F.F.R F.F.R. c f (t)

Transcription

1 r(t) BF i(t) c(t) Quadrature mplitude Modulation - QM - beide it employment a uch to tranmit two independent data flow, the QM i ued a a method of modulation-demodulation of other amplitude, frequency or phae modulation. Quadrature mplitude Modulation QM - the expreion of a QM modulated ignal i: MQ (t)=d (t)coω c t-d (t)inω c t; () where d (T),d (T) are two independent data equence of ymbol-period T, which are amplitudemodulated on two orthogonal carrier that have the ame frequency. - the bloc diagram of the QM modulator i preented in figure. i x (t) i f (t) LF co l t arrier recovery circuit in l t LF c x (t) c f (t) F.F.R Symb-cloc recovery circuit F.F.R - the modulating data ignal may tae value (bit/ymb) or more value (multibit/ymb). The mapping multibitmodulating level i not figured in figure. - the d (T) and d (T) modulating ignal hould be filtered with a RR low-pa characteritic, to enure (after the completion RR-filtering in the receiver) S = and better performance in the preence of noie, according to: ; [, ( )]; E ( ) = R ( ) = ( ) ( ) (- ) () co( - ); [ ( ), ( )]; 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 haping 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 page - of the LM lecture, but it i adapted to the dicreet nature of the modulating ignal -to decribe the adapted demodulation principle we aume that the relation between the phae of the tranmitter and local receiver carrier ignal, with pulation ω c and ω L repectively, i: ω L t=ω c t+δωt+θ =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 chapter. Figure d (t) d (t) Bloc diagram of the QM receiver d (T) d *(T) robing eciion robing eciion d (T) d * (T) - the equation that decribe the demodulation on the two branche i(t), the in-phae branch, and c(t), the quadrature one are (5) and (6). i x (t) and c x (t) denote the ignal at the output of the multiplier and i F (t) and c F (t), denote the ignal at the output of the L-filter of the two branche. - The QM demodulation i imilar to eparate product-coherent demodulation of two SB-S ignal. - d (T) d (T) F.F.E. F.F.E. Gen. urt. d (t) d (t) coω ct inω ct Figure. Bloc diagram of the QM modulator S - MQ + - Σ

2 r(t)colt d(t) d (t) ix(t) = = [co (t) + co(pt + (t))] [-in (t) + in( pt + (t))] K (5) r(t)inlt d(t) d (t) cx(t) = = [in (t) + in(pt + (t))] [co (t) co( pt + (t))] K (6) - the L-filter uppre the pectral component centered around ω c and the ignal at their output are: if(t) = (d(t) co (t) d(t) in (t)) d(t) pt. (t) (7) cf(t) = (d(t)in (t) d(t)co(t)) d(t) pt. (t) (8) - if the carrier recovery circuit enure a phae-hift θ(t), then the ignal at the output of the haping filter would tae, in the probing intant, value proportional 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 paraitic M and the addition of the modulating ignal of the quadrature branch, which alo ha a paraitic M. Therefore, the condition θ(t) hould be impoed. - the ymbol-cloc recovery circuit i intended to extract the phae-reference and ynchronize the local ymbol-cloc, f i and f c, with the demodulated ignal o that they are probed at the right time-intant. - uing the ynchronized ymbol-cloc, the demodulated ignal d (t) and d (t) are read in the probing moment, when they are not affected by S, generating the d and d level, that have contant value during the -th ymbol period. - the probed level are then delivered to the deciion circuit, which decide which of the permitted level i cloer to the received, thu delivering the etimated (decided) level d * and d *. f the tranmiion employ more bit/ymbol, then from the decided level, the correponding multibit are demapped (decoded). SK modulation-demodulation employing the QM modulation (technique) - the expreion of the SK ignal over one (-th) ymbol-period i given by, ee the SK lecture: SK = co( t + )u (t - T); (9) p - by expanding (9) we get (), which reprent a QM ignal in which the two modulating ignal are no longer independent ignal; they fulfill condition (). T SK = co ut(t-t)copt-in ut(t-t)inpt= (T)co pt - Q(T)in pt; (T) = = co ut(t - T) ; Q(T) = Q in ut(t - T); + Q = u T (t-t); () SK modulation generated by the QM technique - a an example we preent the generation of the 4 contellation, figure 3. Table how the phae-hift ΔΦ, the value of the modulating level (, Q ), the input dibit-data a a and of the dibit after the Graynatural converion (G), b b, which i performed according to: 4-SK ontelatia 4 9 b a a; b a; () Figure 3. The 4 ignal contellation Q a a a a b b Q ΔΦ + º + 9º - 8º - - 7º () Table. Signal value in the main point of the SK-4 encoder for c - c - = - thi method generate an abolute-phae modulation, ince the phae-hift of the modulated carrier are referred to the phae of the 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). - to tranform thi modulation into a SK one, the abolute phae of the modulated carrier hould be modified according to (3), which i hard to implement.

