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

Transcription

1 The PSK Modulaion - PSK i a modulaion ha modifie he phae of a carrier ignal, a he beginning of he ymbol period, wih a value ha depend on he mulibi ha ha o be modulaed - i exhibi a good reilience o perurbaion and diorion and ha a raher good pecral efficiency facor, requiring a medium implemenaion complexiy. PSK ignal conellaion - ince he phae i no an abolue magniude, requiring a reference, wo ype of phae modulaion can be defined, acc. o he reference employed: abolue phae modulaion, Abolue PSK (APSK), where he phae-hif of he modulaed ignal, occurring every ymbol period, are referred o he phae of a reference ignal, uually he unmodulaed carrier ignal. differenial phae modulaion, Differenial PSK (DPSK), where he phae-hif of he modulaed ignal, occurring every ymbol period, are referred o he phae of he phae of modulaed carrier during he previou ymbol period. - figure preen boh varian of PSK on a coine carrier, for he phae-hif of, π/, π and 3π/ in advance. The reference ignal, for he APSK, i he non-modulaed carrier. unmodulaed carrier phae refference / / 3π/ APSK / 3/ DPSK /T Figure. Type of PSK modulaion - he daa bi are grouped ino p-bi mulibi; each mulibi i modulaed and ranmied wihin a ymbol period, T. - he relaion beween he ymbol and bi period and beween he bi and ignaling frequency are given by (). Noe ha he ignaling frequency i meaured in Baud, i.e. Baud = ymbol/ec: T pt ; f p f ; () b b p - he number of phae-level (ae or vecor) of he modulaed ignal i: N () -he ignaling rae v, i.e. he number of variaion/econd of he modulaed parameer() of he carrier ignal, i meaured in Baud, ymbol/econd, and i numerically equal he frequency of he ymbol-clock f ; - he bi rae D, which i numerically equal o he bi-clock frequency f b, i expreed by: D = v p [ymb/ bi/ymb = bi/] (3). - he e of vecor ha could be generaed by PSK-modulaing all poible p-bi mulibi are repreened in plane a ignal conellaion. Figure how he conellaion correponding o he PSKmodulaion of able wih heir varian, ee he righ-hand column. - if he vecor are conidered in polar coordinae, hen all have an uniary radiu, being idenified by he differenial phae-hif Δφ k, defined for each mulibi, which i ranmied during a ymbol period. - for p = and, wo ype of coellaion are employed: Type A, conaining he Δφ k = º phae-hif Type B, which do no conain he Δφ k = º phae-hif

2 3 b b ;b b ;b ;b a. -DPSK b. 4-DPSK c. 8-DPSK var. A var. A var. B var. B - he B-ype conellaion are generaed by roaing he A conellaion, in a rigonomerically poiive ene, wih half of he minimum phae-hif beween heir vecor - in figure each vecor i denoed by a decimal label k {,,7}. - he conellaion B8, which Figure. D-PSK ignal conellaion would involve a phae-roaion of 45º/ =,5º of he vecor from conellaion A8, i no ued; he moivaion would be dicued in he ecion dealing wih he ymbol-clock ynchronizaion. Gray mulibi-o-vecor mapping - he mulibi-vecor mapping i made according o he binary Gray code, enuring a minimum Hamming diance of bi beween adjacen vecor. - he maximum Hamming diance occur beween vecor eparaed by π, for boh he 4-vecor and 8- vecor conellaion. - a hown in he PAM lecure, he Gray-mapping i employed o decreae he bi error probabiliy, for a given ymbol error probabiliy. Thi propery i proven for he PSK in he example below - denoe in conellaion A4, he error-probabiliy of vecor ino vecor or 3 by p, and of vecor ino vecor by p ; hen p > p, becaue d(,3) = d(,) < d(,). - For a Gray-mapping ( ; ; ; 3 ) he bi-error probabiliy i expreed by (4).a P bg p p ; a. P bn p p p ; b (4) - for a mapping according o he naural binary code ( ; ; ; 3 ), he probabiliie p and p have he ame value a for he Gray-mapping. The bi-error probabiliy i expreed by (4).b. - comparing he bi-error probabiliie enured by he wo mapping rule for he ame ymbol-error