The Fading Number of Memoryless Multiple-Input Multiple-Output Fading Channels 2 R n 2 =0 = 1 = 1.

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1 65 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY R n =0 = n n0 Cn(ix i +(n 0 i)y 0 n) i=0 = n n0 Cn((x i 0 y)i +ny 0 n) i=0 =(x 0 y) n Cni i +(ny 0 n) n n + (x 0 y)(ny 0 n) Cni: i i=0 i=0 i=0 Fro te fact tat te derivatives of H(Z) wit respect to " are uniforly bounded on [0; =] (see [6], also iplied by Teore. of [4] and te coputation of H " (Z)j "=0), we draw te conclusion tat te second coefficient of H(Z) is equal to H 00 (Z)j "== = : [6] O. Zu, I. Kanter, and E. Doany, Asyptotics of te entropy rate for a idden Marov process, J. Statist. Pys., vol., no. 3 4, pp , 005. [7] O. Zu, E. Doany, I. Kanter, and M. Aizenan, Fro finite-syste entropy to entropy rate for a idden Marov process, IEEE Signal Process. Lett.,, vol. 3, no. 9, pp , Sep ACKNOWLEDGMENT Te autors wis to tan te anonyous reviewer for pointing out te Faa Di Bruno forula, wic greatly siplified te proof of Lea.3. REFERENCES [] D. Blacwell, Te entropy of functions of finite-state Marov cains, in Trans. st Prague Conf. Inforation Toery, Statistical Decision Functions, Rando Processes, Prague, Czecoslovaia, 957, pp [] G. Constantine and T. Savits, A ultivariate Faa Di Bruno forula wit applications, Trans. Aer. Mat. Soc., vol. 348, no., pp , Feb [3] R. Garavi and V. Anantara, An upper bound for te largest Lyapunov exponent of a Marovian product of nonnegative atrices, Teor. Cop. Sci., vol. 33, no. 3, pp , Feb [4] G. Han and B. Marcus, Analyticity of entropy rate of idden Marov cains, IEEE Trans. Inf. Teory, vol. 5, no., pp , Dec [5] T. Holliday, A. Goldsit, and P. Glynn, Capacity of finite state cannels based on Lyapunov exponents of rando atrices, IEEE Trans. Inf. Teory, vol. 5, no. 8, pp , Aug [6] P. Jacquet, G. Seroussi, and W. Szpanowsi, On te entropy of a idden Marov process, in Proc. Data Copression Conf., Snowbird, UT, Mar. 004, pp [7] J. Keeny and J. Snell, Finite Marov Cains. Princeton, N.J.: Van Nostrand, 960. [8] R. Leipni and T. Reid, Multivariable Faa Di Bruno forulas, in Electronic Proc 9t Annu. Int. Conf. Tecnology in Collegiate Mateatics [Online]. Available: ttp://arcives.at.ut.edu/ictcm/ep-9. tl#c3 [9] D. Lind and B. Marcus, An Introduction to Sybolic Dynaics and Coding. Cabridge, U.K.: Cabridge Univ. Press, 995. [0] B. Marcus, K. Petersen, and S. Willias, Transission rates and factors of Marov cains, Contep. Mat., vol. 6, pp , 984. [] E. Ordentlic and T. Weissan, On te optiality of sybol by sybol filtering and denoising, IEEE Trans. Inf. Teory, vol. 5, no., pp. 9 40, Jan [] E. Ordentlic and T. Weissan, New bounds on te entropy rate of idden Marov process, in Proc. Inforation Teory Worsop, San Antonio, TX, Oct. 004, pp. 7. [3] E. Ordentlic and T. Weissan, Personal Counication. [4] Y. Peres, Analytic dependence of Lyapunov exponents on transition probabilities, in Lyapunov s Exponents, Proceedings of a Worsop (Lecture Notes in Mateatics). Berlin, Gerany: Springer-Verlag, 990, vol [5] Y. Peres, Doains of analytic continuation for te top Lyapunov exponent, Ann. Inst. H. Poincaré Probab. Statist., vol. 8, no., pp. 3 48, 99. Te Fading Nuber of Meoryless Multiple-Input Multiple-Output Fading Cannels Stefan M. Moser, Meber, IEEE Abstract In tis correspondence, we derive te fading nuber of ultiple-input ultiple-output (MIMO) flat-fading cannels of general (not necessarily Gaussian) regular law witout teporal eory. Te cannel is assued to be noncoerent, i.e., neiter receiver nor transitter ave nowledge about te cannel state, but tey only now te probability law of te fading process. Te fading nuber is te second ter, after te double-logaritic ter, of te ig signal-to-noise ratio (SNR) expansion of cannel capacity. Hence, te asyptotic cannel capacity of eoryless MIMO fading cannels is derived exactly. Te result is ten specialized to te nown cases of single-input ultiple-output (SIMO), ultiple-input single-output (MISO), and single-input single-output (SISO) fading cannels, as well as to te situation of Gaussian fading. Index Ters Cannel capacity, fading nuber, Gaussian fading, general flat fading, ig signal-to-noise ratio (SNR), ultiple antenna, ultiple-input ultiple-output (MIMO), noncoerent. I. INTRODUCTION It as been recently sown in [], [] tat, wenever te atrixvalued fading process is of finite differential entropy rate (a so-called regular process), te capacity of noncoerent ultiple-input ultipleoutput (MIMO) fading cannels typically grows only double-logaritically in te signal-to-noise ratio (SNR). Tis is in star contrast to bot, te coerent fading cannel were te receiver as perfect nowledge about te cannel state, and to te noncoerent fading cannel wit nonregular cannel law, i.e., te differential entropy rate of te fading process is not finite. In te forer case te capacity grows logaritically in te SNR wit a Manuscript received June, 006; revised Marc, 007. Tis wor was supported by te Industrial Tecnology Researc Institute (ITRI), Zudong, Taiwan, under Contract G Te autor is wit te Departent of Counication Engineering, National Ciao Tung University (NCTU), Hsincu, Taiwan (e-ail: stefan.oser@ieee. org). Counicated by K. Kobayasi, Associate Editor for Sannon Teory. Digital Object Identifier 0.09/TIT /$ IEEE

2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY factor in front of te logarit tat is related to te nuber of receive and transit antennas [3]. In te latter case, te asyptotic growt rate of te capacity depends igly on te specific details of te fading process. In te case of Gaussian fading, nonregularity eans tat te present fading realization can be predicted precisely fro te past realizations. However, in every noncoerent syste te past realizations are not nown a priori, but need to be estiated eiter by nown past cannel inputs and outputs or by eans of special training signals. Depending on te spectral distribution of te fading process, te dependence of suc estiations on te available power can vary largely wic gives rise to a uge variety of possible ig-snr capacity beaviors: it is sown in [4], [5], and [6] tat depending on te spectru of te nonregular Gaussian fading process, te asyptotic beavior of te cannel capacity can be