unmodulated carrier phase refference /2 /2 3π/2 APSK /2 3/2 DPSK t/t s

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1 The PSK Modulaion - PSK is a modulaion ha modifies he phase of a carrier signal, a he beginning of he symbol period, wih a value ha depends on he mulibi ha has o be modulaed - i exhibis a good resilience o perurbaions and disorions and has a raher good specral efficiency facor, requiring a medium implemenaion complexiy. PSK signal consellaions - since he phase is no an absolue magniude, requiring a reference, wo ypes of phase modulaion can be defined, acc. o he reference employed: absolue phase modulaion, Absolue PSK (APSK), where he phase-shifs of he modulaed signal, occurring every symbol period, are referred o he phase of a reference signal, usually he unmodulaed carrier signal. differenial phase modulaion, Differenial PSK (DPSK), where he phase-shifs of he modulaed signal, occurring every symbol period, are referred o he phase of he phase of modulaed carrier during he previous symbol period. - figure presens boh varians of PSK on a cosine carrier, for he phase-shifs of, π/2, π and 3π/2 in advance. The reference signal, for he APSK, is he non-modulaed carrier. unmodulaed carrier phase refference /2 /2 3π/2 APSK /2 3/2 DPSK /T s Figure. Types of PSK modulaions - he daa bis are grouped ino p-bi mulibis; each mulibi is modulaed and ransmied wihin a symbol period, T s. - he relaions beween he symbol and bi periods and beween he bi and signaling frequency are given by (). Noe ha he signaling frequency is measured in Baud, i.e. Baud = symbol/sec: Ts pt b; fb p f s; () p - he number of phase-levels (saes or vecors) of he modulaed signal is: N 2 (2) -he signaling rae v s, i.e. he number of variaions/second of he modulaed parameer(s) of he carrier signal, is measured in Bauds, symbols/second, and is numerically equal he frequency of he symbol-clock f s ; - he bi rae D, which is numerically equal o he bi-clock frequency f b, is expressed by: D = v s p [symb/s bis/symb = bis/s] (3). - he se of vecors ha could be generaed by PSK-modulaing all possible p-bi mulibis are represened in plane as signal consellaions. Figure 2 shows he consellaions corresponding o he PSKmodulaions of able wih heir varians, see he righ-hand column. - if he vecors are considered in polar coordinaes, hen all have an uniary radius, being idenified by he differenial phase-shif Δφ k, defined for each mulibi, which is ransmied during a symbol period. - for p = and 2, wo ypes of cosellaions are employed: Type A, conaining he Δφ k = º phase-shif Type B, which do no conain he Δφ k = º phase-shif

2 2 3 b b ;b b 2 ;b ;b a. 2-DPSK b. 4-DPSK c. 8-DPSK var. A var. A - he B-ype consellaions are generaed by roaing he A consellaions, in a rigonomerically posiive sense, wih half of he minimum phase-shif beween heir vecors - in figure 2 each vecor is denoed by a decimal label k {,,7}. var. B var. B - he consellaion B8, which Figure 2. D-PSK signal consellaions would involve a phase-roaion of 45º/2 = 22,5º of he vecors from consellaion A8, is no used; he moivaion would be discussed in he secion dealing wih he symbol-clock synchronizaion. Gray mulibi-o-vecor mapping - he mulibi-vecor mapping is made according o he binary Gray code, ensuring a minimum Hamming disance of bi beween adjacen vecors. - he maximum Hamming disance occurs beween vecors separaed by π, for boh he 4-vecor and 8- vecor consellaions. - he Gray-mapping is employed o decrease he bi error probabiliy, for a given symbol error probabiliy. - denoe in consellaion A4, he error-probabiliy of vecor ino vecors or 3 by p, and of vecor ino vecor 2 by p 2 ; hen p > p 2. For a Gray-mapping he bi-error probabiliy is expressed by (4).a P bg 2 p p 2 2; a. P bn p p 2 p 2 ; b (4) - for a mapping according o he naural binary code ( ; ; 2 ; 3 ), he probabiliies p and p 2 have he same values as for he Gray-mapping. The bi-error probabiliy is expressed by (4).b. - comparing he bi-error probabiliies ensured by he wo mapping rules for he same symbol-error probabiliy, we ge: p bg p bn p 2 p p bg