Combined Equalization and
Coding Using Precoding*
ECE 492 – Term Project
Betül Arda
Selçuk Köse
Department of Electrical
and Computer Engineering
University of Rochester
*“Combined equalization and coding using precoding” Forney, G.D., Jr.; Eyuboglu, M.V.
Agenda










Introduction
Capacity of Gaussian Channels
Adaptive Modulation
Brief History of Equalization
Equalization Techniques
Tomlinson-Harashima Precoding
Combined Precoding and Coded Modulation
Trellis Precoding
Price’s Result & Attaining Capacity
Conclusion
2
Introduction

What is the paper about?


Recently developed techniques to achieve
capacity objectives
Tomlinson – Harashima precoding: Precoding
technique for uncoded modulation
C
of bandlimited, high-SNR Gaussian
channel  C of ideal Gaussian channel

Precoding + coded modulation + shaping
 Achieves
nearly channel capacity of
bandlimited, high-SNR Gaussian channel


Is precoding approach a practical route to
capacity on high-SNR+bandlimited channel?
Decision feedback equalization structure
3
Agenda










Introduction
Capacity of Gaussian Channels
Adaptive Modulation
Brief History of Equalization
Equalization Techniques
Tomlinson-Harashima Precoding
Combined Precoding and Coded
Modulation
Trellis Precoding
Price’s Result & Attaining Capacity
Conclusion
4
C of Ideal Gaussian channels
Ideal bandlimited Gaussian channel
Gaussian channel model
with power constraint

SNR=Sx/Sn=P/N0W
Ex: Telephone channel SNR~28 to 36 dB & BW~2400 to 3200 Hz


not ideal but C can be estimated by 9 to 12 bits/Hz
or 20,000 b/s to 30,000 b/s
5
C of Non-Ideal Gaussian channels
Determination
of optimum
water-pouring
spectrum
Capacity achieving band:
of telephone channels ~ constant at the center drops at edges



important to optimize B
If B is nearly optimal

typically a flat transmit spectrum is almost as good as water-pouring spectrum6
Agenda










Introduction
Capacity of Gaussian Channels
Adaptive Modulation
Brief History of Equalization
Equalization Techniques
Tomlinson-Harashima Precoding
Combined Precoding and Coded
Modulation
Trellis Precoding
Price’s Result & Attaining Capacity
Conclusion
7
Adaptive BW - Adaptive Rate Modulation


Coded modulation scheme with rate R
bits/symbol (b/s/Hz), as close as possible to C
This scheme is suitable for point-to-point twoway applications: telephone-line modems


To approach capacity: Tx needs to know the channel
Not possible for one-way, broadcast, rapidly timevarying channels unless ch. char.s are known a priori
8
Adaptive BW - Adaptive Rate Modulation

Inherit delay due to long 1/Δf


rules out some modem applications
Multicarrier modulation with few carriers and
short 1/Δf

ISI arises and must be equalized
9
Agenda










Introduction
Capacity of Gaussian Channels
Adaptive Modulation
Brief History of Equalization
Equalization Techniques
Tomlinson-Harashima Precoding
Combined Precoding and Coded
Modulation
Trellis Precoding
Price’s Result & Attaining Capacity
Conclusion
10
History of Equalization

1967: Milgo4400  4800b/s W=1600Hz


1960s: time of considerable research on adaptive modulation






Automatic adaptive digital LE for W=2400Hz and 16-QAM
1970s: modems  more smaller, cheaper, reliable, versatile, but not faster
Fractionally spaced equalizers:


Focused on adaptation algorithms that did not require multiplications
1971: Codec9600C  9600b/s (V.29)


Manually adjustable equalizer  knob on the front panel to zero a null meter
fast-training algorithms for multipoint and half-duplex applications
Even the first 14.4kb/s modem used uncoded modulation, fixed BW, LE
1983: Trellis coded modulation  9600b/s over dial lines
1985: adaptive rate-adaptive BW modem of the multicarrier type
1990: Combined equal., multidimensional TCM and shaping using trellis precoding
11
Modem Milestones
Year
Name
Max.Rate
Sym
Modulation
Eff.
1962
Bell 201
2.4
1.2
4PSK
2
1967
Milgo4400
4.8
1.6
8PSK
3
1971
Codex 9600C
9.6
2.4
16-QAM
4
1980
Paradyne
14.4
2.4
64-QAM
6
1984
Codex 2600
16.8
2.4
Trellis 256-QAM
7
1985
Codex 2680
19.2
2.74
8-D(state) Trellis 7
160-QAM
1984
V.32
9.6
2.4
2D TC
4
1991
V.32 bis
14.4
2.4
2D TC 128-QAM
6
1994
V.34
28.8
2.4-3.4 4D TC 960-QAM
~9
1998
V.90
56
same
same
same
TCM has made possible the development of very high speed modems.
12
Agenda










