Representation of Band-pass Signal

Transcription

1 EGR 544 Counication heory 5. Representation of Diitally Modulated Sinals Z. Aliyaziciolu Electrical and Coputer Enineerin Departent Cal Poly Poona Suary Representation of Band-pass Sinal Band-pass sinal s(t) S( f ) f -f c f c Pre-envelope Sinal s + (t) S + ( f ) f c f Equivalent low-pass Sinal s l (t)=+j S l ( f ) f c f Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR 544-6

2 Representation of Band-pass Sinal s(t) s () t = x() t + jy() t l Band-pass sinals can be represented in three different standard notations Quadrature Notation s() t = x()cos( t π f t) y()sin( t π f t) c where and are real-valued base-band sinals, they are called the inphase and quadrature coponents of s(t) Coplex Envelope Notation j { π fct jπ fct l } { } st () = Re s() te = Re[ xt () + jyt ()] e where s l (t) is coplex envelope of s(t) c Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR Representation Diitally odulated sinal Modulator aps the diital inforation into analo wavefor that atch the characteristic of the channel It takes blocks of k=lo M binary diits at a tie fro the inforation sequence {a n } and represents one of the deterinistic value M= k. he odulated wavefor is {s (t), =1,,,M} for transission over the channel Meoryless Modulation: he appin fro sequence {a n } to the wavefors {s (t)} is perfored without any constraint on previously transitted wavefor. Meory Modulation: he appin fro sequence {a n } to the wavefors {s (t)} is perfored depend on the one or ore previously transitted wavefor. Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR

3 Meoryless Modulation Medhods Pulse-Aplitude Modulation (PAM) sinal PAM is also called Aplitude-shift Keyin ASK) PAM sinal wavefor representation jπ fct { } s () t = Re A () t e = A t ( )cos π ft, = 1,,..., M, 0 t c where { A, = 1,,..., M} denotes the set of M possible aplitude and (t) is sinal pulse shape A takes the discrete values A = ( 1 M) d, = 1,,..., M where d is the distance between adjacent sinal aplitudes Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR Pulse-Aplitude Modulation(PAM) sinal M= 0 1 M= his appin is called Gray Codin M= Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR

4 Pulse-Aplitude Modulation(PAM) sinal If R show the # bit per second {R [bit/s]}. he tie interval will be b 1 = R is called bit interval he sybol rate is R/ k k = = kb R he M PAM sinal eneries ε = sdt 0 1 = A () t dt 0 1 = Aε, then the sybol interval will be Where ε denotes the enery of pulse (t) Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR Pulse-Aplitude Modulation(PAM) sinal Let s define s (t) with unit-enery sinal wavefor s() t = s f() t where f() t = ()cos t π fct ε Unit-enery wavefor 1 s = A ε = 1,,..., M Euclidean distance between any pair of sinal points is d = ( s s ) ( e) n n 1 = ε A An = d ε n he iniu distance d d ε ( e) in = Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR

5 Pulse-Aplitude Modulation(PAM) sinal For SSB PAM can be represented j f t { ˆ } c s () t = Re A [ () t ± ()] t e π Where ˆ( t) is the Hilbert ransfor of (t) he Bandwidth is half of the DSM sinal he siple representation of s (t) s () t = A (), t = 1,,..., M Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR Diital Phase-Modulated sinals Diital PM is also called Phase-shift keyin (PSK) he M sinal wavefors can be represented in PM π ( 1) j jπ fct M s() t = Re () t e e, = 1,,..., M Or π ( 1) s() t = ()cos t π fct+, = 1,,..., M π( 1) π( 1) = ()cos t cos( π fct) ()sin t sin( π fct) Where (t) is the sinal pulse shape and θ =π (-1)/M, =1,,..,M are the M possible phases of the carrier. Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR

6 Diital Phase-Modulated sinals Diital PM sinal has sae enery and ε = 0 sdt 1 1 ε = () t dt = 0 s (t) can be expressed as a linear cobination of two orthoonal sinal s() t = s f () t + s f () t 1 1 where t = [ f1 t f t ] f( ) ( ) ( ) f( t) = ( t)cos π fct ( t)sinπ fct ε ε [ s s ] s = 1 ε π( 1) ε π( 1) s = cos sin Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR Diital Phase-Modulated sinals Sinal Space Diara of PSK M= M= QPSK M= M= π/qpsk Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR

7 Diital Phase-Modulated sinals he Euclidean distance between two sinal points are ( e) dn = s sn 1/ π ( 1) = ε 1 cos he iniu distance ( e) d in = ε 1 cos π M 1/ Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR Quadrature Aplitude Modulation he sinal wavefor is Or jπ fct { } s () t = Re[ A + ja ] () t e c s = A t ()cos π ft A t ()sin π ft, = 1,,..., M, 0 t c c s c jθ jπ fct { } s () t = Re[ V e ] () t e = Vt ( )cos( π ft+ θ ), = 1,,..., M, 0 t c where V = A + A c s θ = tan 1 A A s c Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR

8 Quadrature Aplitude Modulation s (t) can be expressed as a linear cobination of two orthoonal sinal s() t = s f () t + s f () t 1 1 where t = [ f1 t f t ] f( ) ( ) ( ) f( t) = ( t)cos π fct ( t)sinπ fct ε ε [ s s ] s 1 = s ε ε = A A c s he Euclidean distance between two sinal points are d = s s ( e) n n 1 = ε A + A + A + A [( c nc) ( s ns) ] Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR / he iniu distance d = d ε ( e) in Quadrature Aplitude Modulation Sinal space diara of QAM Cal Poly Poona Electrical & Coputer Enineerin Dept. EGR