p ; he Gray mapping is also applied o PAM and ASK (5) bn - a similar reasoning may be applied o consellaion A8, as well. Expression of he PSK modulaed signal - he PSK modulaed signal is expressed by (5); u T (-nt s ) represens a recangular impulse wih uniaryampliude and duraion T s, which indicaes ha he phase of he carrier Acosω c is modified wih Δф n only during he n-h symbol-period. spsk() = A ut( - kts)cos( c + k ); (6) k= - he modulaed PSK signal is a succession of modulaed signals during a symbol period; he phaseshifs of each symbol period do no inerfere wih one anoher. - he phase reference ha is used o compue he phase-shif Δф k indicaes he ype of modulaion: absolue or differenial. - APSK is no employed in pracice because he demodulaion requires he separae phase-reference,. - DPSK is used, because he phase-reference is he carrier s phase during he previous symbol period. Specral disribuion of he QPSK signal - he power specral disribuion of he all PSK consellaions depends on he symbol frequency and on he ype of he carrier signal, i.e. harmonic (cosine or sine) or recangular - a. when he carrier signal is a harmonic signal (having only one specral componen), he specral disribuion of he PSK signal is expressed by (7) for he non-filered modulaing levels, see (6). - he specrum, approximaely represened in figure 3, exhibis a cenral lobe (k = ), wih a bandwidh equaling 2 f s and maximum value S M, around he carrier frequency and side lobes wih maxima S Mk occurring a he f M frequencies, given by (8). These specral lobes are generaed by u T (-kt s ). 7 2

3 2 ( c ) sin 2 2 f s V f f f sin f s A ut(f) = ; Sn( f ) ; f f ( f fc ) s fs Hz f s f s f m = f c +/- kf s k ; f M = f c +/- (kf s + f s /2) k ; S M = A 2 /f s ; S Mk = S M 4/[(2k+)π] 2 ; (8) (7) Figure 3 Power specral disribuion of he QPSK signal nonfilered and RRC filered - he ampliudes of he side lobes decrease raher slowly wih he increase of heir index. - Fig. 4 presens he filered and non-filered power specra of a PSK signal, for f s =.33 khz, α= Figure 4 Power specra of non-filered and RRC-filered PSK signals - snapsho Specrum of u () Ts f/f c f/f c Specrum of PSK modulaed on recangular carrier.5 - b. If he carrier signal is recangular, as for he digial modulaors, he specrum of he modulaing signal is ranslaed around he harmonics of he carrier signal k f c, see (9) (f - c) sin kf f s ut(f - fc) = Ts Ak (9) k= (f - kf c) f s - for a harmonic carrier signal, he sum in (9) conains only one erm, k =, see fig.3 and (7) - figure 5 presens power specra of he non-filered modulaing signal, u T (-mt s ), recangular carrier signal and PSK modulaed signal Fig.5 Specrum of PSK modulaed on a recangular carrier Filering he PSK modulaed signals 2 3 f/f c he consideraions above show ha he modulaed signal has o be filered, so ha i would mach he channel bandwidh. - due o he frequency-band limiaion generaed by filering, he filered signal expands in ime generaing he ISI. - o remove (or a leas decrease) he ISI in he probing momens, he signal has o be filered wih a RC 3

4 characerisic wih a roll-off facor α, see he lecure on Filering he Daa Signals). - for beer performances in he presence of he Gaussian noise, he filering characerisic has o be equally-spli beween ransmier and receiver, as shown here This involves filering wih a RRC characerisic boh in he ransmier and receiver. Basically, he RRC filering may be implemened in wo varians: a band-pass filering placed afer he PSK modulaor; his opion is used by he direc digial PSK modulaors which generae he PSK signal on (f-f c )/f N a recangular carrier signal, as will be shown in he chaper dedicaed o he PSK modulaion and in α = α =,5 α = Annex. The BP-RRC filering characerisic is Figure 6. BP RC characerisics for α =,.5, defined by () and represened in fig.6: Figure 6 BP RC characerisics for α =,.5 and Xα ((f-fp)/fn) ; [ c N( ), c N( )]; N E( c) = N R( c) = N( c) 2 = ( c ) (- ) cos( - ); A; () 4 N 4 A [ c N( ), c N( )] [ c N( ), c N( )]; a low-pass filering applied o he modulaing signal; his approach is used if he PSK signal is generaed