Combined Equalization and Coding Using Precoding* ECE 492 – Term Project Betül Arda Selçuk Köse Department of Electrical and Computer Engineering University

Combined Equalization and Coding Using Precoding*

ECE 492 – Term Project

Betül ArdaSelçuk Köse

Department of Electrical and Computer Engineering

University of Rochester

*“Combined equalization and coding using precoding” Forney, G.D., Jr.; Eyuboglu, M.V.

2

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

3

Introduction What is the paper about?

Recently developed techniques to achieve capacity objectives

Tomlinson – Harashima precoding:Tomlinson – Harashima precoding: Precoding technique for uncoded modulation

C of bandlimited, high-SNR Gaussian channel C of ideal Gaussian channel

Precoding + coded modulation + shapingPrecoding + coded modulation + shaping Achieves nearly channel capacity of

bandlimited, high-SNR Gaussian channel Is precoding approachprecoding approach a practical route to

capacity on high-SNR+bandlimitedhigh-SNR+bandlimited channel? Decision feedback equalization structure

4

Agenda IntroductionIntroduction Capacity of Gaussian Channels Adaptive ModulationAdaptive Modulation Brief History of EqualizationBrief History of Equalization Equalization TechniquesEqualization Techniques Tomlinson-Harashima PrecodingTomlinson-Harashima Precoding Combined Precoding and Coded Combined Precoding and Coded

ModulationModulation Trellis PrecodingTrellis Precoding Price’s Result & Attaining CapacityPrice’s Result & Attaining Capacity ConclusionConclusion

5

with power constraint

C of Ideal Gaussian channels

Gaussian channel modelIdeal bandlimited Gaussian channel

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

SNR=Sx/Sn=P/N0W

6

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 spectrum

Capacity achieving band:

Determination of optimum water-pouring spectrum

C of Non-Ideal Gaussian channels

7

Agenda IntroductionIntroduction Capacity of Gaussian ChannelsCapacity of Gaussian Channels Adaptive Modulation Brief History of EqualizationBrief History of Equalization Equalization TechniquesEqualization Techniques Tomlinson-Harashima PrecodingTomlinson-Harashima Precoding Combined Precoding and Coded Combined Precoding and Coded


8

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 two-way applications: telephone-line modems To approach capacity: Tx needs to know the channel Not possible for one-way, broadcast, rapidly time-

varying channels unless ch. char.s are known a priori

9

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

10

Agenda IntroductionIntroduction Capacity of Gaussian ChannelsCapacity of Gaussian Channels Adaptive ModulationAdaptive Modulation Brief History of Equalization Equalization TechniquesEqualization Techniques Tomlinson-Harashima PrecodingTomlinson-Harashima Precoding Combined Precoding and Coded Combined Precoding and Coded


11

History of Equalization

1967: Milgo4400 4800b/s W=1600Hz Manually adjustable equalizer knob on the front panel to zero a null meter

1960s: time of considerable research on adaptive modulation Focused on adaptation algorithms that did not require multiplications

1971: Codec9600C 9600b/s (V.29) Automatic adaptive digital LE for W=2400Hz and 16-QAM

1970s: modems more smaller, cheaper, reliable, versatile, but not faster Fractionally spaced equalizers:

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

12

Modem MilestonesYear 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 160-QAM

7

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.

13

Agenda IntroductionIntroduction Capacity of Gaussian ChannelsCapacity of Gaussian Channels Adaptive ModulationAdaptive Modulation Brief History of EqualizationBrief History of Equalization Equalization Techniques Tomlinson-Harashima PrecodingTomlinson-Harashima Precoding Combined Precoding and Coded Combined Precoding and Coded


14

Classical Equalization Techniques

Channel is ideal iff:

D transform

&

15

Equalization Tech. – ZF-LE

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

Zero-forcing linear equalization

r(D) is filtered by 1/h(D) to produce an equalized

response

16

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

17

Equalization Tech. – MLSE Optimum equalization structure if ISI exists

xk drawn from M-pt signal set, h(D) has length v Channel can be modeled as Mv-state machine

