ELEG5663 Communication Theory Ch. 4 Bandpass Modulation and Demodulation
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1 Deparmen of Elecrical Engineering Univeriy of Arkana ELEG5663 Communicaion heory Ch. 4 Bandpa Modulaion and Demodulaion Dr. Jingxian Wu wuj@uark.edu
2 OULINE Inroducion Bandpa Modulaion Coheren Deecion Noncoheren Deecion Complex Envelope
3 INRODUCION 3 Bandpa modulaion Recall: baeband modulaion Mapping digial ymbol ono baeband waveform pule hape ha are compaible wih channel. Bandpa modulaion he baeband pule hape are ranlaed o a higher frequency by uing a carrier wave a high frequency inuoid Why bandpa modulae? o ranmi a ignal, anenna are uually ¼ of wavelengh c / f : higher frequency maller wavelengh maller anenna. E.g. 3KHz baeband ignal, 1GHz bandpa ignal, ranlae he ignal o a pre-allocaed channel E.g. frequency diviion muliple acce FDMA
4 INRODUCION 4 Bandpa modulaion and demodulaion
5 OULINE 5 Inroducion Bandpa Modulaion Coheren Deecion Noncoheren Deecion Complex Envelope
6 BANDPASS MODULAION 6 Bandpa modulaion he ampliude, frequency, or phae of a radio frequency RF carrier, or a combinaion of hem, i varied in accordance wih he informaion o be ranmied. A co[ fc ] hrough bandpa modulaion, a baeband ignal i hifed o a higher frequency. Example, ampliude hif keying. ime domain: x x cof c c Frequency domain: X c 1 f X f fc X f fc
7 BANDPASS MODULAION 7 ype of bandpa modulaion Coheren he receiver exploi knowledge of he carrier phae for deecion Phae hif keying PSK Frequency hif keying FSK Ampliude hif keying ASK Coninuou phae modulaion CPM Hybrid: Quadraure ampliude modulaion QAM Noncoheren he receiver operae wihou knowledge of he abolue value of he ignal phae Differenial phae hif keying DPSK Frequency hif keying FSK Ampliude hif keying ASK Coninuou phae modulaion CPM Hybrid: Quadraure ampliude modulaion QAM
8 BANDPASS MODULAION: VECOR REPRESENAION 8 Mo bandpa modulaed ignal can be repreened a I co f Q in f Orhogonal repreenaion of ignal 1 co f in f and are orhonormal 1 Proof: he bandpa modulaed ignal can be repreened a I 1 Q [ I, Q ] : here i a one-o-one relaionhip beween and he wo dimenional vecor I, Q Q I 1
9 BANDPASS MODULAION: VECOR REPRESENAION 9 Vecor repreenaion of bandpa modulaed ignal I Inphae componen: Quadraure componen: Q he inphae carrier,, and quadraure carrier, 1 co are orhonomral. he bandpa modulaed ignal can be equivalenly repreened a vecor [ I, ] Q Complex number repreenaion of bandpa modulaed ignal here i a one-o-one relaionhip beween a D vecor and complex number [, ] j I Q he bandpa modulaed ignal can be equivalenly repreened a a complex number I j Q I Q 1 in he complex number repreenaion i only ued for mahemaical convenience. Q I 1
10 BANDPASS MODULAION: PSK 1 Phae hif keying PSK E i co[ i ] i 1,, M : carrier frequency f i Each digial ymbol i mapped o a differen phae i, i 1,, M M Why E? An example of BPSK
11 BANDPASS MODULAION: FSK 11 Frequency hif keying FSK E i co[ i ] i f i Each digial ymbol i mapped o a differen frequency. M he e of ignal, }, could be orhogonal or non-orhogonal. { i i1 An example of orhogonal FSK. i 1,, M i j d E ij
12 BANDPASS MODULAION: ASK 1 Ampliude hif keying ASK Ei i co[ ] i 1,, M Each digial ymbol i mapped o a differen ampliude An example of BASK on-off keying
13 BANDPASS MODULAION: ASK 13 Ampliude phae keying APK Ei i co[ ] i 1,, M Boh ampliude and phae are alered by he digial ymbol. An example of a 8-ary APK