Introduction
Capacity of Gaussian Channels
Adaptive Modulation
Brief History of Equalization
Equalization Techniques
Tomlinson-Harashima Precoding
Combined Precoding and Coded
Modulation
Trellis Precoding
Price’s Result & Attaining Capacity
Conclusion
13
Classical Equalization Techniques
D transform
Channel is ideal iff:
&
14
Equalization Tech. – ZF-LE
Zero-forcing linear equalization
r(D) is filtered by 1/h(D) to
produce an equalized response

LE can be satisfactory in a QAM modem if the channel has no nulls or near-nulls




If H(θ) ~ const. over {-π < θ ≤ π}  noise enhancement not very serious
|H(θ)|2 has a near-null  noise enhancement becomes very large
|H(θ)|2 has a null  h(D) not invertible, ZF-LE not well-defined
To approach capacity, transmission band must be expanded to entire usable BW
of the channel

Leads to severe attenuation at band edges  LE no longer suffices
15
Equalization Tech. – ZF-DFE
ISI removed and noise is white
||1/h||2 ≥1  SNRZF-DFE ≥ SNRZF-LE
& iff h(D)=1  SNRZF-DFE=SNRZF-LE
16
Equalization Tech. – MLSE

Optimum equalization structure if ISI exists





M -state Viterbi algorithm can be used to
implement MLSE
M and/or v is too large  complex to
implement
If no severe SNR



v
xk drawn from M-pt signal set, h(D) has length v
v
Channel can be modeled as M -state machine
SNR of matched filter bound 
Matched filter bound: bound on the best SNR
achievable with h(D)
If SNR is severe

MLSE fails to achieve this SNR, performance
analysis become difficult
17
Agenda










Introduction
Capacity of Gaussian Channels
Adaptive Modulation
Brief History of Equalization
Equalization Techniques
Tomlinson-Harashima Precoding
Combined Precoding and Coded
Modulation
Trellis Precoding
Price’s Result & Attaining Capacity
Conclusion
18
Tomlinson-Harashima Precoding

Precoding works even if h(D) is not
invertible i.e. ||1/h||2 is infinite.
19
Tomlinson-Harashima Precoding
Key Points


Tx knows h(D)
y(D) = d(D)+2Mz(D) is chosen


Large M, x(D)  PAM seq.


Values continuous in (-M,M]
Rx  symbol-by-symbol


x(D) = y(D)/h(D) is in (-M,M]
Ordinary PAM on ideal channel
Pe same as with ideal ZF-DFE

Same as on an ideal ch. with SNRZF-DFE=Sx/Sn
20
Tomlinson-Harashima Precoding


At first, TH appeared to be an
attractive alternative to ZF-DFE
Its performance is no better than
ZF-DFE under the ideal ZF-DFE
assumption


For uncoded systems ideal ZF-DFE
assumption works well
Therefore, DFE is preferred to TH

DFE does not require CSI at tx
21
Agenda







Introduction
Capacity of Gaussian Channels
Adaptive Modulation
Brief History of Equalization
Equalization Techniques
Tomlinson-Harashima Precoding
Combined Precoding and Coded
Modulation






Using an Interleaver
Combining Trellis Encoder and Channel
Combined Precoding and Coded Modulation
Trellis Precoding
Price’s Result & Attaining Capacity
Conclusion
22
Interleaver
M.V. Eyüboğlu, “Detection of coded modulation signals on linear severely distorted channels using decision-feedback
noise prediction with interleaving,” IEEE Trans. Commun., Vol. 36, No. 4, pp.401-09, April 1988.
23
Interleaver (Cont’d)
Without interleaver
Transmitted message
aaaabbbbccccddddeeeeffffgggg
Received message
aaaabbbbccc____deeeeffffgggg
With interleaver
Transmitted message
aaaabbbbccccddddeeeeffffgggg
Interleaved
abcdefgabcdefgabcdefgabcdefg
Received message
abcdefgabcd____bcdefgabcdefg
De-interleaved
aa_abbbbccccdddde_eef_ffg_gg
24
Combining Trellis Encoder and Channel
MLSE
Algorithm
Finite state machine
representation of trellis
encoder and channel
 Reduced state-
sequence estimation
algorithms are used to
make the computation
faster.
25
Combined Precoding and Coded Modulation


