Gaussian minimum shift keying systems with additive white Gaussian noise

Transcription

1 Indian Journal of ur & Applid hysics Vol. 46, January 8, pp Gaussian minimum shif kying sysms wih addiiv whi Gaussian nois A K Saraf & M Tiwari Dparmn of hysics and Elcronics, Dr Harisingh Gour Vishwavidyalaya, Sagar Rcivd 9 May 6; rvisd 16 May 7; accpd 8 Spmr 7 In his papr, h asic propris of an I-Q Gaussian minimum shif kying (GMSK) wih addiiv whi Gaussian nois hav n invsigad. Th proailiy of rror for non-fading and fading channls of communicaion has n analyzd and discussd. Th hardwar ralizaion is simpl and sraigh forward lik minimum shif kying (MSK) sysm. Som analyical rsuls hav n achivd and compard wih machin compud GMSK rsuls of ohrs. Th modular circuis ar availal for VLSI dsign. Thus, h sysm dvlopd is suial for VLSI dsign of GMSK. Kywords: Gaussian minimum shif kying, Spcral dnsiy, roailiy of rror, Addiiv whi Gaussian nois IC Cod: H1J1/ 1 Inroducion Muroa and Hirad 1 proposd Gaussian minimum shif kying (GMSK) for moil radio lphon srvics in Such modulaion schm is applical o gloal sysm for moil communicaions (GSM) and DCS 18 du o is andwidh fficincy and consan nvlop. Lar on, GMSK was analyzd and sudid y many rsarchrs 3-17 I-Q GMSK modulaor has n sudid 18 and in h prsn papr, som analyical rsuls for spcral dnsiy of in-phas and quadraurphas componn of GMSK signal corrupd wih nois hav n achivd. Th quaion for proailiy of rror for fading channl conaining random anuaion facors has n drivd. Th hardwar ralizaion of GMSK modulaor has n achivd hrough nw inary daa (+1, -1) achivd from h convnional inary daa (,1). Finally, h hardwar ralizaion of GMSK dcor conaining nois signal has n discussd. rliminaris.1 GMSK Modulaion L h signal s() is corrupd y Gaussian nois n i (). Th signal comind wih nois, s() + n i () gos o h inpu of Gaussian filr. Th oupu of Gaussian filr is givn y [s()+n i ()]*h() whr h() is h Gaussian filr rspons and i is givn y: A π ( )β h () β [ ] wih A is h ampliud and is group dlay (1) β π f( BT) () ln() whr B T is dsign paramr and f is h i frquncy. whr h signal s() is a sram of rcangular pulss and i is givn y: (3) n s() ( n) p nt whr p() 1 for (, T ) ohrwis. and T is h symol inrval. [ ] s() + ni() h() s() h() + ni() h() g () + n() Now g() may xprssd as: k n (4) g T s ( nt) h( nt) (5) i.. g() T[() p() h() + (1) p( T) h( T) + () p( T) h( T) + (3) p( 3T) h( 3T) + ] Th phas angl φ () is givn y: (6)

2 66 INDIAN J URE & AL HYS, VOL 46, JANUARY 8 φ π f m g(τ ) dτ + n (τ ) dτ φ () + φ () s n whr f m is h modulad wav frquncy and ω c is carrir wav angular frquncy: f m 1 f 4T 4 Th dsird modulad signal y() is givn y: y () cos [ω c() + φ s()] + cos [ωc+ φ n()] cos(ω c) [ cos φ s() + cos φ n()] sin(ω c) [sin φ s() + sin φ n()] [ I + n ] cos(ω ) [ Q + n ] sin(ω ) c c s c whr φ s () and ϕ n () ar indpndn. Whr I () R A xp jπ fm gτ dτ nc() R xp jπ fm n τ dτ Q () Im A xp jπ fm g τ dτ ns() Im xp jπ fm n τ dτ Susiuing h valu of g(τ) from Eq.(6) and h(τ) from Eq.(1) and ingraing, w hav: g τ dτ β AT p() rf β + p() rf ( β ) { [ β ]} + p T rf T 1 + { p( T) rf [ β ( T ) ]} +... (7). Spcral analysis owr spcral dnsiy of signal I() is givn y h rlaion: { m } + T j π f ( f) cos ( π f ) g(τ ) dτ d T (8) Susiuing h valu of g ( τ )dτ Eq.(8) and ingraing 19-, w hav: 1 ( f) π π β ( f + f AT ) m from Eq. (7) ino sin π f T + πfmβ AT p( T) ( T ) rf [ β ( T ) ] β ( T ) + + p() rf (β ) β π p () rf ( β ) sin πfmβ AT β + + p() rf (β ) (9) β π Similar xprssion can drivd for signal Q ()..3 Error ra Th inpu powr of signal I() is givn y: 1 π S i cos g(τ ) τ d T (1) Th inpu powr of signal Q () is givn y: 1 π S i sin g(τ ) τ d T (11) Th oupu whi nois spcral dnsiy: Ns Nc H( f) η / whr τ is a consan. So, signal o nois raio is givn y: π g(τ ) dτ S j f N A η cos i T α f c α f jf A (η ) roailiy of rror for in-phas signal is givn y: c π cos (τ ) τ 1 g d T rfc α f j f A η 1 (1) (13)

