TELE4653 Lecture 6: Frequency Shift Keying

Transcription

1 ELE4653 Leture 6: Frequeny Shift Keying When frequeny odulation was studied in analogue systes, we found the analysis to be onsiderably ore oplex than that for aplitude odulation, in a large part beause FM is a non-linear odulation tehnique. he sae is true in digital systes the analysis of frequeny shift keying is onsiderably ore involved than that of ASK and PSK. It is a powerful and popular odulation tehnique, however, so it is ertainly worth us taking the tie to address its key issues thoroughly.. Coherent Frequeny Shift Keying As we saw in the earlier hapter on signal spae onepts, two different frequeny pulses; si () t = g( t) os( f it) for i =, will be orthogonal if the frequeny differene, f = f f is an integer ultiple of, where is the sybol period. Most FSK systes ake the frequeny hoie suh that the different frequenies are orthogonal, and bandwidth onsiderations naturally ditate that the frequeny differene is always inial. Binary FSK an be defined as involving the transission, over eah sybol interval, of one of the two possible sinusoids desribed by, Eb si () t = u () t os( f it) for i =, n + i where fi = for i =,, for soe fixed integer n, whih essentially defines the base arrier frequeny, f = n. In requiring this frequeny to be an integer ultiple of the sybol rate, we assure ontinuous phase FSK, as we know that the sinusoids will always oplete a full yle of phase at the sybol transition point. Note here that we have let the aplitude shaping pulse, g ( t), be the unit square pulse, u () t, for sipliity. he effets of using ore spetral effiient shaping pulses an be aounted for using the sae analysis presented in earlier hapters, and we will free ourselves of this additional opliation, and will later find that we an further iprove bandwidth by introduing shaping pulses to sooth our transitions between the different frequenies a property that is unique to FSK. Also, using aplitude shaping pulses will ean our transitted signal will lose the attrative onstant envelope property. We an extend this idea to M-FSK by essentially expanding our set of orthogonal frequeny signals to M different frequenies, Es si () t = u () t os ( n + i) t for i =, K, M keeping the ontinuous phase property, and here the integer n defines the base arrier frequeny.

2 he generation of FSK then easy to envisage erely ap sybols to a hoie of one of M osillators, tuned to the set of frequenies we are transitting. A siple exaple of a binary FSK transitter is shown in the diagra below, after we firstly disuss the for of the signal spae and hene the optial reeiver. aster osillator sybol lok R = frequeny synthesizer f f = = arrier osillator f M f = M s(t) he above shee, however, is not partiularly pratial for the generation of ontinuous phase FSK, as it would require the phase synhronisation of these different frequeny osillators. his is oneptually an easy idea, but in real hardware this is near ipossible. Hene, real FSK transitters are built using onventional analogue FM odulators, feed by the inforation sequene represented by a suitable line ode and then passed through soe frequeny pulse shaping iruitry, as we shall soon see. he basis for the signal spae of FSK is also trivial to onstrut. Sine eah sybol wavefor is orthogonal to every other sybol wavefor, the basis signals are just noralised versions of the signals theselves. si () t ψ i () t = = u () t os ( n + i) t for i =, K, M E s FSK is an orthogonal signal onstellation, in whih eah sybol vetor is orthogonal to every other sybol vetor. he binary ase is skethed below. For larger M the onstellations are diffiult to draw, but the general struture is that eah sybol sits on its own axis, orthogonal to every other signal s axis. Note that FSK has the sae attrative property as PSK that every transitted sybol arries the sae energy. he other iportant insight that an be gathered fro this onstellation is that the distane between every pair of signals in the onstellation is exatly the sae, and equal to Es. Note also that every sybol is a nearest neighbour to every other sybol, so as we inrease the order of the onstellation, M, the nuber of neighbours to eah signal point inreases too. his feature is a signifiant differene between FSK and both QASK and PSK.

3 ψ E s E s ψ We thus have the for of the optial reeiver. It onsists of a bank of orrelators, eah athed to one of the M different frequenies that ould have been transitted. his is generi struture is shown below. ψ ( t) Reeived signal r ( t) = s( t) + n( t) ψ ( t) dt dt r r Disriinator inputs, r k ψ K (t) dt r K Naturally, the shee shown above is a oherent reeiver, sine we have assued that the reeiver an generate synhronous opies of eah of the basis signals. his is pratial for sall onstellations (sall M), for whih we an use a oon osillator to generate a base frequeny and then ix this signal to produe the frequeny-shifted versions. his is not pratial for large onstellations sizes, however. In this ase, we are ore likely to opt for a non-oherent reeiver, disussed shortly, or the trellisbased Maxiu Likelihood Sequene Estiator (MLSE), to be disussed in the ontext of ontinuous phase odulation.. Bit Error Rate Perforane he error perforane of M-FSK is fairly easy to deterine, ared with our understanding of its signal spae struture. 3

