TELE4653 Lecture 3: Quadrature Amplitude Shift Keying (QASK)

Transcription

1 TELE4653 Letue 3: uadatue Amplitude hift Keyin (AK) n this hapte we ll onside Amplitude hift Keyin (AK) as a ommuniation system when thee is the pesene of a aie wave. As we saw in the pevious setion, the diet modulation of a baseband AK ommuniation system onto a hih fequeny aie wave esults in a loss in spetal effiieny of exatly one half, as the bandpass system oupies twie the bandwidth of the baseband system in ahievin the same ommuniation data ate. Howeve, we onstuted a mappin whih showed that a bandpass system of bandwidth B an be mapped to two baseband systems of bandwidth B, whih ae not in eneal independent. We intepeted this esult as meanin that the two independent aie phases pesents us with two available ommuniation hannels the n-phase (-phase the efeene aie wave) and the uadatue Phase(-phase a aie 9 out of phase with the efeene). Any bandpass ommuniation sheme usin AK would thus seek to utilise both aie phases, as failue to do so epesents a sinifiant ost in spetal effiieny. hemes utilisin the two aie phases ae alled uadatue Amplitude hift Keyin (AK) systems. AK shemes ae widespead in ommuniations, and have been applied in a divese ane of ommuniation shemes. Pehaps one of the most well known appliations of AK was in the ealy voieband modems, itial in binin the intenet into homes. These voieband modems took diital data fom a ompute at data ates of 9.6kps, 8.8kbps, o 33.6kbps, fo symmeti modems, and 56.kbps fo asymmeti modems, and modulated this data onto a elatively low fequeny aie wave (in the ode of 1.8kHz), usin a vaiety of AK o hybid-ak shemes, fo tansmission ove the standad telephone line and the Publi withed Telephone etwok (PT). 1. Baseband Equivalent ystem The basi AK ommuniation system is shown in the diaam below. m tansm itte s ( t ) a ( t ) m os ( π f t ) s ( ( t ) m tansm itte s ( t ) a ( t ) m sin os ( π f t ) ( π f t ) Channel esponse h ( ( t ) ( t ) deal LPF ( ( t ) ) ( t deal LPF sin ( π f t ) n ( ( t ) 1

2 Fo an M-AK system, we take K lo M oups of input bits and beak up this K bit wod into two, with K K K. The fist K bits ae modulated onto the -phase aie as a M -AK sinal, whee K lo M, while the othe K bits beome the -phase M -AK sinal. ote that M M M, so that the total numbe of symbols in the -AK system is the podut of the numbe of symbols on the and the phases. The tansmitted sinal is thus, s( a t nt n [ ] [ n] ( ) os( πf a[ n] ( t nt ) sin( πf whee the -phase and -phase amplitudes fo the nth tansmitted symbol ae ± 1, ± 3, ± 5,, ± M 1 a ± 1, ± 3, ± 5, K, ± M 1 espetively, a K and [ ] { ( )} [ n] { ( )} n whih ae used to detemine the amplitude of the shapin pulse modulated onto the aie wave at a fequeny of f. This sinal is then tansmitted ove a passband hannel, with impulse esponse h ( (, and eeived oupted by AWG noise n. The simplest AK system is fo M 4, often alled 4-AK. Hee, eah tansmitted symbol aies two bits, with the -phase and -phase of the aie eah ayin one bit. The amplitudes of the -phase ae thus a [ n] { ±1} and similaly fo the amplitudes of the -phase aie a [ n] { ±1}. Eah tansmitted symbol is then one of fou possible amplitudes, as shown in the diaam below: a 1 1 a n 16-AK we tansmit fou bits pe symbol, two on eah aie phase. The ±1, ± 3 a ±1, ± 3, with the 16 possible symbols amplitudes ae a [ ] { } and [ ] { } shown below: n n

