Discrete Scaling Based on Operator Theory

Discrete Scaling Bas ed on Op erator Theory Aykut Ko¸ c, ∗ Burak Bartan, † Haldun M. Ozaktas, ‡ No v em ber 10, 2017 Abstract Signal scaling is a fundamen tal op eratio n of practical impo rtance in whic h a sig nal is enlarged or shrunk in the co or dinate direction(s). Scaling o r magnifica- tion is not trivial for signals of a discrete v ariable since the sig na l v alues may not fall onto the dis crete co ordinate p oints. One appr oach is to consider the discretely-spa ced v alues as the samples of a signal of a real v ariable, find that signal by in terp olation, scale it, and then re-sample. How ever, this approach comes with complications of in terpretation. W e r e v iew a previously prop osed alter na tive and more eleg ant ap- proach, and then prop ose a new appr oach bas ed o n hyperdifferential op erator theory that we find most satisfactory in terms of obtaining a self-consistent, pure, and elega nt definition of discr ete scaling that is fully consistent with the theor y of the discr ete F ourier transform. 1 In tro duction Signal scaling is a fundament al op era tion in whic h the independent v ariable of the function f ( u ) is scaled by a real num ber M , resulting in the sig nal to be com- pressed or deco mpressed alo ng the u axis in the form f ( u/ M ). With reference to imag es, the terms ma gni- fication/demag nifica tion or zo o ming in/out are more commonly used. Scaling or magnification is not triv- ∗ Aykut Ko¸ c (Co rresp onding Author) is with ASELSAN Researc h Cente r, Ank ar a, T urkey , e-mail : a ykutk oc@aselsan.com.tr. † Burak Bartan is with Electrical Engineering Department, Stanford Univ ersity , Stanford, Californi a ‡ Haldun M. Ozaktas is wi th Electrical Engineering Depart- men t, Bilken t U niv ersity , TR-06800 Bilken t, A nk ara, T urkey ial for signals of a discrete (integer) v ariable since, the signal v alues may not fall onto the discrete co o r- dinate po ints. Given a function f [ n ] defined o n the int egers , the v alue of f [ n/ M ] will b e undefined unless n/ M is an in teger. Nevertheless, discr ete scaling is a necessary opera tion in practice s ince we often wan t to scale sig na ls of contin uous v ariables which a r e rep- resented as functions of discr e te v aria bles in digital computers. A straig htforward a pproach requiring knowledge of no more than elemen tary sig nals and systems [8, 6, 7 ] is to co ns ider the v a lues of f [ n ] as the Nyquist-rate samples of a h ypo thetical bandlimited signal f ( u ). Then, we can use s tandard sinc interpola tion to write an expressio n for f ( u ) in terms of f [ n ]. Now, f ( u ) can be scaled to f ( u/ M ) and then r e-sampled to obtain the v alues of a new signa l of a discrete v ar iable, which can be consider ed the scaled version o f f [ n ]. The v al- ues of the new scaled signal will be linearly r elated to the v alues of the orig inal signal f [ n ]. Of course, scal- ing f ( u ) to o btain f ( u/ M ) will change its bandwidth, which intro duces co mplications in c hoo sing the r e- sampling rate. In an y case, if the re-sampling rate is different, this w ill somewhat complicate interpre- tation of the scaled signa l. If the in teger domain is not defined from −∞ to ∞ , but ra ther ov er a finite int erv al, say fro m 0 to N − 1, and we are w orking in a circ ulant domain, it is p ossible to modify the ap- proach by employing Diric hlet functions [9] instea d of sinc functions as the interp o lation functions. The only more cr eative appr oach to discrete sc al- ing we ar e aw are o f is due to Pei et al., who de vel- op ed a metho d based on “ C e n tered Discrete Dilated Hermite F unctions ” (CDDHFs) [12], which is an im- prov ement of their ea rlier “ n 2 matrix” method [11]. 