Singl-ai ybid ARQ Using Join Iaiv Tx/Rx MMSE-FDE & ISI Canllaion Kazuki TAKEDA and Fumiyuki ADACI Dp. of Elial and Communiaion Engining, Gadua Shool of Engining, Tohoku Univsiy 6-6-5, Aza-Aoba, Aamaki, Aoba-ku, Sndai, 98-8579, Japan kazuki@mobil.i.ohoku.a.jp and adahi@i.ohoku.a.jp Absa Rnly, w poposd a join iaiv ansmi/iv (Tx/Rx minimum man squa o (MMSE fquny-domain qualizaion (FDE & in-symbol infn (ISI anllaion (ISIC fo singl-ai (SC blok ansmissions. In his pap, w xnd h pviously poposd join iaiv Tx/Rx MMSE-FDE & ISIC o h SC hybid auomai pa qus (ARQ pak ass o xploi h ansmission of h sam pak fo impoving h houghpu. opimiz h s of Tx/Rx MMSE-FDE wighs by aking ino aoun h pak ansmission/ombining as wll as ISIC. show by ompu simulaion ha h poposd shm signifianly impovs h pak o a (PER and houghpu pfomans in a sv fquny-sliv fading hannl. Kywods; SC-FDE, ISI anllaion, ARQ I. INTRODUCTION Th boadband wilss hannl ompiss many popagaion pahs having diffn im dlays []. Th minimum man squa o fquny-domain qualizaion (MMSE-FDE povids good bi o a (BER pfoman fo boadband singl-ai (SC ansmission in a sv fquny-sliv fading hannl [-4]. owv, h pfoman of SC using h MMSE-FDE is sill a fw db away fom h mahd fil bound du o sidual insymbol infn (ISI af h MMSE-FDE. Th us of iaiv iv (Rx MMSE-FDE & in-symbol infn (ISI anllaion (ISIC has bn xnsivly sudid [5,6]. Th Rx FDE wigh is updad basd on h MMSE iion by using h liabiliy infomaion of daa dion a h pvious iaion. In [6], iaiv Rx FDE & ISIC was sudid fo SC-hybid auomai pa qus (ARQ. Rnly, w ook diffn appoah fom h iaiv Rx possing and poposd a join ansmi/iv (Tx/Rx MMSE-FDE [7]. A s of Tx and Rx FDE wighs was divd basd on h MMSE iion. Alhough h divd wigh s is subopimal, h poposd shm povids b pfoman han h onvnional Rx MMSE-FDE. Mo nly, w poposd a join iaiv Tx/Rx MMSE-FDE & ISIC ha is an xnsion of join Tx/Rx MMSE-FDE [8]. In his shm, ISIC is inopoad ino join Tx/Rx MMSE- FDE. A h iv, Rx FDE & ISIC is iad. In ah iaion sag, h Rx FDE wigh is updad basd on h MMSE iion. A h ansmi, Tx FDE is aid ou bfo ansmiing h signal basd on h pdid dg of sidual ISI af h ISIC a h iv. Th join iaiv Tx/Rx MMSE-FDE & ISIC povids muh b pfoman han h onvnional iaiv MMSE-FDE & ISIC. In his pap, w xnd h pviously poposd join iaiv Tx/Rx MMSE-FDE & ISIC suiabl o SC-ARQ o xploi h ansmission of h sam pak fo impoving h houghpu. ARQ using h Chas ombining (CC sagy [9,] ansmis h sam pak unil i is oly ivd. Bfo ansmiing h pak, Tx FDE is applid. A h iv, vy im h sam pak is ivd, h Rx FDE, pak ombining, and ISIC a joinly iad. opimiz h s of Tx/Rx MMSE-FDE wighs by aking ino aoun h pak ansmission/ombining as wll as ISIC. In ou shm, h Rx FDE wighs a h sam as h on psnd in [6] xp ha h onanaion of ansmi FDE and hannl is viwd as an quivaln hannl. Th Tx FDE wigh is opimizd fo ah ansmission. Fo h iniial pak ansmission, h Tx FDE wigh is ompud basd on h pdid dg of sidual ISI af h ISIC a h iv. Fo h pak ansmission, h Tx FDE wigh is ompud fo h givn hannl ondiions of