Efficient Time/Frequency Permutation of MIMO-OFDM Systems through Independent and Correlated Nakagami Fading Channels

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1 In. J. Counicaions, Newok and Syse Sciences, 009,, doi:0.436/ijcns Puished Onine Decee 009 (h:// 903 Efficien Tie/Fequency Peuaion of IO-OFD Syses hough Indeenden and Coeaed Nakagai Fading Channes Khod A. SAAIFAN, Ead K. A-HUSSAINI Deaen of Eeconics and Counicaions, Caio Univesiy, Giza, Egy E-ai: Received Augus 6, 009; evised Seee 8, 009; acceed Novee 4, 009 Asac Sace-Tie Fequency (STF codes fo IO-OFD ove ock-fading channe can achieve ae and fu-divesiy which is he oduc of he nue of ansi anennas, eceive anennas, fading ocks and channe as. In his aice, ie euaion is oosed o ovide indeenden ock-fading ove Jake s Doe owe secu channe. oeove, we show he efoance vaiaions of STF code as channe deay sead changes. Theefoe, we inoduce a fequency/ie euaion echnique in ode o eove he fequency coeaion aong su-caies, which susequeny inceases he coding gain and achieves axiu divesiy. Finay, he syo eo ae (SER efoance of he oosed ie/fequency eued STF codes ove indeenden and coeaed IO anenna anches unde Nakagai fading channe is siuaed. We show ha he oosed syses ovide ee efoance and oe ous o age vaues of anennas coeaion coefficiens in coaison wih he un-ineeaved one. Keywods: IO, OFD, Sace-Tie Fequency Coding, Nakagai Fading Channe, Tie/Fequency Ineeaving. Inoducion Achieving high daa ae, fu divesiy gain and highe newok caaciy ecoes he ajo equieens of wieess syse ovides. IO-OFD syse is one of he os aacive echniques o ovide hese caaiiies. Receny, soe aenion has een devoed o design STF codes fo IO-OFD syse wih ansi anennas, eceive anennas, and N-OFD ones hough ui-ah fading channe. Thee ae sevea aes, which discussed he code sucue o ovide fu divesiy gain and high daa ae. In [], W. Su e a. oosed he design of fu divesiy sace fequency ock code (SFBC wih ae- fo any nue of ansi anennas and aiay owe deay ofies. The ae- fu divesiy SFBC was oosed in [] fo any aiay nue of ansi anennas. Howeve, ecause a zeoadding aix has o e used when N is no an inege uie of, he syo ansission ae canno e aways guaaneed. In [3], ee divesiy gains hough ock-fading channes can e oained, ha was done y seading he coding acoss uie fading ocks. In [4], hey sudied he eo efoance esus of STF codes in IO- OFD syses fo a vaiey of syse configuaions and channe condiions. The axiu divesiy is he oduc of ie divesiy, fequency divesiy and sace divesiy as shown in [5]. Receny in [6], W. Zhang e a. oosed a syseaic design of high-ae STF codes fo IO fequency-seecive ock-fading channes. By seading he ageaic coded syos acoss diffeen OFD su-channes, ansi anennas and fading ocks, he oosed STF codes can achieve a ae- and a fu divesiy of, whee is he nue of indeenden fading ocks in he code-wods. To achieve he fu-divesiy efoance of STF code, axiuikeihood ( decoding us e eoyed. In ode o decease he age coexiy of decoding, shee decode can e consideed o achieve nea- efoance [7,8]. Fo ock-fading channes, he efoance of STF-coded OFD is uch ee han SF coding as deonsaed in [9]. In IO-OFD syses, he DFT oeaion inoduces coeaion ino he channe fequency esonse a diffeen su-caies [0,], aking is efoance va- Coyigh 009 SciRes.