1 The CDDHF-based discrete scaling metho d w orks as follows: First, wr ite the signal as a linear sup er po - sition of discrete Hermite-Gaus sian functions. Then, replace the discrete Hermite-Gaussian functions with their dilated (scaled) versions to obtain the sca le d discrete signal [1 2]. In o ther words, the expans io n co- efficients a re kept the same while scaling the discrete functions that form the expansion basis. Although this sounds conceptually simple, the difficulty (and ingenuit y) lies in the developmen t of the the set of dilated discrete Hermite-Ga us sian functions, [12, 4], on which the method rests. This pro cedure provides a mathema tically so und and elegant way of p erform- ing discrete signal scaling . In this work, we present a different a pproach by utilizing hyperdifferential op erator theory [9, 14, 10, 15, 5] to o bta in a discrete scaling matrix. The sc a led version of the sig na l is obtained b y multiplying the unscaled version by this matrix. W e choose to w ork in a fra mework that is not only discrete, but a lso finite. That is, the functions are defined over finite interv als. Our approach employs the basic op erations of differ- ent iation and co o rdinate multiplication. W e b elieve that it pro vides a self-co nsistent, pur e, a nd eleg ant definition of disc rete scaling which is a lso fully com- patible with the theo ry of the dis crete F ourier trans- form and its circulant structure . W e also b elieve that the presented approach of defining a discr e te op era- tion in the context of hyper differential oper ator the- ory ca n set an example tha t can b e a pplied to other problems in signa l theory and analysis. The pap er is orga nized a s follows: preliminaries are given in Section 2; then in Section 3 w e review Pei’s metho d. Our metho d is present ed in Section 4. Numerical results and compar isons are given in Sec- tion 5. Finally we conclude in Section 6. 2 Preliminaries F or simplicit y w e w ork with one-dimensiona l signa ls, although our results can easily b e generalized to higher-dimensiona l s ig nals. Scaling is defined as that op eration which takes f ( u ) to | M | − 1 / 2 f ( u/ M ). The factor | M | − 1 / 2 is included to make the op eration uni- tary , but this will not b e of m uch impo rtance. The real parameter M > 0 ca n b e called the scaling o r zo oming factor or the magnificatio n, depending on context. The sig nal will b e co mpressed/demag nified or decompressed/ ma gnified dep ending on whether M is lesser or gr eater than unity . In o per ator for m we will write M M f ( u ) = | M | − 1 / 2 f ( u/ M ) . (1) where the calligraphic oper ator on the left-ha nd side exhibits the parameter M . O ur con ven tion for the F ourie r transform op erator will b e F f ( u ) = Z ∞ −∞ f ( u ) e − j 2 π uµ du (2) W e define t wo further op erators , the coo r dinate mul- tiplication o p er ator U and the different iation op er a- tor D : U f ( u ) = uf ( u ) (3) D f ( u ) = 1 i 2 π d f ( u ) du , (4) where the ( i 2 π ) − 1 is included so that U and D a r e precisely F ourier duals (the effect of either in one do- main is its dual in the other domain). This duality can b e expr e ssed as follows: U = F DF − 1 . (5) Basically , the ab ove equation sa ys that, instea d of m ultiplying a function f ( u ) with u , we can instea d take its in verse F our ier tr ansform, differentiate it with resp ect to the frequency v ariable, divide b y i 2 π , and tak e its F ourier tra nsform, and we will get the same result. In this pa per we deal with finite-length signals o f a discrete (integer) v ariable. (W e could equiv alen tly think o f them as b eing defined on a circulant doma in, which w ould not make a difference to our arguments.) The length of our signal vectors will be denoted by N . When N is ev en, they will b e defined on the in terv al of integers [ − N 2 , N 2 − 1], a nd when N is o dd, they will be defined on the in terv al of integers [ − N − 1 2 , N − 1 2 ]. W e will also consider a less-co mmon approach based on the device of using “half integers.” In this ap- proach, the domain is defined as the in terv al o f unit- spaced half integers [ − N 2 + 0 . 5 , N 2 − 1 + 0 . 5 ] for even N 2 and [ − N − 1 2 − 0 . 5 , N − 1 2 − 0 . 5] for o dd N . Although not very usual, there is no thing unnatural ab out this wa y of indexing signals of a discrete v ariable; it is merely a particular wa y of b o okkeeping. Note that the in- dices are still spaced b y unit y , and there is merely a shift by 0 . 5 with the purpo s e of making the in ter- v al symmetrical around the origin when N is even (with the consequenc e that symmetr y is lost when N is o dd). A few exa mples o f works considering this wa y of indexing are [2, 1, 3, 13]. Consistent with this literature, we will refer to the fo rmer a pproach as the or dinary DFT and refer to the latter one, in which we use ”half in tegers” , as the c enter e d DFT. It is poss ible to b etter understa nd the choices of indexing by consider ing them in the context of sam- pling a signal of a con tinuous (real) v a riable. The sample v a lues of a function f ( u ) are usually w r it- ten as f ( nh ) where n is the index and h is the sam- pling int erv al. When we us e full integer v alues of the index, which is the usual cas e, w e get a set of samples that includes a sample a t the origin, f (0) for n = 0. F or instance, for N = 4, we would be sampling at u = − 2 h, − 1 h, 0 h, 1 h . Ho wev er, we may also choo se to sample in a manner that does not include the or igin, for insta nc e , w e may choos e our sa mples as f ( nh + 0 . 5), where n are still full int egers , in which case we would b e sampling at u = − 1 . 5 h, − 0 . 5 h, 0 . 5 h , 1 . 5 h . The use of half in- tegers is an alter native w ay of b o okkeeping where we main tain the samples to b e at f ( nh ) r a ther than f ( nh + 0 . 5), but s till get the same sa mples by allowing n to take ha lf in teger v alues. While bo th appr o aches are equiv alent, we find the use of half in tegers (cen- tered) to b e more eleg a nt and unifying. The sampled signa ls c a n b e represented by column vectors with N r ows. The labelling of the rows will follow the same index conv ention as ab ove. In the case of half integers, we may r efer to the “-1.5 th r ow” of the vector, and s o forth. The op erator s a cting o n them can b e represented as ma trices that have N columns and N rows. The matrix representing the F ourie r tr ansformation will be the unitar y discrete F ourie r transform (DFT) matrix F , with appr opriate shifting/circulation of its r ows and columns such that it is consistent with the index r anges we use. The elements F mn of this N - po in t unitary DFT matrix F can be written in terms o f W N = exp( − j 2 π / N ) as follows: F mn = 1 √ N W mn N . Note that this expressio n covers both the ordinar y and cent ered case provided w e remem b er that (i) for the ordina ry cas e m a nd n run through the integers [ − N 2 , N 2 − 1 ] for even N and [ − N − 1 2 , N − 1 2 ] for odd N ; (ii) for the cen tered case m a nd n run through the unit-s pace half integers [ − N 2 + 0 . 5 , N 2 − 1 + 0 . 5] for even N a nd [ − N − 1 2 − 0 . 5 , N − 1 2 − 0 . 