A Direct and Generalized Construction of Polyphase Complementary Set with Low PMEPR and High Code-Rate for OFDM System

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1 1 A Diret and Generalized Constrution of Polyphase Complementary Set with Low PMEPR and High Code-Rate for OFDM System Palash Sarkar, Sudhan Majhi, and Zilong Liu arxiv: v1 [s.it] 16 Jan 019 Abstrat A salient disadvantage of orthogonal frequeny division multiplexing OFDM systems is the high peak-to-mean envelope power ratio PMEPR. The PMEPR problem an be solved by using omplementary sequenes with low PMEPR but it may suffer from low ode rate if the number of omplementary sequenes is small. In this paper, we present a new onstrution of omplementary set CS by using generalized Boolean funtions GBFs, whih generalizes Shmidt s onstrution, Paterson s onstrution and Golay omplementary pair GCP given by Davis and Jedwab. The proposed CS provides lower PMEPR with higher ode rate ompared with Shmidt s method for the sequenes orresponding to higher order 3 GBFs. We obtain omplementary sequenes with maximum PMEPR of k+1 and k+ M k, M are non-negative integers that an be easily derived from the GBF assoiated with the CS. Index Terms Complementary set CS, ode rate, Golay omplementar pair GCP, generalized Boolean funtion GBF, orthogonal frequeny-division multiplexing OFDM, peak-tomean envelope power ratio PMEPR, Reed-Muller RM ode. I. INTRODUCTION Orthogonal frequeny-division multiplexing OFDM is a multiarrier ommuniation tehnique whih is used in several high data rate wireless ommuniation standards suh as Wireless Fidelity Wi-Fi, Mobile Broadband Wireless Aess MBWA, Worldwide Interoperability for Mirowave Aess WiMax, terrestrial digital TV systems, 3GPP Long Term Evolution LTE, et. A major problem of OFDM is its large peak-to-mean envelope power ratio PMEPR for the unoded signals. PMEPR redution through a oding tehnique an be ahieved by designing a large odebook whose odewords, e.g., in the form of sequenes, have low PEMPR values. Golay omplementary pair GCP was introdued by M. J. E. Golay in [1], either sequene from a GCP is known as Golay sequene. The idea of GCP was extended to omplementary sets CSs by Tseng and Liu in [] eah CS onsisting of two or more onstituent sequenes, alled omplementary sequenes, with zero aperiodi autoorrelation funtion AACF property. A PMEPR redution method was introdued by Davis and Jedwab in [3] to onstrut standard h -ary h is a positive integer Golay sequenes of length m m is a positive integer by using seond-order Palash Sarkar is with Department of Mathematis and Sudhan Majhi is with the Department of Eletrial Engineering, Indian Institute of Tehnology Patna, India, palash.pma15; smajhi@iitp.a.in. Zilong Liu is with Institute for Communiation Systems, 5G Innovation Centre, University of Surrey, UK, zilong.liu@surrey.a.uk. generalized Boolean funtion GBF, omprising seond-order osets of generalized first-order Reed-Muller RM odes RM h1, m. By applying the onstruted Golay sequenes to enode OFDM signals with a PMEPR of at most, Davis and Jedwab obtained m! hm+1 odewords, alled Golay-Davis- Jedwab GDJ ode in this paper, for the phase shift keying PSK modulated OFDM signals with good error-orreting apabilities, effiient enoding and deoding. Subsequently, Paterson employed omplementary sequenes to enlarge the ode rate by relaxing the PMEPR of OFDM signal in [4]. Speifially, Paterson showed that eah oset of RM q 1, m inside RM q, m q is an even number no less than an be partitioned into CSs of size k+1 k is a non-negative integer depending only on GQ, a graph naturally assoiated with the quadrati form Q in m variables whih defines the oset and provided an upper bound on the PMEPR of arbitrary seond-order osets of RM q 1, m. The onstrution given in [4, Th. 1] * was unable to provide a tight PMEPR bound for all the ases. By giving an improved version of [4, Th. 1] in [4, Th. 4], Paterson raised question: What is the strongest possible generalization of [4, Th. 1]?. In [4, Th. 4], it was shown that after deleting k verties in GQ, if the resulting graph ontains a path and one isolated vertex then Q + RM q 1, m an be partitioned into CSs of size k+1 instead of k+, i.e., there is no need to delete the isolated vertex. Later, a generalization of [4, Th. 1] was made by Shmidt in [5] to establish a onstrution of omplementary sequenes that are ontained in higher-order generalized RM odes. Shmidt s onstrution states that a GBF also an generate a CS of a given size when the graphs of every restrited Boolean funtions of the GBF are paths. The main drawbak of Shmidt s onstrution is that CS annot be generated orresponding to a GBF without deleting isolated verties from any of the graphs of restrited Boolean funtions if isolated verties are present. As a results CS set size inreases and so the PMEPR. Moreover, a reasonable number of sequenes were exluded from the Shmidt s oding sheme. Hene, an improved version of [5, Th. 5] or, a more generalized version * [4, Th. 1] is given in Lemma 5. [4, Th. 4] is given in Lemma 6. The restrited Boolean funtions of a GBF are obtained by fixing some variables of the GBF. If we restrit a GBF of m variables over k k < m fixed variables, the restrition an be done in k ways. Corresponding to k restrited Boolean funtions, we get k graphs if the restrited Boolean funtions are of order. [5, Th. 5] is given in Lemma 7.