14 OULINE 14 Inroducion Bandpa Modulaion Coheren Deecion Noncoheren Deecion Complex Envelope
15 15 COHEREN DEECION: BPSK Recall: correlaion receiver of binary baeband ignal he oupu of correlaion receiver i he ame a he oupu of mached filer and ampler 1 a a d a d a 1 Bi error probabiliy N E Q E P d d d E 1 ] [
16 16 COHEREN DEECION: BPSK Recall: correlaion receiver of binary baeband ignal Con d Bi error probabiliy wih vecor ignal repreenaion i he quared diance beween he wo modulaion poin. E d a a 1 1 a a d d E 1 ] [ Bipolar ignal N E Q E P b Orhogonal ignal N E Q E P b
17 COHEREN DEECION: BPSK 17 Binary Phae Shif Keying BPSK E E E 1 co[ ] co[ ] co[ ] Orhogonal repreenaion 1 co in E E 1 1 Correlaion receiver 1 co 1 a1 a a d a 1 d
18 18 COHEREN DEECION: BPSK Bi error probabiliy of BPSK N E Q E P d d d E 1 ] [ d d E 1 ] [ N E Q E P b
19 COHEREN DEECION: BPSK 19 BPSK: revii maximum likelihood deecor r i n n d a n 1 z r d i 1 i a 1 E b a Eb Likelihood funcion 1 z a i p z i exp Maximum likelihood deecion rule: If z a1 z a, deec 1 If z a, deec 1 z a Equivalenly: In he orhogonal repreenaion of ignal, chooe he ignal ha ha he malle Euclidean diance wih he received ample
20 COHEREN DEECION: BPSK Example Find he expeced number of bi error made in one day by he following coninuouly operaing coheren BPSK receiver. he daa rae i 1 kbp. he inpu waveform are 1 Aco and Aco where A = mv and he double-ided noie PSD i. 1 1 W / Hz
21 Maximum Likelihood COHEREN DEECION: MAXIMUM LIKELIHOOD Maximum Likelihood Deecion Signal: Rx Signal: Correlaion receiver: i n a j1 ij r i n j 1 n z 1 z n ŝ i 1 Likelihood funcion
22 COHEREN DEECION: MAXIMUM LIKELIHOOD Maximum Likelihood Deecion Chooe ha minimize i n k1 Graphical Inerpreaion z k a ik M-ary modulaion, here are M conellaion poin: m am 1,, amn Afer correlaion deecor, here i an n-dimenion poin: z z 1,, z n i Deecion: chooe uch ha he Euclidean diance i minimized z i
23 COHEREN DEECION 3 Maximum Likelihood deecion: -Dimenion Chooe uch ha he Euclidean diance beween he vecor r and i i are minimized. Deciion region example r i ri ii rq iq Whenever he received ignal i locaed in region 1, chooe ignal Whenever he received ignal i locaed in region, chooe ignal r 1 r
24 COHEREN DEECION: MPSK 4 Muliple Phae Shif Keying MPSK i E i co M Orhogonal repreenaion of MPSK ignal i 1,, i i E co co in M M E in co i i i i E co 1 E in co M M Inphae componen i I E co M i Quadraure componen Q E in M M
25 5 COHEREN DEECION: MPSK MPSK coheren deecion Srucure of receiver Likelihood funcion I ii i n d n X 1 Q iq i n d n Y exp 1 ii i X X p exp 1 QI i Y Y p exp 1, iq ii i Y X Y X p Maximum likelihood deecion Chooe ha minimize i iq ii Y X
26 COHEREN DEECION: MPSK 6 MPSK coheren deecion Con d he diance beween r X, Y and i ii, iq d i X ii Y iq Maximum likelihood deciion rule X r d Y 1 r d Chooe i ha minimize he diance beween r X, Y and i ii, iq Equivalenly, chooe i wih phae i ha i cloe o he phae of he ignal a he oupu of he correlaor: Y Find ha minimize ˆ arcan i ˆ X
27 COHEREN DEECION: MPSK 7 Example: For a yem wih 8PSK, if he received ymbol are:
28 COHEREN DEECION: MFSK 8 Muliple frequency hif keying MFSK i E co he value of i can be choen uch ha co i 1 are muually orhogoanal orhogonal MFSK Orhogonal MFSK i a pecial cae of MFSK We are only going o examine orhogonal MFSK Orhogonal MFSK i i M i co i i j d ij i 1,, Example: Show he following yem i orhogonal BFSK if 1 1 f1 fc f fc M 1 f c
