An Achievable Rate for the MIMO Individual Channel

Transcription

1 A Achievable Rate fo the MIMO Idividual Chael Yuval Lomitz, Mei Fede Tel Aviv Uivesity, Dept. of EE-Systems Eml: Abstact We coside the poblem of commuicatig ove a multiple-iput multiple-output MIMO) eal valued chael fo which o mathematical model is specified, ad achievable ates ae give as a fuctio of the chael iput ad output sequeces kow a-posteioi. This pape exteds pevious esults egadig idividual chaels by pesetig a ate fuctio fo the MIMO idividual chael, ad showig its achievability i a fixed tasmissio ate commuicatio sceaio. I. INTRODUCTION We coside a chael, temed a idividual chael, whee o specific pobabilistic o mathematical elatio betwee the iput ad the output is assumed. This chael is a exteme case of a ukow chael. Achievable ates ae chaacteized usig oly the iput ad output sequeces, which captue the actual a-posteioi) chael behavio. This poit of view is simila to the appoach use uivesal souce codig of idividual sequeces whee the goal is to asymptotically att fo each sequece the same codig ate achieved by the best ecode fom a model class, tued to the sequece. This famewok is a evolutio of those cosidee Shayevitz ad Fede [] ad Eswaa et. al. [] as pesete moe detl i ou papes [3] ad [4], togethe with the elevat backgoud. We will give a bief itoductio below. The settig we coside icludes a sigle ecode eceivig a message to tasmit ad emittig symbols x i X, i,,.., ad a decode eceivig a sequece of symbols y i Y, i,,.., ad attemptig to ecostuct the message. I the peset pape the iput ad output symbols ae eal-valued vectos, i.e. X R t ad Y R. The elatio betwee x [x,..., x ] T ad y [y,..., y ] T is ukow to the ecode ad decode. We coside two commuicatio sceaios: with feedback possibly impefect) ad without feedback. Fo the case i which thee is o feedback the commuicatio system tasmits i a costat ate, ad outage is uavoidable, i.e. oe caot guaatee a small pobability of eo i all cicumstaces. I the case feedback exists, the commuicatio ate may be dyamically adapted ad outage may be peveted. I both cases we assume commo adomess exists i the ecode ad the decode. The esults i the cuet pape exted the pevious esults, yet oly fo the fist case, of tasmissio i a costat ate. The pefomace is measued by a ate fuctio R emp : X Y R epesetig a empiical measue of the achievable ate betwee the chael iput ad chael output, ove chael uses. I examples hee a [3][4], R emp ca be viewed as the mutual ifomatio achieve a cet family of statistical models i the cuet scope, all zeo mea Gaussia chaels), whe the model paametes match the empiical oes. I commuicatio without feedback we say that a give ate fuctio R emp x, y) is achievable with a iput distibutio Qx) if fo lage block size, it is possible to commuicate at ay ate R ad a abitaily small eo pobability is obted wheeve R emp x, y) > R. The commuicatio system is equied to emit blocks with pobability distibutio Qx), which is possible due to the use of adomizatio. By placig this additioal costt we leave aside the questio of adaptig the iput distibutio, so that the cuet famewok attempts at achievig the empiical mutual ifomatio athe tha the empiical capacity. Aothe easo fo the fixed pio is avoidig degeeate systems which may tasmit oly bad sequeces with low o zeo) R emp. This costt is futhe discusse [3], sectio VIII.C. The m esult of this pape is that fo the multipleiput multiple-output MIMO) chael R t R i.e. with t tasmit ad eceive ateas) the ate fuctio defied below is asymptotically achievable, i the fixed ate case: R emp X, Y) log XX ˆRY Y ) XY )XY ) whee the t matix X deotes the chael iput ove symbols, ad the matix Y deotes the output. ˆR XX XT X, ˆRY Y YT Y ad ˆR XY )XY ) [XY]T [XY] ae the iput, the output ad the joit empiical coelatio matices, espectively. This is a geealizatio of the esult ) of [3] whee the ate fuctio R emp log ˆρx,y) was poved to be achievable fo eal valued SISO chael R R ˆρ deotes empiical coelatio). As i [3], the poof is geometically ituitive. The esults easily exted to the complex MIMO case, ad to ate fuctio usig the empiical covaiace athe tha the coelatio), but we focus hee o the simple case. The pape is ogzed as follows: i Sectio II we expl the motivatio fo this ate fuctio ats elatio to the pobabilistic Gaussia chael, i Sectio III we peset i detl the m esult, which is pove i Sectio IV. Sectio V is devoted to commets ad futhe eseach items. We use lowecase boldface lettes to deote vectos, ad uppecase boldface lettes to deote matices. We use the same