using he QAM echnique, see he chaper on PSK modulaion This mehod is preferred in mos applicaions. The LP-RRC filering characerisic is defined by equaion (7) in he lecure on Filering he Daa Signals and can be obained by making c = in () above. Effecs of filering he PSK-modulaed signals - he goals of filering he modulaed signal wih a global RC characerisic are: limiaion of he frequency band occupied by he modulaed signal; removal of ISI in he probing momens - if he modulaing momens are a he beginning of he symbol period, negaive edge of he symbol clock f s, hen, due o he τ g (f) characerisic of he filer, he probing momens are placed in he middle of he symbol periods, i.e. posiive edge of he symbol clock f s, see he daa filering. - o analyze he effecs of filering upon he momenary phase and frequency and upon he envelope of he modulaed signal, we consider ha he signal phase suffers a shif from he phase of he previous symbol, denoed by º, o ΔΦ k. Denoing by Φ() he phase-shif insered by he modulaor, he modulaed signal would be expressed by: s()=acos( p+ ()); ()= kut(+t/2-kt) k; [kt-t/2;ktt/2]; () - heory shows ha afer he filering, he momenary phase of he modulaed signal Φ() has a coninuous variaion described by (), where x() denoes he impulse-response of he RC filer, represened in figure 5 for α =. In figure 5, he ime-reference momen is he probing momen, =, herefore he modulaing momens occur a odd muliples of T s /2. () sin k = arcan ; (2) k k -2 2 k sin x() 2 - fig. 7 presens he variaion of he momenary phase for ΔΦ k = k 2π/8, k {,,4}. - i shows ha he momenary phase has very small (close o zero) values in all probing momens, = k T s, excep for he main probing momen =, (hird axis in fig. 7). - in he main probing momen phase Φ() reaches approximaely he nominal value ΔΦ k of he curren symbol. This shows ha ISI has been significanly decreased in he phase domain, i.e. in every probing momen and only hen, he momenary phase has he nominal value of he phase-shif of ha symbol and is very slighly affeced by he ime-expansion of he phase-shifs of oher symbol-periods. - noe also ha he momenary phase Φ() reaches half of is nominal value a = ±T/2. 4

5 -T s -T s /2 fs +T s /2 +T s - or ΔΦ k = π, he phase variaion is almos recangular, and for ΔΦ k =, he phase variaion is,5 x() zero. - a = ±T/2, he momenary Θ () phase has inflexion poins. ΔΦ n - he variaion of he momenary frequency f in () around f c, afer k=3 k=4 Φ ()/ΔΦ n he filering, can be derived from he Φ() variaion law by: k= k=2,5 d () f in () (3) 2 d Δ f in - m ax Δ f in () - he Δf in () can be derived easier by performing he graphical derivaive of Φ(). The inflexion poins would become exreme poins for he derivae funcion; I min I() he monoony of he derivae Δ f in - m ax funcion can be derived from he I min concaviy/convexiy of Φ(), and,92 he sign of he derivae funcion,, 7,38 from he monoony of he Φ(). The resuled curve, which Figure 7. Effecs of filering he PSK signal -ramane approximaely described he variaion of he momenary frequency around he carrier frequency, is shown on he 5-h axis of fig he deviaion of he momenary frequency has maxima a approximaely = ±T/2. The values of hese maxima depend of he value of phase-shif ΔΦ k = m /4, and can be compued using (4). Δf inmax = m f s /8; m = k, for k =,...,4; m = k - 8, for k = 5,...,7; (4) - he envelope I() of he filered modulaed signal is shown o be no longer consan; i varies in ime according o (5).a, see he 6-h axis of figure 5. I() = A n N sin x()[ - x()] ; a. I min = A cos ; b. (5) (5).a shows ha he envelope of he filered signal has a maximum a he probing momen = and minima a = ±T/2. The values of he minima are expressed by (5).b and depend of he phase-shif value. For ΔΦ n = k 2π/8, k {,,7} hese minima have five possible values,,.92,.77,.38,, which are represened in figure 7 - noe ha for ΔΦ n = º, he variaions of momenary phase, frequency deviaion and of he envelope are zero. This is a major disadvanage of he A ype consellaions. - summarizing, he goals of filering he (D)PSK signals are: decrease of he occupied frequency