Combined Equalization and Coding Using Precoding ECE 492

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 Techniques Tomlinson-Harashima Precoding Combined Precoding and Coded Modulation Trellis Precoding Price’s Result & Attaining Capacity Conclusion 2

Introduction ¡ What is the paper about? l l 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 l Precoding + coded modulation + shaping ¡ Achieves nearly channel capacity of bandlimited, high-SNR Gaussian channel l l 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 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 model with power constraint ¡ SNR=Sx/Sn=P/N 0 W Ex: Telephone channel SNR~28 to 36 d. B & BW~2400 to 3200 Hz l l 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 ¡ l ¡ important to optimize B If B is nearly optimal l typically a flat transmit spectrum is almost as good as water-pouring spectrum 6

Agenda ¡ ¡ ¡ ¡ ¡ Introduction Capacity of Gaussian Channels Adaptive Modulation Brief History of 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 l l 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 l ¡ rules out some modem applications Multicarrier modulation with few carriers and short 1/Δf l ISI arises and must be equalized 9

Agenda ¡ ¡ ¡ ¡ ¡ Introduction Capacity of Gaussian Channels Adaptive Modulation Brief History of Equalization Techniques Tomlinson-Harashima Precoding Combined Precoding and Coded Modulation Trellis Precoding Price’s Result & Attaining Capacity Conclusion 10

History of Equalization ¡ 1967: Milgo 4400 4800 b/s W=1600 Hz l ¡ 1960 s: time of considerable research on adaptive modulation l ¡ ¡ ¡ Automatic adaptive digital LE for W=2400 Hz and 16 -QAM 1970 s: modems more smaller, cheaper, reliable, versatile, but not faster Fractionally spaced equalizers: l ¡ Focused on adaptation algorithms that did not require multiplications 1971: Codec 9600 C 9600 b/s (V. 29) l ¡ 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. 4 kb/s modem used uncoded modulation, fixed BW, LE 1983: Trellis coded modulation 9600 b/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 4 PSK 2 1967 Milgo 4400 4. 8 1. 6 8 PSK 3 1971 Codex 9600 C 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 2 D TC 4 1991 V. 32 bis 14. 4 2 D TC 128 -QAM 6 1994 V. 34 28. 8 2. 4 -3. 4 4 D TC 960 -QAM 1998 V. 90 56 same ~9 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 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 l l l ¡ 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 l 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 l l ¡ ¡ ¡ M -state Viterbi algorithm can be used to implement MLSE M and/or v is too large complex to implement If no severe SNR l l ¡ v xk drawn from M-pt signal set, h(D) has length v Channel can be modeled as Mv-state machine SNR of matched filter bound Matched filter bound: bound on the best SNR achievable with h(D) If SNR is severe l MLSE fails to achieve this SNR, performance analysis become difficult 17

Agenda ¡ ¡ ¡ ¡ ¡ Introduction Capacity of Gaussian Channels Adaptive Modulation Brief History of 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)+2 Mz(D) is chosen l ¡ Large M, x(D) PAM seq. l ¡ Values continuous in (-M, M] Rx symbol-by-symbol l ¡ x(D) = y(D)/h(D) is in (-M, M] Ordinary PAM on ideal channel Pe same as with ideal ZF-DFE l 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 ¡ l ¡ For uncoded systems ideal ZF-DFE assumption works well Therefore, DFE is preferred to TH l DFE does not require CSI at tx 21

Agenda ¡ ¡ ¡ ¡ Introduction Capacity of Gaussian Channels Adaptive Modulation Brief History of Equalization Techniques Tomlinson-Harashima Precoding Combined Precoding and Coded Modulation l l l ¡ ¡ ¡ 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 aaaabbbbccccddddeeeeffffgg gg Received message aaaabbbbccc____deeeeffffgg gg With interleaver Transmitted message aaaabbbbccccddddeeeeffffgggg Interleaved abcdefgabcdefg 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)+2 Mz(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 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. 53 d. B (1. 53 d. B 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 d. B for 4 to 512 states. Binary codes l l l ¡ Sequential decoding of convolution codes Turbo codes Low-density parity check codes. Non-binary codes l 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 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 6 d. B, max 7. 5 d. B • Shaping: can achieve 1 d. B, max 1. 53 d. B • Total: can achieve 7 d. B, max 9 d. B 33

Agenda ¡ ¡ ¡ ¡ ¡ Introduction Capacity of Gaussian Channels Adaptive Modulation Brief History of 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. de. Buda, “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. de. Buda, “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