Nyquist pulse shaping criterion for two-dimensional systems

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1 Nyquist pulse shapig criterio for two-dimesioal systems Citatio for published versio (APA): Yag, H., Padharipade, A., & Bergmas, J. W. M. (8). Nyquist pulse shapig criterio for two-dimesioal systems. I IEEE Iteratioal Coferece o Acoustics, Speech ad Sigal Processig Proceedigs, 8 : ICASSP ; Las Vegas, NV, March 3, 8 - April, 8 (pp ). Piscataway: Istitute of Electrical ad Electroics Egieers (IEEE). DOI:.9/ICASSP DOI:.9/ICASSP Documet status ad date: Published: //8 Documet Versio: Publisher s PDF, also kow as Versio of Record (icludes fial page, issue ad volume umbers) Please check the documet versio of this publicatio: A submitted mauscript is the versio of the article upo submissio ad before peer-review. There ca be importat differeces betwee the submitted versio ad the official published versio of record. People iterested i the research are advised to cotact the author for the fial versio of the publicatio, or visit the DOI to the publisher's website. The fial author versio ad the galley proof are versios of the publicatio after peer review. The fial published versio features the fial layout of the paper icludig the volume, issue ad page umbers. Lik to publicatio Geeral rights Copyright ad moral rights for the publicatios made accessible i the public portal are retaied by the authors ad/or other copyright owers ad it is a coditio of accessig publicatios that users recogise ad abide by the legal requiremets associated with these rights. Users may dowload ad prit oe copy of ay publicatio from the public portal for the purpose of private study or research. You may ot further distribute the material or use it for ay profit-makig activity or commercial gai You may freely distribute the URL idetifyig the publicatio i the public portal. If the publicatio is distributed uder the terms of Article 5fa of the Dutch Copyright Act, idicated by the Tavere licese above, please follow below lik for the Ed User Agreemet: Take dow policy If you believe that this documet breaches copyright please cotact us at: opeaccess@tue.l providig details ad we will ivestigate your claim. Dowload date: 8. Mar. 9

2 NYQUIST PULSE SHAPING CRITERION FOR TWO-DIMENSIONAL SYSTEMS Hogmig Yag, Ashish Padharipade ad Ja W. M. Bergmas Eidhove Uiversity of Techology, Departmet of Electrical Egieerig, 56 MB Eidhove, The Netherlads. Philips Research Europe - Eidhove, High Tech Campus, 5656 AE Eidhove, The Netherlads. h.m.yag@tue.l, ashish.p@philips.com, j.w.m.bergmas@tue.l ABSTRACT We preset Nyquist pulse shapig coditios for twodimesioal (-D) digital commuicatio systems to be free from iter-symbol iterferece, for geeral regular samplig grids. We provide examples of -D pulse fuctios satisfyig these Nyquist coditios. I particular, we show that a family of -D pulse fuctios that we costruct as separable pulse fuctios from oe-dimesioal Nyquist- pulse fuctios obey the -D Nyquist criterio ad moreover iclude the oe with miimum support area i the frequecy domai. Idex Terms -D digital commuicatio systems, -D iter-symbol iterferece, Nyquist pulse shapig criterio, - D Nyquist- pulse fuctios. INTRODUCTION The Nyquist pulse shapig criterio i digital commuicatios provides ecessary ad sufficiet coditios o the pulse fuctio such that o iter-symbol iterferece (ISI) results. I particular, for a oe-dimesioal (-D) digital commuicatios system, the Nyquist criterio [] states that Theorem The sufficiet ad ecessary coditio for the pulse fuctio x(t) to satisfy the zero ISI coditio x(t )=δ() () is that its Fourier trasform X(f) satisfies m= X(f + m/t )=T, () where δ( ) is the Kroecker delta fuctio ad T is the samplig period. The pulse fuctios that satisfy Theorem are kow as -D Nyquist- pulse fuctios. I practice, real-valued pulse fuctios x(t) are of iterest, for which X (f) =X( f). A particular example of a real-valued -D Nyquist- pulse fuctio is the raised cosie fuctio, give by x β,t (t) =sic(πt/t) cos(πβt/t) β t /T, (3) where β, with β, is the roll-off factor. The Fourier trasform of x β,t (t) is T, ( f β X β,t (f)= T + T cos πt β β ( f ), ) β +β < f, elsewhere. () Note that β = results i the miimum badwidth pulse fuctio, which is x T (t) =sic(πt/t), with Fourier trasform X T (f) =T, for f, ad zero, elsewhere. I this paper, we cosider -D systems, ad preset couterpart results to Theorem. Iterest i characterizatio of ISI ad associated sigal processig techiques has resulted from recet work o two-dimesioal storage systems [ 5]. However, to our kowledge, a Nyquist pulse shapig criterio for such -D systems has ot bee addressed i literature. I this paper, we derive -D Nyquist pulse shapig coditios for regular -D samplig grid shapes ad preset families of pulse fuctios that satisfy this criterio. Oe costructio is provided based o -D Nyquist- pulse fuctios, ad we show that such -D Nyquist pulse fuctios have miimum support area i the frequecy domai. We specialize our results to two commoly used samplig grids - rectagular ad hexagoal. Hexagoal samplig, for istace, has attracted attetio i applicatios i storage systems give the storage desity improvemet uder this grid shape. The rest of the paper is orgaized as follows. We develop the Nyquist ISI-free criterio for -D digital commuicatio systems i Sectio. A geeral approach to costruct a class of -D Nyquist- pulse fuctios is preseted i Sectio 3 for geeral samplig grids. Sectio cocludes this paper. I dealig with -D sigal aalysis, we follow the otatios from [6] /8/$5. 8 IEEE 373 ICASSP 8

3 . -D NYQUIST PULSE SHAPING CRITERION I a -D digital commuicatio system, trasmitted symbols are arraged o a ifiite plae, with their locatios prescribed by a regular grid. At the receiver, each symbol results i a pulse cetered at the symbol locatio. There is possible overlap amog the pulses. The receiver i tur samples at the symbol locatios. A -D discrete sigal is represeted by samples {ỹ(, )}, whereỹ(, ) represets the sample value at the locatio described by the ordered iteger pair (, ), where <, <. These samples are obtaied by regular samplig of a cotiuous -D sigal y(t) defied i the cotiuous (t,t ) plae, with t = V, where t =[t,t ] T, =[, ] T,adV =[v, v ] is the samplig matrix, with v ad v beig two liearly idepedet vectors. Amog the umerous choices of the samplig matrix V, two special cases correspodig to rectagular ad hexagoal samplig are of commo iterest. The distortio resultig from overlappig amog differet pulses at the samplig locatio is termed -D ISI. The -D zero ISI criterio correspods to the coditio that the received sampled sigal cotais oly the compoet from oe trasmitted pulse, or equivaletly, samplig a sigle pulse at a samplig locatio results i a -D Kroecker delta fuctio. The -D Nyquist criterio for zero ISI thus is give by the coditio: ỹ() =δ(), (5) where δ() is a -D Kroecker delta fuctio. Now, the respective discrete ad cotiuous -D Fourier trasforms, Ỹ () ad Y (), of sigals ỹ() ad y(t) are related by [6] Ỹ (V T f)= Y (f + V T ), (6) V where V, V T ad V T deote the determiat, traspose ad iverse-traspose of matrix V, respectively. Note that Ỹ (V T f) ca be cosidered to be a periodic extesio of Y (f), with the periodicity specified by the matrix V T.Takig the Fourier trasforms of both sides of (5), we get that (5) is satisfied if ad oly if Ỹ (f) = ad equivaletly Ỹ (V T f)=. The, usig (6), we get that Ỹ (V T f)= Y (f + V T )=. (7) V We thus have the followig -D Nyquist pulse shapig criterio for zero -D ISI. Theorem The sufficiet ad ecessary coditio for the pulse fuctio y(t) to achieve zero -D ISI o a regular grid defied by the matrix V is Y (f + V T )= V. (8) This theorem shows that if the sum of all shifted versios of Y (f) i a directio give by V T gives a flat spectrum, there is o ISI amog the samples of the received sigal. The pulse fuctios satisfyig the coditio i Theorem are amed -D Nyquist- pulse fuctios... -D Nyquist- Pulse Fuctios with Miimum Support Area For -D digital commuicatio systems, Nyquist- pulse fuctios with miimum badwidth are of particular iterest. As metioed earlier i the itroductio, x T (t) =sic(πt/t) is the uique real-valued -D Nyquist- pulse fuctio with miimum badwidth equal to /T. We cosider the otio of miimum support area correspodig to a -D Nyquist- pulse fuctio i order to exted the idea of miimum badwidth to the -D case. Let Y (f) be the Fourier trasform of a pulse fuctio y(t). The support of Y (f) is the defied as the set of (f,f ) such that Y (f) is o-zero; the support area is the area of this support. From Theorem, -D Nyquist- pulse fuctios with miimum support area are those for which there is o overlap amog the regularly shifted versios, specified by V T,ofY (f). The miimum regular shift is determied by V T, ad hece the miimum support area is V T =/ V. Ulike the -D case, the realvalued Nyquist- pulse fuctios with miimum support area for geeral grid shapes i the -D case may ot be uique. 3. -D NYQUIST- PULSE FUNCTIONS 3.. Rectagular Samplig Grid The rectagular samplig grid is obtaied by periodically samplig the (t,t ) plae such that sample poits are spaced T ad T apart alog the t ad t axes respectively. For the rectagular grid, the samplig matrix is give by [ ] T V rec =. (9) T From (8), the Nyquist zero-isi criterio ca be writte as Y (f +[, ] T )=T T. () T T We ow provide costructios of -D Nyquist- pulse fuctios satisfyig the zero ISI criterio, usig -D Nyquist- pulse fuctios. Separable -D Nyquist- pulse fuctios: Cosider the followig separable pulse fuctio y(t,t )=y (t )y (t ). () I this case, coditio () is satisfied if Y (f + T )= T ad Y (f + T )=T,whereY (f) ad Y (f) are the Fourier trasforms of y (t ) ad y (t ), respectively. Now 37

4 3... -D Nyquist- Pulse Fuctios with Miimum Support Area for the Rectagular Grid t (a) y(t,t ) t For the rectagular samplig grid, the pulse fuctio y(t,t ) = x T (t )x T (t ) = sic(πt /T )sic(πt /T ) with Fourier trasform Y (f,f )=X T (f )X T (f ) has a miimum support area of /(T T )=/ V rec.moregeerally, it shows that a separable class of -D Nyquist- pulse fuctios exists which icludes the oe with miimum support area. Note that, ulike the -D case, this -D fuctio with miimum support area is ot uique. The costructio of -D Nyquist pulse fuctios for geeral regular samplig grids ca be doe based o -D Nyquist pulse fuctios o a rectagular grid. We ow provide a costructio of a family of such pulse fuctios for a geeral grid shape by usig pulse fuctios that satisfy the zero ISI coditio uder rectagular samplig f.5 f Geeral Samplig Grids Let y rec (t) be a Nyquist- pulse fuctio that satisfies the zero ISI criterio uder a rectagular grid. Therefore, we must have y rec (V rec )=δ(). For a geeral grid shape characterized by V g,defie a pulse fuctio y g (t) as follows y g (t) =y rec (V rec V g t). (3) (b) Y (f,f ) Now, we have Fig.. Example of Nyquist pulse fuctios for rectagular grid, for T = T =ad β = β =. ote that, if y i (t),i=,, are -D Nyquist- pulse fuctios with zeros at i T i, i,they(t,t ) = y (t )y (t ) is a -D Nyquist pulse fuctio sice it satisfies (5). I fact, y(t,t ) is equal to zero alog the lies t = T ad t = T,for ad. Also, the correspodig Fourier trasform of y(t,t ) ca be writte as Y (f,f ) = Y (f )Y (f ),wherey (f ) ad Y (f ) are the Fourier trasforms of y (t ) ad y (t ), respectively. It is straightforward to see that (8) is satisfied for this Y (f,f ). A particular choice that arises from -D raised cosie fuctios usig (3) is y(t,t )=x β,t (t )x β,t (t ),where β ad β represet roll-off factors. The plots of y(t,t ) ad Y (f,f ) for β = β =ad T = T =are show i Fig.. Note that it is also possible to costruct -D