1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER Capacity of Multiple-Antenna Systes With Both Receiver and Transitter Channel State Inforation Sudharan K. Jayaweera, Student Meber, IEEE, and H. Vincent Poor, Fellow, IEEE Abstract The capacity of ultiple-antenna systes operating in Rayleigh flat fading is considered under the assuptions that channel state inforation (CSI) is available at both transitter and receiver, and that the transitter is subjected to an average power constraint. First, the capacity of such systes is derived for the special case of ultiple transit antennas and a single receive antenna. The optial power-allocation schee for such a syste is shown to be a water-filling algorith, and the corresponding capacity is seen to be the sae as that of a syste having ultiple receive antennas (with a single transitter antenna) whose outputs are cobined via axial ratio cobining. A suboptial adaptive transission technique that transits only over the antenna having the best channel is also proposed for this special case. It is shown that the capacity of such a syste under the proposed suboptial adaptive transission schee is the sae as the capacity of a syste having ultiple receiver antennas (with a single transitter antenna) cobined via selection cobining. Next, the capacity of a general syste of ultiple transitter and receiver antennas is derived together with an equation that deterines the cutoff value for such a syste. The optial power allocation schee for such a ultiple-antenna syste is given by a atrix water-filling algorith. In order to eliinate the need for cubersoe nuerical techniques in solving the cutoff equation, approxiate expressions for the cutoff transission value are also provided. It is shown that, copared to the case in which there is only receiver CSI, large capacity gains are available with optial power and rate adaptation schees. The increased capacity is shown to coe at the price of channel outage, and bounds are derived for this outage probability. Index Ters Adaptive transission, channel capacity, atrix waterfilling, ultiple-antenna systes, outage probability, Wishart distribution. I. INTRODUCTION The capacity of fading channels varies depending on the assuptions one aes about fading statistics and the nowledge of fading coefficients. Over the years, the capacity of single-antenna systes (where both transitter and receiver are equipped with only one antenna each) has been considered for various assuptions on nowledge of fading coefficients. For exaple, ,  have treated the case where the receiver has access to channel state inforation (CSI), ,  have considered the capacity under the assuption that both transitter and receiver have access to CSI, and   have all treated the case when neither the transitter nor the receiver nows the channel fading coefficients. Recently, there has been a surge of interest in ultiple-antenna counications systes. Naturally, this has led to capacity investigation of fading channels with ultiple antennas either at the receiver or at the transitter or at both ends of the counication lin. For exaple, Manuscript received July 9, 22; revised June 25, 23. This wor was supported in part by the Ary Research Laboratory under Contract DAAD 9--2-, and in part by the New Jersey Center for Wireless Telecounications. S. K. Jayaweera was with the Departent of Electrical Engineering, Princeton University, Princeton, NJ 8544 USA. He is now with the Departent of Electrical and Coputer Engineering, Wichita State University, Wichita, KS 6726 USA (e-ail: H. V. Poor is with the Departent of Electrical Engineering, Princeton University, Princeton, NJ 8544 USA (e-ail: Counicated by B. Hassibi, Associate Editor for Counications. Digital Object Identifier.9/TIT   considered the capacity of ultiple transit and receiver antenna systes when CSI is available only at the receiver, and  investigated the capacity of such systes when neither the transitter nor the receiver nows the channel coefficients. The capacity of ultiple receiver antenna systes (with a single transit antenna) when CSI is available at both transitter and receiver has also been previously considered in . When both transitter and receiver have access to the CSI, the optial strategies would ae use of this inforation at both ends of the lin. Intuitively, one would expect the transitter to adjust its power and rate depending on the instantaneous value of the observed CSI. This results in adaptive transission techniques. However, such an optial schee could easily becoe too coplicated to ipleent, for exaple, when the fading is correlated. In order to overcoe this possible