probabiliy, we ge: p bg p bn p p p bg p ; (5) bn - a imilar reaoning may be applied o conellaion A8, a well. Expreion of he PSK modulaed ignal - he PSK modulaed ignal i expreed by (5); u T (-nt ) repreen a recangular impule wih uniaryampliude and duraion T, which indicae ha he phae of he carrier Acoω c i modified wih Δф n only during he n-h ymbol-period. PSK T c k k= () = A u ( - kt )co( + ); (6) - he modulaed PSK ignal i a ucceion of modulaed ignal during a ymbol period; he phaehif of each ymbol period do no inerfere wih one anoher. - he phae reference ha i ued o compue he phae-hif Δф k indicae he ype of modulaion: abolue or differenial. - APSK i no employed in pracice becaue he demodulaion require he eparae phae-reference,. - DPSK i ued, becaue he phae-reference i he carrier phae during he previou ymbol period. Specral diribuion of he QPSK ignal - he power pecral diribuion of he all PSK conellaion depend on he ymbol frequency and on he ype of he carrier ignal, i.e. harmonic (coine or ine) or recangular - a. when he carrier ignal i a harmonic ignal (having only one pecral componen), he pecral diribuion of he PSK ignal i expreed by (7) for he non-filered modulaing level, ee (6). - he pecrum, approximaely repreened in figure 3, exhibi a cenral lobe (k = ), wih a bandwidh equaling f and maximum value S M, around he carrier frequency and ide lobe wih maxima S Mk occurring a he f M frequencie, given by (8). Thee pecral lobe are generaed by u T (-kt ). 7

3 ( c ) in f V f f f in f A u T(f)= ; S n ( f ) ; f f ( f fc ) f Hz f f f m = f c +/- kf k ; f M = f c +/- (kf + f /) k ; S M = A /f ; S Mk = S M 4/[(k+)π] ; (8) (7) Figure 3 Power pecral diribuion of he QPSK ignal non-filered and RRC filered - he ampliude of he ide lobe decreae raher lowly wih he increae of heir index. - hi i a diadvanage in ranmiion chain including non-linear high-frequency amplifier - Fig. 4 preen he filered and non-filered power pecra of a PSK ignal, for f =.33 khz, α=.5..5 Figure 4 Power pecra of non-filered and RRC-filered PSK ignal - napho - b. If he carrier ignal i recangular, a for he direc digial PSK modulaor, he pecrum of he modulaing ignal i ranlaed around he harmonic of he carrier ignal k f c, ee (9) (f - c) in kf f/f c f ut(f - fc) = T Ak (9) k= (f - kf c) f - for a harmonic carrier ignal, he um in (9) conain only one erm, k =, ee fig.3 and (7) - figure 5 preen power pecra of he non-filered f/f c modulaing ignal, u T (-mt ), recangular carrier ignal and PSK modulaed ignal Specrum of PSK modulaed on recangular carrier.5 Fig.5 Specrum of PSK modulaed on a recangular carrier Filering he PSK modulaed ignal he conideraion above how ha he modulaed f/f c ignal ha o be filered, o ha i would mach he channel bandwidh. - due o he frequency-band limiaion, he filered ignal expand in ime generaing he ISI. - o remove (or a lea decreae) he ISI in he probing momen, he ignal ha o be filered wih a RC 3

4 characeriic wih a roll-off facor α, ee he lecure on Filering he Daa Signal). - for beer performance in he preence of he Gauian noie, he filering characeriic ha o be equally-pli beween ranmier and receiver, a hown here Thi involve filering wih a RRC characeriic boh in he ranmier and receiver. Baically, he RRC filering may be implemened in wo varian: a band-pa filering placed afer he PSK (f-f c )/f N modulaor; hi opion i ued by he direc digial PSK modulaor which generae he PSK ignal on α = α =,5 α = a recangular carrier ignal, a will be hown in he Figure 6. BP RC characeriic for α =,.5, chaper dedicaed o he PSK modulaion and in Annex. The BP-RRC filering characeriic i defined by () and repreened in fig.6: ; [ c N( ), c N( )]; N E( c) = N R( c) = N( c) = ( c ) (- ) co( - ); A; () 4 N 4 A [ c N( ), c N( )] [ c N( ), c N( )]; a low-pa filering applied o he modulaing ignal; hi approach i ued if he PSK ignal i generaed uing he QAM echnique, ee he