CHANNEL state information (CSI) has a significant impact

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1 Ffty-second Annual Allerton Conference Allerton House, UIUC, Illnos, USA October 1-3, 2014 Ergodc Capacty Under Channel Dstrbuton Uncertanty Sergey Loyka, Charalambos D. Charalambous and Ioanna Ioannou Abstract The mpact of channel dstrbuton uncertanty on the performance of fadng channels s studed. The compound capacty of a class of ergodc fadng channels subject to channel dstrbuton uncertanty s obtaned, for arbtrary nose and nomnal channel dstrbuton. The saddle-pont property s establshed, so that the compound capacty equals to the worst-case channel capacty, whch s characterzed as 1-D convex optmzaton problem. The propertes of worst-case mutual nformaton and channel dstrbuton are studed. Closed-form solutons are obtaned n the asymptotc regmes of small and large uncertanty, and an error floor effect s establshed n the latter case. The known results for the ergodc capacty of the Gaussan MIMO channel under..d. Raylegh fadng are shown to hold under the channel dstrbuton uncertanty as well. I. INTRODUCTION CHANNEL state nformaton CSI) has a sgnfcant mpact on channel performance as well as code desgn to acheve that performance. Ths effect s especally pronounced for wreless channels, due to ther dynamc nature, lmtatons of a feedback lnk, channel estmaton errors etc. [1][2]. When only ncomplete or naccurate CSI s avalable, performance analyss and codng technques have to be modfed properly. The mpact of channel uncertanty has been extensvely studed snce late 1950s [3]-[5]; see [2] for an extensve lterature revew up to late 1990s. Snce channel estmaton s done at the recever Rx) and then transmtted to the transmtter Tx) va a lmted f any) feedback lnk, most studes concentrate on lmted CSI avalable at the Tx end assumng full CSI at the Rx end. There are several typcal approaches to ths problem. In the compound channel model, the channel s unknown to the Tx but s known to belong to a certan class of channels. A member of the channel uncertanty class s selected at the begnnng and held constant durng the entre transmsson, thus modelng a scenaro wth lttle dynamcs channel coherence tme sgnfcantly exceeds the codeword duraton [1][6]). A more dynamc approach s that of the arbtrary-varyng channel, where the channel s allowed to vary from symbol to symbol beng unknown to the Tx [2]. Incomplete CSI at the Tx end can be addressed by assumng that the channel s not known but ts dstrbuton s known to the Tx, the so-called channel dstrbuton nformaton CDI) [1][6]. However, complete knowledge of CDI can be S. Loyka s wth the School of Electrcal Engneerng and Computer Scence, Unversty of Ottawa, Ontaro, Canada, K1N 6N5, e-mal: sergey.loyka@eee.org I. Ioannou and C.D. Charalambous are wth the ECE Department, Unversty of Cyprus, 75 Kallpoleos Avenue, P.O. Box 20537, Ncosa, 1678, Cyprus, e-mal: aoannak@yahoo.gr, chadcha@ucy.ac.cy questoned on the same grounds as complete CSI: when only a lmted sample set s avalable always a practcalty), channel dstrbuton can be obtaned wth lmted accuracy only especally at the dstrbuton tals); lmted feedback lnk dctates quantzaton of the estmated CDI before transmsson, thus ntroducng the quantzaton nose; presence of nose and channel dynamcs makes any estmate naccurate to a certan degree. Ths motvates us to study the mpact of naccurate channel dstrbuton nformaton on system performance. In the context of non-ergodc fadng channels such study has been reported n [7], where the man performance metrc was the outage probablty. It was demonstrated that naccurate CDI lmts the achevable outage probablty: ncreasng the SNR over a certan threshold does not reduce the outage probablty,.e. an error floor effect. The key parameter characterzng the error floor effect s the dstance between the nomnal estmated) and true dstrbutons as