy ....... 3322110
1h
3h0h
2h
ka 3ka2ka1ka
kykw
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 20
Preliminaries (II)
• PS: z-transform is `shorthand notation’ for discrete-time signals…
…and for input/output behavior of discrete-time systems
)()().()(
hence
....... 3322110
zWzAzHzY
wahahahahy kkkkkk
........)(
........)(
22
11
00
0
22
11
00
0
zhzhzhzhzH
zazazazazA
i
ii
i
ii
)(zAH(z)
)(zW
)(zY
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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...
kkkkkkk wahahahahahy ~....~
.~
.~
.~
.~
... 221101122
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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
kw
0...21 hh
)(1 zH
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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
NOISEISI
3322110 ....... kkkkkk wahahahahy
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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
NOISEISI
3322110 ....... kkkkkk wahahahahy
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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)
)()( 1 zHzC
H(z)
)(zW
)(zY
C(z)
)(zA )(ˆ zA
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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 Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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 :
)(zY
1-H(z)
H(z)
)(zW
)(zA )(ˆ zA
10 h
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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.
)(zY
D(z)
H(z)
)(zW
)(zA
)(ˆ zA
)(1)( zHzD
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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)
)(zY)(zA
H(z)
)(zW
C(z) )(ˆ zA
D(z)
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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.
)(zY
H(z)
)(zW
)(zA
C(z) )(ˆ zA
D(z)
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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)
MFMLSEDFELE
MF
0hih
ih
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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 Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 33
MMSE Equalizers
MMSE Linear Equalizer (LE) :
- combined minimization of ISI and noise leads to
2*
*
**
**
**
)1
().(
)1
(
)()1
().().(
)1
().()(
nWA
A
zHzH
zH
zSz
HzHzS
zHzS
zC
H(z)
)(zW
)(zY
C(z)
)(zA )(ˆ zA
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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)
2*
*
**
**
**
)1
().(
)1
(
)()1
().().(
)1
().()(
WWA
A
zHzH
zH
zSz
HzHzS
zHzS
zC
1)(zSA 2)( WW zS
)()( 1 zHzC )
1(
**
zH
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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
)(zY
H(z)
)(zW
)(zA
C(z)E(z) )(ˆ zA
1-E(z)
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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 Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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 Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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
kkkkk wahahahy ~....~
.~
.~
... 22110
2/122/512/32/12/1~....
~.
~.
~... kkkkk wahahahy
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 39
Fractionally Spaced Equalizers
• Discrete-time matched filter + Equalizer (LE) :
• Fractionally spaced equalizer (LE) :
)(ˆ zA1/2Ts
2MF(z) C(z)
equalizer
)(trC(z) )(ˆ zA
1/2Ts2
Fractionally spaced equalizer
)(trF(f)
F(f)
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 40
Fractionally Spaced Equalizers
• Fractionally spaced equalizer (DFE):
• Theory & formulas : see textbooks & Lecture 6
C(z) )(ˆ zA
D(z)
1/2Ts2
)(trF(f)
Postacademic Course on Telecommunications
Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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 Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA
4/5/00p. 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.
)()( 1 zHzC
22
110 ..)( zhzhhzH
)../(1)( 22
110
zhzhhzC
22/512/32/12/1
22110
...
...
kkkk
kkkk
ahahahy
ahahahy