chaper on PSK modulaion Thi mehod i preferred in mo applicaion. The LP-RRC filering characeriic i defined by equaion (7) in he lecure on Filering he Daa Signal and can be obained by making c = in () above. Xα ((f-f p )/f N ) Effec of filering he PSK-modulaed ignal - he goal of filering he modulaed ignal wih a global RC characeriic are: limiaion of he frequency band occupied by he modulaed ignal; removal of ISI in he probing momen - if he modulaing momen are a he beginning of he ymbol period, negaive edge of he ymbol clock f, hen, due o he τ g (f) characeriic of he filer, he probing momen are placed in he middle of he ymbol period, i.e. poiive edge of he ymbol clock f, ee he daa filering. - o analyze he effec of filering upon he momenary phae and frequency and upon he envelope of he modulaed ignal, we conider ha he ignal phae uffer a hif from he phae of he previou ymbol, denoed by º, o ΔΦ k. Denoing by Φ() he phae-hif inered by he modulaor, he modulaed ignal would be expreed by: () = Aco( p + ()); () = kut( + T / - kt) k; [kt - T / ;kt T / ]; () - heory how ha afer he filering, he momenary phae of he modulaed ignal Φ() ha a coninuou variaion decribed by (), where x() denoe he impule-repone of he RC filer, repreened in figure 5 for α =. In figure 5, he ime-reference momen i he probing momen, =, herefore he modulaing momen occur a odd muliple of T /. () in k = arcan ; () k k - k in x() - fig. 7 preen he variaion of he momenary phae for ΔΦ k = k π/8, k {,,4}. - i how ha he momenary phae ha very mall (cloe o zero) value in all probing momen, = k T, excep for he main probing momen =, (hird axi in fig. 7). - in he main probing momen phae Φ() reache approximaely he nominal value ΔΦ k of he curren ymbol. Thi how ha ISI ha been ignificanly decreaed in he phae domain, i.e. in every probing momen and only hen, he momenary phae ha he nominal value of he phae-hif of ha ymbol and i very lighly affeced by he ime-expanion of he phae-hif of oher ymbol-period. - noe alo ha he momenary phae Φ() reache half of i nominal value a = ±T/. - for ΔΦ k = π, he phae variaion i almo recangular, and for ΔΦ k =, he phae variaion i zero. 4

5 -T -T / f +T / +T ΔΦ n m =3 I min m = m =4 m = Δf in - m ax,7,5 Θ() Φ ()/Δ Φ n,9,5 Δf in () I(),38 Figure 7. Effec of filering he PSK ignal -ramane I min x() Δf in - m ax - a = ±T/, he momenary phae ha inflexion poin. - he variaion of he momenary frequency f in () around f c, afer he filering, can be derived from he Φ() variaion law by: d () f () (3) in d - he Δf in () can be derived eaier by performing he graphical derivaive of Φ(). The inflexion poin would become exreme poin for he derivae funcion; he monoony of he derivae funcion can be derived from he concaviy/convexiy of Φ(), and he ign of he derivae funcion, from he monoony of he Φ(). The reuled curve, which approximaely decribed he variaion of he momenary frequency around he carrier frequency, i hown on he 5-h axi of fig he deviaion of he momenary frequency ha maxima a approximaely = ±T/. The value of hee maxima depend of he value of phae-hif ΔΦ k = m /4, and can be compued uing (4). Δf inmax = m f /8; m = k, for k =,...,4; m = k - 8, for k = 5,...,7; (4) - he envelope I() of he filered modulaed ignal i hown o be no longer conan; i varie in ime according o (5).a, ee he 6-h axi of figure 5. I() = A - 4 n N in x()[ - x()] ; a. I min = A co ; b. (5) - (5).a how ha he envelope of he filered ignal ha a maximum a he probing momen = and minima a = ±T/. The value of he minima are expreed by (5).b and depend of he phae-hif value. For ΔΦ k = m π/8, m {,,7} hee minima have five poible value,,.9,.77,.38,, which are repreened in figure 7 - noe ha for ΔΦ k = º, he variaion of momenary phae, frequency deviaion and of he envelope are zero. Thi i a major diadvanage of he A ype conellaion. - ummarizing, he goal of filering he (D)PSK ignal are: decreae of