Fitting Signals into Given Spectrum Modulation Methods

Transcription

1 S Post-graduate Course i Radio Commuicatios Fittig Sigals ito Give Spectrum Modulatio Methods Lars Maura 41747e Lars.maura@hut.fi

2 Abstract Modulatio is the process where the message iformatio is embedded ito the radio carrier. Message iformatio ca be trasferred i the amplitude, frequecy or the phase compoet of the carrier sigal. Modulatio methods are categorised accordig to which compoet is used for trasmittig the iformatio. To achieve high spectral efficiecy, modulatio schemes eed to have high badwidth efficiecy. Three properties eed to be satisfied, whe digital modulatio techiques are chose for wireless systems. First, compact power desity spectrum with a arrow mai lobe ad fast roll-off of side-lobes is required to miimise the chael iterferece. Secodly a good error rate performace i all eviromets is required. Fially a costat evelope is importat i mobile applicatios, where battery power is a limited sourcead amplifiers are typically o-liear. I this study, differet modulatio methods ad the bit error performace with differet modulatio methods ad sigal sets are evaluated. Oe importat factor i bit error performace is the shape of the pulse. To prevet itersymbol iterferece the selected pulse shape has to satisfy the Nyquist criterio. The ideal Nyquist pulse, however, has slow time decay. Therefore other pulses that satisfy the criterio has to be costructed. Differet modulatio methods are evaluated. The sigal costellatio is a importat factor whe error probability is calculated. I coheret demodulatio of two equally likely sigals trasmitted o AWGN chael the error probability depeds oly o the Euclidea distace betwee the two sigals. Ay digital modulatio aims at realisig the best possible trade-off i a give situatio amog the bit error probability, the badwidth efficiecy, the sigal to oise ratio ad the complexity of the equipmet. I the ed the performace of these modulatio methods are compared. The power desity fuctio is ot i the scope of this study. The backgroud material cosists of three books. All of them descibes digital modulatio methods ad could be used as such. The most part i this study is refers to St über [1], but the presetatio of Nyquist criterio is maily based o Lee [2] ad i the evaluatio of error performace I used Beedetto [3]. Lars.maura@hut.fi 2(39)

3 Table of Cotets Abstract... 3 Table of Cotets... 5 Abbreviatios Digital Modulatio Nyquist Pulse Shapig Error Probability Evaluatio Symbol Error Probability for Biary Sigals Symbol Error Probability for Rectagular Sigal Sets Digital Modulatio Schemes Quadrature Amplitude Modulatio Phase Shift Keyig Offset Quadrature Phase Shift Keyig π/4 -DQPSK Orthogoal Modulatio Orthogoal Frequecy Divisio Multiplexig Multiresolutio Modulatio FFT-based OFDM System Cotiuous Phase Modulatio Full Respose CPM Miimum Shift Keyig Partial Respose CPM Digital Modulatio Trade-Offs Litterature Lars.maura@hut.fi 3(39)

4 Abbreviatios AWGN BER FFT ISI ML LAN PDS Additive White Gaussia Noise Bit Error Rate Fast Fourier Trasform Itersymbol Iterferece Maximum Likelihood Local Area Network Power Desity Spectrum Modulatio methods: π/4-dqpsk π/4-differetial QPSK CPM Cotiuous Phase Modulatio CPFSK Cotiuous Phase Frequecy Shift Keyig DCPSK Differetially Coheret Phase Shift Keyig FSK Frequecy Shift Keyig GMSK Gaussia Miimum Shift Keyig MRM Multiresolutio Modulatio MSK Miimum Shift Keyig OFDM Orthogoal Frequecy Divisio Multiplexig OQPSK Offset QPSK PAM Pulse Amplitude Modulatio PSK Phase Shift Keyig QAM Quadrature Amplitude Modulatio QPSK Quadrature Phase Shift Keyig 4(39)