varied in a large range: it is possible to ave very slow double-logaritic growt, fast logaritic growt, or even exotic situations were te capacity grows proportionally to a fractional power of log SNR. Siilarly, Liang and Veeravalli sow in [7] tat te capacity of a Gaussian bloc-fading cannel depends critically on te assuptions one aes about te tie-correlation of te fading process: if te correlation atrix is ran deficient, te capacity grows logaritically in te SNR, oterwise double-logaritically. In tis correspondence we will only consider noncoerent cannels wit regular fading processes, i.e., te capacity at ig SNR will be growing double-logaritically. To quantify te rates at wic tis poor power efficiency begins, [], [] introduce te fading nuber as te second ter in te ig-snr asyptotic expansion of cannel capacity. Hence, te capacity can be written as C(SNR) = log( + log( + SNR)) + + o() () were o() tends to zero as te SNR tends to infinity, and were is a constant, denoted fading nuber, tat does not depend on te SNR. Explicit expressions of te fading nuber are nown for a nuber of fading odels. For cannels wit eory, te fading nuber of single-input single-output (SISO) fading cannels is derived in [], [] and te single-input ultiple-output (SIMO) case is derived in [8] and []. For eoryless fading cannels, te fading nuber is nown in te situation of only one antenna at transitter and receiver (SISO) (H) = log + log jhj 0 (H) () in te situation of a SIMO fading cannel (H) = ( ^He )+n R log H 0 log 0 (H) (3) (bot are special cases fro te corresponding situation wit eory), and also in te case of a ultiple-input single-output (MISO) fading cannel [], [] (H )=sup log + log jh j 0 (H ) : (4) Te ost general situation of ultiple antennas at bot transitter and receiver, owever, as been solved so far only in te special case of a particular rotational syetry of te fading process: if every rotation of te input vector of te cannel can be undone by a corresponding rotation of te output vector, and vice-versa, ten te fading nuber as been sown in [], [] to be ( ) = log n 0(n R ) + n R log ^e 0 ( ^e) (5) For a precise definition of te notation used in tis corrspondence, we refer to Section II. were ^e n is an arbitrary constant vector of unit lengt, and were n R denotes te nuber of receive antennas. Suc fading cannels are called rotation-coutative in te generalized sense (for a detailed definition see Section V). In tis correspondence, we will extend tese results and derive te fading nuber of general eoryless MIMO fading cannels. Te reainder of tis correspondence is structured as follows. Before we proceed in Section III to introduce te cannel odel in detail, te following section will clarify our notation. We will ten present te ain result, i.e., te fading nuber of te general eoryless MIMO fading cannel in Section IV. Te corresponding proof is found in Section VII. In Section V, te nown fading nubers of SISO, SIMO, MISO, and rotation-coutative MIMO fading cannels are derived once ore as special cases of te new general result fro Section IV. In Section VI, we investigate te situation of Gaussian fading processes. We will conclude in Section VIII. II. NOTATION We try to use uppercase letters for rando quantities and lowercase letters for teir realizations. Tis rule, owever, is broen wen dealing wit atrices and soe constants. To better differentiate between scalars, vectors, and atrices we ave resorted to using different fonts for te different quantities. Uppercase letters suc as X are used to denote scalar rando variables taing value in te reals or in te coplex plane. Teir realizations are typically written in lowercase, e.g., x. For rando vectors we use bold face capitals, e.g., X and bold lowercase for teir realizations, e.g., x. Deterinistic atrices are denoted by uppercase letters but of a special font, e.g., ; and rando atrices are denoted using anoter special uppercase font, e.g.,. Te capacity is denoted by C, te energy per sybol by E, and te signal-to-noise ratio is denoted by SNR. We use te sortand Ha b for (H a;h a+;...;h b ). For ore coplicated expressions, suc as H a a ; H a+ a+ ;...; H b b, b we use te duy variable ` to clarify notation: H` `. `=a Heritian conjugation is denoted by () y, and () stands for te transpose (witout conjugation) of a atrix or vector. Te trace of a atrix is denoted by tr (). We use to denote te Euclidean nor of vectors or te Euclidean operator nor of atrices. Tat is x t= jx (t) j ; x (6) ax ^w: (7) ^w= Tus, is te axial singular value of te atrix. Te Frobenius nor of atrices is denoted by F and is given by te square root of te su of te squared agnitudes of te eleents of te atrix, i.e., F tr ( y ): (8) Note tat for every atrix F (9) as can be verified by upper bounding te squared agnitude of eac of te coponents of ^w using te Caucy Scwarz inequality. We will often split a coplex vector v up into its agnitude v and its direction v ^v (0) v

3 654 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY 007 were we reserve tis notation exclusively for unit vectors, i.e., trougout te correspondence every vector carrying a at, ^v or ^V, denotes a (deterinistic or rando, respectively) vector of unit lengt ^v = ^V =: () To be able to wor wit suc direction vectors we sall need a differential entropy-lie quantity for rando vectors tat tae value on te unit spere in : let denote te area easure on te unit spere in. If a rando vector ^V taes value in te unit spere and as te density p ^V (^v) wit respect to, ten we sall let ( ^V) 0 log p ^V ( ^V) () if te expectation is defined. We note tat just as ordinary differential entropy is invariant under translation, so is ( ^V) invariant under rotation. Tat is, if is a deterinistic unitary atrix, ten ( ^V) = ( ^V): (3) Also note tat ( ^V) is axiized if ^V is uniforly distributed on te unit spere, in wic case ( ^V) = log c (4) were c denotes te surface area of te unit spere in c = 0() : (5) Te definition () can be easily extended to conditional entropies: if W is soe rando vector, and if conditional on W = w te rando vector ^V as density p ^VjW (^vjw) ten we can define ( ^V j W = w) 0 log p ^VjW ( ^VjW) W = w (6) and we can define ( ^Vj W) as te expectation (wit respect to W) of ( ^V j W = w). Based on tese definitions, we