4 Firstly, for binary FSK, the two signals are separated by a distane d = Eb. he exat bit error rate an then be alulated as E b P eb = Q N hus, binary FSK has worse error perforane than binary ASK/PSK a redution in effetive SNR by one half, or 3 db. his was readily apparent fro an inspetion of the signal onstellations binary ASK/PSK ahieves axial separation for an M = onstellation. For M-FSK, we an deterine our upper bound approxiation for the bit error rate as follows. Fro the signal onstellation it is lear that all points are equivalent, and so we an equate the average sybol error probability to the probability of error of any one of the sybols. he interesting thing here is that, if we take one speifi sybol, all other (M-) are equally distant, are nearest neighbours, and, as suh, are all equally likely to be the result of the sybol error. he sybol error upper-bound fored by suing the pair-wise error probabilities over nearest neighbours is thus, E s Pes = M Q N ( ) We annot use a Gray ode here, as we do not have a hierarhy of distanes all pairs of points are equally distant. he bit error rate is then deterined as M E ( ) = b log = M Peb Pes M Q M N or alternatively, sine we have K = log M bits/sybol, we ould write K E b log = M P eb Q N his very different fro the results we obtained for M-QASK and M-PSK. he nature of FSK is that as we inrease M the error perforane atually iproves opletely the opposite to QASK and PSK, for whih error perforane drops substantially as we inrease M. his is shown in the diagra below. As we ll see in the next setion the trade-off here is that the spetral effiieny of M-FSK falls arkedly with inreasing M. Alas, there is no best odulation tehnique. Eah has their own advantages and disadvantages. 4

5 M = P eb..5 M = 4 M = 8 M = 6 M = E b N 3. Spetral Perforane he spetral perforane of FSK an get quite opliated for large M. However, for the binary ase exat forulae an be derived. Let s begin by alulating the power spetral density for binary ontinuous phase FSK or CPFSK). his signal an be written as E t s() t = b u () t os f t ± with the sign deterined by whether the upper or lower frequeny is to be transitted. f here represents the average arrier frequeny. his an be expanded as Eb t Eb t s() t = os os( f t) sin sin( f t) Fro the above expression, it is lear that the In-phase oponent of the CPFSK signal is onstant, and independent of the sybol transitted in that interval. he Q- phase is equivalent to a binary ASK syste with the half-sinusoid shaping pulse. Making the usual approxiations of unorrelated sybol sequene, we an write an expression for the power spetral density of the binary CPFSK signal, in ters of an equivalent baseband PSD, ( ) S f = { S ( f f ) + S ( f + f )} where S B ( f ) E = δ f 4 B + δ f + B 4 8E + ( f ) ( ) os b b f ( ) Notie the for of this spetru, onsisting of two strong peaks at the pair of arrier frequenies, and the ter that oes fro the half-sinusoid behaviour. he halfsinusoidal shaping pulse is extreely spetrally effiient (reall the 99% bandwidth is.8/), so the bandwidth perforane of CPFSK is very good. here is a little bit 5

6 ore to be said here, though, in oparing the distribution of signal power within the oupied band. It is interesting to note the effet on the PSD resulting fro not preserving the ontinuous phase properties at the sybol transitions. he effet of disontinuities in phase is ake the two phases orrelated, and the ross-ters in the resulting PSD result in a net deay of f, not as the fourth power of frequeny as in CPFSK. his represents a onsiderable ost in bandwidth effiieny. he equivalent analysis for M-FSK with higher M is quite involved, so we ll restrit ourselves to aking soe broad, general oents here. he general piture of M- FSK is that we have a set of frequenies equally spaed by /, as shown in the diagra below. f f M f We ould then ake a general approxiation for the bandwidth required to transit this signal as M B = his will in general be an under-estiate of the true signal bandwidth, as we know that while the different frequeny signals are orthogonal they are not spetrally disjoint and there is a signifiant aount of overlap between the spetral oponents. hus, the leakage spetru at the end of the spetru shown in the diagra will be signifiant. However, for large M its ontribution to this bandwidth estiate will be relatively sall. Using the above result we an then obtain an approxiation for the spetral effiieny of M-FSK, R log M η = b = B M While this result is far fro being exat, it learly illustrates that the spetral effiieny of M-FSK drops heavily with inreasing M, in ontrast one again to QASK and PSK. Naturally, we an think of this as a result of the orthogonal struture of the FSK signal onstellation, that inreasing M inreases the diensionality of the onstellation, resulting in the assoiated bandwidth inrease. 6

