3F4 Digital Modulation Course

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1 3F4 Digial Modulaion Course (supervisor copy) 1 3F4 Digial Modulaion Course Nick Kingsbury February 10, 2016 Conens 1 Modulaion, Phasors and Bandlimied Noise Types of Modulaion - AM, FM, PM Imporan Feaures of Modulaion Schemes: Phasor Represenaion of Modulaed Signals Quadraure Demodulaion: Specra of Phasors One-sided and Two-sided Specra: Noise Phasors: Signal/Noise Raios for Signals and Phasors: Digial Modulaion Summary Why Digial (and no Analogue)? Some digial communicaions sysems: The Major Digial Modulaion Techniques Binary Phase-Shif Keying (BPSK) Definiion of BPSK: Power Specrum for Random Daa: Opimum Demodulaor: Bi Error Performance in Noise: Differenial Coding: Pracical Implemenaion of an Opimum BPSK Demodulaor: Approximaion Formulae for he Gaussian Error Inegral, Q(x)

2 2 3F4 Digial Modulaion Course (supervisor copy) 4 Oher Binary Schemes Quadraure PSK (QPSK): Binary Frequency-Shif Keying (BFSK): Muli-level Modulaion M-ary PSK (MPSK): Quadraure Ampliude Modulaion (QAM): M-ary FSK (MFSK): Digial Audio and TV Broadcasing Digial Audio Broadcasing (DAB) Coded Orhogonal Frequency Division Muliplexing (COFDM) Digial TV Recommended Texbooks Key Texbook: L. W. Couch, Digial and Analog Communicaion Sysems, Prenice Hall, 6h ediion, Books covering digial communicaions only: R. E. Zeimer and R. L. Peerson, Inroducion o Digial Communicaion, Prenice Hall, 2nd ediion, J. G. Proakis, Digial Communicaions, McGraw Hill, 4h ediion, I. A. Glover and P. M. Gran, Digial Communicaions, Prenice Hall, Alernaive exbooks on analogue and digial communicaions (do no cover all he course) and seleced opics: K. S. Shanmugam, Digial and Analog Communicaion Sysems, Wiley, 1979 and laer ediions. B. P. Lahi, Modern Digial and Analog Communicaion Sysems, Oxford U P, 3rd ediion, J. D. Gibson, Principles of Digial and Analog Communicaions, Prenice Hall, 2nd ediion, 1993.

3 3F4 Digial Modulaion Course Secion 1 (supervisor copy) 3 1 Modulaion, Phasors and Bandlimied Noise 1.1 Types of Modulaion - AM, FM, PM We define a general sinusoidal wave as: ampliude s() = a cos(ω C + ϕ) angular frequency of carrier phase Ampliude Modulaion (AM) Angle Modulaion Phase Modulaion (PM) Frequency Modulaion (FM) Fig 1.1 shows hese hree forms of modulaion. The riangular wave a he op is he informaion signal, and below i is he carrier wave. Then he hree modulaed waves are shown. If he informaion signal = x(), hen: For Ampl. Mod. (AM): For Phase Mod. (PM): For Frequency Mod. (FM): a = a 0 + K A x() ϕ = ϕ 0 + K P x() dϕ d = K F x() Why use Modulaion? 1. To ransmi informaion via a bandlimied channel. e.g. Compuer daa via a 300 o 3000 Hz elephone circui. 2. To pass many informaion channels via a common medium simulaneously. e.g. Radio and TV signals use differen carrier frequencies o avoid inerference. For coninuous analogue signals (e.g. TV, speech or music) AM or FM are mos common. For digial signals AM and PM are used and are ofen combined.

4 4 3F4 Digial Modulaion Course Secion 1 (supervisor copy) Informaion signal Carrier wave AM wave PM wave FM wave Fig 1.1: Main forms of analogue modulaion 1.2 Imporan Feaures of Modulaion Schemes: There are 3 main crieria for assessing modulaion schemes: 1. Bandwidh required affecs how close adjacen carriers can be wihou inerference occuring. 2. Demodulaor / receiver complexiy affecs he cos of a sysem. 3. Noise rejecion properies affec he ransmier power needed and he maximum range.

5 3F4 Digial Modulaion Course Secion 1 (supervisor copy) Phasor Represenaion of Modulaed Signals I is useful o handle AM, PM and FM in a unified way. Le he modulaed wave be: Noe a() and ϕ() are difficul o combine. s() = a() cos(ω C + ϕ()) (1.1) So we consider he cosine erm as he real par of a complex exponenial: s() = Re[a() e j(ω C+ϕ()) ] = Re[a() e jϕ() e jω C ] = Re[ p() modulaion phasor e jω C carrier wave ] (1.2) where p() = a() ampl. of p() e jϕ() phase of p() (1.3) Now he modulaion is compleely separaed from he carrier wave, and a and ϕ have been combined ino a single phasor waveform, p(). Phasors are very useful for defining any form of modulaion ha is o be applied o a carrier wave, wihou involving he carrier iself. Phasors vary much more slowly han he modulaed wave, s(), and are easier o analyse. See figs 1.2, 1.3 and 1.4 (3D plos of AM, PM and FM). Significance of he Real and Imaginary Pars of Phasors: If p() has real and imaginary pars, i() and q(), hen: Le p() = i() + jq() (1.4)... from (1.3) i() = a() cos ϕ() (1.5) and q() = a() sin ϕ() (1.6) We may obain s() in erms of i and q by subsiuing (1.4) ino (1.2): s() = Re[{i() + jq()} e jω C ] = i() cos(ω C ) q() sin(ω C ) (1.7) Hence i() is he inphase componen of s() and q() is he quadraure componen.

6 6 3F4 Digial Modulaion Course Secion 1 (supervisor copy) x q i Fig 1.2: 3-D plo of AM phasor and inpu signal x q i Fig 1.3: 3-D plo of PM phasor and inpu signal

7 3F4 Digial Modulaion Course Secion 1 (supervisor copy) 7 x q i Fig 1.4: 3-D plo of FM phasor and inpu signal

8 8 3F4 Digial Modulaion Course Secion 1 (supervisor copy) Muliplier 1 i () Lowpass Filer 1 i() s() 2 cos(ω C ) Muliplier 2 q () Lowpass Filer 2 q() 2 sin(ω C ) Fig 1.5: Quadraure Demodulaor. 1.4 Quadraure Demodulaion: Given s(), how do we obain i() and q() for he phasor waveform which generaed s? Fig 1.5 shows he quadraure demodulaor which achieves his. From muliplier 1: i () = s() 2 cos(ω C ) = [i() cos(ω C ) q() sin(ω C )] 2 cos(ω C ) = 2i() cos 2 (ω C ) 2q() sin(ω C ) cos(ω C ) = i() + i() cos(2ω C ) q() sin(2ω C ) Hence he oupu of lowpass filer 1 is i(), since he wo erms modulaed ono carriers a 2ω C are rejeced by he filer. Similarly from muliplier 2: q () = s() [ 2 sin(ω C )] and he oupu of lowpass filer 2 is q(). = [i() cos(ω C ) q() sin(ω C )] [ 2 sin(ω C )] = 2i() cos(ω C ) sin(ω C ) + 2q() sin 2 (ω C ) = q() q() cos(2ω C ) i() sin(2ω C ) The quadraure demodulaor is he basis for demodulaing many ypes of modulaion, alhough ofen only one oupu componen is required, so only half of he demodulaor is needed.