measured by the relatve entropy, regardless of any other channel specfcs e.g. nomnal CDI, nose dstrbuton etc.). In the present paper, we carry out a smlar nvestgaton for ergodc settngs,.e. assumng that the channel s subject to an ergodc fadng process so that the man performance metrc s ergodc capacty [1]. However, snce ncomplete naccurate) CDI s assumed, the standard results on ergodc capacty [1][6] do not apply as certan achevable performance have to be demonstrated for the whole class of dstrbutons, not just for a sngle one. We accomplsh ths usng the standard compound channel approach [1][2] - properly extended to the ergodc settng. Ths allows us to establsh the operatonal meanng of the max-mn ergodc mutual nformaton MI), where mn s over all channel dstrbutons n the uncertanty class and max s over all feasble nput dstrbutons, as the largest achevable rate under the CDI uncertanty. Frst, the worst-case ergodc MI s characterzed as a 1- D convex optmzaton problem; ts propertes are studed and asymptotc analytcal solutons small/large uncertanty regmes) are obtaned n closed forms. An error floor effect s establshed n the large-uncertanty regme: the worst-case MI and thus the compound capacty cannot be ncreased by ncreasng SNR but rather more accurate channel estmaton s requred to accomplsh ths. Our analyss of the smalluncertanty regme answers quanttatvely the queston how accurate s the perfect CDI?. Then, an operatonal meanng of the worst-case MI as the largest achevable rate for a gven nput dstrbuton s establshed and the correspondng compound channel capacty s shown to be the max-mn MI where the mn n over class of /14/$ IEEE 779

2 channels and max s over the nput dstrbuton); a number of ts propertes are also establshed. The saddle-pont property s shown to hold, so that the compound channel capacty equals to the worst-case channel capacty, from whch ts gametheoretc nterpretaton follows. These results are further extended to contnuous fadng dstrbutons and an AWGN MIMO channel s consdered subject to any untary-nvarant fadng of whch..d. Raylegh fadng s a specal case). The optmal sgnalng s shown to be sotropc Gaussan, thus extendng the correspondng result n [11] n several drectons from..d. Raylegh to any untary-nvarant fadng; from a sngle fadng channel to a compound channel settng to accommodate channel dstrbuton uncertanty; the same optmal sgnalng s shown to hold under the total as well as per-antenna power constrants, thus demonstratng that no advantage s ganed by tradng off the power among the Tx antennas). II. CHANNEL MODEL Let x and y be the channel nput and output respectvely, and h be the channel state all can be sequences). Assume that the full channel state nformaton CSI) s avalable at the recever but not the transmtter see e.g. [1][6] for a detaled motvaton of ths assumpton) and that the channel nput x and state h are ndependent of each other. For any channel state and nput dstrbuton px), the channel s characterzed by ts nstantaneous) mutual nformaton MI) Ix; y, h) = Ix; y h), where, followng [1][6], we have augmented the output wth the channel state snce t s known at the Rx) and have used the ndependence of x and h. We wll further assume that the channel s subject to an ergodc fadng characterzed by ts probablty dstrbuton f. For a fnte-state channel, h {h 1,..., h m }, f s the probablty of h = h, and I = Ix; y h ) s the nstantaneous) mutual nformaton supported by channel realzaton h under gven nput dstrbuton px); wthout loss of generalty, assume decreasng orderng I 1 I 2... I m unless otherwse ndcated, we assume that not all I are the same). The ergodc mutual nformaton supported by ths channel s Ix; y f) = f I 1) whch s also a functon of f = {f 1...f m }. When f s known to the Tx, ths s also the largest achevable rate for a gven nput dstrbuton px) [1]. Ergodc channel model s sutable n scenaros wth sgnfcant channel dynamcs