transitter coplexity, it is also of interest to investigate low-coplexity adaptive transission techniques and deterine the capacities under such suboptial adaptive transission techniques. As entioned earlier, this proble has been treated previously in  for the case of single-antenna systes and in  for the case of receiver diversity. With recent interest in ultiple transit antenna systes for wireless counications, it is also of interest to consider this proble in the context of ultiple antennas at both transitter and receiver. In this correspondence, we investigate the capacity of such systes under adaptive transission techniques. First, we obtain the capacity of the optial power and rate allocation schee for a syste having ultiple transit antennas but one receiver antenna and, not surprisingly, this is seen to be identical to the capacity of a receiver diversity schee with axial ratio cobining. We also derive the capacity of a ultiple transit antenna syste with a suboptial adaptive transission technique which is seen to be atheatically equivalent to a receiver diversity syste with selection cobining. Next we consider a general syste with ultiple antennas at both the receiver and the transitter. The capacity of the optial power and rate allocation schee for such a syste is derived and this capacity is evaluated for several representative situations. As we will show, the capacity of such systes could be uch larger than corresponding systes with only receiver channel state inforation. The increased capacity coes at the price of channel outage which we characterize in ters of the outage probability. We also derive siple upper bounds for this outage probability. In all these situations, we also provide approxiate expressions for the capacity, which are easy to evaluate and thus eliinate the need for any nuerical integration or root finding techniques that ight be required otherwise. The rest of this correspondence is organized as follows. In Section II, we outline our syste odel and the assuptions; Section III considers the special case of capacity of ultiple transit antenna systes with a single receiver antenna. Next, in Section IV, we treat the capacity of a general syste having ultiple antennas at both transitter and receiver. We obtain the capacity of such systes under optial power adaptation as well as the cutoff equation associated with the optial transission schee. In Section IV, we also derive siple upper bounds for the outage probability of the optial adaptive transission schee for a ultiple-antenna syste. Finally, in Section V, we give soe concluding rears. II. SYSTEM MODEL DESCRIPTION We consider a single-user flat-fading counications lin in which the transitter and receiver are equipped with N T and N R antennas, /3$7. 23 IEEE
2 2698 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER 23 respectively. The discrete-tie received signal in such a syste can be written in atrix for as y(i) =H(i)x(i) +n(i) () where y(i) is the coplex N R vector of received signals at the N R receive antennas at sybol tie i, x(i) is the (possibly) coplex N T vector of transit signals on the N T transit antennas at tie i and n(i) is the coplex N R vector of receiver noise at tie i. The coponents of n(i) are zero ean, circularly syetric, coplex Gaussian with independent real and iaginary parts having equal variance. It is assued that noise on each receiver antenna is independent of that on all others and thus, Efn(i)n(i) H g = N I N, where I N denotes the N R 2 N R identity atrix. We also assue that n(i) is a sequence of uncorrelated (and thus independent) rando vectors. The atrix H(i) in () is the N R 2 N T atrix of coplex fading coefficients which are assued to be stationary and ergodic. The (n R ;n T )th eleent of the atrix H(i) represents the fading coefficient value at tie i between the n R th receiver antenna and the n T th transitter antenna. These fading coefficients are assued to be slowly varying over the duration of a codeword. We assue that eleents of the atrix H(i) are independent and identically distributed (i.i.d.) coplex Gaussian rando variables with zero ean and =2-variance per diension. Of course, this gives rise to the so-called Rayleigh-fading channel odel, which has often been used to odel land-obile wireless counication channels without a direct line-of-sight path . We assue that the instantaneous value of the fading coefficient atrix H(i) is nown to both the transitter and the receiver. This assuption can be satisfied, for exaple, by eploying a channel estiation schee such as pilot sybol insertion or training bits. The transitter ay be assued to be infored of