The Fading Number of a Multiple-Access Rician Fading Channel

Transcription

1 XTNDD VRSION of a Paper in I TRANSACTIONS ON INFORMATION THORY, VOL. 57, NO. 8, AUGUST The Fading Nuber of a Multiple-Access Rician Fading Channel Gu-Rong Lin and Stefan M. Moser, Senior Meber, I Abstract The su-rate capacity of a noncoherent eoryless ultiple-access Rician fading channel is investigated under three different categories of power constraints: individual per user peak-power constraints, individual per user average-power constraints, or a global power-sharing average-power constraint. Upper and lower bounds on the su-rate capacity are derived, and it is shown that at high signal-to-noise ratio the surate capacity only grows double-logarithically in the available power. The asyptotic behavior of capacity is then analyzed in detail and the exact asyptotic expansion is derived including its second ter, the so called fading nuber. It is shown that the fading nuber is identical to the fading nuber of the single-user Rician fading channel that is obtained when only the user seeing the best channel is transitting and all other users are switched off at all ties. This pessiistic result holds independently of the type of power constraint that is iposed. Index Ters Channel capacity, escaping to infinity, fading nuber, high signal-to-noise ratio SNR, ultiple-access channel MAC, ultiple-input single-output MISO, ultiple users, noncoherent detection, Rician fading, su-rate capacity. I. INTRODUCTION In a noncoherent fading channel where neither transitter nor receiver know the fading realization, it has been shown in 1 that the capacity at high signal-to-noise ratio SNR behaves fundaentally differently fro the usual asyptotics seen in Gaussian channels or in coherent fading channels: instead of a logarithic growth in the SNR, the capacity only grows double-logarithically. To be precise, if the fading process is stationary, ergodic, and has a finite differential entropy rate and a finite expected second oent, then we have CSNR = log1+log1+ SNR+χ+o1 1 where o1 denotes ters that tend to zero as the SNR tends to infinity; and where χ is a constant independent of the SNR that is called fading nuber. The value of χ depends on the exact specifications of the fading law. In the situation of a general eoryless fading process, i.e., a fading process that is independent and identically distributed IID over tie This work was supported in part by the Industrial Technology Research Institute ITRI, Zhudong, Taiwan, under Contracts G and 99-C-17- A ; in part by the National Science Council NSC under Grants NSC MY3 and NSC MY3; and in part by the MediaTek Research Center at National Chiao Tung University. The aterial in this work was presented in part at the I Inforation Theory Workshop ITW, Taorina, Sicily, Italy, October 009. G.-R. Lin was with the Departent of lectrical ngineering, National Chiao Tung University NCTU, Hsinchu, Taiwan. He is now with Hon-Hai Precision Ind. Co., Ltd., Taiwan. S. M. Moser is with the Departent of lectrical ngineering, National Chiao Tung University NCTU, Hsinchu, Taiwan. and of a general law, the fading nuber has been derived for a single-input single-output SISO channel, a single-input ultiple-output SIMO channel, and a ultiple-input singleoutput MISO channel in 1, and the ultiple-input ultipleoutput MIMO channel was solved in. The ore general setup of a stationary, ergodic and regular fading process has been analyzed in 1 for the SISO case, 3 solved the SIMO case, and the ost general MIMO case was addressed in 4. Note that even though the fading nuber is defined only in the it when the available SNR tends to infinity, it has practical relevance also for finite SNR: it is a good estiator for the threshold where the capacity changes fro the noral logarithic growth to the highly inefficient doublelogarithic growth. For ore details we refer to the discussion in Section III-C and to the introduction section in 4. All the above entioned results are restricted to the situation of a single transitter possibly with several antennas and a single receiver. The present work is a first step towards generalizing the setup to a ultiple-user situation. Concretely, we include transitters, each having a certain nuber n i of antennas and trying to counicate to a coon receiver with only one antenna. The fading law is assued to be eoryless both over tie and space and Gaussian distributed with line-of-sight LOS coponents. We will propose upper and lower bounds on the su-rate capacity of this channel and derive the exact asyptotic expansion of the su-rate capacity for the SNR tending to infinity. It will turn out that the asyptotic capacity corresponds to the single-user capacity for the case when all but one user are switched off at all ties. The reainder of this paper is structured as follows. After soe short rearks about notation we will introduce the ultiple-access MAC Rician fading channel and three different power constraints in Section II. In Section III we will derive upper and lower bounds on the su-rate capacity of this odel that are valid for all SNR. These bounds are based on new bounds for the single-user MISO Rician fading channel. We will see there that in contrast to the low-snr regie, at high SNR the capacity only grows double-logarithically in the power. To investigate the threshold between these two regies, in Sections IV and V the asyptotic behavior of the su-rate capacity will be analyzed and stated exactly. The proof of the ain result can be found in Section VI, while the derivations of soe interediate steps have been oved to the appendices. We conclude in Section VII. We try to clearly distinguish rando and constant quantities: while rando quantities are denoted by capital Roan letters, constants are typeset in sall Roans or the Greek alphabet.