7 4. Continuous Phase Frequeny Shift Keying We an further exploit the ontinuous phase struture of FSK to further iprove the bandwidth perforane and error perforane of FSK. he shee we will desribe here is often referred to as Continuous Phase Modulation (CPM), but it is really no ore than a variant of CPFSK with soother phase transitions. he iportant idea here at play is to exploit the inherent eory in the signal sine the phase transitions are ontinuous, our knowledge of the previous arrier phase an help is ake a deision on the urrent arrier phase, and hene the urrent transitted sybol. Let s onsider binary CPFSK. We an write the transitted sybol wavefor as Eb s() t = u () t os[ f t + θ () t ] where the arrier phase deviation is deterined by the binary sybol transitted h θ () t = θ ( ) ± t, for t he paraeter h we will define shortly. Notie that the phase hanges linearly over a sybol period, whih is akin to saying that the frequeny of the wave is onstant over a sybol period. he quantity h is alled the deviation ratio, or alternatively the odulation index (in referene to its siilarity to the analogue ase), and it is related to the differene in the arrier frequenies, h = f = ( f f ) he iniu frequeny separation to aintain orthogonality is /, whih would orrespond to h =. CPFSK with this deviation ratio is alled Miniu Shift Keying (MSK), for this reason. he deviation ratio defines the phase hange of the arrier over the sybol: h, for sybol θ ( ) θ ( ) = h, for sybol For MSK this orresponds to a arrier phase hange of ± eah sybol period. he signifiane of this is that this odulation shee now has eory, in the sense that the arrier phase state over any one sybol depends not solely on the sybol itself, but also on the previous set of transitted sybols, enapsulated by the arrier phase at the end of the previous sybol. his is very different fro QASK and PSK, as these shees had no eory, and so our optial reeiver ould at on a sybol-bysybol basis. CPFSK is ore like DPSK, and as suh our optial reeiver should ake deisions by onsidering ore than a sybol at a tie. he idea of eory in the transitted signal is well represented by the phase trellis. he phase trellis plots the possible evolution of the arrier phase for different possible input sequenes. he phase trellis is shown in the diagra below for MSK, for whih the arrier phase hanges by ± eah sybol. 7

8 3 / 3 / 3 / 3 / / / - / / t = t = t = t = 3 he iportant point in this diagra is that at any sybol period not all arrier phase states are possible for the transitted signal, and knowledge of the previous arrier phase an help us ake a deision on the urrent arrier phase. We will analyse how to use the phase trellis struture to iprove the perforane of reeiver for the ase of MSK in the next setion. CPM beyond the binary ase beoes a for of partial response signalling, whih we will touh on a little later. Our final point in CPM is the idea of a frequeny shaping pulse. It is instrutive to write our sequene of transitted frequenies, for binary-cpm, whih is really the arrier instantaneous frequeny as a funtion of tie, as where { = ±} n f i () t = f + h a g( t n ) n n= a deterines whether the frequeny transitted at that sybol is f = f 4 or f f + 4 g t represents how we transition between frequenies, whih for the signal we have been disussing, is instantaneously, g() t = u () t he instantaneous arrier frequeny as a funtion of tie for this frequeny shaping pulse is shown in the diagra below. =. he funtion ( ) We an integrate the instantaneous arrier frequeny to obtain the arrier phase as a funtion of tie. i t () t = f ( τ ) dτ = f t + h a q( t n ) θ i n n= 8

9 where q() t = g ()dt t is the equivalent phase shaping pulse, equal to the integral of the frequeny shaping pulse. For the square-pulse hoie as our frequeny shaping pulse, orresponding to instantaneous frequeny transitions, the transition pulse is, t t q() t =, < t <, t g(t) q(t) t t For MSK, this orresponds to our failiar phase transitions of ± every sybol period, as illustrated below. h ha ] q( t), a [] [ = 3 t h ha ] q( t ), a[] [ = 3 t ha 3] q( t ), a[3] = [ h 3 t θ (t) 3 t he point is that it is not neessary that we transition the frequenies instantaneously, and an envisage a shee where we transition between the two arrier frequenies ore soothly. Making the frequeny transition soother akes the overall signal soother, and this has the iportant onsequene of reduing the bandwidth of the 9

10 signal still further. A oon frequeny shaping pulse in CPM is the Raised-Cosine (RC) frequeny shaping pulse, t g() t = u () t os he iportant harateristis being erely that the frequeny shaping pulse integrates to be, so that we still aintain the MSK property of induing arrier phase hanges by ± eah sybol. Note that now we an no longer so uh as think of CPM as being us keying the frequeny of the arrier between two frequenies the arrier frequeny instead varies ontinuously over this frequeny interval, effetively averaging at one of those two frequenies. Gaussian MSK (GMSK) is an iportant pratial exaple of CPM with a non-square frequeny shaping pulse, whih was adopted as the odulation solution in the GSM global seond generation obile phone ouniation standard. he shee to generate GMSK is easy to oneive, and illustrates the general pratial way that CPM with non-square frequeny shaping pulses an be generated pass a binary square signal through a suitably designed low-pass filter to produes pulses with the desired for, that an then be used to drive a V.C.O. or another frequeny odulator. he Gaussian filter used in GMSK has the following transfer funtion, and assoiated ipulse response, h () t H = ( f ) = e log f B 3 3 B log B t 3 e log where B 3 is the filter 3 db bandwidth. he output fro this filter when a square pulse is the input an easily be shown to be, () ( ) t t g t = erf B3 erf B3 log log his is shown in the diagra below.