y(D)=d(D)+2Mz(D) where M is a
multiple of 4.
r(D)=y(D)+n(D)
26
Agenda










Introduction
Capacity of Gaussian Channels
Adaptive Modulation
Brief History of Equalization
Equalization Techniques
Tomlinson-Harashima Precoding
Combined Precoding and Coded
Modulation
Trellis Precoding
Price’s Result & Attaining Capacity
Conclusion
27
Trellis Precoding = Shaping+Precoding+Coding

(N  ) then shaping gain1.53dB
(1.53dB is the difference between average energies of
Gaussian and uniform distribution)
 Shaping on regions
 Trellis Shaping
 Shell Mapping
Distribution approaches
truncated Gaussian
28
Trellis Precoding = Coding+Precoding+Shaping


Coding gains of 3 to
6 dB for 4 to 512
states.
Binary codes




Sequential decoding of
convolution codes
Turbo codes
Low-density parity
check codes.
Non-binary codes

Sequential decoding of
trellis codes
29
Trellis Precoding = Precoding+Coding+Shaping



“DFE in transmitter”
It combines nicely with coded
modulation with “no glue”
It has Asymptotically optimal
performance
30
Agenda










Introduction
Capacity of Gaussian Channels
Adaptive Modulation
Brief History of Equalization
Equalization Techniques
Tomlinson-Harashima Precoding
Combined Precoding and Coded
Modulation
Trellis Precoding
Price’s Result & Attaining Capacity
Conclusion
31
Price’s Result



“As SNR  on any linear Gaussian Channel
the gap between capacity and QAM
performance with ideal ZF-DFE is
independent of channel noise and spectra.”
Improved result can be achieved using MSSE
type equalization
Ideal MSSE-optimized tail canceling equalization +
Capacity-approaching ideal AWGN channel coding=
Approach to the capacity of any linear Gaussian
channel
32
Attaining Capacity
•Coding: can achieve 6dB, max 7.5 dB
•Shaping: can achieve 1 dB, max 1.53 dB
•Total: can achieve 7 dB, max 9 dB
33
Agenda










Introduction
Capacity of Gaussian Channels
Adaptive Modulation
Brief History of Equalization
Equalization Techniques
Tomlinson-Harashima Precoding
Combined Precoding and Coded
Modulation
Trellis Precoding
Price’s Result & Attaining Capacity
Conclusion
34
Conclusion



We can approach channel capacity
by combining known codes for
coding gain with simple shaping
techniques for shaping gain.
Can approach channel capacity for
ideal and non-ideal channels.
In principle, on any band-limited
linear Gaussian channel one can
approach capacity as closely as
desired.*
* R. deBuda, “some optimal codes have structure”, IEEE Journal of Selected Areas of Communication, Vol. SAC-7, 893899, August 1989.
35
References





D.Forney and V.Eyuboglu, “Combined Equalization and Coding Using
Precoding,” IEEE Communication Magazine, Vol. 29, pp.24-34, December
1991
R. Price, “Nonlinearly Feedback Equalized PAM versus Capacity for Noisy
Filter Channels,” Proceedings of ICC '72, June 1972
M. V. Eyuboglu and G. D.Forney, Jr., “Trellis Precoding: Combined
Coding, Precoding and Shaping for Intersymbol Interference Channels,”
IEEE Transactions on Information Theory, Vol. 38, pp. 301-314, March
1992.
R. deBuda, “Some Optimal Codes Have Structure”, IEEE Journal of
Selected Areas of Communication, Vol. SAC-7, 893-899, August 1989.
M.V. Eyüboğlu, “Detection of Coded Modulation Signals on Linear
Severely Distorted Channels Using Decision-Feedback Noise Prediction
with Interleaving,” IEEE Transactions on Communications, Vol. 36, No. 4,
pp.401-09, April 1988.
36