3 Φ = (Φ - +ΔΦ ) mod 36 º (3) - becaue all ΔΦ are multiple of 9 º, the abolute phae will be a multiple of 9 ºand (3) may be written a: 9 ( 9 9 ) ( ) (4) mod36 mod4 - but the number and Δ are binary repreented by the dibit c c and b b, (4) my be written a: (b b + c - c - ) mod 4 = c c ; (5) - (4) and (5) how that to obtain a SK, the dibit that i delivered to the circuit that compute the and Q level i obtained by differentially precoding the modulating data-dibit, after the G converion. - the bloc diagram of the SK modulator implemented by the QM technique, i hown in fig. 4. a a. G.. b b - c Σ M odulo 4 c c M G Q F. F. E. F. F. E. coω c t inω c t Σ S S K c - f - 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 previou encoded dibit, when the G converion and the differential encoding are included; by uing a / converter and a circuit that compute the bit which control the / converter - on a S implementation, the G and differential precoding are performed off-line; the and Q level are read from a table, in term of current data and previou encoded-data dibit; thi bloc i called encoder or mapper. - to limit the bandwidth of the modulating ignal and enure S= in the probing moment, the and Q ignal would be L filtered (FFE 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 expreion of the tranmitted modulated ignal i: (t) = co Figure 4. Bloc diagram of the SK modulator implemented by uing the QM technique u T SK (t) = (t)co t - Q(t)in (t - T) after filter; Q(t) = in c c t = u T (t - T) after filter; - the L RR filtering i implemented uing a FR tructure, in which only one ample equaling hould be inerted in every ymbol-period; the ret of the ample of that ymbol period would equal zero, ee S lecture and ata Tranmiion lecture - when implemented on a ignal proceor, the ymbol period i dvided into ampling period. The encoding, multiplication and addition operation are executed for each ample. L flter hould be added at the modulator output, to uppre cuantization noie. - the ample of the carrier ignal would be tored in a table, value per ymbol period; the digital generation of the carrier ignal hould be carefully conidered to decreae ignificantly the TH factor. - thi method can be applied if the frequency of the carrier allow it implementation on a proceor; - for carrier ignal with greater frequencie, the digtally filtered ignal ((t) and Q(t) are multiplied to the carrier ignal by analogue multiplier and the umation i performed by an analogue adder. Filtering the QM-modulated ignal - becaue QM i a LM ignal modulated with rectangular modulating ignal, the band-limitation filtering hould be performed with a RR characteritic. Baically it may be implemented in two variant: a band-pa filtering placed on both branche after the multiplication with the carrier ignal, or placed at (6) 3