5] for o dd N . The ability to write what w ould otherwise b e tw o separate expressions in the familiar form ab ov e is the main adv ant age o f the half-integer indexing s cheme we emplo y . It co uld b e questioned whether it w ould not b e simpler to work with the tra ditional interv al [0 , N − 1] to k eep things simple. This w ould essent ially give the same results, only in s hifted/circulated form. W e choose to work with symmetric in terv als to main tain and reveal as muc h s ymmetry in the problem as there actually is. W e w ork with dimensionless co ordinates; that is, the unit o f u is no t seconds or meters, it is unitless. Say the function ˆ f ( x ) of a contin uo us v ariable x in seconds or meters has an approximate extent lying ov er the int erv al [ − ∆ x/ 2 , ∆ x/ 2], meaning most o f its energy is contained in this in terv al. Likewise, say its e x ten t in the frequency domain lies over the in- terv a l [ − ∆ f / 2 , ∆ f / 2 ], wher e f is the frequency v ari- able in Hz or in verse meter s . Then we can introduce a parameter s , such that u = x/s is a dimensionless nu mber and choos e to w ork with the function f ( u ) = ˆ f ( su ) instead of ˆ f ( x ). If w e c ho ose s = p ∆ x/ ∆ f , then the extent of b oth f ( u ) and its F ourie r tra nsform will lie in the interv al [ − √ ∆ x ∆ f / 2 , √ ∆ x ∆ f / 2 ]. Ac- cording to the sa mpling theo rem, if a sig nal is con- tained within s uch an interv al, it can b e sa mpled with a sampling interv al of 1 / √ ∆ x ∆ f . Thus there will be N = √ ∆ x ∆ f / (1 / √ ∆ x ∆ f ) = ∆ x ∆ f samples in all. The quantit y ∆ x ∆ f is often referred to a s the time-bandwidth or space-ba ndwidth pro duct. Re- expressing in terms of the num ber of samples N , we would b e sa mpling o ver the interv al [ − √ N / 2 , √ N / 2] with a sampling interv al of 1 / √ N for a total of N 3 samples. Should N not hav e an ev en who le squar e ro ot, w e can always choos e ∆ x and ∆ f a little larger than necess a ry to mak e it so, althoug h this will not be impo rtant for o ur discussion. T o put the whole sampling issue together, let us consider the example of N = 16. The interv al ov er which the signal will b e sampled will b e [ − 2 , 2], which is divided in to 16 s a mpling in terv als each of length h = 1 / 4. The re a l is sue now is whether the samples will b e ta ken on the left (or right) edg e of each sampling interv a l, or in the cent er (or yet somewhere else) of each sa mpling interv al. T ak- ing them at the left edge is the familiar case; the sample po in ts will b e [ − 2 , − 1 . 75 , . . . , 0 , . . . , 1 . 5 , 1 . 75]. If we ta ke them at the middle, they will b e [ − 1 . 875 , − 1 . 6 25 , . . . , − 0 . 125 , 0 . 125 , . . . , 1 . 875]. There are t wo ways to b o okkeep the la tter case. W e can contin ue to work with an integer index n , and then the s ample points will b e u = ( n + 1 / 2) h . Alter na - tively , w e can maintain that the sample p oints are still at u = nh , but use half integer v alues of n . W e find grea ter clarity and unity in emphasizing the sam- pling interv als over the sampling p o ints, a nd working with half integer index v alues. 3 P ei’s Metho d In [1 2], Pei et a l. c o nsider a finite signal denoted f , o f length N , to b e sca led. They let f M denote the scaled signal, with M be ing the s caling factor. The signal f can alwa ys b e expressed as a linea r co mb ination of any N linearly indep endent signals. In their method, sp ecial functions called “Cen tered Discrete Dilated Hermite F unctions” (CDDHFs) are utilized as the s et of N linea rly indep endent signa ls s o any f can be expressed as a linear combination of CDDHFs: f = N − 1 X p =0 c p, 1 H p, 1 , (6) where the H p, 1 are the CDDHFs