5 1 Digital Modulatio Modulatio is the process where the message iformatio is embedded ito the radio carrier. Message iformatio ca be trasferred i 1. amplitude, 2. frequecy or 3. the phase of the carrier or a combiatio of these i either aalog or digital form. I digital cellular systems digital modulatio is used because of its badwidth efficiecy. To achieve high spectral efficiecy, modulatio schemes eed to have high badwidth efficiecy, measured i uits of bits per secod Hertz of badwidth (bits/s/hz). Whe digital modulatio techiques are chose for wireless systems followig three properties eed to be satisfied: Compact Power Desity Spectrum: To miimise the effect of adjacet chael iterferece, the power radiated ito the adjacet bad should be 60 to 80 db below that i the desired bad. Hece, modulatio techiques with a arrow mai lobe ad fast roll-off of side-lobes are eeded. Good Bit Error Rate Performace: A low bit error probability must be achieved i the presece of fadig, Doppler spread, itersymbol iterferece, adjacet ad cochael iterferece ad thermal oise. I this presetatio oly itersymbol iterferece ad oise are cosidered. Evelope Properties: Portable ad mobile applicatios typically employ o-liear power amplifiers to miimise battery drai. No-liear amplificatio may degrade the bit error rate (BER) performace of modulatio schemes that trasmit iformatio i the amplitude of the carrier. Also, spectral shapig is usually performed prior to upcoversio ad o-liear amplificatio. To prevet the regrowth of spectral sidelobes durig o-liear amplificatio, relatively costat evelope modulatio schemes are preferred. Two of the more widely used digital modulatio techiques for cellular mobile radio are π/4-dqpsk ad GMSK. I both modulatio methods the iformatio is carried i the phase compoet of the carrier sigal. Lars.maura@hut.fi 5(39)

6 2 Nyquist Pulse Shapig Example: If the chael is a ideal badlimited chael B( jω ) = 1, whe ω < W ad B( jω ) = 0, whe ω W, the the ideally badlimited pulse ca be used which i time domai is a sic pulse as show below. Now cosider two successive symbols with values a = 0 1 ad a 2 1 = these two symbols to the sigal is show below. The cotributio of If the chael is ideally badlimited, the the receiver oly eeds to sample at 0 ad T. Neighborig symbols do ot iterfere with oe aother at the proper samplig time, so there is o itersymbol iterferece (ISI). Cosider a modulatio scheme where the complex evelope has the form s~ () t = A x p( t T ) Where p () t is a shapig pulse, { } x is the complex data symbol sequece, ad T is the baud period. Now suppose the complex evelope is sampled every T secods to yield the y, sample sequece { } Where 0 yk = ~ s 0 0 ( kt + t ) = A x p( kt + t T ) t is a timig offset assumed to lie i the iterval [,T ) ( mt ) p m = p is the sampled pulse 0. Whe t 0 ad 0 = Lars.maura@hut.fi 6(39)

7 yk = A x pk = Axk p0 + A k x p The first term is equal to the data symbol trasmitted at the k th baud epoch, scaled by the factor p 0. The secod term is the cotributio of all other data symbols o the sample y k. This term is called itersymbol iterferece (ISI). To avoid the appearace of ISI, the p must satisfy the coditio sampled pulse respose { } k p p k = δ k 0 0 Therefore, to avoid ISI the pulse p () t must have equally spaced zero crossigs at itervals of T secods. This requiremet is kow as the (first) Nyquist criterio. A equivalet requiremet i the frequecy domai is P Σ 1 = T = T ( f ) ˆ P f + = p0 This allows us to desig pulses i frequecy domai that will yield to zero ISI. First cosider the pulse P ( f ) T rect( ft ) =, k Figure 1 Pulse rect(ft) This pulse yields a flat folded spectrum. I the time domai p(t) = sic(t/t) This pulse achives the Nyquist criterio because it has equally spaced zero crossigs at T secod itervals. Furthermore, from the requiremet of a flat folded spectrum, it achieves zero ISI while occupyig the smallest possible badwidth, hece, it is called a ideal Nyquist pulse. However, the problem with this pulse is that the roll-off of side-lobes is slow. Better roll-off factors are give by raised cosie ad root raised cosie pulses which also achieves Nyquist criterio, see figure 2. Raised cosie pulses are give by Lars.maura@hut.fi 7(39)