4 the output of the final adder; a low-pa filtering placed on both branche applied to the rectangular, Q ignal before the multiplication with the carrier ignal. Thi method i preferred in mot application. emodulation of SK ignal uing the QM technique - the demodulation of the SK ignal by uing the QM technique may be accomplihed in two variant: a variant that employ L filter to uppre the high-frequency pectral component; a variant which employ the Hilbert tranform of the received ignal to uppre the high-frequency pectral component. SK-QM demodulator with L filter - Uing (, ), the expreion of the received SK ignal become (7), where (t) and Q (t) denote the filtered modulating ignal affected by the channel perturbation and ditortion: = '(t) co t - Q'(t) in t; (7) rsk p - the demodulation of the SK ignal may be accomplihed 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)colt '(t) Q'(t) ix(t) = = [co (t) + co(ct + (t))] [-in (t) + in( ct + (t))] K (8) r(t)inlt '(t) Q'(t) cx(t) = = [in (t) + in(ct + (t))] [co (t) - co( ct + (t))] K (9) - by uppreing the pectral component centered on ω c with L filter, the output ignal are: if(t) = ('(t) co (t) Q '(t) in (t)) '(t) for (t) ; ftl f ( ) () cf(t) = ('(t)in (t) Q'(t) co (t)) Q'(t) for (t) ; ftl f ( ) () - the QM demodulation deliver the filtered modulating ignal '(t) and Q'(t) affected by the channel perturbation and ditortion. - then, uing the recovered ymbol cloc, the '(t) and Q'(t) ignal are probed to extract the modulating level of the -th ymbol period 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 alphabet. - 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 probable); therefore the deciion bloc would compute the Euclidean ditance between the received vector and the contellation vector and tore the coordinate of the vector placed at the minimum d E from the received vector. - the decided level are then ued by a decoder to extract the correponding multibit; the decoder perform the invere operation of the three one performed by the tranmitter encoder (,, 5): the bit decoding or demapping, invere to (6), which deliever the etimated c c dibit; thi operation can be implemeted by table reading; the differential decoding, invere of (), deliver the etimated b b dibit in binary-natural code: (c c - c - c - ) mod 4 = b b ; () the natural-gray converion, which finally deliver the data decided dibit a a, (invere of ). - the bloc diagram of a QM-demodulator for the 4-SK ignal, which employ L filter, i hown in figure 5. The ignal employed by the carrier recovery and ymbol-cloc recovery circuit would be decribed later in thi chapter; the cheme doe not include the ouput parallel-erie converter, which i controlled by the bit-cloc. - if the encoding decoding are S-baed implemented the -G converion i not required and the demapping i performed in tabular manner. - the employment of the L filter inert a group-delay ditortion which can affect ignificantly the performance, by inducing S. - SK-QM demodulator with Hilbert tranform will be dicued in the V th year at the ata Tranmiion coure p 4

5 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 E S O * Q* E M G if. E O G b b G a a Figure 5. Bloc cheme of the QM-demodulator for the 4-SK ignal; variant with L filter Recovery and ynchronization of the local carrier - to recover and ynchronize the local carrier, intead of the claical LL circuit which determine the phae-error by comparing an external phae-reference ignal to the locally generated ignal, the QM receiver determine directly the phae error uing: e (t) = Q* +Q' * = (' +Q' ) inθ(t); (3) - the relation (3) can be derived uing (, ); and Q repreent the value of (t), Q'(t), the ignal at the demodulator output, at t = T, i.e. in the ampling moment; *, Q* denote the decided level during the -th ymbol period. Since thi method employ the decided level, it i called eciion irected arrier Recovery R. more elaborated preentation i made in [Bota ch.9]. - the bloc diagram of the receiver which employ thi method i hown in figure 6; it how that the LL cloe acro the demodulator and the hard-deciion bloc. - for mall value of the phae-hift, the error-voltage e (t) may be conidered directly proportional to the magnitude of the phae-hift. For greater value of the phae-hift, but till Θ(t) [-π/, π/], the errorvoltage i no longer directly proportional to the phae-hift, due to the variation law of the ine function, but the ign of the error-voltage till follow the ign of the phae-hift. - therefore, an analogue LL with a proportional phae control would inert error, becaue for great phaehift, the error-voltage i no longer directly proportional to the phae-hift. S ym bol-cloc recovery& ynchro i b l hae o m p e L F V O f r(t) E Q. d e m o d c o ( ω L t i n ( ω l L Q R O B G Q E S O Q * * a a S d e V O L F hae om p. carrier R ecovery & S ynchro Figure 6. Bloc diagram of the QM receiver for the SK ignal - a digital LL with a contant phae-tep, ee the bit-cloc ynchronization in the BB code lecture note, hould be employed intead. t i controlled only by the ign of the error-voltage, which i the ame a the one of the phae-hift for Θ(t) [-π/, π/] - if the value of (t) (-π, -π/) or (+π/, +π), the LL circuit will change the phae of the local carrier o that the error voltage would be minimized; thi lead to the occurrence of a contant phae-hift of -/+ π. - ince ome LL circuit inert a π/ phae-hift between the local and the received carrier (the reference one), a phae hift of π/,, may occur, which can not removed by the carrier recovery circuit. Thi phae-hift i alo called π/ uncertainty - becaue thi uncertainty i contant during a tranmiion, it i removed by the differential encodingdecoding employed to generate the differential phae-modulation. Summarizing, the ynchronization of the local carrier i accomplihed in two tep: - the extraction of the phae-reference ignal, recovery, which i accomplihed in figure 6 by the errorvoltage circuit and by the LF; 5

