Structured Quasi-Gray labelling for Reed-Muller Grassmannian Constellations 2020 IEEE International Symposium on Information Theory 21-26 June 2020 Qin Yi and Renaud-Alexandre Pitaval Huawei Technologies Sweden AB, Stockholm, Sweden qinyi4@huawei.com renaud.alexandre.pitaval@huawei.com
Motivation ▪ In 3GPP LTE and NR system, the data signals are usually transmitted with pilots ➢ Not all resources can be used for data transmission due to pilot overhead ➢ If the pilot overhead can be eliminated, the performance, e.g., block error rate (BLER), spectrum efficiency, may be improved. ▪ Pilot-less multi-dimensional modulation (PMDM) during negligible channel variation is considered a promising technology to eliminate pilot overhead ➢ PMDM is to modulate bits into modulation symbol (sequence) with length (dimension) 𝑀 > 1 . ➢ Demodulation is based on sequence correlation, and therefore channel knowledge is not needed. ▪ In practical wireless communication system, error correction code is employed and transmission performance is characterized by BLER. A good design of labels of modulation symbols can significantly improve BLER performance. ➢ Gray labelling, which guarantees 1 bit difference between the symbol neighbors, is applied for one dimensional modulation, like QAM. ➢ For higher dimension modulation, like PDMD, it is impossible to achieve 1 bit difference between symbol neighbors. ➢ Therefore, quasi-gray labelling for PMDM is desired to achieve good BLER performance 2
Context: Reed-Muller Grassmannian Constellations ▪ Reed-Muller Grassmannian Constellations ➢ Large minimum chordal distance between modulation symbols, i.e., good symbol detection performance ➢ A constellation with 𝑂 = 2 𝑠+2 𝑛 modulation symbols, each of them is a length- 2 𝑛 vector. The 𝑢 -th modulation symbol is generated as: 𝐲 𝑢 = 𝑏 𝑢 0 , 𝑏 𝑢 1 , … , 𝑏 𝑢 𝑚 , … , 𝑏 𝑢 2 𝑛 − 1 1 𝜌𝑗 , 𝐥 (𝒖) is 1 × 𝑛 binary vector, 𝐦 = 𝑒𝑓2𝑐𝑗(𝑚) is an 𝑛 × 1 • 2 𝐦 T 𝐐 (𝑢) 𝐦 + 2𝐥 (𝒖) 𝐦 where 𝑏 𝑢 𝑚 = 2 𝑛 exp 𝑛−1 binary vector indexing the elements in 𝐲 𝑢 , 0 ≤ 𝑠 ≤ is an integer, and 𝑗 = −1 . 2 𝐐 (𝑢) is selected from a set of binary matrices DG(𝑛, 𝑠) called Delsart-Goethals (DG) set, which can be • 𝑢 (mod 2) , where 𝐐 𝑣 𝑢 is generated by linear combination of 𝑛 pre- 𝑠+1 𝐐 𝑣 decomposed as 𝐐 (𝑢) = σ 𝑣=1 (𝑣) in GF(2) [11]. (𝑣) , 𝐑 2 (𝑣) , … , 𝐑 𝑛 defined symmetric binary basis 𝑛 × 𝑛 matrices 𝐑 1 ▪ Generation Label of Reed-Muller Grassmannian Constellations (𝑢) , a vector of length 𝑠 + 2 𝑛 , where ➢ 𝐲 𝑢 can be generated based on its generation label 𝐜 𝑢 1: 𝑛 = 𝐥 (𝒖) • 𝐜 𝑢 = σ 𝑤=1 𝑢 𝑣𝑛 + 𝑤 𝐑 𝑤 The other bits are given by the coordinates of 𝐐 (𝑢) in a basis of DG(𝑛, 𝑠) : 𝐐 𝑣 (𝑣) 𝑛 • 𝐜 [11] A. Jr, P. Kumar, R. Calderbank, N. Sloane, and P. Sol ´ e , “The Z4 -linearity of Kerdock, 3 Preparata, Goethals, and related codes ,” IEEE Trans. Inf. Theory , vol. 40, pp. 301 – 319, 04 1994.