9 3F4 Digial Modulaion Course Secion 1 (supervisor copy) 9 Im[P] -fc Re[P] Im[S] Re[S] f fc Fig 1.6: Phasor specrum, P, and he corresponding signal specrum, S. f 1.5 Specra of Phasors To assess bandwidh, we relae he specrum of any modulaed signal s() o he specrum of is phasor waveform p(). Expanding (1.2): s() = Re[p() e jω C ] = 1 2 [p() ejω C + p () e jω C ] (1.8) If p() P (ω) hen p () P ( ω) By frequency shif: p() e jω C p() e j(ω ω C) d = P (ω ω C ) and p () e jω C p () e j(ω+ωc) d = P ( (ω + ω C ))... aking ransforms: S(ω) = 1 2 [P (ω ω C) + P ( (ω + ω C ))] (1.9)

10 10 3F4 Digial Modulaion Course Secion 1 (supervisor copy) Fig 1.6 shows how an arbirarily shaped phasor specrum, P (ω), is ransformed ino he specrum, S(ω), of he real modulaed signal, s(). Noe ha: 1. Specra are normally ploed in erms of frequency f = ω/(2π) raher han angular frequency ω, since i is usually more convenien o hink of bandwidh in Hz raher han radians per second. 2. If P (ω) is narrowband compared wih ω C, hen S(ω) comprises wo componens cenred on ω C and ω C, such ha P (ω) is shifed o ω C and P ( ω) is shifed o ω C. 3. S(ω) is a linear funcion of P (ω ± ω C ); i.e. if P (ω) comprises a linear sum of cerain componens, hen S(ω) will comprise he same linear combinaion of componens, shifed up and down by ω C. Hence we can easily deermine he specrum of any modulaed signal once we have calculaed P (ω). 4. Since p() is a complex waveform, P (ω) need no possess any symmeries beween posiive and negaive frequencies. Hence he upper sidebands of S(ω) need no be relaed o is lower sidebands. 5. P (ω) depends only on he modulaion mehod and he inpu signal i is independen of he carrier frequency ω C.

11 3F4 Digial Modulaion Course Secion 1 (supervisor copy) One-sided and Two-sided Specra: Two-sided specra include negaive as well as posiive frequencies and are used for complex waveforms (and opionally for real waveforms). One-sided specra consider frequencies as being purely posiive quaniies and may only be used for real waveforms whose 2-sided specra are symmerical abou zero. Noe on Negaive Frequencies A real cosine wave can be wrien: cos(ω) = 1 2 (ejω + e jω ) If ω > 0, hen e jω is he posiive frequency componen of he cosine wave, and e jω is he negaive frequency componen. These wo componens form wo conra-roaing uni phasors which, when summed ogeher, produce he purely real cosine wave. A purely imaginary sine wave is produced if he phasors are subraced, insead of added. Lowpass Noise A real lowpass noise signal conaining frequencies from 0 o B Hz has a 1-sided specrum from 0 o B Hz (fig 1.7a) and a 2-sided specrum from B o +B Hz (fig 1.7b). Hence is 1-sided bandwidh = B Hz, and is 2-sided bandwidh = 2B Hz. If i is a noise signal wih a fla power specrum and is power is P wa, hen is 1-sided power specral densiy (PSD) = P/B wa/hz, and is 2-sided PSD = P/2B wa/hz. (Tuned volmeers or specrum analysers measure 1-sided PSD, since hey respond o posiive and negaive frequency componens, summed ogeher.) Bandpass Noise A real bandpass noise signal is shown in figs 1.7c and 1.7d. Is specrum exends from f C B o f C + B Hz (and equivalen negaive frequencies) and has zero energy elsewhere, so i has a 1-sided bandwidh of 2B Hz. If he oal noise power is P wa, he 1-sided PSD of he noise near f C is P/2B wa/hz and he 2-sided PSD near ±f C is P/4B wa/hz. This bandlimied noise s N () may be represened by a carrier of frequency f C, modulaed by a complex noise phasor p N (): s N () = Re[p N () e jω C ] (1.10) Fig 1.7e shows he 2-sided specrum E{ P N (ω) 2 } of he phasor p N () which represens he bandlimied noise s N (). p N () has a fla specrum from B o +B Hz, and herefore a 2-sided bandwidh of 2B Hz. Since i is a complex phasor waveform, we only use 2-sided descripions for is specrum and bandwidh.

12 12 3F4 Digial Modulaion Course Secion 1 (supervisor copy) (a) Lowpass noise 1-sided PSD P B 0 B (b) Lowpass noise 2-sided PSD P 2B B B (c) Bandpass noise 1-sided PSD N f C B f C f C +B (d) Bandpass noise 2-sided PSD E{ S N (ω) 2 } N 0 4 S N (ω C ω) S N (ω C + ω) f C B f C f C +B f C B f C f C +B N 0 (e) Noise phasor 2-sided PSD E{ P N (ω) 2 } P N ( ω) P N ( ω) B B (f) Real noise componen 2-sided PSD E{ N 1 (ω) 2 } (g) Imag. noise componen 2-sided PSD E{ N 2 (ω) 2 } N 0 2 N 1 ( ω) N 1 ( ω) B N 0 2 N 2 ( ω) N 2 ( ω) B B B Fig 1.7: Power specra of bandlimied noise and noise phasors.

13 3F4 Digial Modulaion Course Secion 1 (supervisor copy) 13 Transmier x() Modulaor s() Transmission Channel Bandpass + Filer B.W.=B Noise R s N () Predeecion Filer Receiver Lowpass y() Demodulaor Filer 0 B X Posdeecion Filer Fig 1.8: Simplified communicaion sysem. x() Transmier Complex Modulaor p() Transmission Channel Noise Phasor p N () + Dual Lowpass Filer Predeecion Filer r() Receiver Complex Demodulaor Fig 1.9: Complex baseband equivalen sysem. Lowpass y() Filer 0 B X Posdeecion Filer 1.7 Noise Phasors: Figs 1.8 and 1.9 show a simplified comms. sysem and is baseband equivalen. Jus as he signal s can be represened by he phasor p, a bandlimied noise waveform s N in he region of f C can be represened by he noise phasor p N. Le: p N () = [n 1 () + jn 2 ()] e jϕ N (1.11) where n 1 () and n 2 () are wo real noise waveforms, and ϕ N is an arbirary consan phase offse. This process is illusraed in figures 1.10 o Figures 1.10 and 1.11 show wo uncorrelaed noise waveforms in real and imaginary direcions, and figures 1.12 and 1.13 show he sum of hese wo waveforms in 2-D and 3-D respecively. Noe he absence of any dominan direcion o he sum, providing visual jusificaion of he use of an arbirary phase offse ϕ N. Figures 1.14 and 1.15 show he effecs of adding noise phasors o a consan phasor and he AM phasor of fig 1.2. We shall now show ha if n 1 () and n 2 () are uncorrelaed Gaussian processes wih idenical 1- sided power specra, hen p N () accuraely represens bandlimied gaussian noise s N (). Le n 1 () and n 2 () each have a 2-sided noise PSD of N 0 /2 wa/hz from B o +B Hz and zero elsewhere (see figs 1.7f and 1.7g). This is expressed using expecaions E{}, since noise is random, as: E{ N 1 (ω) 2 } = E{ N 2 (ω) 2 } = N 0 for ω 2πB 2 0 for ω > 2πB

