Wireless Communication Technologies

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1 Wirlss Commuicatio chologis Rutgrs Uivrsity Dpt. of Elctrical ad Computr Egirig ECE559 (Advacd opics i Commuicatio Egirig Lctur & (Fruary 7 & March 4, Istructor: Dr. araya B. Madayam Summary y Di Wu (diwu@wila.rutgrs.du Lctur Miimum Shift Kyig Miimum shift yig (MSK is a spcial typ of cotiuous phas-frqucy shift yig (CPFSK with h.5. A modulatio idx of.5 corrspods to th miimum frqucy spacig that allows two FSK sigals to cohrtly orthogoal, ad th am miimum shift yig implis th miimum frqucy sparatio (i.. adwidth that allows orthogoal dtctio. MSK has o of two possil frqucis ovr ay symol itrval: S( t πx π Acos[( f c t π π + + x ( x ] ( xcss _ phas I traditioal FSK w us sigals of two diffrt frqucis of ad f to trasmit a mssag m or m ovr a tim of scods, S S f E t cos(πf t (. ( t E t cos(πf t (.3 ( t W assum that f > f >. If w choos th frqucis so that i ach tim itrval thr is a itgr umr of priods,, f ; f Figur. Sigals with diffrt dgrs of discotiuity

2 With ad itgrs, th sigal is guaratd to hav cotiuous phas. Figur shows a xampl of a sigal that is discotiuous, a sigal with discotiuous phas ad a sigal with cotiuous phas. As phas-cotiuous sigals i gral hav ttr spctral proprtis tha sigals that ar ot phas-cotiuous, w prfr to trasmit sigals that hav this proprty. If ithr or f ar chos such that thr is a o-itgr umr of priods th traditioal f FSK modulator will output a sigal with discotiuitis i th phas. I ordr to maitai phas cotiuity, w ca lt th trasmittr hav mmory. W choos th sigals for a gral CPSFSK trasmittr to E S ( t cos(π ft + θ E S ( t cos(π ft + θ ( ( t t (.4 (.5 W p th phas cotiuous y lttig θ ( qual to th argumt of th cosi puls for th prvious it itrval. For th sigals ovr a aritrary it itrval, t < ( +, th gral phas mmory trm is θ (. Figur. Phas trllis for h/ I Figur w dpict th phas variatio ovr tim i a phas trllis, hr, w hav assumd h/ ad θ ( or θ ( π. W s that for vry multipl of th it tim th phas ca oly ta o o of two valus, th valus ig ad π for t, ad t ( +. π ± for CPFSK with dviatio ratio h/ is calld MSK. h frqucy diffrc f f that rsults from choosig h/ is th smallst possil diffrc if th sigals of th two frqucis ar to orthogoal ovr o it itrval. A xampl of a MSK sigal with ad

3 / is giv i Figur 3. Figur 3. Exampl of a MSK sigal BFSK dtctio of MSK is prhaps th most atural first choic for a dtctor pricipl. It has th sam it rror proaility as ordiary BPSK. E P Q( (.6 hat mas MSK is approximat th sam as BPSK i powr fficicy. Gaussia Miimum Shift yig (GMSK Gaussia Miimum Shift Kyig (GMSK is a modificatio of MSK (i.. CPFSK with h /. A filtr usd to rduc th adwidth of a asad puls trai prior to modulatio is calld a pr-modulatio filtr. h Gaussia pr-modulatio filtr smooths th phas tractory of th MSK sigal thus limitig th istataous frqucy variatios. h rsult is a FM modulatd sigal with a much arrowr adwidth. his adwidth rductio dos ot com for fr sic th pr-modulatio filtr smars th idividual pulss i puls trai. As a cosquc of this smarig i tim, adact pulss itrfr with ach othr gratig what is commoly calld itr-symol itrfrc or ISI. I th applicatios whr GMSK is usd, th trad-off tw powr fficicy ad adwidth fficicy is wll worth th cost. BER for GMSK is P αe Q( (.7 whr α is a costat rlatd to B. h valu of B h valus of α al. GMSK paramtr α rlatd to B 3