he occupied frequency BW, o ha i would mach he channel BW; removal of ISI in he probing momen. - he conequence of filering he (D)PSK ignal wih an RC characeriic are: - he probing momen i diplaced wih half of ymbol-period, from he modulaion momen; - a coninuou variaion of he momenary phae, wih maxima, approximaely equaling he nominal phae-hif, a he probing inan; - coninuou variaion of momenary frequency, wih maximum deviaion from he carrier frequency occurring a =±T/; he maximum value of he frequency deviaion depend of he phae-hif value. - he occurrence of a paraiic ampliude modulaion, i.e. he envelope, which ha maxima a he probing momen and minima a approximaely he modulaing momen = ±T/. The minima value depend on he value of he phae-hif. Generaion of he DPSK ignal - he DPSK ignal can be generaed by direc digial mehod or by employing he QAM echnique 5

6 DPSK digial modulaor - wo digial modulaor which produce direcly he APSK and DPSK ignal for all conellaion are preened in Annex ; hey are no included in he examinaion opic DPSK modulaion-demodulaion employing he QAM modulaion (echnique) - he expreion of he (D)PSK ignal over one (k-h) ymbol-period i given by (6), where Φ k denoe he abolue phae of he carrier during he k-h ymbol period: PSK = Aco( p + k) ut( - kt); (6) - by expanding (6) we ge (7), which repreen a QAM ignal in which he wo modulaing ignal are no longer independen ignal; hey fulfill condiion (8). PSK = A co kut( - kt) cop - A in kut( - kt) in p = I(kT) Acop - Q(kT) Ainp; I(kT) = Ik= co kut( - kt) ; Q(kT) = Qk in kut( - kt); wih Ik Qk u T( kt ) - if he modulaing ymbol are wrien a (8).a hen he PSK ignal can be wrien a (8).b : jk m I jq A e ;a. k k k k jck () Re I jq co jin Re A e ; b. PSK k k c c k DPSK modulaion generaed by he QAM echnique 4-PSK Conelaia A4 - a an example we preen he generaion of he A4 conellaion, figure 8. 9 Table how he phae-hif ΔΦ k, he value of he modulaing level (I k, Q Q k ), he inpu dibi-daa a a and of he dibi afer he Gray-naural a a converion (CGN), b b, which i performed according o: 8 b a a; b a; (9) Figure 8. The A4 ignal conellaion I - () and () how ha o obain a DPSK, he dibi ha i delivered o he circui ha compue he I k and Q k level i obained by differenially precoding he modulaing daa-dibi, afer he GN converion. - he block diagram of he DPSK modulaor implemened by he QAM echnique, i hown in fig he I k and Q k level can be obained by wo mehod: by reading he I k and Q k value from a able, in erm of he curren daa dibi and previou encoded dibi, when he GN converion and he differenial encoding are included; by uing a D/A converer and a circui ha compue he bi 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 level are read from a able, in erm of curren daa and previou encoded-daa dibi; hi block i called encoder or mapper. - o limi he bandwidh of he modulaing ignal and enure ISI= in he probing momen, he I k and Q k ignal would be LP filered (FFE block) wih a RRC characeriic wih a roll-off facor of α 6 (7) (8) - hi mehod generae an abolue-phae modulaion, ince he modulaed carrier phae-hif are referred o he phae of he non-modulaed carrier. Mo ofen he lieraure denoe by QPSK he 4- APSK (varian A or B). a a b b I k Q k ΔΦ k + º + 9º - 8º - - 7º Table. Signal value in he main poin of he DPSK-A4 encoder for c k- c k- = - o ranform hi modulaion ino a DPSK one, he abolue phae of he modulaed carrier hould be modified according o (). Φ k = (Φ k- +ΔΦ k ) mod 36 º () - becaue all ΔΦ k are muliple of 9 º, he abolue phae will be a muliple of 9 ºand () may be wrien a: N 9 (N 9N 9 ) N (N N ) () k k k k k mod36 k k k mod 4 - bu he number N k and ΔN k are binary repreened by he dibi c k c k and b k b k ; hen () may be wrien a: (b k b k + c k- c k- ) mod 4 = c k c k ; ()

