Phasor Representation of Sinusoidal Signals

Transcription

1 Phasor Representaton of Snusodal Sgnals COSC 44: Dgtal Communcatons Instructor: Dr. Amr Asf Department of Computer Scence and Engneerng York Unversty Handout # 6: Bandpass odulaton Usng Euler dentty e t t! #"! t + j sn! #"! Inphase (I) Component Quadrature (Q) Component he unmodulated carrer wave c( (! s represented as a unt vector rotatng n a counter-clockwse drecton at a constant rate of! radans/s. opcs:. Phasor Representaton. Dgtal odulaton Schemes: PSK FSK ASK APK (ASK/FSK) 3. Detecton of Sgnal n AWGN (Revew) 4. Coherent Detecton: Bnary PSK -ary PSK FSK Sklar: Sectons (4.4. s not covered). Phasor Representaton of Ampltude odulaton A double sde band ampltude modulated (DSB-A) sgnal s represented by! t " ( + where c( (! s the carrer sgnal and x( ( s the nformaton bearng sgnal. An equvalent representaton of DSB-A sgnal s gven by! t #[ + Re ( e t " { e #[ + ( e + e )]} he phasor representaton of the DSB-A sgnal s shown as + e " )] Phasor Representaton of Frequency odulaton () A frequency modulated (F) sgnal s represented by Assumng that the nformaton bearng sgnal x( ( the above expresson reduces to For narrow band F t + k x( t ] s f ) dt (! [! t + sn(! ] m (! ( sn( )" sn(! sn( sn( ) Re Re (! " # sn(! sn(! # m t # t " { e " e [ e " e ]} t # # " { e [ + e " e ]} he composte sgnal rotates n a counter-clockwse drecton at a constant rate of! radans/s. However the vector expands and shrnks dependng upon the term t. 3 4

2 Phasor Representaton of Frequency odulaton () he phasor representaton of a narrowband F sgnal s gven by s t # # " { e [ + e " e ]} ( Re he phasor dagram of the narrowband F sgnal s shown as Phase Shft Keyng he general expresson for -ary PSK s t + " ( ]! t! E s ( where the phase term " ( #/. he symbol energy s gven by E and s the duraton of the symbol. he waveform and phasor representaton of the -ary PSK (bnary PSK) s shown below. he composte sgnal speeds up or slows down accordng to the term t. 5 6 Frequency Shft Keyng he general expresson for -ary PSK s E t + "]! t! s ( where the frequency term! has dscrete values and phase " s a constant. he symbol energy s gven by E and s the duraton of the symbol. he frequency dfference (! +! ) s typcally assumed to be an ntegral multple of #/. he waveform and phasor representaton of the 3-ary PSK (bnary PSK) s shown below. Ampltude Shft Keyng he general expresson for -ary ASK s where the ampltude term s ( t + "]! t! E ( E ( has dscrete values and frequency! and phase " s a constant. he waveform and phasor representaton of the -ary ASK (bnary ASK) s shown below. 7 8

3 Ampltude Phase Keyng Detecton of Sgnals n Gaussan Nose he general expresson for -ary APK s s ( t + "]! t! E ( Decson Regons: Assume that the receved sgnal r( s gven by r( s ( + n( symbol where both the sgnal ampltude and phase vary wth the symbol. he waveform and phasor representaton of the -ary ASK (bnary ASK) s shown below. 9 r( + n( symbol he task of the detector s to decde whch symbol was transmtted from r(. For equ-probable bnary sgnals corrupted wth AWGN the mnmum error decson rule s equvalent to choosng the symbol such that the dstance d(rs ) r s s mnmzed. Procedure:. Pck an orthonormal bass functons for the sgnal space.. Represent s ( and s ( as vectors n the sgnal space. 3. Connect tps of vectors representng s ( and s (. 4. Construct a perpendcular bsector of the connectng lnes. 5. he perpendcular bsector dvdes D plane n regons. 6. If r( s located n R choose s ( as transmtted sgnal 7. If r( s located n R choose s ( as transmtted sgnal 8. he fgure s referred to as the sgnal constellaton Correlator Recever for -ary ransmsson () Correlator Recever for -ary ransmsson () Approach : Use correlator mplementaton of matched flter. Approach : Use Bass functons {$ (} < < N N < to represent sgnal space Decson Rule: Use sgnal s ( that results n the hghest value of z (. Each sgnal s ( s represented as a lnear combnaton of the bass functons s ( a " ( + a" ( +! + an" N (!! Decson Rule: Pck sgnal s ( whose coeffcent a j best match z j ().

4 Coherent Detecton:Bnary PSK () In coherent detecton exact frequency and phase of the carrer sgnal s known. Bnary PSK:. he transmtted sgnals are gven by s ( s ( $ E E E t + "] t + " + %]! t!! t! t + "]! t!. Pck the bass functon $ ( [ # t ] t + "!! 3. Represent the transmtted sgnals n terms of the bass functon Coherent Detecton:Bnary PSK () 4. Draw the sgnal constellaton for bnary PSK s ( R $ ( 5. Dvde the sgnal space nto two regons by the perpendcular to the connectng lne between tps of vectors s and s. 6. he locaton of the receved sgnal determnes the transmtted sgnal. R s ( s ( s ( " E! ( E! ( 3 4 Coherent Detecton:-ary PSK () -ary PSK:. he transmtted sgnals are gven by Coherent Detecton:-ary PSK () 4. Draw the sgnal constellaton for bnary PSK. he followng llustrates the sgnal constellaton for 4.. Pck the bass functon s ( [ " t + ]! t! E # # ( # ( sn t]! t! t]! t! 3. Represent the transmtted sgnals n terms of the bass functon s ( a! ( + a E ( " ) ( ) sn( "! t + E )! (! ( 5. Dvde the sgnal space nto two regons by the perpendcular to the connectng lne between tps of sgnals vectors. 6. he locaton of the receved sgnal determnes the transmtted sgnal. 7. Note that the decson regon can also be specfed n terms of the angle that the receved vector makes wth the horzontal axs. 5 6

5 Coherent Detecton:-ary PSK (3) 7