of length N , and the c p, 1 are the coefficients. The co e fficien ts are sim- ply the inner pro ducts of the H p, 1 with the signal x : that is, c p, 1 = h x, H p, 1 i . The CDDHFs H p, 1 are the centered discr e te Hermite functions with no sca ling; hence the subscript 1 in the notation. H p,M denotes the p th CDDHF with a sca ling factor of M . Their prop os e d way to s cale the signal f is to scale the ba sis signals H p, 1 , a nd keep the expa nsion co ef- ficient s the same. More explicitly , the sca led signal f M is obtained as follows: f M = N − 1 X p =0 c p, 1 H p,M , (7) where the H p,M are sc a led versions of the H p, 1 with a scaling fac to r of M . The critical task, o f course, is to find the scaled versions of the ba sis s ig nals. Pei et al. develop a metho d for constructing the CDDHFs H p,M , which we now summariz e . The Hermite-Gauss ian functions of a contin uous v ar iable, denoted by ψ p ( u ), are given as follows, [9]: ψ p ( u ) = A p H p ( √ 2 π u ) e − π u 2 , A p = 2 1 / 4 √ 2 p p ! (8) where H p ( u ) denotes the Hermite po lynomials. It is w ell-known that the Hermite-Gaus sian functions, ψ p ( u ), satisfy the differential equation [9] d 2 du 2 ψ p ( u ) − 4 π 2 u 2 ψ p ( u ) = λψ p ( u ) . (9) The time-scaled version of the Hermite-Gaussian function ψ p ( u ) is ψ p ( u/ M ). It is p ossible to find a differential equation for ψ p ( u/ M ) by simply repla c- ing every u b y u/ M in E q. (9): M 2 d 2 du 2 ψ p  u M  − 4 π 2  u M  2 ψ p  u M  = λψ p  u M  . (10) Eq. (10) can b e rewritten in terms o f the co o rdinate m ultiplication op erato r U and the differentiation op- erator D : ( − M 2 4 π 2 D 2 − 4 π 2 M 2 U 2 ) ψ p  u M  = λψ p  u M  . (11) Rearra ng ing the ter ms, we get: ( M 4 D 2 + U 2 ) ψ p  u M  = − M 2 4 π 2 λψ p  u M  . (12) F unctions ψ p ( t/ M ) satisfying E q. (12) ar e the eigen- functions of ( M 4 D 2 + U 2 ). 4 The next step is to find the discrete counterpart o f Eq. (12). This is done by replacing the abstract op- erators (denoted by ca lligraphic letters), by boldface matrix op erato rs that act o n column vectors in the form ( M 4 D 2 + U 2 ). Then, it is p ossible to compute the CDDHFs H p,M as the eigenv ectors of this ma- trix. Here U and D ar e matr ices that a re the finite discrete manifesta tio ns of the abstract op era tors U and D . So the remaining tas k befo re implementing the metho d is to deter mine what U and D should b e. Pei et al. define the matrix U 2 as follows: U 2 mn = (  m − N − 1 2  2 if m = n 0 otherwise , (13) where U 2 mn is the m th row, n th column en try of U 2 , and m, n = 0 , 1 , . . . N − 1. In tuitively , this corre- sp onds to m ultiplying every ent ry in a sig nal by the square of the corres po nding index in a centered ma n- ner (hence the − ( N − 1) / 2 ter m). (It will b e in ter- esting to con trast this with our development of the U matrix later on. W e do not ta ke fo r g r anted that U should be a simple reflection of the form of the contin uous manifestation of the U oper ator, and in- deed sho w that for a for mu lation satisfying complete structural symmetry , it should b e chosen differently .) Once U 2 is defined, w e hav e D 2 = FU 2 F − 1 by using the duality relation giv en in Eq. 5. Being the finite discrete manifestation of the abstra ct o pe r ator F , the matrix F is the standard centered DFT ma- trix. Finally , for any scaling factor M , we can fo r m ( M 4 D 2 + U 2 ), and find its eigenv ectors H p,M , after which we can easily complete the pro ces s. Mo re on the implemen tation details of this appro ach ca n b e found in [12]. 