r(t) BF i(t) c(t) Quadrature mlitude Modulation - QM - beide it emloyment a uch to tranmit two indeendent data flow, the QM i ued a a method of modulation-demodulation of other amlitude, frequency or hae modulation. Quadrature mlitude Modulation QM - the exreion of a QM modulated ignal i: MQ (t)=d (t)coω c t-d (t)inω c t; () where d (),d () are two indeendent data equence of ymbol-eriod, which are amlitudemodulated on two orthogonal carrier that have the ame frequency. - the bloc diagram of the QM modulator i reented in figure. i x (t) i f (t) LF coω l t arrier recovery circuit inω l t LF c x (t) c f (t) R R d (t) Symb-cloc recovery circuit d (t) - the modulating data ignal may tae value (bit/ymb) or more value (multibit/ymb). he maing multibitmodulating level i not figured in figure. - the d () and d () modulating ignal hould be filtered with a RR low-a characteritic, to enure (after the comletion RR-filtering in the receiver) S = 0 and better erformance in the reence of noie, according to: ; ω [0, ω ( α)]; ( ω) = R ( ω) = ( ω) = π( ω) π(- α) () co( - ); ω [ ω ( α), ω ( + α)]; 4ω α 4α - the filtering deliver the modulating ignal of the two branche, d (t) and d (t) in (), which are continuou ignal the bandwidth and frequency band of the modulated ignal i given by (3), for L haing filter of f (+α): BW = f (+α) = f (+α); FB = [f c - f (+α); f c + f (+α)]; (3) - the demodulation ue the method decribed in age 0- of the LM lecture, but it i adated to the dicreet nature of the modulating ignal -to decribe the adated demodulation rincile we aume that the relation between the hae of the tranmitter and local receiver carrier ignal, with ulation ω c and ω L reectively, i: ω L t=ω c t+δωt+θ 0 =ω c t+θ(t); (4) - the bloc diagram of the QM receiver i hown in figure ; it doe not contain the connection of the local carrier and ymbol-cloc recovery bloc, which will decribed later in thi chater. d () d *() robing eciion robing eciion d () d * () Figure Bloc diagram of the QM receiver - the equation that decribe the demodulation on the two branche i(t), the in-hae branch, and c(t), the quadrature one are (5) and (6). i x (t) and c x (t) denote the ignal at the outut of the multilier and i F (t) and c F (t), denote the ignal at the outut of the L-filter of the two branche. - he QM demodulation i imilar to earate roduct-coherent demodulation of two SB-S ignal. - d () d ().. en. urt. d (t) d (t) coω c t inω c t Figure. Bloc diagram of the QM modulator S - MQ + - Σ
r(t)coωlt d(t) d (t) ix(t) = = [co θ(t) + co(ωt + θ(t))] [-in θ(t) + in(ω t + θ(t))] K (5) r(t)inωlt d(t) d (t) cx(t) = = [in θ(t) + in(ωt + θ(t))] [co θ (t) + co(ω t + θ(t))] K (6) - the L-filter ure the ectral comonent centered around ω c and the ignal at their outut are: if(t) = (d(t)coθ(t) + d(t)in θ(t)) d(t) t. θ(t) 0 (7) cf(t) = (d(t)in θ (t) d(t)coθ(t)) d(t) t. θ(t) 0 (8) - if the carrier recovery circuit enure a hae-hift θ(t) 0, then the ignal at the outut of the haing filter would tae, in the robing intant, value roortional to the modulating ignal. - the effect of the incorrect carrier-recovery may be derived from relation (7,8), ee the LM coherent demodulation, and conit, for each branch, in the occurrence of a araitic M and the addition of the modulating ignal of the quadrature branch, which alo ha a araitic M. herefore, the condition θ(t) 0 hould be imoed. - the ymbol-cloc recovery circuit i intended to extract the hae-reference and ynchronize the local ymbol-cloc, f i and f c, with the demodulated ignal o that they are robed at the right timeintant. - uing the ynchronized ymbol-cloc, the demodulated ignal d (t) and d (t) are read in the robing moment, when they are not affected by S, generating the d and d level, that have contant value during the -th ymbol eriod. - the robed level are then delivered to the