pvious ansmissions of h sam pak as wll as h pdid dg of h sidual ISI af pak ombining and h ISIC a h iv. show by ompu simulaion ha h poposd shm signifianly impovs h pak o a (PER and houghpu pfomans. Th s of his pap is oganizd as follows. Sion II dsibs h sysm modl of SC-ARQ using join iaiv Tx/Rx MMSE-FDE & ISIC. In S. III, h s of MMSE- Tx/Rx FDE wighs is divd. Sion IV shows h ompu simulaion suls. Sion V onluds his pap. II. SC-ARQ SYSTEM MODEL Th ansmi/iv suu is illusad in Fig.. Blow, symbol-spad dis-im signal psnaion is usd. onsid SC-ARQ using CC ha ansmis h sam pak unil i is oly ivd. Th (Mh pak ansmission is onsidd (i.., ansmission of h Mh opy of h sam pak. In ou shm, boh ansmi and iv qui h hannl sa infomaion (CSI fo ompuing hi MMSE-FDE wighs. In his pap, w assum h pf knowldg of CSI a boh h ansmi and h iv. 978--444-3574-6//$5. IEEE
Infomaion bis CP-moval N-poin FFT Channl nod Daa modulao Riv FDE wigh (M-, (k Combining pas ansmid paks N -poin FFT Tansmi FDE wigh (M- (k (a Tansmi + N-poin IFFT Rsidual ISI plia gnao LLR alulao N -poin FFT N -poin IFFT D-od Symbol plia gnao (b Riv Fig. Tansmi/iv suu. CP-insion Channl inlav Rivd bis Dlay A. Tansmi signal A pak is gnad by daa-modulaing h odd bi squn. Th daa squn of h pak is goupd in a squn of N -symbol bloks, wh N is h siz of fas Foui ansfom (FFT and invs FFT (IFFT. ihou loss of gnaliy, w onsid on N -symbol blok in a pak (whih onsiss of mulipl bloks and hus, h blok numb in a pak is omid fo h sak of simpliiy. Th N -symbol blok is psnd as d=[d(,,d(,,d(n ] T. N -poin FFT is aid ou on d o obain h fquny-domain ansmi signal D=[D(,,D(k,,D(N ] T, wh D is givn by D = Fd ( wih F = ( ( ( π π N j j N N N (( N N N j π (( j π ( N N ( bing an N N FFT maix. Th Tx FDE wigh, whih is muliplid o h kh fquny omponn D(k of a blok in h (Mh ansmiing pak, is dnod by { (M- (k; k=n. Bfo ansmiing h (Mh ansmiing pak, { (M- (k; k=n is muliplid o {D(k; k=n as S = [ S = (,..., S D, ( k,..., S ( N T ] wh (M- =diag{ (M- (,, (M- (k,, (M- (N is an N N diagonal Tx FDE wigh maix fo h (Mh ansmiing pak. [ (M- (M- ]=N is m o kp h ansmi pow ina. An N -poin invs FFT (IFFT is applid o S o obain a blok s (M- =[s (M- (,,s (M- (,,s (M- (N ] T = (3 F S (M- in h im-domain (Mh ansmiing signal pak. Af h insion of N g -sampl yli pfix (CP ino h guad inval (GI, h pak is ansmid. B. Rivd signal Th popagaion hannl is assumd o b an L-pah fquny-sliv blok fading hannl. Th omplx-valud pah gain and im dlay of h lh pah of h mh ansmiing pak a dnod by h (m l and τ (m l, l=l, m=m, spivly. Th CP-lngh is assumd o b qual o o long han h maximum hannl im dlay τ L. Th ivd signal blok (m =[ (m (,, (m (,, (m (N ] T af h CP-moval in h mh ansmid pak an b xpssd as Es = h s + n, (5 T s wh E s and T s a h avag ansmi symbol ngy and symbol duaion, spivly, h (m is an N N iulan hannl maix givn by h hl h h h ( h m L h =, (6 hl h hl h and n (m =[n (m (,,n (m (,,n (m (N ] T is h nois vo wih n (m ( bing a zo-man addiiv whi Gaussian nois (AGN having vaian N /T (N