2 otatio fo adom vaiables ad thei sample values, ad the distictio should be clea fom the cotext. II. ORIGIN OF THE RATE FUNCTION Coside the chael fom x R t to y R which ae eal valued vectos. Fo the additive white Gaussia oise AWGN) MIMO chael y Hx + v with v N, σ I), ad x N, I) it is well kow that the mutual ifomatio is Ix; y) log I + σ HT H ) see fo example [5][6]. This eflects the maximum achievable ate with the fixed covaiace matix Exx T I, as sometimes temed the ope-loop MIMO capacity, sice equal powe is a easoable choice whe the tasmitte does ot kow the chael. A moe geeal fom of the mutual ifomatio is obted by assumig x, y ae ay joitly Gaussia adom vectos ad witig: hx) log πe covx) 3) hy) log πe covy) 4) hx, y) [ ]) x log πe cov 5) y Theefoe: Ix; y) hx)+hy) hx, y) [ ] covx) covy) log cov [ y]) x 6) whee the factos πe cacel out sice the dimesio of the covaiace matix i the deomiato is the sum of the dimesios i the omiato. The expessio 6) is moe geeal tha ) sice it does ot assume the oise is white, as suitable fo ou pupose sice it expesses the mutual ifomatio though popeties of the iput ad output vectos without usig a explicit chael stuctue. Fo the case of the AWGN MIMO chael it yields the same value as ). Fo the paticula scala case whee x, y ae scalas with vaiaces σx, σ Y ad coelatio facto ρ, Equatio 6) evaluates to Ix; y) log ρ ), as peviously obted fo the SISO case. The empiical ate fuctio we defie ) is a empiical vesio of the mutual ifomatio expessio i 6), except that the covaiace matices ae eplaced by empiical coelatio athe tha covaiace) matices, i.e. we do ot cacel the mea. Whe ˆR XX o ˆR Y Y which leads also to ˆR XY )XY ) ), the ate fuctio will be defied by emovig the colums of X o Y espectively), which ae liealy depedat o the othes, util these detemiats become positive. It is ot impotat which colums ae emoved to beak the liea depedece, due to this fuctio s ivaiace to liea tasfomatio Popety below). Fo the case of Y o X we defie R emp. The ate fuctio has the followig popeties which ae expected fom a empiical metic of the mutual ifomatio : ) No-egativity: R emp X, Y). This is evidet fom the fact R emp X, Y) is the mutual ifomatio betwee two Gaussia vectos with the espective covaiaces. It will also be show i passig as pat of the deivatio i Sectio IV. ) Ivaiace ude liea tasfomatios: Ay ivetible liea matix opeatio o the iput o output fo example, multiplyig ay of the iput o output sigals by a facto, addig sigals, etc) does ot chage R emp X, Y), i.e. R emp XG x, YG y ) R emp X, Y). Poof: Suppose we multiply X ad Y by abitay matices G x,t t ad G y, espectively. Defie X XG x the XX X T X Gx ˆRXX G x XX Gx. Ad likewise fo Y. Sice [X, Y ] [ ] Gx [X, Y] the fom the same cosideatios we will have XY )XY ) G y XY )XY ) G x G y XY )XY ) Gx G y, theefoe the factos cacel out ad R emp X, Y ) R emp X, Y) 3) Symmety: R emp X, Y) R emp Y, X) III. THE MAIN RESULT Theoem No-adaptive, cotiuous MIMO chael). Give the chael R t R, defie the iput ove symbols as a t matix X, ad the output as a matix Y. Let ˆR XX XT X, ˆRY Y YT Y ad ˆR XY )XY ) [XY]T [XY] be the iput, the output ad the joit empiical coelatio matices, espectively. Defie the ate fuctio R emp X, Y) log XX ˆRY Y 7) XY )XY ) The fo evey P e >, a positive defiite t t matix Λ x ad t + thee exists adom ecode-decode p of ate R ove block size, such that the distibutio of the iput sequece is X N, Λ x ) ad fo ay γ < t+ the pobability of eo fo ay message give a iput sequece X ad output sequece Y is ot geate tha P e if: R γ R emp X, Y) + logp e) t / log) logc L) 8) whee C L Γ ) t t / γ) t + + ) t 9) Specifically, fo evey δ > ad γ < thee exists lage eough so that the pobability of eo is ot geate tha P e if: R γ R emp X, Y) δ ) The theoem almost diectly follows fom the ext lemma which we will pove subsequetly:

3 Lemma. Fo ay matix Y, the pobability of R emp X, Y) T whee X is adomly daw X N, Λ x ) is bouded by: P{R emp X, Y) T } C L t / exp γ T ) ) Fo ay γ i the age γ < t+, ad whee C L is defie 9). Note that the boud does ot deped o Λ x. To pove Theoem, the codebook {X m } expr) m is adomly geeated by i.i.d. selectio of each codewod fom the Gaussia matix distibutio N, Λ x ). The commo adomess is the codebook itself. The ecode seds the w-th codewod, ad the decode uses maximum empiical ate decode i.e. chooses: ŵ agmax{r emp X m ; Y)} ) m whee ties ae boke abitaily. By usig Lemma ad the uio boud, the pobability of eo give X w, Y is bouded by: X w, Y) P R emp X m ; Y) R emp X w ; Y)) X w P w) e m w expr) C L t / exp γ R emp X w ; Y)) C L t / exp[r γ R emp X w ; Y))] 3) Theefoe if 8) is satisfied, the P e w) X w, Y) P e, which poves the fist pat of the theoem. The secod pat follows diectly fom the fist pat. Fo ay γ < ad δ > thee is lage eough so that the coditio γ < t+ is satisfied, ad the could be iceased till the edudacy i 8), logpe) t / log) logc L) would be smalle tha δ ote that C L is deceasig i ), theefoe P e w) X w, Y) P e will be satisfief ) is satisfied. IV. PROOF OF LEMMA To pove Lemma we use the Cheoff boud: P{R emp X, Y) T } P{expγR emp X, Y)) expγt )} E expγr empx, Y)) L exp γt ) 4) expγt ) To pove the lemma we eed to calculate γ XX ˆRY Y L E expγr emp X, Y)) E XY )XY ) 5) whee the expected value is take with espect to X. The emde of this sectio is devoted to uppe boudig L. We will fist assume that Λ x I t t, i.e. X N, I), ad the exted to geeal Λ x. Defie Z [Y, X]. We pefom a QR decompositio of X, Y ad Z i ode to obt moe fiedly expessios. As a emide, QR decompositio of a matix A k Q k R k k with Q T Q I ad R uppe tiagula) is pefomed by Gam-Schmidt pocess. We stat fom the left colum of A ad wok ou way to the last oe. At each time we take a colum of A ad split it to the pat which ca be epeseted by a liea combiatio of the colums to the left of it equivaletly, to the colums of Q aleady geeated), ad the iovatio, i.e. the pat which is othogoal to the subspace geeated by the pevious colums. The vecto epesetig the iovatio is omalized, ad becomes the espective colum of Q, ad its powe becomes the diagoal elemet i R. The coefficiets epesetig the pat of the vecto which is i the subspace of pevious colums become the elemets of R above the diagoal. Aothe impotat popety of QR decompositio is that the detemiat of A T A ca be witte i tems of the diagoal elemets i R: A T A R T Q T QR R T R R k i R ii. Now defie the diagoal of the uppe tiagula matix i the QR decompositio of the matices X, Y ad Z espectively to be the vectos a, b ad [c, d]. I.e. if X Q x R x, Y Q y R y ad Z Q z R z the a diagr x ), b diagr y ), ad [c, d] diagr z ). The legths of the vectos c, d ae, t espectively, so that they ovelap with the colums of Y ad X i the matix Z. We have: XX ˆRY Y XT X YT Y t XY )XY ) ZT Z i a i i b i t i c i i 6) Note that the factos cacel out because the matix dimesios ae t ad i the omiato ad t+ i the deomiato. Sice the Gam-Schmidt pocess opeates sequetially fom the fist colum to the last, ad the fist colums of Z ad Y ae equal, we will have b c. Theefoe we ca wite: XX Y Y XY )XY ) i ) 7) Note that a i ad both elate to the same vecto, the i-th colum of X. The atio is the atio betwee the iovatio of the i-th colum of X with espect to the subspace spaed by pevious colums of X aloe omiato) o these colums togethe with the colums of Y deomiato). Obviously fom this easo a i ad theefoe R emp X, Y) - Popety ). The key obsevatio i this deivatio is as follows: coside a sequetial dawig of the colums of X ad calculatio of the factos. Sice the i-th colum of X is chose isotopically adepedetly of the pevious colums, the value of pevious colums does ot affect the distibutio of the iovatios, a i oly the umbe of dimesios i pevious colums does). Usig this obsevatio which we will pove subsequetly, we would be able to beak L epeseted as the expected value of a poduct 7) ito a poduct of expected values equatios 9)-)), ad the poof is completed by a tedious) calculatio of these expected values.

