Index Modulation for 5G Wireless Communications by Miaowen Wen Xiang Cheng & Liuqing Yang

Author:Miaowen Wen, Xiang Cheng & Liuqing Yang
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Bits in lexicographic order

Permutations

Bits in gray code order

Permutations

00

(1 2 3)

00

(1 2 3)

01

(1 3 2)

01

(1 3 2)

10

(2 1 3)

11

(3 1 2)

11

(2 3 1)

10

(3 2 1)

Ranking and Unranking Methods

The look-up table method necessitates a storage of all permutations at both the transmitter and receiver, which becomes impractical for large . For example, 36,28,800 and permutations need to be stored for and 20, respectively. Obviously, it is impractical to store them for current computers. Therefore, it is advisable to create an easy-to-implement one-to-one mapping from the information bits to the corresponding permutations, called unranking, and inverse mapping from the permutations to the corresponding information bits, called ranking.

To begin with, we have to first figure out how to generate permutations in the gray code order. Aiming at this, we resort to the idea presented in [10]. Table 3.5 gives a solution for . From this example, we can expect some important properties of gray code order for a general as follows. For ease of exposition, let us define left and right moving directions for number as and , respectively. Also, define a directional indicator for number as , where stands for the situation of and for the situation of . If the whole swaps between number and the adjacent element in the set for the same direction is said to be one round for number , one will see that the total number of rounds of number , denoted by , is . In addition, it can be concluded that if is odd and if is even, where is a variable indicating that number is in the -th round. Details on the generation rule can be referred to the Trotter-Johnson ranking and unranking algorithms [10]. In what follows, we summarize them in our notations and mainly focus on their application to DSM.Table 3.5Generating permutations of in the gray code order

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