4 ot that th cas whr s symol itrval. B corrspods to MSK (i.. th filtr is allpass for a fixd Rcall th proaility of rror for plai MSK is giv y E P Q( (.8 By comparig it with (.7, w ca coclud that P > P GMSK MSK. his ariss from th trad off tw powr ad adwidth fficit: GMSK achivs a ttr adwidth fficicy tha MSK at th xps of powr fficicy. Error Proailitis o Flat ad Slow Fadig Chal W trasmit a sigal as: E π π (.9 s x : Si ( t cos( f ct + ( i, t < s M I flat fadig chal, th rcivd sigal is modld as: x( t g( t si ( t + w( t (. Whr, g (t is th attuatio paramtr i amplitud of sigal whil w(t is AWG with s zro-ma ad powr spctral dsity of. For slow flat fadig chal, chal chags vry slowly durig a symol itrval, (i.. <<, g(t is ffctivly costat ovr a symol duratio. s c Lt g(t α For a costat x α s x( t α si ( t + w( t (. α, ML dcodig rul still rmai sam. Optimum dtctor should miimiz ovr,,... M. Rciv structur is th sam to proct x(t oto { φ } followd y corrlatio dtctio. ypically, rror: α P M Q d i i ( t i i ( α (. is Rayligh or Rica accordig to LOS or LOS. So, th avrag proaility of P M αd i Q( fα ( dα (.3 i 4

5 Aalysis of BPSK For a spcial cas, BPSK, M, th SR is giv as E α (.4 Lt β α, w ow that β is a xpotial radom varial if α is a Rayligh distriutio. So: P βe Q( f β ( dβ (.5 I ordr to stat th distriutio of, w d its ma: E β (.6 E E[ ] E[ β ] (.7 Aftr ow its ma, w ca writ th distriutio as follows: h, r-writ (.5 as: f ( xp(, (.8 P Q( 443 u 443 dv d (.9 Itgratig y parts usig: du Q( π (. v xp( (. Sustitut: + u (. P / [ Q( ] d (.3 π 5

6 u u du + π ( + / Aalysis of BFSK For Biary Frqucy Shift Kyig (BFSK, sic E P Q( (.4 I slow flat fadig chal, th proaility of it rror is giv y: P ( (.5 + Comparig (.5 to (.3, w ca gt th coclusio that cohrt BPSK is aout 3dB ttr tha BFSK. o-cohrt Dtctio I th aov discussio w hav assumd accurat phas iformatio, howvr w must raliz that i practical coditios fadig actually dstroys all phas iformatio. hus, i practic o-cohrt modulatio may prfral. I this cas, w assum th rasmit sigal is: E Si ( t cos(πf it t (.6 Ad th rcivd sigal is giv as: hat is: E x( t cos(π fi t + θ + w( t (.7 E x( t (cos(π fi t cosθ si(πf itsiθ + w( t (.8 Whr, θ is a uow phas ad w(t is AWG with zro-ma ad assum that θ is uiformly distriutio ovr [,π ]. How to dtct S i (t? his ca accomplishd y usig a Quadratur rcivr.. W usually 6

7 Figur 4. Quadratur Rcivr Also, w ca us a vlop dtctor to achiv this aim: Figur 5. Evlop Dtctor It is asy to prov that a Quadratur rcivr ad a vlop dtctor ca implmtd itrchagaly. o-cohrt Orthogoal Modulatio Figur 6. o-cohrt Rcivr for BFSK h optimum dcisio is giv y comparig th output of th two rachs of th o-cohrt rcivr. W ca gt th P as: 7

8 P E xp( (.9 As for M-ary FSK Systm with o-cohrt dtctig, P M ( i E ( xp( ( M i i i log M (.3 h BER curvs for ocohrt M-ary FSK as a fuctio of M ad SR ar dpictd i Figur 7. Figur 7. o-cohrt FSK BER Lctur Diffrtial phas shift yig (DPSK DPSK is a o-cohrt form of PSK avoidig th d for a cohrt rfrc sigal at th rcivr. Istad, th rcivd sigal of th th symol itrval is compard to th phas of th rcivd sigal of th th ( symol itrval. his mthod of modulatio is appropriat i th prsc of slow fadig whr th diffrc tw two symol itrvals is small. Grat { diffrtially codd squc from {m as follows. Sum d ad m modulo. d } }. St to th complimt of rsult of stp. 8