11 he obvious point here is that this frequeny pulse does not have finite support, and extends well outside the sybol interval, [, ). his anifests itself as ISI. he degree of inter-sybol interferene is deterined by the relative size of the tie extent of the resultant shaping pulse relative to the sybol period. he tie extent of the shaping pulse is inversely proportional the filter bandwidth, B 3, hene the extent of ISI introdued by the Gaussian filter an be easured and expressed by the 3-dB bandwidth sybol period produt, B 3. he saller this value, the larger the pulses are relative to the sybol interval, so the greater the extent of ISI. his then enapsulates the ultiate trade-off in GMSK: a ore spetrally effiient shee is ahieved by reduing the bandwidth of the frequeny pulse shaping filter to redue signal bandwidth, but this has the ounter-effet of introduing ISI in the signal. 5. Miniu Shift Keying Let s now turn our attention firly on MSK, in order to understand how to design a reeiver to ake use of the phase eory of the transitted signal. he MSK signal an be written as, Eb Eb s() t = os( θ () t ) os( f t) sin( θ () t ) sin( f t) where θ () t = θ ( ) ± t. he phase trellis is as shown below. he observation is that the arrier phase an take on values of or at the end of even nubered periods, and values of ± at the end of odd nubered periods, taking the referene initial phase as.

12 3 / 3 / 3 / 3 / / / - / / t = t = t = t = 3 Expressed as above the MSK wavefor resebles that of a QASK wavefor. Looking at the effetive in-phase aplitude, we see that, os( θ () t ) = os( θ ( ) ) os t sin( θ ( ) ) sin ± t Considering the wavefor over the two sybol periods t, we ll take t = to be the representative of the even nubered periods, then θ ( ) = or, and thus sin θ. he effetive in-phase aplitude an thus be expressed as ( ( )) E si with the sign deterined by the arrier phase at = θ =. negative if ( ) b () t = ± os, t θ and t, positive if ( ) = Siilarly, it is easy to show that the quadrature aplitude is deterined only by the arrier phase state at odd ultiples of the sapling period, θ ( ) = ±, and the quadrature aplitude an be expressed as, Eb sq () t = ± sin, t where the aplitude is positive if θ ( ) = +, and negative if θ ( ) =. his iplies that we ay onstrut an alternative basis for the MSK signal, { ψ () t, ψ () t } = os os( f t), sin sin( f t) so that the MSK signal is a linear obination of these, s( t) = sψ ( t) + sψ ( t) he oeffiients an be found as, and = s b θ () t ψ () t dt E os[ ( ) ] s = () t ψ () t dt E sin[ θ ( )] s = s = b Note that here we observe these two aplitudes over two sybol period eah, and that these intervals are staggered. his akes sense, in the sense that we annot ake an aurate estiate of say θ ( ) by only observing for t, but we an if we see the resultant hange over two sybol periods. Inherently we thus inluded eory into our optial reeiver, by arriving at this signalling basis. he

13 perforane gain of MSK oes fro allowing ourselves to extend the signal beyond the siple sybol by sybol ase. sybol over t he four obinations of arrier phase states at the end of even nubered and odd nubered sybol periods, expressed here as θ ( ) and θ ( ), ap to sybols as in the table below: ransitted Phase state at even- Phase state at even- Signal vetor, nubered periods θ ( ) nubered periods θ ( ) ( s, s ) -/ ( E b, E b ) -/ ( E, ) b E b / ( Eb, E b ) / ( E, ) b E b he signal onstellation for MSK, expressed with respet to this basis, is shown in the figure below. he struture of the optial reeiver is shown in the diagra below. he idea is that the reeiver alternatively akes deisions about the arrier phase state at even sybols by looking at the I-phase aplitude over two sybol periods, and for the arrier phase state at odd nubered sybol periods fro the Q-phase aplitude over a the two sybol periods offset fro the previous pair. Fro these pair of deisions the reeiver reonstruts the original bit sequene. 3

14 It is straightforward to show, fro the above signal onstellation, that MSK has the sae bit error rate perforane as BPSK and QPSK, when we utilise the inherent eory in the transitted signal, E b P = eb Q N Our onlusion is that MSK represents then a odulation shee that obines exellent spetral effiieny with good bit error rate perforane. 6. General Continuous Phase Modulation (CPM) We an go further and extend the disussion of the previous two setions to the general ase, and then onto partial-response signalling. he ost general transitted signal ould be written as Es s t = os f t + θ t, a + θ () ( ( ) ) θ is the resultant arrier phase due to the sequene of transitted sybols, a, a,k a i ±, ± 3, K, ± M oes fro an alphabet of M where ( t,a) a = ( ), where { ( )} sybols. he only thing of interest to us is thus this phase funtion, θ ( t,a) onveys the transitted sequene. For a linear frequeny odulation shee, this phase response will be linear obination of the phase response produed by eah sybol:, as it 4

15 θ ( t, a ) = h aiq( t i ) i= where h is the odulation index and ( t) frequeny shaping pulse, q() t = t g n q is the phase shaping pulse, defined by the ( x)dx, as before. Causality iplies that q () t =, t <, and by onvention we require that q( t) =, t L. Note here that we allow the urrent sybol to influene the arrier phase over L onseutive sybols, not siply a single sybol, as did for MSK above requiring ipliitly that L =. he odulation index, h, is generally hosen to be a rational nuber, for purposes of produing a finite state trellis, and also typially h, for reasons of spetral 3 effiieny. he data sybol auses a net arrier phase hange of a i h radians over the interval of L sybols, [ i ( i + L) ),. he two ost oon hoies for frequeny shaping pulses used in CPM are retangular pulses, alled L-REC:, t L g () t = L, otherwise and the L-RC, for raised-osine: t () os =, t L g t L L, otherwise hus, MSK is a speial ase of CPM with -REC and h =. he natural way of visualising CPM is with the phase tree, or phase trellis. A new M- ary sybol influenes the phase trajetory every sybol period,, and its influene N last for L sybol intervals. here are thus M distint paths or odewords (we will see the relationship of CPM to trellis oding later in the ourse) when the trellis is viewed over N sybol intervals. he phase trellis is saled by h. he phase trellis shown in the previous setion for MSK had M =, with the influene of the sybol lasting only for L = sybol interval. A ore general phase trellis, for 3-RC with M = 4 is shown in the diagra below. Note the extree ontinuity of the arrier phase, regardless of the preise trellis path, espeially when opared to soething like M-PSK. his belies the exellent spetral perforane of CPM. Note that, in fat, we ould inorporate our desription of M- DPSK into CPM if we allow disontinuous phase transitions. We get M-DPSK if we set the phase shaping pulse q () t to be the step funtion with height and odulation index h = M, so that the phase in eah interval hanges by one of { ± M, ± 3 M, K, ± ( M ) M }. his is exatly M-DPSK, aside fro a trivial additional phase rotation of M. 5

16 Sine CPM is a non-linear odulation tehnique it s preise spetral analysis is very opliated. However, a ouple of general rearks an be ade. he power spetru generally widens for inreasing h, with spetral lines eerging for integer values of this odulation index. Sall values of h are not in of theselves the seret to bandwidth effiient ouniation, and in fat ost of the inforation ontent of the signal will reside in the low sidelobes (students ould think bak to the analogue ase, for the equivalene of Narrowband FM to AM, so essage ontent oes in the two low power sidelobes). Finally, the phase ontinuity of the CPM signal iplies the 8 spetral sidelobes deay as f f, or 8 db/deade. his rapid spetral deay, while liiting out of band interferene, is not in of itself the end of the story it provides no inforation on how spetral power is distributed within the band. A onvenient tehnique for understanding CPM odulated signals is to think in ters of state diagras and state spae ahines. For rational odulation index, h = p r, then there are a disrete nuber of arrier phases that an exist at the end of eah sybol interval. Moreover, sine n ( t, ) θ + h a q( t n ), n t ( n + ) θ a = n j j= n L+ the entire CPM signal ould be opletely speified eah sybol by the arrier phase at the beginning of the sybol interval, and urrently ative sybols (that is, those reent sybols that are still urrently influening the arrier uulative phase). We 6

17 ould thus speify a state of the arrier as ( θ a, a, K a ) n, n n, n L+, of whih there are L rm possible values, hene possible states. he next input signal auses the CPM wavefor to hange state in a natural way. he state transition diagra for a -REC signal with h = and M = is shown 3 below. We will oe bak to the power of this state spae desription of the CPM signal, partiularly with regard to pratial ipleentation of the optial deoder, later when we eet onvolutional enoding. here are two basi ipleentation of a CPM odulator. he first uses an analogue frequeny odulator, driven by a baseband, pulse-shaped signal fro the inforation sequene. While this is fairly siple and the hardware is readily available, it does suffer fro the aging and toleranes of oponents produing drifts in the preise odulation index, h. { a n } Baseband, partial-response Pulse-shaping n a n g ( t n ) FM odulator (index h) CPM out An alternative is to use the fat that CPM an be redued relative to the QASK basis. As long as we eploy an (adittedly oplex) signal-apper with eory, we ould realise CPM we ould realise CPM in a siilar fashion to QASK, as shown in the diagra below: 7

18 { a n } Signal Mapper (with eory) ( θ ( )) os t,a ( θ ( )) sin t,a os ( f t) sin ( f t) E s CPM out Finally, let s ake a few oents about the optial detetor of CPM, whih will naturally ontinue our previous disussion of detetors that operate over a sequene of reeived sybols, rather than on a sybol-by-sybol basis. Based on our disussion in earlier letures, the optial reeiver for the sequene of N sybols ebedded in L white Gaussian noise would orrelate the reeived wavefor, r(t), against the M possible signal trajetories and hoose the one that produes the largest orrelation, under the assuptions that all sequenes are equally probable and that all signals have equal energy. hese orrelations over the interval N an be broken into a su of N orrelations eah over a interval. he issue here is we would naturally expet the oplexity of suh a reeiver to grow exponentially with N, as a Maxiu Likelihood Sequene Estiator (MLSE). here is an algorith to greatly redue this oplexity, alled the Viterbi algorith, whih exploits the finite state struture of the signal. We will explore this algorith under a different guise later in this ourse, when we apply it to onvolutional deoding. Lastly, a oent about the error perforane of CPM deoders, ipleented using this MLSE struture. An error is aused when a different path through the trellis looks ore like the reeived sequene than the sent path, and so the error perforane of CPM is related to the distane between neighbouring trellis paths. Siilarly as with the Viterbi algorith, we will explore this proble under the guise of onvolutional enoders in a later leture. Our reason for this is that the presentation of these ideas in this ontext is oneptually easier to understand, at least in the opinion of this leturer. 8

19 7. Partial-Response Signalling Partial Response Signalling is a ethod for iproving our transission rate to approah or exeed the Nyquist liit (R > B) by allowing ontrolled ISI into the transitted data sequene. he exaple of duo-binary signalling for binary PAM will be presented in the hand-written leture notes. Partial-response signalling ahieves a odest iproveent in spetral effiieny (inreased data rate for a given bandwidth alloation) at oderate ost in BER perforane (or essentially the signal power budget). 8. Optial Non-oherent Reeiver We ll now onsider deodulation forally when we have a rando arrier phase. he reeived signal is r() t = a g( t) os( f t + γ + θ ) + n( t) where θ is assued to be a rando variable, uniforly distributed as p θ ( θ ) =, < θ he set { a, γ } represent the aplitude and phase odulation that has been given to the arrier to onvey the sybol (be this ASK, PSK, FSK, or whatever). he point is that, no atter what these quantities are, they are known to the reeiver. he proble then is to design the optiu reeiver for the situation when the arrier phase θ is unknown. o otivate our disussion, let s onsider the issues surrounding the arrier phase deterination. Reeiver synhronisation ultiately requires the reeiver and the transitter to synhronise their loal osillators at the beginning of ouniation, and aintain this synhronisation for the duration of transission. In any ases this is opletely ipratial. For instane, a oveent in the relative distane between the transitter and the reeiver by a fration of a wavelength will alter the relative phases. For ouniation at GHz, the wavelength of the assoiated radiation is around.3, and a ulti-pathing radio hannel will easily have hanges in transitter-reeiver distane of this order. Moreover, onsider the effet of a frequeny offset in the osillators. If our GHz has a frequeny differene 9 of only Hz, whih is a phenoenally high quality of one part in, then the relative phases will drift by in one seond. In the next hapter we ll look at tehniques to ahieve synhronisation of the reeiver s osillator using inforation obtained fro the reeived signal the tehniques of phase loked loops and their digital equivalents. Here we ll onsider the ase, whih is a design issue, where we an attept to reover the signal without traking the arrier phase. Naturally, reeivers that have knowledge of the arrier phase ust perfor better than those that do not, but the ay interest here is to develop the tools and analysis for us to be able to forulate this trade-off. he first point is that the variation in the arrier phase eans we need to double the diension of the basis of our signal spae. he signal we reeive an be expanded as 9

20 r () t = a g() t os( f t + γ + θ ) + n( t) = a g() t os( f t + γ ) osθ a g() t sin( f t + γ ) sinθ + n() t With a oherent detetor, with θ =, we would only need a single basis funtion for E g t os f t + γ, but with unknown phase we require this reeived signal, g ( ) ( ) an expanded basis of E g( t) os( f t + γ ), E g( t) sin( f t + γ ) { } { }. g Consequently we ust double the nuber of orrelators in the optial non-oherent reeiver. he reeived signal is then orrelated with the In phase and the Quadrature of eah basis vetor (note that the quadrature of the basis vetor, we forally ean the Hilbert transfor of the basis vetor the basis vetor with its effetive phase shifted). Denote the expanded basis as { ψ ( ) ( )} K i t, ψˆ i t, where K is the diension of i= the signal spae, and the outputs of the orrelators as z = r() t ψ ()dt t z i s i = i r() t ψ ˆ ()dt t We ould then define the onditional distribution for the reeived signal vetor, p( r s,θ ), given the sent essage s and unknown hannel phase. Naturally, we would like to integrate out the unknown hannel phase, and deterine the probability of us reeiving r given s was sent, regardless of the arrier phase, p θ, ( r s ) = p( r s, θ ) p ( θ ) dθ = p( r s θ ) For the ideal AWGN hannel, this probability distribution is deterined only by the noise, sine the noise is the only reason our reeived signal differs fro the transitted signal. hus, K p( r s ) = exp( ( ri si ( θ )) N ) dθ i= N Expanding the exponent we find, K K r N i p( r s ) = e ([ ri si ( θ ) si ( θ )] N ) dθ K ( N ) exp i= i= or in ters of the vetor represented in the expanded signal spae, r N p( r s ) = e exp( [ ( θ ) ( θ ) ] N ) dθ K r s s N ( ) Having expressed the probability distribution in ters of the signal vetors allows us to interpret eah ter ore readily. s ( θ ) is another way of writing the sybol energy, E, whih is independent of θ (and always independent of our hoie of basis for the signal spae). he first ter in the exponent is the orrelation between the reeived signal and the essage signal, i g dθ

21 r s ( θ ) = r( t) s ( t, θ )dt o siplify atters, we ll onsider the ase of an orthogonal signal onstellation, so s t, θ = s ψ t os θ ψ t sin θ, and then that ( ) ( () ( ) ( ) ( )) r s ( θ ) = s ( z osθ z sinθ ) s his siplifiation allows us to express eah signal in ters of one basis signal and not as a su over the basis signals. It is not diffiult to expand this analysis into the general situation of a signal onstellation that is not orthogonal, however this is rarely needed in pratise, and does inrease the level of algebra needed substantially. One an iediately oneive of this siplifiation in ters of the signalling basis. For an orthogonal signal onstellation of diension K the optiu reeiver onsisted of a bank of K orrelators, one for eah eleent signal of the basis. his ould alternatively be realised by a bank of K athed filters. In the equivalent nonoherent detetor, to aount for the unknown arrier phase, we ust expand eah orrelator into a pair, eah orrelating over the alternative arrier phase. For a nonoherent detetor this realisation of the expansion of the signalling basis an be ore diffiult to envisage. For the ase of an orthogonal signalling shee, we an write the probability distribution as r N E N p( r s ) = e e exp E [ z osθ z θ ] N d K s sin ( N ) his is the key step to allow us to integrate out the angular dependene. Define and so that p ( r s ) ( N ) z = z + z s z α = tan z r N E N = e e K ( ) θ s exp ( E z os( θ + α ) N ) dθ Here we have effetively deterined the envelope aplitude of the reeived signal, by taking the orrelation with respet to eah arrier phase and obining. his integral is atually the zero-order odified Bessel funtion of the first kind, defined by I ( x) = [ x β ] dβ exp os hus, ( ) ( ) r N = E z E N p r s e e I K N N Note that I ( x) is a onotoni inreasing funtion of x, as shown in the diagra below.

22 We typially work in log-likelihood ratios, when it oes to deterining deision thresholds on the transitted sybol. For the sybol, the orresponding loglikelihood ratio is, after ignoring ters that are independent of, the sent sybol, and hene will not help us distinguish between sybols z E E l ( ) = log( p( r s )) = log I N N Note that if we followed the analysis through, not aking the assuption of an orthogonal signal onstellation, the log-likelihood ratio would need to be expressed as a su of ters dependent on the quadrature outputs fro the pairs of in-phase and quadrature orrelators, and the assoiated signal oponents relative to those basis vetors, K zisi E l ( ) = log I i N = N he added oplexity that we do not wish to onsider, though, at a fundaental level, will not alter our onlusions. he general for of an orthogonal signal onstellation, suh as FSK whih we are ultiately foussed on in this hapter, though we ai to keep our analysis as general as possible, is that all the sybols have the sae energy, say E g = g. Our hoie of essage, given the reeived vetor r, is one that has the highest probability of r being reeived, over all possible essages sent, = arg ax p( r ) hanks to the onotoni nature of log(x), this is equivalent to axiising loglikelihood ratio. hus, z E = arg ax log I N As log(x) and I ( x) are onotoni, we ould erely axiise z or essage is hosen fro the rule, = arg ax z with z + = z zs. z. hus, our We an oneive the optial non-oherent detetor as a bank of pairs of orrelators, onsisting of a orrelator tuned to eah basis signal and its Hilbert transfor. he

23 reeiver then squares and adds the pairs of aplitudes fro eah orrelator pair, and hoses the essage that orresponds to the largest, One interesting point to note is that deterining the aplitude z is equivalent to deterining the arrier envelope, z os ( f t) zs sin( f t) z os( f t + α ) hus, eah pair of orrelators on eah basis signal ould instead be replaed by an envelope detetor, after an appropriate athed filter has been used to selet that orthogonal oponent fro the others. he three equivalent fors of reeiver ipleentation for a single basis oponent are shown in the diagra below. 3

24 One an see that the non-oherent reeiver, when applied to an orthogonal signalling shee suh as FSK, is siple and heap to ipleent, as it does away with any need of oplex arrier reovery iruits. he iportant question to then ask is: what is the assoiated ost in error perforane in using a non-oherent detetor? For an orthogonal signal onstellation, our sent signal vetor is s = (, K,, Es,, K,) sine we send one of the orthogonal signals eah sybol. Our reeiver then fors a reeived signal vetor by passing the reeived signal r(t) through a pair of orrelators, squaring then adding the outputs, and finally square-rooting, to for eah oponent of the reeived signal vetor, r = ( z, z, K, z, K, z M ) Fro the orthogonal nature of the onstellation, only one of these oponents, z, that orresponding to the sent signal, will depend on the signal energy the others are just due to the noise. he reeiver hoses the essage orresponding to the largest oponent in the reeived signal vetor r. he reeiver will ake an error if one of the other oponents of the reeived signal vetor is larger than z, the one on whih we sent our signal vetor. P = P z > z, for soe k es ( ) k 4