2 904 K. A. SAAIFAN ET A. ies as he deays eween ahs vay. The ouine of he ae is as foows. Secion descies he channe saisics and syse ode. The suggesed ie/fequency euaions of high ae STF codes sucue oosed in [6] fo indeenden and coeaed Nakagai fading ae inoduced in Secion 3. In Secion 4, we ovide siuaion esus fo he efoance of he oosed schee. Finay, soe concusions ae ade in Secion 5.. Channe Saisics and Syse odes Befoe invesigaing euaion schees fo IO- OFD syses equied wih ansi anennas, eceive anennas in oie adio channes, we iefy descie he channe saisics, ehasizing he seaaion oey of oie wieess channes, which is cucia fo siifying ou ie/fequency euaion. In his secion we aso iefy descie a IO-OFD syse... Saisics of oie Radio Channes The channes eween each ai of ansi and eceive anennas ae assued o have indeenden deay ahs and he sae owe deay ofie. The channe iuse esonse eween h ansi anenna and h eceive anenna can e odeed as h, ( ;, ( ( 0 ( whee τ is he deay of he h ah, and, (, coex aiude of he h ah eween h ansi anenna and h eceive anenna. ( s ae odeed as a coex ando fading signas wih Nakagani- disiued fading aiudes and unifo hases. Nakagai -disiuion fading ode [] is one of he os vesaie, in he sense ha i has geae fexiiiy and accuacy in aching soe exeiena daa han Rayeigh, og-noa, o Rician disiuions. The Rayeigh disiuion is a secia case when he fading aaee =. I can aoxiae Rice disiuion fo >. oeove, i is assued ha a ah gains eween any ai of ansi and eceive anennas foow he sae owe ofie, i.e., E (, 0 fo any given (,,. The owes of he ahs ae noaized such ha 0. Using Equaion (, he fequency esonses of he ie-vaying adio channe a ie is H (, f ( ex( jf,, 0 is ( The IO channe is assued o e saiay coeaed fo any (,, whee =,, =,, and indeenden fo any whee, =0, -. e denoes he saia coeaion coefficien eween ( and, ( defined as,, (, T,,, ( (3 The saia coeaion coefficien oseved a he eceive has aso een exensivey sudied in he ieaue and is given as Rx,, (,, ( (4 Given Equaions (3 and (4, he syeica coeaion aices a ansie and he eceive can e defined esecivey as and, R (5 Rx Rx Rx Rx Rx Rx R Rx (6 Rx Rx Rx The saia coeaion aix R of he IO adio channe is he Konecke oduc of he saia coeaion aix a he ansie and he eceive and is given y [3] R R R Rx (7 whee denoes he Konecke oduc. The coeaion funcion of he fequency esonse fo diffeen ies and fequencies is, 0 (, f E H E[ *, *, (, f H, (, f f (8 ( ( ]ex( jf, Assue Jake s Doe owe secu [4], heefoe he coeaion of he h ah is given y [ *,,, 0 D E ( ( ] J (f (9 whee eesens he owe of h ah, f D is he Doe fequency, and J O (x is he zeo ode Besse funcion of he fis kind. Susiue Equaion (9 in Equaion (8, hen Equaion (8 can e ewien as Coyigh 009 SciRes.

3 K. A. SAAIFAN ET A. 905,, (, f J (f ex( jf (0, 0 ( D f 0 ( f whee Φ (Δ is he ie doain coeaion funcion and Φ f (Δf is he fequency doain coeaion funcion. Fo Equaion (0, he ie-fequency doain channe coeaion funcion of H, (,f can e seaaed as he oduc of he saia coeaion coefficien, he ie doain channe coeaion, and he fequency doain channe coeaion, which ae deenden on he anenna seaaion, he Doe fequency, and ui-ah deay sead esecivey. Fo an OFD syse wih ock engh T and one sacing (su-channe sacing Δf=/T, he coeaion funcion fo diffeen ocks and ones can e wien as, f T (, f ( kt ( n (.. IO-OFD Syse ode Conside a STF-coded IO-OFD syse wih ansi anennas, eceive anennas and N su-caies oeaing ove a fequency-seecive ui-ah fading channe. The IO-OFD syse wih code euaions consideed in his ae is shown in Figue. The souce S geneaes N s =N infoaion syos fo he discee ahae A, which ae quadaue aiude oduaion (QA noaized ino he uni owe. Using a aing f: S C, an infoaion syo veco S A Ns is ased ino ocks and aed ono a STF codewod o e ansied ove he ansi anennas and OFD ocks. Each STF codewod C can e exessed as a N aix. C C C C whee he N aix C c c c ( fo =,... denoes he su-codewod eady o e sen duing he ie eoch. The h ( =,... coun of Figue. IO-OFD syse wih code euaion o coa channe coeaion. C denoed y c is sen o he OFD ock a he h ansi anenna duing he ie eoch. Afe invese fas Fouie ansfo (IFFT oduaion and cycic efix (CP inseion, OFD syos ae sen fo a ansi anennas siuaneousy. A he eceive, afe ached fieing, eoving he cycic efix, and aying FFT, he eceived signa a he eceived signa a he h eceive anenna duing he ie eoch is given y whee H Y diag( c H, Z, H, ( 0 H, ( H, ( N Z (3 is he h OFD ock channe fequency esonse veco eween h ansi anenna and h eceive anenna and denoes he coex discee AWGN ocess wih zeo ean and uni vaiance a he h eceive anenna. The faco in Equaion (3 ensues ha he aveage SNR a each eceive anenna is indeenden on he nue of ansi anennas. 3. Tie/Fequency Peued STF Codes STF coding oosed in [6] can achieve ae of and fu divesiy fo any nue of ansi anennas and any aiay channe owe deay ofies. I was consuced y aying he ayeing conce aong wih ageaic code coonens, which was inoduced in he design of headed ageaic sace-ie (TAST code [5]. The STF code sucue seads he ageaic code coonens in adjacen su-caies and adjacen ie sos ha suffe fo high coeaion inoduced y DFT oeaion and ie coeaion esecivey. In his secion, ie/fequency eued STF code sucue is inoduced ino STF code sucue of [6] in ode o eove he effec of channe coeaion aong he code coonens and achieve ee divesiy ode. 3.. STF Codes Sucues e og N, og Nq, and K N N q, hen a ock of N s ansied infoaion syos S=[S,S,...S N ] T ae ased ino J(J=N/K equa size su-ocks. Each su-ock S j AK (j=,, J is esecivey encoded ino an STF code aix Bj of size K hough he foowing ses: Each suock S j (j=,, J ae ased ino N q infoaion veco s A n,,, N N. ( q q T Coyigh 009 SciRes.