9 3. Us d to shift carrir phas (i.. d, θ ; d, θ π. m d d θ π π al. DPSK Carrir Phas Chag Procss Sic w us formula (. to grat d d (. m d So, symol d is uchagd from prvious symol, if th icomig symol is ''. Othrwis, it will chagd. DPSK sigal ovr a itrval ar S ( t E E cos(πf t, t cos(πf t, c c t (. S ( t E E cos(πf t, t c cos(πf t + π, c t (.3 Ovr itrval, S( t S( t, so w ca viw DPSK as a o-cohrt orthogoal modulatio. From (.4, (.5, (.6, w compard th prformac of thr iary sigalig schms, th rsults ar dpictd i Figur 8. P DPSK E E xp( xp( (.4 BPSK E P Q( (.5 P C FSK E xp( (.6 9

10 Figur 8. Compariso of Biary Sigalig Schms Osrv that for a it rror rat of P 3 th diffrc i SR tw BPSK ad BDPSK is lss tha 3 db ad that his diffrc coms lss tha db at a P 5. W ca DPSK BPSK coclud that, at a high SR, P P. Digital Sigalig Ovr Frqucy Slctiv Fadig Chals h iformatio sigal ovr a commuicatio chal is modld as v ( t A ( t, (.7 x For our aalysis, w will rstrict ourslvs to liar modulatio schms (i.. iformatio squc is maipulatd through liar opratios oly: ( t, x xha ( t (.8 { } whr x is th complx symol squc ad h a (t dots th liar modulatio opratio. h iformatio sigal trasmittd through a commuicatio chal th rcivd complx sigal c(t rsults i ( t ω x h( t + z( t (.9

11 whr z(t is a sampl fuctio of a Additiv Whit Gaussia ois (AWG procss with zro ma ad powr spctral dsity ad h(t dots th tim covolutio of th chal impuls rspos ad th liar modulatio: h( t ha ( τ c( t τ dτ (. For causal chals, this itgral is ozro oly for tim t gratr tha zro. W furthr assum that th lgth of th filtr is fiit, maig that oudd tim itrval L: h(t is ozro oly for a h ( t for t ad h( t for t L (. h forgoig procss is dscrid i Figur 9. Figur 9. Matchd Filtr i a Additiv ois Chal I ordr to uild a matchd filtr to corrctly dtct th iformatio w d to hav th chal impuls rspos c(t. If w ow h(t, th matchd filtr ca implmtd as follows: ( t v( y( t x f + t (. whr f (t is th composit puls rspos ad v (t is th filtrd ois. hs two compots ar giv y: * f ( t h ( τ h( τ + t dτ * v ( t h ( τ z( τ + t dτ (.3 (.4 h rcivr th sampls th output of th matchd filtr y(t to gt y giv as:

12 y y( x { f + x f + v { Dsird _ Sigal 443 ISI ( must _ lt _ it ois_ trm o achiv th sam prformac as i AWG, th ISI trm must zro:, i (.5 x f f δ f δ i (.6, i If w mt yquist s critrio, th ISI portio would zro. I ordr to mt this coditio w must ow th chal impuls rspos c(t. Optimum Rcivr ω(t ca rprstd y a st of asis fuctios φ (t as follows: ω ( t lim ω φ ( t (.7 ot that if ω(t is a radom procss. h w should us Karhu-Lov s xpasio ad th limit would i th ma ss. h mai poit is that oc w hav do th mappig from a cotiuous tim fuctio to a coutal st of sampls w ca thm cotiu with our dvlopmts i discrt spac. W thus hav: ω h + z (.8 x h z * h( t φ ( t dt (.9 * z t ( φ ( t dt (. Sic w ar worig i -dimsioal spac, w ca cotiu our dvlopmts usig -dimsioal vctors. ot that ω, ω... ω whr ω is a multivariat Gaussia with PDF ( p( ω x, H xp( ω xh (. π H ] [ h, h... h (. h (... h, 3, h,, h, h,, h,, h,, h, 3... (.3 h optimum rcivr is giv y th coditio