deciion circuit, which decide which of the ermitted level i cloer to the received, thu delivering the etimated (decided) level d * and d *. f the tranmiion emloy more bit/ymbol, then from the decided level, the correonding multibit are demaed (decoded). SK modulation-demodulation emloying the QM modulation (technique) - the exreion of the SK ignal over one (-th) ymbol-eriod i given by, ee the SK lecture: SK = co( ω t + ΔΦ )u (t - ); (9) - by exanding (9) we get (0), which rerent a QM ignal in which the two modulating ignal are no longer indeendent ignal; they fulfill condition (). SK = co ΔΦu(t-)coωt-in ΔΦu(t-)inωt= = () coωt - Q() inωt; () = = co ΔΦ u(t - ) ; Q() = Q = in ΔΦ u(t - ); + Q = u (t-); () SK modulation generated by the QM technique - a an examle we reent the generation of the 4 contellation, figure 3. able how the haehift ΔΦ, the value of the modulating level (, Q ), the inut dibit-data a a 0 and of the dibit after 4-SK ontelatia 4 the ray-natural converion (), b b 0, which i erformed according to: 90 0 b0 = a0 a; b = a; () Figure 3. he 4 ignal contellation Q a a 0 00 able. Signal value in the main oint of the SK-4 encoder for c - c - 0 = 00 0 80 0 - thi method generate an a a 0 b b 0 Q ΔΦ abolute-hae modulation, 00 00 + 0 0º ince the hae-hift of the 70 0 0 0 0 + 90º modulated carrier are - 0-0 80º - 0 referred to the hae of the 0-0 - 70º non-modulated coine carrier. - the SK modulation generated by the QM technique are called QSK. Mot often the literature denote by QSK the 4- vector SK (variant or B). (0)
- to tranform thi modulation into a SK one, the abolute hae of the modulated carrier hould be modified according to (3), which i hard to imlement. Φ = (Φ - +ΔΦ ) mod 360 º (3) - becaue all ΔΦ are multile of 90 º, the abolute hae will be a multile of 90 ºand (3) may be written a: Φ = 90 Φ = ( 90 +Δ 90 ) = ( +Δ ) (4) mod360 mod 4 - but the number and Δ are binary rereented by the dibit c c 0 and b b 0, (4) my be written a: (b b 0 + c - c 0 - ) mod 4 = c c 0 ; (5) - (4) and (5) how that to obtain a SK, the dibit that i delivered to the circuit that comute the and Q level i obtained by differentially recoding the modulating data-dibit, after the converion. - the bloc diagram of the SK modulator imlemented by the QM technique, i hown in fig. 4. - the and Q level can be obtained by two method: by reading the and Q value from a table, in term of the current data dibit and reviou encoded dibit, when the converion and the differential encoding are included; by uing a / converter and a circuit that comute the bit which control the / converter a a 0... b b 0 - c 0 Σ M o d u lo 4 c c 0 M Q.. coω c t inω c t Σ S SK c - f Figure 4. Bloc diagram of the SK modulator imlemented by uing the QM technique - on a S imlementation, the and differential recoding are erformed off-line; the and Q level are read from a table, in term of current data and reviou encoded-data dibit; thi bloc i called encoder or maer. - to limit the bandwidth of the modulating ignal and enure S= 0 in the robing moment, the and Q ignal would be L filtered (FF bloc) with a RR characteritic with a roll-off factor of α. - after the filtering we get the continuou modulating ignal (t) and Q(t). - the exreion of the tranmitted modulated ignal i: (t) = co Δ Φ u SK (t) = (t)coω t - Q(t)in ω (t - ) after filter; Q(t) = in Δ Φ c c t = u (t - ) after filter; - the L RR filtering i imlemented uing a FR tructure, in which only one amle equaling hould be inerted in every ymbol-eriod; the ret of the amle of that ymbol eriod would equal zero, ee S lecture and ata ranmiion lecture - when imlemented on a ignal roceor, the ymbol eriod i dvided into amling eriod. he encoding, multilication and addition oeration are executed for each amle. L flter hould be added at the modulator outut, to ure cuantization noie. - the amle of the carrier ignal would be tored in a table, value er ymbol eriod; the digital generation of the carrier ignal hould be carefully conidered to decreae ignificantly the H factor. - thi method can be alied if the frequency of the carrier allow it imlementation on a roceor; - for carrier ignal with greater frequencie, the digtally filtered ignal ((t) and Q(t) are multilied to the carrier ignal by analogue multilier and the umation i erformed by an analogue adder. (6) 3