8 p () t si π ( πt T ) cos( απt T ) t T 1 ( 2αt T ) = 2 where α is the so called roll-off factor. For α = 0, the pulse is idetical to the ideally badlimited pulse. For other values of α, the eergy rolls off more gradually with icreasig frequecy. The pulse for α = 0 is the pulse with the smallest badwidth that has zero crossigs at multiples of π W ; larger values of α require excess badwidth varyig from 0% to 100% as α varies from 0 to 1. I the time domai, the tails of the pulses are ifiite i extet. However, as α icreases, the size of the tails dimiishes. Figure 2 Raised cosie pulses with differet roll-off factors There are a ifiite umber of pulses that satisfy the Nyquist criterio ad hece have zero crossigs at multiples of π W. Some of these are show i figure 3. Figure 3 The Fourier trasform of some pulses that satisfy the Nyquist criterio Lars.maura@hut.fi 8(39)

9 3 Error Probability Evaluatio It is assumed that the aalog chael coectig the modulator output to the demodulator iput is a additive white Gaussia oise (AWGN chael with a ifiite badwidth. The demodulator is a maximum likelihood (ML) demodulator ad operates accordig to miimum distace rule. Figure 4 Geometry of the miimum distace rule The sigal i figure 4 is a complex sigal with three possible symbols. Whe symbol s I is received, the receiver observes sigal r. While the chael adds oise to the trasmitted sigal, the observed sigal is r = s I + s i as i figure 4. The miimum distace detector chooses the earest value of the possible sigal set. For correct detectio, the received sigal has to be observed i the correct decisio area. Noise is assumed to be zero-mea Gaussia oise with variace N 0 /2. Havig a radom sigal, i.e. all symbols are equally likely, the symbol error probability is expressed as P M 1 () e = P() c = 1 P( c s j ) 1, M j= 1 where P( c s j ) is the probability of a correct decisio give that the sigal vector s j, correspodig to the symbol m j, was trasmitted. Lars.maura@hut.fi 9(39)

10 3.1.1 Symbol Error Probability for Biary Sigals Figure 5 Detectio decisio regios for biary sigals For a biary sigal i figure 5 b), the symbol error probability ca be determied as follows. Sigal s 1 is detected with error, if oise elemet causes the detectio to recogise a value less tha 0. This happes whe the additive oise equals < d /2. Now we ca write the symbol error probability as P () e = P < d 2 Usig the defiitio of error fuctio ( ξ > ) = 1 x P x erfc m, 2 2σ ad rememberig that oise is zero-mea with variace N /2 0, symbol error probability ca be writte 1 () d P e = erfc. 2 2 N 0 Lars.maura@hut.fi 10(39)

11 Figure 6 Error probability for atipodal ad orthogoal biary sigals This meas, that coheret demodulatio of two equally likely sigals trasmitted o the AWGN chael, the error probability depeds oly o the Euclidea distace betwee the two sigals. I.e. the selectio of the set of symbols has a impact o error probability. Comparig atipodal sigals to orthogoal sigals i figure 6 shows, that there is a 3 db pealty i the sigal eergy to be paid with orthogoal sigals with respect to atipodal sigals which are show i figure Symbol Error Probability for Rectagular Sigal Sets The biary represetatio ca be applied to sigal sets that have a rectagular cofiguratio. This meas the cases where the decisio regios are 2D-hyperplaes. Figure 7 Received samples perturbed by additive Gaussia oise form a Gaussia cloud aroud each of the poits i the sigal costellatio Lars.maura@hut.fi 11(39)

12 Figure 8 2D sigal set with 16 sigals, a 16-QAM sigal costellatio By studyig the differet decisio regios i the sigal set i figure 8, we ca see that there are oly three differet types of decisio areas. First we eed to defie the probabilities for correct decisios. ( ) ( ) ( ) p 1 = ˆ P c s 1 p 2 = ˆ P c s 2 p 3 = ˆ P c s 3 Whe differet oise-compoets are idepedet of each other, we ca defie by usig the results of biary case s1 s2 s1 s5 p1 = P 1 < P 2 > 2 2 d d = P 1 < P 2 < = ( 1 p) where p is the symbol error probability for biary sigals with Euclidea distace d betwee the differet symbols. With similar calculatio we ca obtai 2 3 ( 1 2 )( 1 ) p = p p p ( 1 2p) = 2, where p = ˆ 1 2 erfc 2 d N 0 Fially we ca obtai the total symbol error probability for sigal set i figure 8 ( ) P e = 4p + 8p + 4p = ( p) 2 Lars.maura@hut.fi 12(39)