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. Acoω c Ainω c Σ S DPSK c k- D Ck f - afer he filering we ge he coninuou modulaing ignal I() and Q(). - he expreion of he ranmied modulaed ignal i: DPSK () = I() co c - Q()in c = (3) I() = A co kut( - kt) afer filer; Q() = A in kut( - kt) afer filer; - he LP-RRC filering i implemened uing a FIR rucure, in which only one ample of he I k and Q k level hould be inered in every ymbol-period; he re of he ample of ha ymbol period would equal zero, ee DSP lecure and Daa Tranmiion lecure - when implemened on a ignal proceor, he ymbol period i dvided ino N ampling period. The encoding, muliplicaion and addiion operaion are execued for each ample. A BP fler hould be added a he modulaor oupu, o uppre cuanizaion noie. - he ample of he carrier ignal would be ored in a able, N value per ymbol period; he digial generaion of he carrier ignal hould enure a mall THD facor. - hi mehod can be applied if he frequency of he carrier allow i implemenaion on a proceor; - for carrier ignal wih greaer frequencie, he digially filered ignal (I() and Q() are muliplied o he carrier ignal by analogue muliplier and he ummaion i performed by an analogue adder. Generaing oher DPSK conellaion wih he QAM echnique Conellaion A and B - ince hee conellaion involve phae-hif of ΔΦ n =º or 8º and, repecively, ΔΦ n =9º or ΔΦ n =7º, which define he vecor of he wo conellaion, he QAM- expreion of he - DPSK ignal are: PSK-A Figure 9. Block diagram of he DPSK modulaor implemened by uing he QAM echnique () = A n=- n=- co( p ) u T ( - nt); PSK-B () = A in( p ) u T ( - nt); - he value of he modulaing level I k and Q k of he A and B are preened in able. Conellaion Bi I k Q k A + A - B + B - (4) Table. Value of I k and Q k for conellaion A and B - he modulaion and i block diagram remain he ame a he one decribed above for QPSK, excep for he differenial precoding-decoding ha are performed a mod operaion on one bi. Conellaion B4 - QAM generaion of B4 require a modulo-8 differenial precoding-decoding on 3 bi. - he b b dibi obained afer he Gray-naural converion i ranformed in he b b b ribi: b = b ; b = b o ; b = ; (5) - eing he bi b = i equivalen o he 45º roaion impoed by hi conellaion. - hen, afer he differenial precoding, () aplied for hree bi a mod8, he value of he abolue phae would alernaively (in ucceive ymbol period) equal an odd and hen even muliple of 45 º. - he modulaing level would ake aleranively value from he e A = {-,, } au B = {- /, + /}, he e being choen by he value of he c bi, i.e. c = e A, c = e B; he pair of coordinae i hen choen by he bi c c - he c c c ribi i employed o elec he modulaing level I k and Q k a hown in able 3 -ee noe 7

8 Dibi c c c (/) (/) (/) (/) Table 3. Value of he I k and Q k level for conellaion B4 I k + // - //- - // + // - he re of he operaion required by he QAMmodulaion-demodulaion of B4 are imilar o QPSK, Q k + // + // - //- - // bu he differenial decoding hould be a modulo 8 one - noe ha afer he demapping and differenial decoding only he wo mo ignifican bi are employed in he final proceing, becaue he differenial phae hif are muliple of π/. Conellaion A8 - he QAM modulaion-demodulaion of he A8 are implemened imilarly a he one of A4, wih he following difference: - he value of he I k and Q k level, in erm of he daa ribi c c c, are he one of able 4 c c c I k + / + - / - / - + / Q k + / + / + - / - / - Table 4. Value of he I k and Q k level for conellaion A8. - he differenial precoding-decoding hould be made modulo-8 on he hree bi. - Annex preen he block diagram of he DPSK ranmier ha ue he direc digial mehod o generae he modulaed ignal on a recangular carrier and perform a BP-RRC filering on a inermediae frequency f i followed by a frequency ranlaion on he channel carrier frequency- i i no included in he examinaion opic Conideraion Regarding he Implemenaion of he RC filering of he (D)PSK ignal -individual udy - required for examinaion - he RRC characeriic can be implemened