A New Construction of 4q-QAM Golay Complementary Sequences

Quadrature amplitude modulation (QAM) constellation and Golay complementary sequences (GCSs) are usually applied in orthogonal frequency division multiplexing (OFDM) systems to obtain a higher data rate and a lower peak-to-mean envelope power ratio (PMEPR). In this paper, after a sufficient search of the literature, it was found that increasing the family size is an effective way to improve the data rate, and the family size is mainly determined by the number of offsets in the general structure of QAM GCSs. Under the guidance of this idea, we propose a new construction for 4q-QAM GCSs through generalized Boolean functions (GBFs) based on a new description of a 4q-QAM constellation, which aims to enlarge the family size of GCSs and obtain a low PMEPR. Furthermore, a previous construction of 4q-QAM GCSs presented by Li has been proved to be a special case of the new one, and the family size of new sequences is much larger than those previously mentioned, which means that there was a great improvement in the data rate. On the other hand, a previous construction of 16-QAM GCSs presented by Zeng is also a special case of the new one in this paper, when q=2. In the meantime, the proposed sequences have the same PMEPR upper bound as the previously mentioned sequences presented by Li when applied in OFDM systems, which increase the data rate without degrading the PMEPR performance. The theoretical analysis and simulation results show that the proposed new sequences can achieve a higher data rate and a low PMEPR.

Sensors 2022, 22, 7092 3 of 18 of 4 q -QAM GCSs by providing new compatible offsets based on the factorization of the integer q, which had the generalized cases I-V [23,24] as special cases. Since the family size directly determines the data rate, expanding the family size of sequences is an effective way to improve the data rate [32].
As the data rates and mobility supported by the OFDM system increase, the number of subcarriers also increases, resulting in a high PMEPR. However, reducing the PMEPR will increase the computational complexity. To solve this problem, many PMEPR reduction schemes that reduce the computational complexity of OFDM systems have been proposed. In 2013, Rahmatallah and Mohan [33] generated a taxonomy of the available solutions to mitigate the high PMEPR problem in OFDM systems. They also provided complexity analyses for several PMEPR reduction methods to demonstrate the differences in computational complexity between different methods. In 2017, Zhao et al. [34] proposed an improved joint optimization scheme, which combined the partial transmit sequence (PTS) and clipping and filtering (CF) methods with great PMEPR reduction performance. In the same year, Joo et al. [35] proposed two PTS schemes without side information (SI) for reducing the PMEPR of OFDM signals, which did not reduce the BER performance as compared to the conventional PTS with perfect SI. The above-mentioned PMEPR reduction schemes can effectively solve the problem and improve the performance of OFDM systems.
Based on the above literature search, it was found that increasing the family size is an effective way to improve the data rate, and the family size is mainly determined by the number of offsets in the general structure of QAM GCSs. This is the source of the research idea of this paper. The innovation points and main contributions of this paper are summarized as follows:

1.
A new description of a 4 q -QAM constellation with q + 1 independent quaternary variables is presented in this paper, which has one more variable than the previous description and includes it as a special case; 2.
On this basis, a new construction of 4 q -QAM GCSs is proposed, which greatly increases the family size and improves the data rate; 3.
More specifically, the new construction of the QAM sequences includes the construction in [23] as a special case and has a larger family size, which means a higher data rate; 4.
At the same time, the construction of 16-QAM GCSs in [28] is also a special case in this paper when q = 2; 5.
Furthermore, the proposed sequences in this paper have the same PMEPR upper bounds as the known ones, which increase the data rate without degrading the PMEPR performance.
The rest of this paper is organized as follows. In Section 2, some background information is provided, including the definitions of GCSs and generalized Boolean functions (GBFs), the construction of QAM signals from the QPSK constellation, the PMEPR upper bound of GCSs, and some related conclusions. In Section 3, a new description of a 4 q -QAM constellation is first presented. Based on this foundation, a new construction of 4 q -QAM GCSs is proposed, and an example of 64-QAM GCSs is given to verify this conclusion. Then, the family size and the PMEPR upper bound of the new construction are described. In Section 4, the main results are summarized, and the main conclusions are given.

Materials and Methods
In this section, we provide some necessary materials, including the definitions of GCSs and GBFs, the construction of QAM signals from a QPSK constellation, and the PMEPR upper bound of the GCSs as well as several related lemmas.