2 XTNDD VRSION of a Paper in I TRANSACTIONS ON INFORMATION THORY, VOL. 57, NO. 8, AUGUST 011 To distinguish nubers fro vectors, vectors are in bold face..g., X denotes a rando vector and x its realization, while Y is a rando variable and y its realization. There are a few exceptions to this rule. As is custoary we use capital letters to denote atrices, however, in order to be able to distinguish the fro rando variables we use a different font: D. Moreover, C stands for capacity, for the available power, I denotes the utual inforation functional, and Q is a cuulative distribution function CDF of the channel input. The superscript T refers to the transpose operation of vectors and atrices. We use to denote the uclidean nor of vectors. Sets are set in calligraphic font D, and D c denotes the copleent set. All rates specified in this paper are in nats per channel use, i.e., log denotes the natural logarithic function. II. CHANNL MODL AND POWR CONSTRAINTS We consider a ultiple-access channel with transitters users and one receiver. The signals transitted by the users are assued to be independent. The receiver is assued to have only one antenna, whereas each user i has soe nuber n i of transit antennas, i = 1,...,, which yields a total nuber of antennas at the transitter side of n T n i. All channels between one of the n T transit antennas and the receiver antenna are assued to be eoryless and independent Rician fading channels, i.e., the fading is coplex Gaussian distributed with variance 1 and soe ean line-ofsight coponent d l i C. Note that in the following we will use i and soeties j to denote the users, i.e., i = 1,...,, and l to denote the antennas of user i, i.e., l = 1,...,n i. To siplify our notation and because we assue all channels to be IID over tie, we restrain ourselves fro using tie indices. We would like to point out that the assuption of eorylessness has been ade for siplicity. We believe it is possible to extend the results to fading with eory see also the discussion in Section VII. So the channel output Y C can be written as Y = d T i +H T ix i +Z 3 = n i l d i +H l l i x i +Z. 4 l=1 Here x i C ni denotes the input vector for the n i antennas of user i; the coponents of the rando vector d i +H i describe Rician fading H l i +d l i N C d l i,1 5 hence, H l i are zero-ean, unit-variance, circularly syetric, coplex Gaussian rando variables and are assued to be independent H l i H l i, i,l i,l 6 and Z N C 0,σ denotes additive, zero-ean, circularly syetric Gaussian noise, independent fro the fading H 1,...,H. We assue a noncoherent situation, i.e., neither transitters nor receiver have knowledge of the current fading realization, they only know the fading distributions. 1 Note that we do not restrict the receiver and/or transitters to try to gain such knowledge. Any power or bandwidth used for such estiation schees, however, are taken into account for the capacity analysis and are not given for free as in a coherent setup. Neither will it be possible for the receiver to gain perfect channel knowledge. We do not allow cooperation between the users, i.e., we assue that the input vectors of the different users are statistically independent: X i X j, i j. 7 We also ention for copleteness that the users input vectors are assued to be independent fro fading and noise. For siplicity of notation we will soeties collect all LOS vectors d i into one n T -vector d: d d T 1,...,d T T 8 the fading vectors H i into one fading vector H of length n T : H H T 1,...,H T T 9 and the input vectors X i of all users into one n T -vector X: X X T 1,...,X T T. 10 In the given setup we can consider several possible constraints on the power. We will analyze three different scenarios: Peak-Power Constraint: At every tie-step every user i is allowed to use a power of at ost κi : Pr X i > κ i = 0 11 for soe fixed nuber κ i > 0. Average-Power Constraint: Averaged over the length of a codeword, every user i is allowed to use a power of at ost κi : X i κ i 1 for soe fixed nuber κ i > 0. Power-Sharing Average-Power Constraint: Averaged over the length of a codeword all users together are allowed to use a power of at ost κ: X i κ 13 for soe fixed nuber κ > 0. Note that if κ i = 1 for all i, we have the special case where all users have an equal power available. Also note that in 11 and 1 we have noralized the power to the nuber of users. This ight be strange fro an engineering point of view; however, in regard of our freedo to choose κ i, it is 1 Note that the constant line-of-sight LOS vectors d i are part of the distributions and are, therefore, known everywhere.

3 LIN, MOSR: TH FADING NUMBR OF A MULTIPL-ACCSS RICIAN FADING CHANNL 3 irrelevant, and it siplifies our analysis because we can easily connect the power-sharing average-power constraint with the other two constraints. Indeed, if we define κ to be the average of the constants κ i }, i.e., κ 1 κ i 14 then the three constraints are in order of strictness: the peakpower constraint is the ost stringent of the three constraints in the sense that if 11 is satisfied for all i = 1,...,, then the other two constraints are also satisfied; and the averagepower constraint is the second ost stringent in the sense that if 1 is satisfied for all i, then also the power-sharing average-power constraint 13 is satisfied. In the reainder of this paper we will always assue that 14 holds. For soe coents about even ore general types of power constraints, we refer to the discussion in Section VII. It is worth entioning that the slackest constraint, i.e., the power-sharing average-power constraint, iplicitly allows a for of cooperation: while the users are still assued to be statistically independent, we do allow cooperation concerning power distribution. This is not very realistic, however, we include it anyway because it will help in deriving bounds on the su-rate capacity. As a atter of fact, it will turn out that the asyptotic su-rate capacity is unchanged irrespective of which constraint is assued. The su-rate capacity C MAC of the channel 3 is given by C MAC = sup IX 1,...,X ;Y 15 Q X1 Q X power constraint where the supreu is over the set of all probability distributions of the input vectors such that the users are statistically independent of each other 7, and such that one particular power constraint 11, 1, or 13 is satisfied. III. NONASYMPTOTIC BOUNDS ON TH SUM-RAT CAPACITY A. Relationship between MAC and MISO We derive an upper and a lower bound on the su-rate capacity 15 by properly changing the setup to a single-user situation. Firstly, we upper-bound C MAC by dropping the independence-constraint 7, i.e., allowing full cooperation aong all users. Moreover, we choose the ost relaxed power constraint 13: C MAC = sup Q X1 Q X power constraint IX 1,...,X ;Y 16 sup Q X1,...,X Xi κ IX 1,...,X ;Y 17 = sup Q X X κ IX;Y 18 = C MISO,av,d κ. 19 Here C MISO,av,d Υ denotes the single-user capacity of the MISO Rician fading channel withn T transitter antennas and one receiver antenna Y = d T x+h T x+z 0 where d, H, and x are defined in 8, 9, and 10, respectively under the average-power constraint X Υ. 1 On the other hand, obviously the su rate cannot be saller than the single-user rate that can be achieved if all but one user are switched off, assuing the ost stringent type of power constraint 11, and assuing the inial aount of power aong all users. I.e., for an arbitrary i 1,...,}, C MAC = sup Q X1 Q X IX 1,...,X ;Y power constraint sup Q X1 Q X Pr X j > κ in =0, j = sup Pr X i > κ in =0 IX 1,...,X ;Y Xj 0, j i 3 IX i ;Y 4 = C MISO,pp,di κin. 5 Here, C MISO,pp,di Υ denotes the single-user capacity of the MISO Rician fading channel with n i transitter antennas and one receiver antenna under the peak-power constraint and we define Y = d T ix i +H T ix i +Z 6 Pr X i > Υ = 0 7 κ in in i 1,...,} κ i. 8 Hence, we have the following first iportant result. Theore 1: The su-rate capacity 15 of the ultipleaccess Rician fading channel 3 under one of the three power constraints 11, 1, or 13 is bounded as follows: κin axc MISO,pp,di i C MAC C MISO,av,d κ. 9 B. Bounds on Capacity of MISO Rician Fading Channel In order to be able to derive ore explicit bounds on the MAC su-rate capacity, we ake a sall detour and develop soe bounds on the MISO Rician fading channel. We start with an upper bound, which is a generalization of a bound fro 1, based on a dual expression of utual inforation.