Mv-state Viterbi algorithm can be used to implement MLSE

M and/or v is too large complex to implement

If no severe SNR 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

18

Agenda IntroductionIntroduction Capacity of Gaussian ChannelsCapacity of Gaussian Channels Adaptive ModulationAdaptive Modulation Brief History of EqualizationBrief History of Equalization Equalization TechniquesEqualization Techniques Tomlinson-Harashima Precoding Combined Precoding and Coded Combined Precoding and Coded


19

Tomlinson-Harashima Precoding

Precoding works even if h(D) is not invertible i.e. ||1/h||2 is infinite.

20


Tx knows h(D) y(D) = d(D)+2Mz(D) is chosen

x(D) = y(D)/h(D) is in (-M,M] Large M, x(D) PAM seq.

Values continuous in (-M,M] Rx symbol-by-symbol

Ordinary PAM on ideal channel Pe same as with ideal ZF-DFE

Same as on an ideal ch. with SNRZF-DFE=Sx/Sn

Key PointsKey Points

21


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

22

Agenda IntroductionIntroduction Capacity of Gaussian ChannelsCapacity of Gaussian Channels Adaptive ModulationAdaptive Modulation Brief History of EqualizationBrief History of Equalization Equalization TechniquesEqualization Techniques Tomlinson-Harashima PrecodingTomlinson-Harashima Precoding Combined Precoding and Coded

Modulation Using an Interleaver Combining Trellis Encoder and Channel Combined Precoding and Coded Modulation

Trellis PrecodingTrellis Precoding Price’s Result & Attaining CapacityPrice’s Result & Attaining Capacity ConclusionConclusion

23

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.

24

Interleaver (Cont’d)

Transmitted message aaaabbbbccccddddeeeeffffggg

g

Received message aaaabbbbccc____deeeeffffgggg

Transmitted message aaaabbbbccccddddeeeeffffgggg

Interleaved abcdefgabcdefgabcdefgabcdefg

Received message abcdefgabcd____bcdefgabcdefg

De-interleaved aa_abbbbccccdddde_eef_ffg_gg

Without interleaver

With interleaver

25

Combining Trellis Encoder and Channel

Finite state machine representation of trellis encoder and channel

MLSE Algorithm

Reduced state-sequence estimation algorithms are used to make the computation faster.

26

Combined Precoding and Coded Modulation

y(D)=d(D)+2Mz(D) where M is a multiple of 4.

r(D)=y(D)+n(D)

27

Agenda IntroductionIntroduction Capacity of Gaussian ChannelsCapacity of Gaussian Channels Adaptive ModulationAdaptive Modulation Brief History of EqualizationBrief History of Equalization Equalization TechniquesEqualization Techniques Tomlinson-Harashima PrecodingTomlinson-Harashima Precoding Combined Precoding and Coded Combined Precoding and Coded

ModulationModulation Trellis Precoding Price’s Result & Attaining CapacityPrice’s Result & Attaining Capacity ConclusionConclusion

28

Trellis Precoding = Shaping+Precoding+Coding

(N ) then shaping gain1.53dB(1.53dB is the difference between average energies of

Gaussian and uniform distribution) Shaping on regions Trellis Shaping Shell Mapping

Distribution approaches truncated Gaussian

29

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

30

Trellis Precoding = Precoding+Coding+Shaping

“DFE in transmitter”

It combines nicely with coded modulation with “no glue”

It has Asymptotically optimal performance

31


ModulationModulation Trellis PrecodingTrellis Precoding Price’s Result & Attaining Capacity ConclusionConclusion

32

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

33

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

34


ModulationModulation Trellis PrecodingTrellis Precoding Price’s Result & Attaining CapacityPrice’s Result & Attaining Capacity Conclusion

35

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, 893-899, August 1989.

36

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.

Documents

Combined Equalization and Coding Using Precoding* ECE 492 – Term Project Betül Arda Selçuk Köse Department of Electrical and Computer Engineering University