6 - actual ynchronization of the local carrier (actually two quadrature carrier), performed by the OT in figure 6. Thi operation may be performed either by the digital LL, ee BB code, if a contant phae-tep (in term of the ign of the error-voltage) i deired, or by an analog VO, if the phae of the local carrier() i to be changed proportionally to the error-voltage. - though it exhibit good performance, thi carrier recovery method aume an almot perfect ynchronization of the local ymbol-cloc, employed to probe 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 (,) for (t), the ymbol-cloc recovery i affected; o the receiver might enter into viciou circle.- - therefore, thi carrier recovery method may be employed only if the ymbol-cloc i recovered by a method whoe performance do not depend 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 T coure. Generating other SK contellation with the QM technique ontellation and B - ince thee contellation involve phae-hift of ΔΦ n =º or 8º and, repectively, ΔΦ n =9º or ΔΦ n =7º, which define the vector of the two contellation, the QM- expreion of the - SK ignal are: SK- (t) = co( t) u n=- n=- p T (t - nt); SK-B (t) = in( t) u p T (t - nt); ontellation Bit Q - the value of the modulating level and Q of the and B are preented in table. + Table. Value of and Q for contellation and B B B the modulation-demodulation and their bloc diagram remain the ame a the one decribed above for QSK, except for the differential precoding-decoding that are performed a mod. operation on one bit. - the differential precoding-decoding enure the cancellation of only the 8º incertitude inerted by the carrier recovery circuit. The 9º rotation can alo be compenated, but the method would not be decribed here. ontellation B4 - QM generation of B4 require a modulo-8 differential precoding-decoding on 3 bit. - the b b dibit obtained after the Gray-natural converion i tranformed in the c c c tribit: c = b ; c = b o ; c = ; (5) - etting the bit c = i equivalent to the 45º rotation impoed by thi contellation. - the c c c tribit i employed to elect the modulating level and Q a hown in table 3 ibit Q Table 3. Value of the and Q level for contellation B4 (4) - the ret of the operation required by the QM-modulationdemodulation of B4 are imilar to QSK. - note that after the demapping and differential decoding only the two mot ignificant bit are employed in the final proceing. ontellation 8 c c c Q the QM modulation-demodulation of the 8 are implemented imilarly a the one of 4, with the following difference: Table 4. Value of the and Q level for contellation 8. the value of the and Q level, in term of the data tribit c c c, are the one of table 4 the differential precoding-decoding hould be made modulo-8 on the three bit. 6

7 Spectral ditribution of the QSK ignal - the pectral ditribution of the QSK ignal, and of all other SK contellation generated by QM, depend of the ymbol frequency and i expreed by (6) for the non-filtered modulating level: (f fp ) in f V Sn (f ) ; (6) f (f fp ) Hz f - the pectrum, approximately repreented in figure 7, exhibit a central lobe, maximum value S M, around the carrier frequency ( = ) 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 p -5f / f p - f f p -3f / f p - f f p f p+ f f p +3f / f p +f f p +5f / f f p - f (+α)/ f p -f (+α)/ Figure 7 ower pectral ditribution of the QSK ignal - The amplitude of the ide lobe decreae rather lowly with the increae of their index. f m = f c +/- f ; f M = f c +/- (f + f /) ; S M = /f ; S M = S M 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 pectra of meaured unfiltered and filtered QSK ignal, for f =.33, α=.5. Variant of the QSK modulation - a previouly hown in the SK lecture and above how that the abolute phae of the ()QSK ignal modulated with type- contellation exhibit phae-hift of 8º during a ymbol-period. - The filtering of thee ignal with an (R)R characteritic inert an amplitude modulation, ee SK lecture note. The amplitude of the filtered-modulated ignal (t) ha minimum value, occurring at t = T+T/, that depend of the phae-hift ΔΦ a hown in (8). min (t) co ; Q ; (8) 8

