Combined Equalization and Coding Using Precoding*

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

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

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

C of Non-Ideal Gaussian channels Capacity achieving band: Determination of optimum water-pouring spectrum • 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

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

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

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

Modem Milestones TCM has made possible the development of very high speed modems.

Classical Equalization Techniques Channel is ideal iff: D transform &

Equalization Tech. – ZF-LE Zero-forcing linear equalization • 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 r(D) is filtered by 1/h(D) to produce an equalized response

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

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

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

Tomlinson-Harashima Precoding Key Points • 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

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

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

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.

Interleaver (Cont’d) Without interleaver With interleaver

Combining Trellis Encoder and Channel MLSE Algorithm • Reduced state-sequence estimation algorithms are used to make the computation faster. Finite state machine representation of trellis encoder and channel

Combined Precoding and Coded Modulation • y(D)=d(D)+2Mz(D) where M is a multiple of 4. • r(D)=y(D)+n(D)

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

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

Trellis Precoding = Precoding+Coding+Shaping • “DFE in transmitter” • It combines nicely with coded modulation with “no glue” • It has Asymptotically optimal performance

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

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

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.

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.

Combined Equalization and Coding Using Precoding*