13 Choos x if log[ p( ω x, H ] > log[ p( ω x, H ] x x ^ arg{{ max} µ ( x} ω xh (.4 x From aov, w ca ma th followig coclusios. I ordr to implmt th optimum rcivr w must hav owldg of th which will allow us to qualiz th chal. hus, w d to stimat th chal.. A additioal prolm rsults y ispctig ^ f ( t v( y( t x f + t (.5 whr th ois fuctio * v ( t h ( τ z( τ + t dτ (.6 is Gaussia ut ot whit. hus, th ois sampls at th output of th filtr ar corrlatd. o comat th cripplig ffcts of corrlatd ois, w apply a whitig filtr to th sampld squc y. h output of th whit filtr v is giv y v L g x + η (.7 whr g( modis th filtr for th chal ad th whitig filtr. Figur. Whitig Filtr Usig aftr a Matchd Filtr 3. A third poit is that dsig of ISI filtrs is xtrmly ssitiv to timig iformatio. o ovrcom this ssitivity, w itroduc two schms: a. Puls Shapig: I th particular cas of raisd cosi pulss. W ca driv th lgth of puls y samplig at v poits.. Fractioal Samplig: Sampl output at a highr tha / rat ad you achiv lss ssitivity to timig rrors. 3

14 Equalizatio Schms W us a discrt modl for th chal dscrid i th prvious sctio. amly, w will dscri th mmory-limitd chal as a liar comiatio th dlayd chal-iputs { a } wightd y appropriat chal cofficits h }. { Discrt Chal Modl: h discrt chal modl that affcts iformatio iput sigal { a } is giv as r L a h + η (.8 h octiv of a qualizr is to dtrmi a stimat a } of th symol a } that mts a dfid st of critria. his procss is dpictd i low. { ^ { Figur. Discrt im Modl of th AWG liar chal hr ar two mai typs of qualizatio schms. Symol y symol qualizatio. Squc stimatio Symol y Symol Equalizrs Symol y symol qualizrs ca ithr liar or oliar. Zro Forcig Equalizr Equalizr cocpt i which a frqucy rspos is corrctd y procssig a sigal through th ivrs chal rspos, thus forcig itr symol itrfrc to zro ad, thortically, rmovig disprsio impairmt. Lt th output of th chal giv as for r L a h + η (.9 4

15 h output of th qualizr is giv y M aˆ c r (.3 M I a zro forcig qualizr, th qualizr cofficits c ar chos to forc th sampls of th comid chal ad qualizr impuls rspos to zro at all ut o of th spacd sampl poits i th tappd dlay li filtr. By lttig th umr of cofficits icras without oud, a ifiit lgth qualizr with zro ISI at th output ca otaid. Wh ach of th dlay lmts provid a tim dlay qual to th symol duratio, th frqucy rspos ( f H q of th qualizr is priodic with a priod qual to th symol rat rspos of th chal with th qualizr must satisfy yquist s first critrio: H /. h comid ( f H ( f, f < / (.3 ch q whr H ch ( f is th foldd frqucy rspos of th chal. hus, a ifiit lgth, zro, ISI qualizr is simply a ivrs filtr which ivrts th foldd frqucy rspos of th chal. his ifiit lgth qualizr is usually implmtd y a trucatd lgth vrsio. h zro forcig qualizr has th disadvatag that th ivrs filtr may xcssivly amplify ois at frqucis whr th foldd chal spctrum has high attuatio. h ZF qualizr thus glcts th ffct of ois altogthr ad is ot oft usd for wirlss lis. MMSE Equalizr A mor roust qualizr is th LMS qualizr whr th critrio usd is th miimizatio of th ma squar rror (MSE tw th dsird qualizr output ad th actual qualizr output. W dfi th stimatio rror: ε a { a { (.3 st _ symol stimatd _ symol h fuctio to miimizd is giv as M J mi{ E[ ε ]} mi{ E[( a c r ]} (.33 c c M h rror is miimizd y choosig { c }, so as to ma th rror vctor orthogoal to th iput squc: (i.. E [ r ], l l M. I ordr to implmt th MMSE qualizr, typically w us stpst dsct algorithms: E[ ε ( ] C ( + C ( µ C ( + µ Rε x (,, ±, ±... ± M (.34 C C ( + C ( + ε ( r (.35 5

16 I this algorithm, w d us traiig squcs to stimat ε (. Rfrc: []. Madayam: Wirlss Commuicatio chologis. Lctur ots Sprig. th [] J.G. Proais: Digital Commuicatio. 4 Editio. McGraw-Hill Ic. [3] S.G. Wilso: Digtal Modulatio ad codig. Prtic Hall. 998 [4].S.Rappaport: Wirlss Commuicatios Pricipls & Practic. Prtic Hall. 998 [5] A.Papoulis: Proaility Radom Varials ad Stochastic Procsss. McGraw-Hill