29 COHEREN DEECION: MFSK 9 Coheren receiver rucure of MFSK Srucure of a receiver i i 1 1 i im M Oupu of he correlaion deecor for MFSK r im n d i m im n m Oupu of he correlaion deecor of MFSK i coordinae in orhogonal ignal repreenaion i i, i,, 1 im r ri, ri,, r 1 im
30 COHEREN DEECION: MFSK 3 Maximum Likelihood deecion Chooe i ha minimize he Euclidean diance beween i i, i,, 1 im r ri, ri,, r 1 im Bi error probabiliy of BFSK
31 COHEREN DEECION 31 Vecor repreenaion of bandpa communicaion yem Recall vecor repreenaion of bandpa modulaed ignal 1 co f in f i ii 1 iq n ni 1 nq n n r n i I I I Q Q Q r r r I I Q Q r I ii n I r Q iq n Q he bandpa communicaion yem i equivalenly repreened a he ummaion of ignal vecor and noie vecor r i n r r I, r ] i [ ii, iq] n n I, n ] [ Q [ Q
32 COHEREN DEECION 3 Maximum likelihood deecion p r For AWGN wih wo-ided PSD he noie variance per dimenion i Likelihood funcion I ii 1 r exp S I ii n and n I Q are independen p r I, r Q ii, iq N r exp p r Q N iq r 1 I ii Q 4 Maximize p r, r, Minimize 1 r exp S Q iq p I Q ii iq ri ii rq iq Minimize he Euclidean diance beween he vecor r and iq i r i ri ii rq iq
33 OULINE 33 Inroducion Bandpa Modulaion Coheren Deecion Noncoheren Deecion Complex Envelope
34 NONCOHEREN DEECION 34 Why noncoheren deecion? Coheren deecion require he exac knowledge of he phae of he received ignal Example: BPSK he diance beween x and Rx i d A x, he ignal i Eb cof E b b A Rx, he ignal i r cof d cof d d / c : he amoun of ime he ignal ravel from x o Rx f d : he phae of he ignal a he receiver In order o perform coheren deecion, he Rx need o know can be eimaed hrough a circuiry called phae locked loop PLL E
35 NONCOHEREN DEECION 35 Why noncoheren deecion? Con d Wha if he Rx doen have he knowledge of? Example: Aume a yem operae a 1GHz. If he diance beween x and Rx i 4.75m, find ou he phae of he Rx ignal. Noncoheren deecion he Rx doen require he knowledge of he abolue phae of he Rx ignal.
36 NONCOHEREN DEECION: DPSK 36 Differenial PSK he informaion i carried by he phae difference beween he curren ymbol and he previou ymbol Recall: for coheren PSK, he informaion i carried by he abolue phae of one ymbol. Example: he kh Rx ymbol i E r cof k E he k+1h Rx ymbol i r co f k 1 he phae difference beween he wo conecuive ymbol k1 k i he informaion i carried by he phae difference he Rx doen need he knowledge of he abolue phae. he informaion i carried by he phae difference. i i M
37 NONCOHEREN DEECION: DPSK 37 Binary DPSK he eence of differenial deecion i ha he daa i carried in he phae difference beween wo conecuive ymbol x: differenial encoding; Rx: differenial decoding. Binary differenial encoding mk: informaion c k c k 1 m k ck: differenially encoded bi : modulo- addiion : complemen he informaion, mk, i carried by he difference beween ck and ck-1
38 NONCOHEREN DEECION: DPSK 38 Binary DPSK Binary differenial decoding
39 NONCOHEREN DEECION: DPSK 39 DPSK: pro and con Pro: Con: Doen require he abolue value of he ignal phae impler receiver wo noiy ignal are compared o deec he ignal here are wice a much noie a in coheren deecion he performance i wore compared o coheren deecion rade-off beween complexiy and performance
40 NONCOHEREN DEECION: FSK 4 Non-coheren deecion of FSK
41 NONCOHEREN DEECION: FSK 41 Non-coheren deecion of FSK If ha been ranmied Wha are he value a he oupu of he non-coheren deecor of FSK?