Golay Complementary Sequences
Given two sequences of length N with complex elements, A = [A 0 , A 1 , · · · , A N−1 ] and B = [B 0 , B 1 , · · · , B N−1 ], we define [36] to be an aperiodic correlation function (ACF) of A and B. More specifically, we say that C A,A (τ) is an aperiodic autocorrelation function when A = B, which can be simplified as C A (τ). If not, we call C A,B (τ) an aperiodic cross-correlation function when A = B [37]. For two sequences, A and B, if they satisfy [38] we say that (A, B) forms a Golay complementary pair, and that A, B are both Golay complementary sequences (GCSs).
On the basis of the standard GBFs, Davis and Jedwab discovered the connection between 2 h -PSK GCSs and the generalized Reed-Muller codes; this led to the construction of a large class of PSK GCSs, which are well-known as the standard 2 h -PSK GDJ GCSs [40]. Several related conclusions about these sequences are presented below.

Construction of QAM Signals from a QPSK Constellation
The QPSK constellation can be described based on the quaternary symbols Z 4 = {0, 1, 2, 3} using the following set [11]: On the other hand, the 4 q -QAM constellation (positive integer q ≥ 2) can be described with the following set [29]: One of the methods to produce a 4 q -QAM constellation is through the use of QPSK symbols with the shift and rotation operations. Figure 1 shows the construction of a 64-QAM constellation by adding three QPSK symbols [23]. One of the methods to produce a 4 -QAM constellation is through the use of QPSK symbols with the shift and rotation operations. Figure 1 shows the construction of a 64-QAM constellation by adding three QPSK symbols [23]. With the same method, the general 4 -QAM constellation can be expressed as [17]: When the q-dimensional vector ( (0) , (1) , ⋯ , ( −1) ) varies from (0,0, ⋯ ,0) to (3,3, ⋯ ,3), the above equation correspond exactly to the 4 -QAM constellation.

Family Size and Code Rate
The family size of sequences directly determines the data rate. More specifically, when applied in OFDM systems, the family size affects the selection of the number of subcarriers. In [18], the definition of the code rate is provided, which is introduced to guide the selection of subcarriers. The following lemma gives the definition.
Lemma 3 (Ref. [18]). The code rate of a code C consisting of sequences of length N symbols is [18] where | | stands for the family size of the code C.

Family Size and Code Rate
The family size of sequences directly determines the data rate. More specifically, when applied in OFDM systems, the family size affects the selection of the number of subcarriers. In [18], the definition of the code rate is provided, which is introduced to guide the selection of subcarriers. The following lemma gives the definition.
Lemma 3 (Ref. [18]). The code rate of a code C consisting of sequences of length N symbols is [18] where |C| stands for the family size of the code C.

PMEPR Upper Bound of GCSs
Consider an OFDM system that has N subcarriers, f 0 is the carrier frequency, and f i is the frequency of the ith subcarrier, where , and ∆ f is the bandwidth between each sub-channel; hence, the transmitted complex signal S a (t), which is encoded by the sequence a = (a 0 , a 1 , · · · , a N−1 ), is represented as follows [41]: Let C represent the ensemble of all possible codewords (a ∈ C), and p(a) indicate how likely the codeword a is to be transmitted. Thus, the average envelope power of the transmitted signal is written as follows [13]: If the instantaneous envelope power of the transmitted OFDM signal is P(t) = |S a (t)| 2 , then we write the PMEPR of the codeword a as [13] PMEPR(a) = max(P(t)) P av .
The following lemma holds if an OFDM signal is encoded by binary, quaternary, or polyphase GCSs.
However, when it comes to 4 q -QAM GCSs, which are proposed in [23], the following lemma gives their PMEPR upper bound.

Results and Discussion
In this section, we first present a new description of a 4 q -QAM constellation. On this basis, we propose a new construction of 4 q -QAM GCSs and give an example of 64-QAM GCSs to verify this proposal. Then, we describe the family size and the PMEPR upper bound of the new construction.

New Description of 4 q -QAM Constellation
In the description of the 4 q -QAM constellation in Equation (7), there are q independent quaternary variables. Here is a new description of the 4 q -QAM constellation presented by this paper, which is given by the following theorem: This new description has q + 1 independent quaternary variables, which is one more than the description in Equation (7).