14 14 3F4 Digial Modulaion Course Secion 1 (supervisor copy) Taking Fourier ransforms of (1.11): P N (ω) = [N 1 (ω) + jn 2 (ω)] e jϕ N Since n 1 () and n 2 () are uncorrelaed and Gaussian, he phases of N 1 (ω) and N 2 (ω) are uncorrelaed and heir powers add, so he PSD of p N (fig 1.7e) is given by: E{ P N (ω) 2 } = E{ N 1 (ω) 2 } + E{ N 2 (ω) 2 } = Using (1.9) o ge he specrum of s N : N N 0 2 = N 0 for ω 2πB 0 for ω > 2πB S N (ω) = 1 2 [P N(ω ω C ) + P N( (ω + ω C ))] If B < f C, so ha P N (ω ω C ) does no overlap wih P N( (ω + ω C )): E{ S N (ω) 2 } = 1 4 E{ P N(ω ω C ) 2 } E{ P N( (ω + ω C )) 2 } = N 0 4 for ω C 2πB ω ω C + 2πB or ω C 2πB ω ω C + 2πB 0 oherwise This gives he 2-sided PSD of s N (fig 1.7d). Thus is 1-sided PSD is N 0 /2 wa/hz from f C B o f C + B Hz (fig 1.7c). The need for p N () o be complex: For S N (ω) o be he specrum of rue bandlimied noise, S N (ω C ω) mus be uncorrelaed wih S N (ω C + ω) a any ω. Noe he sandard Fourier Transform resul ha, if x() is purely real, is negaive frequency componens are he complex conjugaes of is posiive frequency componens: X( ω) = Hence if p N () were purely real: x() ejω d = ( ) x() e jω d = X (ω) P N ( ω) = P N( ω) and so S N (ω C ω) = S N(ω C + ω) showing undesirable srong correlaion beween S N (ω C ω) and S N (ω C + ω).

15 3F4 Digial Modulaion Course Secion 1 (supervisor copy) 15 Similarly here would be srong correlaion if p N () were purely imaginary: P N ( ω) = P N( ω) and so S N (ω C ω) = S N(ω C + ω). However if p N () = n 1 () + jn 2 () as proposed, P N ( ω) = N 1 ( ω) + jn 2 ( ω) and P N ( ω) = N 1 ( ω) + jn 2 ( ω) = N1 ( ω) + jn2 ( ω) The cross-correlaion beween P N ( ω) and P N( ω) is proporional o E{P N ( ω)p N ( ω)} Now: P N ( ω)p N ( ω) = N 1 ( ω)n1 ( ω) N 2 ( ω)n2 ( ω) +j[n 1 ( ω)n2 ( ω) + N1 ( ω)n 2 ( ω)] = N 1 ( ω) 2 N 2 ( ω) 2 + 2j Re[N 1 ( ω)n2 ( ω)] The expeced (or mean) value of each of he firs 2 erms is N 0 /2 and of he 3rd erm is 0 since n 1 and n 2 are uncorrelaed.... E{P N ( ω)p N ( ω)} = N 0 2 N = 0 Hence here is no correlaion beween P N ( ω) and P N( ω), as desired. Summary: A real noise waveform s N (), bandlimied from (f C B) o (f C + B) Hz and wih a 1-sided PSD of N 0 /2 wa/hz, may be represened by he noise phasor: p N () = [n 1 () + jn 2 ()] e jϕ N where n 1 () and n 2 () are real uncorrelaed Gaussian noise waveforms of equal 2-sided PSD of N 0 /2 wa/hz, bandlimied from B o +B Hz, and p N () has a 2-sided PSD of N 0 wa/hz over he same bandwidh. The phase offse ϕ N may be an arbirary consan.

16 16 3F4 Digial Modulaion Course Secion 1 (supervisor copy) q i Fig 1.10: Real componen of noise phasor. q i Fig 1.11: Imaginary componen of noise phasor.

17 3F4 Digial Modulaion Course Secion 1 (supervisor copy) 17 q i Fig 1.12: Complex noise phasor in 2 D. q i Fig 1.13: Complex noise phasor in 3 D.

18 18 3F4 Digial Modulaion Course Secion 1 (supervisor copy) q i Fig 1.14: Consan signal phasor + (0.3 * Noise phasor). q i Fig 1.15: AM signal phasor + (0.3 * Noise phasor).

19 3F4 Digial Modulaion Course Secion 1 (supervisor copy) Signal/Noise Raios for Signals and Phasors: How do he signal power P S and 1-sided noise PSD N S of he real signals s() and s N () in fig 1.8 relae o he phasor power P 0 and 2-sided noise PSD N 0 for he phasors p() and p N () of fig 1.9? Le P S be he power of s().... P S = (Re[p() e jω C ]) 2 = [i cos(ω C ) q sin(ω C )] 2 if p = i + jq = [i cos(ω C )] 2 + [q sin(ω C )] 2 = 1 2 i q2 = 1 2 p 2 = 1 2 P 0 Le N S be he 1-sided PSD of noise waveform s N (). From he above summary and figs 1.7c o 1.7e: N S = 1 2 N 0... P 0 N 0 = 2P S 2N S = P S N S Hence SNR is unaffeced by conversion from real signals o phasors. Real receiver amplifiers are characerised by heir noise figure F, which is he raio of he acual oupu noise PSD (due o amplifier noise and inpu resisance noise, bu no noise from he anenna) o he ideal oupu noise PSD (assuming he amplifier is ideal and he only source of noise is he inpu resisance). The noise PSD from a resisor (of any value) a absolue emperaure T is kt (where k is Bolzmann s consan), so he 1-sided noise PSD ou of a real receiver amplifier (ignoring anenna noise which is usually small) is given by: N S = GN in = GkT F (joule or wa/hz) where G is he amplifier power gain and N in is he effecive noise PSD a he amplifier inpu. Noe ha here F should be expressed as a power raio, and no be in db. The signal power ou of he amplifier is: P S = GP in (wa) where P in is he receiver inpu signal power.... P 0 N 0 = P S N S = GP in GkT F = P in kt F (Hz) These formulae apply even when here is frequency ranslaion and bandlimiing in he receiver, as long as he bandlimiing does no remove any of he waned signal componens.

20 20 3F4 Digial Modulaion Course Secion 1 (supervisor copy)

21 3F4 Digial Modulaion Course Secion 2 (supervisor copy) 21 2 Digial Modulaion Summary 2.1 Why Digial (and no Analogue)? Main reasons: Less suscepible o cumulaive degradaions. Effecs of addiive noise can be eliminaed a regular inervals in a digial comms link by hreshold deecion and error correcion coding. Ulimae noise levels are deermined by he analogue/digial conversions and no by he channel, giving much beer dynamic range (e.g. audio CDs vs ape cassees). Can be made more secure (encrypion) and less deecable (spread specrum). Can be muliplexed using frequency-division, ime-division or spread-specrum muliple access (FDMA, TDMA or SSMA); whereas FDMA is he only sensible mehod for analogue signals. Digial links can handle a wide range of source maerial (muli-media): E.g. audio, video, sill images and daa, when used wih appropriae source coding / compression echniques. (Noe he variey of maerial now available on CDs and DVDs.) Disadvanages: More bandwidh is needed unless source compression is used. More complex processing is required (OK wih oday s chips).

