Ergodic Mutual Information of Full-Duplex MIMO Radios with Residual Self-Interference

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1 Ergodic Muual Informaion of Full-Duplx MIMO Radios wih Rsidual Slf-Inrfrnc Ali Cagaay Cirik, Yu Rong and Yingbo Hua Dparmn of Elcrical Enginring, Univrsiy of California Rivrsid, Rivrsid, CA, 91 Dparmn of Elcrical and Compur Enginring, Curin Univrsiy, Bnly, WA 6, Ausralia {acirik, Absrac W sudy h horical prformanc of a fullduplx mulipl-inpu mulipl-oupu (MIMO) bi-dircional communicaion sysm. W focus on h ffc of h rsidual slf-inrfrnc du o channl simaion rrors and ransmir impairmns. W assum ha h insananous channl sa informaion (CSI) a h ransmiing nods is no known and h CSI a h rciving nods is imprfc. To maximiz h sysm rgodic muual informaion, which is a non-convx funcion of powr allocaion vcors a h nods, a gradin projcion (GP) algorihm is dvlopd o opimiz h powr allocaion vcors. This algorihm xplois boh spaial and mporal frdoms of h sourc covarianc marics of h MIMO links bwn h nods o achiv highr sum rgodic muual informaion. I is obsrvd hrough h simulaions ha h algorihm rducs o a full-duplx schm whn h nominal rsidual slf-inrfrnc is low, or o a half-duplx schm whn h nominal rsidual slf-inrfrnc is high. I. INTRODUCTION This papr concrns radio frquncy (RF) bi-dircional wirlss communicaion sysms, in which wo radios can b usd o communica dircly wih ach ohr. Currnly, all bi-dircional sysms ar half-duplx, which rquirs wo diffrn channls for wo opposi dircions. A full-duplx bi-dircional sysm uss a singl frquncy a h sam im for boh dircions and is wic as spcrally fficin. Th ponial advanags of full-duplx radios ovr half-duplx radios hav rcnly moivad aciv rsarch in svral diffrn aspcs, ranging from informaion hory and signal procssing basd on mahmaical modls o hardwar xprimnaion and ral sysm dmonsraion 1]-9]. A fundamnal nablr for full-duplx radios is known as h slf-inrfrnc cancllaion (SIC). Whn a full-duplx radio ransmis, i causs slf-inrfrnc which mus b cancld saisfacorily. SIC can b don by diffrn mhods, a diffrn sags, and o diffrn dgrs, along h rciving chain of a full-duplx radio. Rcn xprimnal rsuls 1]- 3] show ha h simulanous ransmission and rcpion in h sam frquncy band is possibl using advancd annna dsigns and passiv/aciv cancllaion chniqus. Th incrasd dgrs of frdom offrd by mulipl-inpu mulipl-oupu (MIMO) sysms lad o a rang of mhods for slf-inrfrnc cancllaion. Mos of hs mhods can b calld ransmi bamforming chniqus 3]-8]. All of h mhods 1]-8] hav hir limiaions, and som lvl of rsidual slf-inrfrnc is sill xpcd for all SIC mhods known o da. In his papr, w assum ha h rsidu slf-inrfrnc is low nough for furhr basband procssing. W will focus on a horical prformanc of h full-duplx radios undr h ffc of h rsidual slf-inrfrnc. W will also rfr o rsidual slf-inrfrnc simply as slf-inrfrnc. Paricularly, w dvlop an algorihm o maximiz a lowr bound on h rgodic muual informaion of a full-duplx bi-dircional MIMO sysm undr a ransmir disorion modl for fas fading channls whr h insananous CSI is no known a h ransmirs and imprfcly known a h rcivrs. W dvlop a Gradin Projcion (GP) mhod o solv his non-convx opimizaion problm. Th mos rlvan prior work is 9]. A main diffrnc bwn his papr and ha on is ha w considr fas fading channls and hy considrd slow fading channls 1. No ha unlik slow fading channls assumd in 9], fas fading channl dos no rquir any insananous CSI fdback from h rcivr. In h absnc of insananous CSI a h ransmiing nods, h knowldg of som saisical propris of h CSI is ncssary for dsigning opimal powr schduls. W will aim o opimiz an rgodic sysm muual informaion wih rspc o h saisical disribuions of h CSI. I is shown hrough numrical simulaions ha a a high slf-inrfrnc powr lvl, h opimal powr schdul rducs o h half-duplx mod and a a low slf-inrfrnc powr lvl, h opimal powr schdul swichs o h full-duplx mod. Th following noaions ar usd in his papr. Marics and vcors ar dnod by bold capial and lowrcas lrs, rspcivly. For marics and vcors, ( ) T and ( ) H dno ranspos and conjuga ranspos, rspcivly. E H { } sands for h saisical xpcaion wih rspc o h channl marix H; I N dnos an N N idniy marix; r{ } sands for marix rac; is h drminan; is h Euclidan norm of a vcor and h Frobnius-norm of a marix; ( ) dnos h firs ordr drivaiv; diag{a 1,,a n } dnos a diagonal marix wih h diagonal lmns givn by a 1,,a n. CN ( μ, σ ) dnos complx Gaussian disribuion wih man μ and varianc σ. W will also rfr o full-duplx as FD and half-duplx as HD. 1 Unlik his work which only considrs h rsidual ransmir disorion, h auhors in 9] considr boh ransmir and rcivr disorion.

