METHODS OF CALCULATION OF DIGITAL SIGNALS SPECTRA

Transcription

1 MEHDS F CALCLAN F DGAL SGNALS SPECA Gusv CEPCANSKY, Ldislv SCHWAZ Drmn lcmmunicins nd Mulimdi, Fculy Elcricl Enginring, nivrsiy Zilin, nivrzin, 6 Zilin, Slvki gccinsky@zznm.sk, schwrz@l.uniz.sk Absrc. h signl is hysicl brr inrmin lcric r icl nrgy, lcrmgnic r ir wvs which chngs in curs im. nly rndm signl h high which in crin im insns cn b nicid wih rbbiliy < < cnvys inrmin. h xrm css rrm h whi nis n n hnd, h vlu which cn n b nicid ll, nd cnsn r ridic drminisic signl n h hr hnd, h vlus which r knwn in ch im insn wih rbbiliy. Such signls d n crry ny inrmin. yicl inrmin brrs in lcmmunicin chniqus r digil signls h cn b clssiid s rndm ns, discr in im nd in mliud. As hy r rrmd by rin ulss wih rndm mliuds, hy cnin ridic drminisic cmnn. Bcus hy r rndm, hy cn nly b dscribd by sisicl chrcrisics s h mn vlu, h disrsin, h wr nd by h mr cmlx chrcrisic h wr scrl dnsiy h wr scrum h cn b drivd ug ls hry rndm rcsss. A simlr cs is digil signl wih ulss wih rndm mliuds wihu ny crrlin mng ulss m PAM cds. s wr scrum cn sily b drivd [], [], [7], []. is mr diicul driv h wr scr h rndm signls rrmd by ulss wih rndm mliuds nd wih crrlin mng riculr ulss. his is h ic which dls his r wih. h siml cnvluin cding, h ML 3 cd nd h AMNZ cd, rqunly usd lcmmunicin brnch [6], [], [9], ll wih h crrlin culing bwn ulss, r cnsidrd s n xml clculin h wr scrum digil signl. Furhr inrmin bu cds nd scrl nlysis cn b und in [3], [], [5]. Kywrds Digil signl, scrum, crsslk, cds, crrlin.. nrducin A digil signl crrying inrmin is crd by h rin ridiclly ring ulss crin sh h mliud which in kh ring rid is rndm vribl A k cquiring discr vlus k wih rbbiliis,, ±, ±,..., ±M/ whr M dns h cun discr ss h nnzr mliuds digil signl. cn b dscribd in h im dmin s: X å k Ak. K KK k k, whr is h uncin wih h mliud which quls nd which shs h ulss digil signl nd is h widh h uls Fig.. k? Fig. : k? X An xml digil signl. k k k h cmlx wr scrl dnsiy S rndm signl ccurring s rin ridiclly ring ulss crin sh is ccrding [7] givn by: F S å n m d v. n n h quin, ò å c v v F d, 3 ADVANCES N ELECCAL AND ELECNC ENGNEENG

2 is h Furir rnsrm h uncin, ò v d c v, is h cmlx cicin h Furir sris h nnrndm cmnn h rndm signl wih ulss which r shd ccrding h uncin, M å m A, 5 M is h mn vlu h rndm mliud A uls nd dv. is h Dirc uls. n gnrl, h crrlin bwn ulss in kh nd knh ring rid cn b xrssd s: M M N n k n å å åki k n, k M M i A A, 6 n k r n,,,..., N whr ki is h rbbiliy ccurrnc h mliud in knh uls n cndiin h n ih mliud hs ccurrd in kh uls: ki ki k n, / i. 7 h quin bwn rids nd is: N. N dns h crrlin rng, i.. h numbr ulss h my hv crrlin culing mng ch hr wihin rid. hr is n crrlin bwn ulss digil signl bu h signl hs ls nnrndm cmnn, rmul gs simlr: s F S m d v, 9 å c v v whr s is h disrsin h rndm mliud digil signl: ki s F S m c. signl cnins nly ur lrn cmnn, hn h quin cn b urhr simliid : s F S. L s cnsidr h siml cnvluin cding, h ML 3 cd nd h AMNZ cd ll wih h crrlin culing bwn ulss s n xml h clculin h wr scrum digil signl.. Cnvluin Cd h min urs cnvluin cds is dc nd crrc rrrs in h rnsmissin digil signls. Ech cding which nbls his brings rdundncy nd nnrndm lmns in h riginl rndm digil signl in h mnnr h h s sm symbls dnds n h s rvius symbls. hnks his rdundncy, hr r mr symbls ssibl r givn inu inrmin srm. h chs h righ symbl mng mr ssibl ulss h cn b kn in cnsidrin is drmind n n lgrihm bsd n h rllis digrm which is knwn r bh h rnsmir nd h rcivr. h rllis digrm in Fig. 3 cchs ll ssibl ss which h ncdr n Fig. cn hv, nd h signl n h uu h ncdr h rlcs hs ss nd h is nlrgd by rdundncy bis. Fig. : x b x x b b x Exml cnvluin ncdr. x x M å M s m A m. n h quins 5 nd, is h rbbiliy ccurrnc uls mliud. hr is nly zr discr cicin, c in h scrum rrsning cnsn cmnn in h rndm signl, hn ADVANCES N ELECCAL AND ELECNC ENGNEENG 9