4 4 XTNDD VRSION of a Paper in I TRANSACTIONS ON INFORMATION THORY, VOL. 57, NO. 8, AUGUST 011 Proposition : The capacity of the MISO Rician fading channel 0 under an average-power constraint 1 is upperbounded as follows: C MISO,av,d Υ inf 0<α 1 >0,ν 0 αlog σ 1+logΓ α, ν + d +1Υ+σ + ν d Υ +1 α log Υ+σ +1 α log i ν σ e ν σ i d Υ Υ+σ ν σ +γ } 30 where i denotes the exponential integral function i ξ e t ξ t dt, ξ > 0 31 and where γ 0.57 denotes uler s constant. Proof: See Appendix A. In order to be able to apply any lower bound on the MISO Rician fading channel to Theore 1, we need to consider a peak-power constraint instead of an average-power constraint. We will derive two different lower bounds. The first bound relies on an input chosen such that the logarith of its agnitude is uniforly distributed in the interval 1 logυ 0, 1 logυ for soe constant 0 < Υ 0 < Υ. The second lower bound is based on a binary input X i Υ Ξ d i d i eiφ 3 with PrΞ = 1 = 1 PrΞ = 0 = p and Φ independent of Ξ being unifor between 0 and π, Φ U0,π. The induced utual inforation is then coputed nuerically. Proposition 3: The capacity of the MISO Rician fading channel 6 under a peak-power constraint 7 is lowerbounded as follows: C MISO,pp,di Υ conv.-hull axc L1,di Υ,C L,di Υ} } where 33 C L1,di Υ Υ ax log log +log d i i d i 0<Υ 0<Υ Υ 0 } 1 log 1+ σ 34 and C L,di Υ ax 0 p 1 0 Υ 0 f R i tlogf R i t dt 1 plog Υ+σ 1 plog σ } 35 with f R i t 1 p t σ e σ + p t+ d i Υ Υ+σ e Υ+σ I 0 d i Υt Υ+σ. 36 Here I 0 denotes the odified Bessel function of order zero, and i is defined in 31. Proof: See Appendix B. C. Discussion Proposition and 3 can be applied directly to Theore 1 to get bounds on the su-rate capacity. Figure 1 depicts an exaple with two users =, each of the having the sae power constraint, i.e., κ 1 = κ = κ = 1. The LOS coponents are assued to be d 1 = 6 and d = 8, such that d = 10. Note that the exact choice of the vectorsd 1 and d including their diensions n 1 and n is irrelevant for the given bounds. The LOS coponents influence the expressions only via their agnitudes. We clearly see that there exist two distinct regies: for SNR values below around 10 db or a rate of about C MAC 5 nats the su-rate capacity grows logarithically in the SNR, while above the threshold the growth changes draatically and becoes very slowly growing. We will show in the next section that this high-snr growth is double-logarithic. We conclude that one should not use this channel at high SNR, and we ask for ore insight about this threshold between the efficient low-snr regie and the highly inefficient high- SNR regie. As described in 4, Sec. I.B it turns out that an asyptotic capacity analysis is the clue to such an investigation. This ight see strange at first sight as we just have concluded that we are not interested in this channel at high SNR. However, it is iportant to realize that around the threshold, the su-rate capacity is doinated by the second constant ter of the asyptotic high-snr expansion of the su-rate capacity and not by the double-logarithic ter!. Indeed, we note that log1+log1+υ nats 37 for Υ 0 db,80 db, and therefore conclude that as a rule of thub the threshold will be around C MAC χ+ nats. Hence, in deriving the asyptotic expansion of capacity one gains iportant understanding of the behavior of the channel at a reasonable and finite SNR. In the reainder of this paper we will investigate the asyptotic behavior of the surate capacity and in particular copute its exact asyptotic expansion. IV. TH ASYMPTOTIC SUM-RAT CAPACITY We will now consider the asyptotic case, i.e., the situation when the available power tends to infinity. We know that for the MISO Rician fading case 1, Theore 4.7 C MISO = C MISO,av,d = C MISO,pp,d = loglog σ +χ MISO,d +o1 38 Note that asyptotically for, log 1 + log 1 + σ = loglog σ +o1.

5 LIN, MOSR: TH FADING NUMBR OF A MULTIPL-ACCSS RICIAN FADING CHANNL 5 8 CMAC nats per channel use χ+ χ Gaussian channel upper bound 30 1 lower bound 33 lower bound 35 lower bound /σ db Fig. 1. Nonasyptotic bounds 9 on the su-rate of a two-user ultiple-access Rician fading channel. The dotted line shows the capacity of an additive Gaussian noise channel with equivalent received SNR. The red horizontal line corresponds to the fading nuber χ as derived in Section V, and the dashed red line is the approxiate threshold χ + nats between the efficient low-snr and the highly inefficient high-snr behavior. whereo1 denotes ters that tend to zero astends to infinity and where χ MISO,d is a constant denoted MISO fading nuber. Note that the value of χ MISO,d is independent of whether we have assued an average-power or a peak-power constraint and is given by 1, Corollary 4.8 χ MISO,d = log d i d 1 39 where i is defined in 31 and where d denotes the LOS vector of the MISO Rician fading channel. We further note that for any constant factor loglog loglog } = 0, > 0 40 i.e., the double-logarithic growth is not influenced by the factors κin or κ in Theore 1. Therefore, we directly get fro 9, 38, and 40 the following result. Corollary 4: The su-rate capacity 15 of the ultipleaccess Rician fading channel 3 under any one of the three power constraints 11, 1, or 13, and irrespective of the values of κ 1,...,κ, grows double-logarithically in the power at high power: } C MAC loglog σ <. 41 We next step out to analyze the second ter of the high- SNR expansion of the su-rate capacity: the MAC fading nuber. Definition 5: The MAC fading nuber is defined as χ MAC C MAC log 1+log 1+ σ }. 4 A priori χ MAC depends on the type of power constraint 11, 1, or 13 that is iposed on the input. However, it will turn out that the value of the MAC fading nuber is identical for all three cases. We therefore take the liberty to use a slightly sloppy notation that does not specify the used power constraint. Fro 9, 38, and 40 we realize that axχ MISO,di χ MAC χ MISO,d 43 i or explicitly by 39 ax log di i d i 1 } i χ MAC log d i d 1 44 where we reind the reader that d i is the LOS vector of user i and d d T 1,...,d T T is the stacked LOS vector of all users. Using the onotonicity of ξ logξ i ξ 1 we now define d MAC 0 such that χ MAC = log d MAC i d MAC Fro 44 we know that ax d 1,..., d } d MAC d = d d. 46 In the reainder we will derive the exact value of d MAC. We would like to point out that in 5 it has been proven that for the two-user case = with n 1 = n = 1 and with κ 1 = κ = 1 the upper bound in 46 cannot be achieved, i.e., d MAC < d 47 with strict inequality.