2 Fig. 1. Sysm Modl of Bi-dircional Full-Duplx MIMO Sysm. II. SYSTEM MODEL In his scion, w dscrib h sysm modl of a FD bidircional MIMO sysm bwn wo nods. Th signals mniond blow ar dfind in complx basband. W considr MIMO wirlss sysms, whr wo nods ar quippd wih mulipl annnas and xchang informaion simulanously in a wo-way communicaion. W assum ha ach nod has h sam numbr of ransmi and rciv annnas, which is dnod by N. W pariion h daa ransmission priod undr considraion or conrol ino wo normalizd quallngh im slos. Th rason of his pariion will b clar in h simulaion rsuls. As illusrad in Fig. 1, h rcivr i {1, } rcivs signals from boh ransmirs via MIMO channls H ij C N N, (i, j) {1, }. All h channl marics ar assumd o b random and indpndn, i.., h nris of ach marix ar indpndn and idnically disribud (i.i.d.) circular complx Gaussian variabls wih zro man and uni varianc. W adop h channl modl usd in 4], ], 6], 7], whr h simaion rror is modld as ΔH ij = H ij H ij, whr H ij and ΔH ij ar uncorrlad, and h nris of ΔH ij ar zro man circularly symmric complx Gaussian wih varianc σ,ij. W will assum ha h channl marics rmain consan ovr wo conscuiv im slos, bu chang randomly ovr an inrval of many mulipls of wo im slos. Pracical RF ransmirs suffr from h prsnc of impairmns rsuling from non-linar disorions in h powr amplifir, phas nois, or from IQ-imbalanc. Th impac of hs impairmns can b rducd hrough calibraion and compnsaion algorihms. Howvr, du o paramr simaion rrors and h mismach bwn h physical RF chain and h modl of h compnsaion algorihm, som rsidual ransmir disorion sill prsiss. Th masurmn rsuls by ] indica ha an i.i.d. addiiv Gaussian nois modl accuraly dscribs h sum of all such rsidual ransmir impairmns. Furhrmor, h auhors assum sufficin dcoupling of h ransmir RF chains such ha h rlvan impairmns ar saisically indpndn across ransmi annnas. Th rsuling Gaussian modl for h ih ransmir nois is givn by c i () CN (,σ I N ), i =1,. Th N 1 rcivd signal vcor a h ih rcivr can b wrin as y i ()= ρ i H ii (x i ()+c i ()) + η i H ij (x j ()+c j ()) + n i () = ρ i Hii x i ()+ ρ i ΔH ii x i ()+ ρ i H ii c i () + η i Hij x j ()+ η i ΔH ij x j ()+ η i H ij c j () + n i (), i,j {1, } and j i (1) whr ρ i dnos h avrag powr gain of h ih ransmirrcivr link, η i dnos h avrag powr gain of h slfinrfrnc channl, x i () C N, i = 1, is h signal vcor ransmid by nod i wihin im slo and is Gaussian disribud wih zro man and covarianc marix E { x i ()x i () H} = Q i (), and n i () C N is h addiiv whi Gaussian nois (AWGN) vcor wih zro man and idniy covarianc marix E { n i ()n i () H} = I N. W assum ha n i () is indpndn of x i (). Th rcivr i {1, } knows h inrfring signal x j () from ransmir j {1, }, j i, so h slf-inrfrnc rm η Hij i x j () can b subracd from y i () 9]. Th inrfrnc canclld signal can hn b wrin as whr ỹ i () = ρ i Hii x i ()+v i () () v i ()= ρ i ΔH ii x i ()+ ρ i H ii c i ()+ η i ΔH ij x j () + η i H ij c j ()+n i () (3) is h oal nois in ỹ i (). Th covarianc marix of v i () can b wrin as Σ i () =ρ i σ,iir {Q i ()} I N + ρ i σ Hii HH ii + σ,iini N + η i σ,ijr {Q j ()} I N +η i σ Hij HH ij +σ,ijni N + I N, i,j {1, } and j i (4) { } whr w hav usd h idniy of E ΔHij ΔHij AΔH H ij = σ,ij r{a}i N, whr h nris of ΔH ij ar i.i.d. wih CN(,σ,ij ) and A CN N is a known marix. III. ACHIEVABLE RATES In his scion, w formula h rgodic muual informaion xprssion for h FD bi-dircional MIMO sysm. As a rsul of h channl simaion rrors and ransmir impairmns in (3), h nois v i () is gnrally non-gaussian. To h bs of our knowldg, h xac muual informaion of MIMO channls wih channl simaion rrors is sill an opn problm vn for poin-o-poin MIMO sysms 11]. Howvr, assuming v i () as Gaussian, which is h wors nois disribuion from h prspciv of muual informaion, w can obain h lowr bound 11], which was also usd in 9]. Th lowr bound of h rgodic sum muual informaion of h sysm avragd ovr wo im slos can b wrin as Ī (Q 1, Q ) () = 1 { IN E Hii, H ij log + ρ i Hii Q i () H H Σ ii i () 1

3 whr Q i Q T i (1), QT i ()] T, i =1,. To driv a closd-form xprssion for h rgodic sum muual informaion (), w us h igndcomposiion of Q i (), which can b wrin as Q i () =U i ()D i ()U i () H, whr U i () is h uniary marix of ignvcors, and D i () = diag {d i1 (),d i (),...,d in ()} is a diagonal marix of all ignvalus. For convninc, w will us h column vcors d i () d i1 (),d i (),...,d in ()] T, i = 1,. Sinc H ij, (i, j) {1, } has i.i.d. Gaussian nris and U j () is uniary, h saisics of Hij U j () is idnical o ha of H ij 1, Lmma ]. W can hn rwri () as Ī (d 1, d ) = 1 { IN E Hii, H ij log +ρ i Hii D i () H H Σ ii i () 1 = 1 E Hi {log Hi Λ i () H H i + I N E Hi { log Hi Λi () H H i + I N }] whr H i = Hii, H ] ij, d i d i (1) T, d i () T ] T, i =1, c i () ρ i σ,ii1 T N d i ()+ρ i σ σ,iin + η i σ,ij1 T N d j () + η i σ σ,ijn +1, i,j {1, } and j i Σ i () ρ i σ,iir {D i ()} I N + ρ i σ Hii HH ii + σ,iini N + η i σ,ijr {D j ()}I N +η i σ Hij HH ij +σ,ijni N + I N, i,j {1, } and j i { } Λ i () diag λ T 1,i(), λ T,i() i =1,. } Λ i () diag { λt 1,i(), λ T,i() i =1,. d i ()+σ 1 N λ 1,i ()=ρ i c i () σ λ 1,i ()=ρ i c i () 1 N., λ,i () =η i σ c i () 1 N. Hr 1 N is an N 1 column vcor of ons. Bcaus of h spac limiaion, w omi h xac closd form xprssion of (6) which can b found in h journal vrsion of his papr. For simpliciy, hrin w considr a much simplr closd form xprssion of Ī(d 1, d ) which is shown o b an accura approximaion vn for sysms wih a small numbr of annnas 13]. This simplificaion is basd on an asympoical form of Ī(d 1, d ) whn N as proposd in 13]. Applying h rsul in 13], h sum rgodic muual informaion in (6) can b approximad as N 1+Nαi,1 ()λ ik Ī (d 1, d ) τ() log + N log ( αi, () α i,1 () 1+Nα i, () λ ik ) ] + N (α i,1 () α i, ()) log whr λ ik Λ i ()] k,k and λ ik Λ i () ] k,k, k = 1,...,N dno h (k, k)h lmn of marix Λ i () and (6) (7) Λ i (), rspcivly. Hr <α i,1 (),α i, () < 1 saisfis h following nonlinar quaion α i,1 ()+ α i, ()+ N N α i,1 ()λ ik =1. (8) Nα i,1 ()λ ik +1 α i, () λ ik =1. (9) Nα i, () λ ik +1 W can us h biscion mhod o compu α i,1 (), sinc h lf hand sid of (8) is monoonically incrasing