4 906 K. A. SAAIFAN ET A. Geneae ageaic code su-ock n q y aying a fuy-divese uniay ansfoaions Θ ino each infoaion veco s n,, N o geneae N q heads y whee, Θs ( n ( q q q, N (, ( N, N (4 N N, and Θ is he fis incia N N uniay aix of he foowing aix whee N og, H Ψ F diag,,, (5 H F N is he discee Fouie ansfo (DFT aix, and ex j 4. 3 Aying he ayeing conce o consuc he encode su-aices n ( n, and,. N T T Nq T n, n, n Nq, n ( ( Nq kn k ( n N k q ( ( N kn k ( n N k ( 6 q ( ( N kn ( kn N k whee / N q and kn 4 Re-aange he eeens of ( n ( N. n y, n n ( q u (, n q : n q, and od, fo, N q ( kn ( k ( n k n Nq T N k ( ( k n kn (7 q ( 3( N kn ( kn N k hen, he K code aix B i is consuced as Bi N N N (8 The STF coding aies he sae coding saegy o evey su-ock B ( j,,, J, hen he ae- STF code CC N j is of he fo T C B B T T B T J (9 I is cea ha, each head of codewod (n (,, Nq and n n,, N is sead ove sace, ie and fequency diensions. Theefoe, he STF code sucue is no oiu in seading he code coonens of each head on adjacen su-caies ha suffe fo high coeaion inoduced y DFT oeaion. Howeve, if he owe deay ofie of he channe is avaiae a he ansie side, fuhe ioveen can e achieved y deveoing an ineeaving saegy (can educe he coeaion eween adjacen su-caies which exiciy consides he owe deay ofie. In addiion, since he STF code sucue ainains is divesiy gain fo sending he OFD ocks hough indeenden fading ocks, we sha inoduce ie euaion o achieve indeenden fading ocks hough IO channes ha suffe fo high coeaion inoduced y Doe owe secu. 3.. Tie/Fequency Peuaion Schees The assuion of indeenden fading a he anches is acceae if he anennas ae saced sufficieny aa wih esec o he adio fequency (RF caie waveengh. In his case, 0, T and T,,hen Equaion ( wi e educed, o he auocoeaion funcion [0], ( k, n J0(f DkT ex( jn / T 0 n q n (0 Oviousy, he souces of channe coeaion ae caused y he ie doain channe coeaion, and he fequency doain channe coeaion. Ou ojecive is o find he seaaion aaees k and n fo IO- OFD syse which oduce zeo ie and fequency coeaions hen eue he ageaic code coonens of B j (j=, J a zeo ie fequency coeaion o axiize he divesiy gain. K c in k J 0(f D kt ( Nc in0 n N ex( jn / T ( 0 The zeos of he Besse funcions (Equaion ( ay a doinan oe in ou aicaions. The Besse funcions have infinie nue of zeos. The axia and inia of J 0 seadiy decease in asoue vaue as k inceases. Coyigh 009 SciRes.

