= a. ; ω N = = b. ; h = = c. ;

Transcription

1 Frequency Shif Keying FSK - he FSK coni of frequency modulaing a coine carrier in erm of he logical value of he bi o be modulaed. - A coine ignal of frequency f i ranmied for bi "" and a frequency f 2 for bi "". - The frequency of he carrier i kep conan for he enire ymbol period. - Since a ymbol carrie bi, he bi-period equal he ymbol period - Thi modulaion i he binary verion of he M-FSK modulaion, where he momenary frequency ha M = 2 n value, correponding o he M combinaion of n bi. Characeriic of FSK Dae CPFSK FSK Modulaing ignal S FSK wih coninuou phae f f f f 2 S f 2 FSK wih phae hif /Tbi necearily be an ineger muliple of an elemenary clock period. - FSK i a frequency modulaion wih a coninuou phae (CPFSK), i.e. he wiching from one frequency o he oher i performed wih keeping he coninuiy of he carrier ignal phae ee figure. Figure B-FSK modulaion rule - i i a non-coheren modulaion, ince i i no performed in he rhyhm of a ymbol-clock ignal, bu he emporary frequency i conrolled by he level of he modulaing ignal. The ymbol duraion hould no Expreion of he CP-FSK ignal - he FSK ignal i expreed in erm of he frequencie f k aociaed o he M=2 n ymbol value by: () = Aco( 2π f +ϕ ) u ( k T ) () FSK k k,k T k= - he momenary frequency of he M-FSK ignal may ake M diinc value (M=2 for B-FSK, fig. ), while, for CPFSK, he ϕkk, hould ake value which would enure he phae coninuiy a he end of he ymbol period (when he modulaing mulibi are changed). T denoe he ymbol period. - for he phae-hif FSK, ϕkk, i conan, depending on he difference of he iniial phae of he wo f 2 and f ocillaor, for each ymbol period, ee (9) below. - aking ino accoun ha he FSK i a frequency modulaed ignal wih a recangular modulaing ignal, i Fourier-erie decompoiion i expreed by (2), where he parameer are defined in (3): () = A C co[( ω + n ω ) + α]; (2) n= n c N ω + ω π ω ω -ω 2 2 M ωc = a. ; ω N = = b. ; h = = c. ; (3) 2 T 2 2 ωn ω /2 - for he "..." daa equence (alo named he : equence), which i he modulaing ignal ha ha he greae fundamenal frequency, he C n coefficien have he value (4): 2h π Cn = in (h + n) ; (4) π( 2-2 h n ) 2 - he ampliude of he pecral componen decreae wih he quared value of heir index; herefore he energy of he modulaed ignal i concenraed cloe o he frequencie f and f 2 ; - he FSK ignal may be conidered a an FM ignal. The f c frequency would correpond o he FM carrier frequency, for U mod = V. - bu, in FSK he modulaing ignal ake only bipolar value, +/-A (ee (5)), o he modulaed ignal would have he momenary frequency equaling f c only during he frequency hif generaed by he change of he bi logical value. - he FSK ignal momenary frequency may be expreed by (5), where S m max = +/-A and S m = (+/-A). Δω

2 Smodulaor + bk = fin = fc + Δf ; Smodulaor = a k A; a k = 2 bk = Smodulaor max bk = (5) - due o he naure of he modulaing ignal he f c i no regarded a he carrier frequency, bu he cenral frequency of he pecrum of he modulaed ignal. - for an arbirary daa equence, he C n coefficien have more complex expreion, he pecrum of he modulaed ignal become coninuou, bu i bandwidh i ill infinie. - he power pecral diribuion of he FSK ignal depend eenially of he modulaing index, denoed here by h (6), where f i he ymbol frequency: f 2- f 2ΔfM ΔfM h = = = ; f f f/2 (6) - recall ha for an FM ignal he modulaion index i: β = Δf M /f mod max ; (7) - comparing (6) and (7) we ge ha h i a modulaing index for a modulaion where he maximum modulaing frequency f mm equal f /2, he Nyqui frequency, and he maximum frequency deviaion Δf M equal half of he difference beween he f and f 2 frequencie. - herefore he B-FSK ignal may alo be expreed a an FM ignal by (9); f c i defined in (3).a; a k in (5). ' B FSK() = A co 2π fc + 2π ΔfM aku( τ kt ) dτ (8) k= - he carrier phae variaion during one ymbol period, i.e. for ε [kt ; (k+)t ), i: ' B FSK c M k c k M () = A co(2π f + 2π Δf a d τ ) = A co(2π f + a 2π Δf ') (9) for ' [kt ;(k + ) T ) - he compuaion of he power pecral deniy i complex and i performed in erm of he raed frequency F, defined by (); S F = ( f - f c) / f ; () Δ=,6 Δ=,65 - he PSD v. F curve of he CPFSK,, 8 Δ=,8 Δ= for a peudorandom daa equence, i hown in figure 2, in i low-pa, 6 equivalen., 4 - for a modulaed ignal he pecrum hould be ranlaed on he, 2 F cenral frequency f c. -,5 - -,5,5,5 - a cae of inere i h =,65: Figure 2. Power pecral deniy of he CP-FSK ignal - chemaical repreenaion he power of he modulaed ignal i concenraed in a relaively narrow BW around he f c (). The widh of he frequency band in () depend on he power percenage of he non-filered FSK ignal ha i o be reained afer filering F [-,5 +,5] f [f c -,5f ; f c +,5f ] or F [-,6 +,6] f [f c -,6f ; f c +,6f ] () he componen ouide hi BW have mall ampliude and he um of heir power i negligible; he diribuion of he componen inide hi BW i relaively fla. - for h (,7;) he pecrum exhibi he ame concenraion of he power diribuion, bu he componen on f and f 2 are very large, leading o complicaion in he demodulaion proce. - for h, he pecral diribuion i more fla, leading o he increae of he BW in which he ignal power i concenraed and o a decreaed immuniy o noie. PSD[dB] f f f 2 c f fc f2 f FSK CPFSK Figure 3. PSD of he FSK ignal wih phae-hif and CP FSK wih phae-hif - he FSK wih phae hif i hown in heory o exhibi a wider frequency pecrum ha lead o ignifican ampliude variaion afer he modulaed ignal i filered. Therefore, hi ype of FSK hould no be ued. I power pecral diribuion i repreened wih blue line in fig. 3. 2

3 Compuaion of he frequencie aigned o he wo logical level - Conidering (3).a and (6) we have he equaion yem: f2+ f 2- = f f f c ; = h (2) 2 f - by chooing he f c in he middle of he available FB and auming h =,65, we ge he frequencie f and f 2 a oluion of he yem (2): h f h f f2= fc+ ; f= fc- ; (3) a an example, le compue he f and f 2 aigned in he 2/6 bp modem for he vocal elephone channel acc. o he ITU-T V.23 Recommendaion. - for he D=2 bp, we chooe f c =7 Hz and h=,65. Uing (3) we ge f 2 '= 29 Hz and f '=3 Hz. For a impler implemenaion he wo value are rounded o f 2 (2)=2 Hz and f (2)=3 Hz, repecively. Wih hee value he modulaion index i h=.66, which generae abou he ame pecral diribuion a h=,65. - for D=6 bp, a imilar compuaion lead o f 2 (6)=9 Hz and f (6)=5 Hz, for h=,66. - hi would involve he implemenaion of a conrolled ocillaor ha would wich beween four frequencie, depending on he bi o be modulaed and he bi rae o be ranmied. For a impler implemenaion we impoe f (6)=f (2)=3 Hz and impoing h=,66 and olving he yem (2), we ge f 2 (6)=7 Hz and f c (6) = 5 Hz; o he ocillaor wiche beween hree frequencie - Table preen he frequency value aigned for he wo logical level, for he deired bi rae. Daa (Bi rae) (6) (2) (6) (2) f or f 2 3 Hz 3 Hz 7 Hz 2 Hz - he andard allow for deviaion of +/- Hz from he nominal value. - he frequency BW of he FSK ignal i compued uing () and ha he value: B FSK (2)=[ Hz;23 Hz]; B FSK (6)=[2 Hz;8 Hz] (4) - if he BW of he modulaed ignal i expreed by (), he pecral efficiency, ee (5), of hi modulaion i β w = bi// Hz, indicaing raher inefficien employmen of he frequency bandwidh. D bp β w = [ ] (5) BW Hz Generaion of he FSK ignal - he FSK modulaion can be generaed boh by analog and digial mehod - he analog FSK modulaor are proven o be le accurae and able and herefore hey will no be dicued hereinafer Digial generaion of FSK by recurive compuing of he carrier