Combined Equalization and Coding Using Precoding* ECE 492 – Term Project Betül Arda Selçuk Köse...

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.

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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

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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

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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

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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

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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

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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

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

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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

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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

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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

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

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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

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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

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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.

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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

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

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Classical Equalization Techniques

Channel is ideal iff:

D transform

&

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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

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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

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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

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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

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

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Tomlinson-Harashima Precoding

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

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Tomlinson-Harashima Precoding

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

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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

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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

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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.

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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

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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.

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Combined Precoding and Coded Modulation

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

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

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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

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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

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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

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Trellis Precoding = Precoding+Coding+Shaping

“DFE in transmitter”

It combines nicely with coded modulation with “no glue”

It has Asymptotically optimal performance

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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 PrecodingTrellis Precoding Price’s Result & Attaining Capacity ConclusionConclusion

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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

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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

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

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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.

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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.