The finite harmonic oscillator and its associated sequences

1 The finite harmonic oscillator and its associated sequences Shamgar Gurevich, Ronny Hadani, and Nir So chen T o appea r in Pr oceeding s of the National Academy of Sciences of the United States of America Abstract — A nov el system of functions (signals) on the fin ite line, called the osc illator system, is described and studied. Applications of this system for discrete radar and digital com- munication theory ar e explained. Index T erms — W eil r epresentation, co mmutative subgroups, eigenfunctions, random beha vior , deterministic construction. I . I N T R O D U C T I O N One-dimen sional an alog sign als are com plex valued func- tions on th e real line R . I n th e same spirit, one- dimensiona l digital signals, also called sequences, might be considered as complex valued functions on the finite line F p , i.e. , the finite field with p elem ents. In both situatio ns the parame ter of the line is denoted by t and is refer red to as time . In th is work , we will consider digital signals only , which wil l be simply referred to as signals. The space o f signals H = C ( F p ) is a Hilbert space with the Her mitian prod uct g iv en by h φ, ϕ i = P t ∈ F p φ ( t ) ϕ ( t ) . A central prob lem is to con struct inter esting and useful systems of signals. Giv en a system S , there are various desired pr operties which appear in th e en gineering wish list. For example, in various situations [1], [2] on e req uires that the signals will be weakly correlated , i.e., that for e very φ 6 = ϕ ∈ S |h φ, ϕ i| ≪ 1 . This p roperty is tri vially satisfied if S is an orthono rmal basis. Such a system cann ot co nsist of mor e than dim( H ) signals, howe ver, for certain application s, e.g ., CDMA (Code Di vision Multiple Access) [3] a larger numb er of signals is desired, in that case the orth ogon ality cond ition is relaxed. During the tran smission process, a signal ϕ might be distorted in various ways. T wo basic types of distortions are time shift ϕ ( t ) 7→ L τ ϕ ( t ) = ϕ ( t + τ ) an d phase shift ϕ ( t ) 7→ M w ϕ ( t ) = e 2 πi p wt ϕ ( t ) , wh ere τ , w ∈ F p . Th e first type appear s in asynchro nous communication and the seco nd type is a Dop pler effect due to relative velocity between the transmitting and r eceiving antenna s. In conclusion, a gen eral distortion is of the typ e ϕ 7→ M w L τ ϕ, sug gesting that fo r ev ery ϕ 6 = φ ∈ S it is n atural to re quire [2] th e fo llowing stronger conditio n |h φ, M w L τ ϕ i| ≪ 1 . Date: January 1, 2007. c  Copyright by S. Gure vich, R. Hada ni and N. Sochen, Janua ry 1, 2007. All rights reserved . Due to technical restrictions in the transmission process, signals are sometim es req uired to admit low peak -to-average power r atio [4], i.e., that fo r ev ery ϕ ∈ S with k ϕ k 2 = 1 max {| ϕ ( t ) | : t ∈ F p } ≪ 1 . Finally , se veral schemes for digital communicatio n r equire that the above pro perties will co ntinue to hold a lso if we replac e signals from S by the ir Fourier transform . In this paper we constru ct a n ovel system of (u nit) signals S O , consisting of order of p 3 signals, where p is an od d prime, called the oscillato r system . These signals con stitute, in an approp riate formal sen se, a finite analog ue for the eigenfun ctions of th e ha rmonic oscillator in the real settin g and, in acco rdance, they sh are many of the nice proper ties of the latter class. In par ticular, the system S O satisfies the following properties 1) Autocorr ela tion (ambiguity function). For ev ery ϕ ∈ S O we have |h ϕ, M w L τ ϕ i| =  1 if ( τ , w ) = 0 , ≤ 2 √ p if ( τ , w ) 6 = 0 . (1) 2) Cr osscorr elation (cr oss-ambigu ity fun ction). For every φ 6 = ϕ ∈ S O we have |h φ, M w L τ ϕ i| ≤ 4 √ p , (2) for every τ , w ∈ F p . 3) Sup r emu m. F or ev ery signal ϕ ∈ S O we have max {| ϕ ( t ) | : t ∈ F p } ≤ 2 √ p . 