ave te following lea. Lea : Let V be a coplex rando vector taing value in and aving differential entropy (V). Let V denote its nor and ^V denote its direction as in (0). Ten (V) =(V)+ ( ^V jv) +(0 ) [log V] (7) = ( ^V) +(Vj ^V) +( 0 ) [log V] (8) wenever all te quantities in (7) and (8), respectively, are defined. Here (V) is te differential entropy of V wen viewed as a real (scalar) rando variable. Proof: Oitted. We sall write X N (; ) if X 0 is a circularly syetric, zero-ean, Gaussian rando vector of covariance atrix (X 0 )(X 0 ) y =.ByX U([a; b]) we denote a rando variable tat is uniforly distributed on te interval [a; b]. Te probability distribution of a rando variable X or rando vector X is denoted by X or X, respectively. Trougout te correspondence e denotes a coplex rando variable tat is uniforly distributed over te unit circle e Unifor on fz : jzj =g: (9) Wen it appears in forulas wit oter rando variables, e is always assued to be independent of tese oter variables. All rates specified in tis correspondence are in nats per cannel use, i.e., log() denotes te natural logaritic function. III. THE CHANNEL MODEL We consider a cannel wit n T transit antennas and n R receive antennas wose tie- output Y n is given by Y = x + Z : (0) Here x n denotes te tie- input vector; te rando atrix n n denotes te tie- fading atrix; and te rando vector Z n denotes te tie- additive noise vector. We assue tat te rando vectors fz g are spatially and teporally wite, zero-ean, circularly syetric, coplex Gaussian rando vectors, i.e., fz g IID N 0; for soe > 0. Here denotes te identity atrix. As for te atrix-valued fading process f g we will not specify a particular distribution, but sall only assue tat it is stationary, ergodic, of a finite-energy fading gain, i.e., F < () and regular, i.e., its differential entropy rate is finite (f g) li n" n ( ;...; n) > 0: () Furterore, we will restrict ourselves to te eoryless case, i.e., we assue tat f g is independetn and identically distributed (IID) wit respect to tie. Since tere is no eory in te cannel, an IID input process fx g will be sufficient to acieve capacity and we will terefore drop te tie index ereafter, i.e., (0) siplifies to Y = x + Z: (3) Note tat wile we assue tat tere is no teporal eory in te cannel, we do not restrict te spatial eory, i.e., te different fading coponents H (i;j) of te fading atrix ay be dependent. We assue tat te fading and te additive noise Z are independent and of a joint law tat does not depend on te cannel input x. As for te input, we consider two different constraints: a pea-power constraint and an average-power constraint. We use E to denote te axial allowed instantaneous power in te forer case, and to denote te allowed average power in te latter case. For bot cases we set E SNR : (4) Te capacity C(SNR) of te cannel (3) is given by C(SNR) =supi(x; Y) (5) were te supreu is over te set of all probability distributions on X satisfying te constraints, i.e. for a pea-power constraint, or for an average-power constraint. X E; alost surely (6) X E (7)

4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY Specializing [, Teore 4.], [, Teore 6.0] to eoryless MIMO fading, we ave li fc(snr) 0 log log SNRg < : (8) SNR" Note tat [, Teore 4.], [, Teore 6.0] is stated under te assuption of an average-power constraint only. However, since a peapower constraint is ore stringent tan an average-power constraint, (8) also olds in te situation of a pea-power constraint. Te fading nuber is now defined as in [, Definition 4.6], [, Definition 6.3] by ( ) li fc(snr) 0 log log SNRg: (9) SNR" Pria facie te fading nuber depends on weter a pea-power constraint (6) or an average-power constraint (7) is iposed on te input. However, it will turn out tat te eoryless MIMO fading nuber is identical for bot cases. IV. MAIN RESULT A. Preliinaries Before we can state our ain result, we need to introduce tree concepts: Te first concerns probability distributions tat escape to infinity, te second a tecnique of upper bounding utual inforation, and te tird concept concerns circular syetry. ) Escaping to Infinity: We start wit a discussion about te concept of capacity-acieving input distributions tat escape to infinity. A sequence of input distributions paraeterized by te allowed cost (in our case of fading cannels te cost is te available power or SNR) is said to escape to infinity if it assigns to every fixed copact set a probability tat tends to zero as te allowed cost tends to infinity. In oter words tis eans tat in te liit wen te allowed cost tends to infinity suc a distribution does not use finite-cost sybols. Tis notion is iportant because te asyptotic capacity of any cannels of interest can only be acieved by input distributions tat escape to infinity. As a atter of fact one can sow tat every input distribution tat only acieves a utual inforation of identical asyptotic growt rate as te capacity ust escape to infinity. Loosely speaing, for any cannels it is not favorable to use finite-cost input sybols wenever te cost constraint is loosened copletely. In te following we will only state tis result specialized to te situation at and. For a ore general description and for all proofs we refer to [8, Sec. VII.C.3], [, Sec..6]. Definition : Let fx;eg E0 be a faily of input distributions for te eoryless fading cannel (3), were tis faily is paraeterized by te available average power E suc tat X E; E0: (30) We say tat te input distributions fx;eg E0 escape to infinity if for every E 0 > 0 li E" X;E(X E 0 )=0: (3) We now ave te following lea. Lea 3: Let te eoryless MIMO fading cannel be given as in (3) and let fx;eg E0 be a faily of distributions on te cannel input tat satisfy te power constraint (30). Let I(X;E) denote te utual inforation between input and output of cannel (3) wen te input is distributed according to te law X;E. Assue tat te faily of input distributions fx;eg E0 is suc tat te following condition is satisfied: I(X;E) li =: (3) E" log log E Ten fx;eg E0 ust escape to infinity. Proof: A proof can be found in [8, Teore 8, Rear 9], [, Corollary.8]. ) An Upper Bound on Cannel Capacity: In [] and [] a new approac of deriving upper bounds to cannel capacity as been introduced. Since capacity is by definition a axiization of utual inforation, it is iplicitly difficult to find upper bounds to it. Te proposed tecnique bases on a dual expression of utual inforation tat leads to an expression of capacity as a iniization instead of a axiization. Tis way it becoes uc easier to find upper bounds. Again, ere we only state te upper bound in a for needed in te derivation of Teore 7. For a ore general for, for ore ateatical details, and for all proofs we refer to [, Sec. IV], [, Sec..4]. Lea 4: Consider a eoryless cannel wit input s n and output T. Ten for an arbitrary distribution on te input S te utual inforation between input and output of te cannel is upper bounded as follows: I(S; T ) 0(T js) + log + log + log 0 ; +( 0 ) log(jt j + ) + jt j + (33) were ; > 0 and 0 are paraeters tat can be cosen freely, but ust not depend on te distribution of S. Proof: A proof can be found in [, Sec. IV], [, Sec..4]. 