Context: Quasi-Gray labelling ▪ Quasi-Gray labelling ➢ Good labelling design for a given constellation is to minimize the bit difference between labels of modulation symbols with high Pairwise Error Probability (PEP) ➢ For PMDM, the channel is unknown at the receiver, it was proved that PEP between two modulation symbols H . 𝐲 𝑢 and 𝐲 𝑘 is large if the chordal distance between them is small, where the chordal distance is 𝑒 𝑑 = 1 − 𝐲 𝑗 𝐲 𝑘 ➢ The quasi-Gray labelling principle is to guarantee that modulation symbols that are close to each other (according to 𝑒 𝑑 ) are assigned labels that have a small Hamming distance. ▪ This principle can guarantee 1 bit difference between neighboring modulation symbols for QAM, which is then known as Gray labelling ▪ For Grassmannian Constellations, each modulation symbol has typically too many neighbors, i.e., more than the number of bits in the label, and it is thus impossible to design a Gray labelling. 4
Quasi-Gray labelling design (1/4) ▪ Simplify the global labelling optimization problem ➢ Labelling design is a global optimization problem, which can be converted to a single point labelling optimization problem by using the following property of generation label: Property 1: Given a Reed-Muller Grassmannian constellation of symbol length 2 𝑛 and constellation size 2 𝑠+2 𝑛 and two arbitrary modulation (𝑢) and 𝐜 (𝑘) , respectively, if there is one modulation symbol ො symbols 𝑦 𝑢 and 𝑦 𝑘 with generation labels 𝐜 𝑦 𝑢 with chordal distance 𝑒 𝑑 from 𝑦 𝑢 and (𝑘) satisfying (𝑢) , there must exist one modulation symbol ො generation label መ 𝑦 𝑘 with the same chordal distance 𝑒 𝑑 from 𝑦 𝑘 and generation label መ 𝐜 𝐜 𝑢 𝑢 𝑘 𝑘 ⨁መ ⨁መ 𝐜 𝑛 + 1: 𝑛 𝑠 + 2 𝐜 𝑛 + 1: 𝑛 𝑠 + 2 = 𝐜 𝑛 + 1: 𝑛 𝑠 + 2 𝐜 𝑛 + 1: 𝑛 𝑠 + 2 ➢ Homogeneity property : According to Property 1, the differences of the last 𝑛(𝑠 + 1) bits of generation labels between an anchor symbol and other symbols have the same set of values for any choice of anchor symbols. 𝑢 and generation labels 𝐜 𝑢 in Galois field ➢ We define a bijective linear mapping between the quasi-Gray labels 𝐜 𝑟 GF(2) to preserve this homogeneity property: 𝐉 𝑛×𝑛 𝟏 𝑛×𝑛(𝑠+1) 𝑢 = 𝐜 𝑟 𝑢 𝐇 (mod 2) , where 𝐇 = 𝐜 𝟏 𝑛 𝑠+1 ×𝑛 𝐇 2 𝐇 2 is a full rank matrix of size 𝑛 𝑠 + 1 × 𝑛(𝑠 + 1) in GF(2) ➢ Then, we need only to minimize the bit difference of the last 𝑛(𝑠 + 1) bits between one given anchor symbol and its neighbors. 5
Quasi-Gray labelling design (2/4) ▪ Determination of bijective linear mapping matrix 𝐇 ➢ We choose without loss of generality the anchor symbol 𝐲 0 with generation and quasi-Gray labels as 𝑢 = 𝐜 𝑟 𝑢 = 0,0, … , 0 𝐜 𝑢 equals to 1 and other elements equal ➢ Let 𝐲 𝑢 (𝑢 = 1,2, … , 𝑛(𝑠 + 2)) be the symbols with the 𝑢 -th element of 𝐜 𝑟 T 1 , 𝐜 𝑟 2 , … , 𝐜 𝑟 𝑛 𝑠+2 to 0, i.e., 𝐜 𝑟 is an identity matrix. Then, the generation labels of 𝐲 1 , 𝐲 2 , … , 𝐲 𝑛(𝑠+2) are: 1 1 𝐜 𝐜 𝑟 𝐉 𝑛×𝑛 𝟏 𝑛×𝑛(𝑠+1) 2 2 𝐜 𝐜 𝑟 = 𝐇 = 𝟏 𝑛 𝑠+1 ×𝑛 𝐇 2 … … 𝑛(𝑠+2) 𝑛(𝑠+2) 𝐜 𝐜 𝑟 ➢ The bit difference of quasi-Gray labels between 𝐲 𝑢 (𝑢 = 1,2, … , 𝑛(𝑠 + 2)) and 𝐲 0 is 1. ➢ In order to reduce the bit difference between 𝐲 0 and its neighbors, we should include as many neighbors of 𝐲 0 as possible in 𝐲 𝑢 (𝑢 = 1,2, … , 𝑛(𝑠 + 2)) 𝑢 equals to 1 and other elements equal to 0, and they are not neighbors of 𝐲 0 . ▪ For 𝑢 = 1,2, … , 𝑛 , the 𝑢 -th element of 𝐜 𝑢 equal to 0. ▪ For 𝑢 = 𝑛 + 1, … , 𝑛(𝑠 + 2) , 𝐲 𝑢 can the neighbors of 𝐲 0 with the first 𝑛 element of 𝐜 ▪ We can obtain 𝐇 2 as the matrix composed by the last 𝑛(𝑠 + 1) bits of generation labels of 𝑛(𝑠 + 1) neighbors of 𝐲 0 6
Quasi-Gray labelling design (3/4) ▪ Determination of bijective linear mapping matrix 𝐇 (cont.) ➢ The following Property 2 shows the feasibility of the method to obtain 𝐇 Property 2: For a Reed-Muller Grassmannian constellation of size 2 𝑛(𝑠+2) with symbol length 2 𝑛 , and the anchor modulation symbol, 𝐲 0 with 0 = [0,0, … , 0] , there are 𝛾 neighbors with generation label with the first 𝑛 bits equal to 0, where 𝛾 is equal to the number generation label 𝐜 of matrices 𝐐 (𝑢) with rank 𝑛 − 2𝑠 . ➢ We can use the generation labels of the neighbors satisfying Property 2 to fill the last 𝑛(𝑠 + 1) rows of 𝐇 ▪ We observed for small values of 𝑛 that 𝛾 ≥ 𝑛(𝑠 + 1) and all last 𝑛(𝑠 + 1) rows of 𝐇 can be selected from the labels of such neighbors. ▪ Otherwise, if 𝛾 < 𝑛 𝑠 + 1 , the remaining rows of 𝐇 can be selected from the generation labels whose first 𝑛 bits are equal to 0. 7
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