22 22 3F4 Digial Modulaion Course Secion 2 (supervisor copy) 2.2 Some digial communicaions sysems: Telegraph Communicaions, Morse code (1830s onwards). (Bell did no inven he analogue elephone unil he 1870s!) Telex and elepriners, 50 o 110 b/s (1930s?). Compuer Modems, up o 2.4 kb/s over elephone nework (1960s). Encryped digial speech a 2.4 kb/s for miliary / sraegic use (1960s). Saellie runk elephone circuis, a 56 or 64 kb/s (1970s). Fax machines, binary images a 2.4 o 9.6 kb/s (1980s). Teleex, digial ex on analogue TV ransmissions (1980s). Digial Telephone Exchanges (sysem X in UK) (1980s). Nicam sereo digial sound on analogue TV (lae 1980s). Fibre-opic inerne links (e.g. Grana Backbone Nework) (lae 1980s) Inerne modems, up o 56 kbi/s over elephone nework (1990s). Digial cellular phones - GSM (Groupe Speciale Mobile) (1990s). Video phones / video conferencing (1990s). Digial Audio Broadcasing - DAB (lae 1990s). See secion 6 of his course. Digial TV (lae 1990s). See secion 6 of his course. Asymmeric inerne modems (asymmeric digial subscriber loop, ADSL), 2 8 Mbi/s one-way (early 2000s). Inerne (WAP and 3rd Generaion) mobile phones (early 2000s). WiFi comms for lapops ec (early 2000s). Digial High Definiion TV (HDTV) (mid 2000s). Analogue TV swich-off in UK, o release more bandwidh / capaciy for digial TV and radio services (2008 onwards, depending on region) (See Couch Chaper 1 for a more complee lis.)

23 3F4 Digial Modulaion Course Secion 2 (supervisor copy) The Major Digial Modulaion Techniques Figure 2.1 shows he modulaed waveforms for various simple forms of digial modulaion, wih he inpu daa and he carrier wave shown a he op. We shall now look a how hese modulaion mehods are defined, and how hey are represened by phasor waveforms. Binary Modulaion: Binary Ampliude Shif Keying (ASK) fig ampliude levels (usually 0 and a 0 ). Phase consan (= ϕ 0 ). Binary Phase-Shif Keying (BPSK) fig 2.3 Ampliude consan (= a 0 ). 2 phase values (usually ϕ 0 and ϕ 0 + π). Binary Frequency-Shif Keying (BFSK) fig 2.4 Ampliude consan (= a 0 ). 2 frequency values ( ω = ±ω D, he frequency deviaion). Muli-level (M-ary) Modulaion: Involves grouping daa bis ino m-bi symbols, and each symbol is ransmied using one of M = 2 m modulaion levels. 4-level Ampliude Shif Keying (4ASK) fig ampliude levels (usually 3a 0, a 0, +a 0 and +3a 0 ). Phase consan (= ϕ 0 ). m = 2 bis per symbol. Quadraure Phase-Shif Keying (QPSK) fig 2.6 Ampliude consan (= a 0 ). 4 phase values (usually ϕ 0 ± π/4 and ϕ 0 ± 3π/4). Equivalen o BPSK on wo quadraure carriers a ϕ 0 and ϕ 0 + π/2. m = 2 bis per symbol. M-ary PSK Ampliude consan (= a 0 ). M phase values (usually ϕ 0 + 2πk M, k = 0 M 1). Quadraure Ampliude Modulaion (QAM) M-level ASK on each of wo quadraure carriers. This gives M 2 saes, represening 2m bis. M-ary FSK Ampliude consan (= a 0 ). M frequency values (usually ω = (k M 1 2 )ω S, where k = 0 M 1 and ω S = 2π(symbol rae) = 2π(bi rae) ). m

24 24 3F4 Digial Modulaion Course Secion 2 (supervisor copy) Modulaing daa Carrier wave ASK wave BPSK wave BFSK wave 4ASK-SC wave QPSK wave Fig 2.1: Basic digial modulaion schemes x q i Fig 2.2: Binary ampliude shif keying (ASK) phasor waveform

25 3F4 Digial Modulaion Course Secion 2 (supervisor copy) 25 x q i Fig 2.3: Binary phase shif keying (BPSK) phasor waveform x q i Fig 2.4: Binary frequency shif keying (BFSK) phasor waveform

26 26 3F4 Digial Modulaion Course Secion 2 (supervisor copy) x q i Fig 2.5: 4-level ASK - suppressed carrier (4ASK-SC) phasor waveform x q i Fig 2.6: Quadraure phase shif keying (QPSK) phasor waveform

27 3F4 Digial Modulaion Course Secion 3 (supervisor copy) 27 3 Binary Phase-Shif Keying (BPSK) BPSK is perhaps he simples modulaion scheme in common use and we shall analyse i in some deail. The resuls for oher schemes can ofen be derived from hose for BPSK. 3.1 Definiion of BPSK: Le k h daa bi b k = +1 or 1 and bi period = T b. The modulaed phasor during he k h bi period is: p k () = b k a 0 e jϕ 0 for kt b < (k + 1)T b To apply his for all, we inroduce a ime limied pulse: a g() = 0 for 0 < T b 0 elsewhere.... p() = e jϕ 0 k b k g( kt b ) (3.1) Noe ha g() is normally a recangular pulse, bu modified forms of BPSK can use oher shapes such as half-sine or raised cosine. 3.2 Power Specrum for Random Daa: We observe ha p() is jus a consan phasor e jϕ 0 muliplied by a polar binary daa sream, in which he daa impulses have been filered (convolved) wih an impulse response g(). Hence we use echniques similar o hose inroduced in he 3F4 Baseband Transmission course for power specra for line codes. The discree auocorrelaion funcion (ACF) of he random daa sream b k, in which b k uncorrelaed wih b k L for any ineger L 0, is: R bb (L) = E{b k b k L } = 1 for L = 0 0 for L 0 is The power specrum (power specral densiy) of he sream of random daa impulses, b() = k b k δ( kt b ), is hen given by: E{ B(ω) 2 } = lim T E{ B T (ω) 2 } T = 1 R bb (L) e jlωt b = 1 T b T b where B T (ω) is he Fourier ransform of b() limied o a ime inerval of T. L

28 28 3F4 Digial Modulaion Course Secion 3 (supervisor copy) Since he sream of daa impulses b() = k b k δ( kt b ) are convolved wih g() and phaseroaed by ϕ 0 in equaion (3.1), he Fourier ransform of p() is given by: So he power specrum of p() is: P (ω) = e jϕ 0 B(ω) G(ω) E{ P (ω) 2 } = e jϕ 0 2 E{ B(ω) 2 } G(ω) 2 = 1 T b G(ω) 2 If g() is a recangular pulse of ampliude a 0 and duraion T b : )]2 G(ω) 2 = [ a 0 T b sinc ( ωtb 2 and so ( ) E{ P (ω) 2 } = a 2 0 T b sinc 2 ωtb 2 For alernaive g() pulse shapes, he power specrum will be proporional o G(ω) 2, so his gives a good way of conrolling he specrum of he modulaed signal. The power specrum of he modulaed signal S(ω) 2 is jus a frequency shifed version of P (ω) 2 as derived in equaion (1.9): S(ω) = 1 2 [P (ω ω C) + P ( (ω + ω C ))] Hence, assuming no overlap beween P (ω ω C ) and P ( (ω + ω C )): E{ S(ω) 2 } = 1 4 [ E{ P (ω ω C) 2 } + E{ P ( (ω + ω C )) 2 } ] Fig 3.1 shows hese specra for recangular daa pulses. a 0 2 T b E{ P(w) 2 } T b T b T b freq E{ S(w) 2 } -f c 0 freq f c Fig 3.1: Power specra of random BPSK phasor and modulaed signal.