2 of [4, Th. 1] is expeted to extend the range of oding options with good PMEPR bound for pratial appliations of OFDM. More onstrutions of CS with low PMEPR an be found in [6] [9]. In, [6], a framework is given to identify known Golay sequenes and pairs of length m m > 4 over Z h diretly in expliit algebrai normal form. In addition to that quaternary Golay sequenes and pairs of length m are also produed by Fiedler et al.. In [7], Paterson et al. established the first lower bound on the PAPR of a onstant energy ode as a funtion of its rate, distane, and length. They also established a lower bound on the ahievable rate of a ode given its distane and peak-to-average power ratio. The results in [6] and [7] provide better upper bound of PMEPR than the results in [4] and [5]. But from the appliation point of view, the results obtained in [4] and [5] seem to be more effiient. Nowadays, besides polyphase omplementary sequenes, the design of quadrature amplitude modulation QAM omplementary sequenes with low PMEPR is also an interesting researh topi. In [8], QAM Golay seuenes were introdued by Li, based on quadrature phase shift keying PSK GDJ-ode. Later, Liu et al. onstruted QAM Golay sequenes by using properly seleted Gaussian integer pairs [9]. Reently, some onstrutions on omplementary or nearomplementary sequenes have been reported in [10] [14]. These sequenes may also be appliable in OFDM system to deal with PMEPR problem, in addition to their appliations in senarios suh as asynhronous ommuniations and hannel estimation. The proposed onstrution generates CS with low PMEPR and high ode rate for OFDM systems. It uses higher order GBF and graphial properties of restrited Boolean funtions of GBF. The onstrution used both Path and isolated verties related graphial properties in higher order GBF to tight the PMEPR upper bound and inrease the ode rate. The PMEPR is related to the number of restrited verties. In the proposed onstrution, we restrit less number of verties to obtain tighter PMPER. For an example, we obtain CS with maximum PMEPR of k+1 and k+ M in the presene of isolated verties as the PMEPR upper bound obtained from Shmidt s onstrution for the same sequenes is at least k+. The proposed ode also provides better ode rate. It is beause of that Shmidt s onstrution whih has more restrition to have CS and thus exlude more number of sequenes. By moving to higher order RM ode, we give a partial answer to question raised by Paterson. We give a onstrution of CSs whih an be used to extend the range of oding options for pratial appliations of OFDM. Thus, we have generalized Patersons onstrution by moving to higherorder GBF and Shmidt s onstrution by introduing onept of isolated verties. It has also been shown that the GDJ is a speial ase of the proposed onstrution. The remainder of the paper is organized as follows. In Setion II, some useful notations and definitions are given. In Setion III, a generalized onstrution of CS is presented. Setion IV ontains some results whih are obtained from our proposed onstrution. Then we ompare our proposed onstrution in Setion V. Finally, onluding remarks are drawn in Setion VI. II. PRELIMINARY A. Definitions of Correlations and Sequenes Let a a 0, a 1,, a L 1 and b b 0, b 1,, b L 1 be two omplex-valued sequenes of equal length L and let τ be an integer. Define L 1 l a i+τ b i, 0 τ < L, Ca, bτ L+l 1 a i b i τ, L < τ < 0, 1 and Aaτ Ca, aτ. The above mentioned funtions are alled the aperiodi ross-orrelation funtion between a and b and the AACF of a, respetively. Definition 1: A set of n sequenes a 0, a 1,, a n 1, eah of equal length L, is said to be a CS if Aa 0 τ+aa 1 τ+ +Aa n 1 nl, τ 0, τ A CS of size two is alled GCP. Sequenes lying in a CS of size n 4 is alled omplementary sequenes in this paper. B. PMEPR of OFDM signal For q-psk modulation, the OFDM signal for the word a a 0, a 1,, a L 1 a i Z q an be modeled as the real part of L 1 Sat ωq aj e πif0+jfst, j0 ω q expπ 1/q is a omplex qth root of unity and f 0 + jf s 0 j < L is jth arrier frequeny of the OFDM signal. We define the instantaneous envelope power of the OFDM signal as [4] P at Sat. From the above expression, it is easy to derive that P at L 1 τ1 L Aaτ expπ 1τf s t Aa0 + Re L 1 Aaτ expπ } 1lf s t, τ1 Rex} denotes the real part of a omplex number x. We define the PMEPR of the signal Sat as PMEPRa 1 n sup P at. 0 f st<1 The largest value that the PMEPR of an n-subarrier OFDM signal is n. 3