momenary phae - auming a ampling frequency f e = p f, p ineger, and a dicree ime = nt e, hen he momenary phae of he B-FSK ignal, ee (9), a = (n+)t e could be wrien a: for Δω M = π h f b Φ ((n + )T e) = [ ωc (n + ) Te + a k ΔωM (n + ) T e] mod2π = (6) = [ ω n Te+ a Δω n T + ( ω + a Δω ) T ] = [ Φ(n T ) +ΔΦ (a )] ; c k M e c k M e mod2π e p k mod2π - (5) how ha every ampling period he correponding phae incremen ΔΦ p (a k ) = (ω c +/-a k Δω M ) T e i compued and added o he carrier phae of he S. modulaor previou ampling period. S FSK faza inananee - hen he coine funcion i applied o he reuled pi ample and he reul i muliplied by he ampliude A o provide (8). The coine funcion migh be applied a uch, if he hoing plaform allow i, or -pi migh be compued by uing numerical mehod. A - he variaion v. ime of he momenary phae of a B-FSK ignal i hown in figure 4 Fig. 4. Time variaion of he momenary phae of B-FSK -A oher digial mehod do no require he on-line /Tbi compuaion of he coine funcion, ee noe on blackboard 3

4 Digial FSK modulaor ha employ he carrier generaion wih he Walh funcion - hi varian of digial implemenaion of he FSK modulaion i baed on he generaion of he inuoidal carrier ignal uing he Walh funcion. Then he frequency of hi ignal i conrolled according o he inpu daa and deired bi rae. - a periodical funcion of period T and ampliude A, may be expanded in Walh erie: () = A al (2π/T) + A c cal (2π/T) + w A wal (2π/T); (7) i i i i i= i= - he coefficien i, c i and w are compued uing: i= (/T) ali(2π/t) d(/t) ; ci= (/T) cali(2π/t) d(/t) ; w= (/T) wal(2π/t) d(/t) ; (8) - he fir 6 Walh funcion are repreened in figure 5. Figure 5. The fir 6 Walh funcion wal cal al al cal cal2 al2 al2 cal2 cal3 al3 al3 cal3 cal4 al4 al4 cal4 cal5 al5 al5 cal5 cal6 al6 cal6 al6 al7 cal7 cal7 al7 al8 cal8 al8 - ince he ine funcion i odd and he cal i funcion of period T are even, he coefficien c i, (8), will equal zero. - he w coefficien, which repreen he d.c. componen of he expanded ignal, i alo zero. - he ine funcion can be repreened a a um of al i, funcion, weighed by he i coefficien. - Bu due o he ymmerie exhibied by he ine funcion, he coefficien of he al i funcion of even order would be zero. Therefore only al i funcion of odd order would be employed. Reaining only four Walh funcion, he ine frequency i approximaed by: 2 in π al (2 π /T) + al (2 π /T) + al (2 π /T) + al (2 π /T); (9) T - if he ampliude of he al i funcion i A, and ha he compued value of he fir four coefficien i are hown in (9) we ge he ampliude of he reuling aircae inuoid o be of abou.977a =,636; 3 = -,265; 5 = -,52; 7 = -,28; (2) - he ignal generaed by (9) and (2) -,8,6,4,2, 977 T/4 T/2 3T/4 T T i /2 Figure 6. Sinuoid ynhei wih 4 Walh funcion T/4 T/2 3T/4 T i preened in figure 6. - he runcaion of he infinie um (7) o he finie um (9) lead o he occurrence of harmonic diorion, which are no very high. The addiional pecral componen are placed a muliple of 6f i and are eay o filer. The filering ge impler wih he increae of he number of erm of he runcaed um. - he generaion of he coine funcion i imilar, bu employ he cal i funcion, becaue he coine funcion i even; he cal i funcion are obained by hifing he al i wih T/4 al al 3 al 5 al 7 Sinu inheized w ih W a l h f u n c io n T i 4

