S-7.330 Rdio rnsmission nd nework ccess Exercise 1 - P1 In four-symbol digil sysem wih eqully probble symbols he pulses in he figure re used in rnsmission over AWGN-chnnel. s () s () s () s () 1 3 4 ) Conver he pulse wveforms ino signl vecors using recngulr pulses s bsis funcions. b) Skech he opimum receiver using mched filers. Drw he filer impulse responses. c) Using he vecor represenion drw he signl consellion in he r1 -r-coordine sysem nd derive he SEP-expression. d) Derive he exc BEP-expression when Gry-coding is used P In bsebnd binry rnsmission he symbol vlues -1 nd +1 hve equl probbiliy. he decision circui inpu noise is Lplce-disribued (symmericlly exponenilly disribued): 1 n pn ( ) = exp σ σ where σ is he r.m.s. noise mpliude. ) Derive he BEP-expression when he signl smples re ±s. b) Clcule how mny db beer he signl o noise rio mus be in he Lplce-chnnel hn in he Gussin chnnel on he BEP-levels 3 6 9 10, 10 nd 10. 3 6 9 Noe! Q(3.09) 10 =, Q(4.75) = 10, Q(6.00) = 10
P3 In bipolr binry sysem recngulr pulse (durion ) sequences re rnsmied hrough n AWGN-chnnel wih he pulse re 1/, nd he receiver uses mched filer. here is iming error Δ in decision smpling cusing ISI. How lrge my Δ/ be, h he degrdion should no exceed 1.0 db on he BEP-level 10-3? P4 Invesige how mny db ML-reference receiver is degrded compred o MAP-reference receiver s funcion of he occurrence probbiliy of binry 0 in he AWGN-chnnel. Give numericl resuls for he bi error probbiliies 10-3 nd 10-6. P5 In bipolr binry opimum PAM-sysem he decision hreshold wnders s funcion of ime s given in he figure below. he consellion poins re +d nd d. ) Derive he expression of he verge BEP. b) Clcule he lrges llowble vlue of he prmeer, if degrdion cused by decision hreshold wndering mus be less hn 1 db on he BEP-level 10-9. Usble formuls: x e Q( x) dx = x Q( x) + C, Q(6.00)=10-9 π x e 1 3 xq( x) 1 +... π 4 x x
P6 In bipolr binry rnsmission sysem he rnsmi filer generes recngulr wveform, nd he receiver filer is sub-opiml RC-lowpss filer wih he ime consn τ. ) Opimize he filer in single symbol rnsmission nd esime he degrdion compred o mched filer. b) In he recepion of symbol sequence iner-symbol inerference (ISI) is genered. By using gurd inervls of durion Δ he impc of ISI cn be reduced. How much mus he d re be reduced, if he pek ISI should be lower hn 1% of he signl smple vlue using he ime consn obined in pr )? c) Opimize he receiver filer king ino ccoun wih respec o pek ISI, nd clcule he degrdion compred o he single symbol mched filer receiver d) Clcule how much he d re should be reduced wih he requiremen in pr b) wih he filer ime consn from pr c). P7 A bipolr binry PAM-signl is rnsmied using recngulr pulses over n echo chnnel he impulse response of which is hc() = δ() + 0,4 δ(-), where = 1/Rb. he receiver filer is mched o he rnsmied pulse ssuming he chnnel o be idel wih he F impulse E response hc() = δ(), giving he bi error probbiliy Pb = Q x H G I No K J, where Ex is he recngulr pulse energy nd No/ he wo-sided power specrl densiy of he AWGN. he receiver mkes symbol by symbol decisions. ) Skech he wveform of he decision circui inpu pulse. b) Derive he BEP-expression in he echo chnnel. c) Esime he performnce degrdion in db cused by he echo. P8. A digil rnsmission sysem in n AWGN-chnnel is power-limied bu no bndwidh-limied. he received power is 1 pw nd he onesided noise power specrl densiy is 10-0 W/Hz. Using orhogonl signlling where he number of symbols M is ineger power of wo: ) Deermine he minimum M o rnsmi 10 Mbi/s if he BEPrequiremen is 10-6. b) Clcule how mny imes mus he signl bndwidh be incresed from he vlue in ) when he bi re is doubled bu he BEP requiremen remins unchnged?
P9 A rised-cosine specrum pulse hs hlf mpliude bndwidh of 100 Hz nd he specrum upper limi is 1500 Hz. he pulse is used for rnsmission of 4-se digil signl. ) Deermine he roll-off fcor. b) Wh is he bi re of he rnsmied signl? he signl is rnsmied over muliple echo chnnel wih he impulse response h ( ) = 0. δ ( + ) + δ( ) + 0.4 δ( ) 0.3 δ( ) where is he inverse of he symbol re. c) Which is he symbol sequence cusing he mximum inersymbol inerference (ISI), nd wh is he mgniude of he mximum ISI? Smpling is ssumed o ke plce sme ime insn s in he idel chnnel. d) Wh is he occurrence probbiliy of he symbol sequence cusing he mximum ISI? Homework 1. Submi your soluion les on 15 h December, 005 3 s 1 () s () s 3 () s 4 () 3 / - - s 5 () s 6 () s 7 () s 8 () / - - - -3-3 In he rnsmission of symbol king 8 differen vlues, he pulse shpes shown in he bove figure re used. ) Selec he minimum number (=) of bse funcion nd deermine nd drw he consellion digrm including he receiver decision res, when ll symbol vlues hve equl probbiliy of occurrence. b) Derive he expression of symbol error probbiliy s funcion of verge Eb/No.
c) How mny db beer is his rnsmission sysem beer hn bipolr 8PAM-sysem when he required verge energy/bi for chieving he symbol error probbiliy 10-6 re compred? Homework δ 1 hx( ) = exp( τ ) u() τ he figure shows he bsic block digrm of bipolr binry rnsmission sysem in n AWGN-chnnel. Only single binry symbol is rnsmied. ) Presen he expression nd grph of he impulse response of mched receiver filer. b) he receiver filer is mde implemenble by removing he pr on he negive ime xis. Derive expressions for he mpliude of he signl smple nd he vrince (verge power) of he noise smple. c) How lrge mus he vlue of he smpling insn be, h he degrdion compred o n idel mched filer should no exceed 1 db? Γιϖε τηε ρεσυλτ ωιτη ηελπ οφ τηε τιμε χονσταντ τ.