so that a sngle codeword spans many dfferent channel realzatons and an encoder can take advantage of t [1][6]. However, n many practcal scenaros, complete knowledge of channel dstrbuton f may be not avalable at the transmtter, due to e.g. naccuracy n estmatng f at the recever due to fnte sample sze or estmaton nose); lmted quantzed) feedback lnk quantzaton nose); outdated estmate, so that the true channel dstrbuton f dffers from ts estmate f 0 avalable at the transmtter. To model ths CDI uncertanty naccuracy), consder the scenaro where the transmtter has only partal CDI. Namely, t knows that the true f s wthn a certan dstance of the nomnal estmated) known f 0. We use the relatve entropy as a measure of the dstance between two dstrbutons, so that all feasble dstrbutons f satsfy the followng nequalty: f = {f 1...f m } : Df f 0 ) = f ln f f 0 d, 2) where f 0 = {f } s a nomnal known) dstrbuton and d 0 determnes the sze of the dstrbuton uncertanty set. Smlar approach has been adopted n [7] to characterze the mpact of channel dstrbuton uncertanty on the performance of non-ergodc quas-statc) fadng channels, where the man performance metrcs are outage probablty for a gven target rate) or outage capacty for a gven outage probablty). Whle the value of relatve entropy as a measure of dstance between two dstrbutons s well-known [12], t wll become clear from the present study that d s a crtcal parameter that characterzes the loss n performance due to the channel dstrbuton uncertanty as well. We wll not assume any partcular nose or channel dstrbuton except for examples) so that our results are general and apply to any such dstrbuton. III. WORST-CASE ERGODIC MUTUAL INFORMATION Under a gven px), the worst-case ergodc mutual nformaton for the CDI uncertanty set n 2) s gven by I w = mn Ix; y f) 3) Df f 0) d Its operatonal meanng wll be establshed n the next secton: when the nomnal dstrbuton f 0 and radus d are known at the transmtter, ths s the largest achevable rate under the worst-case fadng channel for a gven px) and s a functon of f 0 and d). The Theorem below gves ts characterzaton as a 1-D convex optmzaton problem. Theorem 1: For a gven nput dstrbuton px) and arbtrary nomnal fadng dstrbuton f 0, the worst-case ergodc mutual nformaton I w n 3) can be expressed as a scalar convex optmzaton problem: I w = max s ln s 0 ) f 0 e I/s + d and the maxmzng s can be found as a unque soluton of the followng equaton F s) = f 0 I s ei/s f ln f 0e I/s 0 e I/s = d 5) f d ln 1. The worst-case mnmzng) fadng dstrbuton s f 4) f = f 0e I/s f 0e I/s, 6) 780

3 so that I w = If d ln 1, s = 0 and f 0I e I/s f 0e I/s 7) f 1...f m 1 = 0, f m = 1, I w = I m 8).e. all the probablty mass s on the weakest channel and the worst-case ergodc MI equals to that of a weakest channel realzaton. If d = 0, then f = f 0 and the correspondng worst-case MI s that under the nomnal dstrbuton: I w = I 0 = f 0I, so that n general I m I w I 0 9) Proof: see Appendx. We now proceed to establsh a number of propertes of F s) n 5), whch reflect on correspondng solutons. Proposton 1: The functon F s) has the followng propertes: 1) F s) s ncreasng: F s) 0, wth strct nequalty unless s = or 0 or all I are the same. 2) Its lmtng values are F ) = 0, F 0 ) = ln 1, so that 3) 0 F s) ln 1 for s 0. Note that d = ln 1 s the threshold radus, beyond whch the worst-case ergodc capacty equals to the pontwse nstantaneous) worst-case capacty and the worst-case fadng dstrbuton puts all the mass on the weakest channel realzaton. A. Asymptotc regmes Let us now study the worst-case MI n 2 asymptotc regmes, where more nsghts can be obtaned. Proposton 2: Consder the small uncertanty regme d 0. The worst-case ergodc MI can be approxmated as follows: I w = I 0 2dσ I + o d) 10) where σi 2 = f 0I 2 I2 0 s the varance of the nstantaneous MI under the nomnal fadng dstrbuton. Proof: Based on the standard tools of