3) Capacity-Acieving Input Distributions and Circular Syetry: Te final preliinary rear concerns circular syetry. We say tat a rando vector W is circularly syetric if W L = W e (34) were U([0; ]) is independent of W and were L = stands for equal in law. Note tat tis is not to be confused wit isotropically distributed, wic eans tat a vector as equal probability to point in every direction. Circular syetry only concerns te pase of te coponents of a vector, not te vector s direction. Te following lea says tat for our cannel odel an optial input can be assued to be circularly syetric. Lea 5: Assue a cannel as given in (3). Ten te capacity-acieving input distribution can be assued to be circularly syetric, i.e., te input vector X can be replaced by Xe, were U([0; ]) is independent of every oter rando quantity. Proof: A proof is given in Appendix A. Rear 6: Note tat te proof of Lea 5 relies only on te fact tat te additive noise is assued to be circularly syetric. B. Fading Nuber of General Meoryless MIMO Fading Cannels We are now ready for te ain result, i.e., te fading nuber of a eoryless MIMO fading cannel. Teore 7: Consider a eoryless MIMO fading cannel (3) were te rando fading atrix taes value in n n and satisfies ( ) > 0 (35)

5 656 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY 007 and F < : (36) Ten, irrespective of weter a pea-power constraint (6) or an average-power constraint (7) is iposed on te input, te fading nuber ( ) is given by (37) sown at te botto of te page. Here denotes a rando vector of unit lengt and denotes its probability law, i.e., te supreu is taen over all distributions of te rando unit-vector. Note tat te expectation in te second ter is understood jointly over and. Moreover, tis fading nuber is acievable by a rando vector X = R were is distributed according to te distribution tat acieves te fading nuber in (37) and were R is a nonnegative rando variable independent of suc tat log R U([log log E; log E]) : (38) Proof: A proof is given in Section VII. Note tat even if it igt not be obvious at first sigt it is not ard to sow tat te distribution tat acieves te supreu in (37) is circularly syetric. Tis is in agreeent wit Lea 5. Te evaluation of (37) can be pretty awward ainly due to te first ter, i.e., te differential entropy wit respect to te surface area easure. We terefore will derive next an upper bound to te fading nuber tat is easier to evaluate. To tat goal firstly note tat for an arbitrary constant nonsingular n R n R atrix and an arbitrary constant nonsingular n T n T atrix ( )=( ); (39) see [, Lea 4.7], [, Lea 6.4]. Second, note tat for an arbitrary rando unit vector ^Y n ( ^Y) log c n = log n (40) 0(n R) were c n denotes te surface area of te unit spere in n as defined in (5) and were te upper bound is acieved wit equality only if ^Y is uniforly distributed on te spere, i.e., ^Y is isotropically distributed. Using tese two observations we get te following upper bound on te fading nuber. Corollary 8: Te fading nuber of a eoryless MIMO fading cannel as given in Teore 7 can be upper bounded as follows: ( ) n R log 0 log 0(n R) +infsup ; n R log 0 ( ) (4) were te infiu is over all nonsingular n R n R coplex atrices and nonsingular n T n T coplex atrices. Proof: Using te two observations (39) and (40), we iediately get fro Teore 7 + n R [log j = ] 0 ( j = ) : (4) Te result now follows by noting tat (4) can always be acieved by coosing in (4) to be te distribution wic wit probability taes on te value tat acieves te axiu in (4). Tis upper bound is possibly tigter tan te upper bound given in [, Lea 4.4], [, Lea 6.6] because of te additional infiu over. V. SOME KNOWN SPECIAL CASES In tis section we will briefly sow ow soe already nown results of various fading nubers can be derived as special cases fro tis new ore general result. We start wit te situation of a fading atrix tat is rotation-coutative in te generalized sense, i.e., te fading atrix is suc tat for every constant unitary n T n T atrix t tere exists an n R n R constant unitary atrix r suc tat r L = t (43) were L = stands for as te sae law ; and for every constant unitary n R n R atrix r tere exists a constant unitary n T n T atrix t suc tat (43) olds [, Definition 4.37], [, Definition 6.37]. Te property of rotation-coutativity for rando atrices is a generalization of te isotropic distribution of rando vectors, i.e., we ave te following lea. Lea 9: Let be rotation-coutative in te generalized sense. Ten te following two stateents old. If n is an isotropically distributed rando vector tat is independent of, ten n is isotropically distributed. If ^e; ^e 0 n are two constant unit vectors, ten ^e L = ^e 0 ; ^e = ^e 0 = (44) ( ^e) =( ^e 0 ); ^e = ^e 0 =: (45) Proof: For a proof see, e.g., [, Lea 4.38], [, Lea 6.38]. Fro Lea 9 it iediately follows tat in te situation of rotation-coutative fading te only ter in te expression of te fading nuber (37) tat depends on is Tis entropy is axiized if is uniforly distributed on te surface of te n R -diensional coplex unit spere, wic can be acieved according to Lea 9 by te coice of an isotropic distribution for. Ten according to (4) and (5) : ( ) inf sup ; n R log 0 log 0(n R ) = log n 0(n R ) : (46) ( ) = sup + n R log 0 log 0 ( j ) : (37)