4) F ourier in va riance. For every signal ϕ ∈ S O its Fourier transform b ϕ is (up to multiplication by a unitary scalar) also in S O . Remark 1: Explicit algorith m that gen erate the (split) os- cillator system is given in the supp orting text. The oscillator system c an be extended to a much larger system S E , consisting of order of p 5 signals if one is willing to comprom ise Properties 1 and 2 f or a we aker co ndition. The extended system con sists of all signals of th e form M w L τ ϕ for τ , w ∈ F p and ϕ ∈ S O . It is n ot hard to show that # ( S E ) = p 2 · # ( S O ) ≈ p 5 . As a consequence of (1) and (2) for e very ϕ 6 = φ ∈ S E we have |h ϕ, φ i| ≤ 4 √ p . The characteriz ation and construction of th e oscillator sys- tem is representation theoretic and we d ev ote the rest of 2 the paper to an intuitive explanatio n of th e main underlyin g ideas. As a sugg estiv e model example we explain first the construction o f the well k nown system of chirp (H eisenberg) signals, deliberately takin g a representation theore tic point of view ( see [5], [2] f or a more compr ehensive treatment). I I . M O D E L E X A M P L E ( H E I S E N B E R G S Y S T E M ) Let us denote by ψ : F p → C × the character ψ ( t ) = e 2 πi p t . W e consider the p air o f orthonorm al bases ∆ = { δ a : a ∈ F p } and ∆ ∨ = { ψ a : a ∈ F p } , where ψ a ( t ) = 1 √ p ψ ( at ) , and δ a is the Kron ecker delta function, δ a ( t ) = 1 if t = a an d δ a ( t ) = 0 if t 6 = a. A. Characterization of the bases ∆ and ∆ ∨ Let L : H → H be th e tim e shif t o perator L ϕ ( t ) = ϕ ( t + 1) . This operato r is unitary and it ind uces a h omom orphism of group s L : F p → U ( H ) given by L τ ϕ ( t ) = ϕ ( t + τ ) fo r any τ ∈ F p . Elements of the ba sis ∆ ∨ are character vectors with respec t to the action L , i.e., L τ ψ a = ψ ( aτ ) ψ a for any τ ∈ F p . In the same fashion, the basis ∆ con sists of cha racter vectors with respect to the homomor phism M : F p → U ( H ) given by the phase shift ope rators M w ϕ ( t ) = ψ ( w t ) ϕ ( t ) . B. The Heisenberg repr esentation The homomor phisms L and M can be combined into a single map e π : F p × F p → U ( H ) wh ich sends a p air ( τ , w ) to the unitary o perator e π ( τ , ω ) = ψ  − 1 2 τ w  M w ◦ L τ . The p lane F p × F p is called th e time-frequency plan e and will be deno ted by V . The map e π is not an homomorp hism since, in general, the o perators L τ and M w do not commute. This d eficiency can be corr ected if we consider the group H = V × F p with multiplication given by ( τ , w , z ) · ( τ ′ , w ′ , z ′ ) = ( τ + τ ′ , w + w ′ , z + z ′ + 1 2 ( τ w ′ − τ ′ w )) . The map e π extends to a h omom orphism π : H → U ( H ) g iv en by π ( τ , w , z ) = ψ  − 1 2 τ w + z  M w ◦ L τ . The g roup H is called the Heisenber g group an d the homo- morph ism π is called the Heisenberg repr esentation . C. Maximal commuta tive subgr oups The Heisenb erg group is n o lon ger commu tativ e, ho wever , it contains various commu tati ve subgrou ps which can be easily described. T o ev ery line L ⊂ V , tha t pass thr ough the origin, one can associate a maximal commu tativ e subgro up A L = { ( l , 0) ∈ V × F p : l ∈ L } . I t will be conv enient to identify th e subgrou p A L with the line L . D. Bases associated with lines Restricting the Heisenberg represen tation π to a sub grou p L y ields a decomp osition of the Hilbert space H into a direct sum of one-dimensio nal subspaces H = L χ H χ , where χ run s in the set L ∨ of (c omplex valued) ch aracters o f the group L . Th e subspace H χ consists o f vectors ϕ ∈ H such that π ( l ) ϕ = χ ( l ) ϕ . In other words, th e space H χ consists of common eigenvectors with respect to th e commu tativ e system of unitary operators { π ( l ) } l ∈ L such that the o perator π ( l ) has eigenv alue χ ( l ) . Choosing a unit vector ϕ χ ∈ H χ for ev ery χ ∈ L ∨ we obtain an ortho normal basis B L =  ϕ χ : χ ∈ L ∨  . In partic- ular , ∆ ∨ and ∆ are rec overed as the ba ses associated with the line s T = { ( τ , 0) : τ ∈ F p } an d W = { (0 , w ) : w ∈ F p } respectively . For a general L the signa ls in