3 3 C. Generalized Boolean Funtions Let f be a funtion of m variables x 0, x 1,, x m 1 over Z q. A monomial of degree r is defined as the produt of any r distint variables among x 0, x 1 x m 1. There are m distint monomials over m variables listed below: 1, x 0, x 1,, x m 1, x 0 x 1, x 0 x,, x m x m 1,, x 0 x 1 x m 1. A funtion f is said to be a GBF of order r if it an be uniquely expressed as a linear ombination of monomials of degree atmost r, the oeffiient of eah monomial is drawn from Z q. A GBF of order r an be expressed as Q r m 1 f Q + g i x i + g, 4 p 0 α 0<α 1< <α p 1<m and g i, g, a α0,α 1,,α p 1 Z q. D. Quadrati Forms and Graphs a α0,α 1,,α p 1 x α0 x α1 x αp 1, 5 Let f be a rth order GBF of m variables over Z q. Assume x x j0, x j1,, x jk 1 and 0, 1,, k 1. Then f x is obtained by substituting x jα α α 0, 1,, k 1 in f. If f x is a quadrati GBF, then Gf x denotes a graph with V x 0, x 1,, x m 1 }\x j0, x j1,, x jk 1 } as the set of verties. The Gf x is obtained by joining the verties x α1 and x α by an edge if there is a term q α1α x α1 x α 0 α 1 < α < m, x α1, x α V in the GBF f x with q α1α 0 q α1α Z q. For k 0, Gf x is nothing but Gf. E. Sequene Corresponding to a Boolean Funtion Corresponding to a GBF f, we define a omplex-valued vetor or, sequene ψf, as follows. ψf ω f0 q, ω f1 q,, ω f m 1 q, 6 f i fi 0, i 1,, i m 1 and i 0, i 1,, i m 1 is the binary vetor representation of integer i i m 1 α0 i α α. Again, we define ψf x as a omplex-valued sequene fi0,i1,,im 1 with ωq as ith omponent if i jα α for eah 0 α < k and equal to zero otherwise. Definition Effetive-Degree of a GBF [5]: The effetivedegree of a GBF f : 0, 1} m Z h, is defined as follows. max [deg f mod i+1 i]. 7 0 i<h Let Fr, m, h be the set of all GBFs f : 0, 1} m Z h. Then Fr, m, h, denotes the number of GBFs in Fr, m, h and given by [5] log Fr, m, h r m h i m h i. 8 r + i h 1 + Definition 3 Effetive-Degree RM Code [5]: For 0 r m, the effetive-degree RM ode is denoted by ERMr, m, h and defined as ERMr, m, h ψf : f Fr, m, h}. 9 Definition 4 Lee Weight and Squared Eulidean Weight: Let a a 0, a 1,, a L 1 be a Z h-valued sequene. The Lee weight of a is denoted by wt L a and defined as follows. L 1 wt L a mina i, h a i }. 10 The squared Eulidean weight of a when the entries of a are mapped onto a h -ary PSK onstellation is denoted by wt E a and given by L 1 wt Ea ωq ai Let d L a, b wt L a b and d E a, b wt E a b be the Lee and squared Eulidean distane between a, b Z L, h respetively. The symbols d L C and d E C will be used to denote minimum distanes taken over all distint sequenes of a ode C Z L. h Next, we present some lemmas whih will be used in our proposed onstrution. Lemma 1 [4]: Let f, g be GBFs of m variables. Consider 0 j 0 < j 1 < < j k 1 < m, whih is a list of k indies and 0, 1,, k 1 and d d 0, d 1,, d k 1 are two binary vetors. Write x x j0, x j1,, x jk 1 and onsider 0 i 0 < i 1 < < i l 1 < m, whih is a set of indies whih has no intersetion with j 0, j 1,, j k 1 }. Let y x i0, x i1,, x il 1, then C ψf x, ψg xd τ 1, C ψf xy1, ψg xyd τ. 1 Lemma [15]: Suppose that there are two GBFs f and f of m-variables x 0, x 1,, x m 1 over Z q, suh that for some k m 3 restriting variables x x j0, x j1,, x jk 1, x and f x is given by x P + L + g l x l + g, f x P + L + g l x l + q x a + g, 13 L m k α0 g α x α, both Gf x and Gf x onsist of a path over m k 1 verties, given by GP, x a be the either end vertex, x l be an isolated vertex, and g l, g Z q. Then for fixed and d 1 d d 1, d 0, 1}, Cf xxl d 1, f xxl d τ + Cf xxl d 1, f xxl d τ ω d1 dg l q m k, τ d d 1 l 14 Lemma 3 [16]: Let d, 1, 0, 1} k. If 1, 1 d d