6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY Te expression of te fading nuber (37) ten reduces to (5) ( )=log n 0(n R ) 0 log + nr log ^e 0 ( ^e) (47) we get (H) = log + log jhj 0 log 0 (H) (55) = log + log jhj 0 (H): (56) were ^e is an arbitrary constant unit vector in n. In case of a SIMO fading cannel, te direction vector reduces to a pase ter e 8. Fro Lea 5 we now tat an optial coice of e 8 is circularly syetric, suc tat (37) becoes (H) = ( ^He )+n R log H 0 log 0 (H): (48) Before we continue wit te MISO case, we would lie to rear tat te only ter in (37) tat depends on te distribution of te pase of eac coponent of X is Since we now fro Lea 5 tat is circularly syetric, we can terefore equivalently write : = e : (49) Turning to te MISO case now note tat te distribution of H jh j e is identical to te distribution of e, independently of te distribution of H and. Hence, H jh j e = (e ) = log : (50) Te fading nuber (37) ten becoes (H )=sup log + log jh j 0 log 0 (H j ) (5) VI. GAUSSIAN FADING Te evaluation of te fading nuber is rater difficult even for te usually sipler situation of Gaussian fading processes. However, we are able to give te exact value for soe iportant special cases, and we will give bounds on soe oters. Trougout tis section we assue tat te fading atrix can be written as = + ~ (57) were all coponents of ~ are independent of eac oter and zeroean, unit-variance Gaussian distributed, and were denotes a constant line-of-sigt atrix. Note tat for soe constant unitary n R n R atrix and soe constant unitary n T n T atrix te law of ~ is identical to te law of ~. Terefore, witout loss of generality, we ay restrict ourselves to atrices tat are diagonal, i.e., for n R n T or, for n R >n T =( ~ n (n 0n ) ) (58) = ~ (n 0n )n (59) were ~ is a infn R ;n T ginfn R ;n T g diagonal atrix wit te singular values of on te diagonal. A. Scalar Line-of-Sigt Matrix We start wit a scalar line-of-sigt atrix, i.e., we assue ~ = d were denotes te identity atrix. Under tese assuptions te fading nuber as been nown already for n R = n T =, in wic case te fading atrix is rotation coutative [], []: ( )=g (jdj ) 0 0 log 0(): (60) = sup sup log + log jh j j = 0 (H = ) (5) log + log jh j 0 (H ) (53) Here g () is a continuous, onotonically increasing, concave function defined as sown in (6) at te botto of te page, for, were Ei () denotes te exponential integral function defined as Ei (0x) 0 x e 0t and () is Euler s psi function given by t dt; x > 0 (6) wic can be acieved for a distribution of tat wit probability taes on te value tat acieves te fading nuber in (53). Finally, te SISO case is a cobination of te arguents of te SIMO and MISO case, i.e., using (e ) = log (54) () j= j (63) wit 0:577 denoting Euler s constant. Tis function g () is te expected value of te logarit of a noncentral ci-square rando g () 0 log() 0 Ei (0) + (0) j e 0 (j 0 )! 0 (0)! j(00j)! j= j ; > 0 (); =0 (6)

7 658 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY 007 variable, i.e., for soe Gaussian rando variables fu jg j= IID N (0; ) and for soe coplex constants f j g j= we ave Proposition : Assue n R >n T and a Gaussian fading atrix as given in (57). Let te line-of-sigt atrix be given as were log j= s ju j + jj = g (s ) (64) j= j j j (65) (see [9], [, Lea 0.], [, Lea A.6] for ore details and a proof). We would lie to epasize tat g () is continuous for all 0, i.e., in particular li log() 0 Ei (0) + #0 0 (0) j j= e 0 ( 0 )! (j 0 )! 0 j( 0 0 j)! for all. Moreover, for all and 0 j = () (66) log 0 Ei (0) g () log( + ): (67) A derivation of (67) can be found in Appendix B. We now consider te case were n R n T : Corollary 0: Assue n R n T and a Gaussian fading atrix as given in (57). Let te line-of-sigt atrix be given as Ten = d ( n n (n 0n ) ) : (68) ( )=n R g n (jdj ) 0 n R 0 log 0(n R ) (69) were g () is defined in (6). Proof: We write for te unit vector were 4 n and 4 0 n 0n. Ten were ~ H N (0; n ). Hence = (70) = + ~ = d4 + ~ H (7) ( j ) =( ~ H)=n R log e (7) n R log = n Rg n (jdj 4 ) n Rg n (jdj ) (73) log n 0(n R ) : (74) Here, te equality in (73) follows fro te fact tat d4 + ~ H is noncentral ci-square distributed and fro (64); te inequality in (73) follows fro te onotonicity of g () and is tigt if 4 =, i.e., 4 0 = 0; and te inequality in (74) follows fro (4) and (5) and is tigt if 4 is uniforly distributed on te unit spere in n so tat is isotropically distributed. Te result now follows fro Teore 7. Te case n R >n T is ore difficult since ten (74) is in general not tigt. We will only state an upper bound. Ten = d n (n 0n )n : (75) ( ) n T log + jdj n T + n R log n R 0 n R 0 log 0(n R ): (76) Proof: Tis result is a special case of Proposition 3 and as been publised before in [, Eq. (8)], [, Eq. (6.4)]. B. General Line-of-Sigt Matrix Next we assue Gaussian fading as defined in (57) wit a general line-of-sigt atrix aving singular values d ;...;d infn ;n g. Hence, ~,defined in (58) and (59), is given as ~ =diag d;...;d infn ;n g (77) were jd jjd j jd infn ;n g j > 0. We again start wit te case n R n T. Corollary : Assue n R n T and a Gaussian fading atrix as given in (57). Let te line-of-sigt atrix ave singular values d ;...;d n, were jd jjd jjd n j > 0. Ten ( ) n R g n ( ) 0 n R 0 log 0(n R ) (78) were g () is given in (6) and were = jd j. Proof: A proof is given in Appendix C. Te situation n R >n T is again ore coplicated. We include tis case in a new upper bound based on (4) wic olds independently of te particular relation between n R and n T. Proposition 3: Assue a Gaussian fading atrix as given in (57) and let te line-of-sigt atrix be general wit singular values d ;...;d infn ;n g, were jd jjd jjd infn ;n g j > 0. Ten te fading nuber is upper bounded as follows: ( ) infn R ;n T g log were infn R;n Tg +n R log n R 0 n R 0 log 0(n R ) (79) jd j jd infn ;n g j = infn ;n g + jd j + + jd infn ;n g j : (80) Proof: A proof is given in Appendix D. VII. PROOF OF THE MAIN RESULT Te proof of Teore 7 consists of two parts. First, we derive an upper bound to te fading nuber assuing an average-power constraint (7) on te input. Te ey ingredients ere are te preliinary results fro Section IV-A. In a second part we ten sow tat tis upper bound can actually be acieved by an input tat satisfies te pea-power constraint (6). Since a pea-power constraint is ore restrictive tan te corresponding average-power constraint, te teore follows.