is h on-sidd nois pow spum dnsiy. An N -poin FFT is aid ou on M opis of h sam pak, (m, m=m, o obain h fquny-domain ivd signals R (m, m=m. R (m =[R (m (,,R (m (k,,r (m (N ] T is givn as Es ( m R = F = D + N, (7 wh N (m =Fn (m and (m =Fh (m F. Du o h iulan popy of h (m, h hannl gain maix (m of siz N N is diagonal. Th kh diagonal lmn of (m is givn by L l= l l ( k = h xp( jπkτ / N. (8 C. Iaiv Rx MMSE-FDE, pak ombining, and ISIC A h iv, MMSE-FDE, pak ombining, and ISIC a aid ou in ah iaiv sag. Th numb of iaions is dnod by I(>. Blow, h ih iaion sag (<ii is dsibd. M opis of h fquny-domain ivd signals R (m, (m, m=m, a muliplid by h Rx FDE wighs =diag{ (m, (,, (m, (k,, (m, (N, m=m, and a pak ombind in fquny-domain. Thn, h sidual
ISI plia, gnad using h dision sul a h (h iaion sag, is subad fom h fquny-domain signal af pak ombining. Th sulan fquny-domain (, ivd signal blok, ˆ M i D =[ ˆ, D i (,, ˆ (, D M i ( k,, ˆ (, D ( N ] T af aying ou h pak ombining and ISIC is givn as ˆ M = D R, (9 m=, wh i is h fquny-domain sidual ISI plia., In Eq. (9, i is ompud as ( M (, (, ( ( (, = Es m i m m ID m= wh D is h fquny-domain sof symbol plia givn as D = Fd ( i i i i wih d =[ d (,, d (n,, d (N ] T bing h sof symbol plia blok. Th nh lmn d ( n of d is givn as [6] ( d ( n ( ( λ n ( λn ( anh + j anh fo QPSK, ( ( λ λ n ( n ( = anh + anh ( ( j λ ( λ (3 n + anh n + anh fo 6QAM, ( ( i wh λ n ( x is h log-liklihood aio (LLR assoiad wih h xh bi of h nh daa symbol in a blok, ompud using h dod oupu in h (h iaion sag (no ha x=log M and n=n (s h modulaion lvl and ( d M, = fo h fis iaion sag. ˆ D of Eq. (9 is ansfomd ino h im-domain signal blok by IFFT. Th doding is aid ou using h sulan im-domain signal blok. Af h doding, h oupu LLR is usd o ompu h updad sidual ISI plia o b usd in h nx iaion sag. III. SET OF TX/RX MMSE-EIGTS In his sion, w div h s of Tx and Rx FDE wighs assuming ha M opis of h sam pak hav bn ivd. Fis, w div h Rx FDE wighs fo pak ombining, fo h givn Tx FDE wigh. Thn, h Tx FDE wigh is divd assuming ha h divd iv FDE wighs a usd. A. Rx MMSE-FDE wigh A onanaion of h ansmi FDE and h popagaion hannl is ad as an quivaln hannl. dfin h xpandd ivd signal vo R of siz MN as ( Es [ R R ] = D N R = +, (3 wh ( ( ( N =, N =. (4 N Using Eq. (3, Eq. (9 an b win as ˆ E D, (5 ( s ( { i = R I D ( (, wh [,...,,...,, i = ]. hn M opis of h sam pak a ivd, h o vo (M-, =[ (M-, (,, (M-, (,, (M-, (N ] T (, (, bwn d and ˆ M i ˆ M i d = F D a h ih iaion sag is givn as ˆ = d d / E s / T s ( = F { I{ D D + γ F ( (6 N, wh γ=(e s /N. Th oal MSE, i, i =[E( ] is givn as = ρ + γ [{ ( ( i ( [ { I{ ], ( i I ] (7 wh [6] ρ I = E{( D D ( D D. (8 Fom Eq. (8, w obain h MMSE soluion of minimizs, i as ( ( ha = { + ( γρ I. (9 Using h maix invsion lmma [], h MMSE soluion, of ( m i an b divd as M ( m ( m ( m ( m, i = { + ( γρ I m = {. ( If o is dd af h Ih iaion, h sam pak ansmission is qusd. Thn, af iving h ansmid pak, iv ompus Rx FDE wighs fo all h sam paks o ombin hm basd on h MMSE iion.