13 4 Digital Modulatio Schemes 4.1 Quadrature Amplitude Modulatio Whe usig quadrature amplitude modulatio (QAM) the data iformatio is trasmitted i the amplitude compoet of the sigal. The quadrature represetatio is a special case of pulse amplitude modulatio (PAM). I QAM, two PAM-sigals are combied i, ad the combiatio of these determies the trasferred sigal. With QAM, the complex evelope is where ~ s () t = A b( t T, x ) () b ( t, x ) = x h ( t) a ha t is the amplitude shapig pulse ad x xi, jxq, = + is the complex data symbol that is trasmitted at epoch. It is apparet that with the amplitude ad the phase of a QAM sigal deped o the complex symbol. QAM has the advatage of high badwidth efficiecy, but amplifier oliearities will degrade its performace due to the o-costat evelope. Figure 9 Complex sigal-space diagram for square QAM costellatio A variety of QAM sigal costellatios may be costructed. Square costellatios ca be costructed whe M is a power of 4, as show i figure 9. Whe M is ot a power of 4, the sigal costellatio is ot a square. Usually, the costellatio is give the shape of a Lars.maura@hut.fi 13(39)

14 cross to miimise the average eergy i the costellatio for a give miimum Euclidea distace betwee sigal vectors. Error probability for 16-QAM was calculated i previous chapter. 4.2 Phase Shift Keyig I PSK modulatio the iformatio is sigalled i the phase-compoet. The complex evelope is where ~ s () t = A b( t T, x ) jθ ( t, x ) h ( t) e b = The carrier phase takes o values a 2π θ = x + θ 0 M Figure 10 Complex sigal-space diagram QPSK, OQPSK ad π/4-dqpsk The source biary symbols are Gray-coded. As a cosequece, adjacet phase sigals differ by oly oe biary digit Offset Quadrature Phase Shift Keyig QPSK or 4-PSK is equivalet to 4-QAM. The QPSK sigal ca have either ±90 or ±180 phase shifts from oe baud iterval to the ext. With offset QPSK (OQPSK) sigals the possibility of ±180 phase shifts is elimiated. I fact the phase ca chage by oly ±90 every T b secods. This correspods to the bit rate of the sigal. Lars.maura@hut.fi 14(39)

15 Figure 11 I-phase ad quadrature basebad compoets i QPSK, OQPSK ad MSK sigals With OQPSK two bits are trasferred every baud epoch as i QPSK, but the quadrature compoet is delayed by half of the baud rate. With this shift, the two separate compoets ever chage at the same time. The differece betwee QPSK ad OQPSK is show i figure 9. The i-phase compoets are the same, but the quadrature compoet is shifted by half of a symbol i OQPSK. Note from figure 10 that the phase trajectories do ot pass through the origi. This property reduces the peak-to-average ratio of the complex evelope, makig the OQPSK sigal less sesitive to amplifier o-liearities tha the QPSK sigal. What is gaied from OQPSK with respect to QPSK: Both methods have same error performace, sice sigal costellatio is equal. The gai of choosig OQPSK is o power desity spectrum. I OQPSK the ±180 phase shifts are elimiated ad hece the pds is more compact π/4 -DQPSK QPSK trasmits 2 bits/baud by trasmittig siusoidal pulses havig oe of 4 absolute phases. π/4-dqpsk also trasmits 2 bits/baud, but iformatio is ecoded ito the Lars.maura@hut.fi 15(39)