8 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY Because te proof is rater tecnical, we will give a sort overview to clarify te ain ideas. Te upper bound relies strongly on Lea 3 wic says tat te input can be assued to tae on large values only, i.e., at ig SNR te additive noise will becoe negligible so tat we can bound I(X; Y) I(X; X): (8) Tis ter is ten split into a ter tat only considers te agnitude of X and a ter tat taes into account te direction: I(X; X) =I(X; X) +I X; X : (8) For te first ter wic is related to MISO fading we ten use te bounding tecnique of Lea 4. Because Lea 3 only olds in te liit wen E tends to infinity, we introduce an event X > E 0 for soe fixed E 0 0 and condition everyting on tis event. To derive a lower bound on capacity we coose a specific input distribution of te for X = R (83) were te distribution of R is suc tat it acieves te fading nuber of an SIMO fading cannel and were te distribution of is independent of R and will be only specified at te very end of te derivation (it will be cosen to axiize te fading nuber). We ten split te utual inforation into two ters: I(X; Y) =I(R; Y j ) +I( ; Y): (84) Te first ter (alost) corresponds to an SIMO fading cannel wit side-inforation for wic te fading nuber is nown. Te second ter is treated separately. A. Derivation of an Upper Bound In te following we will use te notation R X to denote te agnitude of te input vector X, i.e., we ave X = R. Note tat in tis section we are not allowed to assue tat R is independent of. Fro Lea 3, we now tat te capacity-acieving input distribution ust escape to infinity. Hence, we fix an arbitrary finite E 0 0 and define an indicator rando variable as follows: Let E ; if X E 0 0; oterwise. (85) p Pr [E =]=Pr X E 0 (86) were we now fro Lea 3 tat We now bound as follows: li p =: (87) E" I(X; Y) I(X;E; Y) (88) = I(E; Y) +I(X; Y j E) (89) = H(E) 0 H(E j Y) +I(X; Y j E) (90) H(E)+I(X; Y j E) (9) = H b (p)+pi(x; Y j E =) +(0p)I(X; Y j E =0) (9) H b (p)+i(x; Y j E =)+(0 p)c(e 0) (93) were H b () 0 log 0 ( 0 ) log( 0 ) (94) is te binary entropy function. Here, (88) follows fro adding an additional rando variable to utual inforation; te subsequent two equalities follow fro te cain rule and fro te definition of utual inforation (notice tat we use entropy and not differential entropy because E is a binary rando variable); in te subsequent inequality we rely on te nonnegativity of entropy; and te last inequality follows fro bounding p and fro upper bounding te utual inforation ter by te capacity C for te available power wic conditional on E =0 is E 0. We rear tat even toug C(E 0) is unnown, we now tat it is finite and independent of E so tat fro (87) we ave li E" fh b(p)+(0 p)c(e 0)g =0: (95) We continue wit te second ter of (93) as follows: I(X; Y j E =)=I(X; X + Z j E =) (96) I(X; X + Z; Z j E =) (97) = I(X; X; Z j E =) (98) = I(X; X j E =) + I(X; Z j X;E =) (99) = I(X; X j E =) (00) X = I X; X; E = (0) X = I X; X; = I(X; X je =) + I X; I(X; X;e j E =) + I X; = I(X; Xe j E =) + I X; E = (0) X;E = (03) X;E = (04) X;E = : (05) Here, (97) follows fro adding an additional rando vector Z to te arguent of te utual inforation; te subsequent equality fro subtracting te nown vector Z fro Y; te subsequent two equalities follow fro te cain rule and te independence between te noise and all oter rando quantities; ten we split X into agnitude and direction vector and use te cain rule again; (04) follows fro adding a rando variable to utual inforation: we introduce e tat is independent of all te oter rando quantities and tat is uniforly distributed on te coplex unit circle; and te last equality olds because fro Xe we can easily get bac X and e. We next apply Lea 4 to te first ter in (05), i.e., we coose S = X and T = Xe. Note tat we need to condition everyting on te event E =. We get I(X; Xe j E =) 0( + log 0 ; Xe j X;E = ) + log + log

9 660 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY 007 +(0 ) log X + E = + X E = + (06) were ; > 0, and 0 can be cosen freely, but ust not depend on X. Notice tat fro a conditional version of Lea wit = follows tat ( Xe j X = x;e =) = (e j X = x;e =) + ( X je ; X = x;e =) + [log X jx = x;e =] (07) = log + ( X jx = x;e =) + [log X jx = x;e =] (08) were we ave used tat e is independent of all oter rando quantities and uniforly distributed on te unit circle. Taing te expectation over X conditional on E =we ten yield ( Xe j X;E =) = log + ( X jx;e =) + [log X je = ] (09) = log + ( R j ;R;E =) + log R E = (0) = log + ( j ;R;E =) + log R E = + log je = + [log RjE = ] () = log + ( j ;E =) + [log R j E =] + log E = () were te second equality follows fro te definition of R X; were te tird equality follows fro te scaling property of entropy wit a real arguent; and were te last equality follows because given, is independent of R. Next we assue 0 <<suc tat 0 >0. Ten we define sup x E suc tat ( 0 ) log( X + ) E = log x + 0 log x (3) =(0 ) log( X + ) 0 log X E = +(0 ) log X E = (4) ( 0 ) sup x E log( x + ) 0 log x +(0 ) log X E = (5) =(0 ) +(0 ) log X E = (6) +(0 ) log X E = : (7) Note tat in te first inequality we ave ade use of te fact tat E =, i.e., tat X E 0. Finally, we bound = X E = sup = sup R E = (8) R E = (9) R E = (0) sup E () p were we ave used te fact tat R needs to satisfy te average-power constraint (7) to get te following bound: E R () = p R E = +(0 p) R E =0 (3) p R E = : (4) Plugging (), (7), and () into (06) we yield I(X; Xe j E =) 0log 0 ( j ;E =)0 [log R j E =] 0 log E = + log + log 0 ; +(0 ) log X E = + + sup E p + : (5) Next, we continue wit te second ter in (05): I X; X;E = = X;E = 0 R; ;R;E = (6) = X;E = 0 ; ;R;E = (7) E = 0 ; ;E = : (8) Here, te last inequality follows because conditioning cannot increase entropy and because given and, te ter = does not depend on R.

