EGR 544 Communication Theory

Transcription

1 EGR 544 Commuicaio heory 7. Represeaio of Digially Modulaed Sigals II Z. Aliyazicioglu Elecrical ad Compuer Egieerig Deparme Cal Poly Pomoa Represeaio of Digial Modulaio wih Memory Liear Digial Modulaio wih Memory NRZ NRZ Daa NRZ modulaio is equivale o biary PAM or PSK. I is memoryless sysem NRZ is called differeial ecodig. If A is pulse ampliude, A chage whe daa= ad A does chage whe daa= b = a b k k k where {a k } s biary iformaio sequece, {b k } is he oupu sequece, addiio modulo Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR 544-7

2 Liear Digial Modulaio wih Memory Sae diagram for he NRZ represes he ecoder ad modulaor operaios. s() represes he waveform o rasmi he biary iformaio Daa s() for Daa ad s() for Daa /-s() /s() /s() Ipu bi/chael symbol Sae diagram, I is also called Markow Chai Sae rasiio marix for ipu S = S = S /-s() = = = /-s() /-s() /-s() /-s() Sae rasiio marix for ipu S = /s() /s() /s() /s() /-s() /-s() /-s() /-s() /s() /s() /s() /s() he rellis diagram for NRZ Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR Liear Digial Modulaio wih Memory Delay Modulaio Miller Code s () s () NRZ Sae S S S 4 S 3 S S 4 S S 4 S 3 Miller Code Daa s 4 () s 3 () s 4 ()= -s () for << S /s () /s () s 3 ()= -s () for << /s 3 () /s () S S 3 Basic Waveform for Miller Code /s () /s 4 () /s 4 () S 4 /s 3 () Sae Diagram for Miller Code Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR

3 Liear Digial Modulaio wih Memory Sae rasiio marix S () S ( ) S() S( ) = S3() S3( ) S () S ( ) 4 4 /s () S /s 3 () S /s () /s () /s 4 () /s 4 () S 4 /s () S 3 /s 3 () ps ( S, I= ) ps ( S, I= ) ps ( 3 S, I= ) ps ( 4 S, I= ) p( S S, I = ) p( S S, I = ) p( S3 S, I = ) p( S4 S, I = ) = p( S S3, I = ) p( S S3, I = ) p( S3 S3, I = ) p( S4 S3, I = ) p( S S4, I = ) p( S S4, I = ) p( S3 S4, I = ) p( S4 S4, I = ) = = Similarly Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR Liear Digial Modulaio wih Memory Sae or chael Symbol) rasiio marix Le p ij =P(S i S j ). he he marix P=[p ij ] is called he rasiio probabiliy marix which is give for delay modulaio as P p p p p p p p p = p3 p3 p33 p34 p p p p P = P( I = ) + P( I = ) Example: he rasiio marix wih equal likely symbols P()=P()=/ / / / / P = + = / / / / Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR

4 No-Liear Modulaio wih Memory Coiuous Phase FSK (CPFSK) Firs, le s look a he FSK s -(M-) () s -(M-) () s (M-) () Modulaio selecor s (), s (),.. Chael Source Oupu Chael Ecoder FSK ε s () = cos π f π( f I ) m + c where I = ±, ± 3,... ± ( M ) Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR No-Liear Modulaio wih Memory FSK sigal is geeraed by shifig carrier frequecy o represe he digial iformaio. f = f I, I =±, ± 3,..., ± ( M ) Needs o have M= k separae oscillaor o o represe each sigal. he sudde swichig from oe sigal frequecy o aoher eeds large badwidh for rasmissio o avoid he problem, coiuous-phase FSK is used Sige carrier whose frequecy is chaged coiuously his ypes of FSK sigal has memory because he phase of he carriers is cosraied o be coiuous Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR

5 Coiuous Phase FSK (CPFSK) o represe CPFSK, le s use PAM sigal for each k-bis block d () = Ig ( ) Where I represe he ampliude values ±, ±3,, ±(M-) ad each of hem maps k-bi blocks of iformaio sequece g() is recagular pulse, ampliude / ad duraio secod. d() sigal is used o frequecy modulae he carrier ad he carrier -modulaed sigal is ε s() = cos π fc+ 4 πfd d( τ) dτ is called dela fucio f d : peak frequecy deviaio he fucio d( τ ) dτ No cosiss of jump, his makes he resul phase shif wih memory Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR Coiuous Phase FSK (CPFSK) For simpliciy Le g() be a recagular pulse of ampliude / a [.] where ε s() = cos fc+ (; I) φ(; I) = 4 πf d( τ) dτ d = 4 πfd Ig( τ ) dτ ( ) = 4 πfd Ik + I, [,( + ) ] [ π φ ] = π f I + π f( ) I d k d Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR 544-7

6 Coiuous Phase FSK (CPFSK) he phase of carrier i (+) is φ(; I) = πf I + π f ( ) I d k d = θ + πhi q( ) Special case of CPM explaied i ex slide where h = Modulaio idex f d θ = πh I k Represes he accumulaor (memory) up o ime (-) ( < ) q () = / ( ) / ( > ) / / Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR Coiues-phase Modulaio (CPM) he carrier phase of coiuous-phase modulaed sigal is φ(; I) = π I h q( k) ( + ) k k where {I k } ±, ±3,, ±(M-) sequece of M-ary iformaio symbols {h k } is a sequece of modulaio idex q() is ormalized waveform shape. Whe h k is o fixed, he CPM sigal is called muli-h he ormalized waveform q() ca be represeed as q () = g( τ ) dτ CPM sigal is called full respose CPM, if g()= for > CPM sigal is called parial respose CPM, if g() for some > Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR 544-7

7 Coiues-phase Modulaio (CPM) Example of full respose CPM; Some shapes of g() ad q() / g() / q() /4 g() / q() L g () = L ( ow. ) Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR Coiues-phase Modulaio (CPM) Example of parial respose CPM;Some shapes of g() ad q() π q() g () = ( cos ) / / π q () = si 4π π g () = ( cos ) 4 q() / / π q () = si 4 4π π ( cos ) L g () = L L ( ow. ) Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR

8 Coiues-phase Modulaio (CPM) Represeaio of Coiuous-Phase Modulaio Phase rajecory of phase ree rellis Phase rellis Phase cylider Phase sae rellis Phase sae diagram Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR Coiues-phase Modulaio (CPM) Example. he case of Biary CPFSK wih I = ± ad g() is a full respose recagular fucio. he se of phase rajecories sarig = φ(; I) = πh I + πhi q( ) k Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR

9 Coiues-phase Modulaio (CPM) Example : he case of Quaerary CPFSK wih I = ± ±3 ad full respose recagular fucio, he se of phase rajecories sarig = φ(; I) = πh I + πhi q( ) k Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR Coiues-phase Modulaio (CPM) Example: A phase rajecories geeraed by he sequece I =(,-,-,-,,,,) for he parial respose π g () = ( cos ) Raised cosie of legh Phase rajecories for biary CPFSK (dashed) ad biary, parial respose CPM based o raised cosie pulse of legh 3 (solid). [From Sudberg (986), 986 IEEE.] Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR

10 Phase Sae rellis Coiues-Phase Modulaio (CPM) Simple way o represe he phase rajecories is cocer oly hose phase values a =. Rage from φ = o φ =π. πm πm ( p ) πm Θ s =,,,..., p p p πm πm (p ) πm Θ s =,,,..., p p p he maximum umber of phase sae is pm { } φ(, I) Θ =, hπ, hπ,3 hπ,... s Example: For a full respose CPM ad h=m/p m eve For m is eve here are p ermial phase sae For m is odd here are p ermial phase sae L M is alphabe size S = L pm m odd L is a ieger # exeds he pulse shape Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR Coiues-phase Modulaio (CPM) Example: he phase sae of he biary CPFSK (full respose) wih h=/ ad S =4 Differece bewee phase sae rellis ad phase rellis is: he coecio bewee sae are made by drawig sraigh lies for phase sae rellis. his is o rue for phase rajecories from oe sae o aoher Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR 544-7