B L are certain kind of chirp s. Con cluding, we associated with every line L ⊂ V an orthonorm al basis B L , and overall we con structed a system of signals consisting of a union of ortho normal bases S H = { ϕ ∈ B L : L ⊂ V } . For obvious re asons, the system S H will be called th e Heisenber g system . E. Pr operties of the Heisenber g system It will be co n venient to intro duce the following g eneral notion. Giv en two signals φ , ϕ ∈ H , their m atrix coefficient is the function m φ,ϕ : H → C g iv en b y m φ,ϕ ( h ) = h φ, π ( h ) ϕ i . In coor dinates, if we write h = ( τ , w , z ) th en m φ,ϕ ( h ) = ψ  − 1 2 τ w + z  h φ, M w ◦ L τ ϕ i . When φ = ϕ the fun ction m ϕ,ϕ is called th e amb iguity func tion o f the vector ϕ and is denoted by A ϕ = m ϕ,ϕ . The system S H consists of p + 1 o rthono rmal bases 1 , alto- gether p ( p + 1 ) signa ls and it satisfies the follo wing properties [5], [2] 1) Autocorr ela tion . For ev ery signa l ϕ ∈ B L the fu nction | A ϕ | is the char acteristic function of the line L , i.e. , | A ϕ ( v ) | =  0 , v / ∈ L, 1 , v ∈ L. 2) Cr osscorr elation . F o r every φ ∈ B L and ϕ ∈ B M where L 6 = M we hav e | m ϕ,φ ( v ) | ≤ 1 √ p , for ev ery v ∈ V . If L = M then m ϕ,φ is the characteristic fun ction of some tr anslation of the lin e L . 3) Sup r emu m . A signal ϕ ∈ S H is a unimodular f unction , i.e., | ϕ ( t ) | = 1 √ p for every t ∈ F p , in par ticular we hav e max {| ϕ ( t ) | : t ∈ F p } = 1 √ p ≪ 1 . Remark 2: Note the main differences between th e Heisen- berg and the oscillator systems. The oscillator system con sists 1 Note that p + 1 is the number of lines in V . 3 of ord er of p 3 signals, while th e Heisenb erg system con sists of order of p 2 signals. Sig nals in the oscillator system admits an ambiguity function concentr ated a t 0 ∈ V (th umbtack pattern) while signals in the Heisenb erg system admits am biguity function concen trated on a line. I I I . T H E O S C I L L ATO R S Y S T E M Reflecting b ack on the Heisenb erg system we see that each vector ϕ ∈ S H is characterized in terms of a ction of the additive grou p G a = F p . Rough ly , in co mparison , ea ch vector in the oscillator system is char acterized in ter ms of actio n of the multiplicative group G m = F × p . Our n ext goal is to expla in the last assertion. W e begin by giving a model example. Giv en a multiplicative character 2 χ : G m → C × , we define a vector χ ∈ H by χ ( t ) =  1 √ p − 1 χ ( t ) , t 6 = 0 , 0 , t = 0 . W e consider th e system B std =  χ : χ ∈ G ∨ m , χ 6 = 1  , wh ere G ∨ m is the dual gr oup of characters. A. Characterizing the system B std For each element a ∈ G m let ρ a : H → H be the unitary operator acting by scaling ρ a ϕ ( t ) = ϕ ( at ) . This collection of operator s form a homomorp hism ρ : G m → U ( H ) . Elements of B std are character vectors with respect to ρ , i.e., the vector χ satisfies ρ a  χ  = χ ( a ) χ for every a ∈ G m . In more co nceptual ter ms, the action ρ yields a de composition of the Hilbert space H into character spaces H = L H χ , where χ runs in the g roup G ∨ m . The system B std consists of a represen tativ e un it vector for each space H χ , χ 6 = 1 . B. The W eil r epr e sentation W e would like to generalize the system B std in a similar fashion like we g eneralized the bases ∆ and ∆ ∨ in the Heisenberg setting. In o rder to do this we need to introdu ce se veral auxiliary operator s. Let ρ a : H → H , a ∈ F × p , be the oper ators acting b y ρ a ϕ ( t ) = σ ( a ) ϕ ( a − 1 t ) (scaling), where σ is the uniq ue quadra tic c haracter o f F × p , let ρ T : H → H be the operator acting by ρ T ϕ ( t ) = ψ ( t 2 ) ϕ ( t ) (qu adratic modulatio n), and finally let ρ S : H → H be the op erator of Fourier transform ρ S ϕ ( t ) = ν √ p P s ∈ F p ψ ( ts ) ϕ ( s ) , where ν is a normalization constant [6]. The operators ρ a , ρ T and ρ S are unitar y . Let us consider the sub group of u nitary operator s g enerated by ρ a , ρ S and ρ T . This group turns out to be isomorph ic to the fin ite gr oup S p = S L 2 ( F p ) , therefor e we o btained a ho momor phism ρ : S p → U ( H ) . The representation ρ is called the W eil r epr esentation [7] and it will play a pr ominent role in this pa per . 