6 6 XTNDD VRSION of a Paper in I TRANSACTIONS ON INFORMATION THORY, VOL. 57, NO. 8, AUGUST 011 V. MAIN RSULT: TH MAC FADING NUMBR Theore 6: Consider a ultiple-access Rician fading channel as defined in 3. Then, irrespective of which power constraint 11, 1, or 13 is iposed on the input and irrespective of the values of κ 1,...,κ, the MAC fading nuber χ MAC 4 is given by with χ MAC = log d MAC i d MAC 1 48 d MAC ax d 1,..., d }. 49 This shows that the lower bound in 46 is tight, which is a rather pessiistic result. It eans that if the agnitude of the LOS vector of one user is strictly larger than the LOS vectors of the other users, then the asyptotic su-rate capacity can only be achieved if all but this strongest user are switched off at all ties. If there are several users with LOS vectors of identical largest agnitude, the su-rate capacity can also be achieved by tie sharing aong those best users. Note that the result holds even if we allow for power sharing aong the users. VI. PROOF OF MAIN RSULT The proof of Theore 6 consists of two parts. The first part is given already in Section IV: it is shown in 46 that ax i d i is a lower bound to d MAC. Note that this lower bound can be achieved using an input that satisfies the strictest constraint, i.e., the peak-power constraint 11. The second part will be to prove that ax i d i also is an upper bound to d MAC. We will prove this under the assuption of the slackest constraint, i.e., the power-sharing averagepower constraint 13. Since the peak-power constraint 11 and the average-power constraint 1 are ore stringent than the power-sharing average-power constraint 13, the result will follow. Before we start with the actual derivation of this upper bound, we need to generalize a concept that has been introduced in 1 and 3. Proposition 7 Input Distributions that scape to Infinity: Let Q } >0 be a faily of joint input distributions of the ultiple-access Rician fading channel 3, paraetrized by the available power > 0, satisfying the power-sharing average-power constraint 13, and satisfying IQ loglog = 1 50 where IQ denotes the utual inforation between input and output of this channel induced by the input distribution Q. Then at least one user s input distribution ust escape to infinity, i.e., for any fixed 0 > 0, Q X 1 0 } X } 0 = Proof: See Appendix C. To put it in an engineering way, Proposition 7 says that in the it when the available power tends to infinity, at least one user ust use a coding schee where every used sybol uses infinite energy. Or in other words, if all users use one or ore sybols with finite energy, the asyptotic growth rate of the su-rate capacity cannot be achieved. Definition 8: We define A to be the set of failies of joint input distributions of all users such that the users are independent 7, such that the power-sharing average-power constraint 13 is satisfied, and such that the input distribution of at least one user escapes to infinity when the available power tends to infinity 51, i.e., A Q X } >0 : 7, 13, and 51 are satisfied }. 5 We are now ready for the derivation of an upper bound on the MAC fading nuber. The following bound is derived fro a duality-based bound on utual inforation. Lea 9: The MAC fading nuber 4 is upper-bounded as follows: χ MAC sup Q A log i dt X X dt X X } Proof: See Appendix D. Noting that ξ logξ i ξ 1 is a onotonically increasing function and using our definition of d MAC in 45, we hence conclude that d MAC sup Q A dt X X. 54 We would like to point out that without the constraint 51 the right-hand side RHS of 54 actually equals to d, i.e., to the RHS of 46, fro which we already know that it is at least in soe cases strictly loose. So we see that the presented generalization of the concept of input distributions that escape to infinity Proposition 7 is crucial to this proof. We now continue as follows: dt X sup Q A X d T = sup 1 X 1 + +d T X Q A X X 55 d T sup 1 X d T X Q A X X d T i + sup X i d T j X j X X 56 j=1 Q A j i sup Q A + d1 X d X j=1 j i X X di X i d j X j sup X X Q A 57 where in 56 we split the supreu into any separate suprea, and where 57 follows fro the Cauchy-Schwarz inequality d T ix i d i X i. 58

7 LIN, MOSR: TH FADING NUMBR OF A MULTIPL-ACCSS RICIAN FADING CHANNL 7 We next upper-bound the first ter in 57 as follows: d1 X d X sup Q A X X d1 r1 + + d r } sup r 1,...,r r r r T Dr = sup r r 60 = λ ax D 61 = ax d 1,..., d } 6 where we have defined the vector r r 1,...,r T and the atrix D diag d 1,..., d. 63 The equality 61 follows fro the Rayleigh-Ritz Theore 6, Theore 4... To address the reaining ters in 57 we note that by definition of A in 5 at least one user s input ust escape to infinity. Without loss of generality assue that X 1 is aong the. Then we can separate the reaining ters in 57 into two kinds: sup Q A d1 d i X 1 X i X X, i,...,} 64 and di d j X i X j sup Q A X X, i,j,...,}, i j. 65 Our proof is concluded once we can show that sup d1 d i X 1 X i Q A X X = 0 66 sup di d j X i X j Q A X X = 0 67 for i,j,...,}, i j. We start with 66 and note that by dropping soe ters in the denoinator we have d1 d i X 1 X i sup Q A Next we define X X d 1 d i sup Q A X 1 X i X 1 + X i X 1 69 and recall that if then 1 by our assuption that user 1 escapes to infinity. Note further that r 1 r i r 1 +r i 1 70 and that r 1 r1ri is onotonically decreasing if r r1 1 > r i. +r i Therefore, for an arbitrary choice of a > 1, we define the