3 3 E / E / 3 E / E / E / 3 E / E / 3 E / t is possible to have AK systems in whih the numbe of bits sent on eah aie phase is unequal, thouh these ae less ommon in patise. Fo instane, 8-AK would involve bits sent on the -phase, say, and one sinle bit on the -phase. a a Thus, in 8-AK eah tansmitted symbol onsists of one of eiht pais of amplitudes, a, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1 1, 1, 3, 1, 3, 1 ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) a { } n the last hapte we onstuted a mappin of this bandpass ommuniation system into a baseband equivalent sheme, shown in the diaam below. The mappin poeeded by findin baseband equivalent sinals fo eah bandpass sinal. m tansmitte s Equivalent Channel Filte: h Filte: h - oise n m tansmitte s Filte: h Filte: h n oise 3

4 The ompliation hee is the tem h ( in the baseband equivalent of the hannel impulse esponse, whih intodues mixin between the two equivalent baseband hannels. This is essentially a onsequene of the phase esponse of the hannel, intoduin a mixin between the aie phases, and hene, the two ommuniation hannels. t is instutive fist to look at the ase whee this quadatue tem in the hannel impulse esponse vanishes, h (. Reall that this would imply that the hannel tansfe funtion possesses onjuate symmety about the aie fequeny. This is a featue of many ommon hannel models, and, amon othes, the flat ain hannel (with H ( f ) A, fo f B f f B ) has this popety. n the ase when h () t, the baseband equivalent system beomes two ompletely independent baseband AK systems. m tansmitte s Equivalent Channel Filte: h oise n m tansmitte s Filte: h n oise Hene, we an avoid intesymbol intefeene () in this sheme by desinin the system to satisfy the yquist Citeion. We an eove the sinal on eah phase usin a mathed filte, () ( h )( t E to detemine the eoveed amplitudes of the sinal on eah phase, (, ) ( s (, (, s (, ( ) Then the lo-likelihood estimates an then be alulated on eah of these eoveed sinals as befoe, dependin on whethe MAP o ML detetion is bein used. Let s onside this detetion poedue in a little moe detail.. Optimal Reeive n the ase of ML estimation, fo the onjuate symmeti hannel, the demodulation involves usin the same mathed filte fo eah phase sinal, () ( h )( t E to detemine a pai of eeived amplitudes, 4

5 ( ) ( a n a n ) (, ) s () t, (, s (, (, whee the noise samples on eah phase ae statistially independent samples fom idential White Gaussian distibutions. The deision ule fo the two phase hannels is ompletely independent. Fo ML estimation, we have: ( ) am m a max m σ m a max m ( a ) To obtain the most likely sent messae pai ( m, m ). Fo MAP estimation the poedue is the same exept that these lo-likelihood atios ae adjusted by lo ( P m ), whee P m is the pobability of that messae ouin. ote that in the MAP ase this deision ule may not be independent, as the pobably of the tansmitte sendin patiula ombinations of amplitudes on the -phase and -phase may not be independent. ow, it is easy to see how the two eeived sinals an patially be eoveed usin a pai of mixes and LPFs as in the taditional oheent deteto. m σ ( LPF h( T t ( ) t nt os( πf () t LPF h( T t ( ) t nt sin( πf This shows the mathed filte implementation, thouh one should bea in mind that thee is the equivalent oelato, moe suitable fo diital implementations. t is instutive hee to aise the issue of synhonisation, whih will be teated as a topi in its own iht late in the ouse. The is then what happens if the s loal osillato has a phase offset of φ say, elative to the loal osillato of the tansmitte. Then, the sinal eoveed on the -phase is: ˆ t t os πf t φ and similaly fo the -phase, () () ( ) ( () t os( πf () t sin( πf ) os( πf t φ) () t osφ () t sinφ (hih fequeny tems) 5