asymptotc analyss [14]. Note that, n ths regme, the worst-case MI decreases proportonally to the standard devaton of the nstantaneous MI under the nomnal fadng dstrbuton), the proportonalty coeffcent beng 2d, and that ncreasng I results n smaller f.,.e. weaker channels get larger weghts. Large uncertanty regme: ths corresponds to d ln, whch s consdered n Theorem 1 n 8). Note that n ths regme further ncrease n d beyond ln ) does not result n any decrease n I w, as the lower bound n 9) s already acheved. If I m = 0 for any SNR.e. zero-gan channel realzaton) and d ln, then I w = 0 regardless of the SNR, so that the worst-case MI and thus the compound channel capacty, whch cannot exceed the worst-case MI under the optmal nput dstrbuton) cannot be ncreased by ncreasng the SNR n the large-uncertanty regme,.e. there s an error floor effect nduced by the channel dstrbuton uncertanty. More accurate channel estmaton.e. smaller d) s requred to ncrease the worst-case MI n ths case. B. Propertes of the worst-case channel dstrbuton and MI We study below the propertes of the worst-case MI. Snce the proofs follow mostly n a standard way from Theorem 1, they are omtted due to the page lmt. Proposton 3: The worst-case MI I w d) as a functon of radus d has the followng propertes: 1) I w d) s a convex functon of d, strctly so unless d ln 1. 2) I w d) s a decreasng functon of d, strctly so unless d ln 1, I w d 1 ) > I w d 2 ) d 1 < d 2 < ln 1. 11) 3) Its boundary values are as follows: I w 0) = I 0, I w d ln 1 ) = I m. 12) Proposton 4: The worst-case MI s an ncreasng functon of I, = 1...m, strctly so f d < ln. Proposton 5: Under the assumed nstantaneous MI orderng I 1 I 2.. I m, the normalzed worst-case fadng dstrbuton α = f /f 0 s ncreasng n : α 1 α 2.. α m. If I < I j and d < ln, then α > α j. Corollary 5.1: If the nomnal fadng dstrbuton s unform, f 01 = f 02 =.. =, the worst-case fadng dstrbuton s ncreasng n : f 1 f 2.. f m. If I < I j and d < ln, then f > f j. Corollary 5.2: If f 0 = 0, then f = 0. If d < ln, then f = 0 f and only f f 0 = 0. IV. OPTIMIZING OVER THE INPUT DISTRIBUTION The next step s to optmze the worst-case MI over the nput dstrbuton to obtan the compound channel capacty. The followng Theorem establshes the operatonal meanng of ths max-mn MI. Ths corresponds to exstence of a sngle code operatng over the whole class of fadng dstrbutons. Theorem 2: Consder an ergodc fadng channel, whose dstrbuton f s not known at the Tx, but s known to belong to a convex set S and assume that the set of all feasble nput dstrbutons px) s convex. Its compound channel capacty C c s the same as the worst-case channel capacty C w, = sup nf C c a) px) f S Ix; y f)b) = nf sup f S px) Ix; y f) = C w 13) Proof: The proof s done n 4 steps, as outlned below: 1) Assume frst that S s of fnte cardnalty. In ths case, a) follows from Han s compound channel capacty theorem see theorems and 5 n [8]) by consderng fadng dstrbuton f as a channel state. 2) When S s a convex polyhedron, a) follows from 1) and the fact that any code that works for fnte-cardnalty set {f } also works for ts convex envelope α f. 781

4 3) When S s an arbtrary convex set, evoke 2) and use a sequence of ncreasngly fner nner/outer polyhedral approxmatons as n e.g. [13]. 4) b) follows from Von Neumann mn-max Theorem [9][10]. Applyng ths theorem to the settng n the prevous secton, one obtans the followng. Theorem 3: Consder the compound ergodc fadng channel n 2) when the transmtter knows f 0 and d but not f, and the recever has full CSI. Assume that the set of feasble nput dstrbutons px) s convex and compact e.g. average or maxmum power constrant). The compound ergodc channel capacty n ths settng s gven by C = max px) mn Df f 0) d Ix; y f) = mn max Df f 0) d px) Ix; y f) = C w 14).e. the compound capacty equals to the worst-case channel capacty C w and the saddle-pont property holds for any feasble