funcions of α i,1 (). Sam argumn also holds for α i, (). IV. MAXIMIZATION OF THE SUM ERGODIC MUTUAL INFORMATION In his scion, w aim a maximizing h sum rgodic muual informaion () by choosing h ransmi covarianc marics Q i (), (i, ) {1, } subjc o pr nod avrag powr consrains 1 =1 r {Q i()} P i, whr P i is h avragd ransmi powr from h ih ransmir. No ha w considr fas fading channls in which h insananous CSI is assumd o b unknown a h ransmiing nods. Whn h knowldg of h insananous CSI is absn, saisical propris of h CSI is ncssary for dsigning opimal powr schduls. Paricularly, w will dsign h powr schdul o maximiz an rgodic sysm muual informaion which is avragd ovr h saisical disribuion of h channl marics. Th opimizaion problm can b formulad as max Ī(d 1, d ) () d 1,d s.. d i 1 =P i, i =1, (11) d i, i =1, (1) whr Ī(d 1, d ) is givn in (7). Hr. 1 dnos h sum norm (or l 1 norm) of a vcor. For a vcor x, x mans ha ach nry of x is nonngaiv. W now dvlop a numrical algorihm o obain a locally opimal soluion o h problm ()-(1). by mploying GP mhod. Thr ar wo imporan sps in h GP algorihm: h compuaion of h gradin of h objciv funcion, and h projcion of h updad opimizaion variabl ono h convx s spcifid by consrain funcions. To apply h GP mhod o solv h problm ()-(1), w firs ak gradin sps for d 1 and d, and hn projc h updad d 1 and d ono h consrain s spcifid by (11) and (1). Th gradin of h objciv funcion () wih rspc o d lm (), l = 1,, m =1,...,N, =1,, is givn by Ī (d 1, d ) = 1 N Nαi,1 ()λ ik d lm () 1+Nα i=1 i,1 ()λ ik Nα i,() λ ] ik (13) 1+Nα i, () λ ik whr λ ik and λ ik ar dfind a h boom of h following pag. L us firs considr h gradin sps of h ih ransmirrcivr pair, i {1, }, and dno h N 1 vcor of

4 Ī(d 1,d ) d i1(1) ] T gradin as g i,..., Ī(d1,d) d in (). Thn aking a sp along h posiiv gradin dircion, h powr allocaion vcor is updad as ˆd i = d i +sg i,i=1, whr s is a scalar of sp siz. Th nx sp of h GP algorihm is o projc ˆd i ono h fasibl rgion of powr vcor consrains (11)-(1). Th projcion opraion is basically sarching for a poin d i in h rgion of (11)-(1), which has a minimum Euclidan disanc o h poin ˆd i. Thus, h opimizaion problm for h projcion opraion can b wrin as min di ˆd i (14) d i s.. d i 1 =P i, di, i =1,. () Th problm (14)-() is convx and can b fficinly solvd by h Lagrang muliplir mhod. I urns ou ha h problm (14)-() has a war-filling soluion which is givn ˆdik μ ] +, k = 1,...,N, i = 1,, whr by d ik = μ is h Lagrang muliplir, and for a ral scalar x, x] + max{x, }. Th Lagrang muliplir μ can b obaind by subsiuing d ik back ino () and solving h nonlinar quaion N ˆdik μ ] + =Pi, i =1,. Th lf hand sid of his quaion is a picwis linar funcion and monoonically dcrasing wih rspc o μ, so i can solvd using h biscion mhod. Afr compuing h wo mos imporan sps of h GP algorihm, h dails of h rs of h GP algorihm can b found in 9]. V. SIMULATION RESULTS In his scion, w sudy h prformanc of h proposd FD MIMO bi-dircional communicaion sysm hrough numrical simulaions as a funcion of h avragd SNR, h nominal inrfrnc-o-nois raio (INR), h