4 To show the idepedece of a i, i peviously daw values, deote by X i m a matix icludig the colums m to i of X, ad by x i the i-th colum. Defie a a uitay matix Q whose fist i colums spa the subspace spaed by the fist i colums of X, its ext colums exted this subspace to cove also the colums subspace of Y, ad the ext i ) colums complete it to a othoomal basis. This matix does ot deped o X t i ad specifically o the colum i. We assume that the colums of Y ae liealy idepedet we will elax this assumptio late). Also, i pobability oe, assumig t +, the colums of X i ae liealy idepedet of each othe ad of the colums of Y. To see this, it is easy to show that the pojectio of each colum i ay diectio othogoal to the subspace aleady spaed by pevious oes icludig Y), is also Gaussia theefoe has pobability to be, as log as thee exists such a othogoal vecto, i.e. the umbe of peviously geeated vectos is smalle tha. Now defie z Q T x i. Sice x i N, I ) also z N, I ). The fist i elemets of z epeset the pojectios of x i to the subspace spaed by pevious colums of X, ad the ext elemets epeset the pojectios to the subspace spaed by colums of Y. So a i collects the eegy of all elemets except the fist i, ad d i collects the eegy of all elemets except the fist i +. To see this fomally, i the Gam-Schmidt pocess the coefficiets of the pojectio of x i o the subspace spaed by X i ae Q i T xi, ad the pojectio itself is Q i Q i T xi, theefoe the iovatio is v i x i Q i Q i T xi. Sice Q i T vi ad Q T i v i Q T i x i we have a i v i Q T v i [ ] Q i T [ ] Q T v i Q i i T z x i ad similaly, i d i x i Q i + Q i + T xi z i+. Theefoe a i, ae idepedet of Y ad the pevious colums of X, ad ca be give by oms ove pats of a Gaussia i.i.d. vecto of legth. Defiig D i E ) γ ) X i E ) γ ) 8) Whee the equality is due to the idepedece show above, we have fo ay k,,..., t: E k i E E [ k i ) γ E [ [ k i E k ) γ E ) γ D k ] d i i ak E d k k ) γ i Xk )] ) γ ) ] X k ) γ ) D k 9) Theefoe by iductio: γ XX ˆRY Y ) γ L E 9) E D i d XY )XY ) i i i ) Now we boud D i usig the peviously defied Gaussia vecto z): ) a γ/ D i E i d E i + h s b) c E a) E h χ ) z i z i+ i+ ji zj ) γ/ ) γ/ + ji+ z j s χ i +) ) γ h e h / Γ + h s ) γ/ ) s i + e s Γ )dsdh i + i + i+)/ Γ ) Γ i + ) }{{} c s + h) γ h s γ) i e s+h ds dh w γ ) v v + w e w/ w v + ) dw dv c v + w w i w ) γ) i e w/ dw } {{ } c w ) γ) i+ v } v + {{ } c v v γ) i dv ) whee i a) we usedepedet Chi-Squaed distibuted h, s, a b) we chaged vaiables fom s, h to w s + h, v s/h, with ivese tasfomatio s v v+ w, h v+ w ad Jacobia J w,v s,h /h s/h s+h h v+) w. The fist itegal i the expessio above evaluates to: ) c w i+ i + Γ ) By defiitio of Γ). The secotegal behaves like v γ) i ea v ad like v γ) i γ) i+ v at v. Theefoe it will exist covege) iff the powe of v ea is lage tha ad at is smalle tha. The fist coditio is γ) i > γ) > i+ γ < i+. The othe coditio always holds sice >. Note that sice the powe of v+ is lage by moe tha tha the powe of v it is positive whe the fist coditio holds). Theefoe we ca boud:

5 c v < v maxv, ) ) γ) i+ v γ) i dv γ) i + + γ) t + + 3) Combiig ), ) ad 3) we obt: Γ ) i+ D i < Γ ) Γ i + ) γ) t + + ) 4) Sice L esults i a ate loss of log L, ad Γ ) i+ is supeexpoetial i, we would like to expess moe explicitly the depedece o. Usig Γt + ) tγt) with t t+ i, i,,.. / we ca obt the boud Γ ) t+ ) / Γ ) t+ 5) theefoe ) / D i < Γ ) γ) t + + ) 6) ad ) t / L D i < Γ ) t γ) t + + ) t i C L t / 7) Substitutig the above ito 4) poves Lemma fo Λ x I. The two assumptios o the paametes of the poblem we have made i ode fo L to be bouded ae a) t + which was eede ode that each ew colum of X would