4 Hyp erdifferen tial Op erator Based M atrix Metho d It is an established fact that the scaling o pe r ator M M can be written in h yp erdifferential for m as follows in terms of the U and D op erato rs [14, 9, 15, 5]: M M = exp  − i 2 π ln ( M ) U D + D U 2  . (14) Our appro a ch is bas ed on requiring that all the dis- crete entit ies we de fine obs erve the same op e rational prop erties a nd re lationships as they do in abstract op erator form. Therefo re, we will require the dis - crete manifestations of Eq. (5) and Eq. (1 4) to have the same structure, with the abstract ope r ators be- ing replaced by ma trix op erator s. As a co nsequence, Eq. (5) will hold for finite differe nc e a nd matrix ver- sions of the D and U o pe r ators and the matrix op er - ator counterpart of M M will be M M = exp  − i 2 π ln ( M ) UD + DU 2  . (15) Thu s, to s cale a function o f a discrete v ariable, w e need to write it as a column vector and m ultiply it with the scaling ma trix M M . In or der to obtain the scaling matr ix , we need the first-order differentiation and co ordina te mult iplication matr ic es D and U and then co mpute the matrix expo ne ntial of the expr es- sion inside the pa rentheses. Therefor e our first task is to obtain the D and U matrices. F or s ignals of discre te v ariables, the closest thing to differen tiation is finite differencing. Consider the following definition: ˜ D h f ( u ) = 1 i 2 π f ( u + h/ 2) − f ( u − h/ 2) h . (16) If h → 0, then ˜ D h → D , since in this case the r ight- hand side approaches ( i 2 π ) − 1 d f ( u ) /du . Therefore, ˜ D h can be interpreted as a finite difference op erator . Now, using f ( u + h ) = exp( i 2 π h D ) f ( u ), which is another es tablished result in o p e r ator theory [1 4, 9], we express Eq. (16) in h yp erdifferential form: ˜ D h = 1 i 2 π e iπh D − e − iπh D h = 1 i 2 π 2 i sin( π h D ) h = sinc( h D ) D . (17) Note tha t if w e let h → 0 in the last equation and take the limit, we can verify that ˜ D h → D from here as well. Now, we turn our a tten tion to the task of defining ˜ U h . It is tempting to define the discrete version of the co ordinate m ultiplication matrix by simply form- ing a diagonal matrix with the diag onal ent ries b eing 5 equal to the co ordina te v a lues, with due adjustment for cen tering and discreteness, muc h a s in Eq. (1 3). How ev er, up on c loser insp ection we have decided that this could not b e taken for granted. In order to obtain the most elegant and purest formulation po s sible, we m ust be sure to main tain the structural symmetry betw een U and D in all their manifestations. There- fore, w e c ho ose to define ˜ U h such that it is related to U , in exactly the same wa y as ˜ D h is related to D : ˜ U h = sinc( h U ) U , (18) from whic h we can obser ve that a s h → 0, w e hav e ˜ U h → U , as should be. But beyond that, it is also po ssible to s how that, ˜ U h , when de fined lik e this, sa t- isfies the same dualit y expres s ion Eq. (5) satisfied b y U and D : ˜ U h = F ˜ D h F − 1 (19) T o see this, s ubstitute ˜ D h in this equation: ˜ U h = F  1 i 2 π 2 i sin( π h D ) h  F − 1 = 1 i 2 π 2 i sin( π h U ) h = sinc( h U ) U . (20) When acting on a con tin uous signal f ( u ), the opera - tor U bec omes ˜ U h f ( u ) = 1 π sin( π hu ) h f ( u ) . (21) W e o bs erve the effect is not merely multiplying with the co o rdinate v a riable. Had we de fined ˜ U h such that it corr esp onds to multiplication with the co ordinate v ar iable, we would hav e des tr oy ed the symmetry and duality b etw een U and D in passing to the discr ete world. Now, b y sa mpling Eq . (21), w e can obtain the ma- trix op era tor to act on finite discrete signals. The sample po in ts will b e ta ken as u = nh with the r a nge of n