set D as D x 1 : 0 x 1 a x i } 71 and bound sup X 1 X i Q A X 1 + X i sup sup X 1 X i X 1 + X i 7 = sup sup x 1 x i x 1 + x i dq X1 x 1 d x i 73 sup sup x 1 x i x 1 D x 1 + x i dq X1 x 1 d x i +sup sup x 1 x i x 1 D x c 1 + x i dq X1 x 1 d x i. 74 Here in the first inequality 7 we define A 1 as the set of all input distributions of the first user that escape to infinity, and take the supreu over all without any constraint on the average power and no dependence on Q X1. The last inequality 74 then follows fro splitting the inner integration into two parts and fro the fact that the supreu of a su is always upper-bounded by the su of the suprea. Next, let s look at the first ter in 74 and use 70: sup x 1 D sup 1 x 1 D sup 1 = 1 1 Q X1 A 1 1 sup Q X1 A 1 x 1 x i x 1 + x i } } 1 dq X1 x 1 d x i dq X 1 x 1 d x i 75 x 1 D x 1 D dq X1 x 1 dq X1 x 1 d x i 76 d x i 77 = sup 1 Pr X 1 a x i d x i 78 = 0 d x i = Here, 75 follows fro 70; the subsequent inequality 76 follows by taking the supreu into the first integral which can only enlarge the expression; in 77 we exchange it and integration which needs justification: define 1 g 1 x i sup Q X1 A 1 1 sup Q X1 A 1 x 1 D dq X1 x 1 80 dq X1 x 1 81 = 1 g upperx i 8 and then note that 1 g upper x i d x i = dq X i x i = 1 83

8 8 XTNDD VRSION of a Paper in I TRANSACTIONS ON INFORMATION THORY, VOL. 57, NO. 8, AUGUST 011 i.e., g upper is independent of 1 and integrable. Thus, by the Doinated Convergence Theore 7 we are allowed to swap it and integration. Finally, 79 follows fro Proposition 7 because Q X1 escapes to infinity. Continuing with 74 we now have: sup X 1 X i Q A X 1 + X i sup 1 sup Q X1 A 1 x 1 D c sup sup x 1 D c x 1 x i x 1 + x i dq X1 x 1 d x i 84 a xi x i a x i + x i dq X1 x 1 d x i 85 = sup sup a x 1 D a +1 c dq X1 x 1 d x i 86 sup sup a a +1 dq X 1 x 1 d x i 87 a = sup a +1 dq X i x i 88 = a a +1 < ǫ 89 for any ǫ > 0 if we choose a large enough. Here 85 follows because r 1 r1ri is onotonically decreasing if r r1 1 > r i. +r i Since a > 1 is arbitrary, we obtain: sup X 1 X i Q A X 1 + X i = This proves 66. To prove 67, we again drop soe ters in the denoinator: di d j X i X j X X d i d j X i X j X 1 + X i + X j We once ore use definition 69 and note that r i r i r j r1 +r i +r j r1 1 +r i and that r 1 r i is onotonically decreasing. r1 +r i For an arbitrary choice of a > 1, we use the set D fro 71 to derive sup X i X j Q A X 1 + X i + X j sup Q Xj 1 sup Q X1 A 1 x i x j x 1 + x i + x j dq X1 x 1 d x i dq Xj x j 93 sup sup x i x 1 + x i dq X1 x 1 d x i 94 sup sup x i x 1 D x 1 + x i dq X1 x d x i x1 Dc +sup sup x i x 1 + x i dq X1 x d x i. 95 Here in 93 we define A 1 as the set of all input distributions such that the first user escapes to infinity, and take the supreu over all joint distributions of and Q Xj without any restriction on the average power. In the subsequent inequality 94 we apply 9 to replace x j by x i. In the last inequality we split the inner integration into two parts using 71. For the first ter in 95, we have sup x 1 D x i x 1 + x i } } 1 dq X1 x 1 d x i 1 x 1 D dq X 1 x 1 d x i 96 = 0 97 where 97 follows fro a derivation analogous to The second ter in 95 can be bounded as follows: x1 Dc sup sup x i x 1 + x i dq X1 x 1 d x i x1 Dc sup sup x i a x i + x i dq X1 x 1 d x i 98 1 sup a + dq X i x i 99 = 1 a + < ǫ 100 for any ǫ > 0 if we choose a large enough. Here in the first inequality we use that r 1 r i is onotonically r1 +r i decreasing. Since a is arbitrary, this proves 67 and concludes the proof. VII. CONCLUSIONS In this paper we have derived a new upper and lower bound on the su-rate capacity of a noncoherent eoryless ultiple-access Rician fading channel with transitters with a different nuber of antennas each and one receiver with only one antenna. We have shown that while the surate capacity at low SNR behaves norally with a logarithic growth in the available power, at high SNR it is highly power-inefficient and only grows double-logarithically. It is therefore advisable not to operate such a channel at high SNR. These bounds rely on novel bounds on the capacity of a singleuser MISO Rician fading channel that are valid for any SNR. In a second step we then derived the exact asyptotic high- SNR expansion of the su-rate capacity, which has the for C MAC = loglog σ +χ MAC +o1. 101

9 LIN, MOSR: TH FADING NUMBR OF A MULTIPL-ACCSS RICIAN FADING CHANNL 9 We have shown that this asyptotic su-rate capacity is ited by the asyptotic capacity of the user seeing the best channel and can only be achieved if all users with a channel that is strictly worse than the best channel are always switched off and cannot counicate. Note that this should not be confused with the idea of tie sharing where at any given tie only one user is allowed to counicate. In the presented setup, as long as the channel odel does not change, the best user will reain the best user, i.e., all other users can never counicate. 