Cyclic Coded Integer-Forcing Equalization Or Ordentlich Joint work with Uri Erez EE-Systems, Tel Aviv University ACC workshop, February 15th, 2011 Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
The Gaussian ISI channel n k ˆ m x k y k m encoder decoder H ( D ) Σ CSI � y k = x k + h m x k − m + n k m � =0 = x k + ISI k + n k n k is AWGN with unit power. � � 1 + S x ( e j ω ) | H ( e j ω ) | 2 � 1 Mutual Info: I ( S x ( · )) = ω log d ω 2 π Assume (for simplicity) white input: S x ( e j ω ) = const = σ 2 x CSI@Rx only Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
Reliable communication over the ISI channel We are interested in schemes where decoding is decoupled from equalization. -Turbo equalization not considered. Multi-carrier (frequency domain) - OFDM/DMT Transforms the ISI channel into parallel AWGN subchannels - simplifies equalization Coding over a channel with varying SNR may incur an unbounded gap-to-capacity PAPR Non-applicable to channels with finite alphabet (magnetic etc.) Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
Reliable communication over the ISI channel Single carrier (time domain) (Tomlinson-Harashima Precoding (THP) requires complete CSI@Tx - inapplicable...) Linear equalizers: ZF-LE, MMSE-LE Decision-feedback equalization (DFE) Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
Closer look at DFE n k y ′ ˆ m x k y k x k ˆ m k encoder Σ Σ decoder A ( D ) H ( D ) (FFE) − G ( D ) DFE equivalent channel: postcursor or equivalently: z k y ′ m x k ˆ x k m ˆ encoder k G ( D ) decoder Σ − G ( D ) − 1 Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
“Optimality” of DFE MMSE-DFE is known to be “optimal” assuming correct detection of past symbols (CDEF) 1 2 log (1 + SNR DFE − MMSE − U ) = C But how can one get error-free decisions ? Must replace slicer with decoder Possible solution: Guess-Varanasi interleaving We pursue different solution: Move decoder before feedback loop Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
Preview of Suggested Approach Equalize the channel to integer-valued impulse response Add Zero-Padding/Cyclic prefix (as in OFDM) so that Linear Convolution → Cyclic Convolution Use linear cyclic code ⇒ closed under integer-valued cyclic convolution Decode convolved codeword which is also a codeword Apply DFE Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
Integer-Forcing Equalization n k y ′ x ′ ˆ ˆ x k , ˆ m x k y k m I ( D ) encoder H ( D ) k decoder k Σ Σ H ( D ) − I ( D ) − 1 J. Zhan, B. Nazer, U. Erez, M. Gastpar ISIT 2010: Integer-Forcing Equalization proposed ⇒ FFE ( D ) = I ( D ) H ( D ) such that I ( D ) = FFE ( D ) H ( D ) is a monic polynomial with integer coefficients More general than Zero-Forcing where I ( D ) = 1 Less general than DFE since coefficients have to be integers FFE part is reminiscent of partial response equalization by lattice reduction (R. Fischer & C. Siegl 2005) Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
DFE- IF Equalization w n y ′ x n y n n FFE H ( D ) Σ Y ′ ( D ) = FFE ( D ) H ( D ) X ( D ) + FFE ( D ) W ( D ) � �� � I ( D ) L � y ′ n = x n + i n − k x k + z n k =1 For DFE-IF choose I ( D ) so as to maximize SNR DFE − IF = σ 2 σ 2 x x = � 1 /π σ 2 | I ( e j ω ) | 2 1 | H ( e j ω ) | 2 d ω z 2 π − 1 /π Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
Cyclic Codes Let C = { x k } 2 NR k =1 be a linear code over Z q C is cyclic if any cyclic shift of codeword is also a codeword ⇒ Cyclic linear code is closed under integer-valued cyclic convolution with operations performed over Z q . x ∈ C ⇒ x ′ = [ x ⊗ i ] ∈ C Examples of cyclic codes: ”Most” algebraic codes: BCH, RS over prime field,. . . Classes of LDPC codes: type-I EG, type-I PG (Kou, Lin, Fossorier 2001), codes by Shibuya and Sakaniwa (2003) Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
Finding I(D) We would like to maximize SNR DFE − IF = σ 2 σ 2 x x = � 1 /π σ 2 | I ( e j ω ) | 2 1 | H ( e j ω ) | 2 d ω z 2 π − 1 /π σ 2 z can be written in matrix form k 0 k − 1 k − 2 . . . k − L k 1 k 0 k − 1 . . . k − ( L − 1) i T = i ˜ σ 2 K i T z = i . . . . ... . . . . . k L k L − 1 k L − 2 . . . k 0 � π 1 | H ( e j ω ) | 2 e − jmw d ω . 1 where k m = 2 π − π A shortest lattice vector problem Can use LLL as an approximate solution Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
Performance How much do we lose w.r.t. “ideal” DFE by integer forcing? Theorem The noise enhancement caused by the IF equalizer is upper bounded by � 1 � 1 � n � | z 0 z 1 . . . z p − 1 | 2 p n 2 (1 . 4 π n ) n σ 2 z ≤ σ 2 ZF − DFE · min � µ,ν | z ∗ π e µ z ν − 1 | n ≥ p +1 where z 0 , z 1 , . . . , z p − 1 are the maximum-phase zeros of H ( D ) H ∗ ( D −∗ ), and p + 1 is the channel’s length. Bound is based on Minkowski bound for shortest lattice vector Not tight in general Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
Simulation results 8-PAM constellation is used in a TCM-like manner bits cyclic LDPC n=255, k=175 ( R = 2 . 6862 cahnnel use ) code for IF-DFE and MMSE-LE uncoded transmission for MMSE-DFE Channel is 1 + 0 . 894 D + 0 . 814 D 2 + 0 . 239 D 3 − 0 . 070 D 4 + 0 . 036 D 5 − 0 . 022 D 6 0 10 IF−MMSE−DFE MMSE−LE DFE−MMSE −1 10 Block Error Rate −2 10 −3 10 −4 10 2 3 4 5 6 7 8 9 10 SNR norm [dB] Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
Summary, Extensions and Open Questions Integer-Forcing equalization allows channel decoding before applying the DFE loop. Gains: No error propagation Channel coding is much more effective Penalties: DFE coefficients must be integers Code must be cyclic Method is effective for channels of moderate lengths, and high SNR Extension to MMSE exists Explore specific channel models for which IF is advantageous Or Ordentlich and Uri Erez Cyclic Coded Integer-forcing equalization
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