TELE4653 Lecture 5: Phase Shift Keying

Transcription

1 TELE4653 Leture 5: Phase Shift Keyin n this leture we ll examine in detail Phase Shift Keyin(PSK). PSK is a very popular diital modulation tehnique, applied in many pratial systems. We ll address eah of the harateristis that mae it suitable for a wide variety of appliations.. Coherent Phase Shift Keyin n M-ary Phase Shift Keyin (M-PSK), the transmitted sinal durin a symbol period onsists of the arrier with one of M possible phases, spaed equally over the interval to π: s t = t os π f t + θ ( ) m ( ) ( ) [ m] ( M ) π π 4π where θ[ m],,, K. M M M This an be expanded usin the usual trionometri identities, and we an easily see that M-PSK an be represented with exatly the same sinal basis as we used for ASK. where () t s m () t = a E ψ ( t) + a E ψ ( t) m m () t = os( πf t) and () t ψ E ( t) ψ = sin E ( πf t) and the amplitudes of the relative phases are ( a m a m ) os θ [ m ] ( ( ), sin ( θ [ m ] ), =. The sinal onstellations for Binary PSK, 4-PSK, and 8-PSK are shown in the diaram below. ψ ψ ψ ψ ψ ψ -PSK: phase ambiuity = nπ 4-PSK: phase ambiuity = ½ nπ 8-PSK: phase ambiuity = ¼ nπ The first obvious point is that the sinal onstellation of Binary PSK is idential to that of Binary-ASK. t is not diffiult to see from this that in fat these two sinallin shemes are idential. Note that the two arrier phases for binary PSK are θ [ m ] {,π }, and the arrier amplitudes are thus, ( a, a ) { ( +, ), (,) }. The m m transmitted sinal is thus,

2 s m ( t) = ± ( t) os( πf t) exatly idential to bandpass binary ASK. Consequently all the results we have derived for binary ASK arry over into binary PSK, so we need not reprodue them here. The more observant will also realise that the sinal onstellation of 4-PSK, often alled PSK, is idential that of 4-ASK. The onstellation shown above is really a rotated version of the sinal onstellation of 4-PSK, if one was to about the onvention that the transmitted phase set is θ[ m ] {, π, π,3π }. Rotated versions of our sinal onstellation are merely the result of different hoies of the sinal basis, and we now that the preise seletion of the sinal basis is irrelevant in assessin the performane of a sinallin sheme. Hene, we an onlude that 4-ASK and 4- PSK are one and the same. However, for M > 4 the struture of the PSK onstellations are fundamentally different from that of ASK. There are a few points to mae reardin M-PSK in eneral. Every symbol has exatly the same transmitted enery, E, reardless of the assined disrete phase. This maes PSK robust to the distortin effets of non-linear amplifiers. We ll develop this idea later when we disuss later when we address Offset-PSK. The identifiation of the sinal basis for PSK as that of ASK ives us the struture of a PSK modulator and of the optimal demodulator. These are illustrated in the diarams below.

3 The deision reions an be learly interpreted from the sinal onstellation. The deision reions are setors of the plane, bisetin eah pair of sinal points. They are shown below for M = 8. ψ s 3 s 4 s θ 4 θ 3 s 5 θ s ψ E s 6 s 8 s 7 The bit error rate performane of M-PSK will be the subjet of the next setion.. Bit Error Rate Performane As disussed in the previous hapter on sinal spaes it is very diffiult to obtain exat expression for the BER for omplex onstellation lie that of PSK for M > 4. Consequently we attempt to establish error bounds, in partiular bounds, as performane metri. The error situation is illustrated below: ψ s 3 E s 4 s s 5 s ψ s 6 s 8 s 7 E 8 3