11 Coiues-phase Modulaio (CPM) Phase sae diagram Oly possible phase saes ad heir rasiios are displayed Oly cocer hose phase values a = ime o appear explicily as a variable φ = φ = π φ = 3 3π φ = π Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR Miimum-shif Keyig (MSK) MSK is special case of CPFSK ad modulaio idex h=/. he phase of he carrier i he ierval (+) is φ(;i) = π Ik + πiq( ) = θ + πi( ), ( + ) / g() (+) he modulaed carrier sigal is s () = Acosπ f c + θ + πi = Acos π fc + I πi + θ 4 Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR 544-7

12 Miimum-shif Keyig (MSK) he biary CPFSK sigal will have wo frequecies i he ierval (+) = f = f c f f c he biary CPFSK sigal ca be also wrie as s A π f θ π i i i() = cos i + + ( ), =, he frequecy separaio f = f f = / his he miimum frequecy separaio ha is ecessary o esure he orhogoaliy of he sigal s() ad s() over a sigalig ierval of he legh. ha is why biary CPFSK wih h=/ is MSK Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR Miimum-shif Keyig (MSK) Also, MSK map be represes as a form of four-phase PSK. he equivale low-pass digially modulaed sigal is [ + ] v () = A I g ( ) + ji g ( (+ ) ) = π where si, [, ) g () =, ow.. Viewed as a four-phase PSK sigal wih pulse shape is oe-half cycle of siusoidal. he eve umbered biary valued symbols I of he iformaio sequece are rasmied via cos of he carrier. he odd-umbered symbols {I + } are rasmied via he si carrier Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR

13 Miimum-shif Keyig (MSK) rasmissio rae for each is / bis/ss, combie rasmissio rae will be / bi/s. MSK [ π c + π c ] s() = A I g( )cos f + I g( (+ ) )si f = s(), a cosa ampliude ad frequecy modulaed sigal Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR Miimum-shif Keyig (MSK) [ ( )cosπ c ] A I g f = I phase sigal compoe [ + ( ( + ) )siπ c ] A I g f = Quadraure sigal compoe s(), a cosa ampliude ad frequecy modulaed sigal Frequecy a [,( + ) ) fc + I 4 Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR

14 Compariso of MSK ad QPSK MSK [ π c + π c ] s() = A I g( )cos f + I g( (+ ) )si f = where π si, [, ) Coiuous phase g () =, ow.. Offse QPSK s() = A I g( )cos π f + I g( (+ ) )siπ f [ c + c ] = where, [9, ) g () = Possibly ±9 degree phase jump, ow.. a each QPSK I + I+ I I+ s() = g( )cos π fc+ g( )siπ fc = where Possibly ±9 or ±8 degree phase Cal Poly Pomoa Elecrical & Compuer jump Egieerig a each Dep. EGR Miimum-shif Keyig (MSK) QPSK ( I, I, I, I3) = ( +,, +, ) MSK : fc =.5( f = ; f =.5) -9 phase shif 9 phase shif ( I, I, I, I3) = ( +, +,, ) OQPSK : fc =.5 ( I, I, I, I3, I4, I5) = ( +, +,,, + ) QPSK : fc =.5 9 phase shif 9 phase shif 8 phase shif -9 phase shif Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR

15 φ(;i) = θ+ πi( ) for = θ + πi + πi( ) = θ ( ) + π π = π + π + θ φ(;i) = θ + πi ( ) for 3 = θ + πi + πi+ πi( ) π π π = θ + + ( ) π = π + θ π φ( ; I) φ( ; I) = π φ(3 ; I) φ( ; I) = Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR Sigal space diagram of CPM Coiues phase sigal ca o be represeed by discree pois i sigal space, like PAM, PSK I ca be described by he rajecories from oe phase sae o aoher Here is sigal phase rajecories diagram for CPFSK sigal for h=/4,h=/3,h=/, ad h=/3 H=/4 H=/3 H=/ H=/3 Cal Poly Pomoa Elecrical & Compuer Egieerig Dep. EGR