5 Generaion of he Sal i funcion - he fir four al i funcion may be generaed noing ha he hore level ha a duraion of T l /6. - conidering a ignal of frequency 8f l and a couner wih hree range ha generae ignal wih frequencie 4f l, 2f l and f l, he Walh funcion have he expreion: al = f l; al3 = 4fl 2fl f 8fl l al5 = 8fl 4fl f l; (2) 4fi al7 = 8fl 2fl f l; 2fl - he ynhei of he al 5 funcion i hown in figure 7. fl - he diagram of he circui ha generae he ine funcion, uing he fir four odd-order al Sal5 Ti funcion i preened in figure 8. - he weighed ummaion of (9) can be Figure 7. Synhei of he al 5 funcion accomplihed in wo way: - if he Walh funcion are generaed a bipolar funcion of ampliude A, hen he addiion i performed by mean of an Op Amp, configured a an invering adder, ee fig. 8. Figure 8. Diagram of he ine generaion uing he ynhei wih Walh funcion The oupu volage of he Op Amp i: 7 = - R r Vo ali ; (22) i= R i - Idenifying he coefficien beween (2) and (22), we may compue he value of he R i reior by uing (23), provided ha he value of he feedback reior R r i choen. = R r R i ; (23) i - noing ha he coefficien i poiive (2), o obain a negaive value for he produc of al, we need he al funcion o be negaive; hi i accomplihed by he inverion of he al funcion (a bipolar funcion). - if he Walh funcion are unipolar funcion, generaed by TTL circui wih level of V and +A V, he generaed inuoid i negaive and hifed on a d.c. componen ha equal -A/2. o obain a inuoid cenered on a null d.c. componen, he addiion of a poiive d.c. volage equaling +A/2 i required. - Thi i accomplihed by adding a volage A by mean of he reior R d = R r /2; he R d reior, repreened in figure 8 i required only if he Walh funcion are unipolar. FSK modulaor uing he Walh funcion - he FSK modulaion i accomplihed by changing he value of he 8f i frequency, beween 8f and 8f 2, by mean of a frequency divider conrolled by he inpu daa ignal. - hi divider would change he diviion facor, by which i divide he frequency of he pilo ocillaor wih f oc = l.c.m. (8f, 8f 2 ), beween n and n 2, in erm of he daa bi. - he reuled inuoidal ignal would have he frequency f or f 2. If he modulaor of a V.23 modem i o be implemened, hen f oc = l.c.m. (8f, 8f 2 (6), 8f 2 (2)), and he dividing facor would be changed according o he bi rae conrol ignal and o he daa bi value. - he change of he frequency diviion facor hould be performed in uch a manner ha i would no affec he phae of he 8f l ignal; hi enure he mainenance of he phae coninuiy of he reuled inuoidal ignal. - he block diagram of he FSK ranmier baed on hi approach i hown in figure 9. - he ranmiion conrol block can enable/diable he modulaor, depending on he RTS ignal ae, by enabling/diabling he conrolled divider. 5

6 P ilo O c. f o c C D o n l e d i v i d e r 8 f l 4 f l 2 f l f l W a l h F u n c. S y n h. B l o c k S a l 7 S a l 5 S a l 3 S a l n W e i g h e d A d d e r T x L P F F ile re d F S K S i g n a l D A T A B I T R A T E R T S C -A h a n n e l L. U. L. A. Figure 9. Block diagram of he FSK ranmier wih Walh ynhei of he carrier ignal - he generaed FSK ignal ha a main pecrum cenered around f c, ee figure 2, and imilar pecra cenered around he 6f c and around i harmonic; hi i becaue he inuoid ha a period equaling T l and i generaed in 6 ep. - he value of he pecral componen placed ouide he deired pecrum are mall, conaining a negligible par of he modulaed ignal power, o heir filering would no induce ignifican harmonic diorion; due o heir poiion a a large diance from he deired pecrum, he filering can be accomplihed wih a imple LP filer. - he elegraphic diorion, ee nex paragraph, of he modulaed ignal generaed by hi mehod, i maller han he one generaed by he analog modulaor, if he ame receiver i employed. - ome oher digial mehod of generaing he FSK ignal are decribed in lieraure. Filering he FSK ignal and i effec upon he modulaed ignal - filering i required o fi he pecrum of he FSK ignal, an infinie one, ino he channel limied BW - he frequency band ha hould be reained ou of he FSK modulaed ignal, wihou affecing ignificanly i qualiy, i given by () - he fulfillmen of () i equivalen o he reaining of he componen wih indexe -,, ou of he modulaed