Proof of Theorem 1 ([29]
). We divide the proof into two parts: (1) each symbol in the set of Equation (14) must be included in the 4 q -QAM constellation Ω 4 q −QAM ; (2) each 4 q -QAM symbol in Ω 4 q −QAM can be produced by the set in Equation (14).
the description of Equation (14) produces the following symbol: It is obvious that n 0 + n 1 + n 2 + n 3 = 2 q − 1, which is an odd integer, so the values of the four integers n 0 , n 1 , n 2 , and n 3 only have two cases: "one odd and three evens" or "one even and three odds". Apparently, both cases have the same conclusion, which is that the values of the integers n 0 − n 1 − n 2 + n 3 and n 0 + n 1 − n 2 − n 3 are both odd. Based on this conclusion, we clearly have To sum up, we reach the following conclusion: S ∈ Ω 4 q −QAM .
(2) For ∀a + jb ∈ Ω 4 q −QAM , combined with Equation (14), we can obtain the following equation: We then need to prove that there is at least one (q that satisfies Equation (18). We chose the vector by using the following strategy.
Step 1: We discretionarily chose a+b 2 "0s" or "2s" in this vector, depending on the sign of a+b 2 .
Step 2: We discretionarily chose b−a 2 "1s" or "3s" in the remaining items aside from the chosen part in Step 1, depending on the sign of b−a 2 .
Step 3: We discretionarily chose "0 and 2" or "1 and 3" in pairs in the remaining items aside from the chosen part in Steps 1 and 2, so that all the powers of j from these unused items add up to zero.
From the above steps, we can draw the following conclusion. For each symbol in Ω 4 q −QAM , we can find at least one (q to ensure this 4 q -QAM symbol can be generated by Equation (18).
Thus, summarizing the above, Theorem 1 has been proved. More specifically, if we set then Equation (14) can be converted into Obviously, Equation (20) is equivalent to Equation (7). Therefore, Equation (7) is a special case of Theorem 1.

Construction of New QAM Sequences
Based on the description of Equation (14), we propose a new construction of 4 q -QAM GCSs in this section. (3), then the obtained functions f (x) are quaternary GBFs. Hence, we let By means of Equation (14), the 4 q -QAM sequences A = (A 0 , A 1 , · · · , A N−1 ) and B = (B 0 , B 1 ,· · · , B N−1 ) with length N = 2 m can be constructed as follows: Then, the obtained 4 q -QAM sequences A and B are 4 q -QAM GCSs when the offsets s (p) (x) (1 ≤ p ≤ q) and the corresponding pairing difference are as in one of the following cases: Case I I I : Particularly, if s (q−1) (x) = s (q) (x), we have which is the same as the construction of Theorem 2 in [23]. Therefore, the construction of Theorem 2 in [23] is a special case of the one constructed by Theorem 2 in this paper. Figure 2 clearly depicts the process of how to construct the QAM GCSs from the QAM constellation and shows the relationship between [23] and this paper.
Particularly, if ( −1) ( ) = ( ) ( ), we have which is the same as the construction of Theorem 2 in [23]. Therefore, the construction of Theorem 2 in [23] is a special case of the one constructed by Theorem 7 in this paper. Figure 2 clearly depicts the process of how to construct the QAM GCSs from the QAM constellation and shows the relationship between [23] and this paper.

Figure 2.
The process from the QAM constellation to the QAM GCSs, and the relationship between [23] and this paper. Figure 2. The process from the QAM constellation to the QAM GCSs, and the relationship between [23] and this paper.
Proof of Theorem 2. For ∀τ > 0, the aperiodic autocorrelation function of the sequence A can be expressed as follows: Combined with Equation (22), we can calculate Equation (27) into For the sequence B in Equation (22), using the same method, we can get Sensors 2022, 22, 7092 10 of 18 By employing Lemma 2, we can get that sequences a (p) and b (p) (0 ≤ p ≤ q) form GCSs, and then we can obtain Hence, in order to ensure that sequences A and B are GCSs, the following equation would be a sufficient condition [41]: There are two QPSK GCS pairs involved in Equation (31), represented as (a p , b p ) and (a p , b p ). Then, we have the following equation: Let i = (i 1 , i 2 , · · · , i m ) denote the binary representation of i, i.e., i = ∑ m k=1 i k 2 m−k . Let f i , a i , b i , µ i , and s i denote the ith elements of the sequences generated from f (x), a(x), b(x), µ(x), and s(x) over Z 4 . From Equation (32), we have Therefore, The last summation in Equation (34) was verified to equal its own negation in [23], so Equation (34) must be zero. From this conclusion, it can be concluded that Equation (31) is valid.
Because Equation (31) was proved to be true, we can get which can prove that sequences A and B are GCSs, and then the proof of Theorem 2 is complete. An example is given below to verify this conclusion and make it easier for readers to understand.
Sequence A's autocorrelation function C A (τ), sequence B's autocorrelation function C B (τ), and their sum C A (τ) + C B (τ) were calculated, and the results are depicted in Figure 3.
Part 2: Consider the offset pair s (q−1) (x), s (q) (x) s (p) (x) ∈ Z 4 , q − 1 ≤ p ≤ q . By using the same method in [18], we can group the possible offsets into five groups, which satisfy the empty pairwise intersections, as below. It is known that the permutations of the offset coefficients d Table 1 [23], and we can get
Sequence A's autocorrelation function ( ) , sequence B's autocorrelation function ( ), and their sum ( ) + ( ) were calculated, and the results are depicted in Figure 3.
Employing the above strategy, (a) for S 1 , we calculated the possible offset pairs with 3 + 2 + 1 = 6; (b) for S 2 , we calculated the possible offset pairs with 11 + 10 + · · · + 1 = 66; (c) for S 3 , this used the same situation as the previous one; (d) for S 4 , there existed only one offset pair. Furthermore, the parameter ω can vary from 2 to m − 1, so the number of possible offset pairs is m − 2 in total; (e) for S 5 , we calculated the possible offset pairs with 9 + 8 + · · · + 1 = 45. In addition, the parameter ω can vary from 1 to m − 1 in each offset pair. Hence, there are a total of 45(m − 1) possible offset pairs in this case. By adding up (a) to (e), we can obtain 46m + 91 possible offset pairs in total.
Combining Part 1 and Part 2, Case II has a total of (46m + 91)·4 q−2 possible offset pairs. By summing all the possible offset pairs in Case I and Case II, the results show that there are (46m + 91)·4 q−2 + (m + 1)·4 2(q−1) − (m + 1)·4 q−1 + 2 q−1 different offset pairs in total. Thus, by employing Lemma 1, the theorem below gives the family size of the new QAM sequences.
More specifically, let q = 2 in Equation (43), and we can obtain that the number of 16-QAM GCSs is (58m which is exactly the result of Theorem 6 in [28]. Hence, the conclusion of [28] is a special case in this paper when q = 2. Figure 4 depicts the comparison of the family sizes of 16-QAM GCSs when q = 2 between [23] and this paper; it can be seen that the number of sequences increased significantly. Table 2 shows the comparison of the code rates and family sizes between [23,28] and this paper, providing a visual representation of the data rate improvement.
which is exactly the result of Theorem 6 in [28]. Hence, the conclusion of [28] is a special case in this paper when = 2. Figure 4 depicts the comparison of the family sizes of 16-QAM GCSs when = 2 between [23] and this paper; it can be seen that the number of sequences increased significantly. Table 2 shows the comparison of the code rates and family sizes between [23,28] and this paper, providing a visual representation of the data rate improvement.    The PMEPR upper bound of the new 4 q -QAM GCSs are represented by the following theorem.