4 4 Lemma 4 [3]: Suppose f : 0, 1} m Z q be a quadrati GBF of m variables. Suppose further that Gf is a path with h 1 being the weight of every edge. Then for any hoie of, Z h, the pair f +, f + h 1 x a + forms a GCP. Lemma 5 [4, Th. 1]: Suppose f : 0, 1} m Z q be a quadrati GBF of m variables. Suppose further that Gf ontains a set of k distint verties labeled j 0, j 1,, j k 1 with the property that deleting those k verties and orresponding their edges results in a path. Then for any hoie of g i, g Z q f + q k 1 } d α x jα + dx a : d α, d 0, 1} 15 α0 is a CS of size k+1. Lemma 6 [4, Th. 4]: Suppose f : 0, 1} m Z q be a quadrati GBF of m variables. In addition, suppose that Gf ontains a set of k distint verties labeled j 0, j 1,, j k 1 with the property that deleting those k verties and all their edges results in a path on m k 1 verties and an isolated vertex. Suppose further that all edges in the original graph between the isolated vertex and the k deleted verties are weighted by q/. Let x a be the either end vertex in this path. Then for any hoie of g i, g Z q f + q k 1 } d α x jα + dx a : d α, d 0, 1} 16 α0 is a CS of size k+1. Lemma 7 [5, Th. 5]: Let f : 0, 1} m Z q be a GBF of m variables. Suppose further that for eah 0, 1} k, the Gf x is a path in m k verties. Suppose further that q/ is the weight of eah edge and x be the either end vertex of the path Gf x. Then for any hoie of g i, g Z q f + q k 1 } d α x jα + dx : d α, d 0, 1} 17 α0 is a CS of size k+1 and hene ψf lies in a CS of size k+1. Lemma 8 [5, Th. 9]: d L ERMr, m, h m r, d EERMr, m, h m r+ sin π h. III. PROPOSED CONSTRUCTIONS 18 In this setion, we present a generalized onstrution of CS. For ease of presentation, whenever the ontext is lear, we use Cf, gτ to denote Cψf, ψgτ for any two GBFs f and g. Similar hanges are applied to restrited Boolean funtions as well. Theorem 1: Let f be a GBF of m variables over Z q with the property that there exist M number of suh for whih Gf x is a path over m k verties and there exist N i number of suh for whih Gf x onsists of a path over m k 1 verties and one isolated vertex x li suh that M, N i 0, M + p N i k. Suppose further that all the relevant edges in Gf x for all have idential weight of q/. Then for any hie of g i, g Z q, ψf lies in a set S of size k+1 with the following aperiodi auto-orrelation property. p m+1 N i + m+1 M, τ 0, ASτ ω g l i q m ω L S Ni ω g l i q m l i q, τ li,,,, p, S Ni q, τ li,,,, p, 19, g li Z q, i 1,,, p is the oeffiient of x li in f, S Ni ontains all those for whih Gf x onsists of a path over m k 1 verties and one isolated vertex labeled l i l i 0, 1,, m 1} \ j 0, j 1,, j k 1 }, and l 1, l,, l p are all distint, and L li k ϱ li r1 0 i 1<i < <i r<k i 1,i,,i r i1 i ir ϱ li i 1,i,,i r Z q. Proof: Let x x j0, x j1,, x jk 1 and d k 1 d 0, d 1,, d k 1. Then d x d α x jα. Define α0 S f + q } d x + dx : d 0, 1} k, d 0, 1}, 0 x is an end vertex of the path whih is ontained in Gf x. Now for any τ 0, the sum of AACF of sequenes from the set S an be written as A f + q d x + dx τ L 1 + L, 1 and dd L 1 dd L dd 1 C A f + q x d x + dx τ, f + q d x + dx 1 x1, f + q d x + dx x τ. 3 We first fous on the term L 1, whih an be written as p L 1 T + T i, 4 and T dd T i dd S M A S Ni A f + q x d x + dx τ, 5 f + q d x + dx x τ, 6

5 5 S M is the set of all those for whih Gf x is a path over m k verties. To find L 1, we first start with T. Sine, Gf x is a path over m k verties for all S M, we have [4] d A f + q x d x+dx τ Therefore, T dd S M A m k+1, τ 0, 7 f + q x d x + dx τ m+1 M, τ 0, To find L 1, it remains to find T i i 1,,, p T i A f + q x d x + dx τ. dd S Ni We an express eah of T i, as T i dd dd + dd S Ni A S Ni β 0,1} f + q x d x + dx τ S Ni β 0,1} 8 A f + q d x + dx τ xxli β C f + q d x + dx xxli, β f + q d x + dx xxli 1 β τ. 9, L li x k ϱ li r1 0 i 1<i < <i r<k i 1,i,,i r x ji1 x ji x jir. To simplify the ross-orrelation term in 9, we have the following equality by Lemma and 3. d C f + q d x + dx xxli β, f + q d x + dx τ xxli 1 β ω β 1g l i q ω β 1Ll i q m k, τ β 1 li, β 0, 1}. Therefore, the ross-orrelation term of 9 is simplified as dd S Ni β 0,1} ω g l i q m ω g l i q m C f + q d x + dx xxli, β ω L l i S Ni f + q d x + dx τ xxli 1 β q, τ li, S Ni q, τ li, 33 Sine, for all S Ni, Gf x onsists of a path over m k 1 verties and one isolated vertex labeled l i, Gf xxli β is a path over m k 1 verties. Therefore A d f + q d x + dx xxli β m k, τ 0, τ 30 Hene, the first auto-orrelation term in 9 an be expressed From 9, 31 and 33, we have m+1 N i, τ 0, ω g l i q m ω L l i q, τ li, S Ni T i ω g l i q m q, τ li, S Ni 34 as dd S Ni β 0,1} m+1 N i, τ 0, A f + q d x + dx τ xxli β 31 Sine, for all S Ni, Gf x onsists of a path and one isolated vertex x li, the only term involving x li is with the variables of the deleted verties. Thus the only term in x li in f an be expressed as follows. k ϱ li r1 0 i 1<i < <i r<k i 1,i,,i r x ji1 x ji x jir x li L li x x li, 3 From 4, 8 and 34, we have L 1 T + p T i p m+1 N i + m+1 M, τ 0, ω g l i q m ω L S Ni ω g l i q m l i q, τ li,,,, p, S Ni q, τ li,,,, p, 35

6 6 To find L, we start with C f + q d x + dx x1 1, d d C 0 τ. f + q d x + dx 1 d 1+ C f + q dx 1 f + q dx x1 1, f + q dx x1 x, τ f + q dx x τ x Therefore, from 3 and 36, we have L C dd 1 0 τ. τ d f + q d x + dx 1 x1, 1 d f + q d x + dx x τ 37 By substituting 35 and 37 in 1, we omplete the proof. We have introdued M and N i i 1,,, p in Theorem 1 p with the ondition M + N i k, M, N i 0. Therefore, M and N i s range from 0 to k. Note 1 Constrution of GBFs as Defined in Theorem 1: The GBFs f, as defined in Theorem 1, an be expressed as q S M + p m k k k 1 x πix πi+1 α0 + q p δ1 S Nδ ϱ l δ δ1 r1 0 i 1<i < <i r<k + k k 1 α0 r 0 i 1<i < <i r<k x α j α 1 x jα 1 α m k 3 x π δ ix π δ i+1 x α j α 1 x jα 1 α i1,i,,i r x ji1 x ji x xir x lδ α i1,i,,i r x ji1 x ji x xir g i x i + g, m π δ are N δ permutations of 0, 1,, m 1} \ j 0, j 1,, j k 1, l δ } δ 1,,, p, π are M permutations of 0, 1,, m 1} \ j 0, j 1,, j k 1 }, and α i1,i,,i r s are belongs to Z q. Below, we have illustrated the Theorem 1 by an example. x 1 x 3 x 1 a b x 3 Fig. 1. The G x0 0 and Gf x0 1. Example 1: Let f be a GBF of 4 variables over Z, given by fx 0, x 1, x, x 3 x 0 x 1 x 3 + x 0 x 3 x + x 0 x x 1 + x 1 x. 39 The restrited Boolean funtions x00 and f x01 are given by f x x00 1x, f g x 40 x01 1x 3 + x 3 x, respetively. Fig. 1 a andx Fig. 1 b x 4 represents Gf x01 and Gf respetively. It is lear that Gf is a x00 x01 path over the variables x 1, x, and x 3. Gf ontains x00 a path over the variables x 1, x and one isolated vertex x 3. Therefore, M 1, N 1 1, S N1 0}, ϱ 3 0 0, and L respetively. By using Theorem 1, we obtain the set S given by S f + d 0 x 0 + dx : d 0, d 0, 1}} The AACF of S is given by 64, τ 0, AS 16, τ ±. x x 41 4 Remark 1: Let f be a quadrati GBF with the property that for all 0, 1} k, Gf x is a path in m k verties. Then from Therorem 1, we have M k and m+k+1, τ 0, ASτ 43 Hene, S is a CS of size k+1 and therefore, Paterson s onstrution [4, Th. 1] turns to be a speial ase of our proposed one. Remark : From Remark 1, for k 0, S is a CS of size, i.e., S is a GCP and thus the GDJ ode in [3] is also a speial ase of Theorem 1. Remark 3: Let f be a quadrati GBF with the property that for all 0, 1} k, Gf x ontains a path in m k 1 verties and one isolated vertex x l1. We also assume that all edges in the original graph between the isolated vertex and the

7 7 k deleted verties are weighted by q/. Then, from Therorem 1, we have N 1 k, S N1 0, 1} k, L l1 q k 1 α0 α, and m+k+1, τ 0, ω g l 1 q m+k ASτ ω g l 1 q m+k S N1 ω L l 1 q, τ l1, ω L l 1 S N1 q, τ l1, m+k+1, τ 0,, 44 Therefore, ψf lies in a CS of size k+1 and the result given by Paterson in [4, Th. 4] is a speial ase of Theorem 1. Remark 4: Let f be a GBF with the property that for all 0, 1} k, Gf x is a path in m k verties. Then from Therorem 1, we have M k and m+k+1, τ 0, ASτ 45 From 45, it is lear that ψf lies in a CS of size k+1 and hene the PMEPR of ψf is atmost k+1. Therefore, the result given by Shmidt in [5, Th. 5] is a speial ase of Theorem 1. IV. CONSTRUCTION OF COMPLEMENTARY SEQUENCES WITH LOW PMEPR In this setion, we present two onstrutions of CSs whih are derived from Theorem 1 and provide tighter PMEPR upper bound than the PMEPR bound introdued in Shmidt s onstrution [5, Th. 5]. Corollary 1: Let f be a GBF with the property that for all 0, 1} k, Gf x ontains a path over m k 1 veries, all the relevant edges have idential weight q/, x is either end vertex of the path and one isoltaed vertex x l1. Also, let ϱ l1 0 x j 0 +ϱ l1 1 x j 1 + +ϱ l1 k 1 x j k 1, ϱ l1 0, ϱl1 1,, ϱl1 k 1 0, q/}, annot be all zero at the same time be the only term present in f, involving with x l1. Then for any hoie of g i, g Z q, f + q } d x + dx : d 0, 1} k, d 0, 1}, 46 is a CS of size k+1. Proof: Let S f + q } d x + dx : d 0, 1} k, d 0, 1}. 