29 3F4 Digial Modulaion Course Secion 3 (supervisor copy) 29 Monosable r() Re[ ] G g( kt b ) e jϕ 0 C R Mached Filer (Inegrae-and-Dump) Threshold Deecor (Comparaor) + 0v Sampling Regiser D Clk Q Daa Clock a... kt b, (k + 1)T b,... b k Daa Ou Fig 3.2: Mached correlaor demodulaor for BPSK. 3.3 Opimum Demodulaor: An opimum demodulaor for equiprobable signals in gaussian noise is one which selecs he daa value ha resuls in he minimum mean squared error (MSE) beween he received signal plus noise and he signal corresponding o ha daa value. (Bayesian analysis can be used o prove his.) If he received signal+noise phasor is r() = p() + p N () he opimum demodulaor for binary daa during he k h bi period measures he MSEs: (k+1)tb r() g( kt b )e jϕ 0 2 d and kt b } {{ } +1 phasor (k+1)tb r() ( g( kt b )e jϕ 0 ) 2 d kt b } {{ } 1 phasor and selecs whichever gives he smaller resul. Bu we can simplify his. r g 2 = (r g)(r g ) = r 2 2Re[rg ] + g 2 If g = ±g( kt b )e jϕ 0, r 2 and g 2 are he same for boh inegrals, so we may ignore hese erms and simply calculae y(b, k) = (k+1)tb kt b Re[r() b g( kt b )e jϕ 0 ]d for b = +1 and 1, and selec he value of b according o which b gives he larger resul. Since b can be aken ou of he inegral, we find ha y( 1, k) = y(1, k); and so, deecing he larger of y(1, k) and y( 1, k) is equivalen o deecing he polariy of jus y(1, k).

30 30 3F4 Digial Modulaion Course Secion 3 (supervisor copy) Thus b k = polariy of y(k), where (k+1)tb y(k) = G Re[r() g( kt b )e jϕ 0 ]d (3.2) kt b and G is an arbirary posiive gain consan. This is known as a mached correlaion demodulaor, because i correlaes r() wih a waveform g( kt b )e jϕ 0 ha is mached o he daa pulse being deeced. To work correcly, he demodulaor mus know he reference phase ϕ 0 of he received phasors. This is normally exraced by a phase-locked loop (see secion 3.6). A demodulaor using phasors for he inpu and reference signals is shown in fig 3.2. I implemens equaion 3.2. The correc limis for he inegraion are achieved by discharging he capacior a he sar of each bi period (a = kt b using a shor pulse from he monosable), and by sampling he polariy of he inegraor oupu a he end of each bi period (a = (k + 1)T b ) jus before he monosable discharges he inegraor for he nex bi. Noe ha if he daa pulses g() are recangular and consan over he period of inegraion, we may replace g( kt b ) by a 0 in equaion 3.2, and he inpu muliplier in fig 3.2 simply produces a phase shif ϕ 0 and an arbirary gain Ga 0. A more pracical implemenaion of an opimum demodulaor, using real bandpass signals raher han phasors, is described in secion 3.6.

31 3F4 Digial Modulaion Course Secion 3 (supervisor copy) Bi Error Performance in Noise: The signal+noise phasor received during he k h bi period is r() = p() + p N () = b k g( kt b ) e jϕ 0 + [n 1 () + jn 2 ()] e jϕ N where n 1 () and n 2 () are real uncorrelaed noise waveforms of equal PSDs. If we choose he arbirary noise reference phase ϕ N o equal ϕ 0 (as discussed in secion 1.7), he oupu of he mached correlaor is: (k+1)tb y(k) = G Re[{b k g( kt b ) + n 1 () + jn 2 ()} e jϕ 0 g( kt b ) e jϕ 0 ] d kt b Tb = Gb k g 2 () d } 0 {{ } signal (k+1)tb + G n 1 () g( kt b ) d kt } b {{ } noise Now he noise inegral is equivalen o convolving he noise wih a filer whose impulse response h() equals g(t b ) from 0 o T b and is zero elsewhere. This is in fac he mached filer for he daa. i.e. (k+1)tb kt b n 1 () g( kt b ) d = n 1() h((k + 1)T b ) d Le he 2-sided PSD of p N () be N 0 wa/hz. Hence n 1 and n 2 will need o be real uncorrelaed whie noise waveforms wih 2-sided PSDs of N 0 /2 wa/hz = N 0 /(4π) wa s/rad (see secion 1.7). If h() H(ω): Mean noise power from filer = G 2 = G 2 = G2 N 0 2 N 0 4π H(ω) 2 dω = G2 N 0 2 Tb 0 g() 2 d = G2 N 0 E b 2 E{ N 1(ω) H(ω) 2 } dω h() 2 d (by Parseval s Theorem) where E b = P 0 T b is he energy of he g() pulse, or he energy per bi of he signal phasor. Hence rms noise volage σ in y(k) = G N 0E b 2 Tb Signal volage ampliude v s in y(k) = Gb k g 2 () d = GE b 0 The deecor hreshold = 0 since he signal volage = ±GE b, which is symmeric abou zero.... Volage SNR a hreshold deecor = v s 0 σ = GE b G N 0 E b /2 = 2E b N 0

32 32 3F4 Digial Modulaion Course Secion 3 (supervisor copy) -1 Signal Volage +1 Signal Volage Gaussian PDF Sd. dev. = σ Threshold - v s 0 v s Probabiliy of error = Q(v s / σ) Fig 3.3: Probabiliy densiy funcions of deeced signal+noise. We can now calculae he probabiliy of bi error see fig 3.3. We assume he noise PDF is gaussian, since i is heavily bandlimied and assume equal probabiliy of 1 and 1 in b k. ) PDF of noise wih rms volage σ = 1 σ f ( v σ where f(x) = e x2 /2 2π is he PDF of a uni variance zero mean gaussian process and x = v σ. Probabiliy of error = ( ) vs = 0 ( ) 1 v + σ f vs σ dv = v s /σ f(x) dx = Q ( vs σ ) ( ) 1 v v s σ f dv σ where Q is he gaussian inegral funcion (see secion 3.7 on Approximaion Formulae for σ Q(x)) and v s is he volage SNR a he hreshold deecor. σ... Probabiliy of a bi error P E = Q 2E b N 0 This funcion is ploed in fig 3.5, as he BPSK (and QPSK) curve. Noe ha his resul is independen of he shape of he signal pulse g(). Only he energy E b of he pulse and he PSD N 0 of he noise are imporan. For a recangular pulse of ampliude a 0 and duraion T b, E b = a 2 0 T b.