6 ˆ () t () t sin( πf t φ) ( () t os( πf () t sin( πf ) sin( πf t φ) () t osφ () t sinφ (hih fequeny tems) Afte the oelato, o mathed filte, the eoveed amplitudes ae: ˆ osφ sinφ whee ( ) ˆ osφ sinφ, ae the eoveed amplitudes with no phase eo (i.e. when φ ). These eeived amplitudes ae the oiinal amplitudes, otated by the anle φ in the, -plane. ote that, fo a that is not pefetly synhonised, we will have ( ) oss-hannel intefeene. ynhonisation is thus a impotant desin issue in any AK system. Let s now onside the eneal ase, when we do not impose onjuate symmety of the hannel tansfe funtion about the aie fequeny. m tansmitte s Equivalent Channel Filte: h Filte: h - oise n m tansmitte s Filte: h Filte: h n oise Fo this ase it is not diffiult to show that we equie two mathed filtes on eah hannel, () ( h )( t E () t ( h )( The eeived sinal fo eah phase omponent is t a h t a h () ( )( ) ( )( n ( () t a ( h )( a ( h )( n ( The fou oelations then ae: t ( a E, a E, () () E () n, () () t ( a E, a E, n, () t ( a E, a E, n, () () t ( a E, a E, n, 6

7 t is vey easy to see, usin the independene of the noise distibutions, that ou best a, is fomed by estimates of the tansmitted amplitude pai ( ) a () E E () E E An appopiate linea ombination of oelato outputs. The deisions an then be made on these deteted amplitudes as desibed above, usin eithe MAP o ML methodoloy. The onlusion hee is that, even fo the eneal bandpass hannel eadless of symmety, as lon as we an estimate the hannel impulse esponse (usin an equalise) and build appopiate mathed filtes, the bandpass AK system is still equivalent to two independent baseband AK systems, in ead to modulation, demodulation, deision makin, and of ouse equiements. The additional ompliation is to ahieve synhonisation. Thee is moe to be said eadin BER pefomane and the powe spetal density of the AK shemes. These ideas will enompass the followin two setions. 3. Bit Eo Rate The pevious setion showed that the deisions on eah of the phase an be made independently. Howeve, this doesn t mean that the pobabilities of makin eos on eah of the phases ae neessaily independent. To see this onside the detetion poblem in veto fom. Takin the simplest ase of the onjuate symmeti hannel implemented with a mathed filte on eah phase, the eeived sinal is (, ) ( a n, a n ) ( a, a ) ( n, n ) The fomulation an easily be extended to the eneal hannel ase, howeve the additional touble need not woy us hee. ote that the sinal detetion poblem an be undestood as a veto poblem. The eeived amplitude veto is the sent sinal amplitude plus a two dimensional Gaussian noise veto, n ( n, n ). This noise veto is a sample fom a two dimensional Gaussian distibution whose omponents ae unoelated and independent. We tend to then model the detetion poblem in tems of deision eions deide on the eeived symbol based on what eion of the plane the eeived amplitude veto in elation the tansmitted points. Fo example, some deision eions ae shown in the diaam below fo 4-AK and 8-AK. These ideas will be developed futhe when we look at sinal spaes in the next hapte. We ould then find the aveae symbol eo pobability by aveain ove the symbol the pobability that eah symbol is in eo. The pobability that a patiula symbol is in eo is then the pobability that the noise n pushes the eeived symbol outside of its deision eion. This is quite ompliated fo eneal AK sinallin shemes fo two majo easons. Fistly, the deision eions ae in eneal not the same fo all points of the onstellation. This is obvious fom the deision eions skethed below fo 8-AK. eondly the pobability that noise pushes the sinal 7

8 outside of these deision eions is not eneally expessible in tems of simple - funtion statements. a 1 1 a Deision eion fo symbol (, ) ( 1, 1) m in 4-AK m a a ome deision eions fo 8-AK Let s fistly alulate the bit eo ate fo 4-AK, fo whih it is possible to obtain an exat expession easily. All the symbols ae equally likely, and makin a oet deision on a patiula symbol equies us to make oet deisions on both bits. Let P es be the pobability of makin an eo on the symbol, and P ~ eb be the pobability of makin an eo in the deision on one of the bits alone on one of the phases. Then, 1 P ( 1 ~ ) es Peb o, ~ ~ P es Peb Peb ow, ~ E P eb σ 8