BW, so ha i would mach he channel BW; removal of ISI in he probing momens. - he consequences of filering he (D)PSK signals wih an RC characerisic are: - he probing momen is displaced wih half of symbol-period, from he modulaion momen; - a coninuous variaion of he momenary phase, wih maxima, approximaely equaling he nominal phase-shif, a he probing insans; - coninuous variaion of momenary frequency, wih maximum deviaion from he carrier frequency occurring a =±T/2; he maximum values of he frequency deviaion depend of he phase-shif values. - he occurrence of a parasiic ampliude modulaion, i.e. he envelope, which has maxima a he probing momens and minima a approximaely he modulaing momens = ±T/2. The minima s values depend on he values of he phase-shifs. Generaion of he DPSK signals - he DPSK signals can be generaed by direc digial mehods or by employing he QAM echnique 5

6 - he digial mehods - wo digial modulaors which produce direcly he APSK and DPSK signals for all consellaions are presened in Annex ; hey are no included in he examinaion opics DPSK modulaion-demodulaion employing he QAM modulaion (echnique) - he expresion of he (D)PSK signal over one (k-h) symbol-period is given by (6), where Φ k denoes he absolue phase of he carrier during he k-h symbol period: spsk = Acos( p + k) ut( - kt); (6) - by expanding (6) we ge (7), which represens a QAM signal in which he wo modulaing signals are no longer independen signals; hey fulfill condiion (8). spsk = A cos kut( - kt) cos p - Asin kut( - kt) sin p = I(kT) Acosp - Q(kT) Asinp; 2 2 I(kT) = Ik= cos kut( - kt) ; Q(kT) = Qk sin kut( - kt); wih Ik Qk u T( kt s) - if he modulaing symbols are wrien as (8).a hen he PSK signal can be wrien as (8).b : jk mk Ik jqk Ak e ;a. (8) j ck s () Re I jq cos jsin Re A e ; b. PSK k k c c k DPSK modulaion generaed by he QAM echnique 4-PSK Conselaia A4 - as an example we presen he generaion of he A4 consellaion, figure 8. 9 Table shows he phase-shifs ΔΦ k, he values of he modulaing levels (I k, Q Q k ), he inpu dibi-daa a a and of he dibi afer he Gray-naural a a conversion (CGN), b b, which is performed according o: 8 b a a; b a; (9) Figure 8. The A4 signal consellaion I (7) - his mehod generaes an absolue-phase modulaion, since he modulaed carrier s phase-shifs are referred o he phase of he non-modulaed carrier. Mos ofen he lieraure denoes by QPSK he 4- APSK (varian A or B). a a b b I k Q k ΔΦ k + º + 9º - 8º º Table. Signal values in he main poins of he DPSK-A4 encoder for c k- c k- = - o ransform his modulaion ino a DPSK one, he absolue phase of he modulaed carrier should be modified according o (2). Φ k = (Φ k- +ΔΦ k ) mod 36 º (2) - because all ΔΦ k are muliples of 9 º, he absolue phase will be a muliple of 9 ºand (2) may be wrien as: N 9 (N 9N 9 ) N (N N ) (2) k k k k k mod36 k k k mod4 - bu he numbers N k and ΔN k are binary represened by he dibis c k c k and b k b k, (2) may be wrien as: (b k b k + c k- c k- ) mod 4 = c k c k ; (22) - (2) and (22) show ha o obain a DPSK, he dibi ha is delivered o he circui ha compues he I k and Q k levels is obained by differenially precoding he modulaing daa-dibi, afer he GN conversion. - he block diagram of he DPSK modulaor implemened by he QAM echnique, is shown in fig he I k and Q k levels can be obained by wo mehods: by reading he I k and Q k values from a able, in erms of he curren daa dibi and previous encoded dibi, when he GN conversion and he differenial encoding are included; by using a D/A converer and a circui ha compues he bis which conrol he D/A converer - on a DSP implemenaion, he CGN and differenial precoding are performed off-line; he I k and Q k levels are read from a able, in erms of curren daa and previous encoded-daa dibis; his block is called encoder or mapper. - o limi he bandwidh of he modulaing signal and ensure ISI= in he probing momens, he I k and Q k signals would be LP filered (FFE blocks) wih a RRC characerisic wih a roll-off facor of α 6