4 Naturally all symbols are equivalent, in the sense that they all have the same set of distanes to neihbours and the same eneries, by the symmetry of the onstellation. Thus we an find the averae symbol error probability by onsiderin the symbol error probability for a sinle symbol, say s, P es = P e The probability of symbol s bein in error is merely the probability that the reeived sinal vetor lies in the shaded reion in the above plot. The would represent a fairly ompliated two dimensional interal to ompute exatly for the -D Gaussian noise distribution, so we an use the union bound approximation to estimate this area. The union bound approximates the probability of the symbol s bein in error as the sum of the pair-wise probabilities of the symbol s bein inorretly deteted as the differin symbol, s. where P e M P ( s s ) = = = d represents the distane between s and M d N s in the sinal spae. However, as was disussed in the Sinal Spae hapter, the union bound is usually exessive, and a tihter and simpler approximate bound an be found by restritin the sum to just inlude nearest neihbour points. That this is still an upper bound is well illustrated in the above fiure it is equivalent to addin toether the probability measure of the two half-planes, formed as the bisetors of the neihbourin points. Clearly there is a sinifiant reion of area that is double ounted in sum, but at relatively hih sinal to noise ratios the ontribution this area maes will be small. P e d + d8 N N The distane between nearest neihbour points in the sinal onstellation an be alulated as, π d n. n. = Es sin M Hene, we find the followin approximate expression for the symbol error probability for M-PSK, (for M > 4) π ( ) Es sin M P es N We note that the PSK onstellation does lend itself readily to the appliation of a Gray ode, as there is a lear hierarhy of distanes between points in the onstellation. Usin the Gray ode idea, we obtain the BER for M-PSK, π ( ) lo M Eb sin M P eb lo M N The interest is then to ompare the BER performane of PSK to that of ASK. Both shemes, for a iven value of M have the same sinal basis, and onsequently an be 4

5 assumed to oupy the same bandwidth. Thus, for a iven value of M both shemes should have idential spetral effiieny. The raph below plots the BER for PSK and ASK for different values of K = lo M, when the SNR per bit is db. t is evident from the plots that ASK has superior BER performane. This is beause, for a iven enery budet, whih is equivalent to fixin the averae distane of sinal points from the oriin, ASK will ahieve reater averae separation between neihbourin points. Essentially, ASK is a more effiient way of pain points in a two dimensional plane than the PSK onstellation, whih onstrains all points to lie on a irle entred at the oriin PSK P eb ASK P eb = lo M K 3. Spetral Performane t is a straihtforward matter to show that the Power Spetral Density (PSD) of M- PSK is idential to that of M-ASK, where ( f ) ( S { } ( f ) = R ( E E ) G( f f ) + G( f f ) S + s G is the spetrum of the shapin pulse. Here we mae the usual assumption that the transmitted symbol sequene is unorrelated. The bandwidth of a M-PSK an then be approximated as γ B = T where the fator of omes from the fat that the sinal is in the passband, and the parameter γ is a numerial onstant that is determined both by the hoie of shapin pulse and preise definition of bandwidth (null to null, 99% power, 3dB, et.) For instane, for the half-sinusoid shapin pulse, γ =. 8, while for the Nyquist squareroot Raised-osine overall transfer funtion we have γ = ( + β ), with roll-off fator β, when we use the 99% enery definition of bandwidth. The spetral effiieny M-PSK an then be alulated as, 5

6 R lo η M = b = B γ We find the same results for spetral effiieny as we saw previously for ASK. As we inrease M, the spetral effiieny improves at the expense of error performane and power budet. The error performane of M-PSK is worse than that of M-ASK, however, there are a few additional features that an be introdued into M-PSK to mae it an attrative hoie for a modulation sheme. We ll address these fators now. 4. Offset - PSK This is a pratial modifiation that an be done to a PSK sheme to improve its robustness to distortions introdued by non-linear effets in real amplifiers. An ideal amplifier is linear in its voltae response, that is, the input voltae v in is related to the output voltae v out by, v out = Kv in where K is the voltae ain, whih may be frequeny dependent but ertainly not amplitude dependent. However the response of a pratially implementable amplifier must loo more lie the diaram below it must saturate at some finite voltae level. Output voltae, v out nput voltae, v in The point is that a real amplifier will not amplify the low amplitude parts of the sinal the same as the hih voltae parts, whih will alter the balane of the sinal and possibly distort its information ontent. Not only do amplifiers suffer from voltae non-linearities lie this, so do antennae, waveuides, and other eletrial omments and sub-systems in our ommuniation lin. Offset PSK is a lever way main the sinal robust to suh non-linear effets in the hannel and the ommuniation system in eneral. To understand this, let s write the PSK sinal as s ( ) () t ( t nt ) os π f t + θ m[ n] = n= 6