Theorem 4.
Consider a code C whose codewords are made up of 4 q -QAM GCSs constructed by Theorem 2, then the PMEPR upper bound of the code C satisfies Proof of Theorem 4. For ∀A ∈ C, let A be a 4 q -QAM GCSs with length N constructed by Theorem 2, then the peak envelope power (PEP) of sequence A satisfies PEP(A) ≤ 2 ∑ N−1 i=0 |A i | 2 due to Equation (26) in [23]. Combined with Equation (22), and in order to ensure that the average squared magnitude is equal to one, we have Then, we can get and the proof is complete.
As a result, the new QAM GCSs in this paper have the same PMEPR upper bound as the sequences in [23], which means that there was no degradation in the PMEPR performance.

Conclusions
In this paper, a new description of a 4 -QAM constellation has been presented. Based on this foundation, we proposed a new construction of 4 -QAM GCSs of length 2 using GBFs, which resulted in the enlargement of the family size of GCSs and allowed us to obtain a low PMEPR. A previous construction of 4 -QAM GCSs presented by Li and another previous construction of 16-QAM GCSs presented by Zeng were proven to be special cases of ours. The family size of the new sequences was calculated to be [(46 + 91) • 4 −2 + ( + 1) • 4 2( −1) − ( + 1) • 4 −1 + 2 −1 ] • ( !/2)4 +1 . This result shows that the new sequences have a larger family size, which means that there was a great improvement in the data rate. When applied in OFDM systems, the new sequences As a result, the new QAM GCSs in this paper have the same PMEPR upper bound as the sequences in [23], which means that there was no degradation in the PMEPR performance.

Conclusions
In this paper, a new description of a 4 q -QAM constellation has been presented. Based on this foundation, we proposed a new construction of 4 q -QAM GCSs of length 2 m using GBFs, which resulted in the enlargement of the family size of GCSs and allowed us to obtain a low PMEPR. A previous construction of 4 q -QAM GCSs presented by Li and another previous construction of 16-QAM GCSs presented by Zeng were proven to be special cases of ours. The family size of the new sequences was calculated to be (46m + 91)·4 q−2 + (m + 1)·4 2(q−1) − (m + 1)·4 q−1 + 2 q−1 ·(m!/2)4 m+1 . This result shows that the new sequences have a larger family size, which means that there was a great improvement in the data rate. When applied in OFDM systems, the new sequences have the same PMEPR upper bound of 6(2 q − 1)/(2 q + 1) as the sequences presented by Li, which means we increased the data rate without degrading the PMEPR performance. Our next research directions will be to propose the PMEPR reduction schemes by reducing the computational complexity of OFDM systems, and to focus on the future challenges of a lower PMEPR by improving or reducing the computational complexity of OFDM MIMO systems.

Data Availability Statement:
The data used to support this study will be available from the corresponding author on reasonable request.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: