New Sequences Design from Weil Representation with Low Two-Dimensional Correlation in Both Time and Phase Shifts

New Sequences Design from W eil Represen tation with Lo w Tw o-Dimensional Co rrelation in Both Time and Phase Shifts Zilong W ang ∗ 1 and Guang Gong 2 1 Sc ho ol of T elecomm unication Engineering, Xidian Univ ersit y , Xi’an, 710071, P .R.CHINA 2 Departmen t of Electrical and Computer Engineering, Unive r sit y of W aterlo o W aterlo o, On t ario N2L 3G1, CANAD A Email: wzlmath@gmail.com ggong@calliop e.u w aterlo o.ca Septem b er 1 4, 2018 Abstract A new elementar y expression of the construction fi rst prop osed by Gurevic h, Hadani, and So chen is given, which av oids the ex plicit u se of the W eil represen tation. The sequ ences in this signal set are given by b oth multiplicativ e character and additive character of fin ite field F p . Suc h a signal set consists of p 2 ( p − 2) time-shift d istinct sequences, the magnitud e of the tw o-d imensional autocorrelation function (i.e., the ambiguit y function) in b oth t ime and phase of each sequence is upp er b ounded by 2 √ p at any shift not equ al to ( 0 , 0). F urthermore, the magnitud e of their F ourier transform sp ectrum is les s than or equal to 2. F or a subset consisting of p ( p − 2) phase-shift distinct sequences in this signal set, the magnitude of th e am biguity function of any p air is upp er b ounded by 4 √ p . A pro of is given through finding a new expression of the sequences in the fi nite harmonic oscillator system. A n open problem for directly establishing these assertions without inv olving th e W ei l represen tation is addressed. Index T erms. Seq uence, auto correlation, cross correlation, ambiguit y fun ction, F ourier trans- form, and W eil representatio n . 1 In tro d uction Sequence design for goo d correlation finds man y impor tant applica tions in v arious tra ns mission s ystems in communication networks, and rada r sy stems. A. L ow Corr elation In co de division multiple access (CDMA) applications of spr ead sp ectral communication, multiple users share a common channel. Each user is a ssigned a different spreading sequence (or spread co de) for transmission. At an intended r eceiver, desprea ding (reco vering the original data) is accomplished b y 0 ∗ The w or k w as conducte d when Zilong W an g was a vis i ting Ph. D studen t at the Departmen t of ECE in Univ ers it y of W aterloo fr om Se ptember 2008 to August 2009. 1 the cor r elation of the re c eived spr ead signal with a synchronized replica of the spr eading sequence used to spread the information where the sprea ding sequences used by o ther use r s a re treated as interference, which is referr ed to as multiple ac c ess interfer enc e . This type o f interference, which is different from int e r ference tha t arises in radio -frequency (RF) commun ication channels, can b e reduced by prop er design of a sprea ding signal set. The pe r formance of a signal (or sequence) set used in a CDMA sys tem is measured by the parameters L , the length or p erio d of a sequence in the set, r , the num be r of time-shift distinct sequences, and ρ , the maxim um magnitude of the out-o f- pha se a uto c o rrelatio n of any sequence a nd cro ss correla tion of a ny pair of the se q uences in the set. This is r eferred to a s an ( L, r , ρ ) signal set . The trade-off of these three pa rameters is b ounded by the W elch b o und, establishe d in 1974 by W elc h [43]. The res earch for cons tr ucting goo d signal sets has flourished in the litera ture. The reader is referr ed to [6, 1, 39, 32, 5, 9] for polyphase sequences with large alphabet sizes, [26, 20] for Z 4 sequences, [12, 30] for in terleaved sequences, and [36, 2 4, 13] in ge neral, for ex a mple. B. Minimize d F ourier Sp e ctrum The orthog onal frequency division multiplexing (OFDM) utilizes the concept o f pars ing the input data into N symbol strea ms, and each o f which in turn is used to mo dulate para lle l, s ynchronous sub c arrier s. With an OFDM system having N sub channels, the symbol rate on ea ch sub carrier is reduced by a factor o f N relative to the s ymbol r a te on a single ca rrier system that employs the en tire bandwidth and transmits data at the same ra te as OFDM. An OFDM signal can b e implemented b y computing an inv erse F ourier tra nsform a nd F our ier tra nsform at the transmitter side and receiver side, resp ectively . A ma jo r pro blem with the multicarrier mo dulatio n in general and OFDM system in particular is the high peak-to - av era ge p ow er ratio (P APR) that is inherent in th e transmitted signal. A bo und on P APR through the magnitude of the discrete F ourier transform (DFT) sp ectrum of emplo yed signals is shown in [28, 31]. (See [4 0] for details of F o urier transform.) One way to achiev e lo w P APR is to emplo y Gola y complement a ry sequences, a s first sho wn b y Da v is and Jedwab in [8]. A tremendous amount of work has b een done along this line since then. C. L ow V alue d A mbiguity F unctions In rada r o r so nar applications, a sequence should be designed in such a way that the ambiguity function (the t wo-dimensional a uto correla tion function in b oth time and freq ue nc y o r equiv alently phase, will b e formally defined later), having the v alue of the leng th of the sequence at (0 , 0), and small v alues at any shift not ( 0 , 0). The ambiguity function is requir ed for determining the r ange (prop ortiona l to the time-shift) and Doppler (the v elo city to or from the o bserver, prop o r tional to the frequency shift) of a targ et. Sequences with low ambiguit y function can be achiev ed by Costas arr ays, which yield the so-called ide al o r thumb-tack ambiguit y function (which only takes the v alues 0 o r 1 at any shift not (0, 0)) [7, 14]. 2 It is int eresting to see whether ther e exists a s ignal set which simultaneously satisfies the require men ts that a rise from the ab ov e thr ee transmission scenarios, i.e., having low correlatio n, low P APR, low ambiguit y function, but with la rge size and mo dera te implementation cost. It is anticipated that employing those se q uences will improv e the per formance of communication systems with multi-carrier CDMA transmissio n [33], radar netw orks , and tra nsmission systems in future co gnitive radio netw or ks [34]. Gurevich, Hadani, and So chen [16, 18] pr op osed a signal s et c alled finite oscillator system S which gives a p o sitive answ er to the ab ov e question except for the implementation cost. Their construction makes use o f the g roup-theor e tic W eil represe ntation a nd the s equences ar e describ ed in a lgorithmic terms by the end of [1 6, 17]. The main contribution of this pa per is to pro po se a simple elementary expression for those sequences, which av oids the need to explicitly emplo y relatively cos tly g roup- theoretic computations. It is interesting to observe that to date, almost all sequences with low c o rrelatio n in the literature are related to the use of additive o r multiplicative characters of the finite field or Galois rings together with functions. Recently , inspir e d by mutually un bias ed bases discussed by How e in [22], Ho ward, Calderbank, a nd Mo ran [21] in vestigated sequences constructed fro m the Heisenberg representation in 2006, then Gurevich, Hadani, and So chen [16, 18] introduced sequences from the W eil representation in 200 8, which a re referr ed to as a finite os cillator system S . In fact, sequences from the Heisenberg repre s ent ation are related to extended a F r ank-Za do ff-Chu (FZC) se quence [10, 6, 1 1], b eing complex v alued sequences with perio d p . After normaliz a tion by the energy , the v alues of their ambiguit y functions (pr e c isely defined in the next section) is b ounded by 1 √ p except for some sp ecial c a se. While the sequences fro m the W eil r epresentation, whic h will be int r o duced la ter, hav e the desired prop er ties in the a b ove men tioned three application s cenarios, but hav e a complicated form. Gurevich, Hadani, and Sochen in vestigated how to implemen t their sequences in ter ms of an algorithm. The g oal of this pap er is to find a s imple e lement a ry expre ssion for the finite harmonic oscillator s ystem. W e s how that there are t wo types of the sequences in the finite ha rmonic oscillator system o f the splitting case (we will formally define it later). Sequences of the first type can be given as pro duct sequences using both multiplicativ e characters and additive characters of the finite field F p , and sequences of the second type are involv ed the summations of sequences of the fir st t y pe . W e construct a new signa l set from the set consisting of the s equences of the firs t type with some extension. The rest of the pap er is o rganized as follows. In Se c tion 2, we in tr o duce some basic concepts and notations. In Se c tio n 3 , we prese nt o ur new constructions and the main results. In Section 4 , w e int r o duce W eil repres ent a tion a nd the finite oscilla to r system constructed by Gure vich, Hadani and So chen in [16, 18]. W e s how a simple element ary expression for this finite oscillator system, and pr esent 3 a pro of for the new constructions in Section 5. Compa risons of the new constructions with some kno wn constructions a re ma de in Section 6. Section 7 is for concluding remarks a nd addressing a n op en problem. 2 Basic Concepts and Definitions In this section, w e in tro duce some basic concepts and notations whic h are frequently used in this pap er. F or a given prime p , let θ and η denote the ( p − 1)th and p th primitiv e ro ots of unit y in the complex field resp ectively , i.e., θ = exp  2 π i p − 1  and η = exp  2 π i p  . W e denote F p as the finite field with p elements, and F ∗ p as the multiplicativ e gro up of F p with a generator α . Then for every element β ∈ F ∗ p , there exists i with 0 6 i 6 p − 2 , such that β = α i . In other words, i = log α β . W e set θ log α 0 = 0 throughout this paper . Every s equence with per io d p can b e denoted b y ϕ = ( ϕ (0) , ϕ (1) , · · · , ϕ ( p − 1)), and also conside r ed as a vector in the Hilb ert space H = C ( F p ) with the inner pro duct given by the standard fo rmula: < ϕ, ψ > = P i ∈ F p ϕ ( i ) ψ ( i ) wher e x is the co mplex conjugate of x . W e denote U ( H ) (App endix 7.3 ) as the gro up o f unitary op erators on H . L e t L t , M w and F be unitary op erators of the time-shift, phase-shift and DFT res p ectively , which a re defined a s follows, L t [ ϕ ]( i ) = ϕ ( i + t ) M w [ ϕ ]( i ) = η wi ϕ ( i ) and F [ ϕ ]( j ) = 1 √ p X i ∈ F p η j i ϕ ( i ) , ϕ ∈ H . (1) W e also use the no tation b ϕ for F [ ϕ ] for simplicity . If ψ = L t ϕ or ψ = M w ϕ , then we say tha t ϕ a nd ψ are time-shift e quivalent or phase-shift e quivalent . Other wise, they are t ime-shift di s tinct or phase-shift distinct . W e denote C ϕ ( t ) and C ϕ,ψ ( t ) their res pec tive auto c orr elation and cr oss c orr elation functions, which are defined b y C ϕ ( t ) = X i ∈ F p ϕ ( i ) ϕ ( i + t ) and C ϕ,ψ ( t ) = X i ∈ F p ϕ ( i ) ψ ( i + t ) . (2) Definition 1 We say that S is a ( p, r, σ, ρ ) signal s e t if e ach se quen c e in S has p erio d p , ther e ar e r time-shift distinct se quenc es in S , and the maximum magnitude of out-of-phase auto c orr elation values and cr oss c orr elation values ar e upp er b ounde d by σ and ρ r esp e ctively, i.e., | C ϕ ( t ) | 6 σ, t 6 = 0 , ϕ ∈ S, | C ϕ,ψ ( t ) | 6 ρ, t ∈ F p , ϕ 6 = ψ ∈ S. (3) 4 In t his paper , w e also say that auto and cros s co r relation o f S is upp er b o unded b y σ and ρ res p ectively . W e sa y that a sequence ϕ is a p erfe ct se quen c e if C ϕ ( t ) = ( p t ≡ 0 mod p, 0 t 6≡ 0 mod p. The auto and cr oss ambiguity fu nctions of sequences are defined as t wo-dimensional autocorr elation and cross c orrela tio n functions in both time and phase, and a re given by A ϕ ( t, w ) = < ϕ, M w L t ϕ > and A ϕ,ψ ( t, w ) = < ϕ, M w L t ψ > . (4) The definitions of the auto and cross corre lation functions are equa l to their resp ective auto and cross ambiguit y functions for the case w = 0. Definition 2 We say t hat S is a ( p, r , σ , ρ ) a m biguity signal set if e ach se quenc e in S has p erio d p , ther e ar e r time-s hift distinct se quenc es in S , and the maximum magnitude of ambiguity out- of-phase auto c orr elation values and cr oss c orr elation values ar e u pp er b ounde d by σ and ρ r esp e ctively, i.e., | A ϕ ( t, w ) | 6 σ , ( t, w ) 6 = (0 , 0) , | A ϕ,ψ ( t, w ) | 6 ρ, ϕ 6 = ψ ∈ S. (5) Prop ert y 1 L et S 1 b e a ( p, r , σ, ρ ) ambiguity signal set, and S 2 = { M w ϕ | w ∈ F p , ϕ ∈ S 1 } . Then S 2 is a ( p, pr, σ , ρ ) signal set. Remark 1 All the definitions and notations are s tated for sequences with per io d p in this section. How ever, they a r e als o v alid for sequences with p erio d n when p and F p are replaced by n and Z n resp ectively . 3 Main Results There are tw o types of seq uences in the set of the fi nite osci l lator system S [16]. One is from the split case, denoted as S s , and the other is fro m the no n-split case, denoted a s S ns . In o ther words, S = S s ∪ S ns . Gurevich, Hadani, and Sochen investigated how to implemen t the sequences in S s by an algorithm [1 6]. Here we found a simple elementary constr uction for the s equences in S s , which is presented as follows. Theorem 1 L et α b e a gener ator of F ∗ p . S s = { ϕ x,y ,z | 1 6 x 6 p − 2 , 0 6 y 6 p − 1 , 0 6 z 6 ( p − 1) / 2 } 5 wher e ϕ x,y ,z = { ϕ x,y ,z ( i ) } is a normalize d se quenc e with p erio d p whose elements ar e give n by ϕ x,y , 0 ( i ) = 1 √ p − 1 θ x · log α i η y i 2 , and ϕ x,y ,z ( i ) = η y i 2 p p ( p − 1) p − 1 X j =1 θ x · log α j η − (2 z ) − 1 ( j − i ) 2 for z 6 = 0 . If z 6 = 0, it is clear ly every elemen t in ϕ x,y ,z has co mplicated form and do es not lie on the unit cir c le , so we only co nsider the se quences where z = 0. Construction of Ω 0 . Let α b e a generato r of F ∗ p . F or a given prime p ( p > 5), n ∈ Z a nd 0 6 n < p ( p − 2), n has a p -a dic decomp osition giv en by: n = ( x − 1) p + y where 1 6 x 6 p − 2 , 0 6 y 6 p − 1 . Let ϕ n = { ϕ n ( i ) } b e a sequence whose elements ar e defined as ϕ n ( i ) = θ x · log α i · η y i 2 , 0 6 i 6 p − 1 , and Ω 0 = { ϕ n : 0 6 n < p ( p − 2) } . Then from the main results of [16] (also Theo rem 3 in this paper ), we hav e Theorem 2 The Signal set Ω 0 satisfies the fo l lowing pr op erties. (a) Ω 0 is a ( p, p ( p − 2) , 2 √ p, 4 √ p ) ambiguity signal set. (b) DFT of ϕ is b ounde d by | b ϕ ( i ) | < 2 , for ϕ ∈ Ω 0 , i ∈ F p . (c) The elements of e ach se quenc e ϕ in Ω lie on the unit cir cle of the c omplex plane exc ept ϕ (0) = 0 . W e can extend Ω 0 by the phase shift o p erator as follows. Construction of Ω . Let α be a genera tor of F ∗ p . F or a given prime p ( p > 5), n ∈ Z and 0 6 n < p 2 ( p − 2), n ha s a p -adic decomp osition given b y: n = ( x − 1) p 2 + y p + z where 1 6 x 6 p − 2 , 0 6 y , z 6 p − 1 . Let ϕ n = { ϕ n ( i ) } b e a sequence whose elements ar e defined as ϕ n ( i ) = θ x · log α i · η y i 2 + z i , 0 6 i 6 p − 1 , and Ω = { ϕ n : 0 6 n < p 2 ( p − 2) } . Then from P r op erty 1, we hav e 6 Corollary 1 The signal set Ω satisfies the fol lowing pr op erties. (a) Ω is a ( p, p 2 ( p − 2) , 2 √ p, 4 √ p ) s ignal set. (b) DFT of ϕ is b ounde d by | b ϕ ( i ) | < 2 , for ϕ ∈ Ω , i ∈ F p . (c) The elements of e ach se quenc e ϕ in Ω lie on the unit cir cle of the c omplex plane exc ept ϕ (0) = 0 . (d) The magnitude of auto ambiguity function of every se quenc e in Ω is u pp er b ounde d by 2 √ p at any shift not e qual to (0 , 0) . Example 1 F or p = 5, a = 2 is a genera tor of F 5 , the elements of the seque nc e s ϕ x , ϕ y , and ϕ z are defined as ϕ x ( i ) = θ x · log α i , ϕ y ( i ) = η y i 2 , and ϕ z ( i ) = η z i resp ectively , which are g iven as follows. x ϕ x ( i ) = θ x · log α i 1 { 0 , 1 , θ , θ 3 , θ 2 } 2 { 0 , 1 , θ 2 , θ 2 , 1 } 3 { 0 , 1 , θ 3 , θ , θ 2 } y ϕ y ( i ) = η y i 2 0 { 1 , 1 , 1 , 1 , 1 } 1 { 1 , η , η 4 , η 4 , η } 2 { 1 , η 2 , η 3 , η 3 , η 2 } 3 { 1 , η 3 , η 2 , η 2 , η 3 } 4 { 1 , η 4 , η , η , η 4 } z ϕ z ( i ) = η z i 0 { 1 , 1 , 1 , 1 , 1 } 1 { 1 , η , η 2 , η 3 , η 4 } 2 { 1 , η 2 , η 4 , η , η 3 } 3 { 1 , η 3 , η , η 4 , η 2 } 4 { 1 , η 4 , η 3 , η 2 , η } Then the elemen ts o f eac h sequence in the signal set Ω are construc ted by term-b y-ter m pro ducts of the elements of ϕ x , ϕ y , and ϕ z . The first thr ee sequences and last tw o sequences are given as fo llows. ϕ 0 = ϕ 1 , 0 , 0 = (0 , 1 , θ, θ 3 , θ 2 ) , ϕ 1 = ϕ 1 , 0 , 1 = (0 , η , θη 2 , θ 3 η 3 , θ 2 η 4 ) , ϕ 2 = ϕ 1 , 0 , 2 = (0 , η 2 , θ η 4 , θ 3 η , θ 2 η 3 ) , . . . . . . ϕ 73 = ϕ 3 , 4 , 3 = (0 , η 2 , θ 3 η 2 , θ , θ 2 η ) , ϕ 74 = ϕ 3 , 4 , 4 = (0 , η 3 , θ 3 η 4 , θ η 3 , θ 2 ) . In the rest of the sections, we fir st prove that Theore m 1 is the split ca s e of the finite oscillator system, and then complete pr o ofs for Theo r em 2 and Coro lla ry 1. In order to do so, in the next se c tion, we first introduce some basic co ncepts and definitions on W eil repre sentations, and then present the oscillator system s ig nal set. 7 4 The W eil Represen tation and Finite Oscillator System F or mor e details ab out the r epresentation theory and the W eil r epresentation, we refer the rea der to [16, 21, 2 2] a s well a s the a pp endix in this pap er. 4.1 W eil Represen tation The W eil r e presentation is a unitary representation from S L 2 ( F p ) to U ( H ) (see the deta ils in App endix). S L 2 ( F p ) can b e generated by g a = a 0 0 a − 1 ! , g b = 1 0 b 1 ! , and W e y l element w = 0 1 − 1 0 ! where a ∈ F ∗ p and b ∈ F p . The W eil representations for g a , g b and w are given in [17] as follows ρ ( g a )[ ϕ ]( i ) = σ ( a ) ϕ ( a − 1 i ) (6) ρ ( g b )[ ϕ ]( i ) = η − 2 − 1 bi 2 ϕ ( i ) (7) ρ ( w )[ ϕ ]( j ) = 1 √ p X i ∈ F p η j i ϕ ( i ) (8) where σ : F ∗ p → {± 1 } is the Legendre character, i.e., σ ( a ) = a p − 1 2 in F p . Obviously , ρ ( w ) is equal to F de fined in (1). Here we de no te ρ ( g a ) = S a , ρ ( g b ) = N b , ρ ( w ) = F for simplicity . F o r g = a b c d ! ∈ S L 2 ( F p ), if b 6 = 0, we have g = a b c d ! = a b ( ad − 1) b − 1 d ! = b 0 0 b − 1 ! 1 0 bd 1 ! 0 1 − 1 0 ! 1 0 ab − 1 1 ! . Thu s the W eil repr esentation of g is giv e n by ρ ( g ) = S b ◦ N bd ◦ F ◦ N ab − 1 . (9) If b = 0, then g = a b c d ! = a 0 c a − 1 ! = a 0 0 a − 1 ! 1 0 ac 1 ! . Hence the W eil representation of g is as follows ρ ( g ) = S a ◦ N ac . (10) 8 4.2 The Finite Oscillator System In this s ubsection, we introduce the main results of [16]. A. M axim al Algebraic T ori A m ax imal algebr aic torus [4] in S L 2 ( F p ) is a ma ximal comm utative subgr o up which b ecomes diago- nalizable ov er the original field or quadr atic extension of the field. One example of a maxima l alg e braic torus in S L 2 ( F p ) is the s tandard diago nal tor us A = ( a 0 0 a − 1 ! : a ∈ F ∗ p ) . Up to conjuga tion, there are tw o classes of the maximal algebra ic tor i in S L 2 ( F p ). The first class , called split tori , consists o f those to ri which are diagona liz able over F p . Every split torus T is conjugated to the standard diagonal torus A , i.e., there exists an elemen t g ∈ S L 2 ( F p ) suc h that g · T · g − 1 = A . The second class, called non- split tori , consists of those tor i whic h are not diago na lizable ov er F p , but bec ome dia gonalizable ov er the quadr atic extension F p 2 . In fact, a s plit torus is a cyclic subgroup of S L 2 ( F p ) with or der p − 1 , while a non-split to r us is a cyclic subgr oup o f S L 2 ( F p ) with or der p + 1 . All split (no n- split) tori are conjuga ted to one another , so the num b er of split (non-s plit) tor i is the nu m b er o f elemen ts in the coset space S L 2 ( F p ) / N ( S L 2 ( F p ) / M ) (see [41] for basics o f group theory), where N ( M ) is the normalizer gr oup of a non-split tor us A . Thus #( S L 2 ( F p ) / N ) = 1 2 p ( p + 1) and #( S L 2 ( F p ) / M ) = 1 2 p ( p − 1) . (11) Remark 2 A dir ect calculation shows tha t the n umber of non-split tor i is equal to 1 2 p ( p − 1) instead of p ( p − 1), which is a mistak e made in [16]. B. Decomp osition of W eil Re presen tatio n Asso ciated w i th Maxim al T ori Because ev ery maximal torus T ∈ S L 2 ( F p ) is a cyclic group, restricting the W eil repr esentation to T : ρ | T : T → U ( H ), we obtain a decompo s ition of ρ | T corres p o nding to an o r thogonal decompo sition of H . ρ | T = M χ ∈ Λ T χ and H = M χ ∈ Λ T H χ (12) where Λ T is a collection of all the o ne dimensiona l subrepres ent a tions (characters) χ : T → C in the decomp osition of the W eil r epresentation restr ic ted on the tor us T . The decomp ositio n (12) depends on the type of T . In the case where T is a s plit torus, χ is a character given b y χ : Z p − 1 → C . W e hav e dim H χ = 1 unless χ = σ where σ is the Legendre character 9 of T , a nd dim H σ = 2. In the cas e wher e T is a non-s plit torus, χ is the c haracter given by χ : Z p +1 → C . There is only one c ha racter σ with order 2 that do es not appe a r in the decomp o sition. F or the other characters χ 6 = σ , dim H χ = 1 . C. Sequences Asso ciated with Fini te Oscill ator Syste m F or a given torus T and ea ch c hara cter χ ∈ Λ T , choo s ing a vector ϕ χ ∈ H χ of unit no rm, we obtain a collection o f orthonormal vectors B T = { ϕ χ : χ ∈ Λ T , χ 6 = σ } . (13) Considering the union of these collections, then the finite os c illator system S = { ϕ ∈ B T : T ⊂ S L 2 ( F p ) } . (14) S is naturally separa ted int o t wo sub-systems S s and S ns which co rresp ond to the split tori and the non-split tori r esp ectively . The sub-system S s ( S ns ) consists of the union o f B T , where T runs through all the split tori (non-s plit tor i) in S L 2 ( F p ). T otally there are 1 2 p ( p + 1) ( 1 2 p ( p − 1)) to ri co ns isting of p − 2 ( p ) or tho normal sequences. Hence # S s = 1 2 p ( p + 1)( p − 2) and # S ns = 1 2 p 2 ( p − 1) . (15) Theorem 3 Se quenc es in the set S satisfy the fol lowing pr op erties. F o r ϕ, ψ ∈ S and ( t, w ) ∈ V = F p × F p , (a) S is a ( p, p ( p 2 − p − 1) , 2 √ p p − 1 , 4 √ p p − 1 ) ambiguity signal s et . (b) S u pr emum of ϕ is given by max {| ϕ ( i ) | : i ∈ F p } 6 2 √ p . (c) F or every se quenc es ϕ ∈ S , its DFT ˆ ϕ is (u p to multiplic ation by a unitary sc alar) also in S . 5 Pro of of Main Results An efficien t metho d to specify the decomp osition (12) is b y c ho o sing a ge nerator t ∈ T , the c ha r acter which is genera ted b y the eigenv alue of linear op er ator ρ ( t ), and the character space H χ that natura lly corres p o nds to the eige nspace. Below are three s teps to construct the sequences in the split cas e of finite oscilla tor s y stem S s . Step 1 Compute the generator g α for the standard torus A and B A . In other w or ds, the c o llection o f the eigenv e ctors of ρ ( g a ) which do not c o rresp ond to eigenv alue − 1. 10 Step 2 Compute all r epresentativ e e le men ts g in the coset { g N ( A ) : g ∈ S L 2 ( F p ) } where N ( A ) is the normalizer gr o up of A . Step 3 Compute all s equences ρ ( g ) ϕ wher e g is the representative elemen t presented in Step 2 a nd ϕ ∈ B A is calculated in Step 1. Considering { δ i : i ∈ F p } as the Kroneck er delta function of Hilbert space H = C ( F p ) (i.e., δ i is defined as δ i ( j ) = δ ij for ∀ j ∈ F p ), every sequence ϕ = { ϕ ( i ) } with p erio d p can b e written a s ϕ = P p − 1 i =0 ϕ ( i ) δ i . Recall that S L 2 ( F p ) can b e g enerated by g a = a 0 0 a − 1 ! , g b = 1 0 b 1 ! and w = 0 1 − 1 0 ! where a ∈ F ∗ p and b ∈ F p , then their resp ective W eil representations (6), (7), and (8) of g a , g b , and w can b e rewritten a s follows ρ ( g a ) δ i = S a δ i = σ ( a ) δ ai (16) ρ ( g b ) δ i = N b δ i = η − 2 − 1 bi 2 δ i (17) ρ ( w ) δ j = F δ j = 1 √ p X i ∈ F p η j i δ i . (18) Lemma 1 L et α b e a gener ator of F ∗ p , and A = ( a 0 0 a − 1 ! : a ∈ F ∗ p ) b e the st andar d diagonal torus. Then B A = ( ϕ x = 1 √ p − 1 p − 1 X i =1 θ x · log α i δ i : 1 6 x 6 p − 2 ) . Pr o of. The set B A is a collection of ϕ χ with unit norm wher e ϕ χ ∈ H χ for every character χ 6 = σ . In other words, the s et B A is a collection of unit eigenv ectors (not b elonging to eigenv alue − 1) of ρ ( g α ) where g α is a genera tor of T or us A . Let α b e a genera to r of F ∗ p . Then g α = α 0 0 α − 1 ! is a g enerator of torus A . F rom (16), we have ρ ( g α ) δ i = σ ( α ) δ αi = − δ ai . The eigenfunction of ρ ( g α ) is ( x + 1)( x p − 1 − 1), so the eigenv alues of ρ ( g α ) are − 1 , θ 0 , θ 1 , θ 2 · · · · · · θ p − 2 . Obviously , − 1 = θ p − 1 2 o ccurs twice in the eig env alues set. W e asser t that P p − 1 i =1 θ ( p − 1 2 − j ) log α i δ i is an 11 eigenv ector asso cia ted to the eig e n v alue θ j (0 6 j 6 p − 2), and it can b e v erifie d a s follows ρ ( g α )( p − 1 X i =1 θ ( p − 1 2 − j ) log α i δ i ) = − p − 1 X i =1 θ ( p − 1 2 − j ) log α i δ ai = − p − 1 X i =1 θ ( p − 1 2 − j ) log α ( a − 1 i ) δ i = θ p − 1 2 p − 1 X i =1 θ ( p − 1 2 − j )(log α i − 1) δ i = θ p − 1 2 θ j − p − 1 2 p − 1 X i =1 θ ( p − 1 2 − j ) log α i δ i = θ j p − 1 X i =1 θ ( p − 1 2 − j ) log α i δ i . Let x = p − 1 2 − j . Then { P p − 1 i =1 θ x · log α i δ i (1 ≤ x ≤ q − 2) } is a set of the e igenv ectors cor resp onding to all the eige nv alues not equal to − 1. By nor malizing the eigen vectors, we complete the proo f.  Lemma 2 L et A = ( a 0 0 a − 1 ! : a ∈ F ∗ p ) b e t he standar d diagonal torus, and N ( A ) b e the normal- izer gr oup of A . Then R = ( 1 b c 1 + b c ! : 0 6 b 6 p − 1 2 , c ∈ F p ) is a c ol le ction of c oset r epr esentatives of { g N ( A ) : g ∈ S L 2 ( F p ) } . Pr o of. Denote B = ( 0 − b b − 1 0 ! : b ∈ F ∗ p ) . The n it’s not hard to v erify N ( A ) = { g : g Ag − 1 = A, g ∈ S L 2 ( F p ) } = AB . Thu s every r epresentativ e ele men t g can b e written as the for m g = 1 b c 1 + bc ! b, c ∈ F p . 12 Note that g = 1 b c 1 + b c ! and g ′ = 1 b ′ c ′ 1 + b ′ c ′ ! are in the sa me coset, i.e., g − 1 g ′ ∈ N ( A ), if and only if 1 b ′ c ′ 1 + b ′ c ′ ! = 1 b c 1 + bc ! 0 − b b − 1 0 ! = 1 − b b − 1 + c − bc ! = 1 − b b − 1 + c 1 + ( − b )( b − 1 + c ) ! if and only if b ′ = − b and c ′ = b − 1 + c . Therefo re R contains all repr esentativ e elemen ts in the c o set { g N ( A ) : g ∈ S L 2 ( F p ) } .  By Lemmas 1 and 2, we can now pro ve Theor em 1, whic h is a direct consequence of the following result. Prop ositi o n 1 Ther e ar e two t yp es of ve ctors in S s . The first typ e is ϕ x,y , 0 = 1 √ p − 1 p − 1 X i =1 θ x · log α i η y i 2 δ i wher e 1 6 x 6 p − 2 , 0 6 y 6 p − 1 . The se c ond typ e is ϕ x,y ,z = 1 p p ( p − 1) p − 1 X i =0 p − 1 X j =1 θ x · log α j η y i 2 − (2 z ) − 1 ( j − i ) 2 δ i wher e 1 6 x 6 p − 2 , 0 6 y 6 p − 1 , 1 6 z 6 p − 1 2 . Pr o of. Every split tor us T ⊂ S L 2 ( F p ) can b e written as the form g Ag − 1 where A is the dia gonal torus and g = 1 b c 1 + b c ! ∈ R in Lemma 2 . Then B T = B gAg − 1 = { ρ ( g ) ϕ : ϕ ∈ B A } , and S s = [ g ∈ R B gT g − 1 = { ρ ( g ) ϕ : g ∈ R, ϕ ∈ B A } . 13 If b = 0, g = 1 b c 1 + bc ! has the for m 1 0 c 1 ! (0 6 c 6 p − 1), then from (1 7), w e ha ve ρ ( g ) ϕ x = N c ( 1 √ p − 1 p − 1 X i =1 θ x · log α i δ i ) = 1 √ p − 1 p − 1 X i =1 θ x · log α i N c δ i = 1 √ p − 1 p − 1 X i =1 θ x · log α i η − 2 − 1 ci 2 δ i . If b 6 = 0, g has the follo wing decomp ositio n g = 1 b c 1 + bc ! = b 0 0 b − 1 ! 1 0 b (1 + bc ) 1 ! 0 1 − 1 0 ! 1 0 b − 1 1 ! . Then applying (1 6),(17), and (18), for 1 6 x 6 p − 1, we hav e ρ ( g ) ϕ x = S b ◦ N b (1+ bc ) ◦ F ◦ N b − 1 ( 1 √ p − 1 p − 1 X j =1 θ x · log α j δ j ) = S b ◦ N b (1+ bc ) ◦ F ( 1 √ p − 1 p − 1 X j =1 θ x · log α j η − 2 − 1 b − 1 j 2 δ j ) = S b ◦ N b (1+ bc ) ( 1 p p ( p − 1) p − 1 X i =0 p − 1 X j =1 θ x · log α j η − 2 − 1 b − 1 j 2 η ij δ i ) = S b ( 1 p p ( p − 1) p − 1 X i =0 p − 1 X j =1 θ x · log α j η − 2 − 1 b − 1 j 2 η ij η − 2 − 1 b (1+ bc ) i 2 δ i ) = σ ( b )( 1 p p ( p − 1) p − 1 X i =0 p − 1 X j =1 θ x · log α j η − 2 − 1 b − 1 j 2 η ij η − 2 − 1 b (1+ bc ) i 2 δ bi ) = σ ( b )( 1 p p ( p − 1) p − 1 X i =0 p − 1 X j =1 θ x · log α j η − 2 − 1 b − 1 j 2 η b − 1 ij η − 2 − 1 b − 1 (1+ bc ) i 2 δ i ) = σ ( b ) p p ( p − 1) p − 1 X i =0 p − 1 X j =1 θ x · log α j η − (2 b ) − 1 ( j − i ) 2 − 2 − 1 ci 2 δ i . Let y = − 2 − 1 c, z = b . Then y ra nges ov er F p as c ranges o ver F p . Note that σ ( z ) = ± 1 is a constant , so 1 √ p − 1 P p − 1 i =1 θ x · log α i η y i 2 δ i and 1 √ p ( p − 1) P p − 1 i =0 η y i 2 P p − 1 j =1 θ x · log α j η − (2 z ) − 1 ( j − i ) 2 δ i with 1 6 x 6 p − 2 , 14 0 6 y 6 p − 1, 1 6 z 6 p − 1 2 are all the v ector s in S s , which completes the pro of.  Thu s, we have found a simple elementary r e presentation for the split ca s e of the finite o scillator system. The following lemma gives the re la tionship of the c orrela tio n function, am big uit y function, and unitary op era tor L t , M w , F defined in (1 ), which is easy to verify . Lemma 3 ∀ ϕ, ψ se qu enc es with p erio d p , ∀ t, w , z ∈ F p , and wher e L t , M w , F ar e define d in (1), we have: (a) C ϕ ( t ) = < ϕ, L t ϕ > and C ϕ,ψ ( t ) = < ϕ, L t ψ > . (b) | < ϕ, π ( t, w , z ) ψ > | = | < ϕ, M w · L t ψ > | = | < ϕ, L t · M w ψ > | . (c) L t · F = F · M t and F L − t = M t · F . (d) < b ϕ, L t b ψ > = < ϕ, M − t ψ > and < b ϕ, M w b ψ > = < ϕ, L t ψ > ( P ar sev al F orm ul ae ) . Now we extend s ig nal s e t from S to S by the phase shift opera tor, i.e., S = { M w ϕ : ∀ ϕ ∈ S , w ∈ F p } . Then S satis fy the follo wing prop erty . Prop ert y 2 With t he ab ov e notation, (a) S is a ( p, p 2 ( p 2 − p − 1) , 2 √ p p − 1 , 4 √ p p − 1 ) signal set. (b) S u pr emum of ψ is given by max {| ψ ( i ) | : i ∈ F p } 6 2 √ p , ψ ∈ S . (c) DFT of ψ is b ounde d by | b ψ ( i ) | 6 2 √ p , ∀ i ∈ F p , ψ ∈ S . Pr o of. (a) By Pro p er ty 1, S is a ( p, p ( p 2 − p − 1) , 2 √ p p − 1 , 4 √ p p − 1 ) am big uity signa l s et, so S is a ( p, p 2 ( p 2 − p − 1) , 2 √ p p − 1 , 4 √ p p − 1 ) signal set. (b) ∀ M w ϕ ∈ S , it is clear that the ma g nitude of M w ϕ ( i ) is a s same as tha t of ϕ ( i ). (c) Applying Lemma 3-(c), the DFT of M w ϕ can b e written a s F · M w ϕ = L w · F ϕ . W e can see that F ϕ is also in S from Lemma 3- (c), and | F ϕ ( i ) | 6 2 √ p from Theorem 3-(b). Thus | F · M w ϕ ( i ) | = | L w · F ϕ ( i ) | 6 2 √ p , which completes the pro of. 15  Pr o of of The or em 2. Co nsidering Ω 0 and S s , it is obvious that Ω 0 is a subset of S s up to multiplication by √ p − 1. Th us Ω 0 is a ( p, p 2 ( p − 2) , 2 √ p, 4 √ p ) a mbiguit y sig nal set, and the DFT of ϕ ∈ Ω 0 is bo unded by | b ϕ ( i ) | 6 2 q p − 1 p < 2 ∀ ϕ ∈ Ω and ∀ i ∈ F p .  F ro m Theo rem 2 , Pr op erty 1 and Lemma 3, Cor ollary 1 holds. 6 Comparisons of the New Constructions with Some Kno wn Constructions The split case of the finite oscillator system S s and the extended co ns truction S can be efficiently implemen ted for mo derate p . How ever, for la rge p , since one needs to c o mpute the ex po nent ial sum of p elements, they a re not so efficient. Therefore, in this se c tion, we only make some c omparisons for the set Ω or Ω 0 with some known constructions. A. Com pared with Complex V alued Sequences w i th Go o d Ambiguit y F unction or DFT Let n b e a p ositive integer and ω n be an n th primitive ro o t of unit in the co mplex field, i.e., ω n = e − 2 πj n where j = √ − 1. F or fixed 0 < y < n, 0 6 z < n where y is r elatively prime to n , a F ra nk -Zadoff-Chu (FZC) sequence [10, 6, 11] { ϕ y ,z ( i ) } is g iven by ϕ y ,z ( i ) =        ω (1 / 2) y i 2 + z i n n is even , ω (1 / 2) y i ( i +1)+ zi n n is o dd . (19) An y FZC seq ue nc e is a pe r fect sequence, i.e., its out-of-phas e auto co rrelation is zer o. Note that ω 1 / 2 n is a (2 n )th primitiv e ro ot of unit in the complex field. F o r n odd, ϕ y ,z ( i ) can b e given by an eq uiv alen t expression: ω y ′ i 2 + z ′ i n where 0 < y ′ < n, 0 6 z ′ < n . This form will be use d b elow. 1. F or a fixed z , a FZC signal set is a set co nsisting o f the ϕ ( n ) seq uences defined by (19) w he r e ϕ ( n ) is the E uler function. When n = p a prime, the FZC s et is a ( p, p − 1 , 0 , √ p ) signal set. The magnitude of the DFT of these sequences is b ounded by 1. 2. The ele men ts in Alltop cubic sequences [1] with p erio d p ar e given by ϕ y ( i ) = ω i 3 + y i p where 0 6 y 6 p − 1 . The a uto and cr o ss ambiguit y function ca n reach p with 1 p probability , and the magnitude of the DFT of these sequences is b ounded by 2. 3. Sequences fro m Heisenberg re pr esentation: The elements in a sequence from the Heisenberg re p- resentation [21] hav e the form ϕ y ,z ( i ) = ω y i 2 + z i p where 0 6 y, z 6 p − 1. (Note that the sequences 16 from Heisenber g repres ent a tion are the same a s the FZC sequences with p er io d n = p , a prime.) Here the magnitude o f the auto ambiguity function of such sequences can reach p with 1 p prob- ability , while the upp e r b ound o f the cro ss ambiguit y function b etw een tw o phase-shift distinct sequences is given by √ p , and the magnitude of the DFT o f these sequences is b o unded by 1. 4. Modulatable orthogo nal seq uences [39]: An h × h discr ete F our ie r tra nsform (DFT) ma trix is defined by the j th r ow and the k th column elements of d z ,j,k = ω z j k h (20) where z is a fixed num b er with 0 < z < h and gcd( z , h ) = 1, and 0 6 j, k < h . Let a sequence { a z ( i ) } b e given by co ncatenation of the r ows of DFT matrix star ting from the fir st row, second row, and so o n, i.e., a z ( i = j h + k ) = d z ,j,k , 0 6 j, k < h . (Note that { a z ( i ) } can b e consider ed as a n interleav ed sequence as so ciated with the DFT matrix [13].) Let { b ( i ) } i ≥ 0 be a co mplex v alued sequence with p erio d h and | b i | = 1, i.e., the magnitude of b i is equa l to 1. A mo dulatable orthogo nal (MO) seq uence { c z ( i ) } of p erio d n = h 2 is given by c z ( i ) = a z ( i ) b ( i ) , i = 0 , 1 , · · · . F or each h , an MO sequence is a p erfect seq uence. An MO sig nal set consists of the sequences for all z . When h = p , a prime, this set is a sig nal set with parameters ( p 2 , p − 1 , 0 , p ). 5. Generalized chirp-lik e (GCL) sequences [32]: Pop ovi ´ c, genera lized the constructio n of the mo d- ulatable or tho gonal sequences in 1992 a s follows. Let { ϕ y ,z ( i ) } b e a FZC sequence with p erio d n = th 2 where b oth t and h are a rbitrary po sitive integers, and { b ( i ) } b e the sa me a s defined for MO sequences. A genera lized chirp-like sequence { c y ,z ( i ) } is g iven b y c y ,z ( i ) = ϕ y ,z ( i ) b ( i ) , i = 0 , 1 , · · · where the index i of ϕ y ,z ( i ) is re duced mo dular n and the index of b ( i ) is r educed mo dular h . Each generalize d chirp-like sequence sequence is a perfect sequence. F or a fixed z , a GCL signal set consists o f all { c y ,z ( i ) } for GCD( y , n ) = 1. When n = p 2 where p is a prime, a GCL signal set is a ( p 2 , p − 1 , 0 , p ) signal se t. Note that their resp ective auto/ cross ambiguit y functions and the DFT of MO and GCL se quences are not repo r ted in the literature. A more recent work [5] us ing the Zak transform show ed that the a bove p er fect s e quences, i.e., FZC, MO and GCL sequences, can b e considered as subsets of the sequences constr ucted fr om the Zak transfor m fo r some sp ecial pa rameters. 17 6. P ower residue sequences [38, 27, 35]: Let k be a prop er facto r of p − 1. A p ower residue sequence { ϕ x ( i ) } of p erio d p is defined a s ϕ x ( i ) = ω x · log a i k , i = 0 , 1 , · · · , (21) where 0 < x < k . A pow er r e sidue sequence is a p olyphase sequence with p erio d p and k different phases, which is repr esented by m ultiplica tive character s. A k -ar y p ower re s idue se quenc e of per io d p has the out-of-phas e auto corr elation mag nitude o f at most 3, which is a lso studied in [15]. Mo r eov er , it is shown in [2 3] that the ma gnitude of the cr oss-co rrelation of distinct k -ar y power residue sequences of p erio d p is b ounded b y √ p + 2. Thus, the set consisting of the power residue sequences defined by (21) for a ll x : 0 < x < k is a s ignal s et w ith parameters ( p, k − 1 , 3 , √ p + 2) where k can b e up to k = p − 1. When k = p − 1, it can b e seen that this is a subset of Ω, the new expr ession of the sequences fro m the W eil representation. Thus the ambiguit y a nd the DFT ar e b ounded with the sa me v alues as for Ω. F urther mo re, this signal set can b e enlar ged using the shift-a nd-add op erato rs. F o r details, see a r ecent pap er [45]. 7. F or the new co nstruction Ω, there ar e p 2 ( p − 2) time-shift distinct seque nc e s , and the element s in every sequence have the expressio n ϕ x,y ,z ( i ) = ω x · log a i p − 1 · ω y i 2 + z i p (note θ = ω p − 1 and η = ω p in the previous notatio n for the new construction. The magnitude o f auto a nd cr o ss correla ton of sequences in the set are upper b ounded by 2 √ p and 4 √ p , resp ectively , and the magnitude of the DFT of thes e sequences is upp er bo unded b y 2. The subset Ω 0 where z = 0 is an ambiguit y signal set with par ameters ( p, p ( p − 2) , 2 √ p, 4 √ p ). How ever, there is a p ossible drawbac k of those sequences in pra c tice . The alpha b et for a sequence o f length p grows roughly as O ( p 2 ). W e summarize the ab ove discussions in T able 1. W e use the notation η = ω p as we used in the previous sections exc e pt for the case o f GCL where we use ω p 2 . B. Signal Sets wi th Sizes i n the Order o f p 3 and Low Correlation Signal sets with family size in the o rder of p 3 , and with low cor relation are known in the literature and a re s hown in T able 2. The b ounds of auto and cr oss co r relation function for construc tio n Ω are better than or a s go o d as the sequences in [3], Z 4 sequences S (2 ) [2 5], and the sequences in [44], while the maximum magnitudes of DFT are o nly known for Ω, and Z 4 sequences S (2). C. Implem en tation Cost Note that the i th element of a seq uence in Ω is a pr o duct of the i th element of a ( p − 1)-ary power re sidue sequenc e of p erio d p and the i th element of a n FZC sequence of p erio d p . Thus, the implemen ta tion cos t of co nstruction Ω is equal to the sum of the cost of those tw o t yp es of sequences. Since b oth p ow er r esidue sequences a nd FZC sequences can b e im plement e d efficien tly a t bo th hardware 18 T able 1 : The Comparison with W e ll-known Complex V alued Sequences F amily i th element Period L Size Am big uit y and DFT FZC (1) ϕ y ( i ) = η y i 2 | AA | : p . [10] [6] [11] (0 6 y 6 p − 1) p p | C A | 6 √ p . | D F T | 6 1. ϕ y ( i ) = η i 3 + y i 2 | AA | : p . Alltop cubic [1] (0 6 y 6 p − 1) p p | C A | : p . | D F T | 6 2. Sequences from ϕ y ( i ) = η y i 2 + z i | AA | : p . Heisenberg (0 6 y 6 p − 1) p p | C A | 6 √ p . representation [21] | D F T | 6 1. MO (1) [39] c z ( ip + k ) = η z ik b ( k ) p 2 p − 1 AA, CA, DFT (1 6 z 6 p − 1 ) are unknown. GCL (1) [32] c y ( i ) = ω y i 2 + z i p 2 b ( i ) p 2 p − 1 AA, CA, DFT (1 6 y 6 p − 1 ) are unknown. Po wer residue sequences ϕ x ( i ) = θ x · log a i p p − 2 The sa me as Ω 0 . [38][23] (1 6 x 6 p − 2) Sequences from ϕ x,y ,z ( i ) = θ x · log a i η y i 2 | AA | 6 2 √ p. W eil represent ation (1 6 x 6 p − 2 , p p ( p − 2) | C A | 6 4 √ p. Ω 0 (this pa per ) 0 6 y 6 p − 1) | D F T | 6 2 - AA =Auto am biguity , CA = Cross amb iguity . - (1) Those are p erfect sequences. and softw are level, so do the sequences in Ω. F urthermore, the new expressio n of W eil representation sequences provides a trade-off among the alphab et size and go o d ambiguit y . 7 Concluding Remarks and An O p en Problem W e ha ve discovered a simple elemen tary representation of the sequences in the finite oscillator system from the W eil r epresentation, in tr o duced by Gure v ich, Hadani, a nd So chen. F rom this, we hav e s hown a construction Ω of families of complex v alued sequences of perio d p having low v alued cor relation functions. This co nstruction pro duces a sig nal set with p 2 ( p − 2 ) shift distinct sequences. The magnitude of the auto and cross correlation functions are upper b ounded by 2 √ p and 4 √ p , resp ectively . The DFT of ev er y se quence in the sig nal set is upp er bo unded b y 2. The signal set Ω 0 , a subset of Ω, p ossess e s all the prope rties of Ω as w e ll a s the mag nitude o f their a uto and cross am big uit y functions are bounded 19 T able 2 : T he C o mparison with Sequence s with Low Correla tion F amily Period L Size Correla tion DFT Am big uit y Blake and Mar k [3] (2) p − 1 ( L + 1) 3 4 √ L + 1 + 1 N N Z 4 sequences S (2 ) [2 5] 2 k − 1 L 3 + 4 L 2 + 5 L + 2 4 √ L + 1 + 1 5 [31] N Y u a nd Go ng [44] 2 k − 1 ( L + 1) 3 2 2 . 5 √ L + 1 N N Ω p L 2 ( L − 2) 2 √ L, 4 √ L 2 Not go o d Ω 0 p L ( L − 2) 2 √ L, 4 √ L 2 | AA | 6 2 √ L , | C A | 6 4 √ L - (2) This f amily can b e easily extende d to sequences ov er the finite field F p with peri od p n − 1 and the s ame correlation property fr om the work i n [29]. - AA =Auto am biguity , CA = Cross amb iguity . - N: no reported results in the literature. by 2 √ p and 4 √ p . Ho wever, there is a dra wba ck of this construction in practice, sinc e the alphab et for a sequence of length p grows ro ug hly as O ( p 2 ). If we lo ok at the constr uction Ω ag ain, we find that each sequence ϕ n = { ϕ n ( i ) } i > 0 is the term-by- term pro duct o f s e q uences { θ x log α i } i > 0 and { η y i 2 + z i } i > 0 which ar e rela ted to pow er residue s equences and FZC sequences, respe c tively . Go ing back to the literature , all the known constructions only in volv e one t yp e of c har acter from fin ite field F p , While here w e use both mu ltiplicative a nd additive characters of finite field F p . The pr o of of thos e results req uires very deep mathematics, i.e., the representation theory and l -adic algebraic geometry . This suggests that it is worth looking for a direct pro of for the construction, which will have a tw o-fold effect. One is for b etter promo tion