ignal () (for a : modulaing daa paern); - knowing ha C - = -C, (4) and neglecing he conan phae α, he FSK filered ignal ha he expreion: f-fsk() = Ccoωc + C-co( ωc- ωn)t + Cco( ωc+ ωn) = R()co φr(); C R() = + 4 ; () = + arcg( C C in ω φ ω in ω ); (24) N r C - (24) how ha he momenary phae of he filered ignal φ r varie around he linear phae of he cenral frequency, wih he Nyqui frequency ω N /(2π), i.e. half of he fundamenal frequency of he modulaing ignal. - performing he derivaive of he momenary phae of he modulaed ignal, wih repec of ime, we ge he expreion of he momenary pulaion: d ϕr() 2 C C ωn coωn ωin() = = ωc + ; d C +4C in ω N - he fundamenal of he modulaing ignal, he variaion of he momenary frequency of he non-filered and filered B-FSK ignal are preened in figure. f2 c N fi-filra fi fundam enala f i (25) fc f /T Figure Momenary frequencie of B-FSK before and afer filering and he fundamenal of he modulaing ignal - he momenary pulaion ha a coninuou variaion in ime and ha a he middle of he bi period he value ω in = ω c ; i i no longer a dicree variaion a he one of he non-filered modulaed ignal; - he pulaion maximum deviaion around he cenral carrier equal (2C /C ) ω N.46ω bi, value aained a he beginning of each bi period, = k T. - recalling ha hi expreion i obained for he : modulaing daa paern, hen ω N i he fundamenal of he modulaing ignal, regarded a a periodical recangular ignal of period 2T. 6

7 - Concluding, he filered FSK ignal i imilar o a FM ignal modulaed wih he fundamenal of he daa ignal, i.e. he deviaion of he momenary frequency around he cenral frequency of he filered FSK i approximaely proporional o he level of he fundamenal componen of he daa ignal - relaion (24) alo how ha he filered FSK ignal i ampliude-modulaed; - he envelope alo varie in he rhyhm of he fundamenal of he modulaing ignal, a hown by he expreion (26) of he envelope R() of (24), which i repreened in figure : R()= C+4 C in N = C+2 C -2 C co2 N = C+2 C -2 C co ω ω ω ; (26) Figure. Filered B-FSK envelope variaion he po-filering envelope ha i maxima a he middle of he bi-period and minima a he margin of he bi-period. Summarizing, he effec of he FSK ignal filering are: coninuou variaion of he momenary phae of he FSK filered ignal; coninuou variaion of he momenary frequency of he FSK filered ignal; i reache i exreme value a he bi-period margin; he deviaion of momenary frequency, around he cenral frequency, i approximaely proporional o he level of he fundamenal frequency of he modulaing daa ignal; he occurrence of a paraiic ampliude modulaion, proporional o he bi frequency ha ha minima a margin of he bi period and maxima a heir middle; Telegraphic Diorion - anoher conequence of he aenuaion of ome pecral componen of he FSK ignal i he occurrence of he elegraphic diorion; - i coni in he variaion of he duraion of he demodulaed daa bi, around he nominal value T, ee figure 2. Figure 2. Schemaic repreenaion of he elegraphic diorion Te T Te T T - denoing by ΔT he maximum change of demodulaed bi duraion and by BW he bandwidh of he receiver inpu filer, he elegraphic diorion i defined by (27).a, and i approximae value may be compued by (27).b; - conidering he bandwidh given by (), hi diorion i δ %. The meauremen of he elegraphic diorion i performed uing a daa paern. BW 2 ΔT π- 2in T δ [%] = ; a. δ[%] = 2 ; b. (27) T BW BW T - in T T he value of δ given by (27).a i an ideal one; he real value depend boh of he harmonic diorion inered by he modulaor and of he demodulaion mehod employed. - noe ha he elegraphic diorion hould no be confued wih he jier; i i generaed by he ignal proceing in he modulaor and demodulaor, while he jier i generaed by he channel diorion and he clock ynchronizaion yem. 7