3 ,,,,,,,,, b, b B,,,,,,,,,,,,,,, Fig. 3: rllis digrm h ncdr n Dig..,,,,,,,, cn b und u by h nlysis rnsiins nd ss in Dig. 3 h h cul bi is bund wih h rvius bi nd s hy cr ghr bis rm which rm ll 6 ssibl cmbinins bis r nly llwd ccrding b.. As i cn b sn rm his bl, ll cmbinins,,, cn ccur n h nd nd 3 rd lc in h bi. Bu i r ccur n h s lc, hs r r ls n h h lc. h mns h h sris bis cn b dcmsd in 3 indndn bi sris ccrding Fig. :, rndm bi sris h includs bi n h nd lc in h bi, b rndm bi sris h includs bi n h 3 rd lc in h bi, c rndm bi sris h includs bis n h s nd h lc in h bi which, cring h signl c, r crrld. c,,, b c b.: Allwd ss n h uu h ncdr s n h Fig.. Numbr Bi cmbinins ssibl llwd h mliud digil signl is rndm vribl A k which cquirs vlus: k wih h rbbiliy k / nd k wih h rbbiliy k /. hn h mn vlu h mliud will b: m A, 3 nd h disrsin: k å k k s åk k m. L s cnsidr rcngulr sh h rndm uls illing h whl rid, Fig. 5. hn: K KK. 5 h Furir rnsrm 3 h uls is: Fig. : Dcmsiin bi sris r cnvluin ncding. ADVANCES N ELECCAL AND ELECNC ENGNEENG

4 Fig. 5: cngulr uls. / / Fr n, h ur bundry h irs sum N n nd k ki k, k i A A, k ååå s hr is n crrlin bwn dcn ulss wihin rid. Similrly, r n, h ur bundry h irs sum N n nd, k ki k, k i A A, k ååå F ò d, 6 nd Furir cicins: ì ï ò v v v K c v d í N. 7 v ï î Kv ¹ h signl hs nly h cnsn cmnn c. Alrn Cmnn h Signl h quin 6 rrms h Furir rnsrm h uls. And hus h dubl sid wr scrl dnsiy h lrn cmnn h signl will b givn by h quin : S. Alrn Cmnn h Signl b his cmnn is h sm s h cmnn nd S. 9 S 3 Alrn Cmnn h Signl c hr is h crrlin culing bwn h s nd h in h rid. hrr h crrlin cicins n, n,,, 3 will b clculd ccrding h quin 6 whr N is h numbr ulss in h rid h crrlin rng. Fr n : Ak Ak Ak s m, nd s h cmlicd clculin ccrding 6 cn b vidd r n. s hr is n crrlin h s uls wih h 3 rd n, nd h nd uls wih h h n in h rid. Finlly, r n 3, h ur bundry h irs sum N n 3 nd i is h sm s bm n, s h irs sum cn b l u: A A 3 åå i i 3 3 i 3. 3 h in rbbiliy i, i,, dns h rbbiliy h simulnus ccurrnc mliuds i in h s uls nd 3 in h h uls h rid. h rndm vribls A k k, 3 nly cquir vlus k wih h rbbiliy k / nd k wih h rbbiliy k / r k, 3 wihin h rid. hn ccrding quin 7, i is: i i 3. Nw h wr scrl dnsiy h lrn cmnn cn lrdy b xrssd ug quin : 3 S å n3 n 3 n 6 6 ADVANCES N ELECCAL AND ELECNC ENGNEENG

5 6 6 6 cs 3 cs. 5 Scrum h Alrn Cmnn h Whl Signl Whn h xmind signl is cmsd 3 indndn signls, b, c, h wr scrl dnsiy h whl signl is cmsd rm h sum wr scrl dnsiis h riculr signls. h dubl sid wr scrl dnsiy h lrn cmnn h signl will hn b: S S S S3 S S S cs 3. ML 3 Cd c 3. 9 d h ncding digil signl by h ML 3 cd is usd in Ehrn n mllic shildd r unshildd wisd irs Mbs bi rs. Any 3 lvls,, cn b ssignd h lgicl ss r ccrding h llwing rul: h lgicl ccurs, hn h rnsiin rm h cul lvl h nx lvl lwys bcms. h lgicl ccurs, hn n rnsiin ks lc. h ncding is clsr xlind in Fig. 6 nd in b.. X 3 cs Fig. 6: ML 3 Cd. cs 5 Cnsn Cmnn h Signl 3. 6 As N, h scrl cmnn c is givn by h sum h riculr cmnns h signl, b nd c: c, 7 nd h scrum h cnsn cmnn h signl will b: S c m c d. 6 h Enir Scrum his is givn by h sum h wr scrl dnsiy h lrn cmnn nd h scrum h cnsn cmnn h signl. hn h rl scrum will b: b.: Ss h ML 3 cd. Prvius lvl Nx rnsiin Nw lvl r r h rndm mliud A h uls cquirs 3 discr vlus in h ML 3 cd: wih ADVANCES N ELECCAL AND ELECNC ENGNEENG