7 a k a k C. G. N. k b k b k- c Σ M o d u lo 4 c k c k M A P P I N G I k Q k F. F. E. F. F. E. Acosω c Asinω c Σ S D P S K c k- D Ck f s - afer he filering we ge he coninuous modulaing signals I() and Q(). - he expression of he ransmied modulaed signal is: s DPSK () = I()cosc-Q()sinc = (23) I() = A cos kut( - kt) afer filer; Q() = A sin kut( - kt) afer filer; - he LP-RRC filering is implemened using a FIR srucure, in which only one sample of he I k and Q k levels should be insered in every symbol-period; he res of he samples of ha symbol period would equal zero, see DSP lecures and Daa Transmissions lecures - when implemened on a signal processor, he symbol period is dvided ino N sampling periods. The encoding, muliplicaion and addiion operaions are execued for each sample. A LP fler should be added a he modulaor s oupu, o suppress cuanizaion noise. - he samples of he carrier signals would be sored in a able, N values per symbol period; he digial generaion of he carrier signals should ensure a small THD facor. - his mehod can be applied if he frequency of he carrier allows is implemenaion on a processor; - for carrier signals wih greaer frequencies, he digially filered signals (I() and Q() are muliplied o he carrier signal by analogue mulipliers and he summaion is performed by an analogue adder. Generaing oher DPSK consellaions wih he QAM echnique Consellaions A2 and B2 - since hese consellaions involve phase-shifs of ΔΦ n =º or 8º and, respecively, ΔΦ n =9º or ΔΦ n =27º, which define he vecors of he wo consellaions, he QAM- expression of he 2- DPSK signals are: s () = A cos( ) u ( - nt); s () = A sin( ) u ( - nt); (24) PSK-A2 Figure 9. Block diagram of he DPSK modulaor implemened by using he QAM echnique n=- n=- p T PSK-B2 - he values of he modulaing levels I k and Q k of he A2 and B2 are presened in able 2. Table 2. Values of I k and Q k for consellaions A2 and B2 Consellaion Bi I k Q k A2 + - he modulaion and is block diagram remain he same as he A2 - ones described above for QPSK, excep for he differenial B2 + precoding-decoding ha are performed as mod.2 operaions on B2 - one bi. Consellaion B4 - QAM generaion of B4 requires a modulo-8 differenial precoding-decoding on 3 bis. - he b b dibi obained afer he Gray-naural conversion is ransformed in he c 2 c c ribi: c 2 = b ; c = b o ; c = ; (25) - seing he bi c = is equivalen o he 45º roaion imposed by his consellaion. - he c 2 c c ribi is employed o selec he modulaing levels I k and Q k as shown in able 3 Dibi c 2 c Table 3. Values of he I k and Q k levels for consellaion B4 I k + 2/2-2/2-2/2 + 2/2 - he res of he operaions required by he QAMmodulaion-demodulaion of B4 are similar o QPSK. Q k + 2/2 + 2/2-2/2-2/2 - noe ha afer he demapping and differenial decoding only he wo mos significan bis are employed in he final processing. 7 p T

8 Consellaion A8 - he QAM modulaion-demodulaion of he A8 are implemened similarly as he ones of A4, wih he following differences: - he values of he I k and Q k levels, in erms of he daa ribi c 2 c c, are he ones of able 4 c 2 c c I k + 2/ /2-2/ /2 Q k + 2/2 + 2/ /2-2/2 - Table 4. Values of he I k and Q k levels for consellaion A8. - he differenial precoding-decoding should be made modulo-8 on he hree bis. - Annex 2 presens he block diagram of he DPSK ransmiers ha use he direc digial mehod o generae he modulaed signal on a recangular carrier and perform a BP-RRC filering on a inermediae frequency f i followed by a frequency ranslaion on he channel carrier frequency- i is no included in he examinaion opics Consideraions Regarding he Implemenaion of he RC filering of he (D)PSK signal -individual sudy - required for examinaion - he RRC characerisic can be implemened eiher wih analog or digial mehods. - he analog implemenaion provide accepable (no good!) accuracy only for roll-off facor α > o ensure high accuracy and smaller roll-off facors, he RCC characerisic should be implemened digially using a FIR filering srucure. For a brief presenaion of his approach, see he Annex of he lecure on Filering he Daa Signals. This opic will be deal wih in he laboraory classes of he daa Transmissions course in he IV h year. - he RRC characerisic could be implemened eiher as