33 3F4 Digial Modulaion Course Secion 3 (supervisor copy) 33 b k Differenial Encoder b k p() Modulaor Demodulaor b k Differenial Decoder b k Fig 3.4: Differenial encoding and decoding. 3.5 Differenial Coding: A real BPSK receiver canno deec which of he wo received carrier phases is 0 and which is 180, since here is nohing o disinguish hem in he presence of an arbirary phase offse ϕ 0 (see fig 2.3). Hence he receiver can lock o he wrong signal phase, and hus generae invered daa.... Differenial Coding is used (see fig. 3.4). A he Diff. Encoder: b k = b k 1 b k Hence b k changes sae if b k = 1, and remains in is previous sae if b k = 1. A he Diff. Decoder: b k = b k /b k 1 = b k b k 1 since b k 1 = ±1 Hence b k = 1 if b k and b k 1 differ, and b k = 1 if hey are he same. If b k and b k 1 are boh invered, b k will sill be decoded correcly phase ambiguiy does no maer. BUT differenial coding increases he oupu bi error rae. If he probabiliy of error in b k is P E for all k, hen b k will be in error if: b k is in error and b k 1 is correc, or b k 1 is in error and b k is correc. Each even has probabiliy P E (1 P E ) and hey are muually exclusive,... Probabiliy of error in b k = 2P E (1 P E ) 2P E if P E is small. Hence he error rae is approximaely doubled by differenial coding and errors end o occur in pairs. However only a small increase in SNR ( 0.5dB) is needed o compensae for his see fig 3.5, curves for BPSK wih and wihou differenial coding.

34 34 3F4 Digial Modulaion Course Secion 3 (supervisor copy) Bi error rae QAM 64-QAM BPSK wih BPSK differenial and QPSK coding Energy per bi / Noise PSD, Eb / No (db) Fig 3.5: Bi error rae curves for PSK and QAM.

35 3F4 Digial Modulaion Course Secion 3 (supervisor copy) 35 Inpu Signal s() Quadraure Deecor cos(ω c + ϕ 0 ) C R Mached Filer (Inegrae-and-Dump) Dual Lowpass Filer sin(ω c + ϕ 0 ) Carrier Phase-Locked Loop i q Comparaor + 0v Monosable Mached Correlaor Demodulaor Sampling Regiser D Clk Comparaor Daa Clock + Raw Daa sgn(i) 0v Q Oupu Daa b k Carrier VCO Loop Filer Loop Error, q sgn(i) Transiion Deecor Early/Lae Deecor Daa Clock Phase-Locked Loop Loop Filer Delay Daa Clock VCO Fig 3.6: Pracical implemenaion of an opimum demodulaor for BPSK. 3.6 Pracical Implemenaion of an Opimum BPSK Demodulaor: Fig 3.6 shows he block diagram of a pracical demodulaor. The received bandpass signal s() is demodulaed ino real and imag pars by he quadraure deecor and is bandlimied by he dual lowpass filers o a bandwidh approximaely equal o he bi rae. This generaes he signals i() and q(). The carrier phase-locked loop (PLL) error signal is q sgn(i) (see fig 3.7), in order o produce a characerisic which repeas a muliples of π. This allows he loop o lock up a he posiive zero-crossings of he error characerisic such ha he phase error beween s() and he Carrier VCO (volage conrolled oscillaor) is eiher 0 or π. Thus he Carrier VCO will eiher be in

36 36 3F4 Digial Modulaion Course Secion 3 (supervisor copy) phase or in aniphase wih he carrier of s(), and he i() signal will be a smoohed version of he original daa or is complemen. Differenial decoding (no shown) mus be used o eliminae his ambiguiy. The dual lowpass filers provide coninuous oupus o allow he Carrier and Daa Clock PLLs o operae correcly, bu i is difficul o obain he ideal recangular impulse response (required for opimum mached-correlaion deecion) from real coninuous filers. Therefore he daa is deeced via an alernaive discree-ime filer which is an Inegrae-and-Dump circui. This circui akes is inpu from he inphase (real) componen of he quadraure deecor, and provides opimum mached-correlaion filering for daa deecion. The clock for he sampling regiser and he rese pulse for he Inegrae-and-Dump is provided from he Daa Clock PLL. This loop is conrolled by an Early/Lae deecor, which generaes a posiive pulse if a daa ransiion occurs early wih respec o he neares delayed clock edge, and a negaive pulse if a ransiion occurs lae. These pulses are inegraed by he loop filer and used o increase or decrease he clock rae very slighly, so as gradually o lock he delayed clock edges o he daa ransiions. The Dual LPF inroduces delay o he iming of he ransiions, so a compensaing delay is included beween he Daa Clock VCO and he Early/Lae deecor. The clock iming will hen be correc for he Inegrae-and-Dump and sampling regiser. The Monosable generaes pulses which are shor compared wih T b, bu long enough o discharge he inegraor capacior fully a he sar of each bi period. i q sgn(i) q sgn(i) 0 π 2π 3π 4π Phase error beween s() and Carrier VCO Fig 3.7: Phase deecor characerisic of he Carrier Phase-Locked Loop.

37 3F4 Digial Modulaion Course Secion 3 (supervisor copy) Approximaion Formulae for he Gaussian Error Inegral, Q(x) A Gaussian probabiliy densiy funcion (PDF) wih uni variance is given by: f(x) = 1 2π e x2 /2 The probabiliy ha a signal wih a PDF given by f(x) lies above a given hreshold x is given by he Gaussian Error Inegral or Q funcion: Q(x) = x f(u) du There is no analyical soluion o his inegral, bu i has a simple relaionship o he error funcion, erf(x), or is complemen, erfc(x), which are abulaed in many books of mahemaical ables. Q(x) is abulaed in Appendix A-10 of Couch, Digial and Analog Communicaion Sysems, 3rd ediion, and in Appendix D of Shanmugam, same ile. erf(x) = 2 x e u2 du and erfc(x) = 1 erf(x) = 2 π π 0... Q(x) = 1 2 erfc ( x 2 ) = 1 2 [ ( )] x 1 erf 2 Noe ha erf(0) = 0 and erf( ) = 1, and herefore Q(0) = 0.5 and Q(x) 0 very rapidly as x becomes large. I is useful o derive simple approximaions o Q(x) which can be used on a calculaor and avoid he need for ables. Le v = u x:... Q(x) = 0 x e u2 f(v + x) dv = 1 e (v2 +2vx+x 2 )/2 /2 dv = e x2 e vx e v2 /2 dv 2π 2π 0 Now if x 1, we may obain an approximae soluion by replacing he e v2 /2 erm in he inegral by uniy, since i will iniially decay much slower han he e vx erm... /2. Q(x) < e x2 2π 0 e vx dv = e x2 /2 2π x This approximaion is an upper bound, and is raio o he rue value of Q(x) becomes less han 1.1 only when x > 3, as shown in fig 3.8. We may obain a much beer approximaion o Q(x) by alering he denominaor above from ( 2π x) o (1.64x x 2 + 4) o give: 0 du Q(x) e x2 /2 1.64x x This improved approximaion (developed originally by Borjesson and Sundberg, IEEE Trans. on Communicaions, March 1979, p 639) gives a curve indisinguishable from Q(x) in fig 3.8 and is raio o he rue Q(x) is now wihin ±0.3% of uniy for all x 0 as shown in fig 3.9. This accuracy is sufficien for nearly all pracical problems.

38 38 3F4 Digial Modulaion Course Secion 3 (supervisor copy) Q(x) simple approximaion x Fig 3.8: Q(x) and he simple approximaion exp(-x*x/2) / sqr(2*pi)*x x Fig 3.9: The raio of he improved approximaion of Q(x) o is rue value

39 3F4 Digial Modulaion Course Secion 4 (supervisor copy) 39 q 10 j i 11 -j 01 Fig 4.1: QPSK phasor diagram. 4 Oher Binary Schemes 4.1 Quadraure PSK (QPSK): QPSK is equivalen o BPSK on wo quadraure carriers. Even bis b 2k modulae he inphase carrier. Odd bis b 2k+1 modulae he quadraure carrier.... p k () = [ b 2k + j b 2k+1 ] g( kt s ) e jϕ 0 where g() is as for BPSK excep ha i is now non-zero from = 0 o T s, he 2-bi symbol period (T s = 2T b ). Hence p() can have one of 4 values: See fig 4.1 for QPSK phasor diagram. (±1 ± j) g( kt s ) e jϕ 0 QPSK can be regarded as 4-level modulaion, bu i is usually easier o rea i as wo independen 2-level (binary) processes. See Fig. 4.3 for symbol iming.