7 { } where θ m[ n] is the sequene of transmitted phases. The sinal power as a n= funtion of time is found by squarin the sinal, s ( ) () t ( t nt ) os π f t + θ m[ n] = n= sine, neletin S introdued by the hannel, the shapin pulses sent at different symbol intervals should be disjoint. We find the power profile by averain out the rapidly varyin arrier terms, () = ( t nt ) P t n= The shapin pulse is in eneral not onstant over a symbol interval, and hene the power profile of the transmitted sinal will flutuate at the symbol rate R = T. This is shown in the diaram below. The exeption to this is naturally the unit retanular shapin pulse, () t = ut () t, however this pulse has very poor bandwidth effiieny and as suh is not popular in pratise. For any other type of shapin pulse the power profile will vary with time, and the sinal is thus suseptible to distortions aused by non-linear amplifiers. (t) P(t) T t Offset PSK is a simple tehnique to obtain a more onstant power profile by offsettin the in-phase and quadrature data streams by half a symbol period. Denotin T = T b, where T b is the bit period (or time to send one bit, nowin that for PSK we are transmittin two bits per symbol), we write the O-PSK sinal as s () t = a ( t nt ) os( πf t) a ( t ( n + ) T ) sin( πf t) [ ] [ ] b b ] m n m n [ n= The power profile for this sinal is, () t = ( t nt ) + ( t ( n + ) T ) n= b P n= The offsetted -phase power profile fills in the aps of the -phase power profile, to produe an overall smoother power profile. This is illustrated in the diaram below. b t 7

8 For the ase when the shapin pulse is a half-sinusoid, πt () t = ut () t sin T the power spetrum of the O-PSK sinal is onstant. π ( t ntb ) π t ( n + ) Tb P() t = sin + sin n= Tb Tb ( ) ( ) π t ntb π t ntb = sin + os n= Tb Tb = [ ] n= - ( ) The half-sinusoid also has exellent spetral effiieny, so O-PSK with this shapin pulse is a popular solution in ommuniation systems. 5. Non-oherent detetion of PSK We ll address the eneral problem of non-oherent detetion in the followin hapter. Non-oherent detetion refers to detetion in the ase where the reeiver annot obtain a synhronised version of the arrier, as this may be too ostly, or alternatively the desiner may hoose to disreard the phase information at the expense of some deradation in noise performane. Partiularly for narrowband wireless ommuniation systems, the arrier phase an be very diffiult to reover, as there an be transmission over a multitude of paths of different and variable lenths, and rapidly varyin delays in the propaatin medium from transmitter to reeiver may ause the reeived phase to vary in a way that it is hard for the reeiver to follow. The first omponent of a non-oherent PSK system is differential enodin. Differential enodin maes the transmitted sinal dependent not on the preise arrier phase, but instead the information is enapsulated in the hanes in arrier phase. Differential enodin of PSK, nown as DPSK, an be ahieved by enodin the transmitted phases as π θ θ m n m = + [ n] m[ n ] [] M 8

9 where θ m[ n ] represents the arrier phase at the last transmitted symbol, and m n,, K M is the messae to be transmitted at the nth symbol. [] { } DPSK lends itself to reovery with a very simple and eleant reeiver struture. The reeiver for DPSK is shown in the diaram below. How this reeiver differs from reeiver strutures that we have previously onsidered is that it has memory, in the sense that the deisions are made usin the previous reeived sinal values, not solely on a symbol by symbol basis. Our urrent ommuniation model assumed both the data sinal and the hannel to memory-less, and onsequently the transmission and detetion of a partiular symbol was not at all effeted by what was transmitted or reeived before. Thus, our optimal reeiver has always been implemented in a symbol by symbol manner. n DPSK, we have introdued a dependene of the urrent transmitted symbol on the previous symbol, and onsequently the form of optimal reeiver must hane. T ( t) ( ) dt E r [n] [ n ] delay r os ( π f t +φ ) r(t) loal osillator mixer r phase detetor phase Shift, 9 º sin ( π t +φ ) f T ( t) ( ) dt E delay r [n] [ n ] r r The feature of this reeiver is that it will suessfully reover the transmitted symbol reardless of the phase error of the loal osillator, φ, by omparin the previous symbol to urrent symbol and in doin so, removin the dependene of the reeived symbol on φ. The reeiver reovers two amplitudes, by orrelatin the reeived symbol with both phases of the loal osillator, r n = E os θ φ n n The noise terms, [] n r n and [ ] [] s m[ n] [] n Es sin θ m[ n] ( ) [ ] ( ) n [ n] + = φ + n n, are two samples from independent and identially distributed Gaussian white noise proesses. The mixer then ombines these reeived amplitudes with those of the previous symbol as follows: r = r n r n + r n r n [ ] [ ] [ ] [ ] [ n] r [ n ] r [ n] r [ ] r = r n One an easily show that these amplitudes are ompletely independent of the loal osillator phase, r E os θ θ + n ( m[ n] m[ n ]) ( θ m n θ m n ) + n = s = Es sin r [ ] [ ] 9

10 Sine, for DPSK, our information is assoiated with the differene in arrier phase, the deision rule is same then as for M-PSK, on the reovered amplitude pair ( r, r ). The deision rule is to hoose the transmitted messae m that is losest to the reeived amplitude sinal vetor, πm πm m = ar min r Es os + r Es sin m M M The deision reions are idential to those of M-PSK. ψ Δs 3 Δs 4 Δs Δs 5 y Δs ψ deision thresholds E Δs 6 Δs 8 Δs 7 The noise performane of this non-oherent DPSK reeiver is in eneral more ompliated than for oherent DPSK, as the noise variables affetin the reovered amplitudes, n and n are proportional to the produt of Gaussian random variables, and as suh are not Gaussian distributed. For the binary ase an exat expression for the BER is easily able to found. n the next hapter we ll loo at the error performane of the optimal non-oherent demodulator for binary orthoonal sinal. The BER we ll obtain is, Eb N Peb = e The point here is that we an visualise binary-dpsk as a binary orthoonal system lastin T. The last symbol ats lie us transmittin a phase referene, so in effet our symbols are one of either s t t os π f t + θ, t os πf t + θ () () ( ) ( ) ( ) () t () t os( π f t + θ ), ( t) os( πf t + θ ) s Note that eah new symbol is transmitted every T seonds, with the previous symbol beomin the phase referene. These symbols are equivalent to the pair of orthoonal vetors {( +, + ), ( +, ) }. Usin that eah bit effetively taes two symbols to transmit (sine we must transmit the phase referene too), we find the expression for the BER of non-oherent detetion of DPSK, n.-dpsk Eb / N P = e eb t is instrutive to ompare this result to that obtained for oherent BPSK, and also DPSK when oherent detetion is used. Note that DPSK an also be reovered with the oherent BPSK reeiver, with the additional ompliation that the reovered

11 messae is obtained by tain the differene of the reeived symbol with the previous reeived symbol. The BER of DPSK with oherent detetion is twie that of nondifferentially enoded BPSK, sine for differential enodin any error will affet two reovered symbols the urrent and the next. BPSK E b P = eb N -DPSK E b P = eb N The three ases are ompared in the diaram below. The important result is that BPSK -DPSK n.-dpsk P eb < Peb < Peb meanin that differential enodin a ost in error performane in main transmitted symbols dependent, while the non-oherent detetor a further ost in error performane by doublin the number of orrelators required, reatin more soures of noise on our reovered sinal. - - P e E b N (db) Fiure: Comparison of the BER for (i) binary PSK with oherent detetion (lower urve); (ii) binary DPSK with oherent detetion (middle urve); and (iii) binary DPSK with non-oherent detetion (upper urve). The BER for hiher M-PSK with non-oherent detetion is fairly ompliated to alulate, but as expeted the demodulator s la of phase nowlede always osts in enery effiieny. Fundamentally this is beause in DPSK detetion two noise vetors influene the reovery of the phase differene however exat results are diffiult as produts of Gaussian variables means hi-squared statistis. The omparative performane of non-oherent detetion as a funtion of M for M-DPSK is shown in the fiure below. Due to the poor enery effiieny of non-oherent M-DPSK for M > 8, partiularly when ompared to their oherent ounterpart, these are rarely used in pratise.

12 6. Optimal Linear Reeiver Revisited At this point it is worth revisitin our disussion of the optimal linear reeiver. As we saw in the above setion, when we introdue orrelation between the symbols transmitted at different symbols our optimal reeiver is no loner a symbol-by-symbol detetor, and must instead involve some memory. The other soure of orrelation between different symbols at the reeiver is of ourse the hannel, by suh effets as multi-pathin, ulminatin in ntersymbol nterferene (S). t is natural then to as: what is the optimal form of the reeiver when S is present? Earlier, we determined the form of the reeiver that was optimal when we onsidered the effet of noise alone our mathed filter. The reeived sinal is, tain the ase simple baseband ASK system, r( t) = a ( t T ) + n( t) where () t = ( h )() t is the form of the pulse after it has been passed throuh the hannel with impulse response h ( t). The linear reeiver we an then thin of a filter with impulse response () t. We pass the reeived sinal throuh this filter and sample the output at time instants t = T (or equivalently realise this reeiver with a orrelator). The reovered symbols are y T = ξ + n ( )

13 where ξ a ( τ ) ( T nt τ ) dτ n n = is the reeived sinal amplitude and n is the noise at the output. We desire naturally that for perfet operation that we have ( T ) a y =. The deviation of our obtained output from that desired is, nown as our error, e = ξ + n a For the Gaussian distributed noise samples, our optimal reeiver is obtained by desinin it to minimise the mean square of this error, J = [ ] E e The fator of one half is introdued for onveniene. This is J = E ξ + E n + E a + E ξ n E ξ a E a n [ ] [ ] [ ] [ ] [ ] [ ] The from the independene of the noise from our sinal we have, E[ ξ n ] = E[ a n ] = The other expetation values in the above mean squared error must be diretly alulated. Firstly, E [ ξ ] = E[ ala ] ( τ ) ( τ ) ( lt τ ) ( T τ ) dτ dτ l We ll onsider that the transmitted symbol sequene is unorrelated, so E[ a al ] = δ [ l], and treat when the transmitted symbols are orrelated separately, as we did for DPSK above, and will aain for Continuous Phase Modulation(CPM) in the next hapter. Thus, E [ ξ ] = R ( τ, τ ) ( τ ) ( τ ) dτ dτ, = τ where R ( τ τ ) ( lt τ ) ( ) T in some way aptures the S introdued by the hannel. Similarly, one easily finds that E [ ξ a ] ( τ ) ( τ )τ d = N E[ n ] = δ( τ τ ) ( τ ) ( τ ) dτ dτ Substitutin these into the form of the mean squared error, N J = + R( t ) + ( t ) ( t) ( ) dt d ( t) ( τ δ τ τ τ t)dt To minimise this mean square error, with the best hoie of reeiver, we use the alulus of variations methods to define the funtional hoie of the reeiver filter t to minimise the above expression. The result is the equation () N ( t τ ) + δ ( t τ ) ( τ ) d = ( t) R τ 3

14 Note for the previous ase, when we neleted S, so R ( t τ ) = E δ ( t τ ) the mathed filter result, () t ( t) eneral form of the optimal filter is, ( f ) = N. Expressed in the frequeny domain, the + T G ( f ) C G f + T Note the similarity of the above expression that of an R filter, C( z) = N. + z =, we et This is important, as we an realise this optimal linear filter as a mathed filter, followed by a tapped delay line diital filter, as shown below. This form of reeiver is nown as a linear equaliser. t is usually required to mae this reeiver filter adaptive, to aount for the hanin hannel harateristis. The priniples of adaptive equalisation will be overed in other ourses. This type of reeiver, whih we will explore aain when we onsider partial response sinallin, is now as the Maximum Lielihood Sequene Estimator (MLSE). 7. Case Study S-95 CDMA S-95 CDMA, short for nterim Standard-95 Code Division Multiple Aess, was the G mobile phone system used in the USA, Japan, and Korea, and formed the basis for the urrent 3G mobile networs that are in the proess of bein rolled out and implemented. The modulation tehnique hosen for this standard was π/4-odpsk, or π/4 Offset Differential uadrature Phase Shift Keyin. The only feature of this modulation sheme that we have not yet disussed is the π/4 aspet. This is the idea to mae the phase shifts introdued into the transmitted arrier wave are multiples of π/4. That is, eah input symbol introdues a hane in arrier phase as desribed in the followin table: 4

15 nput Bit Pair Phase hane of arrier π/4 3π/4-3π/4 -π/4 The motivation for this that every symbol will ause a hane in the phase of the arrier, main symbol synhronisation easier. The priniples of symbol synhronisation, where the reeiver must determine when transmitted symbols bein and end will be overed in a later hapter. Symbol synhronisation is an important and ritial issue in Diret Sequene Spread Spetrum Multiple Aess tehniques, however disussion of the priniples of CDMA is left for later ourses. Note that for DPSK, the phase hanes of the arrier are one of {,, π, π } π, and thus the transmitted arrier phase over any symbol must be one of {, π, π, π }, and we see that the sinal onstellation of DPSK is the same as that of PSK. However, for π/4-dpsk there are two alternative sinal onstellations: if we all the initial arrier phase state as θ =, then for all even symbols, the arrier phase is one of {, π, π, π }, while for all odd numbered symbols the arrier phase state is one of { π / 4,3π 4, 3π / 4, π 4}. This alternatin sinal onstellation is shown in the diaram below. The ey parameters of the S-95 CDMA system are listed below. Uplin Frequeny Band: MHz Downlin Frequeny Band: MHz Number of arriers/band: Bandwidth per arrier:.5 MHz Number of users per arrier: 6 Chip rate:.88 Mps Pulse-shapin: Overall Nyquist Square-root Raised Cosine Channel Codin: onvolutional enodin, Uplin (3,,9), Downlin (,,9) Modulation (Uplin): 64-ary orthoonal Hadamard, with π/4-odpsk for spreadin. (non-oherent detetion) Modulation (Downlin): BPSK for data with π/4-odpsk for spreadin (oherent detetion) Speeh Data rate: 96, 48, 4, bps Speeh odin: Variable rate Code-Exited Linear Predition (CELP) 5

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