EITG05 Digital Communications

Transcription

1 Week, Lecure ETG5 Digial Communicaions Week, Lecure Signal Consellaions (. 55) ichael Lenmaier Thursday, Augus, 7 Chaer : odel of a Digial Communicaion Sysem. Signal consellaions.. Pulse amliude modulaion (PA).. Phase shif keying (PSK)... Freuency shif keying (FSK).. Pulse osiion modulaion (PP)..5 Quadraure amliude modulaion (QA).. Pulse widh modulaion (PW)..7. ulione ing: OFD Pages 55 (excluding...) Exercises: Problems.,.,.,.a,.8,.5 Augus, 7 ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure Signal consellaions n case of -ary ing, one of = k messages m[i] is ransmied by is corresonding alernaive Pulse Amliude odulaion (PA) n ulse amliude modulaion he message is maed ino he amliude only: s`() {s (),s (),...,s ()} s`()=a` g(), ` =,,..., When he message euals m[i]=j hen s j ( it s ) is sen s()=s m[] ()+s m[] ( T s )+s m[] ( T s )+ The consellaion is he se of ossible alernaives The maing defines which message is assigned o which Quesion: how should we choose he differen s? PA is a naural generaliaion of binary on-off ing and aniodal ing, which are secial cases for = A common choice are euidisan amliudes locaed symmerically around ero: A` = + + `, ` =,,..., Examle: =, A =, A =, A =+, A =+ The same consellaion {A`} `= = {±,±,±5,...,±( )} could also be used wih oher maings ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure

2 Examle of -ary PA.. Signal Consellaions Symbol Energy of PA The symbol energy of a PA is s()/a E` = Using / s` () d = Eg = 8 A` g () d s` () d we can wrie he average symbol energy as Es = Eg  P` A` `= Ofen he messages are eually likely, i.e., P` = he symmeric consellaion from above we ge Figure.7: An examle of a seuence s() of -ary PA alernaives (see A` = +, ` =in,,...,wih amliude ()` (defined (D.)) A and (.)). The ulse shaeis+ ghcs duraion T = /. The ime axis shows he ime inerval 8. Wha is he message seuence m[i]? Es = Eg assumed ha (.9) alies. Hence, he se of ossible amliudes {A }= is (from (.9)), Digial Communicaions: Week, Lecure {A }= = {±, ±, ±5,..., ±(.. Bandass Signals: Basic Conces )} = Di,j and Wih P` = = n= (s () n= i ( n= a WLP ) sj ()) d = Eg (Ai ( ) ( n = Di,j = Eg (i disance beween wo PA Aj ) (.7) ) j) a) sb () = s() cos( fc ) =  Am[i] g( (.8) Examle: been used. Assume ha s() is a -ary PA, s() =has n= Am[n] grec ( is illusraed below in he ime inerval 7. minimum Euclidean We will laer see ha he disance min Di,j = Eg (i j) (.9) i,j Di,j srongly influences he error robabiliy of he receiver s()/a ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure s() above is consan. The xdsb EXAPLE. Comare his wih Examle.7 on age 8 For his reason, euidisan consellaions are ofen used i ) cos( fc ) i= n his secial case he suared Euclidean disance Di,j in (.) is, WLP wan o ransmi s a high n many alicaions we a/ freuencies, cenered around a carrier freuency fc f [H] f [H] WLP fc fc To achieve his, we can mulily he original PA s() wih b) Lower sideband Uer sideband a sinusoidal (more abou his we see in Cha. ) Figure.: llusraing he freuency conen in x () and xdsb sc (). a) X (f ) ; b) Xdsb sc (f ). Chaer. nformaion Transmission wih Carrier odulaion... and A` = + + ` his becomes Xdsb-sc (f) = F{x ()cos(πf c)} X (f) where is used ha (see [5]), The isuared Euclidean alernaives is Bandass -ary PA (.) sc ()/A g Digial Communicaions: Week, Lecure n Examle. below, digial informaion is ransmied wih -ary PA echniue and DSB-SC modulaion (i.e. -ary bandass PA). ( + + `) = Eg E = E Euclidean disances of PA s s. ransmission (VSB), one sideband and only a fracion of he oher sideband is ransmied, see [], [], [59]. We hen obain, = k, and for ichael Lenmaier, Fall 7 (.5) xdsb ichael Lenmaier, Fall 7 / 5 7 cos( π f c ) is shown below where fc = / ichael Lenmaier, Fall 7 echniue: s() above is ransmied wih DSB-SC sc () n ), and s() / 5 7 Digial Communicaions: Week, Lecure As usual he energy Edsb imoran arameers, sc, and he energy secrum of xdsb Z sc () in (.), are