42 NONCOHEREN DEECION: FSK 4 Non-coheren deecion of FSK he minimum one pace for non-coheren orhogonal FSK
43 NONCOHEREN DEECION: FSK 43 Minimum one Spacing for Coheren Orhogonal FSK
44 NONCOHEREN DEECION 44 Coheren deecion v.. non-coheren deecion
45 OULINE 45 Inroducion Bandpa Modulaion Coheren Deecion Noncoheren Deecion Complex Envelope
46 COMPLEX ENVELOPE 46 Complex repreenaion of bandpa modulaed ignal I co f Q in f here i a one o one relaionhip beween and Complex envelope Re ~ ~ j f [ I jq ] e ~ he complex baeband ignal g i called complex envelope he envelope of he bandpa ignal. g j I he complex envelope i he ame a he vecor repreenaion of he ignal up o a caling facor, j I Q Q I Q
47 47 COMPLEX ENVELOPE Complex envelope j e g g Polar repreenaion jy x g Ampliude y x g Phae an 1 x y
48 COMPLEX ENVELOPE 48 Bandpa modulaion can be divided ino wo ep Conellaion 1. Baeband modulaion: ranfer informaion 1 and ino complex envelope Example: 8PSK Frequency upconverion Muliply he complex envelope wih j f Re{ g e } e jf
49 COMPLEX ENVELOPE 49 Quadraure implemenaion of a modulaor Baeband modulaion: Mapping and 1 o he value of x and y. Frequency upconverion: Upconvering he frequency of he baeband ignal hrough quadraure modulaion.
50 COMPLEX ENVELOPE 5 Quadraure implemenaion of a demodulaor Frequency downconverion: Downconvering he frequency of he bandpa ignal. Baeband demodulaion: Mapping he baeband ignal o and 1
51 COMPLEX ENVELOPE: D8PSK 51 D8PSK Differenial 8PSK Baeband modulaion Frequency upconverion
52 COMPLEX ENVELOPE: D8PSK 5 D8PSK Demodulaion Frequency downconverion Baeband demodulaion
53 OULINE 53 Inroducion Bandpa Modulaion Coheren Deecion Noncoheren Deecion Complex Envelope Error Probabiliy
54 54 ERROR PROBABILIY: BINARY MODULAION Comparion of error performance of binary yem BER of BPSK N E Q P b B BER of coherenly deeced, differenially encoded binary PSK 1 N E Q N E Q P b b B BER of differenially deeced, differenially encoded binary PSK DPSK exp 1 N E P b B
55 ERROR PROBABILIY: BINARY MODULAION 55 Comparion of error performance of binary yem BER for coherenly deeced binary orhogonal FSK P B Q E b N BER for non-coherenly deeced binary orhogonal FSK P B 1 exp E b N
56 ERROR PROBABILIY: BINARY MODULAION 56 Coheren deecion PSK > coheren deecion of differenially encoded PSK > Differenial deecion of differenially encoded PSK DPSK > Coheren deecion of orhogonal FSK > Noncoheren deecion of orhogonal FSK.
57 57 ERROR PROBABILIY: MPSK Symbol error rae for MPSK Fig Symbol error rae SER: # of error ymbol/# of ymbol ranmied M N E Q M P E in : ymbol energy M E E b log
58 ERROR PROBABILIY: MFSK 58 SER for MFSK P E M M 1 Q E N
59 ERROR PROBABILIY: BER V.S. SER 59 Relaionhip beween BER and SER for MPSK Fig Gray encoding: wo adjacen ymbol differ in 1 bi A high SNR, Mo of he error are he confuion beween adjacen ymbol A high SNR, 1 ymbol error approximaely correpond o 1 bi error P 1 log M B P E
60 ERROR PROBABILIY: BER V.S. SER 6 Relaionhip beween BER and SER for orhogonal ignal P M M he relaionhip i exac / 1 B P E
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