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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

SLIDE 1

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

SLIDE 2

▪ 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.

Motivation

▪ 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

SLIDE 3

▪ 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-2𝑛 vector. The 𝑢-th modulation symbol is generated as:

𝐲𝑢 = 𝑏𝑢 0 , 𝑏𝑢 1 , … , 𝑏𝑢 𝑚 , … , 𝑏𝑢 2𝑛 − 1

where 𝑏𝑢 𝑚 =

1 2𝑛 exp 𝜌𝑗 2 𝐦T𝐐(𝑢)𝐦 + 2𝐥(𝒖)𝐦

, 𝐥(𝒖) is 1 × 𝑛 binary vector, 𝐦 = 𝑒𝑓2𝑐𝑗(𝑚) is an 𝑛 × 1 binary vector indexing the elements in 𝐲𝑢, 0 ≤ 𝑠 ≤

𝑛−1 2

is an integer, and 𝑗 = −1.

𝐐(𝑢) is selected from a set of binary matrices DG(𝑛, 𝑠) called Delsart-Goethals (DG) set, which can be

decomposed as 𝐐(𝑢) = σ𝑣=1

𝑠+1 𝐐𝑣 𝑢 (mod 2), where 𝐐𝑣 𝑢 is generated by linear combination of 𝑛 pre-

defined symmetric binary basis 𝑛 × 𝑛 matrices 𝐑1

(𝑣), 𝐑2 (𝑣), … , 𝐑𝑛 (𝑣) in GF(2) [11].

▪ Generation Label of Reed-Muller Grassmannian Constellations

➢ 𝐲𝑢 can be generated based on its generation label 𝐜𝑕

(𝑢), a vector of length 𝑠 + 2 𝑛, where

𝐜𝑕

𝑢 1: 𝑛 = 𝐥(𝒖)

The other bits are given by the coordinates of 𝐐(𝑢) in a basis of DG(𝑛, 𝑠) : 𝐐𝑣

𝑢 = σ𝑤=1 𝑛

𝐜𝑕

𝑢 𝑣𝑛 + 𝑤 𝐑𝑤 (𝑣)

Context: Reed-Muller Grassmannian Constellations

[11] A. Jr, P. Kumar, R. Calderbank, N. Sloane, and P. Sol´e, “The Z4-linearity of Kerdock, Preparata, Goethals, and related codes,” IEEE Trans. Inf. Theory, vol. 40, pp. 301 – 319, 04 1994.

SLIDE 4

▪ 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 𝐲𝑢 and 𝐲𝑘 is large if the chordal distance between them is small, where the chordal distance is 𝑒𝑑 = 1 − 𝐲𝑗𝐲𝑘

➢ The quasi-Gray labelling principle is to guarantee that modulation symbols that are close to each other (according to 𝑒𝑑) 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.

Context: Quasi-Gray labelling

SLIDE 5

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: ➢ 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. ➢ We define a bijective linear mapping between the quasi-Gray labels 𝐜𝑟

𝑢 and generation labels 𝐜𝑕 𝑢 in Galois field

GF(2) to preserve this homogeneity property: 𝐜𝑕

𝑢 = 𝐜𝑟 𝑢 𝐇 (mod 2), where 𝐇 =

𝐉𝑛×𝑛 𝟏𝑛×𝑛(𝑠+1) 𝟏𝑛 𝑠+1 ×𝑛 𝐇2 𝐇2 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.

Property 1: Given a Reed-Muller Grassmannian constellation of symbol length 2𝑛 and constellation size 2 𝑠+2 𝑛 and two arbitrary modulation symbols 𝑦𝑢 and 𝑦𝑘 with generation labels 𝐜𝑕

(𝑢) and 𝐜𝑕 (𝑘), respectively, if there is one modulation symbol ො

𝑦𝑢 with chordal distance 𝑒𝑑 from 𝑦𝑢 and generation label መ 𝐜𝑕

(𝑢), there must exist one modulation symbol ො

𝑦𝑘 with the same chordal distance 𝑒𝑑 from 𝑦𝑘 and generation label መ 𝐜𝑕

(𝑘) satisfying

𝐜𝑕

𝑢

𝑛 + 1: 𝑛 𝑠 + 2 ⨁መ 𝐜𝑕

𝑢

𝑛 + 1: 𝑛 𝑠 + 2 = 𝐜𝑕

𝑘

𝑛 + 1: 𝑛 𝑠 + 2 ⨁መ 𝐜𝑕

𝑘

𝑛 + 1: 𝑛 𝑠 + 2

SLIDE 6

Quasi-Gray labelling design (2/4)

▪ Determination of bijective linear mapping matrix 𝐇

➢ We choose without loss of generality the anchor symbol 𝐲0 with generation and quasi-Gray labels as 𝐜𝑕

𝑢 = 𝐜𝑟 𝑢 = 0,0, … , 0

➢ Let 𝐲𝑢 (𝑢 = 1,2, … , 𝑛(𝑠 + 2)) be the symbols with the 𝑢-th element of 𝐜𝑟

𝑢 equals to 1 and other elements equal

to 0, i.e., 𝐜𝑟

1 , 𝐜𝑟 2 , … , 𝐜𝑟 𝑛 𝑠+2 T

is an identity matrix. Then, the generation labels of 𝐲1, 𝐲2, … , 𝐲𝑛(𝑠+2) are: 𝐜𝑕