3 Phase Shif Keying (PSK) Examle of QPSK We have seen ha wih PA ing he message modulaes he amliude A` of he s`() The idea of hase shif keying ing is o modulae insead he hase n` of s`() s()/a.8... f we choose s`()=g() cos( f c + n`), ` =,,...,, f c = nr s for some osiive ineger n, hen n full cycles of he carrier wave are conained wihin a symbol inerval T s = : binary PSK (BPSK) wih n = and n = is euivalen o binary PA wih A =+ and A = = : -ary PSK is also called uadraure PSK (QPSK)....8 /T s f c = R s, n =, n = /, n =, and n = / Wha is he message seuence m[i]? ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure Symmeric -ary PSK Normally, he hase alernaives are locaed symmerically on a circle n` = ` + n cons, ` =,,...,, where n cons is a conan hase offse value f P` =, and f c R s, hen he average symbol energy is E s = E g and D i,j = E g ( cos(n i n j )) PSK has a consan symbol energy Freuency Shif Keying (FSK) nsead of amliude and hase, he message can modulae he freuency f` s`()=a cos( f` + n), ` =,,..., Amliude A and hase n are consans n many alicaions he freuency alernaives f` are chosen such ha he s are orhogonal, i.e., f n = or n = R s f` = n + ` R s s i () s j () d =, i = j / (ofen used), hen we can choose def = f + ` f D, ` =,,...,, where n and are osiive inegers ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure

4 Chaer. odel of a Digial Communicaion Sysem inerval. n, he `:h alernaive, s (), may be exressed as, Examle of -ary FSK Chaer. odel of a Digial Communicaion Sysem s () = c( ` / ) ` =,,..., (.8) -ary FSK where he ulse c() euals ero ouside he ime inerval /. Noe.8 ha alernaive s () euals ero ouside he ime slo ` /. (` + ) /. So, he message modulaes only he osiion of he ulse c(). hrough he delay alernaives ` /. Observe ha he alernaives are. orhogonal since he roduc of wo alernaives euals ero, Energy and Disance of -ary FSK.. si ()sj ()d = /T, s i = j Es = (.8) Observe ha Es is he same as for -ary PSK wih Eg = A A secial roery of FSK is ha he Euclidean disance Di,j is he same for any air (i, j) of s This means ha we can increase (and hus he bi rae Rb ) wihou increasing he error robabiliy n=, ( +) / c ( ` / )d = Ec f = Rs, f= /Rs, f = Rs, and f = Rs (.8) menioned here ha a so-called Simlex consella should alsobe ion (see refs. [8], []) is obained by subracing he average a() = P E = Ec (.85) E s = P smessage () from each of he (orhogonal) FSK alernaives s () in Wha is seuence m[i]? = he = (.8). Hence,Z{s () a()} = is hen a Simlex consellaion. Di,j = ichael Lenmaier, Fall 7 (si () sj ()) d = Ei + Ej = Ec Communicaions: Week, Lecure... -dimensional Digial vecor reresenaion of orhogonal -ary FSK n ulse osiion modulaion he message modulaes he osiion s () s () = E = s, + + (), ` =,,..., (.78) of a shor EXAPLE.ulse c()e wihin he symbol inerval + Assume ha we have sa, binary seuence of messages m[i] ha is,,,,,,. Skech + () he ransmied s() 7T s` ()in=(.) c in` he ime, inerval ` =,,..., b if binary PP (BPP)The ing is oin used. here Assume in alernaive (.8) has sa()recangular shae wih can be exressed imoran is haha eachc() by amliude eual o., The duraion of,...,s he,ulse c() wih hasobasis saisfy T Tresecively. combined funcions As coordinaes, s / before, he energy in each basis funcion is eual o one, and wo basis funcions The ulses are orhogonal and we ge Soluion: are orhogonal (see (.7)). Since he basis funcions are known, he secific The i:h message m[i] is reresened by he waveform sm[i] ( i), and = ktb = Tb. comleely defined byi + he alernaive s () sis=,,...,s,. Dand E ETc /). sso, c, = c() j =coordinaes i,j =se we find ha The wo alernaives Eare se() () = c( Noe ha for alernaive s (), all coordinaes exce ss, + are eual o s()examle: is, ero, and s = E., + Conseuenly, orhogonal -ary FSK alernaives may be described by s() using an -dimensional vecor. This is illusraed in Figure 5. on age () () () () () () () (relace he -dimensional vecor j by sj = (sj,, sj,,..., sj, )). Tb Tb Tb Tb 5 Tb Tb 7 Tb Digial Communicaions: Week, Lecure ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure Pulse Widh odulaion (PW) n ulse widh modulaion he message modulaes he duraion T of a ulse c() wihin he symbol inerval s` () = c, ` =,,..., ` The duraion of he ulse c() is eual o T = follows ha s` () is ero ouside he inerval ` is assumed ha ` < Average symbol energy: E s = Ec ` Examle: s() A Used for low-ower oical links (e.g. R remoe conrols) ichael Lenmaier, Fall 7 Wha haens wih he bandwidh W if increases? (.8) Hence, all alernaives have he same energy since a ime delayed version alernaives c( ) of a waveform c() does no change he energy conen. Signal alernaives As PP for heare revious consellaions is convenien o useused an alernaive for -ary very differen from he i alernaives in orhogonal descriion of orhogonal -ary FSK alernaives. Since wo differen sig-ary FSK. However, in erms of energy- and Euclidean disance-characerisics, nal alernaives by definiion are orhogonal, we can scale each alernaive Posiion (PP) hepulse woand consellaionsodulaion are idenical (le E c = A / and comare wih hereby obain basis funcions. So, (.75) (.77)). - A Di,j = Ei + Ej = A. in he ulse c(). Then he following resuls are Le Ec denoe he energy.8 obained, Z E = s ()d = Choosing f` Rs and he orhogonal freuency alernaives form above we ge Used in conrol alicaions, no much for daa ransmission (e.g., seed of CPU fan, LED inensiy) ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure

5 Quadraure Amliude odulaion (QA) Energy and Disance of -ary QA Wih QA ing he message modulaes he amliudes of wo orhogonal s (inhase and uadraure comonen) s`()=a` g() cos( f c ) B` g() sin( f c ), ` =,,..., Choosing f c R s i can be shown ha = A E` ` + B` E g We can inerre s`() as he sum of wo bandass PA s oivaion: We can ransmi wo s indeendenly using he same carrier freuency and bandwidh The s`() can also be exressed as s`()=g() A ` + B` cos( f c + n`) follows ha QA is a generaliaion of PSK: selecing A ` + B` = we can u he informaion ino n` and ge A` = cos(n`), B` = sin(n`) D i,j = (A i A j ) +(B i B j ) E g A common choice are euidisan amliudes locaed symmerically around ero: (wo -ary PA wih k/ bis each) n {A`} `= = {B`} o `= = ±,±,±5,...,± For eually likely messages P` =, his resuls in he average energy E s = Â `= E` = ( ) E g ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure Geomeric inerreaion is ossible o describe QA s as wo-dimensional vecors in a so-called sace For his he is wrien as s`()=a` g() cos( f c ) B` g() sin( f c ) s`()=s`, f ()+s`, f () Here s`, = A` Eg / and s`, = B` Eg / are he coordinaes The funcions f () and f () form an orhonormal basis of a vecor sace ha sans all ossible ransmi s: f ()= g() cos( f c ) Eg / This looks absrac, bu can be very useful!, f ()= g() sin( f c ) Eg / Signal sace reresenaion of QA Now we can describe each alernaive s`() as a oin wih coordinaes (s`,,s`, ) wihin a consellaion diagram -QA -QA -QA s`, = A` E g /, s`, = B` E g / The energy E` and he Euclidean disance D i,j can be deermined in he sace ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure ichael Lenmaier, Fall 7 Digial Communicaions: Week, Lecure