Filtering the QM-modulated ignal - becaue QM i a LM ignal modulated with rectangular modulating ignal, the band-limitation filtering hould be erformed with a RR characteritic. Baically it may be imlemented in two variant: a band-a filtering laced on both branche after the multilication with the carrier ignal, or laced at the outut of the final adder; a low-a filtering laced on both branche alied to the rectangular, Q ignal before the multilication with the carrier ignal. hi method i referred in mot alication. emodulation of SK ignal uing the QM technique - the demodulation of the SK ignal by uing the QM technique may be accomlihed in two variant: a variant that emloy L filter to ure the high-frequency ectral comonent; a variant which emloy the Hilbert tranform of the received ignal to ure the high-frequency ectral comonent. SK-QM demodulator with L filter - Uing (0, ), the exreion of the received SK ignal become (7), where (t) and Q (t) denote the filtered modulating ignal affected by the channel erturbation and ditortion: rsk = '(t) coω t - Q'(t) inω t; (7) - the demodulation of the SK ignal may be accomlihed uing the QM demodulation decribed by relation (5...8), i.e. a coherent SB-S demodulation, and i hown in figure 5. - rewriting thee equation for the SK ignal we get: r(t)coωlt '(t) Q '(t) ix(t) = = [co θ(t) + co(ωct + θ(t))] [-in θ(t) + in(ω ct + θ(t))] K (8) r(t)inωlt '(t) Q '(t) cx(t) = = [in θ(t) + in(ωct + θ(t))] [co θ(t) - co(ω ct + θ(t))] K (9) - by ureing the ectral comonent centered on ω c with L filter, the outut ignal are: if(t) = ('(t) co θ (t) + Q'(t) in θ(t)) '(t) for θ(t) 0; ft L > f ( +α ) (0) cf(t) = ( '(t) in θ(t) Q'(t) co θ(t)) Q'(t) for θ(t) 0; ft L > f ( +α ) () - the QM demodulation deliver the filtered modulating ignal '(t) and Q'(t) affected by the channel erturbation and ditortion. - then, uing the recovered ymbol cloc, the '(t) and Q'(t) ignal are robed to extract the modulating level of the -th ymbol eriod and Q, which are affected by the channel. - thee ignal are inerted in the deciion bloc which deliver the two etimate * and Q* of the tranmitted level, ee (); note that * and Q* belong to the modulating alhabet. - the decided level * and Q* hould be the coordinate of the contellation-vector which i the cloet to the received vector (and hence it the mot robable); therefore the deciion bloc would comute the uclidean ditance between the received vector and the contellation vector and tore the coordinate of the vector laced at the minimum d from the received vector. - the decided level are then ued by a decoder to extract the correonding multibit; the decoder erform the invere oeration of the three one erformed by the tranmitter encoder (,, 5): the bit decoding or demaing, invere to (6), which deliever the etimated c c 0 dibit; thi oeration can be imlemeted by table reading; the differential decoding, invere of (), deliver the etimated b b 0 dibit in binary-natural code: (c c 0 - c - c 0 - ) mod 4 = b b 0 ; () the natural-ray converion, which finally deliver the data decided dibit a a 0, (invere of ). - the bloc diagram of a QM-demodulator for the 4-SK ignal, which emloy L filter, i hown in figure 5. he ignal emloyed by the carrier recovery and ymbol-cloc recovery circuit would be decribed later in thi chater; the cheme doe not include the ouut arallel-erie converter, which i controlled by the bit-cloc. 4