40 40 3F4 Digial Modulaion Course Secion 4 (supervisor copy) db BPSK QPSK QAM Frequency relaive o carrier / bi rae 64-QAM Fig 4.2: Power specra of BPSK, QPSK, 16-QAM and 64-QAM for a given bi rae. Power Specrum of QPSK: If each quadraure carrier of ampliude a 0 is BPSK modulaed by recangular daa pulses a a symbol rae of 1/T s, hen, using he BPSK resul, he specrum of each carrier is given by: E{ P I (ω) 2 } = E{ P Q (ω) 2 } = a 2 0 T s sinc 2 (ωt s /2) The daa on he wo carriers are uncorrelaed, so he power specra add o give a oal specrum of: E{ P (ω) 2 } = 2 a 2 0 T s sinc 2 (ωt s /2) = 4 a 2 0 T b sinc 2 (ωt b ) since T b = T s /2... he QPSK specrum is half as wide as he BPSK specrum for a given daa rae - a big advanage! See fig 4.2 for specra.

41 3F4 Digial Modulaion Course Secion 4 (supervisor copy) 41 Bi Error Performance of QPSK: Since he wo carriers are in quadraure, hey may be demodulaed independenly as wo BPSK phasor componens. Power in each componen = 1 2 oal power. Bi rae for each componen = 1 oal bi rae. 2 signal power Bu E b = (signal power) (bi period) = bi rae so E b is he same for each componen as i is for he oal QPSK signal. N 0 is unaffeced by he modulaion mehod.... applying he BPSK resul of secion 3.4 o each componen: Probabiliy of bi error P E = Q 2E b N 0 he same as BPSK. Differenial Coding is normally used wih QPSK o overcome carrier phase ambiguiy, as for BPSK, again approx. doubling P E. Offse QPSK (OQPSK) is a varian of QPSK (see fig 4.3): 1. The keying of he wo carriers is saggered by T b = T s /2. 2. The performance and specrum are he same as QPSK. 3. OQPSK has less ampliude flucuaion afer bandlimiing han QPSK. I is herefore beer suied o non-linear bandlimied channels, such as hose involving saellie onboard ransmiers. QPSK Demodulaor Design: This may be based on he BPSK design of fig 3.6 wih he following addiions: Daa is deermined from he polariies of boh he i and q oupus of he Quadraure Deecor, using wo Mached Correlaor Demodulaors o give he odd and even daa bis respecively. The Carrier PLL error signal is q sgn(i) i sgn(q) in order o give a phase error characerisic which repeas a muliples of π/2 and has lock poins a odd muliples of π/4.

42 42 3F4 Digial Modulaion Course Secion 4 (supervisor copy) Re(p) b 0 b 2 b 4 b 6 QPSK Im(p) b 1 b 3 b 5 b 7 Re(p) b 0 b 2 b 4 b 6 Offse QPSK Im(p) b 1 b 1 b 3 b 5 b 7 0 T s 2T s 3T s 4T s Fig 4.3: QPSK and Offse QPSK symbol iming. Phase(p) π 0 π Phase Trellis 2π 2π 3π 3π Im(p) b 0 b 2 b 4 b 6 MSK Re(p) b 1 b 1 b 3 b 5 b 7 0 2T b 4T b 6T b 8T b Fig 4.4: MSK phase rellis and symbol iming.

43 3F4 Digial Modulaion Course Secion 4 (supervisor copy) Binary Frequency-Shif Keying (BFSK): Daa causes a shif beween 2 frequencies, ω D and +ω D w.r.. he carrier. f D = ω D /2π is he frequency deviaion (in Hz). Hence he phasor during he k h bi period is: p k () = a 0 e j(b k ω D ( kt b ) + ϕ k ) ϕ k is he iniial phase value a he sar of each bi period, and i is updaed a each bi boundary so as o preven abrup phase changes in p(). This has he advanage of reducing specral sidelobes and i can also improve performance agains noise. FSK wih his consrain is known as Coninuous Phase FSK (CPFSK). See fig 2.4. The FSK mod. index is given by m F SK = difference beween he 2 ransmi freqs. bi rae = 2f D T b As wih Analogue FM, FSK is difficul o analyse in general. Is bandwidh is equal o he peak-o-peak frequency deviaion, 2f D, plus an amoun ha is approximaely equal o he rae of signalling (ie he symbol rae), 1/T b. Hence (similar o Carson s rule for FM). Bandwidh required 2f D + 1 T b = m F SK + 1 T b Minimum (Freq.) Shif Keying (MSK) a special case: m F SK = f D = m F SK 2T b = 1 4T b so he phase of p increases or decreases by 90 in each bi period, as in fig 2.4. This is he minimum m F which gives an error performance ha is opimum in some sense and is he reason for he name MSK. Fig 4.4 shows he phase plo (phase rellis) for MSK and below his are he i and q componens of he phasor waveform. Noe he similariy o o OQPSK excep ha he g() pulses for MSK are half-sine pulses insead of recangles. The broad line shows a ypical phase rajecory and is componens. The power specrum of MSK is proporional o he square of he Fourier ransform of he half-sine pulse shape, raher he han he recangular pulse shape of QPSK. The righhand halves of hese wo specra are shown in fig. 4.5 and labelled MSK and QPSK respecively. Noe ha MSK has a 50% wider main lobe han QPSK, bu his is compensaed for by MSK s significanly lower sidelobe levels (due o MSK being a coninuous phase process).

44 44 3F4 Digial Modulaion Course Secion 4 (supervisor copy) If he daa deermines he slope of he phase rajecory, hen he polariies of he half-sine pulses are obained by differenially encoding he daa, afer invering alernae bis. Hence he error performance for MSK is equivalen o OQPSK, QPSK and BPSK, each wih differenial coding. Gaussian-filered MSK (GMSK) In many pracical sysems, such as in he mobile phone sysem GSM, subsanially lower levels of specral sidelobes are required han even MSK is able o give. In hese cases i is common o apply smoohing o he binary daa pulses before hey are applied o he MSK modulaor. For GMSK, he smoohing lowpass filer has a Gaussian frequency response, whose 3 db bandwidh is ypically 0.3 imes he bi rae. This filer bandwidh is chosen o give a good radeoff beween narrow ransmied bandwidh and low inersymbol inerference. This scheme is known as 0.3R GMSK. Figures 4.5 and 4.6 illusrae his radeoff. Fig. 4.5 shows one side of he power specrum of a QPSK signal, a pure MSK signal, and hree GMSK signals wih gaussian filer bandwidhs of 0.5R b, 0.3R b and 0.2R b respecively. Fig. 4.5 shows he eye diagrams a he hreshold deecor of an ideal MSK demodulaor for he hree filer bandwidhs. We can see ha he 0.3R b filer represens a good radeoff beween a well conained specrum and a good eye opening. In pracise an equaliser would probably be used o improve he eye opening furher QPSK db MSK R GMSK 0.3R GMSK 0.5R GMSK Frequency relaive o carrier / bi rae Fig 4.5: Power specra of QPSK, MSK, 0.5R GMSK, 0.3R GMSK and 0.2R GMSK.