3 SARAF & TIWARI: GAUSSIAN MINIMUM SHIFT KEYING SYSTEMS 67 roailiy of rror for quadraur phas signal is givn y: 1 π sin (τ ) τ 1 g d T s rfc (14) α f j f A η whr rfc() is a complmnary rror funcion and i is dfind as rfc( u) 1 rf ( u) For slowly fading channl 1 : c π α' cos (τ ) τ 1 g d T rfc α f j f A η whr α is a consan. whn α is random hn 1 c 1 (15) rfc α'γ p γ dγ (16) Sic whr γ N 1 γ γ and p(γ ) (17) γ E and γ E ( α' ) N whr E is signal nrgy and E(α ) is h avrag valu of α. Ingraing Eq. (16), w g proailiy of rror: mainly dpnds on h valu of h ingral of g() which is inhrnly dpnds on h paramr β i.. B T. Th naur of h curv dpnds on cosin of h ingrad valu of g(). B B 3dB /π is calld normalizd 3 db andwidh. Th largr valu of B rsuls in a largr signal andwidh. Th valus of E /N indicad y h rsarchrs 7 for B T c, whr T c γt wn.1 o.3 shows ha i dcrass firs and hn incrass as B T c incrass. So, h naur of curv rsmls o ha of ours. owr spcral dnsiy is numrically compud from Eq.(9) and h rsul is shown in Fig.. Th variaion of powr spcral dnsiy wih normalizd frquncy for diffrn valus of B T is shown in Fig.. Our rsuls hav n compard wih ohrs. Th dark coninuous lin shows our rsul for B T.7. Th dark rokn lins for B T.7 ar du o h rsarchrs 1,18 and h naur of hs curvs ar sam. Th powr spcral dnsiy dcrass as normalizd frquncy incrass. Our numrical valus of spcral dnsiy ar clos o h valus of rsarchrs 1. For B T 1., h rsuls agr wih h rsarchrs 18 as shown y h dod lin. Th rsuls for B T.3 agr wih h rsul as givn in Rf. 18 (no shown for simpliciy). roailiy of rror ( c ) is compud from Eq. (13). Th rsul is shown in Fig. 3. Th coninuous lin shows our rsuls and h rokn lin shows h rsul as givn in Rf. 1 I indicas h variaion of proailiis of rror wih signal o nois raio for B T. and B T.5. Th proailiy of rror dcrass wih h incras of SNR and h naur of curvs agr wih h rsuls as givn in Rf. 1. Th numrical valus of c and h SNR ar larg in comparison wih hos y rsarchrs 1. Th rason hind his is ha h ffc of group dlay o c 1 α'γ 1 1+ α'γ (18) 3 Rsuls and Discussion Signal o nois raio (SNR) is compud from Eq. (1) and h rsuls ar shown in Fig. 1. Th variaion of signal o nois raio (S/N) wih h common dsign paramr B T is shown in Fig. 1. I dcrass sharply wih B T from.1 o. hn slowly from. o.3 and hn incrass wn.3 o.5 and dcrass wn.5 and.7 and finally, incrass and coms almos consan. Th SNR Fig. 1 Variaion of signal o nois raion wih B T