band-pass filers, cenered on he inermediae or channel frequency, or as low-pass filers, by filering he modulaing I k and Q k signals in base-band. - he BP-RRC approach should be used for he ransmiers ha direc digial (D)PSK modulaors, as he ones described in Annex of his maerial. Their posiion in he ransmier is shown in Annex 2. - he ransmiers ha implemen he (D)PSK modulaion using he QAM approach, as described above, could use boh he BP or LP varians of he RRC filering. Sill, almos all implemenaions use he LP varian, by LP filering he I k and Q k modulaing signals, as also shown above. Consideraions Regarding he Frequency Translaion of DPSK signals - individual sudy- required for he examinaion - he frequency ranslaion from an inermediae-carrier frequency f i o he channel-carrier frequency f c or he oher way around, can be accomplished by muliplying he signal o a ranslaion signal of frequency f followed by a BP or LP filering which would selec he desired frequency band and aenuae he undesired specral componens resuled from muliplicaion. - he muliplicaion can be made by using a muliplier or a chopper, see he LM and FM lecures - he consideraions below are made for he case when f c > f i, which is me in almos all pracical applicaions. - as shown by equaion (52) a he end of he FM lecure, which is repeaed here for convenience, if he f > f c, he sign of he addiional phase insered by he modulaing signal Φ() is changed, while for f < f c i is no changed. c c i and s( ) k fv 'cos( i ( )); a. c i (26) c c i and s() k fv 'cos( i ()); b. - his conclusion shown here for he downwards ranslaion, also holds for he upwards f i f c ranslaion. - his sign change ha occurs for f > f i, (case b.) should be compensaed in he modulaor - due o he changed sign of he phase-shif, he resuled phase-shif would correspond o a daa mulibi which equals he modulo (2 n ) complemen of he modulaed daa mulibi (boh in binarynaural represenaion), for all consellaions excep for 2-PSK varian A. This sysemaic error should be compensaed in he ransmier, afer he G-N converer, by delivering a he modulaor s inpu he complemen modulo (2 n ) of he mulibi ha has o be ransmied. - he case when f c < f i is me only for ransmissions over he elephone channel, in he so called dialup modems. For his siuaion similar resuls are obained if we exchange f i and f c wih each oher. 8

9 Annex - no required for examinaion APSK and DPSK direc digial modulaors buil wih an arihmeic adder and a couner - AAC - due o he finie number of phase-shifs required and o he ime-discree characer of he modulaion, he digial mehods ensure a higher accuracy and a beer sabiliy. - he DPSK can be obained boh by DPSK modulaors and by APSK modulaors, preceded by a differenial encoding of daa mulibi. - he digial modulaors will employ a serial-parallel converer, o build he mulibis. This converer would acquire serially he inpu bis, using he bi-clock f b, and he mulibis will be read by he modulaor using he symbol-clock f s. - hen, he digial modulaors would perform he Gray-binary naural conversion, CGN; he inpu mulibi is looked-upon as Gray-coded combinaions (o decrease he bi-error probabiliy) and hen hey are convered ino naural-binary combinaions o mach he modulaor (as will be described laer) - he bi-mapping may lead, someimes, o a significan decrease of he bi-error probabiliy, for he same symbol-error probabiliy (he same SNR), sill using he same implemenaion complexiy. - he conversion beween Gray and naural-binary codes is performed according o (6), g i indicaing he bis of he Gray-coded combinaion. a g gg 2; a gg 2; a2 g 2; g a a ; g aa 2; g2 a 2; (A..) - he A8 consellaion requires a modified (compleed!) conversion rule, because he sandard employs a differen mapping rule, see fig. 2.c. Homework: esablish how relaions Error! Reference source no found. should be modified o accomplish he GNC for A8 - for a p-bi mulbi, he modulaor is implemened using a p-bi arihmeic adder a p-sage couner; i is shown in figure A., for p = 3. The signal diagram is shown in figure A.2. f bi d i g 2 g :3 f s a 2 a D ifferenial precoding A 2 A Σ pd Σ 2 Σ b 2 b A 2 A APSK Modulaor Σ m Σ 2 s D P S K g a A B 2 B B Σ b A B B B 2 4f i 2f i f i Accumulaor 8f i :8 Figure A.. Elecric diagram of he APSK and DPSK modulaors wih arihmeic adder and couner 8f i 4f i 2f i f i b i = ΔΦ k = 45º b i = ΔΦ k = 4 45º b i = ΔΦ k = 2 45º b i = ΔΦ k = 45º Figure A.2. Signal diagram of he APSK modulaor of figure 3. 9

