Sklar: Sections (4.4.2 is not covered).

Transcription

1 COSC 44: Dgal Councaons Insrucor: Dr. Ar Asf Deparen of Copuer Scence and Engneerng York Unversy Handou # 6: Bandpass Modulaon opcs:. Phasor Represenaon. Dgal Modulaon Schees: PSK FSK ASK APK ASK/FSK) 3. Deecon of Sgnal n AWGN Revew) 4. Coheren Deecon: Bnary PSK M-ary PSK FSK Sklar: Secons s no covered).

2 Phasor Represenaon of Snusodal Sgnals Usng Euler deny e jω cosω #"! + j sn ω #"! Inphase I) Coponen Quadraure Q) Coponen he unodulaed carrer wave c) cosω ) s represened as a un vecor roang n a couner-clockwse drecon a a consan rae of ω radans/s.

3 Phasor Represenaon of Aplude Modulaon A double sde band aplude odulaed DSB-AM) sgnal s represened by s ) cosω + cosω) where c) cosω ) s he carrer sgnal and x) cosω ) s he nforaon bearng sgnal. An equvalen represenaon of DSB-AM sgnal s gven by s ) cosω Re [ + jω jω jω { e [ + e + e )]} he phasor represenaon of he DSB-AM sgnal s shown as e jω + e jω )] he copose sgnal roaes n a couner-clockwse drecon a a consan rae of ω radans/s. However he vecor expands and shrnks dependng upon he er ω. 3

4 Phasor Represenaon of Frequency Modulaon ) A frequency odulaed FM) sgnal s represened by [ ] ω + k x s ) cos f ) d Assung ha he nforaon bearng sgnal x) cosω ) he above expresson reduces o s ) cos cos [ ] k ω + sn ω ) ω ) ) ) ) k k ω cos sn ω ) sn ω sn sn ω ) f ω f ω f For narrow band FM s ) cos Re Re ω ) βsn ω ) sn ω ) β jω β jω jω jω { e e [ e e ]} { [ ]} j ω β j ω β j ω e + e e k ω f 4

5 Phasor Represenaon of Frequency Modulaon ) he phasor represenaon of a narrowband FM sgnal s gven by s { [ ] jω β jω β jω e + e e } ) Re he phasor dagra of he narrowband FM sgnal s shown as he copose sgnal speeds up or slows down accordng o he er ω. 5

6 Phase Shf Keyng he general expresson for M-ary PSK s [ ω + φ ) ] M E s ) cos where he phase er φ ) π/m. he sybol energy s gven by E and s he duraon of he sybol. he wavefor and phasor represenaon of he -ary PSK bnary PSK) s shown below. 6

7 Frequency Shf Keyng he general expresson for M-ary PSK s E [ ω + φ] M s ) cos where he frequency er ω has M dscree values and phase φ s a consan. he sybol energy s gven by E and s he duraon of he sybol. he frequency dfference ω + ω ) s ypcally assued o be an negral ulple of π/. he wavefor and phasor represenaon of he 3-ary PSK bnary PSK) s shown below. 7

8 Aplude Shf Keyng he general expresson for M-ary ASK s where he aplude er s E ) ) cos [ ω + φ] M E ) has M dscree values and frequency ω and phase φ s a consan. he wavefor and phasor represenaon of he -ary ASK bnary ASK) s shown below. 8

9 Aplude Phase Keyng he general expresson for M-ary APK s s ) E ) [ ω + φ] M cos where boh he sgnal aplude and phase vary wh he sybol. he wavefor and phasor represenaon of he -ary ASK bnary ASK) s shown below. 9

10 Deecon of Sgnals n Gaussan Nose Decson Regons: Assue ha he receved sgnal r) s gven by r ) s ) + n ) sybol r ) s ) + n ) he ask of he deecor s o decde whch sybol was ransed fro r). For equ-probable bnary sgnals corruped wh AWGN he nu error decson rule s equvalen o choosng he sybol such ha he dsance drs ) r s s nzed. Procedure:. Pck an orhonoral bass funcons for he sgnal space.. Represen s ) and s ) as vecors n he sgnal space. 3. Connec ps of vecors represenng s ) and s ). 4. Consruc a perpendcular bsecor of he connecng lnes. 5. he perpendcular bsecor dvdes D plane n regons. 6. If r) s locaed n R choose s ) as ransed sgnal 7. If r) s locaed n R choose s ) as ransed sgnal 8. he fgure s referred o as he sgnal consellaon sybol

11 Correlaor Recever for M-ary ranssson ) Approach : Use correlaor pleenaon of ached fler. Decson Rule: Use sgnal s ) ha resuls n he hghes value of z ).

12 Correlaor Recever for M-ary ranssson ) Approach : Use Bass funcons {ψ )} < < N N < M o represen sgnal space Each sgnal s ) s represened as a lnear cobnaon of he bass funcons s ) a ψ ) + a ψ ) + + a ψ ) M % Decson Rule: Pck sgnal s ) whose coeffcen a j bes ach z j ). N N

13 Coheren Deecon:Bnary PSK ) In coheren deecon exac frequency and phase of he carrer sgnal s known. Bnary PSK:. he ransed sgnals are gven by. Pck he bass funcon s s ) ) E E [ ω + φ] [ ω + φ + π] [ ω + φ] 3. Represen he ransed sgnals n ers of he bass funcon E cos cos cos [ ω + φ] ψ ) cos s ) Eψ ) s ) Eψ ) 3

14 Coheren Deecon:Bnary PSK ) 4. Draw he sgnal consellaon for bnary PSK s ) s ) ψ ) R R 5. Dvde he sgnal space no wo regons by he perpendcular o he connecng lne beween ps of vecors s and s. 6. he locaon of he receved sgnal deernes he ransed sgnal. 4

15 5 Coheren Deecon:M-ary PSK ) M-ary PSK:. he ransed sgnals are gven by. Pck he bass funcon 3. Represen he ransed sgnals n ers of he bass funcon [ ] M s M E cos ) + ω π [ ] [ ] ω ψ ω ψ sn ) cos ) ) ) ) sn ) cos ) ) ) E E M a a s M M ψ + ψ ψ + ψ π π

16 Coheren Deecon:M-ary PSK ) 4. Draw he sgnal consellaon for bnary PSK. he followng llusraes he sgnal consellaon for M Dvde he sgnal space no wo regons by he perpendcular o he connecng lne beween ps of sgnals vecors. 6. he locaon of he receved sgnal deernes he ransed sgnal. 7. Noe ha he decson regon can also be specfed n ers of he angle ha he receved vecor akes wh he horzonal axs. 6

17 Coheren Deecon:M-ary PSK 3) 7