channl simaion rrors σ,ij and h ransmir impairmns σ.for simpliciy, w considr σ,ij = σ, i, j {1, }, ρ i = ρ and η i = η and P i = N, i = 1,. Thus, w hav h sam avragd SNR for all dsird links SNR i =SNR= ρn, i =1, and h sam nominal INR for all inrfring links INR i =INR=ηN, i =1,. Th Armijo paramrs ar slcd as σ =.1, θ =., and h sopping hrshold of h GP algorihm is chosn as ɛ =. To show h FD Approx FD1 Approx HD INR (db) Fig.. Ergodic muual informaion comparison of FD, FD1, and HD sysms vrsus INR. HrN =3, SNR = db, σ =.1, σ = db. imporanc of using wo im slos, w compar our FD sysm using wo daa ransmission slos (FD) wih h FD sysm using only on daa ransmission slo (FD1). Sinc h GP algorihm only convrgs o a locally opimal soluion, w us h oupu of h HD schm as h iniializaion of h FD schm. In our firs xampl, w invsiga h impac of INR on h rgodic muual informaion of h FD, FD1, and HD schms. As xpcd, i can b obsrvd from Fig. ha h HD schm is invarian o INR. For h low-o-mid valus of INR, h FD schm has h FD sysm bhavior and i swichs o h HD schm a h high valus of INR. Th FD1 schm prforms similar o h FD schm a low-omid valus of INR, bu is prformanc drops blow ha of h HD schm for largr valus of INR. Th us of wo disinc daa im slos givs h frdom o swich o h HD signaling whn h powr of h slf-inrfrnc channl is high (whr h HD schm is opimal), whil h FD1 sysm forcs FD signaling a ach im slo, rgardlss of h srngh of h slf-inrfrnc channl. This is similar o FD sysms for slow fading channls 9]. In h scond xampl, w invsiga h rol of channl simaion rrors on h lowr bound of h rgodic muual informaion (7). I can b sn from Fig. 3 ha as h channl simaion rrors incrass, h rgodic muual informaion of ρ i c i() σ,ii dik ()+σ, l = i and k m and k N ρ i λ c i() ρ i c i() σ,ii dik ()+σ, l = i and k = m and k N ik = ρ i η i c i () σ,ii σ, l = i and k>n. ρ i η i c i () σ,ij dik ()+σ, l i (l = j) and k N ηi c i() σ,ij σ, l i (l = j) and k>n ρ i c i() σ,ii σ, l = i and k N λ ρ i η i c i () σ,ii ik = σ, l = i and k>n ρ i η i c i () σ,ij. σ, l i (l = j) and k N ηi c i() σ,ij σ, l i (l = j) and k>n

5 σ =.1, FD Approx 3 18 σ =., FD Approx σ =.3, FD Approx σ =.4, FD Approx σ =., FD Approx σ =.6, FD Approx σ =.1, HD Approx σ =., HD Approx σ =.3, HD Approx σ =.4, HD Approx σ =., HD Approx σ =.6, HD Approx σ = 4dB, FD Approx σ = db, FD Approx σ = db, FD Approx σ = db, FD Approx σ =db, FD Approx INR (db) SNR (db) Fig. 3. Ergodic muual informaion comparison of h FD and HD sysms wih diffrn channl simaion rrors vrsus INR. Hr N =, SNR = db, σ = db. Fig.. Ergodic muual informaion comparison of h FD sysm wih diffrn σ valus vrsus SNR. HrN =3, INR = db, σ =.1. 