10 - if he inpu ribi b i is kep o, he number (expressed in naural-binary code) is added o he inpus of he adder; so, he Σ 2 oupu will deliver a recangular signal of frequency f i affeced by a phase shif ΔΦ = 45º, compared o he reference signal, i.e. he signal a he couner oupu. - if he ribi =, hen he number 4 is added and he sum will suffer a shif of four unis, which equivalens o a phase-shif ΔΦ = 4 45º = 8º, compared o he reference signal. - similarly, for b i =, a phase-shif ΔΦ = 2 45º= 9º, and for b i =, a phase-shif of ΔΦ = 45º= 45º is obained. - noe ha bi b 2 conrols he phase-shif of 8º, bi b he one of 9º and bi b he one of 45º. - he combinaions of he hree bis generae all numbers k {,,7} ha correspond o all he phaseshifs equaling k 45º, which compose he A8 consellaion. - he conversion of he inpu ribi from he Gray-code o he naural binary code is required because he number k is he represenaion in he naural binary code of he daa ribi. - a he beginning of each symbol period, i.e. a he negaive edge of he symbol-clock, he ribi a i changes, generaing he phase-shif corresponding o ha period. - he phase-reference is he phase of he carrier f i, so his modulaor generaes APSK. - o generae he DPSK modulaed signal, he inpu daa ribi a i is differenially precoded, generaing he ribi b i which is applied o he APSK modulaor, see figure 3. For he DPSK, he absolue phase of he carrier signal during he n-h symbol period may be wrien as: abs Φ n = (Φ abs n- +ΔΦ n )modulo 36º; (A.2) - since all he phase-shifs are muliples of 45º, his can be simplified as: k n abs = (k n- abs +Δk n )modulo 8; (A.3) - so, he differenial precoding consiss of a modulo-2 p arihmeic addiion, on p bis, of he previous mulibi b n- o he curren mulibi a n ha comes from he CGN. - his operaion is performed using a p-bi arihmeic adder and a p-bi shif regiser, as a memory elemen, see figure A.. - if his modulaor is o generae he A4 or A2 consellaions, i should operae on 2 or bi (he MSB ones); he differenial precoding should be performed modulo 4 or, modulo2. The inpus corresponding he unemployed bis should be conneced o, i.e. b = and b = b =. - he generaion of he B-ype consellaions involves a roaion of 45º, for he B4, or of 9º, for he B2. This is accomplished by seing b =, regardless he daa dibi, for B4, or seing b = and b =, for B2. For he consellaions using 2 or 4 vecors, he significances of he hree bis b i are summarized in able A., where d denoes daa bis. Bi ; Consellaion A8 A4 B4 A2 B2 b 2 d d d d d b d d d b d Table A.. Values of ribi-bis for differen consellaions - he frequency of he modulaed carrier-signal f i, may be eiher he channel carrier signal f c or an inermediae frequency, higher han he channel-carrier frequency, depending of he mehod employed o filer he modulaed signal, see he PSK-filering paragraph. - he AAC modulaor insers advance phase-shifs. Someimes, delay phase-shifs should be insered; o generae his ype of phase-shifs we employ he periodiciy of he carrier-signal and ge (A..4), where he backwards phase-shis are marked by. ΔΦ' n = +(36º-ΔΦ n ) modulo 36 ; Δk' n = (8 - Δk n ) modulo 8; (A.4) - o ge hese phase-shifs he modulaor should be provided wih he 8-complemen of he ribi ha corresponds o he advance desired phase-shif. The block ha performs he complemen should be placed beween he CGN and accumulaor. - for consellaions wih 4 vecors he 4-complemen should be employed; for B2, he modulaing bi should be invered and for A2 his operaion is no required. DPSK modulaor buil wih an arihmeic adder and a shif-regiser - AASR - is elecric diagram is shown in figure A.3 and he signal diagram is displayed in figure A.4 - noe ha if he modulaing funcions F i =, he assembly AA-SR acs like an 8-couner.