Nyquist- fuctios that are o-separable. For istace, a family of - D o-separable fuctios ca be writte as y(t,t )=t y (t )+t y (t )+y (t )y (t ). () However, we restrict our attetio i this paper to separable pulse fuctios sice they iclude the oes with miimum support area, as show as follows. y g (V g )=y rec (V rec )=δ(). Thus y g (t) is a -D Nyquist- pulse fuctio for the grid specified by V g. Moreover, if y rec (t) is a separable pulse fuctio o the rectagular samplig grid, the y g (t) is also separable i the directios of the liearly idepedet vectors specified by V g. To see this, suppose y rec (t) is separable. The y rec (t) = y (t )y (t ).Now,deoteV rec Vg =[u, u ] T,whereu ad u are liearly idepedet vectors. The, y g (t) = y rec (V rec Vg t) = y (u T t)y (u T t), () where the last equatio holds sice y rec (t) is separable D Nyquist- Pulse Fuctios with Miimum Support Area for Geeral Grids We kow that the separable fuctio, y rec (t) = x T (t )x T (t ), is the oe with miimum support area, i.e. / V rec, for the rectagular grid. We ow show that y g (t) =y rec (V rec Vg t) also has miimum support area, i.e. / V g, for the regular grid characterized by V g. The Fourier trasforms Y g (f) ad Y rec (f) of y g (t) ad y rec (t) respectively are related as Y g (f) = Y rec((v rec Vg ) T f). V recvg 375

5 grid is give by t f (a) y(t,t ) t.5 f (b) Y (f,f ) Fig.. Example of Nyquist- pulse fuctios for hexagoal grid, for T = T = T =ad β = β =. The zeros of y(t,t ) are alog the directios defied by V hex. Thus the support area of Y g (f) is = (V rec Vg ) T.miimum support area of Y rec(f) = V rec V g. V rec = V g. (5) 3... Hexagoal Samplig Grid As a example of a geeral grid, we cosider the hexagoal grid, which is characterized by the matrix V hex, V hex = [ 3T/ 3T/ T/ T/.5 ], (6) where T is the distace betwee eighborig ode o the hexagoal grid. We follow the approach itroduced i Sectio 3. to costruct a -D Nyquist- pulse fuctio for a hexagoal grid. Usig (), a -D Nyquist- pulse fuctio for a hexagoal y(t,t )=y ( T 3T t + T T t )y ( T 3T t T T t ). (7) A particular case with y (t ) = x β,t (t ),ady (t ) = x β,t (t ), with β = β = ad T = T = T =is show i Fig.. Note that β = β =yields a real-valued -D Nyquist pulse fuctio with miimum support area.. CONCLUDING REMARKS We preseted a Nyquist pulse shapig criterio to achieve zero -D ISI i -D digital commuicatio systems. We provided families of -D Nyquist- pulse fuctios for the rectagular samplig grid ad exteded them to geeral samplig grids. I particular, a family of separable -D Nyquist- pulse fuctios was costructed from -D Nyquist- pulse fuctios, ad was show to iclude the oe with miimum support area. Thus, it is sufficiet to restrict attetio to separable pulse fuctios to achieve zero -D ISI, while havig miimum support area. A iterestig ext step would be to develop pulse fuctios that satisfy the -D Nyquist criterio ad have other desirable properties specific to practical applicatios such as storage systems. 5. REFERENCES [] J. G. Proakis, Digital Commuicatios, McGraw Hill,. [] S. Tosi, M. Power, ad T. Coway, Multitrack digital data storage: A reduced complexity architecture, IEEE Tras. Mag., vol. 3, pp., Mar 7. [3]W.M.J.Coee,D.M.Bruls,A.H.J.Immik,A.M. va der Lee, A. P. Hekstra, J. Riai, S. Va Beede, M. Ciacci, J.W.M. Bergmas, ad M. Furuki, Twodimesioal optical storage, i IEEE Iteratioal Coferece o Acoustics, Speech ad Sigal Processig (ICASSP), 5, pp [] S. Va Beede, J. Riai, ad J. W. M. Bergmas, Characterizatio of two-dimesioal digital storage systems, i IEEE Iteratioal Coferece o Acoustics, Speech ad Sigal Processig (ICASSP), 6, pp [5] S. Va Beede, J. Riai, ad J. W. M. Bergmas, Cacellatio of liear itersymbol iterferece for twodimesioal storage systems, i IEEE Iteratioal Coferece o Commuicatios (ICC), 6, pp [6] D. E. Dudgeo ad R. M. Mersereau, Multidimesioal Digital Sigal Processig, Pretice-Hall,