16 differetial carrier phase ad siusoidal pulses havig oe of 8 absolute carrier phases are trasmitted at each baud epoch. The phase differeces are ±π/4 ad ±3π/4. The absolute carrier phase durig the eve ad odd baud itervals belogs to the sets {0, π/2, π, 3π/2} ad {π/4, 3π/4, 5π/4, 7π/4}. With π/4-dqpsk the amplitude shapig pulse is ofte chose to be the root raised cosie pulse. The sigal space diagram is show i figure 10, where the dotted lies show allowable phase trasitios. Note that the phase trajectories do ot pass through the origi. Like i OQPSK, this property reduces the peak-to-average ratio of the complex evelope, makig the π/4-dqpsk sigal less sesitive to amplifier o-liearities. The error performace is equal to QPSK, sice sigal costellatio durig oe symbol is same (or shifted by π/4). 4.3 Orthogoal Modulatio Orthogoal modulatio schemes trasmit iformatio by usig a set of waveforms, 1 { ( )} M sm t m= 0 that are orthogoal i time. FSK modulatio techique provides simple meas of geeratig a orthogoal sigal set. Orthogoal M-ary frequecy shift keyig (MFSK) modulatio uses a set of M waveforms that have differet frequecies. For coheret demodulatio orthogoality is met whe the correlatio coefficiets of the real sigal is zero. This coditio is fulfilled whe the frequecy separatio betwee adjacet sigals is such that m 2 ft= d, m ay iteger. 2 Thus the miimum frequecy separatio for orthogoality with coheret detectio is such that 2fT= d 0,5. The demodulatio of of these types of sigals icreases the complexity i the receiver. The eed for a bak of perfectly coheret oscillators reders it rather impractical. The bit error performace is though differet from amplitude ad phase modulatio techiques. There exists a improvemet i performace whe M is icreased, which is exactly the opposite behaviour of PAM ad PSK sigals. However, this improvemet is obtaied at the expese of a larger badwidth. Icreasig M requires more frequecies ad therefore more badwidth. For icoheret demodulatio the orthogoality coditio eed to be fulfilled idepedetly of the phases of the sigals. The coditio is fulfilled whe the frequecy separatio betwee adjacet sigals is Lars.maura@hut.fi 16(39)

17 2 ft d = m, m ay iteger which is twice as much frequecy separatio as of coheret demodulatio. The performace is somewhat iferior to the coheret case, but this is traded off by the easier implemetatio. 4.4 Orthogoal Frequecy Divisio Multiplexig Orthogoal Frequecy divisio multiplexig (OFDM) is a modulatio techique that has bee suggested for use i cellular radio, digital audio broadcastig, digital video broadcastig ad wireless LAN systems. OFDM is a block modulatio scheme where data symbols are trasmitted i parallel by employig a (large) umber of orthogoal subcarriers Multiresolutio Modulatio Multiresolutio modulatio (MRM) refers to a class of modulatio techiques where multiple classes of bit streams are trasmitted simultaeously that differ i their rates ad error probabilities. Figure QAM embedded MRM sigal costellatio, defiig two priority classes Figure 12 shows ad example of a 16-QAM MRM sigal costellatio, that ca be used to trasmit two diferet classes of bit streams, called low priority (LP) ad high priority (HP). Two HP bits are used to select the quadrat of the trasmitted sigal poit, while two LP bits are used to select the sigal poit withi the selected quadrat. I order to cotrol the l h relative error probability betwee the two priorities, a parameter λ = d d is used. I geeral, λ should be less tha 0,5, sice the MRM costellatio becomes symmetric 16- QAM at this poit. As λ becomes smaller, more power is allocated to the HP bits ad they are received with a smaller error probability. Lars.maura@hut.fi 17(39)

18 4.4.2 FFT-based OFDM System A key advatage of usig OFDMis that the modulatio ad demodulatio ca be achieved i the discrete-domai by usig a discrete Fourier traform. The fast Fourier trasform (FFT) algorithm efficietly implemets the discrete Fourier trasform. Whe FFT is used, the rectagular shapig pulses tur ito o-causal pulses i figure 13. Figure 13 Time domai OFDM amplitude shapig pulse Aother key advatage of OFDM is the ease by which the effects of ISI ca be mitigated. This ca be doe, by usig a cyclic guard iterval. The guard is appeded to the geerated sigal i the trasmitter. O the receiver it is assumed that the first sample is corrupted by ISI ad therefore replaces the ISI-compoet with the guard. 4.5 Cotiuous Phase Modulatio Cotiuous Phase modulatio (CPM) refers to a broad class of frequecy modulatio techiques where the carrier phase varies i a cotiuous maer. CPM schemes are attractive because they have costat evelope ad excellet spectral characteristics, i.e. arrow mai lobe ad fast roll-off of side lobes. The complex evelope of a geeral CPM waveform has the form ~ j( φ () t + θ ) s () t = Ae 0 where φ () t is called the excess phase ad defied as () t φ t 0 k = 0 () t = π h x h ( τ kt ) 2 dτ k k f h f is the frequecy shapig fuctio, that is zero for t < 0 ad t > LT. A full respose CPM has L = 1, while partial respose CPM has L > 1. The phase is cotiuous i CPM sigals so log as the frequecy shapig fuctio does ot cotai impulses, which accouts for all practical cases. h k is the modulatio idex. Lars.maura@hut.fi 18(39)