6 ADVANCES N ELECCAL AND ELECNC ENGNEENG 3 rbbiliy /, wih rbbiliy / nd wih rbbiliy / whr rbbiliy h lgicl is / nd rbbiliy h lgicl is /. h mn vlu h mliud h signl wih h ML 3 cd is zr bcus: å m, 3 nd h disrsin mliuds h zr mn vlu is: å s. 3 cn b sn rm b. h hr is ls crrlin culing bwn dcn ulss. h crrlin cicin hs h sm vlu s in 3. h crrlin cicin shll b clculd. As hr is nly h crrlin bwn dcn ulss, h crrlin rng N, h ur bundry h irs sum in quin 6 N n, s h his sum cn b mid. hn lik in 3, i is: åå i i i A A,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,. 3 Nw h rbbiliis,i,, i,,, shll b clculd. h rnsiins rm nd rm r n llwd ccrding b.. hn h rbbiliis,, nd,, r zr:,,,,. 33 An cul lvl, r sys h sm, i h bi chng rm ccurs r h bi is n chngd, h rbbiliis which r:,,,,,3,,. 35 An cul lvl, r will b chngd, i h bi chng rm ccurs r h bi sys h sm, h rbbiliis which r:,,,,,36,,,,. 37 hus h crrlin cicin will b:. 3 As h signl ds n cnin ihr nnrndm r cnsn cmnn, uing 3 w bin r h nir scrum h cd: å n n n S. cs cs. 39 h rl scrum will b: S.

7 . AMNZ Cd his cd is usd minly n h S bus h SDN bsic ccss nd slighly mdiid s h HDB 3 cd is usd in im divisin mulilx rnsmissins. Lik h ML 3 cd, i is h cd wih 3 ss whr h rndm mliud A h uls ls cquirs 3 discr vlus: wih rbbiliy /, wih rbbiliy / nd wih rbbiliy / whr rbbiliy h lgicl is / nd rbbiliy h lgicl is /. Bu hr is h crrlin culing bwn h mliuds nd, nmly, whn n mliud wih h siiv vlu hs ccurrd in h s, hn h nx mliud which will ccur r h will hv h ngiv vlu nd cnvrsly, whn n mliud wih h ngiv vlu hs ccurrd in h s, hn h nx mliud which will ccur r h will hv h siiv vlu Fig. 7, b. 3. X Fig. 7: AMNZ cd. Similrly s h ML 3 cd, h mn vlu m is ls zr h AM cd nd h disrsin mliuds s nd h cvrinc K r givn by 3. b.3: Ss h AM cd. Prvius lvl Nx rnsiin Nw lvl r L s clcul h rbbiliis,i,, i,,,. is n ssibl r h ss r ccur nx ch hr ccrding b. 3. hn h rbbiliis,, nd,, r zrs: Fuhr:.,,,,,,,,,,,,.,,. 3 h rnsiin rm h lvl h lvl r cnvrsly rm bcms whn h bi wih h lgicl ccurs. hn,,,, / /. S h crrlin cicin ccrding 3 will b:. 5 As h signl ds n cnin ny ihr nnrndm r cnsn cmnn, uing 5 in w bin r h nir scrum h cd: S å n n n cs. 6 h rl scrum will b: S S. 7 ADVANCES N ELECCAL AND ELECNC ENGNEENG