eiher wih analog or digial mehod. - he analog implemenaion provide accepable (no good!) accuracy only for roll-off facor α > o enure high accuracy and maller roll-off facor, he RCC characeriic hould be implemened digially uing a FIR filering rucure. For a brief preenaion of hi approach, ee he Annex of he lecure on Filering he Daa Signal. Thi opic will be deal wih in he laboraory clae of he daa Tranmiion coure in he IV h year. - he RRC characeriic could be implemened eiher a band-pa filer, cenered on he inermediae or channel frequency, or a low-pa filer, by filering he modulaing I k and Q k ignal in bae-band. - he BP-RRC approach hould be ued for he ranmier ha direc digial (D)PSK modulaor, a he one decribed in Annex of hi maerial. Their poiion in he ranmier i hown in Annex. - he ranmier ha implemen he (D)PSK modulaion uing he QAM approach, a decribed above, could ue eiher he BP or LP varian of he RRC filering. Sill, almo all implemenaion ue he LP varian, by LP filering he I k and Q k modulaing ignal, a alo hown above. Conideraion Regarding he Frequency Tranlaion of DPSK ignal - individual udy- required for he examinaion - he frequency ranlaion from an inermediae-carrier frequency f i o he channel-carrier frequency f c or he oher way around, can be accomplihed by muliplying he ignal o a ranlaion ignal of frequency f followed by a BP or LP filering which would elec he deired frequency band and aenuae he undeired pecral componen reuled from muliplicaion. - he muliplicaion can be made by uing a muliplier or a chopper, ee he LM and FM lecure - he conideraion below are made for he cae when f c > f i, which i me in almo all pracical applicaion. - a hown by equaion (5) a he end of he FM lecure, which i repeaed here for convenience, if he f > f c, he ign of he addiional phae inered by he modulaing ignal Φ() i changed, while for f < f c i i no changed. c c i and () k fv 'co( i ()); a. c i (6) c c i and () k fv 'co( i ()); b. - hi concluion hown here for he downward ranlaion, alo hold for he upward f i f c ranlaion. - hi ign change ha occur for f > f i, (cae b.) hould be compenaed in he modulaor 8

9 - due o he changed ign of he phae-hif, he reuled phae-hif would correpond o a daa mulibi which equal he modulo ( n ) complemen of he modulaed daa mulibi (boh in binarynaural repreenaion), for all conellaion excep for -PSK varian A. Thi yemaic error hould be compenaed in he ranmier, afer he G-N converer, by delivering a he modulaor inpu he complemen modulo ( n ) of he mulibi ha ha o be ranmied. - he cae when f c < f i i me only for ranmiion over he elephone channel, in he o called dialup modem. For hi iuaion imilar reul are obained if we exchange f i and f c wih each oher. Annex - no required for examinaion APSK and DPSK direc digial modulaor buil wih an arihmeic adder and a couner - AAC - due o he finie number of phae-hif required and o he ime-dicree characer of he modulaion, he digial mehod enure a higher accuracy and a beer abiliy. - he DPSK can be obained boh by DPSK modulaor and by APSK modulaor, preceded by a differenial encoding of daa mulibi. - he digial modulaor will employ a erial-parallel converer, o build he mulibi. Thi converer would acquire erially he inpu bi, uing he bi-clock f b, and he mulibi will be read by he modulaor uing he ymbol-clock f. - hen, he digial modulaor would perform he Gray-binary naural converion, CGN; he inpu mulibi i looked-upon a Gray-coded combinaion (o decreae he bi-error probabiliy) and hen hey are convered ino naural-binary combinaion o mach he modulaor (a will be decribed laer) - he bi-mapping may lead, omeime, o a ignifican decreae of he bi-error probabiliy, for he ame ymbol-error probabiliy (he ame SNR), ill uing he ame implemenaion complexiy. - he converion beween Gray and naural-binary code i performed according o (6), g i indicaing he bi of he Gray-coded combinaion. a