6 n Figure 5. examles of hese consellaions are shown in sace, for differen valuesof. Noe ha for -ary PSK he oins lie on a circle wih radius E s = Eg /. -PA -PA 8-PA Signal sace reresenaion of PSK and PA PSK 5 7 and PA can be seen as a secial cases of QA: -PSK -PSK (QPSK) (ν ) = π 8-PSK s s s s AP Receiver s for hesawgn s The 5.. Channel s s s7 s55 s consellaions are shown in n Figure 5. examles of hese sace, s 7 for differen valuesof. Noe ha for -ary PSK he oins lie on a circle wih radius E s =-FSK E -FSK g /. s`, = cos(n` ) Eg /, -PA s s -PA s s s s`, = sin(n` ) Eg / s -QA ( ν = π ) s = ( + + `) Eg Digial Communicaions: Week, Lecure 8-PA s s s s55 s s77 -QA 5 7 -FSK -FSK Examle of an OFD symbol N =5, -ary QA in Chaer each subcarrier 5) Communicaion Sysem. odel of(. a Digial An OFD symbol Figure 5.: Examles of -ary PA, -ary PSK, -ary FSK and -ary QA consellaions in sace. See also he corresonding subsecions in -QA -QA -QA Chaer..average. he..8symbol received To be able o calculae energy for he con/ sellaion { ()}=, he energy of he individual s j () firs has o be calculaed, = j ()d (5.5) E j However, here exiss an alernaive way o obain he energy Ej direcly from N An g() examle of a secific symbol x() given x()figure = Â.: (a [n] cos( fn ) OFD aq [n] g() sin( fn ))in,(.5). The Figure 5.: Examles of -ary PA,inerval -ary PSK, FSK and -ary QA figure shows x()/a over he symbol -ary. n= consellaions in sace. See also he corresonding subsecions in Chaer. n his examle he symbol x() carries = bis common roery of his kind of s is ha relaively high eaks may occur (consrucive and desrucive addiion of he N s wihin a symbol To Lenmaier, be able othis calculae hedigial received energy he ichael Fall 7 Communicaions: Week,symbol Lecure raio inerval). is referred oaverage as he eak-o-average (PAR)for roblem asso- con sellaion { ()} energy of (see he ref. individual ciaed wih mulicarrier []). s j () firs has o be =, hemodulaion calculaed, EXAPLE.5 Z Disadvanage: only one freuency can be used a he same ime Orhogonal Freuency Division ulilexing (OFD): use QA a N orhogonal freuencies and ransmi he sum OFD is widely used in modern communicaion sysems: WLAN, LTE, DAB (radio), DVB (TV), DSL Examle: N = 9 -ary QA a each freuency (carrier) Then an OFD carries 9 = 57 bis ichael Lenmaier, Fall 7 How can such a sysem be realied in racice? ) OFD will be exlained in deail in he advanced course 8-PSK Wih FSK ing, orhogonal alernaives are ransmied a differen freuencies How does a yical OFD look like? -QA ichael Lenmaier, Fall 7 s -PSK (QPSK) `, -PSK s s ulione Signaling: OFD Digial Communicaions: Week, Lecure