19 4.5.1 Full Respose CPM Cotiuous phase frequecy shift keyig (CPFSK) is a special type of full respose CPM obtaied by usig the rectagular frequecy shapig fuctio with L = 1. h f 1 2LT () t = u () t LT Figure 14 Phase tree of biary CPFSK with modulatio idex h. CPM sigals ca be visualised by sketchig the evolutio of the excess phase Φ(t) for all possible data sequeces. This plot is called a phase tree ad a typical phase tree is show i figure 14 for biary CPFSK. I each baud iterval, the phase icreases by πh if the data symbol is +1 ad decreases by πh if the data symbol is Miimum Shift Keyig Miimum shift keyig (MSK) is a special case of CPFSK, with modulatio idex h = 1 2 ad umber of levels M = 2. The MSK sigal ca be described i terms of the phase tree i figure 14 with h = 1 2. At the ed of each symbol iterval the excess phase φ () t takes o values that are iteger multiplies of π 2 ad a phase trellis may be plotted. Figure 15 Phase trellis diagram for MSK. Lars.maura@hut.fi 19(39)

20 Cosider the MSK bad-pass waveform i the iterval [ T ( 1) T ], +, give by () = x π π s t Acos 2π f c t xk x. 4T 2 k = 0 2 Observe that the MSK sigal has oe of two possible frequecies f L 1 = f c ad 4T f U = f c 1 + 4T The differece betwee these frequecies is f = fu f L = 1 2T. This is the miimum frequecy separatio to esure orthogoality betwee two co-phased siusoids of duratio T ad, hece, the ame miimum shift keyig. By viewig figure 15 this type of modulatio ca be thought of as a special case of OQPSK i which the rectagular waveform is replaced by a siusoidal pulse, which is show i figure 11. Figure 16 Performace of differet CPFSK sigals Figure 16 shows performaces of differet CPFSK sigals compared to MSK. By icreasig the umber of levels, the badwidth efficiecy is icreased clearly Partial Respose CPM Partial respose CPM sigals have a frequecy shapig pulse h f (t) with duratio LT where L > 1. Partial respose sigals have better spectral characteristics tha full respose CPM sigals, i.e., a arrower mai lobe ad faster roll-off of side lobes. Lars.maura@hut.fi 20(39)

21 5 Digital Modulatio Trade-Offs Ay digital modulatio aims at realisig the best possible trade-off i a give situatio amog the bit error probability P b (e), the badwidth efficiecy R s /W, the ratio ε b /N 0 ad the complexity of the equipmet. Followig results are take from Beedetto [3]. Figure 17 Compariso of differet modulatio methods o the badwidth-efficiecy plae for a bit error probability P b (e) = A compariso of differet modulatio methods is illustrated i figure 17 where a bit error probability P b (e) =10-5 has bee fixed. The Shao capacity boud shows the maximum badwidth efficiecy, which ca teoretically be achieved. The graph shows the fact that amplitude modulatio (ASK), ad phase modulatio (CPSK ad DCPSK) systems are badwidth-efficiet sigallig techiques, sice they cover the regio of the plae where R s /W > 1. I this regio, the system badwidth is limited ad it ca be traded for power (i.e. ε b /N 0 ). I fact, for a fixed badwidth, the badwidth efficiecy ca be icreased with a icrease i the umber of levels M. The price paid to achieve the same P b (e) is a icrease i ε b /N 0. O the other had, FSK sigals make a iefficiet use of badwidth, sice they cover the regio of the plae where R s /W < 1. But these systems trade badwidth for a reductio of the ε b /N 0 required to achieve the same P b (e). Lars.maura@hut.fi 21(39)

22 Litterature [1] Stüber, G. Priciples of Mobile Commuicatio. Secod editio. Norwell, Massachusetts, USA. Kluwer Academic Publishers p [2] Lee, E.A. Messerschmitt, D.G. Digital Commuicatio. Secod editio. Norwell, Massachusetts, USA. Kluwer Academic Publishers p [3] Beedetto, S. Biglieri, E. Castellai, V. Digital Trasmissio Theory. Eglewood Cliffs, New Jersey, USA. Pretice-Hall, Ic p (39)