5 K. A. SAAIFAN ET A. 907 The fis five zeos of J 0 ae.4048, 5.50, ,.795, and The ineva eween he as wo is 3.394, which is aeady cose o π. The age oos ae aoxiaey v, whee v is he nue of 4 he oo. To eak he ie coeaion of he channe, veify indeenden fading ock and eaize high-ae fu-divesiy STC of [6], he -OFD ocks of STC aix B j (j=, J shoud e ansied a ie diffe ence of Kc. Fo age coheence ie o f DT equivaeny ow Doe sead of he fading, high ineeaving size is equied o eak he eoy of he channe. The oiu su-caies seaaion faco N c (see Equaion ( can e easiy found via ow-coexiy coue seach. Howeve, cosed-fo souions fo secific cases ae eoed in []. Based on he knowedge of channe seaaions facos Nc and K c, ie/fequency eued STF code can e inoduced using he foowing ses: Disiue he STC ocks ove indeenden fading ocks y euing he u -OFD ocks of STC aix B j (j=, J wih hose ocks a ie uk c, ( u,. Ay fequency euaion ino each ai of code aices B j and B j, whee j j N, N Nc K, j [,, N ] ( n N and n,,..., J N y euing ows K,, K of B j wih he ows,, K of B j. 3 Fuhe euaion shoud e done o eak he es of channe fequency coeaion y euing each ai of ows ( n, n, whee n, K and n K, K fo a code aices B j ( j, J wih he coesonding ai of ows a ock disances u ( Kc whee ( u,, K. By efoing he aove ses as shown in Figue, he code coonens ( n n (,, Nq and n n,, N of each head of code aix B j ae affeced y indeenden fading ocks which susequeny achieve axiu divesiy gain. Exaes of STF codes and eued STF codes fo =, = ae shown Figues 3 and 4. Fo =, STF codes wi e, in fac, he SF codes of [6]. The ae- SF code sucue and he suggesed ie/fequency euaion (anenna is shown ony ae shown in Figue 3. The ae- STF code sucue and he suggesed ie/fequency euaion fo = ae shown in Figue Siuaion Resus In his secion, we siuaed he oosed euaion Figue. The suggesed ie/fequency euaion of STF codes. Figue 3. Rae- ie/fequency eued SF code (T/FP- S F. Figue 4. Rae- ie/fequency eued STF code (T/FP-STF. schee and coaed wih he non-eued STF codes fo diffeen owe deay ofies of he channe. We esen aveage syo-eo ae (SER cuves as funcions of he aveage SNR. Then we iusae he efoance of he oosed euaion fo SF codes hough coeaed Nakagai fading channes. To invesigae he efoance of he oosed ie/fequency euaion of STF codes ove fequency-seecive fading channes, we efo he siuaion exeiens and coae wih he STF codes [6] fo IO-OFD syses. In he siuaion, we use a syse wih 8 OFD ones and 4QA ansission schee, hus he seca efficiency is 4 i/s/hz, ignoing he cycic efix. The andwidh of OFD syse is Hz and he engh of he cycic efix is 3, i.e., 3μs. Hence he duaion of one OFD syo (cycic efix excuded is T=8μs. A wo-ay Nakagai fading channe sais- Coyigh 009 SciRes.