those receiver estiated CSI via a delay- and error-free feedbac path. This is a reasonable assuption when the channel varies at a uch slower rate copared to the data rate of the syste. In a tie-duplexed syste, the transitter ay also estiate its own CSI values using the reverse lin received signals. As we will see shortly, the capacity will be dependent on the nuber of transitter and receiver antennas only through the relative paraeters defined as n =axfn R;N T g and =infn R;N T g. III. SINGLE RECEIVER ANTENNA SYSTEMS We start by considering the capacity of a ultiple transit and single receiver antenna syste with adaptive transission techniques; i.e., N T = n and N R = =. In this case, the received signal y is a scalar which we denote as y. Note that, for convenience, we will drop the tie index i whenever this causes no confusion. In general, we ay decopose the fading coefficient atrix H using the singular value decoposition ,  H = U 3V H (2) where U, 3, and V are atrices of diension N R 2 N R, N R 2 N T, and N T 2N T, respectively. The atrices U and V are unitary atrices satisfying UU H = U H U = I N and V V H = V H V = I N. The atrix 3 = [ i; j] is a diagonal atrix with diagonal entries being equal to the nonnegative square roots of the eigenvalues of either HH H or H H H, and, thus, are uniquely deterined. For later use, we ay also define the following 2 atrix: W = HHH ; if N R N T H H (3) H; if N R >N T : Note that W can have at ost nonzero eigenvalues and thus correspondingly at ost diagonal entries of the atrix 3 are nonzero. It is also worth entioning that the distribution of the atrix W is given by the well-nown Wishart distribution . In the present case of N T = n and N R = =, it is easily seen that a singular value decoposition of H is defined by U = where = 3 = p ; ;...; V = [v v 2 v n] (4) n i= jh i; jj 2 and v = HH p : Defining the transforations ~y = U H y, ~x = V H x, and ~n = U H n, we see that the channel in () is equivalent to ~y =3~x + ~n: (5) If the average transit power is constrained as Efx H xg =tr[efxx H g]=p then we also have that Ef~x H ~xg =tr Ef~x~x H g = P: (6) Fro (4) and (5) we see that, for the present case of N R =, the channel is equivalent to the following scalar channel: ~y = p ~x +~n (7) where ~x is the first coponent of the vector ~x. Hence, we see that only that energy contained in the coponent ~x is useful in detecting the signal, and thus we ay as well set ~x 2 =~x 3 = =~x N =. Then, fro (6) we have that Ef~x 2 g = P: (8) A. Optial Adaptive Transission Let us define the received signal-to-noise ratio (SNR) (i) for a given value of the channel coefficient atrix as (i) = (i) P : (9) N We let the transitter adapt its instantaneous transit power P ((i)) according to the channel variations, subject to the average power constraint P ()f () d P () where P () denotes the tie-varying instantaneous adaptive power and f () is the probability distribution function of (i). Note that the channel (5) with this adaptive transission is atheatically equivalent to the scalar channel treated in  with the sae average received SNR. Thus, observing that the instantaneous received SNR is given by P ((i)) (i), the average capacity of the channel in P (7), and also in (), can be defined siilarly to  as C = ax log + P () f ()d: () P (): P ()f () d=p P Thus, the coding theore and converse proven in  apply directly to the equivalent channel odel in (5), and the axiizing power adaptation rule is thereby easily shown to be the water-filling algorith ,  given as P () P = ; if (2) ; if where the cutoff value is chosen to satisfy the power constraint () as f () d =: (3)
3 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER Fro (), the capacity of the ultiple transit antenna syste under this optial power and rate allocation schee is then given by C = log f () d: (4) In order to evaluate the above capacity, we recall fro above that is the su of n squared coplex Gaussian rando variables. Hence, fro (9), is an n-erlang rando variable having the distribution function  f () = (n )! n exp ; > ; (5) where we have defined the average SNR as = P. N Fro (5), we see that this channel is, in fact, equivalent to a single transitter antenna syste with receiver antenna diversity and axial ratio cobining. However, in this case, we are transitting fro ultiple antennas. For exaple, given H, the actual transission schee is such that if the instantaneous value of (i) defined in (9) is greater than satisfying (3), then the transitted signals on the N T antennas are given by x = H H ~x, where ~x is the capacity-achieving ). On the other hand, if the instantaneous value of (i) defined