r(t) BF i x (t) coω L t inω L t c x (t) LF i f (t) Recovery + Synchro Localarrier LF c f (t) RR-R RR-R (t) Recovery + Synchro Symbol & Bit loc Q (t) robing f f robing Q S * Q* M 0 if. b b 0 a a 0 Figure 5. Bloc cheme of the QM-demodulator for the 4-SK ignal; variant with L filter - if the encoding decoding are S-baed imlemented the - converion i not required and the demaing i erformed in tabular manner. - the emloyment of the L filter inert a grou-delay ditortion which can affect ignificantly the erformance, by inducing S. - SK-QM demodulator with Hilbert tranform i to be dicued in the V th year at the ata ranmiion coure Recovery and ynchronization of the local carrier - to recover and ynchronize the local carrier, intead of the claical LL circuit which determine the hae-error by comaring an external hae-reference ignal to the locally generated ignal, the QM receiver determine directly the hae error uing: e (t) = Q* +Q' * = (' +Q' ) inθ(t); (3) - the relation (3) can be derived uing (, ); and Q rereent the value of (t), Q'(t), the ignal at the demodulator outut, at t =, i.e. in the amling moment; *, Q* denote the decided level during the -th ymbol eriod. Since thi method emloy the decided level, it i called eciion irected arrier Recovery R. more elaborated reentation i made in [Bota ch.9]. - the bloc diagram of the receiver which emloy thi method i hown in figure 6; it how that the LL cloe acro the demodulator and the hard-deciion bloc. S ym bol-cloc recovery& ynchro t t i b l hae o m e L F V f r(t) Q. dem od c o ( ω L t i n ( ω l L Q R B Q S * Q * a a 0 S d e V L F hae om. carrier R ecovery & S ynchro Figure 6. Bloc diagram of the QM receiver for the SK ignal - for mall value of the hae-hift, the error-voltage e (t) may be conidered directly roortional to the magnitude of the hae-hift. For greater value of the hae-hift, but till Θ(t) [-π/, π/], the error-voltage i no longer directly roortional to the hae-hift, due to the variation law of the ine function, but the ign of the error-voltage till follow the ign of the hae-hift. - therefore, an analogue LL with a roortional hae control would inert error, becaue for great hae-hift, the error-voltage i no longer directly roortional to the hae-hift. - a digital LL with a contant hae-te, ee the bit-cloc ynchronization in the BB code lecture note, hould be emloyed intead. t i controlled only by the ign of the error-voltage, which i the ame a the one of the hae-hift for Θ(t) [-π/, π/] - if the value of θ(t) (-π, -π/) or (+π/, +π), then the LL circuit will change the hae of the local carrier o that the error voltage would be minimized; thi would lead to the occurrence of a contant hae-hift of -/+ π. 5
- ince ome LL circuit inert a π/ hae-hift between the local and the received carrier (the reference one), a hae hift of π/, 0, may occur, which can not removed by the carrier recovery circuit. hi hae-hift i alo called π/ uncertainty - becaue thi uncertainty i contant during a tranmiion, it i removed by the differential encoding-decoding emloyed to generate the differential hae-modulation. Summarizing, the ynchronization of the local carrier i accomlihed in two te: - the extraction of the hae-reference ignal, recovery, which i accomlihed in figure 6 by the error-voltage circuit and by the LF; - actual ynchronization of the local carrier (actually two quadrature carrier), erformed by the in figure 6. hi oeration may be erformed either by the digital LL, ee BB code, if a contant hae-te (in term of the ign of the error-voltage) i deired, or by an analog V, if the hae of the local carrier() i to be changed roortionally to the error-voltage. - though it exhibit good erformance, thi carrier recovery method aume an almot erfect ynchronization of the local ymbol-cloc, emloyed to robe the correctly the * and Q * level. - if the local ymbol-cloc i not correctly recovered (and ynchronized), the * and Q * level may tae wrong value, thi lead to a wrong error-voltage obtained by (3), which lead to an incorrect demodulation (wrong carrier ynchronization), ee (0,) for θ(t) 0, the ymbol-cloc recovery i affected; o the receiver might enter into viciou circle.