47 By Theorem 1, we have p 1, M 0, N 1 k, S N1 0, 1} k, L l1 k 1 α0 ϱl1 α α, and m+k+1, τ 0, ω g l 1 q m ω L l 1 q, τ l1, S N1 ASτ ω g l 1 q m ω L l 1 48 q, τ l1, S N1 Sine, ϱ l1 0, ϱl1 1,, ϱl1 k 1 0, q/} and at least one of ϱl1 u s u 0, 1,, k 1 takes the value q/, by using Lemma 3, we have 48, we have S N1 ω L l 1 q 0, ASτ l 1 ωq L S N1 m+k+1, τ 0, 0. Therefore from 49 From 49, we have S is a CS of size k+1 and hene at most PMEPR of eah sequenes lying in S is k+1. Remark 5: The onstrution of CSs given in Corollary 1 is based on GBFs of any order. It is observed that Corollary 1 an provide tighter upper bound of PMEPR than that given by Shmidt [5, Th. 5] for a sequene orresponding to a GBF whih satisfy the property given in Corollary 1. Below, we present an example to illustrate Corollary 1. Example : Let f be GBF of 5 variables over Z 4, given by fx 0, x 1, x, x 3, x 4 x 0 x 1 x + x 0 x 1 x 3 + x 1 x 3 +x 3 x + x 0 x 4 + x 1 + x + x 3 + x From the GBF f, we obtain the restrited Boolean funtions as follows. x00 x 1x 3 + x 3 x + x 1 + x + x 3 + x 4 + 3, x01 x 1x + x x 3 + x 1 + x + x From 51, it is observed that G x00 and Gf x01 both ontain a path over the verties x 1, x, x 3 and one isoltaed vertex x 4. Thererfore, p 1, N 1, S N1 0, 1}, ϱ 4 0, L 4 0 0, and L 4 1. By Corollary 1, S x 0 x 1 x +x 0 x 1 x 3 + x 1 x 3 + x 3 x + x 0 x 4 + x 1 +x + x 3 + x d 0 x 0 + dx 1 : d 0, d 0, 1}}, 5 is a CS of size 4. Therefore, the PMEPR of ψf is at most 4 and from Shmidt s onstrution, the PMEPR upper bound of ψf is 8. Corollary : Let f be a GBF as defined in Theorem 1 and unlike the GBF as defined in Corollary 1. Then for any hoie of g i, g Z q, } f + q p d x+d x li +dx : d 0, 1} k, d, d 0, 1}, 53 is a CS of size k+ with at most PMEPR k+ M. Proof: The set S an be expressed as S S 1 S, S 1 f + q } d x+dx : d 0, 1} k, d, 0, 1}, } S f + q p d x+ x li +dx :d 0, 1} k, d, 0, 1}. 54

8 8 By Theorem 1, we have p m+1 N i + m+1 M, τ 0, AS 1 τ and AS τ ω g l i q m ω L S Ni ω g l i q m l i q, τ li,,,, p, S Ni q, τ li,,,, p, p m+1 N i + m+1 M, τ 0, ω q +g l i q m ω L l i S Ni ω q +g l i q m p m+1 N i + m+1 M, τ 0, ω g l i q m ω g l i q m 55 q, τ li,,,, p, S Ni ω L l i S Ni q, τ li,,,, p, q, τ li,,,, p, S Ni q, τ li,,,, p, From 55 and 56, we have AS 1 τ + AS τ m+k+, τ 0, Therefore, S is a CS of size k+. From 55 and 56, it is observed that the PMEPR of the sequenes that lies in S 1 and S, is at most k+1 + p N i or, k+ M. Sine, the set S is union of two sets S 1 and S, the PMEPR of S is at most k+ M. Remark 6: It is observed that Corollary an provide more tight upper bound of PMEPR than that of [5, Th. 5] for a sequene orresponding to a GBF whih satisfy the properties introdued in Corollary. For different values of M and p, we ompare the PMEPR upper bounds obtained from Corollary 1 and Corollary, with the PMEPR upper bound obtained from [5] in TABLE I. Example 3: Let f be a GBF of 5 variables x 0, x 1, x, x 3, x 4 over Z 4, given by fx 0, x 1, x, x 3 x 0 x 1 x 3 + x 0 x 3 x 4 + x 1 x 3 + x 3 x. 58 The restrited Boolean funtions x00 and f x01 are x00 x 1x 3 + x 3 x, x01 x 4x 3 + x 3 x, 59 respetively. From 59, it is lear that G x00 ontains a path over the verties x 1, x, x 3 and x 4 as isolated vertex, TABLE I PMEPR UPPER BOUND FOR DIFFERENT VALUES OF M AND p. k Proess M p PMEPR upper bound Proposed [5] Corollary Corollary Corollary Corollary and Gf ontains a path over the verties x x01, x 3, x 4 and x 1 as isolated vertex. Hene p, M 0, N 1 1, N 1, S N1 0}, S N 1}, ϱ 4 0 0, ϱ 1 0 0, L 4 0 0, and L By using Corollary, the set f + q } d 0x 0 + d x 1 + x 4 + dx 1 : d 0, d, d 0, 1} 60 is a CS of size 8. Hene, by using Corollary, the PMEPR upper bound for ψf is 8 as Shmidt s onstrution provides a PMEPR upper bound of 16. Example 4: Let f be a GBF of 6 variables over Z 4, given by fx 0, x 1,, x 5 x 0 x x 3 + x 0 x 3 x 4 + x 0 x 4 x 5 + x 0 x x 4 + x 0 x 1 x 4 + x 0 x 1 x 3 + x 0 x 3 x 5 + x x 4 + x 4 x 1 + x 1 x 3 + x 3 x The restrited Boolean funtions x00 and f x01 are given by x00 x x 4 + x 4 x 1 + x 1 x 3 + x 3 x 5, x01 x x 3 + x 3 x 4 + x 4 x 5, 6 respetively. It is lear that Gf x00 is a path and Gf ontains a path and the isolated vertex x x01 1. Therefore, p 1, M 1, N 1 1, S N1 1}, and L By using Corollary, the set S f + d 0 x 0 + d x 1 + dx : d 0, d, d 0, 1}}, 63 is a omplementary set of size 8 and the PMEPR upper bound of the sequenes lying in S is 6. The Gf x00 and Gf x01 are represented by Fig. a and Fig. b respetively. Sine, Gf ontains the isolated vertex x x01 1, Shmidt s onstrution suggests to delete the isolated vertex x 1. After deleting the isolated vertex or restriting x 1, we obtain the following restrited Boolean funtions f, x0,x 10,0 f, f x0,x 10,1 x0,x 11,0 and f. The x0,x 11,1 Gf, Gf x0,x 10,0 x0,x 10,1, are represented by