3 This very pessiistic result fits to the already rather pessiistic double-logarithic behavior and strengthen the conviction that these channels should not be used at high SNR, but only at low SNR where the channel will behave norally like a coherent fading channel. At first sight our results sees very siilar to a result by Knopp and Hublet 8, 9, 10, who showed that the strategy of one user at a tie also is optial for the MAC with full channel state inforation both at the transitter and receiver side. In 8 a continuous-tie syste is considered and it is shown that if the transitter and receiver have full knowledge of the fading, then it is best if the users are assigned separate frequency and tie slots corresponding to best fading realizations orthogonal signaling. However, we would like to point out that in this setup, each user can transit regularly and has a strictly nonzero average counication rate, while in the channel odel considered here, it turns out that optially ost users have a zero transission rate. So these two results are not properly coparable. We reark further that fro the fact that the su-rate capacity is achieved in a corner where only one user has positive rate, one can deduce that the asyptotic capacity region has the shape of an -diensional siplex. In the analysis of the channel we have allowed for any different types of power constraints. We grouped the into three categories: an individual peak-power constraint for each user, an individual average-power constraint for each user, and a cobined power-sharing average-power constraint aong all users. The power-sharing constraint does not ake sense in a practical setup as it requires the users to share a coon battery, while their signals still are restricted to be independent. However, the inclusion of this case helps with the analysis. Moreover, it turns out that the pessiistic results described above even hold if we allow for such power sharing. Within a category of constraints, we do allow for different power settings for different users as long as the constraints scale linearly see the constants κ i and κ in It would be possible to extend the shown results to situations where the power constraints aong the different user differ exponentially, i.e., if every user i is allowed to use a power of at ost κ i ϑi for soe κ i,ϑ i > 0. In this case, however, ϑ i will influence the MAC fading nuber 4 via an additive ter logϑ i. This 3 The only exception is if there are several users having the sae best channel. In this case these equivalent best users can use tie sharing to alternatively counicate. 4 The double-logarithic ter in the asyptotic expansion will reain unchanged. then eans that in the evaluation of the MAC fading nuber 48 not only d i is iportant, but also this additive ter logϑ i has to be taken into account. While in this paper we have restricted the channel odel to be eoryless, a generalization to a fading process with eory is possible. Again, one has to be careful as the eory will influence the MAC fading nuber and thereby affect the search for the best channel. As already discussed in Section III-C, we would like to ephasize once ore that the analysis of the asyptotic surate capacity of this channel is of practical interest in spite of the fact that we will not use the channel at high SNR. The reason is that the MAC fading nuber χ MAC is a good indicator for the threshold between the efficient low-snr and the highly inefficient high-snr regie. As a rule of thub, the MAC Rician fading channel can be used up to a su-rate of about χ+ nats see Figure 1 for an exaple. It is oinous that the fading nuber and ergo also the threshold does not vary with the type of the used power constraint. This eans that once a su rate of around χ+ nats is achieved, the channel behavior will becoe very poor and cannot be iproved by any optiization of the power allocation. Instead the syste has to be changed in a ore fundaental fashion in order to achieve a change in the channel odel. An iportant clue to the derivations is a generalization of the concept of input distributions that escape to infinity. To put it in engineering words, the concept says that one cannot achieve the su-rate capacity asyptotically unless at least one of the users always uses input sybols of infinite power. Note that while we have stated this concept here specifically for the ultiple-access Rician fading channel at hand, it can be further extended to ore general ultiple-user channels. APPNDIX A UPPR BOUND ON MISO RICIAN FADING CAPACITY The upper bound 30 on the MISO Rician fading channel 0 is a generalization of an upper bound on the SISO Rician fading channel presented in 1, q It is based on a duality-based upper bound on the utual inforation taken fro 1, q. 5: IX;Y hy X+logπ +αlog +logγ α, ν +1 α log Y +ν + 1 Y + ν 10 where α, > 0 and ν 0 are free paraeters. We start with an upper bound on the fifth ter on the RHS of 10. To that goal we assue 0 < α < 1 such that 1 α > 0 and define ǫ ν sup x log Y +ν X = x log Y X = x } 103

10 10 XTNDD VRSION of a Paper in I TRANSACTIONS ON INFORMATION THORY, VOL. 57, NO. 8, AUGUST 011 such that 1 α log Y +ν = 1 α log Y +1 α log Y +ν log Y α log Y +1 αsup log Y +ν X = x x log Y X = x } 105 = 1 α log Y +1 αǫ ν. 106 Next we apply 10 to the MISO Rician fading channel 0. We note that conditional on X = x and copute and Y N C dt x, x +σ 107 hy X = x = logπ +1+log x +σ 108 Y X = x = dt x + x +σ 109 log Y X = x dt x x = log x +σ i dt x +σ + log x +σ 110 ǫ ν = sup log 1+ ν x Y X = x 111 = log 1+ ν Y X = 0 11 ν = log σ e ν/σ i ν σ +γ. 113 Here, in 110 we evaluate the expected logarith of a noncentral chi-square rando variable as derived in 11, 1, Lea 10.1, 1, Lea A.6; and 11 follows fro a stochastic ordering arguent by noting that the function ξ log ξ is onotonically decreasing and that the distribution of Y conditional on X = x is stochastically larger