in (9) is less than, then transission fro all antennas are cut off; i.e., no signal is transitted fro any antenna. Fro the equivalence of the scalar syste in (7) to the axial ratio cobining receiver diversity syste, it then also follows that the properties of the cutoff value and capacity expression given in  for the receiver diversity syste holds for this transitter diversity syste verbati. In fact, substituting (5) into (3) we get the equation that the cutoff value o ust satisfy to be signal for the syste in (7) with ~x 2 = P ( (n; ) (n ; )=(n )! (6) where (n; ) denotes the copleentary incoplete gaa function (a; x) = x et x a dt and = : (7) It was shown in  that there exists a unique, and thus a unique, that satisfies (6), and that this always satisfies. Specifically, li! =and li! =. In general, solution of (6) requires nuerical root finding techniques. However, it can be shown that for large (i.e., for large SNR), a reasonable approxiation for is given by n ; for : (8) + n Substituting (5) into (4) and following the sae steps as in  we can also obtain the equivalent capacity forula for the ultiple transit antenna syste with optial power adaptation to be C = log2 (e) n = P () + E() bits/channel use (9) where is given in (7), E() is the exponential integral function ,  defined as E() = and the Poisson su P () is e t dt t P () =e j= j j! : In Fig., we have shown the exact cutoff value and its approxiation given by (8). Fro Fig. it is clear that the above approxiation is indeed good for large and the approxiation becoes tighter as N T = n increases. In fact, if N T = n>3, the approxiation (8) becoes tight for all SNR > db, as one can see fro Fig.. Fig. 2 plots the exact capacity of the optial adaptive transission schee for the ultiple transit antenna syste with a single receiver antenna. Also shown in Fig. 2 is the capacity of a siilar syste with only receiver channel state inforation as derived in . Note that the asyptotic capacity of the receiver CSI only syste tends to log( + P N ) for large N T as shown in . We observe that when the CSI is available at both ends of the counication syste, large capacity gains are possible copared to a syste with only the receiver CSI. Of course, the price one has to pay for these large capacity gains is the outage probability deterined by the cutoff value. Still, when the delay caused by the outage is within acceptable liits it is possible to gain large capacity iproveents in ultiple transit antenna systes with the optial adaptive transission schee proposed above. B. Maxial Gain Transission Suppose now that instead of the above schee we eploy the sipler technique of choosing the transitter antenna corresponding to the largest channel gain coefficient H;n and then transit only on that particular antenna. We call this strategy the axial gain transission technique. In this subsection, we derive the capacity of this schee, the adaptive power allocation rule that achieves it, and show that, in fact, this schee is equivalent to a receiver diversity syste with selection cobining , . We also provide siple approxiations to the capacity and the cutoff value in this case. With this new transission schee received signal can be written as y(i) =h(i)x(i)+n(i) (2) where x(i) is the transitted signal at tie i (which can be on any antenna) and h(i) is the corresponding fading coefficient. Analogously to the previous case, we ay define (i) = P jh(i)j2 : (2) N Since h(i) = axfh; (i); H; 2(i);...; H;n(i)g, it is easily shown that the probability density function (pdf) of is given by f () = n exp exp n : (22) Coparing the pdf in (22) with the pdf of the received SNR of a receiver diversity syste with selection cobining, given in , we observe that, in fact, they are identical. Thus, substituting (22) into (3), we obtain an equation that ust be satisfied by the cutoff value of the adaptive transission rule that achieves the capacity in the axial gain transission schee to be n () n = exp(( + )) ( + ) E(( + )) = n (23) where is given by (7) and E() is the exponential integral defined earlier. Again, (23) is identical to the equation that deterines the cutoff for the selection cobining receiver diversity syste obtained in . As a result, properties of the cutoff value given in  directly applies to this syste as well. Specifically,. Substituting the series representation ,  (x) E(x) =E log(x) :! =