9 9 a x x 4 x 1 x 3 x 5 x x 3 b x 4 x 5 x 1 x x 4 x 3 x 5 d x x 3 x 4 x 5 omplementary sequenes with PMEPR at most 8 of length m. By using Corollary, We get at least [ m! + m!m!m 3 4 ] q m q 1, omplementary sequenes with PMEPR at most 6 and at least [ ] m! mm q m 3 q 1, omplementary sequenes with PMEPR at most 8. Now we define three oodbooks S 1, S, S 3 S 1, S, and S 3 ontain odewords of length m over Z q with PMEPR at most 4, 6, and 8 respetively. The ode rate of a ode- e x x 4 g x x 4 Fig.. The graphs of the restrited Boolean funtions obtained from f. Fig. and G x0,x 11,0, Gf x0,x 11,1 are represented by Fig. d. Again, deletion proess need to be performed by following Smidt s onstrution. But, after performing another deletion of verties, the resulting graphs of restrited Boolean funtions will be represented by Fig. e and Fig. g. The deletion proess need to be ontinued until the graphs of every restrited Boolean funtions are path or ontain single vertex. Therefore, from Shmidts onstrution, the PMEPR upper bound of ψf is 64 as from Corollary, the PMEPR upper bound of ψf is 6. V. CODE RATE COMPARISON WITH EXISTING WORK In this setion, we ompare our proposed onstrution with the onstrutions given in [4] and [5] in terms of ode rate and PMEPR. A. Comparison With [4] In this subsetion, we give a omparison of our proposed onstrution with [4] to show that the proposed onstrution an generate more sequenes, i.e., higher ode rate with tighter PMEPR upper bound. It is observed that by using Corollary 1, we get at least m! [ ] m! 1 q m 3 q 1, omplementary sequenes with PMEPR at most 4 and [ ] 3m! m 3! 1 q 3m 8 q 1, 4 TABLE II PMEPRS AND ENUMERATIONS FOR CODEBOOKS S 1, S, S 3 Codebook PMEPR upper bound S 1 4 S 6 S 3 8 Size of Codebook ] q m 3 q 1 [ m! 1 m! m! + m!m!m 3 [ ] q 4 m q 1 [ ] 3m! m 3! 1 q 4 3m 8 q 1 [ ] m! +mm q m 3 q 1 keying OFDM is defined as RC : log C L, C and L denote the set size of odebook C and the number of subarriers respetively. In TABLE III and TABLE V, oderate omparisons with [4] is given. TABLE IV ontains oderates for binary and quaternary ases with PMEPR at most 6. TABLE III CODE-RATE COMPARISON WITH CODEBOOK IN [4] WITH PMEPR AT MOST 4 OVER Z q L m Z q q q 4 Proposed [4] Proposed [4] m m m m m m TABLE IV CODE-RATE FOR OFDM CODES WITH PMEPR AT MOST 6 OVER Z q L m Z q q q 4 m m m m m m m

10 10 TABLE V CODE-RATE COMPARISON WITH CODEBOOK IN [4] WITH PMEPR AT MOST 8 OVER Z L m Z q q Proposed [4] m m m m B. Comparison With [5] In this subsetion, we present a omparison between our proposed onstrution with [5] to show that the proposed onstrution an provide higher ode rate and PMEPR upper bound. For 0 k < m, 0 r k + 1, and h 1, a linear ode Ak, r, m, h [5] is defined to be the set of odewords orresponding to the set of polynomials m k 1 x α g i x m k,, x m 1 + gx m k,, x m 1 : g 0,, g m k 1 Fr 1, k, h, g Fr, k, h}. 64 The number of odewords in Ak, r, m, h is equal to s k, s k m k log Fr 1, k, h + log Fr, k, h. Now, Rk, m, h [5] is defined to be the set of odewords assoiated with the following polynomials over Z h h 1 0,1} k m k k 1 j0 x πix πi+1 x j m k+j 1 x m k+j 1 j, 65 π are k permutations of 0, 1,, m k 1}. For m k > 1 and r > h, the set Rk, m, h ontains [m k!/] minr+h 3,k} odewords orresponding to a GBF of effetive-degree at most r. The sequenes in the osets of Ak, r, m, h with oset representatives in Rk, m, h have PMEPR at most k+1 and the ode has minimum Lee and squared Eulidean distane equal to m r and m r+ sin π respetively. h 1 Code Constrution by Using Corollary 1: For 0 k < m 1, 0 r k + 1, α l 1 and h 1, we drfine a linear ode A 1 k, r, m, h orresponding to the set of polynomials m k 1 x α g i x m k,, x m 1 + gx m k,, x m 1 : g 0,, g m k 1 Fr 1, k, h, g Fr, k, h}. 