6 908 K. A. SAAIFAN ET A. ics ode is consideed wih he equa gain, Doe sead f D =00Hz, and fading deh = 0.5, and. I is o e noed ha = 0.5 eesens he wos fad- ing siuaion ha can e eesened y Nakagai disiuion. This case can e couneed in ad uan oie adio. When =, we oain Rayeigh fading channe. Finay, = eesens he es consideed siuaion in which he fading is ess han ha of Rayeigh. 4.. Pefoance Coaison fo Diffeen Deay Seads The fis se of exeiens is conduced o coae he efoance of he oosed schee wih STF codes fo diffeen ah deay of he wo-ay ode. A sie wo-ay, equa-owe deay ofie, wih a deay τ icoseconds eween he wo ays is assued. Siuaion is caied ou fo wo cases: 8μ sec (oiu euaion N c =8 and 0μ sec (oiu euaion N c =6. Fo Doe sead f D =00Hz he oiu ie seaaion is 4 OFD syos o ensue indeenden fading ocks, heefoe he ineeaved STF code is sanned ove 56 OFD syos. Figues 5, 6 and 7 deic he ioveen in SER efoance offeed y he oosed ie/fequency euaions hough indeenden Nakagai fading channe wih diffeen. The vaues of he fading deh consideed ae = 0.5,, and esecivey. I can e oseved fo hese figues ha he SER efoance of STF codes [6] vaied as he deay sead of he channe changed. The SER efoance of STF codes is fuhe ioved as deay sead of he channe inceased. Such an ioveen is aiued o he age coding gain induced y ui-ah fading channes wih a age deay sead. The efoance of he STF code degaded significany fo he 0μs case o he 8μs case, wheeas he efoance of he STF code using ie/ fequency euaion was aos he sae fo he wo deay ofies. We can see ha he T/FP-STF codes have ee SER efoance han he non-eued STF codes. Fo τ=8μs case, hee is an ioveen of aou 3. db fo SF codes and an ioveen of aou.8 db fo he STF codes a a SER of 0-4 when =. Theefoe; he oosed ineeaving ehod offeing highe code gains aking i oe ous o sa deay sead. This confis ha y caefu ineeave design, he efoance of he STF codes can e significany ioved. Fo Tae, i is cea ha he SNR deceases wih he incease of. The efoance of he ineeaved codes is no sensiive o he vaiaion in he channe ie deay sead. In a of cases consideed, he equied SNR of he ie/fequency ineeaved codes is owe han ha needed fo he un-ineeaved one o achieve he sae SER. Figue 5. Aveage SER vesus SNR of, IO-OFD syse hough indeenden Nakagai fading channe =0.5 wih diffeen deay sead. Coyigh 009 SciRes.

7 K. A. SAAIFAN ET A. 909 Figue 6. Aveage SER vesus SNR of, IO-OFD syse hough indeenden Nakagai fading channe = wih diffeen deay sead. Figue 7. Aveage SER vesus SNR of, IO-OFD syse hough indeenden Nakagai fading channe = wih diffeen deay sead. Coyigh 009 SciRes.

8 90 K. A. SAAIFAN ET A. Tae. SNR equied o oain a SER=0-4 fo STFC and T/FP-STFC a diffeen ie deay sead. SFC T/FP-SFC STFC T/FP-STFC 8μsec 0μsec 8μsec 0μsec 8μsec 0μsec 8μsec 0μsec db 9.4 db 8 db 6.3 db 5.8 db 4.7dB 8. db 6. db 4.9 db 4.9 db 3.8 db 3.3 db 7.4 db 5. db 3.9 db 4.3 db 3. db.53 db 4.. Pefoance Coaisons ove Coeaed Nakagai Fading Channes IO syse wih cosey saced anenna eeens is consideed hee. Ou ai is o anayze he infuence of he Nakagai- fading aaee and he effec of anenna coeaion on he SER efoance of he ae- SF code, and he oosed T/FP-SF code deiced in Figue 3. Figue 8 shows he SER degadaion as he coeaion coefficiens eween he ansiing anenna anches ρ vay fo 0 u o 0.8. Siia coeaion is assued eween eceiving anenna anches. Siuaion is caied ou fo wo cases: Tansie coeaed Nakagai IO fading channe case: R, and 0 R, and Douy coeaed Nakagai 0 IO fading channe case: R, and R. The vaues of he fading deh consideed ae =0.5,, and esecivey. I is cea ha he SER inceases wih he incease of coeaion coefficien ρ. A ρ=0, he eceived signas ae indeenden and he codes acicay achieves fu divesiy eceion gain. I is cea ha he oaiiy of eo deceases wih he incease of, which is wih he decease of he seveiy of fading. Fo hese figues, i is cea ha he syses unde consideaion aeciay doinae he syses consideed in [6]. 5. Concusions In his ae, he iiaion fo achieving fu-divesiy of STF-coded OFD is inoduced. The iiaion aises due o he fac ha he ageaic code coonens ae sead in adjacen su-caies ha suffe fo high coeaion inoduced y DFT oeaion. Assuing ha he owe deay ofie of he channe is avaiae a he ansie, we oosed an efficien ie-fequency ineeaving schee o fuhe iove he efoance. Based on siuaion esus, we can daw he foowing concusions. Figue 8. Aveage SER vesus coeaion coefficiens fo IO-OFD syses a SNR=4dB. Fis, he oosed ie/fequency euaions STF codes offe consideae efoance ioveen ove eviousy eoed esus. Second, he aied ineeaving schee can have a significan effec on he ovea efoance of he STF code hough coeaed and indeenden Nakagai fading channes. Coyigh 009 SciRes.

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