g gg ; a gg ; a g ; g aa ; g aa ; g a ; (A..) - he A8 conellaion require a modified (compleed!) converion rule, becaue he andard employ a differen mapping rule, ee fig..c. Homework: eablih how relaion (A..) hould be modified o accomplih he GNC for A8 - for a p-bi mulbi, he modulaor i implemened uing a p-bi arihmeic adder a p-age couner; i i hown in figure A., for p = 3. The ignal diagram i hown in figure A.. f bi d i g g :3 f a a D ifferenial precoding A A Σpd Σ Σ b b A A APSK M odulaor Σm Σ D P S K g a A B B B Σ b A B B B 4f i f i f i A ccum ulao r 8f i :8 Figure A.. Elecric diagram of he APSK and DPSK modulaor wih arihmeic adder and couner - if he inpu ribi b i i kep o, he number (expreed in naural-binary code) i added o he inpu of he adder; o, he Σ oupu will deliver a recangular ignal of frequency f i affeced by a phae hif ΔΦ = 45º, compared o he reference ignal, i.e. he ignal a he couner oupu. - if he ribi =, hen he number 4 i added and he um will uffer a hif of four uni, which equivalen o a phae-hif ΔΦ = 4 45º = 8º, compared o he reference ignal. - imilarly, for b i =, a phae-hif ΔΦ = 45º= 9º, and for b i =, a phae-hif of ΔΦ = 45º= 45º i obained. - he combinaion of he hree bi generae all number k {,,7} ha correpond o all he phaehif equaling k 45º, which compoe he A8 conellaion. 9

10 8f i 4f i f i f i b i = ΔΦ k = 45º b i = ΔΦ k = 4 45º b i = ΔΦ k = 45º b i = ΔΦ k = 45º 3 Figure A.. Signal diagram of he APSK modulaor of figure 3. - he converion of he inpu ribi from he Gray-code o he naural binary code i required becaue he number k i he repreenaion in he naural binary code of he daa ribi. - a he beginning of each ymbol period, i.e. a he negaive edge of he ymbol-clock, he ribi a i change, generaing he phae-hif correponding o ha period. - he phae-reference i he phae of he carrier f i, o hi modulaor generae APSK. - o generae he DPSK modulaed ignal, he inpu daa ribi a i i differenially precoded, generaing he ribi b i which i applied o he APSK modulaor, ee figure 3. For he DPSK, he abolue phae of he carrier ignal during he n-h ymbol period may be wrien a: Φ n ab = (Φ n- ab +ΔΦ n )modulo 36º; - ince all he phae-hif are muliple of 45º, hi can be implified a: k n ab = (k n- ab +Δk n )modulo 8; (A.) (A.3) - o, he differenial precoding coni of a modulo- p arihmeic addiion, on p bi, of he previou mulibi b n- o he curren mulibi a n ha come from he CGN. - hi operaion i performed uing a p-bi arihmeic adder and a p-bi hif regier, a a memory elemen, ee figure A.. - if hi modulaor i o generae he A4 or A conellaion, i hould operae on or bi (he MSB one); he differenial precoding hould be performed modulo 4 or, modulo. The inpu correponding he unemployed bi hould be conneced o, i.e. b = and b = b =. - he generaion of he B-ype conellaion involve a roaion of 45º, for he B4, or of 9º, for he B. Thi i accomplihed by eing b =, regardle he daa dibi, for B4, or eing b = and b =, for B. For he conellaion uing or 4 vecor, he ignificance of he hree bi b i are ummarized in able A., where d denoe daa bi. Bi ; Conellaion A8 A4 B4 A B b d d d d d b d d d b d Table A.. Value of ribi-bi for differen conellaion - he frequency of he modulaed carrier-ignal f i, may be eiher he channel carrier ignal f c or an inermediae frequency, higher han he channel-carrier frequency, depending of he mehod employed o filer he modulaed ignal, ee he PSK-filering paragraph. - he AAC modulaor iner advance phae-hif. Someime, delay phae-hif hould be inered; o generae hi ype of phae-hif we employ he periodiciy of he carrier-ignal and ge (A..4), where he backward phae-hi are marked by. ΔΦ' n = +(36º-ΔΦ n ) modulo 36 ; Δk' n = (8 - Δk n ) modulo 8; (A.4) - o ge hee phae-hif he modulaor hould be provided wih he 8-complemen of he ribi ha correpond o he advance deired phae-hif. The block ha perform he complemen hould be placed beween he CGN and accumulaor.