8 - for ΔΦ = 8º the amplitude would tae value around zero and it trip would be the maximum poible, from to during half a ymbol-period. - but in radio tranmiion employing non-linear RF amplifier, the characteritic M-M and M-M of thee amplifier, generate the preading of the frequency pectrum of the modulated ignal. - thi lead to the pectral regrowth, i.e. the remaing of the ide lobe, outide the permitted band, which have been filtered before the final amplification; it alo ditort the pectral component within the ueful bandwidth. - therefore the SK or QSK ignal uing type contellation are recommended to be ued only on channel that employ linear amplifier. - for channel employing non-linear amplifier, radio lin, mobile lin, variant of QSK were developed which hould have ide lobe a mall a poible and exhibit mall amplitude variation of the filtered ignal. - thee requirement can be fulfilled if the abolute phae of the modulated ignal would not exhibit hift of 8º, and o the amplitude of the modulated ignal would not vary between and. - another important parameter of the modulated ignal affecting it behaviour when paed through a nonlinear RF amplifier, i the R (ea-to-verage ower Ratio during a ymbol period). - to decreae the pectral regrowth, R hould be a cloe to ( db) a poible. - the mot employed variant of QSK that filfill the requirement above are Offet QSK (OQSK) and π/4-qsk. Offet QSK OQSK - to avoid the 8º phae hift from one ymbol to another, which occur in the QSK, the OQSK doe not change the value of the modulating level and Q in the ame time intant, at the beginning of the ymbol period. - intead, the modulating moment of the and Q level are delayed to one another with half a ymbolperiod, a hown in figure Q T / T QSK and OQSK 3T / T OQSK 5T / 3T 4T QSK 7T / Fig. 9. Relative poitioning of the and Q modulating level, to generate the QSK and OQSK modulation OQSK - ΔΦ =9º ΔΦ =9º Q ΔΦ=8º QSK t t - thi way the 8º hift of the abolute phae are achieved in two tep, ee fig. for the abolute phae hift from 45º to 5º: in the firt half of the ymbol period the vector (- ;+ ) i tranmitted, generating a 9º phae-hift; during the econd half of the ymbol period the (- ;- ) vector i tranmitted, generating a econd 9º phae-hift. - by avoiding the 8º phae-hift, the amplitude of the modulated ignal would not reach zero and the amplitude variation during a ymbol period decreae ignificantly. The R value alo decreae ignificantly, from the one of the QSK. Fig. Generation of the 8º phae-hift by the QSK and - the bloc cheme of the OQSK tranmitter i almot identical to the one of the QSK tranmitter, figure 4; the only difference i that the and Q level are delivered to multiplier at time intant eparated by half a ymbol-period, i.e. there i one ymbol-cloc for each branch. The Q ymbol-cloc i inverted referred to the one. 9