B. Tx MMSE-FDE wigh Th ansmi assums ha h sidual ISI will b paially dud by a fao of ρ a h iv, wh x ρ indias h liabiliy of h ISI anllaion a h iv, whih is pdid by h ansmi, and aks a ( valu bwn. Assuming ha is usd in h iv, h o vo osponding o Eq. (6 an b givn as ( = ρ F { I D + γ F ( N. ( Simila o h divaion of Eq. (8, h MMSE soluion ha, minimizs x =[E( ] an b divd as ( = { + ( γρ I. ( Subsiuing Eq. ( ino Eq. (, w obain = γ = N k= [{ ρ γ + ( γρ M m = ρ ( m ] ( ( k m ( k. + (3 ( div h MMSE soluion M using h Lagang mulipli mhod [] und h ansmi pow onsain (m [ { ] = N fo h givn { ; m=m (m and { ; m<m. Th soluion is givn as (divaion is omid fo h sak of bviy / γ γ μ ( k ρ ( k ( k = max, ( k ( k m< M ( k,(4 wh μ is hosn so as o saisfy [ (M- (M- ]=N., Th paam appas in h sond m of h fis omponn. If h ansmi blivs ha h iv (, an almos pfly anl h sidual ISI, ρ M x. On h oh hand, if h ansmi blivs ha h iv anno anl h sidual ISI a all (o whn h iaiv ISIC, is no mployd, is s o b. I is qui diffiul o, analyially find and hn, in his pap, by ompu, simulaion, w find suh ha h avag PER (o h houghpu is minimizd (o maximizd fo h givn avag ansmi E s /N. IV. PERFORMANCE EVALUATION Th pfomans of SC-ARQ using h join iaiv Tx/Rx MMSE-FDE & ISIC a valuad by ompu simulaion. Th numb of iaions is s o I=6. N =56 and CP lngh of N g =3 a onsidd. QPSK and 6QAM a assumd fo daa modulaion. Th hannl is assumd o b an L=6-pah fquny-sliv blok Rayligh fading hannl having unifom pow dlay pofil (E[h l (m ]=/L. Indpndn hannl is assumd fo ah ansmission. A ubo nod [] wih h oiginal oding a /3 using wo (3,5 usiv sysmai onvoluional nods is usd. 48 bi lngh odwod wih h oding a R=/ is gnad by punuing h paiy bi squns. Th dod onsiss of wo log-map dods. assum idal ACK/NACK ansmissions. A. Avag PER pfoman Th PER pfoman whn h sam pak has bn ansmid M ims (i.., h numb s fixd h is plod in Fig. as a funion of avag ansmi symbol ngy-o-nois pow spum dnsiy aio (E s /N wih M=4 as a paam. Fo ompaison, h PER pfoman wih h onvnional iaiv Rx MMSE-FDE & ISIC is also plod. I an b sn fom Fig. ha h poposd shm oupfoms h onvnional on. As nass, h poposd shm povids muh b PER pfoman han h onvnional on. This is baus, bfo ansmission of h sam pak in ou poposd shm, h ansmi FDE fo M> is ompud fo h givn hannl ondiions of pvious ansmissions of h sam pak as wll as h pdid dg of h sidual ISI af pak ombining and h ISIC a h iv. Fo xampl, whn M= (4 and QPSK is usd, h join iaiv Tx/Rx MMSE-FDE & ISIC an du h quid E s /N fo ahiving PER= by abou.8db (.8dB fom h onvnional iv MMSE-FDE. B. Thoughpu pfoman Figu 3 plos h ahivabl houghpu pfoman of SC-ARQ using join iaiv Tx/Rx MMSE-FDE & ISIC. Th houghpu pfoman using h onvnional iaiv Rx MMSE-FDE & ISIC is also plod fo ompaison. I an b sn fom Fig. 3 ha h poposd shm always povids b houghpu pfoman han h onvnional shm. In h low E s /N gion, h pak is mo likly ansmid. owv, h poposd shm offs high pak ombining gain han h onvnional shm and hfo, h houghpu signifianly impovs. In h high E s /N gion, h fis pak ansmission is mos likly sussful. Evn in his gion, h poposd shm povids high houghpu han h onvnional shm. V. CONCLUSION In his pap, w poposd h join iaiv Tx/Rx MMSE-FDE & ISIC fo SC-ARQ. divd a s of Tx/Rx MMSE-FDE wighs. In h poposd shm, h Rx MMSE-FDE wigh fo pak ombining and h sidual ISI plia fo ISIC a updad basd on h MMSE iion. Bfo ansmission of h sam pak, Tx MMSE-FDE is don basd on h pdid dg of sidual ISI af join MMSE-FDE, pak ombining, and ISIC. Th ompu simulaion suls showd ha h join iaiv Tx/Rx MMSE-FDE & ISIC povids signifianly b pfoman han h onvnional iaiv Rx MMSE-FDE & ISIC.