66 A 1 k, r, m, h ontains s 1,k odewords, s 1,k m k 1 log Fr 1, k, h + log Fr, k, h. Sine, A 1 k, r, m, h Ak, r, m, h, the minimum distanes of A 1 k, r, m, h an be lower bounded by m r and m r+ sin π h. Now, we assume R 1 k, m, h be the set of odewords assoiated with the following polynomials h 1 k 1 j0 0,1} k m k 3 x πix πi+1 x j m k+j 1 x m k+j 1 j + h 1 x l1 e 0 x m e k 1 x m k, 67 π are k permutations of 0, 1,, m k 1} \ l 1 and e 0,, e k 1 0, 1}, but all an not be zero at the same time. For m k > and r > h, it an be shown that the set R 1 k, m, h ontains k 1[m k 1!/] minr+h 3,k odewords orresponding to a GBF of effetive degree at most r. Note : Assume m k >. Let r k + when h 1, 1 r k + 1 when h > 1 and r minr, k + 1}. By using Corollary 1, it an be shown that any oset of A 1 k, r, m, h with oset representatives in R 1 k, m, h have PMEPR at most k+1. Now take the union of k 1[m k 1!/] minr+h 3,k distint osets of A 1 k, r, m, h, eah ontaining a word in R 1 k, m, h with effetive degree at most r. The PMEPR of the orresponding polyphase odewords in this ode is at most k+1. Sine the ode is a subode of ERMr, m, h, its minimum Lee and squared Eulidean distanes are lower bounded by m r and m r+ sin π respetively. h Code Constrution by Using Corollary : In this setion, we onsider the ase p, M 0 of Corollary. Consider R k, m, h be the set of odewords assoiated with the following polynomials h 1 p α1 α S Nα k 1 j0 m k 3 x πα ix πα i+1 x α j m k+j 1 x m k+j 1 α j, 68 α α 0,, α k 1, π α are N α permutations of 0, 1,, m k 1} \ l α and p α1 N α k. For m k > and r > h, it an be shown that the set R k, m, h ontains [m k 1!/] minr+h 3,N 1 [m k 1!/] minr+h 3,N [m k 1!/] minr+h 3,N p odewords orresponding to a GBF of effetive degree at most r. Note 3: Assume m k >. Let r k+ when h 1, 1 r k + 1 when h > 1 and r minr, k + 1}. By using Corollary, it an be shown that any oset of Ak, r, m, h with oset representatives in R k, m, h have at most PMEPR

11 11 k+. It is also observed that the minimum Lee and squared Eulidean distanes of the ode a + Ak, r, m, h a R k,m,h are lower bounded by m r and m r+ sin π respetively. h 3 Code Constrution With Maximum PMEPR 4 and 8: In this part, we onstrut odes with maximum PMEPR 4 and 8 by using the above disussed odes. Corollary 3 Code With Maximum PMEPR 4: Assume m > 3. Let r 3 when h 1, 1 r when h > 1 and r minr, }. Now, onsider C a 1 +A1, r, m, h a 1 R1,m,h a +A 1 1, r, m, h. a R 11,m,h 69 The ode C ontains odewords or sequenes with at most PMEPR 4. Hene, the maximum PMEPR of C is 4. We denote the number of odewords or sequenes in the ode by C, s1 [m 1!/] minr+h 3,1} C + s1,1 [m!/] minr+h 3,1. 70 Sine C is a subode of ERMr, m, h, the minimum Lee and squared Eulidean distanes of the ode are lower bounded by m r and m r+ sin π h respetively. TABLE VI CODE-RATE COMPARISON WITH CODEBOOK IN [5] WITH MAXIMUM PMEPR 4 OVER Z h For the ase r 3 when h 1, 1 r when h > 1 and r minr, 3}, we onsider the ode C 1, defined by C 1 b 1 +A, r, m, h b +A 1, r, m, h b 1 R,m,h b R 1,m,h b 3 +A1, r, m, h, b 3 R 1,m,h 71 C 1 s [m!/] minr+h 3,} + 3 s1, [m 3!/] minr+h 3, + s1 [m!/] minr+h 3,1} 7 Sine C 1 is a subode of ERMr, m, h, the minimum Lee and squared Eulidean distanes of the ode are lower bounded by m r and m r+ sin π respetively. h For r 4 when h 1 and r 3 when h > 1, we onsider the ode C, defined by C b 1 +A, r, m, h b +A 1, r, m, h, b 1 R,m,h b R 1,m,h 73 C s [m!/] minr+h 3,} + 3 s1, [m 3!/] minr+h 3,. 74 Sine C is a subode of ERMr, m, h, the minimum Lee and squared Eulidean distanes of the ode are lower bounded by m r and m r+ sin π h respetively. m h r Proposed [5] d L d E From 70, it is lear that the set size of the sequenes with maximum PMEPR 4 obtained from our proposed onstrution is larger than the set size given in [5]. In TABLE VI, we have ompared the ode rate of sequenes with maximum PMEPR 4 obtained from our proposed onstrution with that of the onstrution given in [5]. Corollary 4 Code With Maximum PMEPR 8: Suppose m > 4. Let r 4 when h 1, 1 r 3 when h > 1 and r minr, 3}. TABLE VII CODE-RATE COMPARISON WITH CODEBOOK IN [5] WITH MAXIMUM PMEPR 8 OVER Z h m h r Proposed [5] d L d E From 7 and 74, it is lear that our proposed onstrution an provide more number of sequenes than the onstrution given in [5]. In TABLE VII, we have ompared the ode rate of sequenes with maximum PMEPR 8 obtained from our proposed onstrution with that of the onstrution given in [5].

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