10 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY Hence, using (8), (5), and (05) in (93), we get I(X;Y) H b (p) +(0 p)c(e 0) 0 log 0 ( j ;E =) 0 [log R j E =]0 log E = + log + log 0 ; +(0) log X E = = li sup I(X; Y) 0 log + log + E (38) E" li E" sup 0 ( j ) 0 log + + sup E p + + E = + sup + n R log + log 0 ; E p H b (p) 0 ; ;E = (9) = H b (p)+(0 p)c(e 0 ) 0 log 0 ( j ;E =) +(n R 0 ) log E = 0 log R E = +(0 p)c(e 0)+(log 0 log E 0 0 ) 0 log + log + E (39) 0 log je = + log + log 0 ; + [log X je =]0 log X j E = + + sup E p + + E = (30) = li E" sup 0 ( j ) 0 log + n R log + log 0 ; = E = 0 ( j ;E =) + n R log je = 0 log + log 0 ; + sup E p H b (p)+(0 p)c(e 0 ) + log 0 log X E = (3) E = 0 ( j ;E =) + n R log E = 0 log + log 0 ; = sup + sup E p H b (p) +(0p)C(E 0)+(log 0 log E 0 0 ) 0 log + log + E (40) + n R log 0 ( j ) 0 log + sup E p H b(p)+(0 p)c(e 0) + (log 0 log E 0 0 ): (3) Here, (30) follows again fro a conditional version of Lea siilar to (07) () wic allows us to cobine te fourt and te last ter in (9); in te subsequent equality we aritetically rearrange te ters; and te final inequality follows fro te following bound: log X E = inf x E = log E 0 + inf log x (33) log (34) log E 0 + (35) were te last line sould be taen as a definition for. Notice tat 0 << (36) as can be argued as follows: te lower bound on follows fro [, Lea 6.7f)], [, Lea A.5f)] because ( ) > 0 and F <. Te upper bound on can be verified using te concavity of te logarit function and Jensen s inequality. Note tat (3) does not depend on te distribution of R anyore, but only on! Hence, we can get an upper bound on capacity by taing te supreu over all possible distributions. Tis ten gives us te following upper bound on te fading nuber: + li E" log 0 ; = sup 0 log + sup E p H b(p) +(0 p)c(e 0 )+(log 0 log E 0 0 ) + log 0 log + log + E (4) 0 ( j ) +n R log 0 log + log( 0 e 0 )+ + 0 log : (4) Here, te first two equalities follows fro te definition of te fading nuber (9); te subsequent inequality fro (3); (40) follows because te paraeters,, and ust not depend on te input distribution (owever, note tat we are allowed to let te depend on E); te subsequent equality follows since te first four ters do not depend on E; and in te last equality we ave used (95) and ade te following coices on te free paraeters and ( )= li E" C(E) 0 log + log + E (37) (E) = log E + log sup [ ] (43)

11 66 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY 007 (E) = (E) e= (44) for soe constant 0. For tis coice, note tat li E" log 0 ; 0 log = log( 0 e0 ) li E" (45) li E" (log 0 log E 0 0 ) = (46) sup E p + =0 (47) li E" log 0 log + log + E = 0 log : (48) (Copare wit [, App. VII], [, Sec. B.5.9].) To finis te derivation of te upper bound, we let go to zero. Note tat! 0 as # 0 as can be seen fro (3). Note furter tat Terefore, we get ( ) sup liflog( 0 e 0 ) 0 log g =0: (49) #0 0( j ) +n R log 0 log : (50) B. Derivation of a Lower Bound To derive a lower bound on capacity (or te fading nuber, respectively) we coose a specific input distribution. Let X be of te for X = R : (5) Here n is assued to be a rando unit-vector tat is circularly syetric, but wose exact distribution will be specified later. Te rando variable R + 0 is cosen to be independent of and suc tat were we coose x in as log R U [log x in; log E] (5) x in log E: (53) Note tat tis coice of R satisfies te pea-power constraint (6) and terefore also te average-power constraint (7). Using suc an input to our MIMO fading cannel we get te following lower bound to cannel capacity: C(E) I(X; Y) (54) = I(R; ; Y) (55) = I( ; Y) +I(R; Y j ) (56) = I( ; Y) +I(R; Ye j ) 0 I(R; Ye j ) + I(R; Y j ) (57) = I( ; Y) +I(R; e ; Ye j ) 0 I(e ; Ye j ;R) 0 I(R; Ye j ) +I(R; Y j ): (58) Here we ave introduced a new rando variable U ([0; ]) wic is assued to be independent of every oter rando quantity. Te last two ters can be rearranged as follows: 0I(R; Ye j ) +I(R; Y j ) = 0(Ye j ) +(Ye j ;R)+(Y j ) 0 (Y j ;R) (59) = 0(Ye j ) +(Ye j ;R)+(Ye j ;e ) 0 (Ye j ;R;e ) (60) = 0I(e ; Ye j ) +I(e ; Ye j ;R): (6) Here te second equality follows because e is independent of everyting else so tat we can add it to te conditioning part of te entropy witout canging its values, and because differential entropy reains uncanged if its arguent is ultiplied by a constant coplex nuber of agnitude. Cobining tis wit (58), we yield C(E) I( ; Y) +I(R; e ; Ye j ) 0 I(e ; Ye j ) = I( ; Y) +I(Re ; Ye j ) 0 I(e ; Ye j ) (6) (63) were te last equality follows because fro Re te rando variables R and e can be gained bac. We continue wit bounding te first ter in (63) I( ; Y) =I( ; Y; Z) 0 I( ; Z j Y) (x ) (64) I( ; Y; Z) 0 (x in ) (65) = I( ; R) 0 (x in ) (66) = I ; = I ; ; R 0 (x in ) (67) + I ; R 0 (x in): (68) Here te first equality follows fro te cain rule; in te subsequent inequality we lower bound te second ter by 0(x in ) wic is defined in Appendix E and is sown tere to be independent of te input distribution X and to tend to zero as x in "; in te subsequent equality we use Z in order to extract R fro Y and ten drop (Y; Z) since given R it is independent of te oter rando variables; and te last equality follows again fro te cain rule. Siilarly, we bound te tird ter in (63) I(e ; Ye j ) I(e ; Ye ; Ze j ) (69) = I(e ; Xe ; Ze j ) (70) = I(e ; Xe j ) +I(e ; Ze j Xe ; ) (7) = I(e ; Xe j ) (7) = I e ; R; e (73)

12 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY = I e ; e 0I e ; e 0 (x in ): (8) + I e ; R Hence, plugging tese results into (63), we get e ; : (74) C(E) I(Re ; Ye j ) +I ; + I ; R 0 I e ; e 0 I e ; R e ; 0 (x in ): (75) We next bound te tird and fift utual inforation ter in (75) I ; R = R 0 R + R = R 0 R + R = R R = R 0 I e ; R 0 R e ; e ; ; e ; ;e (76) 0 R e ; ; ; (77) 0 R 0 R 0 R e ; e (78) (79) (80) =0: (8) Here, te inequality follows fro conditioning tat reduces entropy; and te second last equality olds because we ave assued to be circularly syetric, i.e., destroys te rando pase sift of e. Terefore, we are left wit te following bound: Now, we rewrite te second and tird ter as follows: I ; = 0 + = + = 0 I e ; 0 e e e ;e (83) 0 0 e (84) 0 e (85) were te second equality follows fro (3) wit a coice = e 0 n and fro te fact tat e is independent of all oter rando quantities. Tis leaves us wit C(E) I(Re ; Ye j ) + 0 e 0 (x in ): (86) Next, we let te power grow to infinity E! and use te definition of te fading nuber (9). Since Re is circularly syetric wit a agnitude distributed according to (5), we now fro [, Eq. (08) and Teore 4.8], [, Eq. (6.94) and Teore 6.5], tat Re acieves te fading nuber of a eoryless SIMO fading cannel wit partial side-inforation. In our situation we ave I(Re ; Ye j ) =I(Re ; Re + Z j ) (87) = I(Re ; Re + Z; ) (88) were serves as partial receiver side-inforation (tat is independent of te SIMO input Re ). Note tat a rando vector A is said to contain only partial side-inforation about B if (BjA) > 0, i.e., in our case we need ( j ) > 0 (89) wic is satisfied since we assue tat ( ) > 0 and F < (see [, Lea 6.6], [, Lea A.4]). Hence ( ) li I(Re ; Re + Z j ) + E" 0 e 0 (xin) 0 log + log + E (90) C(E) I(Re ; Ye j ) +I ; = li E" I(Re ; Re + Z j )