11 - for conellaion wih 4 vecor he 4-complemen hould be employed; for B, he modulaing bi hould be invered and for A hi operaion i no required. DPSK modulaor buil wih an arihmeic adder and a hif-regier - AASR - i elecric diagram i hown in figure A.3 and he ignal diagram i diplayed in figure A.4 - noe ha if he modulaing funcion F i =, he aembly AA-SR ac like an 8-couner. - Conidering he adder operaional equaion (A.5), he Bi inpu of he adder increae heir value wih one uni in he rhyhm of he 8f i -clock ignal, due o he inpu-carry c =. Then he Σ ignal would have a period equaling 8 period of he 8f i ignal, i.e. a frequency equaling f i. i Ai Bi Ci ; (A.5) F =F =F = Σ Σ Σ F =;F =F = Σ Σ Σ F =F =;F = Σ Σ Σ () (A) ΔΦ º 45º 9º 8º No.8f i -Ck Per. F =F =;F = Σ Σ Σ Table A. Operaing principle of he AASR DPSK modulaor - if a he beginning of he ymbol-period, for a period of he 8f i ignal, 8 fi > f, he modulaing funcion F =, he adder oupu increae i value wih 4 uni and he phae of ignal from he Σ oupu uffer a d i C S P g g g : 3 Im p. M od. 8f i a F M F A C C O u Σ R D n a U L c F G A i T o A Σ I n N a F G N A B Σ D D P S K B B Figure A.3. Block diagram of he AASR DPSK-modulaor Sar of h e ym bol-period 8f i F i = Δ Φ = º Σ Σ F = Δ Φ = 45º A F = Δ Φ = 9º Σ Σ F = ΔΦ = 8º M od u la in g im p ule Figure A.4. Signal diagram of he AASR DPSK-modulaor phae-hif, in advance, of ΔΦ = 4 45º = 8º, ee able A. and figure A.4. The phae-hif appear

12 obviou if i i conidered from he end of he fir ymbol period of he carrier ignal (f i ), marked by poin A in figure A.4 and in able A.. - Similarly, for F =, we ge ΔΦ = 45º = 9º, and for F = we ge ΔΦ = 45º. - by combining he value of he hree modulaing funcion, all he phae-hif equaling k 45º, wih k {,,7} can be obained. - for hi modulaor, he modulaing daa ribi i applied only during he modulaion impule, ee figure 6; for he re of he ymbol period, he value of he modulaing funcion F i, are forced o. So he ribi a i hould be proceed by he block ha generae he modulaing funcion, which allow i acce o he modulaor only during he modulaion impule. -he modulaing funcion alo perform he 8-complemen (or 4-complemen) if hi modulaor hould iner backward phae-hif. - he conellaion wih 4 or vecor can be produced in he ame manner a he one decribed for he AAC DPSK modulaor, by aking ino accoun able A.. -becaue he phae-hif i referred o he phae of he carrier ignal during he previou ymbol period, conidered o have Δ = º, hi modulaor generae a DPSK modulaion. DPSK modulaor wih conrolled diviion CD - hi modulaor i baed on he phae-hif by conrolled diviion decribed in he dynamic ynchronizaion yem (ee ynchronizaion in he BB lecure noe). Annex Block diagram of a PSK ranmier - no required for examinaion PSK ranmier wih modulaion on he inermediary frequency - he block diagram of hi ranmier i hown in figure A.. TxCk OSC. DIV. f Var. D IM P.M OD. 8f i PSK dr. PSK c o. f TxD CTS RTS SCR. Conrol E m iie b d i C.S.P. b C.G.N. b Block Synhei of M odulaing Funcion M odulaor F F F Modu- Laor DPSK f i F. F. E. f i TR. FR. f p PSK co. B.P.F. LF/HF. f p EQ. Compr. Line Am plif. Line Uni Channel Figure A... Block diagram of he PSK ranmier wih modulaion on he inermediary frequency - he daa o be ranmied TxD, are inered ino a crambler SCR, ha randomize he daa o be modulaed. The crambler i employed only for conellaion wih 8 or more vecor. - he crambled daa are hen en o he erie-parallel converer, hen o he CGN generaing he mulibi which i delivered o he modulaor. - he modulaor generae he modulaed ignal PSK-fi on a recangular carrier f i, uing he auxiliary ignal of frequencie 8f i and f, obained from he ocillaor-divider block, OSC-DIV. - he modulaed ignal i filered wih a BP RRC filer wih a roll-off facor α, by he ranmiion haping-filer, FFE, generaing a modulaed ignal on a coine carrier ignal, -PSK,c,f i. - hi ignal i hen ranlaed on he channel-carrier frequency f c by he frequency ranlaion block TR.FR., which employ a recangular ignal of frequency f, provided by he OSC.-DIV block. The band-pa low-frequency filer BPF-LF, reain only he inferior ideband generaed by he freq. ran., which i he modulaed ignal on a coine carrier of frequency f c, -PSK,c,f c. - he level of hi ignal i eablihed by he line amplifier; hen he ignal i en o he line-uni which enure he adapaion wih ranmiion channel. - he ranmiion conrol circui manage he enable/diable of he ranmier, uing he RTS and CTS ignal. Some conrucive varian include a compromie equalizer.