b e ing determined by whether the n umber of sam- ple N is even or o dd, and by whether w e use the or - dinary or centered s ampling scheme, a s explained in detail in Sectio n 2. Finally , we are able to wr ite the elements of the matrix U : U mn = ( √ N π sin  π N n  , for m = n 0 , for m 6 = n . (22) The next step is to obtain the D matrix. T o do so, first reca ll that Eq. 5 can also be written as D = F − 1 U F . (23) Since we wan t the finite discrete manifestations of these a bs tract op erators to also exhibit the s a me structure, we write D = F − 1 UF , (24) where F was defined in Eq. (2). Thus, we have no w obtained disc rete matr ix forms U and D o f the c o or- dinate multip lication a nd differentiation op erator s so we are finally in a p osition to calculate the discrete scaling op erato r defined in Eq. (15). Before we mov e on to n umerical results and inter- pretations, several comments will b e in or der. Fir st of all, it will be worth r ecapitulating what we did and why . As mentioned, it is tempting to define the discrete version of the co ordina te multiplication ma- trix b y simply forming a diag onal matr ix with the diagonal ent ries b eing equal to the co ordinate v alues . Then one could also hav e easily obtained the discrete version of the differentiation matr ix by using dualit y , without having to go thro ugh the circuitous route we follow ed. How ever, due to the circulant structure of the finite/p erio dic lattice asso ciated with the DFT, we susp ected this ma y not b e true and decided to beg in with the differentiation ma trix instead. The simplest wa y to de fine the finite difference op erator would be, instea d of Eq. (16), ˜ D h f ( u ) = 1 i 2 π f ( u + h ) − f ( u ) h . (25) How ev er, when discretized, the corres po nding differ- ent iation matrix would have v a lues of − 1 along the primary diagona l and v alues of 1 alo ng the diagona l adjacent to the primary , leaving us with a matrix that is not sy mmetr ic. W e r ejected this option since it would clearly not give us a pure and eleg ant for- m ulation, opting for Eq. (16) instead. How ev er, this definition, while symmetric, did not a llow us to im- mediately write a differentiation matrix, beca use it inv olv ed sample points in the middle o f the sampling int erv als, r ather than the ends. F o r tunately , the rela- tionship Eq. (1 7) betw een ˜ D h and D that we der ived 6 show ed us the wa y to define ˜ U h . The operato r ˜ U h did not exhibit the same problem of in volving sam- ple p oints in the middle that ˜ D h did, a nd could b e discretized without difficulty . It was a ls o symmetri- cal, as we desired it to b e. Once we obtained the U matrix, it was p ossible to use duality to obtain the D matrix as well. W e b elieve tha t the pr esented wa y of defining the finite matr ix forms of the co or dinate mu ltiplication and differentiation op erator s is the only w ay co ns is- ten t with the circulant structure of the DFT a nd the dual nature of these o p er ators. 5 Quan titat iv e Discussion In this section, we examine our formulation fro m a numerical pe rsp ective. W e c o nsider tw o differ- ent functions: a chirped pulse function exp( − π u 2 − j πu 2 ), deno ted by F1 , and the trap ezo ida l function 1 . 5tri( u/ 2) − 0 . 5tr i(2 u ), deno ted by F2 (tri( u ) = rect( u ) ∗ rec t( u )). W e cons idered three different v al- ues for the s c ale parameter M : 0 . 