Rise-Time Distortion of Signal without Carrying Signal

Transcription

1 Journal of Physics: Conference Series PAPER OPEN ACCESS Rise-Time Disorion of Signal wihou Carrying Signal To cie his aricle: N S Bukhman 6 J. Phys.: Conf. Ser View he aricle online for udaes and enhancemens. This conen was downloaded from IP address on //7 a 7:57

2 5h Inernaional Conference on Mahemaical Modeling in Physical Sciences (IC-MSquare 6) IOP Publishing Journal of Physics: Conference Series 738 (6) 8 doi:.88/ /738//8 Rise-Time Disorion of Signal wihou Carrying Signal N S Bukhman Samara sae universiy of archiecure and civil engineering, Russian Federaion, 443, Samara, 94 Molodogvardeiskaya sr. nik34rambler@rambler.ru Absrac. The aricle deals wih one-dimensional roblem of rise-ime disorion signal wihou carrying signal, ha aears in he saring oin inermienly, ha is signal disorion a fron edge or one of is derivaive. The auhors show ha fron edge of signal isn disored in case of roagaion in unresriced (including absorbing) area (amliude of saring signal se or of one of is derivaives doesn change) and move wih he accuracy of vacuum ligh seed. The aer roves ha i is he ime inerval shorage ha causes signal loss wih he roue exension, bu no he reducion of is saring amliude, during which fron edge of signal reains is saring value. The research resens new values for his ime inerval.. Inroducion The maer of rise-ime disorion of he signal as i roagaes in a maerial medium has go a long hisory, bu i is sill researched oday (see [-7]). The aers quoed above deal wih signal wih carrier frequency ; his aer deals wih he rising edge of video signal, i.e. signal "wihou carrier frequency". Le s consider he signal E() as i roagaes in a medium wih comlex dielecric consan () and refracive index n ( ). I s eviden ha E ( ) E( ) ex( i) d, E E( )ex( i) d, where E () is he frequency secrum of he signal in he saring oin ( z ). As he signal roagaes, wih increase of ah lengh z, boh is secrum E(, and is ime deendence E (, change. Afer he signal goes a ah he lengh of which is z in a homogeneous medium wih wave number k ( c) n( ), we have he frequency secrum of he signal: E(, F(, E( ) where F(, ex( i( c) Fdef (, where ex( i( c) is he frequency characerisic of he ideal delay line (for he ime of ligh delay"; furher, o reduce wriing, we usually imly his delay, F def (, ex i( c) n( ) is he addiional (wih bu do no exlicily wrie i), and resec o he laer) ar of frequency characerisic of he above-menioned filer, ha describes he deformaion (disorion) of he signal as i roagaes in he medium. To be secific, le s consider he case of collisional lasma; as i urns ou, basically he obained resuls are of much more general characer. For he lasma, we have he dielecric consan [8.9] Conen from his work may be used under he erms of he Creaive Commons Aribuion 3. licence. Any furher disribuion of his work mus mainain aribuion o he auhor(s) and he ile of he work, journal ciaion and DOI. Published under licence by IOP Publishing Ld

3 5h Inernaional Conference on Mahemaical Modeling in Physical Sciences (IC-MSquare 6) IOP Publishing Journal of Physics: Conference Series 738 (6) 8 doi:.88/ /738//8 i ( ), where 4 m ne - is he lasma frequency, - is he effecive collision frequency. The firs hree exansion erms F def (, in series in owers of are F def ik z k z k z 8, where k c (,. To undersand furher analysis i is necessary o recall ha in he heory of comlex variable funcions here is a heorem on he uniqueness of analyical funcion (see []) which saes ha wo analyic funcions coinciding on any finie secion of real axis, coincide on all real axis. In aricular, he analyical funcion which is idenically zero on some secion of he real axis is idenically zero on all real axis. In our case, his means ha he signal E (), which is he analyical funcion of real variable in oin z and is no idenically zero, can no become on any finie secion of real axis - in aricular, i can no saisfy he condiion E if. This means ha such signal exised, exiss and will always exis (if ), and basically i can no be used o ransmi informaion. Indeed, in any oin in sace z a any given ime he ar of he signal received a all revious imes is already available for analysis. By his ar of he signal, in accordance wih he abovemenioned heorem on he uniqueness of analyical funcion, is ime deendence a any as or fuure oin of ime can be basically reconsruced. Therefore, no signal used o ransmi informaion can be he analyical funcion on all real axis; i can coincide wih he analyical funcion only if, where - is he ime of he signal s emergence. The oin here is he oin of breach of analyiciy, which usually manifess iself as a breach of he ime deendence of he signal and (or) is derivaives. The main urose of his aer is o research, how exacly his breach of he ime deendence of he signal in he saring oin and is near neighborhood look afer he signal goes a ah he lengh of which is z in a subsance.. Resuls Le s suose he signal E () wih shar rising edge roagaes in he medium, i emerges in he saring oin z a he oin of ime ( E if, E if ). Then for he asymoics (if ) of he secrum of such signal in he saring oin z, wih hird order accuracy in owers of, we have (see Aendix) i E( ) i E( ) i E ( i 3 E ex ), where E, E( ), E - is he value of he signal and is firs derivaives on he edge, fully describing (See Aendix) he relevan erms of is secrum s high-frequency asymoics. The resul for he signal s secrum in oin z (wih hird order accuracy in owers / ) is: E( i E(, i E(, i E (, z i, ex ) (, E( ) E( ) k z E, E( ) k z E( ) k z k, E 8 3, where E, E( ), ( E ( z - are he values of he amliude of he signal and is wo firs derivaives on he edge (if ) in oin z. Le us discuss he characerisics of he changes of he rising edge of he signal, ha emerges as a jum in oin z a he oin a ime ( E if, E ( ) ), as i roagaes in he medium, ha is le us make some obvious conclusions from he above formulae.. In he case of signal amliude juming from o final value E( ) a he oin of ime in saring oin z, is amliude also jums from o he same final value E ( ) in any oher