45 3F4 Digial Modulaion Course Secion 4 (supervisor copy) R GMSK R GMSK R GMSK Fig 4.6: Eye diagrams for GMSK a he deecor of an ideal MSK demodulaor.

46 46 3F4 Digial Modulaion Course Secion 4 (supervisor copy)

47 3F4 Digial Modulaion Course Secion 5 (supervisor copy) 47 q i Fig 5.1: 16-level PSK phasor diagram 5 Muli-level Modulaion M-ary modulaion uses one of M signals during each symbol inerval T s, hereby ransmiing m = log 2 M bis of informaion per T s (M = 2 m ). By varying M we may rade bandwidh for performance in noise. 5.1 M-ary PSK (MPSK): Transmis one of M phases, usually 2πi/M for i = 0... M 1. For example: BPSK uses 2 phases (m = 1) QPSK uses 4 phases (m = 2) 8-PSK uses 8 phases (m = 3) ec. Bandwidh 1 T s = 1 mt b Noise immuniy disance beween adjacen poins = 2a 0 sin(π/m), if ampliude = a Increasing m reduces bandwidh; Bu i rapidly worsens he performance in noise. Fig 5.1 shows he phasor diagram of 16-PSK.

48 48 3F4 Digial Modulaion Course Secion 5 (supervisor copy) q j j i -j 10-3j 11 Fig 5.2: 16-QAM phasor diagram 5.2 Quadraure Ampliude Modulaion (QAM): QAM achieves a greaer disance beween adjacen poins by filling he p-plane more uniformly han MPSK. However he consan ampliude propery is hen los. QAM is he basic modulaion mehod of choice for almos all bandwidh criical sysems of oday, such as inerne modems and digial TV broadcasing. QAM is similar o QPSK excep ha i uses mulilevel ASK-SC (wih zero mean) on he wo quadraure carriers. See figs 5.2 and 5.3 for 16-QAM and 64-QAM. 8-QAM and 32-QAM are possible, bu are more complicaed due o he problem of dividing an odd number of bis beween he wo carriers. Generalising he QPSK case, he QAM phasor during he k h symbol period is: p k () = [s 2k + j s 2k+1 ] g( kt s ) e jϕ 0 (5.1) where s 2k or s 2k+1 = 2i + 1 M, and he sae i is chosen from 0... M 1 according o he daa. To analyse QAM noise performance, we consider each carrier separaely. We shall analyse he s 2k componen and hen assume he same performance for he oher componen. For M 2 -QAM, he number of levels for each carrier M = 2 m, conveying m bis per symbol on each carrier. Hence he oal capaciy is 2m bis per symbol.

49 3F4 Digial Modulaion Course Secion 5 (supervisor copy) q i Fig 5.3: 64-QAM phasor diagram Noise performance of QAM Le he signal level expeced a he receiver inphase hreshold deecor for he k h symbol be s 2k v s where: s 2k = (2i + 1 M) for i = 0... M 1. e.g. if M = 8, s 2k = 7, 5, 3, 1, 1, 3, 5, or 7. Hence he signal levels are separaed by 2v s and he opimum hreshold levels are v s above and below each expeced signal level, assuming equiprobable daa symbols. Le he rms noise vols a he hreshold deecor be σ. Consider a signal componen in sae i, where 0 < i < M 1 (so i is no an ouer sae): Probabiliy of sae i becoming i + 1 = Q( v s σ ) Probabiliy of sae i becoming i 1 = Q( v s σ )... Probabiliy of error from sae i = 2Q( v s σ ) There are M 2 saes in his caegory. For he wo ouer saes, where i = 0 or M 1, here is only one error direcion:

50 50 3F4 Digial Modulaion Course Secion 5 (supervisor copy)... Probabiliy of error from sae 0 or M 1 = Q( v s σ ) There are 2 saes in his caegory. Hence he mean probabiliy of symbol error is P SE = M 2 M 2Q(v s σ ) + 2 M Q(v s σ ) = 2(1 1 M ) Q(v s σ ) (5.2) If we use m-bi Gray (uni disance) coding for he M levels, each symbol error o an adjacen sae will only cause a single bi error in each m-bi word. We ignore errors o non-adjacen saes (unlikely excep a poor SNR). Since here are m bis for every symbol:... Mean probabiliy of bi error, P BE = P SE m = 2 m (1 1 M ) Q(v s σ ) (5.3) Now we evaluae v s /σ in erms of E b /N 0. The opimum receiver is as for BPSK (i.e. i correlaes he inpu wih g( kt s )e jϕ 0 ), excep ha M 1 hreshold deecors are used o deec he M levels of each quadraure componen. Modifying he BPSK resul from secion 3.4 o ge he waveform a he inphase hreshold deecors gives: y(k) = Gs 2k Ts 0 = Gs 2k E g + where he energy of he g() pulse is: g 2 (k+1)ts () d + G kt s n 1 () g( kt s ) d noise of mean power G2 N 0 E g 2 E g = Ts 0 g 2 () d The signal levels Gs 2k E g are he same as hose we assumed above o be s 2k v s. (5.4)... v s = GE g The rms noise vols were assumed o be σ, so from (5.4):... v s σ = σ = G N 0E g 2 GE g G N 0 E g /2 = 2E g N 0 (5.5) Finally we need o express E g in erms of he energy per bi E b (hese are no longer he same wih mulilevel modulaion).

51 3F4 Digial Modulaion Course Secion 5 (supervisor copy) 51 Using (5.1), he average symbol energy on he s 2k carrier is: E s = 1 M M 1 i=0 (2i + 1 M) 2 T s g 2 () d 0 Bu M 1 i=0 (2i + 1 M) 2 = M(M 2 1) 3... E s = M Since here are m bis per symbol, E s = me b. (Prove by inducion) E g... E b = E s m = M 2 1 3m E g (5.6) Subsiuing (5.6) ino (5.5): v s σ = 3m M 2 1 2E b N 0... from (5.3) he mean probabiliy of bi error is given by: P BE = 2 m (1 1 M ) Q 3m M 2 1 2E b (5.7) N 0 Fig 3.5 (repeaed on nex page) shows he error performance curves of 16-QAM and 64-QAM, compared wih BPSK & QPSK. We see ha 16-QAM is approximaely 4 db worse han BPSK and QPSK, and ha 64-QAM is anoher 4 db worse han 16-QAM. 3m This difference in performance is almos enirely due o he facor in he argumen M 2 1 of he Q funcion in he above equaion, because Q is so seep on a log scale in he region of ineres ha he scaling of Q by 2 (1 1 ) has minimal effec. The values of he former facor, m M as a volage raio (and in dbs), are given below for various m and M = 2 m : m M 2 (BPSK/QPSK) 4 (16-QAM) 8 (64-QAM) 16 (256-QAM) 3m M (0dB) 6 15 ( 3.98dB) 9 63 ( 8.45dB) ( 13.27dB) We see ha he prediced degradaions by 3.98 db and 8.45 db agree very closely wih he spacing of he curves in fig 3.5. Fig 5.4 shows his resul graphically for QPSK, 16-QAM and 64-QAM, whose consellaions are all ploed o he same scale of E b (energy per bi). This is achieved by scaling he unis of he axes in proporion o E g in eq(5.6), as E b is held consan. In fig 5.4, he separaion of he consellaion poins direcly shows he resilience of each modulaion scheme o noise.

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