11 - Considering he adder operaional equaion (A.5), he Bi inpus of he adder increase heir value wih one uni in he rhyhm of he 8f i -clock signal, due o he inpu-carry c =. Then he Σ 2 signal would have a period equaling 8 periods of he 8f i signal, i.e. a frequency equaling f i. i Ai Bi Ci ; (A.5) F =F =F 2 = Σ 2 Σ Σ F =;F =F 2 = Σ 2 Σ Σ F =F 2 =;F = Σ 2 Σ Σ () (A) ΔΦ º 45º 9º 8º No.8f i -Ck Per. F =F =;F 2 = Σ 2 Σ Σ Table A.2 Operaing principle of he AASR DPSK modulaor d i C S P g 2 g g : 3 Im p. Mod. 8f i a 2 F 2 M F A 2 C C O u Σ 2 R D n a U L c F G A i T o A Σ I n N a F G N s A B 2 Σ D s DPSK B B Figure A.3. Block diagram of he AASR DPSK-modulaor Sar of he sym bol-period 8f i F i = ΔΦ = º Σ 2 Σ F = ΔΦ = 45º A F = ΔΦ = 9º Σ 2 Σ 2 F 2 = ΔΦ = 8º M odulaing im pulse Figure A.4. Signal diagram of he AASR DPSK-modulaor - if a he beginning of he symbol-period, for a period of he 8f i signal, 8 fi > f s, he modulaing funcion F 2 =, he adder oupu increases is value wih 4 unis and he phase of signal from he Σ2 oupu suffers a phase-shif, in advance, of ΔΦ = 4 45º = 8º, see able A.2 and figure A.4. The phaseshif appears obvious if i is considered from he end of he firs symbol period of he carrier signal (f i ), marked by poin A in figure A.4 and in able A.2. - Similarly, for F =, we ge ΔΦ = 2 45º = 9º, and for F = we ge ΔΦ = 45º. - by combining he value of he hree modulaing funcions, all he phase-shifs equaling k 45º, wih k {,,7} can be obained.

12 - for his modulaor, he modulaing daa ribi is applied only during he modulaion impulse, see figure 6; for he res of he symbol period, he values of he modulaing funcions F i, are forced o. So he ribi a i should be processed by he block ha generaes he modulaing funcions, which allow is access o he modulaor only during he modulaion impulse. -he modulaing funcion also perform he 8-complemen (or 4-complemen) if his modulaor should inser backwards phase-shifs. - he consellaions wih 4 or 2 vecors can be produced in he same manner as he one described for he AAC DPSK modulaor, by aking ino accoun able A.. -because he phase-shif is referred o he phase of he carrier signal during he previous symbol period, considered o have Δ = º, his modulaor generaes a DPSK modulaion. DPSK modulaor wih conrolled division CD - his modulaor is based on he phase-shif by conrolled division described in he dynamic synchronizaion sysem (see synchronizaion in he BB lecure noes). Annex 2 Block diagram of a PSK ransmier - no required for examinaion PSK ransmier wih modulaion on he inermediary frequency - he block diagram of his ransmier is shown in figure A.2. TxCk OSC. DIV. f s Var. D IM P.M OD. 8f i s PSK dr. s PSK cos. f TxD CTS RTS SCR. Conrol Emisie b 2 d i C.S.P. b C.G.N. b Block Synhesis of M odulaing Funcions M odulaor F 2 F F Modu- Laor DPSK f i F. F. E. f i TR. FR. f p s PSK cos. B.P.F. LF/HF. f p EQ. Compr. Line Amplif. Line Uni Channel Figure A.2.. Block diagram of he PSK ransmier wih modulaion on he inermediary frequency - he daa o be ransmied TxD, are insered ino a scrambler SCR, ha randomizes he daa o be modulaed. The scrambler is employed only for consellaions wih 8 or more vecors. - he scrambled daa are hen sen o he series-parallel converer, hen o he CGN generaing he mulibi which is delivered o he modulaor. - he modulaor generaes he modulaed signal s PSK-fi on a recangular carrier f i, using he auxiliary signals of frequencies 8f i and f s, obained from he oscillaor-divider block, OSC-DIV. - he modulaed signal is filered wih a BP RRC filer wih a roll-off facor α, by he ransmission shaping-filer, FFE, generaing a modulaed signal on a cosine carrier signal, s-psk,c,f i. - his signal is hen ranslaed on he channel-carrier frequency f c by he frequency ranslaion block TR.FR., which employs a recangular signal of frequency f, provided by he OSC.-DIV block. The band-pass low-frequency filer BPF-LF, reains only he inferior sideband generaed by he freq. rans., which is he modulaed signal on a cosine carrier of frequency f c, s-psk,c,f c. - he level of his signal is esablished by he line amplifier; hen he signal is sen o he line-uni which ensures he adapaion wih ransmission channel. - he ransmission conrol circui manages he enable/disable of he ransmier, using he RTS and CTS signals. Some consrucive varians include a compromise equalizer. 2

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