Dirty Paper Coding using Sign-bit Shaping and LDPC Codes

ISIT 2010, Austin, Texas, U.S.A., June 13 - 18, 20101000 100110111010111011111101110000000001001100100110011101010100-15/2 -13/2 -11/2-9/2-7/2-5/2-3/2-1/21/23/25/27/29/211/213/215/2Fig. 1.16-PAM constellation.n − sParity plus MessageBitsk ′MessageBits✲H −1Ts = nlog2MBits✲❄Shaping CoderViterbiAlgorithmminimizes theenergy of(v − αS) mod MsShapedBits✲LDPCEncoderof rate(k − k ′ + s) /n✲ Delay ✲n − (k − k ′ + s)ParityBits✲k − k ′Message bits∏✲a b cMappingtoM − PAMmap (zabc)v−αS✓✏ ❄✲ ∑✒✑❄mod M✲(k − k ′ )MessageBits✻sShaped Bitsz❄(v − αS) mod MFig. 2.Encoder structure.and p T −1 are rearranged by a permutation Π to form thevectors a j , 2 ≤ j ≤ l. This permutation is necessary in animplementation of BICM [11].Note that both the shaping and coding objectives havebeen met at the encoder. The transmitted symbols v − αSmod M have minimal energy in the lattice defined by signbitshaping using the convolutional code. Selected bits insuccessive blocks of symbols form codewords of the LDPCcode. In summary, the encoder structure achieves DPC shapingand LDPC coding with bit-interleaved modulation.B. Decoder StructureThe decoder for the proposed scheme is as shown in Fig.3.Because of the one-codeword delay, parity bits of the T +1-th block and message plus shaped bits of the T -th blockform a valid LDPC codeword. Therefore, an LDPC decoderis first run on these symbols corresponding to a codeword. Thedecoded shaped bits are passed through a syndrome former toget message bits used for shaping. Iterations between shapingdecoder and channel decoder are not needed. The demappercomputes log likelihood ratios (LLRs) for the bits from thereceived symbols in Ŷ = αY + U. The LLRs of the (k − k′ )message bits after a delay of one time step, and the LLRs of then−(k − k ′ + s) parity bits are de-interleaved. The s =nlog 2MLLRs of the sign bits after a delay on one time step, and then−s output LLRs of the de-interleaver are given as the input tothe LDPC decoder. The LDPC decoder outputs k−k ′ messagebits and s bits of the sign bit vector of the previous block. Now,the s-bit sign vector is passed through the syndrome formerto recover the remaining k ′ message bits.The demapper function at the receiver has to calculate LLRstaking into account the modulo M operation at the encoder [4].Therefore, the received constellation A R is a replicated versionof the M-PAM constellation A used at the transmitter (assumingthat scaling factors have been corrected at the receiver).That is, A R = {A−rM, ··· ,A−M, A, A+M,··· ,A+rM}.The number of replications r is chosen so that the averagepower of A R is approximately equal to the total averagepower P X + P S , and the bit mapping of the symbol a + jM(a ∈ A, 1 ≤ j ≤ r) is the same as that for a. The LLR forthe i-th bit in the j-th symbol Ŷj is computed according tothe constellation A R using the following formula:⎛2⎞∑(Ŷj⎜exp ⎝− 1 − a)⎟⎠2 αP NL i =a∈A R:bit i=0∑a∈A R:bit i=1⎛⎜exp ⎝− 1 22⎞.(Ŷj − a)αP N⎟⎠925