- - therefore, thi carrier recovery method may be emloyed only if the ymbol-cloc i recovered by a method whoe erformance do not deend of the quality of the carrier recovery. Recovery and ynchronization of the local ymbol-cloc - one of the bet ymbol-cloc recovery method i the energetic method to be dicued in the V th year at the coure. enerating other SK contellation with the QM technique ontellation and B - ince thee contellation involve hae-hift of ΔΦ n =0º or 80º and, reectively, ΔΦ n = 90º or ΔΦ n = 70º, which define the vector of the two contellation, the QM- exreion of the - SK ignal are: SK- (t) = ± n=- n=- co( ω t) u (t - n); SK-B (t) = ± in( ω t) u (t - n); ontellation Bit Q - the value of the modulating level and Q of the and B are reented in table. 0 + 0 able. Value of and Q for contellation and B - 0 - the modulation-demodulation and their bloc diagram remain B 0 0 + the ame a the one decribed above for QSK, excet for the differential recoding-decoding which are erformed a modulo B 0 - oeration on one bit. - the differential recoding-decoding enure the cancellation of only the 80º incertitude inerted by the carrier recovery circuit. he 90º rotation can alo be comenated, but the method would not be decribed here. ontellation B4 - QM generation of B4 require a modulo-8 differential recoding-decoding on 3 bit. - the b b 0 dibit obtained after the ray-natural converion i tranformed in the c c c 0 tribit: (4) c = b ; c = b o ; c 0 = ; (5) - etting the bit c 0 = i equivalent to the 45º rotation imoed by thi contellation. - the c c c 0 tribit i emloyed to elect the modulating level and Q a hown in table 3 ibit 00 0 0 + - - + Q + + - - able 3. Value of the and Q level for contellation B4 - the ret of the oeration required by the QMmodulation-demodulation of B4 are imilar to QSK. - note that after the demaing differential decoding only the two mot ignificant bit are emloyed in the final roceing. 6
ontellation 8 - the QM modulation-demodulation of the 8 are imlemented imilarly a the one of 4, with the following difference: the value of the and Q level, in term of the data tribit c c c 0, are the one of table 4 the differential recoding-decoding hould be c c c 0 000 00 00 0 00 0 0 made modulo-8 on the three bit. + + - 0 - - + 0 Q + 0 + + - 0 - - able 4. Value of the and Q level for contellation 8. Sectral ditribution of the QSK ignal - the ectral ditribution of the QSK ignal, and of all other SK contellation generated by QM, deend of the ymbol frequency and i exreed by (6) for the non-filtered modulating level: π(f f ) in f V Sn (f ) = ; (6) f π(f f ) Hz f - the ectrum, aroximately rereented in figure 7, exhibit a central lobe, maximum value S M0, around the carrier frequency ( = 0) with a bandwidth equaling f, and ide lobe with maxima S M occurring at the f M frequencie, given by (7). S/S Mo [db] on-filtered RR(α) - filtered -3 S M -8 S M f -5f / f - f f -3f / f - f f f + f f +3f / f +f f +5f / f f - f (+α)/ f -f (+α)/ Figure 7 ower ectral ditribution of the QSK ignal he amlitude of the ide lobe decreae rather lowly with the increae of their index. f m = f c +/- f 0; f M = f c +/- (f + f /) 0 ; S M0 = /f ; S M = S M0 4/[(+)π] ; (7) - to match the channel bandwidth, the QSK ignal i filtered with an RR(α) characteritic, which attenuate the ide lobe; the bandwidth of the filtered ignal i given by (3). Figure 8 below how the ectra of meaured unfiltered and filtered QSK ignal, for f =0.33, α= 0.5. 7