ot be spaed by the pevious colums ad the colums of Y i pobability, ad b) i t : γ < i+ γ < t+, is eeded fo the existece of {D i } t i. Suppose ow that X N, Λ x ). Usig the Cholesky decompositio we ca defie a coloig matix W, W T W Λ x so that X W X w ad X w N, I). Sice by Popety the ate fuctio is ivaiat to a liea tasfomatio of X we would have R emp X w, Y) R emp X, Y), theefoe if Lemma holds with espect to the white sigal X w it also holds with espect to X. With egad to the assumptio that the colums of Y ae liealy idepedet: if they ae ot, the the ate fuctio is defied with espect to a smalle matix Y cotig oly the idepedet colums. Compaig with a full ak Y, the adom vaiables icease i.e. d i ) due to the smalle dimesio of Y, theefoe L L ad the lemma still holds. V. COMMENTS AND EXTENSIONS Compaiso with the SISO case: Compaig Lemma with Lemma 4 of [3] fo the SISO case t, which is pove by a diect calculatio, the boud hee is slightly wose due to the limitatio γ < t+ )/ which stems fom the use of the Cheoff boud. Compaiso with MIMO capacity: The scheme above achieves the mutual ifomatio of a Gaussia MIMO chael but ot its capacity. Achievig the capacity equies adaptatio of the iput distibutio, which fo the kow AWGN chael y Hx + v is pefomed by SVD ad wate pouig [5]. The stegth of the scheme is i the lack of ay assumptios about the pobability distibutio, which make it applicable fo example fo o Gaussia oise o oe that depeds o the tasmitted sigal. Exploitig tempoal coelatio: I the cuet esults, as i pevious oes [3], the ate fuctio depeds o the zeo ode empiical pobability, ad lacks the ability to exploit tempoal coelatio. Howeve the esults ca be used to exploit such coelatio i the SISO o MIMO chael, i a cude way, by applyig the scheme o blocks of k chael uses. The ate fuctio ove blocks is always supeio to the sigle lette case, ad the pealty is a icease i the fixed edudacy. Usig empiical covaiace istead of coelatio: Whe the matices ˆR i ) ae eplaced with the empiical coelatio Ĉ whee ĈX X T X) T X T X)), the deivatio is simila, except pojectio o a additioal dimesio the all-oes vecto) pecedes the othe pojectios. The esults ae the same with a loss of oe dimesio: γ < t+ ad > t + ae equied, ad thee is a small vaiatio i C L. The complex MIMO chael: The esults easily exted to the complex-valued MIMO chael, usig the same techique. The m diffeece is a double umbe of degees of feedom i the deivatio of D i, which doubles the ate compaed to Equatio. Adaptivity: I [3][4] we peseted a commuicatio scheme usig a low ate feedback, which dyamically adapts the tasmissio ate ad achieves the ate fuctios without outage. Such schemes ae of highe pactical iteest. It is possible to show that the adaptive scheme of [3][4] achieves R emp of ) up asymptotically vshig edudacy, ad up to a set of x sequeces havig vshig pobability. REFERENCES [] O. Shayevitz ad M. Fede, Achievig the Empiical Capacity Usig Feedback: Memoyless Additive Models,IEEE Tasactios o Ifomatio Theoy, Vol.55 No.3, Mach 9, pp [] K. Eswaa, A.D. Sawate, A. Sah, ad M. Gastpa, Zeo-ate feedback ca achieve the empiical capacity, IEEE Tasactios o Ifomatio Theoy, Vol.58, No., Jauay [3] Y. Lomitz ad M. Fede, Commuicatio ove Idividual Chaels, axiv:9.473v [cs.it], Ja 9, [4] Y. Lomitz ad M. Fede, Feedback commuicatio ove Idividual Chaels, IEEE Iteatioal Symposium o Ifomatio Theoy ISIT), 9, pp.56-5, Jue 8 9-July 3 9 [5] I. Telata, Capacity of Multi-atea Gaussia Chaels, AT&T Techical Memoadum, Jue 995 [6] A. Goldsmith, S.A. Jafa, N. Jidal, ad S. Vishwaath, Capacity Limits of MIMO Chaels, IEEE Joual o Selected Aeas i Commuicatios, Vol., No. 5, Jue 3