than the distribution of Y conditional on X = 0 1, Sec. IV.B. The final step 113 follows by a direct calculation. Plugging 106 and into 10 then yields IX;Y 1+αlog α log X +σ +logγ + X +σ + d T X + ν dt X +1 α log X +σ i dt X +ǫ X +σ ν 1+αlog σ +logγ α, ν + X +σ + d X + ν α, ν α log i d X X +σ d X +ǫ ν X +σ 115 where for the last step we have lower-bounded log X +σ logσ ; used the onotonicity of ξ logξ i ξ together with the Cauchy-Schwarz inequality 58; and applied Jensen s inequality to the concave function d ξ ξ log ξ +σ i d ξ ξ +σ. 116 The upper bound 30 now follows fro the average-power constraint 1. APPNDIX B LOWR BOUND ON MISO RICIAN FADING CAPACITY The first lower bound 34 on the capacity of the MISO Rician fading channel 6 with peak-power constraint 7 is based on the following lea that has been proven in 1, Lea 4.9. Lea 10 1: Let the rando vector X take value in C nt and satisfy Pr X x in = for soe x in > 0. Let H be a rando n R n T atrix having finite entropy hh > and finite expected squared Frobenius nor H F <. Let Z NC 0,σ I and assue that X, H, and Z are independent. Then IX;HX+Z IX;HX sup h Hˆx+ Z } hhˆx. 118 ˆx =1 x in We apply this lea to the situation of the MISO Rician fading channel 6 and choose the following distribution on X i : d i X i R d i eiφ 119 where Φ and R are statistically independent, Φ is unifor between 0 and π, Φ U0,π, and R is such that logr UlogΥ 0,logΥ 10 for soe fixed Υ 0. This choice satisfies the peak-power constraint 7 and also Hence by Lea 10, we get C MISO,pp,di Υ Pr X i Υ 0 = IX i ;d T ix i +H T ix i +Z 1 IX i ;d T ix i +H T ix i h sup ˆx =1 } Z T diˆx+ht iˆx+ hd T iˆx+ht iˆx Υ0 13 = IX i ;d T ix i +H T ix i log 1+ σ. 14 Υ 0

11 LIN, MOSR: TH FADING NUMBR OF A MULTIPL-ACCSS RICIAN FADING CHANNL 11 We introduce a rando variable H H T i d i d i eiφ N C 0,1 15 and rewrite the first ter on the RHS of 14 as follows: IX i ;d T ix i +H T ix i = hd T ix i +H T ix i hd T ix i +H T ix i X i 16 = h d i e iφ R+ HR log πe X i 17 di = h e iφ R+ HR 1 logr 18 h R di e iφ + H Φ, H 1 logr 19 = h R + log d i e iφ + H 1 logr. 130 Here, 18 follows fro the fact that for a circularly syetric rando variable U we have 1, Lea 6.16 hu = h U +logπ. 131 In 19 we condition the differential entropy which cannot increase its value; and 130 follows fro the scaling property of differential entropy 13, Th Next, we again evaluate the expected logarith of a noncentral chi-square rando variable 11, 1, Lea 10.1, 1, Lea A.6: di log e iφ + H = log di + H 13 = log d i i d i 133 where the first equality follows because H is circularly syetric and use the following identity 1, Lea 6.15: h logr = h R logr. 134 The lower bound 34 now follows by plugging 130, 133, and 134 into 14 and noting that because of 10 h logr Υ = loglog. 135 The second lower bound 35 follows fro 1 with the choice 3: C MISO,pp,di Υ hd T ix i +H T ix i +Z hd T ix i +H T ix i +Z X i 136 = hd T ix i +H T ix i +Z log πe X +σ 137 = hr i 1 plogυ+σ 1 plogσ. 138 Here in the last step we have used 131 together with the fact that d T ix i +H T ix i +Z = L d i Υe iφ Ξ+ Υ HΞ+Z 139 L = d i ΥΞ+ Υ HΞ+Z e iφ 140 L = d i ΥΞ+ Υ HΞ+Z e iφ 141 }} R i Υ 0 where = L denotes equal in probability law. Hence, we see that d T i X i + H T i X i + Z is circularly syetric with a agnitude R i that, conditional on Ξ = 0, is Rayleigh and, conditional on Ξ = 1, Rician distributed. The probability density function of Ri is given by 36. APPNDIX C PROOF OF PROPOSITION 7 Fix soe 0 > 0 and let 1 if i: X i 0 U 0 if X i < 0, i. 14 Further we define µ PrU = To prove Proposition 7, we need to show that To that goal, note the following: µ = IQ = IX 1,...,X ;Y 145 = IX 1,...,X,U;Y 146 = IU;Y+IX 1,...,X ;Y U 147 = IU;Y+IX 1,...,X ;Y U = 0PrU = 0 +IX 1,...,X ;Y U = 1PrU = log+ix 1,...,X ;Y U = 0 +µix 1,...,X ;Y U = κ log+c MISO,av,d 0 +µc MISO,av,d. 150 µ Here 149 follows because U is a binary rando variable and because PrU = 0 1. To justify the subsequent inequality 150, we note that because conditional on U = 0, X i < 0 for all i, i.e., X i U = we can upper-bound the MAC situation by full-cooperation MISO: IX 1,...,X ;Y U = 0 C MISO,av,d Moreover, by total expectation, X i = µ X i U = µ X i U = 0 }} 0 µ X i U = 1 fro which follows that X i U = 1 X i κ µ µ

12 1 XTNDD VRSION of a Paper in I TRANSACTIONS ON INFORMATION THORY, VOL. 57, NO. 8, AUGUST 011 and hence, again allowing full-cooperation, IX 1,...,X ;Y U = 1 C MISO,av,d κ µ. 156 Next, let n be a sequence with n, let Q n } n be a faily of joint input distributions for the ultiple-access Rician fading channel 3 such that n IQ n loglog n = and define µ n Q n X 1 } 0 X } By contradiction, assue µ n µ < 1. Then there ust exist soe µ 0 < 1 such that Fro 150 we have IQ n loglog n }} 1 µ n < µ 0, n sufficiently large. 