9 - the demodulation i imilar to the QSK one, ee figure 5, but two ymbol-cloc are employed for probing the demodulated ignla on the two branche; the two cloc are phae-hifted with 8º by inverting the cloc on the quadrature branch. - note that probing in two different moment complicate the ymbol-cloc recovery and increae the ymbol-error probability of the OQSK, compared to the one of QSK. - the frequency band of the OQSK ignal i the ame a the one of the QSK ignal and it pectrum loo imilar, ee figure 7, but the ide lobe have maller level. Thi fact mae the pectral regrowth performed by the non-linear RF amplifier generate maller ide lobe; due to thi feature the OQSK, i employed in radio tranmiion that include non-linear RF amplifier. π/4-qsk modulation Q 4 B4 4 to B4 B4 to 4 Figure.The poible trajectorie of the abolute-phae of the π/4-qsk tarting from a vector of 4 and B4 - the 8º phae-hift of the abolute phae during one ymbol period can alo be avoided by uing the four poible phae-hift of (p + ) 45º, imilarly to the B4 contellation, ee SK lecture. - the abolute phae would pa from contellation 4 to B4 or vice vera in every ymbol period, a hown in figure. - the amplitude of the current vector would not reach the null value at any time intant. - figure only how the poible trajectorie tarting from one vector (º) of contellation 4 and from one vector (35º) of contellation B4. - the phae-hift of thi ind can be implemented in a QM abolute-phae modulator by uing alternatively the mapping table of the 4 and B4 contellation, a hown by the complete diagram of the phae-hift performed by π/4-qsk, which i preented in figure ; Figure. oible phae hift produced by the π/4-qsk - thi can be accomplihed by a 3-bit addreing, where the LSB i alternatively or. So, the differential precoding i no longer required to generate the SK. - the frequency pectrum of π/4-qsk i imilar to the one of QSK and OQSK and the amplitude of the ide lobe are maller than the one of QSK, but greater than the one of OQSK. The bandwidth of the main pectral lobe i equal to the one of QSK, ee (3). - - the value of R for π/4-qsk, decreae to 3. db - compared to the 4 db of QSK. - the amplitude variation of the QSK, OQSK and π/4-qsk are preented in figure 3 below. 3pi/ pi pi/ QSK OQSK /4-Q SK (t) the figure how that, a expected, the QSK modulated ignal ha the greatet amplitude variation, the OQSK ha maller amplitude variation, while π/4-qsk ha the mallet amplitude variation. - recall that a the amplitude variation of the modulated ignal get maller, the non-linear RF amplifier ditort le the amplified modulated ignal. - but the better reilience of OQSK and π/4-qsk at the effect of the RF amplifier are accompanied by increaed BER value a hown below.

10 onideration regarding the error-performance of the QSK modulation - the ymbol-error probability of the QSK ha the ame value a the one the 4-SK. - the ymbol-error probabilitie of -SK and 8-SK generated by QM are imilar to the one of thee modulation generated directly. - the conideration regarding the bit-error probabilitie of different SK contellation preented in the SK lecture till hold for the ame contellation generated by QM. - the ymbol-error and bit-error probabilitie of QSK are the ame a the one of the SK modulation. - the OQSK modulation exhibit a greater ymbol-error probability than the one of QSK. The increae i greater for maller roll-off factor of the R filtering, e.g. for α =.6, the SR required by OQSK i greater with about db than the one of QSK to enure the ame ymbol-error probability. Thi i the price paid for the better pectral compoition of the OQSK. - the π/4-qsk ha about the ame error-performance a the QSK. pplication of the QSK modulation - QSK i one of the mot employed modulation in digital tranmiion, due to it reilience to noie and ditortion. - in modem deigned for atellite and radio channel including non-linear RF amplifier, the QSK i quite often replaced by it variant OQSK or π/4-qsk. onideration regarding the definition of the ignal to noie ratio - the error-probabilitie are expreed in term of the ignal to noie ratio. - in thi ratio, the noie power i computed by multiplying the power pectral denity of the noie, e.g. the contant [V /RHz] (or [dbm/rhz] in logarithmic reprezentation) for the Gauian noie, to the bandwidth of the input filter. - onidering the two bandwidth uually employed for the R filtered ignal, (9.a), the power can be expreed by (9.b), alo uing it variance σ. BW = f for R α = - ideal; BW = f (+ α) for R α - real; (9.a) z = σ /R = f /R for R α = ; z = σ /R = f (+α)/r for R α ; (9.b) - but the noie power and of the SR are depending of the bandwidth of the modulated ignal. - to avoid thi dependency, the SR may be expreed by the ratio of the energy/bit and the noie power pectral denity, denoted by E b /. The relation between the two ratio i derived in (3.a) for α = and in (3.b) for α. n both relation E and E b denote the average energy per ymbol and per bit, of the modulated ignal; denote the carrier amplitude and n the number of bit/ymbol, while ρ denote the ignal/noie ratio in linear value and SR, the ignal/noie ratio in logarithmic value, i.e. in db. R T E n E b E b ; SR[dB] [db] lg n[db]; for (3.a) R T z z E SR[dB] b R T ( ) E n E b ; R T ( ) ( ) ( ) n [db] lg [db]; ( ) for (3.b)