25 o deterine this probability we ust find the distribution of eah of these oponents. We have already perfored ost of the analysis for this on pages 5 and 6, when we established the probability distribution for the reeived signal vetor. For the probability that the reeived aplitude is z, given the essage sent was, we find a Riian distribution, z z E s p( z ) = I exp( ( z + Es ) N ) N N while the other oponents of the reeived signal follow a Rayleigh distribution, zk p( zk k) = exp( zk N ) N his intuitively akes sense, sine for k, the resultant distribution is fored by taking the root-squared su of two zero ean Gaussian rando variables, with is known to be a Rayleigh distribution. When the Gaussian rando variables do not have zero ean, as in the ase for the oponent the essage was sent on, the result of the root-squared su is a Riian distribution. o find the probability of a sybol error, it is easiest to find the probability that a orret deision is ade. A orret deision is ade if, for our reeived oponent z, all other oponents are less than this, then average over all possible values of z, weighted by the likelihood of that value ourring. ( orret ) = P( zk < z for all k ) P( z ) dz P Using the distributions above, and the fat that eah oponent is independent, 5

26 P z N M z N ( orret ) = e dz P( z ) = z M z N z z E s [ e ] I exp ( z + Es ) dz ( N ) dz N N he above integral looks fairly involved, however it is atually quite siple. When the ter to the exponent (M-) using the binoial theore, it redues to a su of integrals of the for xe x dx, eah of whih is straightforward to opute. One finds that the sybol error probability is then, M ( ) ( ) = = j+ Pes P orret j= j + M exp j je ( j + ) N Being an orthogonal onstellation, the assoiated bit error rate for this non-oherent detetion shee is, with K = log M bits per sybol, M = ( ) ( ) j+ M M jeb log M ( ) P eb exp M j + + = j j j N One should note that, for binary signalling, this has the quite siple for, Eb N P = e eb s 9. Non-oherent FSK Having ade the general struture of the optial non-oherent detetor for an arbitrary orthogonal signalling shee, we ll ake a few speifi oents for its appliation in FSK. he first point above that was glossed over was that it was inherently assued that, while the basis set of signals is orthogonal, so is the expanded basis of the signal basis and the Hilbert transfor of eah basis signal. his is not trivial, and in the ase of FSK, would ean requiring that, for arbitrary frequenies with separation f = f f, ( f t) sin( f t) dt os = Perforing the integral this requires that os ( f ) = f whih is true if the frequeny spaing is an integer ultiple of, exatly double the frequeny separation needed to aintain orthogonality in oherent FSK. his is the ase, that non-oherent detetion of FSK involves a redution in spetral effiieny of one-half. Not aintaining orthogonality would involve a uh too greater ost in bit error rate and reeiver oplexity. 6

27 Aepting this inrease in frequeny trade-off, a non-oherent detetion shee ould then be ipleented for M-FSK, following the priniples explained in the earlier setion. he for of the reeiver is as shown, ψ ( t) = osft dt r Reeived signal r ( t) = s( t) + n( t) ψ s( t) = sin ft dt r s Disriinator inputs, r k,r ks ψ Ms ( t) = sin f M t dt r Ms or else, the alternative struture using envelope detetors previously desribed. he bit error rate resulting fro using suh a detetor is, M = ( ) ( ) j+ M M jeb log M P eb exp M j + + = j j j N ( ) he bit error rate perforane of non-oherent FSK is worse than that of oherent FSK, with the sae reason as we saw in the ase of DPSK, resulting fro the effet of squaring or ultiplying the noise ters. It is worth oparing the perforane of the different shees in the binary ase. For non-oherent detetion for binary FSK, the bit error rate is n.. FSK Eb N Peb = e For basi CPFSK with oherent detetions, we have. FSK E b P = eb Q N Finally, for MSK, with the optial detetor (in turn the sae as BPSK), we find MSK E b P = eb Q N hese are plotted together in the figure below. 7

28 . Case Study - GSM GSM (originally Groupe Speiale Mobile, but later Global Standard for Mobile Couniation) was the European solution for the seond generation of obile phone ouniation networks. he seond generation referred to a fully digital network, as the first generation used analogue odulation tehniques. he GSM was widely adopted around the world, and was the leading G obile standard worldwide, and as suh, is the basis for any of the urrent 3G obile phone networks. A suary of the iportant features of the GSM standard with regard to Radio layer of ouniations is as follows: Uplink frequeny band: MHz Downlink frequeny band: MHz Channel bandwidth: khz Multiple Aess: ie Division (DMA) 8 tieslots/frae Modulation: GMSK - ie-bandwidth produt B 3 =.3 Data rate: kbps (Data rate for individual user varies) Speeh oding: Regular Pulse Exited Long er Predition (RPE-LP, really LPC) Channel Coding: Convolutional Coding, plus blok interleaving. Diversity: frequeny hoping and tie interleaving. he iportant point for us is the use of GMSK, Gaussian Miniu Shift Keying, hosen for its very good bit error rate perforane and exellent bandwidth onservation. Blok diagras illustrating the odulation and deodulation proess for a GSM syste are shown below: 8

29 9