4 5h Inernaional Conference on Mahemaical Modeling in Physical Sciences (IC-MSquare 6) IOP Publishing Journal of Physics: Conference Series 738 (6) 8 doi:.88/ /738//8 oin in sace z a he oin of ime z / c corresonding o he ligh delay ime of he signal s edge. In oher words, he amliude of he iniial jum of he signal roagaing in an arbirary medium (including an absorbing or an amlifying one) does no change, and he delay ime of is rising edge in any medium is exacly equal o he ligh delay ime. We would like o emhasize ha he delay ime of he iniial jum (ha is, he rising edge of he signal) can no be less or more han "he ligh delay", ha is, no "early" or "lae" aearance of he rising jum is ossible. The fac ha his conclusion does no refer o any aricular medium (for examle, collisional lasma), bu i refers o any arbirary maerial medium, is conneced wih he asec ha in he mos general case (see [8]), if, he dielecric consan of an arbirary medium wih an accuracy u o erms of order ( 4ne m where n - is he oal conen of free and bound elecrons in he medium) is and, if, i ends o. Of course, in any disersive medium signal deformaion occurs - i is disored (usually sreched) and is amliude decays; in an absorbing medium, in addiion o his, he signal is aenuaed (i.e., is energy is los), bu all his is no relaed o he amliude of he signal s rising edge he laer never changes. As an examle, he figure below shows he numerical resuls for he ime deendence of square wave signal (a) afer i goes ah z k in collisionless lasma (b) and in collisional lasma (c). "Ligh delay" is no shown. Line (d) shows he "linear" aroximaion, used o assess he ime inerval during which he iniial amliude of he signal remains he same. Figure. Time deendence of he signal. To esimae he duraion of he ime inerval during which he signal which emerges as a jum mainains he amliude close o ha of he iniial jum in any arbirary medium, for he energy of he signal s rising ulse we have he following aroximae esimaes: fr k z, W z fr def ( fr E ) 3 E ( ) 3 k z. You can ge evaluaion formula mc ne ulse for he characerisic weak disorion disance of he square-wave ulse. 3. Similarly le us consider he signals, on he rising edge of which here s a breach of ime deendence no of he signal as such, bu only of is derivaives of order k and higher: E if ( ) E k, E if, E ( ),, E k ( ), ( ).I is easy o show ha in E k his case, as he signal roagaes in he medium, i is he value of he jum of he derivaive ( ) ha does no change; as a resul in he arbirary medium he amliude of he rising ulse of he signal 3

5 5h Inernaional Conference on Mahemaical Modeling in Physical Sciences (IC-MSquare 6) IOP Publishing Journal of Physics: Conference Series 738 (6) 8 doi:.88/ /738//8 ( k) k k slowly decays in accordance wih he law E k fr ~ E ( k ~ k z sill deermined by he law ~. fr k z, and he duraion is Aendix Le us research he Fourier secrum E () ha exiss for a limied eriod of ime (wihin he range [, ] ) of smooh signal E() wih shar rising edge and falling edge ( E if and, E if ). Obviously E( ) E( ) ex( i) d, E E( )ex( i) d. Afer n-fold inegraion by ars ( n ), he firs of hese formulae can be wrien as an asymoic formula E( ) ex( i) i k k k E laer formula demonsraes a direc link beween he breaches of he signal on he rising and falling edges and he asymoic behaviour (if ) of is secrum. Of course, he noed connecion beween he asymoics of he signal secrum, if, and he characer of is breaches can be used "reverse" oo - he resence of relevan erms in he asymoics of he signal s secrum indicaes ha he signal iself (or is derivaives of he corresonding order) are exeriencing breaches of he corresonding rank and amliude a he relevan oins of ime. This fac allows, while researching he rising edge of he signal, limiing yourself o he minimum amoun of informaion abou he roeries of he medium, i.e. he firs few exansion erms of is refracive index in series in owers of /. References [] Sommerfeld A 94 Ann. Phys [] Brillouin L 94 Ann. Phys [3] Maske B and Segard B Phys. Rev. A [4] Maske B and Segard B 3 Phys. Rev. A [5] Akulshin A M, McLean R J Journal of Oics 4 [6] Malykin G B, Romanes E A Oics and Secroscoy 9 [7] Bukhman N 7 J. Commun. Technol. Elecron [8] Landau L D and Lifshis E M 96 Elecrodynamics of Coninuous Media (Oxford: Pergamon Press). [9] Vinogradova M B, Rudenko O V and Sukhorukov A P 979 The Wave Theory (Moscow: Nauka) [] Smirnov V I The higher mahemaics course 974 vol 3 ar (Moscow: Nauka) 7. The 4