.E+ QPSK QPSK M= M=.E- Avag PER.E- M=3 Thoughpu (bps/z.5 M=4 Pop. (Rx only.e-3-8 -6-4 - 4 6 Avag ansmi E s /N (db (a QPSK.E+ 6QAM Pop. (Rx only -5 5 Avag ansmi E s /N (db (a QPSK 6QAM M= M=.5.E- Avag PER.E- M=3 Thoughpu (bps/z.5 M=4 Pop. (Rx only.e-3-4 6 8 Avag ansmi E s /N (db (b 6QAM Fig. Codd PER pfoman. REFERENCES [] J. G. Poakis, Digial Communiaions, 4h diion, MGaw-ill,. []. Sai, G. Kaam, and I. Janlaud, An analysis of ohogonal fquny-division muliplxing fo mobil adio appliaions, Po. IEEE Vhiula Thnology Confn (VTC, Vol. 3, pp. 635-639, Jun 994. [3] D. Falon, S. L. Aiyavisakul, A. Bnyamin-Sya, and B. Eidson, Fquny domain qualizaion fo singl-ai boadband wilss sysms, IEEE Commun., Mag., Vol. 4, No. 4, pp. 58-66, Ap.. [4] F. Adahi, D. Gag, S. Takaoka, and K. Takda, Boadband CDMA hniqus, IEEE ilss Commun., Mag., Vol., No., pp. 8-8, Ap. 5. [5] R. Dinis, P. Silva, and T. Aaujo, Join ubo qualizaion and anlaion of nonlia disoion ffs in MC-CDMA signals, Po. Innaional Confn on Signal and Imag Possing, onolulu, awaii, USA, 6. Pop. (Rx only 5 5 Avag ansmi E s /N (db (b 6QAM Fig. 3 Thoughpu pfoman. [6] K. Takda and F. Adahi, ARQ houghpu pfoman of muliod DS-CDMA wih MMSE ubo qualizaion, Po. IEEE Vh. Thnol. Conf., Dublin, Iland, Ap. 7. [7] K. Takda,. Tomba, and F. Adahi, Muliod DS-CDMA wih join ansmi/iv fquny-domain qualizaion, Po. h IEEE Psonal Indoo and Mobil Radio Communiaions (PIMRC, Tokyo, Japan, Sp. 9. [8] K. Takda and F. Adahi, submid o ICMC.(Should b modifid [9] D. Chas, Cod ombining-a maximum-liklihood doding appoah fo ombining an abiay numb of noisy paks, IEEE Tans, Commun., Vol. 33, No. 5, pp. 385-393, May 985. [] D. Gag and F. Adahi, Pak ass using DS-CDMA wih fquny-domain qualizaion, IEEE Jounal of Sl. Aas in Commun., Vol. 4, No., pp.6-7, Jan. 6. [] D. Raphali and Y. Zaai, Combind ubo qualizaion and ubo doding, Po. IEEE Global Tlommuniaions Confn (GLOBECOM, vol., no. 3-8, pp. 639-643, Nov. 997.