13 664 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY log + log + E 0 (x in ) + = ( j ) + 0 e (9) 0 e (9) = e + n R log 0 log 0 ( j ) + 0 e (93) = + n R log 0 log 0 ( j ): (94) Here in (9), we ave used te fact tat our coice (53) guarantees tat (x in ) tends to zero as E!(see Appendix E) and tat we acieve te SIMO fading nuber for a cannel wit input Re and output Re + Z; te subsequent equality follows fro te fading nuber of a eoryless SIMO fading cannel were te receiver as access to soe partial side-inforation [, Eq. (08)], [, Eq. (6.94)]: (HjS) = ( ^He j S) +n R log H 0 log 0 (HjS): (95) Te result now follows by coosing te distribution suc as to axiize te lower bound (94) to te fading nuber. VIII. CONCLUSION We ave derived te fading nuber of a MIMO fading cannel of general fading law including spatial, but witout teporal eory. Since te fading nuber is te second ter after te double-logaritic ter of te ig-snr expansion of cannel capacity, tis eans tat we ave precisely specified te beavior of te cannel capacity asyptotically wen te power grows to infinity. Te result sows tat te asyptotic capacity can be acieved by an input tat consists of te product of two independent rando quantities: a circularly syetric rando unit vector (te direction) and a nonnegative (i.e., real) rando variable (te agnitude). Te distribution of te rando direction is cosen suc as to axiize te fading nuber and terefore depends on te particular law of te fading process. Te nonnegative rando variable is suc tat (38) is satisfied. Tis is te well-nown coice tat also acieves te fading nuber in te SISO and SIMO case and is also used in te MISO case were it is ultiplied by a constant bea-direction. All tese special cases follow nicely fro tis new result. We ave ten derived soe new results for te iportant special situation of Gaussian fading. For te case of a scalar line-of-sigt atrix (68) assuing at least as any transit as receive antennas n R n T we ave been able to state te fading nuber precisely = n R g n (jdj ) 0 n R 0 log 0(n R ) (96) were g () denotes te expected value of a noncentral ci-square rando variable (see (6)). We see tat te asyptotic capacity only depends on te nuber of receive antennas and is growing proportionally to n R log jdj. For a general line-of-sigt atrix, we ave sown an upper bound tat grows lie infn R ;n T g log were is a certain ind of average of all singular values of te line-of-sigt atrix (see (79) and (80)). We would lie to epasize tat even toug all results on te fading nuber are asyptotic results for te teoretical situation of infinite power, tey are still of relevance for finite SNR values: it as been sown tat te approxiation C(SNR) log( + log( + SNR)) + (97) olds already for oderate values of te SNR. Actually, pulling ourselves by our bootstraps, let us consider for te oent tat (97) starts to be valid for an SNR soewere in te range of 30 to 80 db. In tis case log( + log( + SNR)) will ave a value between and 3 nats. Hence, once te capacity is appreciably above +nats, te approxiation (97) is liely to be valid [0], []. Terefore, te fading nuber can be seen as an indicator of te axial rate at wic power efficient counication is possible on te cannel. For a furter discussion about te practical relevance of te fading nuber we refer to [0] and []. APPENDIX A PROOF OF LEMMA 5 Assue tat U([0; ]), independent of every oter rando quantity. Ten I(X; Y) =I(X; Y j e ) (98) = I(Xe ; Ye j e ) (99) = I(Xe ; Xe + Z j e ) (00) = I( ~ X; ~ X + Z j e ) (0) = ( ~ X + Z j e ) 0 ( ~ X + Z j ~ X;e ) (0) = ( ~ X + Z j e ) 0 ( ~ X + Z j ~ X) (03) ( ~ X + Z) 0 ( ~ X + Z j ~ X) (04) = I( X; ~ X ~ + Z): (05) Here te first equality follows because is independent of every oter rando quantity; te tird equality follows because Z is circularly syetric; in te subsequent equality we substitute X ~ = Xe ; and te inequality follows since conditioning reduces entropy. Hence, a circularly syetric input acieves a utual inforation tat is at least as big as te original utual inforation. APPENDIX B DERIVATION OF BOUNDS (67) In tis appendix we will derive te bounds (67) on g (). We start wit te upper bound wic follows directly fro (64) and (65) and fro Jensen s inequality: g (s )= log = log log j= j= j= ju j + jj (06) ju j + jj (07) ( + j j j ) (08) = log( + s ): (09) For te lower bound we also start wit (64) and coose = s and = = =0. Ten we get g (s )= log j= ju j + j j (0)

14 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 7, JULY log ju + j () = g (s ) () = log s 0 Ei 0s : (3) Here, () follows fro dropping soe nonnegative ters in te su; and in te subsequent two equalities we use te definition of g (). APPENDIX C PROOF OF COROLLARY We coose a constant n T n T atrix diag d ;...; as follows: ; ;...; (4) d n d d and ten we note tat for a unit vector =( () ;...; (n ) ) = + ~ = () were ~ H N 0; () n wit () j () j j jd j + + jd n j + j jd j. + ~ + H ~ (5) (n ) (n ) j (n +) j and were n wit. Terefore + + j (n ) j jd j (6) ( j = ) =n R log e () (7) log = log () +g n (were te last equality follows fro (64)) and ence n R log 0 ( j ) () = n R g n j () j + + j (n ) j ( ) (8) 0 n R log e: (9) Te upper bound on te fading nuber now follows fro (39); fro Teore 7 by upper bounding te -ter by log c n ; and fro te additional observations tat g () is a onotonically increasing function, tat and tat j () j + + j (n ) j (0) ( ) = j () j jd j + j (n +) j jd j j () j jd j j (n ) j + + jd n j j (n ) j + + () jd j j (n ) j + + () jd j = jd j j () j + + j (n ) j (3) = jd j = (4) were te inequality follows since jd jjd j jd n j. APPENDIX D PROOF OF PROPOSITION 3 Tis upper bound is based on te upper bound given in Corollary 8 for a coice of = n.ifn R >n T we coose for a a diag ;...; ;b;...;b (5) d d n wit b (6) n T for as given in (80), and wit a suc tat det =, i.e., a (d d n ) b : (7) For suc a coice we note tat so tat = a 0 jaj jaj + N 0; ;...; N 0; ; jd j jd n j N 0;b ;...; N 0;b (8) = b +(n R 0 n T )b (9) = n R : (30) n T Hence, using Jensen s inequality and te fact tat det =we get n R log 0 ( ) n R log 0 log det 0 ( ) (3) = n R log n R n T n =n 0 n R log e: (3) Plugging tis into te upper bound (4) of Corollary 8, we yield n R log 0 log 0(n R)+n R log n R + n T log = n T log If n R n T we coose for wit a suc tat det n T 0 n R log e (33) n T + n R log n R 0 log 0(n R ) 0 n R : = diag =, i.e., For suc a coice we note tat = a () ;...; (n ) (34) a a ;...; (35) d d n a (d d n ) : (36) + N 0; jaj jaj ;...; N 0; (37) jd j jd n j