of those sequences in pr a ctice without in tro ducing the W eil represent ation theor y . The other is that it may lea d to more discov eries of new sig nal sets with go o d auto and cross a mbiguit y functions as well as with lo w magnitude of the DFT sp ectrum. Op en Problem. F or Ω = { ϕ n | 0 6 n 6 p 2 ( p − 2) } , directly show that Ω is a ( p, p 2 ( p − 2) , 2 √ p, 4 √ p ) signal set and that the DFT of e very sequence is upp er b ounded by 2 without intro ducing W eil r epre- sentation and finite oscillator sy stem. Ac kno wledgmen t The authors would like to thank Grevich, Hadani and So chen for their help dur ing the co urse of conducting this work. The authors a re dee ply grateful to the Asso ciate Editor and reviewers for their many v aluable and helpful sug g estions which gr eatly impr ov ed the presentation of the work. 20 App endix The Heisen b erg Representation Let V = F 2 p be a tw o-dimensio nal vector space over the finite field F p . Then ( V , ω ) is symplectic if the symplectic for m ω is given by ω (( t 1 , w 1 ) , ( t 2 , w 2 )) = t 1 w 2 − t 2 w 1 , for ( t i , w i ) ∈ V , i = 1 , 2 . Considering V as a n Ab elian group, it a dmits a non-trivial cen tra l extension called the Heisenb er g gr oup H ( p 6 = 2). The g roup H can b e pres ent ed as H = V × F p with the multiplication given by ( t 1 , w 1 , z 1 ) · ( t 2 , w 2 , z 2 ) = ( t 1 + t 2 , w 1 + w 2 , z 1 + z 2 + 2 − 1 ω (( t 1 , w 1 ) , ( t 2 , w 2 ))) . It is easy to verify tha t the center of H is Z = Z ( H ) = { (0 , 0 , z ) : z ∈ F p } . Theorem 4 (Stone-V on Neuman) U p to isomorph ism, ther e exists a unique irr e ducible unitary r epr e- sentation π : H → U ( H ) with c en tr al char acter φ , that is, π | Z = φ · I d H . The represe n tation π in the a bove theo rem is called the Heisenb er g r epr esentation . In this pap er , we take a character o f Z as φ ((0 , 0 , z )) = η z . Then the unique ir reducible unita r y r epresentation π corres p o nding to φ has the follo wing fo r mula π ( t, w , z )[ ϕ ]( i ) = η 2 − 1 tw + z + w i ϕ ( i + t ) (22) for ϕ ∈ C ( F p ), ( t, w, z ) ∈ H . Consequently , we hav e π ( t, 0 , 0 )[ ϕ ]( i ) = ϕ ( i + t ) π (0 , w, 0)[ ϕ ]( i ) = η wi ϕ ( i ) π (0 , 0 , z )[ ϕ ]( i ) = η z ϕ ( i ) . Thu s π ( t, 0 , 0) , π (0 , w , 0 ) are equal to the unitary o p er ators time-shift L t and phase-s hift M w , res pe c - tively , defined in (1). The W eil Represen tation The symplectic group S p = S p ( V , ω ), which is isomorphic to S L 2 ( F p ), acts by a utomorphism of H through its action on the V -co o r dinate, i.e., ∀ ( t, w , z ) ∈ H and a matrix g = a b c d ! ∈ S L 2 ( F p ), the action g on ( t, w, z ) as g · ( t, w , z ) = ( at + bw , ct + dw , z ) . (23) 21 Due to W eil [42], a pro jective unitary representation e ρ : S p → P GL ( H ) is c o nstructed as follows. Considering the Heisenberg representation π : H → U ( H ) and g ∈ S p , a new representation is defined as: π g : H → U ( H ) by π g ( h ) = π ( g ( h )). Because both π and π g hav e the same central character φ , they are isomorphic b y Theor em 4. By Sch ur’s Lemma [37], H om H ( π , π g ) ∼ = C ∗ , so there exist a pro jective representation e ρ : S p → P GL ( H ). This pro jective representation e ρ is characterized by the for m ula: e ρ ( g ) π ( h ) e ρ ( g − 1 ) = π ( g ( h )) (24) for every g ∈ S p and h ∈ H . Moreover, e ρ ( g ) uniquely lifts to a unitary r epresentation ρ : S p → U ( H ) that satisfies equation (24). The existence o f ρ follows from the fa ct [2] that any pr o jective r e pr esentation of S L 2 ( F p ) can b e lifted to a n honest representation, while the uniqueness o f ρ follows from the fact [19] that the group S L 2 ( F p ) has no no n-trivial characters fo r p 6 = 3. Thu s the W eil repr esentation, sp ecified in Section 4.1 , follows. Notion of an Unitary R epresen tation Let H be a Hilb ert spa ce. A unitar y o p er ator on H is an op era tor A : H → H which pr e s erves the inner product, that is, < Aϕ, Aψ > = < ϕ, ψ > for every ϕ, ψ ∈ H . The s et o f unitar y opera tors for ms a group under compositio n o f op era tors, which is denoted by U ( H ). Definition 3 A unitary repres entation of a gr oup G on the Hilb ert sp ac e H is a homomorphism π : G → U ( H ) , i.e., π is map which satisfies the c ondition π ( g · h ) = π ( g ) · π ( h ) , ∀ g , h ∈ G. Definition 4 A unitary r epr esentation π : G → U ( H ) is c al le d irreducible if ther e is no pr op er subsp ac e H ′ ⊂ H invari ant under G, i.e., such that π ( g ) ϕ ∈ H ′ , ∀ ϕ ∈ H ′ . All unitary representations π : G → U ( H ) ca n b e decomp osed into a direct s um of irreducible unitary representations. In other words, there exists is a decompo sition of the Hilb ert spa ce H in to a direct sum H = M i ∈ I H i , such that each subspace H i is closed under the action of G , tha t is π ( g ) ϕ ∈ H i , ∀ ϕ ∈ H i , and such that the restr ic ted unitar y repres e n tations π i : G → U ( H i ) are ir reducible. 22 A unitary op erator A : H → H can b e diago nalized, which mea ns that there e x ists an orthog onal decomp osition of H into a direc t s um of eige nspaces H = M λ i H λ i , where dim( H λ i )= 1, and ∀ ϕ ∈ H λ i , Aϕ = λ i ϕ . Now we consider a unitary representation π : G → U ( H ) of a commutativ e gr oup G , which yields a comm utative gro up { π ( g ) : g ∈ G } of unitar y opera tors. Then the unitary op er ators { π ( g ) : g ∈ G } can b e diago nalized simult a neously , i.e., there e x ists a n orthogo nal decomp osition of H into common eigenspaces H = n M i =1 H χ i . Here the eig enspaces are indexed b y the characters χ i : G → S 1 where we have π ( g ) ϕ = χ i ( g ) ϕ for every g ∈ G , ϕ ∈ H χ i . Then w e achiev e the decomp osition of the W eil repr esentation asso ciated with max imal tori in Section 4.2 . P A PR and Discrete F ourier T ransform The following theorem g ives a r elationship among the contin uo us F ourier tra nsform, discre te F ourie r transform and P APR. 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