2 A mult iplica ti ve chara cter is a function χ : G m → C × which s atisfies χ ( xy ) = χ ( x ) χ ( y ) for e very x, y ∈ G m . C. Systems associated with maxima l (split) tori The grou p S p consists of various type s of co mmutative subgrou ps. W e will be interested in maximal diagona lizable commutative subg roup s. A subgroup o f this type is called maximal split to rus. The standar d examp le is the su bgrou p consisting of all diag onal matrices A =  a 0 0 a − 1  : a ∈ G m  , which is called the stand ar d to rus . The r estriction of the W eil representatio n to a sp lit torus T ⊂ S p yields a d ecompo sition of the Hilbe rt spac e H into a direct sum of ch aracter spaces H = L H χ , where χ runs in the set of characters T ∨ . Choos- ing a unit vector ϕ χ ∈ H χ for every χ we obtain a collection of orth onorm al vectors B T =  ϕ χ : χ ∈ T ∨ , χ 6 = σ  . Over- all, we constructed a system S s O = { ϕ ∈ B T : T ⊂ S p split } , which will be re ferred to as the split oscillator system. W e note that our initial system B std is recovered a s B std = B A . D. Systems associated with max imal (non-sp lit) tori From the point of view of this pa per, the most interesting maximal co mmutative subg roups in S p are th ose wh ich ar e diagona lizable over an extension field rather than over the base field F p . A subgroup of this type is called max imal non - split torus. It might be sugg esti ve to first explain the analog ue notion in the mor e familiar settin g of the field R . Here, the standard example of a maximal non -split tor us is the circle group S O (2 ) ⊂ S L 2 ( R ) . In deed, it is a maximal c ommutative subgrou p which becomes diagonalizable when considered over the extension field C of complex number s. The above analogy su ggests a way to construct examples of maximal non-split tori in th e finite field settin g as well. L et us assume for simplicity that − 1 does n ot admit a square roo t in F p . The gr oup S p acts natur ally o n the plane V = F p × F p . Consider the symmetric biline ar form B o n V gi ven by B (( t, w ) , ( t ′ , w ′ )) = tt ′ + w w ′ . An example of ma ximal non- split torus is the subgroup T ns ⊂ S p consisting of all elements g ∈ S p p reserving the form B , i.e., g ∈ T ns if and only if B ( g u, g v ) = B ( u, v ) for every u, v ∈ V . In the same fashion like in the split case, restrictin g the W eil representation to a n on-split to rus T yield s a decomp osition into character spaces H = L H χ . Choosing a unit vector ϕ χ ∈ H χ for every χ ∈ T ∨ we ob tain an orthon ormal b asis B T . Overall, we constru cted a system of signals S ns O = { ϕ ∈ B T : T ⊂ S p no n-split } . The system S ns O will be referred to as the no n-split oscillato r system . The co nstruction o f the system S O = S s O ∪ S ns O together with the f ormulation of some of its p roper ties ar e the main contribution of this paper . 4 E. Behavior under F ourier transform The oscillator system is closed un der the oper ation of Fourier tr ansform, i.e., for every ϕ ∈ S O we have b ϕ ∈ S O . The Fourier transform on the space C ( F p ) appea rs as a specific oper ator ρ (w) in the W eil re presentation , wh ere w =  0 1 − 1 0  ∈ S p . Giv en a sign al ϕ ∈ B T ⊂ S O , its Fourier transfo rm b ϕ = ρ (w) ϕ is, up to a unitary scalar , a signal in B T ′ where T ′ = w T w − 1 . In fact, S O is closed under all the oper ators in the W eil representation ! I ndeed, given an element g ∈ S p and a signal ϕ ∈ B T we have, up to a unitar y scalar , that ρ ( g ) ϕ ∈ B T ′ , where T ′ = g T g − 1 . In addition , the W eyl element w is an elemen t in som e maximal torus T w (the split type of T w depend s on the characteristic p of th e field) and as a r esult signals ϕ ∈ B T w are, in particular , eigenvectors of th e Fourier