px) and f, Ix; y f ) C = Ix ; y f ) Ix ; y f) 15) where x denotes the nput under ts optmal dstrbuton p x) and p, f ) s a saddle pont. The nequaltes n 15) have a well-known game-theoretc nterpretaton: the Tx chooses p x) and the adversary nature) chooses f ; nether player can devate from ths optmal strategy wthout ncurrng a penalty. We are now n a poston to obtan the compound channel capacty n the asymptotc regmes. Proposton 6: Consder the large-uncertanty regme d ln. The compound channel capacty n ths regme s gven by C = max I m 16) px).e. desgnng a sngle code for the whole class of fadng channels s equvalent to desgnng a code for a weakest channel realzaton n ths regme. Proof: Follows from 14) and 8). Proposton 7: Consder the small-uncertanty regme as n Proposton 2. The compound channel capacty n ths regme s gven by C = max px) {I 0 2dσ I } + o d) 17) max px) I 0 18) where 2nd approxmaton holds when d 1 ) 2 I0. 19) 2 Proof: Follows from 10) and 14). In fact, 19) answers the queston how accurate s the perfect CDI? : when 19) holds, the CDI uncertanty s neglgble and thus the CDI can be consdered perfect. Note that optmzng desgnng a code for) the nomnal MI I 0 s σ I not optmal n general as t does not necessarly optmze σ I ), but s optmal when uncertanty s neglgble as n 19), so that one can recycle known optmal dstrbutons codes) n ths small-uncertanty regme. On the other hand, one can recycle known dstrbutons codes) for a weakest channel realzaton n the large uncertanty regme. Usng the general nequalty I m I w I 0, one obtans the general bounds on the compound ergodc capacty. Proposton 8: The compound ergodc capacty of a fntestate fadng channel can be bounded as follows max I m C max I 0 20) px) px) and the bounds are tght: the lower bound s attaned n the large uncertanty regme d ln, and the upper bound s attaned n the small-uncertanty regme d 1 2 I 0/σ I ) 2. We would lke to pont out that the above results are general enough to apply to arbtrary nomnal fadng dstrbuton and arbtrary nose not necessarly Gaussan). V. CONTINUOUS FADING DISTRIBUTIONS Here we consder a contnuous fadng dstrbuton. The results follow from the fnte-state case by usng ntegrals nstead of the sums and calculus of varatons to establsh optmalty). In partcular, the worst-case MI can be characterzed as n Theorem 1 wth ntegrals nstead of the sums and a number of ts propertes mmc those for the fnte-state channels. In the asymptotc regmes, one obtans the followng. Proposton 9: Consder the small uncertanty regme d 0. When all moments of Ih) are bounded, the worst-case ergodc MI can be approxmated as follows: I w = I 0 2dσ I + o d) 21) where σ 2 I = f 0 h)i 2 h)dh I 2 0 s the varance of the nstantaneous MI under the nomnal fadng dstrbuton. Large uncertanty regme: ths corresponds to d ln when there s a pont mass at h = h m so that I w = I m f d ln. When there s no such mass, I w I m as d, whch corresponds to 0. A. An example: Gaussan MIMO channel In ths secton, we consder an example of ergodc Gaussan MIMO fadng channel when the nomnal fadng dstrbuton s untary-nvarant. In the specal case of..d. Raylegh fadng, ts capacty has been establshed n [11] and the optmal sgnallng s sotropc Gaussan. Our example extends ths n two drectons: ) we consder a class of fadng channels thus allowng channel dstrbuton uncertanty, and ) we allow the nomnal dstrbuton to be any untary-nvarant one, of whch..d. Raylegh fadng s a specal case. The key result s that the optmal sgnalng s stll sotropc Gaussan, exactly as n [11]. The channel model s y = Hx + ξ 22) where x, y are the nput and output sgnals, ξ s AWG nose, ξ CN0, I), where I s the dentty matrx, and H s the 782