8 5. Evluin suls n Fig., h scr h cnvluin ncdd signl 9 dd lin, h ML 3 cd dshd lin nd h AMNZ cd dd nd dshd lin r cmrd wih h bsic rndm digil signl wih h sm uls sh ull lin which cn b ccrding nd xrssd s: Sr d. h whl wr signl cn b clculd by wys ihr in h im dmin: P ò X A d ò d, 9 r in h rquncy dmin by h ingrin h wr scrl dnsiy S nd/r lying Prsvl hrm. P ò S d m c å c n. 5 n Alying hs quins h wrs ll xmind signls cn b clculd: cn b inrg rm h rnsmissin in viw hw much nrgy is in h min lb which rrms h bndwidh h riculr signl. cn b shwn by h numricl cmuin h ingrl in 5 in h rng rm h 95 % h whl wr is scrd in h min scrum lb in cs h bsic rndm digil signl, h cnvluin ncdd signl nd h ML 3 cd, nd 6 % in cs h AM NZ cd. As rnsmid nrgy in runkd scrum cncrns, h AM cds r lss suibl r rnsmissin hn h hr cds xmind hr. 6. Cnclusin Knwldg scr digil signls crrying inrmin is imrn bh in rdi cmmunicins h rquncy scrum mngmn, nd in lcmmunicins drmin scrl cmibiliy vrius rnsmissin sysms cnvying inrmin hrugh hysicl circuis in mllic nwrks s h hy r inluncd ch hr by crsslks s lss s ssibl. h scrum clculin rndm digil signl lying r is qui siml. h gl his cnribuin ws shw h scrum clculin in mr cmlicd css, whn hr is crrlin culing mng ulss signl. P. 5 h mns h nir wr h xmind signls is n chngd by cding. n hl nrgy h bsic digil rndm signl nd h cnvluin ncdd signl is cncnrd in cnsn cmnn c which is: P, 9 s i cn b dducd rm quins 9, nd 5. h hr hl nrgy is scrd vr h lrnd scrum. ndm bsbnd digil signls r chrcrisd by cninuus wr scrum dnsiy curv. Du h ridic drminisic cmnn rcngulr ulss, h scrum mniss yicl min lb nd diminishing sid lbs. hr r ld nly min lbs h xmind digil signls n Fig.. h min lb n h cnvluin signl is drmd by cding. h wr scrl dnsiis curvs gin zr vlus scrl rquncis h qul ring rquncy nd is mulilis. Fig. : Scr cmrisn. Acknwldgmns h uhrs grully cknwldg sur rm h VEGA rc N. /655/ Algrihms r curing, rnsmissin nd rcnsrucin 3D img r 3D P lvisin. ADVANCES N ELECCAL AND ELECNC ENGNEENG 5

9 rncs [] PCHAL, J.: ri rvdědbnsi v sdělvcí chnic.. vydání. NADAS Prh 975, 39. [] PCHAL, J.: Signály susvy.. vydání. SNL ALFA Prh, 97, 33. [3] PAKS, G., J.: Digil Cmmunicins. 3 rd diin, McGrwHill, nc., 995, 9. SBN [] PEÁSEK, M., PCHAL, J.; ŠKP, M.,: Digiální lkmunikční chnik,. díl: Digiální zrcvání signálů. 3. vydání. C Mrcni Prh, 996,. SBN [5] NEVŘVA, P.: Anlýz signálů susv.. vydání. BEN Prh,, 67. SBN [6] FANEKVÁ, M.: Mdlvni kmunikčných sysémv v rsrdí Mlb. Cmmunicins lbx Simulink. Žilinská univrzi 3, 6. [7] XNG, F. Digil Mdulin chniqus. Arch Hus Bsn Lndn,, 653. SBN [] GAGNAE, M.: Brdbnd Lcl Ls r HighSd nrn Accss. Arc Hus Bsn Lndn, 3, 9. SBN [9] CABALLE, J. M. Gigbi Ehrn ll u. rnd Cmmunicins Ld., 5, 5. SBN [] ČEPČANSKY, G., VACLÍK, M. Skráln nlýz digiálnych signálv. EDS Žilin, 5,. Abu Auhrs Gusv CEPCANSKY ws brn n 7 h My Msr scincs M.Sc. in lcmmunicins n nivrsiy Zilin mlyd Slvk lcm s chnicl dvlmn scilis. 976 mlyd Slvk lcm s h hd h rinl rsrch cnr. Acully s n xrnl lcurr n h lcmmunicin nd Mulimdi Drmn h Elcricl Fculy, nivrsiy Zilin. Nx rmin: 9 h s dcr dgr CSc h sudy sy chnicl nivrsiy Sulc in Pris. 99 h nd dcr dgr Ph.D.. 5 h sscid rssr Elcricl Fculy, nivrsiy Zilin. Ldislv SCHWAZ ws brn n 95 in Zilin. n 97 grdud n h nivrsiy Zilin, Drmn lcmmunicins wih h Msr s dgr M.Sc.. Frm 97 99, h wrkd in h srch nsiu Cmur chniqus in Zilin s h hd h Drmn D Cmmunicins. n 96, h ws wrdd h il Ph.D. nd in 99, h bcm ull im chr nd rsrch wrkr h nivrsiy Zilin. n 999, h ws ind n ssci rssr. His min civiis r cusd n rin, rlibiliy, scuriy lcmmunicin nd cmur nwrks nd d cmmunicin. ADVANCES N ELECCAL AND ELECNC ENGNEENG 6