transfo rm. As a con sequences a signal ϕ ∈ B T w and its Four ier transform b ϕ differ b y a un itary constant, ther efore are p ractically the ”same” for all essential matter s. These prope rties might be relev an t for application s to OFDM ( Orthogo nal Frequency Division Multip lexing) [8] where on e r equires g ood p roperties both from the signal a nd its Fourier transform. F . Relation to the harmon ic oscillator Here we gi ve th e explanatio n why fu nctions in the non- split oscillator system S ns O constitute a finite an alogue of the eigenfun ctions of the harmon ic oscillator in the r eal setting. The W e il representation estab lishes the dictiona ry betwe en these two, seemingly , un related objects. The argument works as follows. The one-dimen sional harm onic oscillator is giv en by the differential op erator D = ∂ 2 − t 2 . The o perator D ca n be exponentiated to give a u nitary representation of the circle group ρ : S O (2 , R ) − → U  L 2 ( R  ) where ρ ( θ ) = e iθD . Eigenfu nctions of D are natur ally identified with char acter vectors with respect to ρ . The crucial point is that ρ is the re striction of th e W eil rep resentation o f S L 2 ( R ) to the maximal non -split torus S O (2 , R ) ⊂ S L 2 ( R ) . Summarizin g, the eigenf unction s o f th e harmo nic oscillator and fu nctions in S ns O are g overned by the same mechanism, namely b oth are character vectors with resp ect to the restric- tion of the W eil r epresentation to a maxima l no n-split torus in S L 2 . Th e only difference appears to be the field of definition, which for the harmonic oscillator is the reals and for the oscillator functio ns is the finite field. I V . A P P L I C A T I O N S T wo ap plications of th e oscillator system will be described . The first app lication is to the th eory of discrete r adar . The second ap plication is to CDMA systems. W e will give a br ief explanation of these prob lems, while emphasizing the relatio n to the Heisenberg representatio n. A. Discr ete Ra dar The theor y of discrete radar is closely re lated [2] to the finite Heisenberg gr oup H . A r adar sends a signal ϕ ( t ) and obtains an echo e ( t ) . The goal [9] is to recon struct, in maximal accuracy , the target r ange and velocity . The signa l ϕ ( t ) and the echo e ( t ) are, principally , related by the transform ation e ( t ) = e 2 πiw t ϕ ( t + τ ) = M w L τ ϕ ( t ) , where the time shift τ encod es the distance of the target fr om the r adar and the phase shift encod es the velocity o f th e target. Equiv alently saying, the transmitted signal ϕ an d the r eceived echo e are related by an action of an element h 0 ∈ H , i.e. , e = π ( h 0 ) ϕ. The p roblem of discre te rad ar can be described as follows. Given a signal ϕ and an echo e = π ( h 0 ) ϕ extract the value of h 0 . It is easy to sh ow th at | m ϕ,e ( h ) | = | A ϕ ( h · h 0 ) | and it obtains its maximum at h − 1 0 . Th is sug gests that a desired signal ϕ for discrete r adar should adm it an ambigu ity fun ction A ϕ which is highly concentr ated arou nd 0 ∈ H , which is a proper ty satisfied by signals in the oscillator system (Pro perty 2). Remark 3: It should be noted that the system S O is ”large” consisting of aprox imately p 3 signals. This property becomes importan t in a jamming scenario . B. Code Division Multiple Access (CDMA) W e are co nsidering the following setting. • Ther e exists a co llection of u sers i ∈ I , each holdin g a bit of inform ation b i ∈ C (usually b i is taken to be an N ’th root of unity). • Each user transmits his bit of information , say , to a central antenna. In or der to do that, he multip lies his bit b i by a priv a te signal ϕ i ∈ H and fo rms a message u i = b i ϕ i . • The transmission is carried thr ough a single chan nel (for example in the case of cellular commu nication the chan- nel is th e atmosphere), th erefore the message rec eiv ed b y the antenna is the sum u = P i u i . The main problem [3] is to e xtract the individual b its b i from