159 log+c MISO,av,d 0 loglog } n } 0 κ C MISO,av,d µ n n + κ log log µ n n }} 1 µ n loglog κ µ n n loglog n.160 Here the iting behavior of the left-hand side LHS follows fro 157; the iting behavior of the first ter on the RHS is because C MISO,av,d 0 < ; the second ter on the RHS tends to one because n iplies κ n /µ n and because of 38. Hence, when n we obtain κ µ n loglog µ n n n loglog n κ µloglog sup µ 16 µ 0,µ 0 loglog where the first inequality follows fro 160 and the second inequality follows fro 159. This, however, is a contradiction to the fact that µloglog κ µ µ 0,µ 0 loglog < 1, 0 < µ 0 < Hence, we ust have that µ n 1, which proves the clai. APPNDIX D PROOF OF LMMA 9 The derivation of this result is based on 114. We start by bounding the following expressions: log X +σ logσ 164 X +σ + d T X κ+σ + d κ αǫ ν ǫ ν 166 and dt X log X +σ i dt X X +σ γ. 167 Here 165 follows fro Cauchy-Schwarz 58 and the fact that the input needs to satisfy the power-sharing average-power constraint 13; and 167 follows because logξ i ξ γ where γ 0.57 denotes uler s constant. Plugging these bounds into 114 and applying once ore Jensen s inequality to ξ logξ i ξ now yields IX;Y log dt X X i +α log logσ +γ +logγ + ν + 1 dt X X 1 α, ν +ǫ ν 1+ d κ+σ. 168 We will now ake the following choices of the free paraeters α and : ν α log1+ d κ+σ α e ν α 170 for soe constant ν 0. This leads to the following asyptotic behavior: logγ α, ν } 1 log = log 1 e ν 171 α α log logσ +γ = ν d κ+σ + ν } = and 1 log log 1+log 1+ σ } α = logν. 174 Copare with 1, Appendix VII, 1, Sec. B.5.9. Hence, using the definition of the MAC fading nuber 4 and the definition of the su-rate capacity 15, we have derived the following upper bound on the MAC fading nuber: χ MAC = C MAC log 1+log 1+ σ } 175 = sup IX;Y Q X independent users power constraint 13 log 1+log 1+ σ 176 = sup IX;Y log 1+log 1+ σ } Q A 177

13 LIN, MOSR: TH FADING NUMBR OF A MULTIPL-ACCSS RICIAN FADING CHANNL 13 sup Q A log } 1 dt X dt X X i X +α log logσ +γ +logγ α, ν +ǫ ν + ν d κ+σ log 1+log 1+ } σ 178 dt X dt X = sup log Q A X i X } 1 +ǫ ν +ν +log 1 e ν logν. 179 Here, in 177 we ake use of Proposition 7; 178 follows fro 168; and the last equality is due to By letting ν tend to zero which akes sure that ǫ ν 0 as can be seen fro 103 and 113 the clai follows. ACKNOWLDGMNTS Many fruitful discussions with Aos Lapidoth are gratefully acknowledged. The authors are also indebted to the anonyous reviewers for their inspiring coents. RFRNCS 1 A. Lapidoth and S. M. Moser, Capacity bounds via duality with applications to ultiple-antenna systes on flat fading channels, I Trans. Inf. Theory, vol. 49, no. 10, pp , Oct S. M. Moser, The fading nuber of eoryless ultiple-input ultiple-output fading channels, I Trans. Inf. Theory, vol. 53, no. 7, pp , Jul A. Lapidoth and S. M. Moser, The fading nuber of single-input ultiple-output fading channels with eory, I Trans. Inf. Theory, vol. 5, no., pp , Feb S. M. Moser, The fading nuber of ultiple-input ultiple-output fading channels with eory, I Trans. Inf. Theory, vol. 55, no. 6, pp , Jun A. Lapidoth and S. M. Moser, On the Ricean fading ulti-access channel, in Proc. Winter School Cod. and Inf. Theory, Monte Verità, Ascona, Switzerland, Feb. 4 7, R. A. Horn and C. R. Johnson, Matrix Analysis. Cabridge University Press, H. A. Priestley, Introduction to Integration. Oxford University Press, R. Knopp and P. A. Hublet, Multiuser diversity, Institut ureco, France, Tech. Rep. URCOM+607, Jan R. Knopp and P. A. Hublet, Inforation capacity and power control in single-cell ultiuser counications, in Proc. I Int. Conf. Co., Seattle, WA, USA, Jun. 18, 1995, pp R. Knopp and P. A. Hublet, Multiple-accessing over frequencyselective fading channels, in Proc. I Int. Syp. Pers., Indoor and Mob. Radio Coun., Toronto, Canada, Sep. 7 9, 1995, pp A. Lapidoth and S. M. Moser, The expected logarith of a noncentral chi-square rando variable, website. Online. Available: c.nctu.edu.tw/explog.htl 1 S. M. Moser, Duality-based bounds on channel capacity, Ph.D. dissertation, TH Zurich, Oct. 004, Diss. TH No Online. Available: 13 T. M. Cover and J. A. Thoas, leents of Inforation Theory, nd ed. John Wiley & Sons, 006. Gu-Rong Lin was born in Taiwan. He received a Bachelor s degree in counication engineering in 006 and a Master s degree in counication engineering in 009, both fro National Chiao Tung University NCTU, Hsinchu, Taiwan. During 007 to 009, he was a eber of the Inforation Theory Laboratory advised by Dr. Stefan Moser. He studied inforation theory, and focused on the field of channel capacity. His research interests include inforation theory, digital counications and wireless counications. Since 010, Gu-Rong Lin has been a counication software engineer with Hon-Hai Precision Industry. Stefan M. Moser S 01 M 05 SM 10 was born in Switzerland. He received the diploa M.Sc. in electrical engineering with distinction in 1999, the M.Sc. degree in industrial anageent M.B.A. in 003, and the Ph.D. degree Dr. sc. techn. in the field of inforation theory in 004, all fro TH Zurich, Switzerland. During 1999 to 003, he was a Research and Teaching Assistant, and fro 004 to 005, he was a Senior Research Assistant with the Signal and Inforation Processing Laboratory, TH Zurich. Since 005, he has been an Assistant Professor, and since 008, an Associate Professor with the Departent of lectrical ngineering, National Chiao Tung University NCTU, Hsinchu, Taiwan. His research interests are in inforation theory and digital counications. Dr. Moser received the Best Paper Award for Young Scholars by the I Counications Society Taipei and Tainan Chapters and the I Inforation Theory Society Taipei Chapter in 009, the National Chiao Tung University Outstanding Researchers Award in 007, 008, and 009, the National Chiao Tung University xcellent Teaching Award and the National Chiao Tung University Outstanding Mentoring Award both in 007, the Willi Studer Award of TH in 1999, the TH Medal for an excellent diploa thesis in 1999, and the Sandoz Novartis Basler Maturandenpreis in 1993.