5 , 2 , 3. Analytically- derived sca le d v ersions for our t wo functions are taken as the co mparison reference. W e calc ulated normal- ized mean-squar e errors (MSE) b etw een the following vectors: (i) Reference: Samples of the co n tinuously scaled functions f ( u/ M ); (ii) Discrete scaling: The pro duct of the sa mples of the or iginal function f ( u ) with the discr e te s c aling ma tr ix. As e x plained in Sec- tion 2, results are calc ula ted for bo th c enter e d and or dinary sampling reg imes. The num ber of samples N a re taken as 128, 256, and 512. Results are ta bu- lated as pe rcentages in T ables 1 to 2. The results confirm that o ur a pproach for dis- crete scaling for mulation a pproximates the con tinu- ous scaling reasonably well. If higher a ccuracies are needed, o ne can alwa ys increa se N and make a b etter approximation as the MSE decreases with increa sing N . This is because la rge N means larger e xtent s in b oth the time and frequency domains, so that a smaller p ercentage o f the signal is left outside of these extents. Mor eov er, as expe c ted, the MSE v alues also depe nd on the signa l that is being scaled. Recall- ing the informa tion theoretic consideratio ns giv en in Section 2 , the accuracy obtained dep ends on what T able 1: Percentage MSE Sco res - Chirp ed Pulse Parameter M N Ce ntered Ordinary 2 128 1 . 22 × 10 − 2 1 . 22 × 10 − 2 256 3 . 1 × 10 − 3 3 . 1 × 10 − 3 512 7 . 75 × 10 − 4 7 . 75 × 10 − 4 3 128 6 . 36 × 10 − 2 6 . 36 × 10 − 2 256 1 . 62 × 10 − 2 1 . 62 × 10 − 2 512 4 . 1 × 10 − 3 4 . 1 × 10 − 3 0.5 128 2 . 68 × 10 − 2 2 . 68 × 10 − 2 256 6 . 9 × 10 − 3 6 . 9 × 10 − 3 512 1 . 8 × 10 − 3 1 . 8 × 10 − 3 T able 2: Percentage MSE Sco res - T r ap ezoid Parameter M N Ce ntered Ordinary 2 128 0 . 33 0 . 313 256 9 . 47 × 10 − 2 0 . 103 512 2 . 91 × 10 − 2 2 . 92 × 10 − 2 3 128 1 . 62 1 . 59 256 0 . 51 0 . 53 512 0 . 15 0 . 15 0.5 128 4 . 21 × 10 − 2 4 . 75 × 10 − 2 256 1 . 69 × 10 − 2 3 . 16 × 10 − 2 512 1 . 2 × 10 − 2 7 . 4 × 10 − 3 per centage of the signal energy is confined within the chosen ex ten ts in the time and fre quency domains. F or example, MSE v a lues for F2 are relatively higher than those of F1. This is caus e d by the fact that its frequency domain conten t is sprea d over a rela- tively g r eater extent, leading to a greater per c en tage of its energy to fall outside the chosen extents. As can b e o bserved, use of either the cen tered or o rdi- nary appro aches gives similar results, as this c hoice do es not make any es sential difference w ith regards to accura cy . 7 6 Conclusion In this paper, a form ulation for s caling of discr ete- time signa ls based on hyperdiffere n tial op erator the- ory is pr op osed. F or finite-length sig nals of a dis- crete v a r iable, a unita r y scaling matrix is obtained so that the scaled version can b e obtained by a di- rect matrix multiplication. Giv en the vector holding the samples of the unsca led signal, this sc aling ma- trix m ultiplies the input vector to obtain the samples of the sca le d s ignal. W e a lso discussed tw o different approaches to indexing the discrete signals, namely or dinary indexing and c enter e d indexing. These in- dexing approaches a re fully consistent with the well- known ordinary a nd centered discr ete F ourier trans- form (DFT) definitions. F urther mo re, the pro po sed formulation is mathematica lly elegant, pure and uses self-consistent co ordinate multiplication and differen- tiation op er ations. If needed, depending on the a ppli- cation, the accura cy o f the r esulting metho d can be improv ed by using co or dinate multiplication and dif- ferentiation matrices tha t are obta ined by brute for ce nu merical a pproximations to the co n tinuous do main. How ev er, in this paper our purp ose was to demon- strate these matric es in their purest for ms without any numerical approximation. 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