the message u . The bit b i can b e estimated b y c alculating th e inner produc t h ϕ i , u i = P j h ϕ i , u j i = P j b j  ϕ i , ϕ j  = b i + P j 6 = i b j  ϕ i , ϕ j  . The last expression above sh ould be considere d as a sum of the in formation bit b i and an a dditional n oise cau sed b y the interferen ce of the other m essages. This is th e standard sce- nario also called the Synchr o nous scenario. In practice, more complicated scenarios appear, e.g., asynchr ono us scen ario - in whic h each message u i is allowed to acquir e an a rbitrary time shift u i ( t ) 7→ u i ( t + τ i ) , phase sh ift scen ario - in which each message u i is allowed to acqu ire an arbitra ry phase shift u i ( t ) 7→ e 2 πi p w i t u i ( t ) and pr obably also a comb ination of the two where each message u i is allowed to acquire a n ar bitrary distortion of the for m u i ( t ) 7→ e 2 πi p w i t u i ( t + τ i ) . 5 The previous discussion suggests that what we are seeking for is a large system S of signals which will en able a reliable extraction of each bit b i for as many users transmitting through the chan nel simultaneously . Definition 4 (Sta bility conditions): T wo unit signals φ 6 = ϕ are called stably cross-correlated if | m ϕ,φ ( v ) | ≪ 1 for ev ery v ∈ V . A unit signal ϕ is called stably auto correlated if | A ϕ ( v ) | ≪ 1 , for every v 6 = 0 . A system S of signals is called a stable system if every signal ϕ ∈ S is stably autocorr elated and any two different signals φ, ϕ ∈ S are stably cross-corr elated. Formally what we require for CDMA is a stable system S . Let u s explain why this correspo nds to a reasonable solu tion to ou r problem . At a certain time t th e anten na r eceiv es a message u = P i ∈ J u i , which is transmitted from a subset of users J ⊂ I . Eac h message u i , i ∈ J, is of the form u i = b i e 2 πi p w i t ϕ i ( t + τ i ) = b i π ( h i ) ϕ i , wher e h i ∈ H . In or der to extract the bit b i we compute the matrix coefficient m ϕ i ,u = b i R h i A ϕ i + #( J − { i } ) o (1) , where R h i is the operator of r ight translation R h i A ϕ i ( h ) = A ϕ i ( hh i ) . If the cardinality o f the set J is not too big then by ev aluatin g m ϕ i ,u at h = h − 1 i we can reconstruct the bit b i . It follows from (1) and (2) that the oscillator system S O can support orde r o f p 3 users, enabling reliab le reconstruction when order of √ p users are transmitting simultan eously . Remark about field extensions. All the results in this paper were stated for th e b asic finite field F p for th e re ason of making the termino logy more accessible. H owe ver, they are valid for any field extension o f th e form F q with q = p n . Complete proo fs appear in [6]. Acknowledgement. The authors would like to thank J. Bernstein f or h is interest and gu idance in the mathematical aspects of this work. W e are gratef ul to S. Golomb and G. Gong for their interest in this project. W e thank B. Sturmf els for encour aging u s to pro ceed in this line of research. Th e authors would like to than k V . An antharam , A. Gr ¨ unbau m and A. Sah ai for inte resting discussions. Finally , the second author is indebted to B. Porat for so many discussions where each tried to un derstand the cryptic termino logy o f the other . R E F E R E N C E S [1] Golomb S.W . and Gong G., Signal design for good correlati on. For wireless communication, cryptogra phy , a nd rada r . Cambridge Unive rsity Press, Cambridg e (2005). [2] Ho ward S. D., Calde rbank A. R. and Moran W ., The finite Heise nberg- W eyl groups in radar and communic ations. URASIP J ournal on Applied Signal P r ocessing V olume 2006 (2006), Articl e ID 85685, 12 pa ges. [3] V iterbi A. J., CDMA: Principles of Spread Spectrum Communicat ion. Addison-W esle y (1995) . [4] Pate rson, K.G. and T arokh V . , On the exist ence and construct ion of good codes with low peak-to-a verage po wer ratios. IEE E T rans. Inform. Theory 46 (2000) 1974-1987. 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