4 68 INDIAN J URE & AL HYS, VOL 46, JANUARY 8 Fig. owr spcral dnsiy vrsus normalisd frquncy a diffrn B T Fig. 3 roailiy of rror ( c ) vrsus signal o nois raio (γ ) a B T. and B T.5 and h narrow and spcrum hav n akn ino accoun in h calculaion of proailiy of rror. Th proailiy of rror ( ) for GMSK for slowly Rayligh fading channl is numrically calculad from Eq. (18). Th variaion of c (γ) wih fadd valu of signal o nois raio (SNR) γ for diffrn channl anuaion consans α' is shown in Fig. 4. Our rsuls hav n compard wih h rsuls as givn in Rf. 1 shown y dod lin for fixd α.68. From Fig.4, i is ovious ha for ach α' h proailiy of rror dcrass wih γ. Fig. 4 roailiy of rror ( c ) vrsus signal o nois raio ( γ ) a diffrn anuaion consans (α ) for fadd channl 4 Hardwar Implmnaion 4.1 GMSK modulaor Th asic circui diagram of GMSK modulaor wih addiiv Gaussian nois is shown in Fig. 5. In his sysm, inary daa a() firs of all convrd ino (+1, 1) squnc and hn a sram of rcangular pulss s() is achivd. Th nois n i () is now addd wih h signal s(). Th nois corrupd signal is filrd y using Gaussian filr. Th filrd oupu conains g() + n (). Now h filrd signal wih nois is ingrad in h phas accumulaor. Thus, h

5 SARAF & TIWARI: GAUSSIAN MINIMUM SHIFT KEYING SYSTEMS 69 Fig. 5 Hardwar ralizaion of GMSK modulaor wih addiiv nois Fig. 6 Hardwar ralizaion of GMSK dcor wih addiiv nois phas angls ϕ S and ϕ n ar gnrad. Th quadraur signal componns I and Q ar oaind from a look up al for h sin and cosin implmnd in LA's. Th sin funcion is gnrad using wo sag approximaion chniqu. Th carrir wav cos(ω c ) is gnrad y local oscillaor and muliplid y inphas and quadraur phas signals and finally addd o g h rquird modulad signal y(). 4. GMSK dcor Th circui diagram of GMSK dcor is shown in Fig. 6. Th rcivd signal is muliplid y h sam carrir using carrir rcovry sysm using wo muliplirs, on for in-phas componn of signal and ohr for quadraur phas componn of signal. Th highr and ohr carrir rms ar soppd y using low pass filrs. Th oupu of filrs gos o phas dcors. Th oupu of phas dcors producs h phas angls of in-phas and quadraur componns of nois corrupd signals. Th phas angls ar allowd o go hrough diffrniaor and hn o h diffrniaing filr which givs h oupu of rcangular pulss. Th lvls of pulss ar oaind y nvlop dcor and finally h inary daa is achivd y dcision dvic. 5 Conclusion In his papr, Gaussian minimum shif kying (GMSK) in addiiv whi Gaussian nois (AWGN) is xamind y implmning in-phas I and quadraur phas Q signals oaind from complx phas funcion. Th consrucion and working of modulaor and dmodulaor has n discussd. Th fundamnal propris lik spcral analysis and i rror ra prformanc (BER) in prsnc of AWGN for non-fading and fading channls hav n analyzd and sudid in dail. Th rsuls so oaind, hav n compard wih ohrs and found o saisfacory. Rfrncs 1 Muroa K & Hirad K, IEEE Trans Commun, 9 (1981) 144. Sl R al., IEEE Commun Mag, (1995) 1. 3 Haspslagh J al., IEEE J Solid - Sa Circuis, 5 (199) Korn I, IEEE Trans Commun, 39 (1991) Varshny & Kumar S, IEEE Trans Vh Tch l, 4 (1991) Vankka, Jouko al, IEEE J Sl Aras in Commun, 19, 6, (1) Asa D K & asupahy S, IEEE Trans Informaion Thory, 48, 1 () Tllado-Mourlo J, Wsl E K & Cloffi J M, IEEE J Sl Aras Commun, (1996) 49.

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