3 INR= db, FD Approx INR= 4dB, FD Approx INR= 6dB, FD Approx INR= 7dB, FD Approx HD SNR (db) Fig. 4. Ergodic muual informaion comparison of h FD and HD sysms vrsus SNR. HrN =3, σ =.1, σ = db. boh h FD and HD sysms dcrass. Th gap bwn rgodic muual informaion curvs diminishs as h σ incrass. In our hird xampl, w xamin h rgodic muual informaion of h FD and HD sysms vrsus SNR for various fixd valus of INR. I can b obsrvd from Fig. 4 ha a low INR, h sysm opras in h FD mod for all valus of SNR, sinc SNR mosly dominas INR. Ahigh INR, h sysm opras in h HD mod a low valus of SNR (sinc INR dominas SNR), bu swichs o h FD mod as SNR incrass, sinc SNR sars o domina INR. In our las xampl, w xamin h rgodic muual informaion of h FD sysms vrsus SNR for various valus of σ. I can b obsrvd from Fig. ha as σ dcrass, h rgodic muual informaion incrass and h gap bwn h curvs diminishs. VI. CONCLUSION In his work, w sudid h maximizaion of h asympoic rgodic muual informaion for FD MIMO bi-dircional sysm ha suffrs from a rsidual slf-inrfrnc. Sinc h globally opimal soluion is difficul o obain du o h nonconvx naur of h problm, a gradin projcion algorihm is dvlopd o opimiz h powr allocaion vcors a wo nods wih h knowldg of saisical CSI a h ransmirs. I is shown hrough numrical simulaions ha a a high slfinrfrnc powr lvl, h opimal powr schdul rducs o h HD mod and a a low slf-inrfrnc powr lvl, h opimal powr schdul swichs o h FD mod. REFERENCES 1] J. I. Choi, M. Jain, K. Srinivasan, P. Lvis, and S. Kai, Achiving singl channl, full duplx wirlss communicaion, in Proc. Mobicom, pp. 1-1, Sp.. ] M. Duar and A. Sabharwal, Full-duplx wirlss communicaions using off-h-shlf radios: Fasibiliy and firs rsuls, in Proc. Asilomar Conf. Signals, Sys. Compurs, pp. 8-6, Nov.. 3] Y. Hua, P. Liang, Y. Ma, A. Cirik and Q. Gao, A mhod for broadband full-duplx MIMO radio, IEEE Signal Procssing Lrs, Vol. 19, No. 1, pp , Dc 1. 4] T. Riihonn, S. Wrnr, and R. Wichman, Miigaion of loopback slfinrfrnc in full-duplx MIMO rlays, IEEE Trans. Signal Procss., vol. 9, no. 1, pp , Dc. 11. ] T. Riihonn, S. Wrnr, and R. Wichman, Rsidual slf-inrfrnc in full-duplx MIMO rlays afr null-spac projcion and cancllaion, in Proc. 44h Annual Asilomar Confrnc on Signals, Sysms, and Compurs, Novmbr. 6] P. Lioliou, M. Vibrg, M. Coldry, and F. Ahly, Slf-inrfrnc supprssion in full-duplx MIMO rlays, in Proc. Asilomar Conf. Signals Sys. Compu., (Pacific Grov, CA), pp , Oc.. 7] B. Chun, E. Jong, J. Joung, Y. Oh, and Y. H. L, Pr-nulling for slf-inrfrnc supprssion in full-duplx rlays, in Proc. Asia- Pacific Signal and Informaion Procssing Associaion Annual Summi Confrnc (APSIPA ASC), Sapporo, Oc. 9. 8] Y. Hua, An ovrviw of bamforming and powr allocaion for MIMO rlays, in Proc. IEEE Miliary Commun. Conf., (San Jos, CA), pp , Nov.. 9] B. P. Day, D. W. Bliss, A. R. Margs, and P. Schnir, Full-duplx bidircional MIMO: Achivabl ras undr limid dynamic rang, IEEE Trans. Signal Procss., vol. 6, no. 7, pp , July 1. ] C. Sudr, M. Wnk, and A. Burg, MIMO ransmission wih rsidual Tx-RF impairmns, in Inrnaional ITG Workshop on Smar Annnas, pp , Fb.. 11] B. Hassibi and B. M. Hochwald, How much raining is ndd in mulipl-annna wirlss links? IEEE Trans. Inf. Thory, vol. 49, pp , Apr. 3. 1] I. E. Tlaar, Capaciy of muli-annna Gaussian channls, Europ. Trans. Tlcommun., vol., pp. 8-9, Nov ] A. Lozano and A. M. Tulino, Capaciy of mulipl-ransmi muliplrciv annna archicurs, IEEE Trans. Inf. Thory, vol. 48, pp , Dc..