5 channel matrx. Under gven channel dstrbuton fh), ts ergodc capacty, under the total power constrant trr P T, s [11] Cf) = max fh) ln I + HRH + dh 23) trr P T where R = xx + s the covarance of x, ) + denotes Hermtan conjugaton and denotes determnant; we have also used the fact that Gaussan sgnalng s optmal snce the nose s Gaussan. When the channel dstrbuton s uncertan and belongs to the class n 2), the compound capacty becomes C = max mn trr P T Df f 0) d fh) ln I + HRH + dh 24) The followng Proposton characterzes t for a broad class of nomnal fadng dstrbutons. Proposton 10: Consder an ergodc-fadng AWGN MIMO channel as n 22) whose fadng dstrbuton belongs to the class n 2) and assume that the nomnal fadng dstrbuton f 0 H) s rght untary nvarant,.e. f 0 H) = f 0 HU) for any untary U. The compound channel capacty s { C = max s ln s 0 } I + γhh + s 1 f 0 H)dH + d 25) where m s the number of transmt antennas, γ = P T /m s the per-antenna SNR,.e. an optmal covarance R = γi, so that sotropc Gaussan sgnalng s optmal. Ths holds under the total as well as per-antenna power constrants: trr P T or r P T /m, where r s -th dagonal entry of R. Whle ths optmal sgnalng s the same as n the case of..d. Raylegh-fadng channel n [11], the present result extends [11] n three drectons: a class of fadng dstrbutons s consdered, rather than a sngle one, thus allowng fadng dstrbuton uncertanty typcal n wreless communcatons;..d. Raylegh fadng n [11] s extended to any rght-untary-nvarant dstrbuton, of whch any sphercallysymmetrc and thus..d. Ralegh fadngs are just specal cases; the same optmal sgnalng and capacty are shown to hold under the total as well as the per-antenna power constrants; snce the per-antenna power constrant r P T /m mples the total power constrant trr P T but not vce-versa, ths ndcates that nothng s ganed by allowng transmtters to trade-off the power under an ergodc, untary-nvarant fadng. Ths may have mportant applcatons n mult-user systems. VI. APPENDIX: PROOF OF THEOREM 1 The Lagrangan for the optmzaton problem n 3) s L = ) ) f I + λ f ln f d + µ f 1 f 0 26) and the correspondng KKT condtons are L = I + λ ln f ) µ = 0, 27) f f ) 0 λ f ln f d = 0, λ 0, f = 1. 28) f 0 It s straghtforward to see that the problem s convex snce the objectve Ix; y f) n 1) s lnear n f and the constrant n 2) s convex) and the Slater s condton holds for any d > 0), so that the KKT condtons are suffcent for optmalty [9]. Combnng 27) wth the constrant f = 1 one obtans, after some manpulatons, the mnmzng dstrbuton f = f 0e I/λ f, 29) I/λ 0e Usng ths n 26), one obtans, after some manpulatons, the Lagrange dual functon Lλ): ) Lλ) = λ ln f 0 e I/λ + d, λ 0 30) Snce the dualty gap s zero, the problem n 3) s equvalent to ts dual, I w = max Lλ) 31) λ 0 Changng the dual varable s = λ results n 4). To prove 5), let Qs) = s ln ) f 0 e I/s + d and observe that F s) = d Qs). Furthermore, 32) Qs) = F s) 0 33) Ths clearly demonstrates that Qs) s concave and, thus, the problem n 4) s convex strctly so, unless s = 0 or all I are the same, so that the soluton s unque), and that F s) s ncreasng unless s = or 0 ), so that the equaton n 5) has a unque soluton f d ln, whch corresponds to the maxmzer n 4) note that Qs) = 0 F s) = d) and can be easly found numercally usng any sutable algorthm e.g. bsecton or Newton decent method [9]). 5) can also be obtaned from complementary slackness n 28) when λ > 0. If d > ln, then Qs) = d F s) > 0 from Proposton 1) so that s = 0 and 8) follows. The same soluton apples when d = ln. REFERENCES [1] E. Bgler, J. Proaks, and S. Shama, Fadng Channels: Informaton- Theoretc and Communcatons Aspects, IEEE Trans. Inform. Theory, vol. 44, No. 6, pp , Oct [2] A. Lapdoth and P. Narayan, Relable Communcaton Under Channel Uncertanty, IEEE Trans. Inform. Theory, vol. 44, No. 6, Oct [3] L. Dobrushn, Optmal nformaton Transmsson through a channel wth unknown parameters, Radotekhnka Electronka, vol. 4, pp , [4] D. Blackwell, L. Breman, and A. J. Thomasan, The capacty of a class of channels, Ann. Math. Statst., vol. 30, pp , December

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