9 and all symbols have the same eney, Es E b AK an thus be witten as, E E. Eah symbol aies two bits, s 1 Peb, and a Gay ode an be used to ive Pes. The bit eo ate fo 4- P 4AK E 1 b eb f the sinal to noise atio elatively lae, the -funtion will be small and we an nelet the squaed tem to ive, 4AK E b P eb This is exatly the same expession as we obtained fo the BER of binay AK. This shows a lot of equivalene between binay baseband AK and 4-AK: they have idential BER and idential spetal effiienies. The next simplest AK onstellation is fo M 8. t is not woth pusuin the exat expession hee, and instead we will look fo an expession that is valid in the limit of lae R. n this ase we only need to onside the pobability of noise ausin the deteted symbol to be pushed ove the neaest deision theshold, as all othes ae fa less likely. The neaest deision theshold is a distane of fom the symbol of inteest. n 8-AK thee ae two lasses of points the fou inne points, that have thee neaest neihbous; and the fou oute points that only have two neaest neihbous. The pobability that noise pushes a point ove its neaest neihbou bounday is ~ Pe P( n ) ( σ ). The appoximate expession fo the aveae symbol eo ate is then, σ σ 5 E P es 8 4 1E 4 E The aveae symbol eney is Es 6E. ine eah symbol 8 aies thee bits, and a Gay ode an be applied to make sue that eah neihbouin symbol diffes in only one bit position, we an find the BER fo 8-AK, 8AK 5 E b P eb 6 The same piniples used to estimate the BER fo 8-AK an also be applied to estimate the BER fo the lae onstellation AK vaiants. E b 4. Powe petal Density The tansmitted AK sinal is 9

10 s( n [ a ] [ n] ( t nt ) os( πf a[ n] ( t nt ) sin( πf This is in eneal not a stationay poess, due to the pesene of the aie tems, the statistis ae peiodi. One an detemine the autooelation funtion of this andom sinal, and find the time aveae by inteatin ove a peiod of the aie. The powe spetal density is then the Fouie tansfom of the autooelation funtion, whee ( f ) ( ( f ) R ( τ ) 1 { ( f f ) ( f f )} e jπfτ dτ is an equivalent baseband powe spetal density, iven by f f f ( f ) R G( f ) A A A A R R R n eneal, we assume that the messae symbol sequene is unoelated, fo whih A ( f ) A ( f ) Es E and A A ( f ) The PD of the AK sinal is, fo an unoelated messae symbol sequene, ( 1 ( f ) R ( E E ) G( f f ) G( f f ) s { } 5. Case tudy The Diital Voieband Modem tandad As a patial example of an appliation of AK let s biefly look at the V.3 modem standad, fo diital data ommuniation ove the analoue PT. This is a standad fo a symmeti modem, in whih the data ate in both dietions is the same. The basi paametes of the V.3 modem standad is as follows: Caie fequeny: 18 Hz ymbol ate: 4 baud Data ate: 96 bps Thee ae two diffeent hoies fo modulation in this standad, dependin on the level of hannel odin desied to potet the data. The two onstellations ae 16- AK and 3-AK, and involve a tade-off between odin ain and sinal to noise atio. The two shemes ae shown in the diaam below, followed by thei sinal onstellations. The odin stutues will make moe sense when we ve disussed hannel odin late in the ouse. Foussin on the non-edundant odin standad, the 16-AK onstellation shown is a ood example of the appliation of a Gay ode. ote also, that when the sinal to noise atio is poo, both shemes edue to a 4-AK sheme, whose sinal onstellation is illustate with the lettes A, B, C, and D. 1

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