𝐜𝑕

… 𝐜𝑕

𝑛(𝑠+2)

= 𝐜𝑟

𝐜𝑟

… 𝐜𝑟

𝑛(𝑠+2)

𝐇 = 𝐉𝑛×𝑛 𝟏𝑛×𝑛(𝑠+1) 𝟏𝑛 𝑠+1 ×𝑛 𝐇2 ➢ The bit difference of quasi-Gray labels between 𝐲𝑢 (𝑢 = 1,2, … , 𝑛(𝑠 + 2)) and 𝐲0 is 1. ➢ In order to reduce the bit difference between 𝐲0 and its neighbors, we should include as many neighbors of 𝐲0 as possible in 𝐲𝑢 (𝑢 = 1,2, … , 𝑛(𝑠 + 2))

▪ For 𝑢 = 1,2, … , 𝑛, the 𝑢-th element of 𝐜𝑕

𝑢 equals to 1 and other elements equal to 0, and they are not neighbors of 𝐲0.

▪ For 𝑢 = 𝑛 + 1, … , 𝑛(𝑠 + 2), 𝐲𝑢 can the neighbors of 𝐲0 with the first 𝑛 element of 𝐜𝑕

𝑢 equal to 0.

▪ We can obtain 𝐇2 as the matrix composed by the last 𝑛(𝑠 + 1) bits of generation labels of 𝑛(𝑠 + 1) neighbors of 𝐲0

SLIDE 7

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 ➢ 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.

Property 2: For a Reed-Muller Grassmannian constellation of size 2𝑛(𝑠+2) with symbol length 2𝑛, and the anchor modulation symbol, 𝐲0 with generation label 𝐜𝑕

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

f matrices 𝐐(𝑢) with rank 𝑛 − 2𝑠.

SLIDE 8

Quasi-Gray labelling design (4/4)

▪ Hamming Distance Discussion

➢ The Hamming distances from 𝐲0 to the following two kinds of neighbors of 𝐲0 are significantly reduced. ▪ Neighbor type 1: According to Property 2, there are min 𝛾, 𝑛(𝑠 + 1) neighbors of 𝐲0 whose quasi-Gray labels have 1 bit difference from 𝐲0.

Since the first 𝑛 bits of their generation labels are 0, these neighbors belong to different orthogonal

sets of modulation symbols. ▪ Neighbor type 2: For each neighbor in neighbor set 1, there are 2𝑛−2𝑠 − 1 other neighbors of 𝐲0 in the same orthogonal modulation symbol set, whose quasi-Gray labels have at most 𝑛 + 1 bit difference from 𝐲0

It was pointed out in [7] that if a symbol is a neighbor of 𝐲0, there exists 2𝑛−2𝑠 − 1 other neighbors
f 𝐲0 in the same orthogonal set
Note that the last 𝑛(𝑠 + 1) bits of the generation/quasi-Gray labels of symbols in the same orthogonal set

are the same.

➢ To conclude, the Hamming distance of at least 2𝑛−2𝑠 × min 𝛾, 𝑛(𝑠 + 1) neighbors from each modulation symbol are significantly reduced by using the proposed quasi-Gray labelling.

[7] R. Calderbank, S. Howard, and S. Jafarpour, “Construction of a large class of deterministic sensing matrices that satisfy a statistical isometry property,” IEEE J. Sel. Topics Sig. Proc., vol. 4, no. 2, pp. 358–374, April 2010.

SLIDE 9

Comparison and evaluations(1/2)

▪ Average Hamming distances between neighbors

𝐜𝑕

1 = 0,0,0,1,0,0,1,0,0 ; 𝐜𝑕 2 = 1,0,0,1,0,0,1,0,0

𝐜𝑕

3 = 0,0,0,1,1,0,0,1,0 ; 𝐜𝑕 4 = 1,1,1,1,1,0,0,1,0

𝐜𝑕

5 = 0,0,0,0,0,1,1,1,0 ; 𝐜𝑕 6 = 0,0,1,0,0,1,1,1,0

𝐜𝑕

7 = 0,0,0,1,0,1,0,0,1 ; 𝐜𝑕 8 = 1,0,1,1,0,1,0,0,1

𝐜𝑕

9 = 0,0,0,0,1,1,1,0,1 ; 𝐜𝑕 10 = 0,1,0,0,1,1,1,0,1

𝐜𝑕

11 = 0,0,0,1,1,1,0,1,1 ; 𝐜𝑕 12 = 1,1,0,1,1,1,0,1,1

𝐜𝑕

13 = 0,0,0,0,1,0,1,1,1 ; 𝐜𝑕 14 = 0,1,1,0,1,0,1,1,1

➢ In this example, we consider 𝑛 = 3 and 𝑠 = 1, for which the modulation symbol length is 2𝑛 = 8 and the constellation size is 2𝑛(𝑠+2) = 512 (9 bits). ➢ The generation and quasi-Gray labels of anchor symbol 𝐲0 are 𝐜𝑕