4 27 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER 23 Fig.. Approxiation to the optial cut-off value versus SNR (in decibels). N =. of the exponential integral function, where E is the Euler s constant (E = : ) , and after soe anipulations, we ay obtain the following nth-order approxiation to (23): n n + n = () n log( + ) + n n(n ) : Hence, for large, an (n )th-order approxiation to the cutoff value can be given as M where we have defined the constant M n as n M n = n () + n = +; for (24) log( + ): Thus, the capacity of a axial gain transission syste with N T = n antennas and a single receiver antenna is equal to that of a selection cobining receiver diversity syste with N R = n receiver antennas. However, it should be noted that in this case the codewords are transitted fro different transitter antennas at each tie instant depending on which antenna corresponds to the largest fading gain. This is soewhat siilar to a particular ipleentation of Bell Labs layered space tie (BLAST) architecture  where one periodically rotates the transit antennas. However, BLAST does not assue nowledge of fading coefficients at the transitter and thus there is no associated cutoff value, and the order of antenna rotation is predeterined. Using the results derived in  for the selection cobining receiver diversity schee, we have the capacity of axial gain transission schee n C = n () n = J (( + )); (25) where is given in (7) and the integral J p () is defined as in  to be J p () = t p log(t)e t dt; for p =; 2;... : (26) Using the fact that J () = E (), we ay write the capacity in (25) as n C = n () + n = E (( + )) : (27) + It can also be shown that for n and, the above capacity is well approxiated by C log() n E + log( )+n () n = log( + ) + for n ; and : (28) In particular, for >, the capacity asyptote for large N T is given by li n! C = log() [E + log( )] log(); for : (29) Fig. 3 plots the exact capacity expression in (27), evaluated with both exact and approxiate cutoff value. The figure shows that unless ;
5 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER Fig. 2. Exact and approxiate capacities for the optial adaptive transission versus SNR (in decibels). N =. both SNR and NT = n are sall, the capacity deviation due to the use of the approxiate cutoff value given by (24) is not significant. In fact, fro these plots it is evident that for N T as low as 4, the error due to the approxiation is negligible. Also, coparison of Fig. 3 with Fig. 2 illustrates the capacity loss due to the suboptial transission schee. IV. CAPACITY WITH MULTIPLE ANTENNAS AT BOTH TRANSMITTER AND RECEIVER We now turn to the situation in which there are ultiple antennas at both transitter and receiver ends. In this case, applying singular value decoposition of the atrix H in (2), we still have the equivalent channel odel given in (5). In analogy with (9) we ay define 3 P (i) = 3(i) (3) N and, as before, let the transit power vary with the observed channel state inforation subject to the average power constraint P. If we define Q ~ = ~x~x H, then the instantaneous transit power can be written as ~x H ~x = tr[ Q], ~ and the average power constraint becoes Eftr[ Q]g ~ P: Hence, in this case the adaptive transission strategy based on the observed channel state inforation can be achieved by letting Q ~ be a function of 3 (i). Thus, we denote the instantaneous value of Q(i) ~ as Q(3 ~ (i)). Then, we ay define the average capacity of the vector, tie-varying channel with adaptive transission schee to be C = ax E 3 log det I +3 Q(3 ~ ) ~Q(3 )>; tr(ef Q(3 ~ )g)=p (P=) 3 : (3) It can be shown that the above axiization is achieved by a diagonal ~ Q(3 ) and that the diagonal entries are given by a atrix waterfilling forula to be, for i =;...; ~Q i; i (P=) = where i for i =;...;are defined as ; if i i; (32) ; if i i; i = i (33) and i are the eigenvalues of the Wishart distributed atrix W defined in (3). We have also redefined as = P : (34) N The cutoff values i; in (32) are chosen to satisfy the power constraint P = tr(ef Q(3 ~ )g) = P i= i; i f ( i) d i (35) where f ( i) denotes the pdf of the ith nonzero eigenvalue of the Wishart atrix W. If we let f () denote the pdf of any unordered i, for i =;...;, then (35) leads to f () d = (36) where is the cutoff transission value corresponding to any eigenvalue. The probability distribution function p () of an unordered eigenvalue of a Wishart distributed atrix was given in , and can be
6 272 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER 23 Fig. 3. Capacity of the axial gain transission schee evaluated with both exact and approxiate cutoff values versus SNR (in decibels). N =. written as p () = e n = ( )! (n + )! L n () 2 (37) where the associated Laguerre polynoial of order, L n (), for, is defined by ,  L a () =! e (a) d d e a+ : (38) Then fro the definition in (33) we have that f () = p : (39) Substituting (39) in (36) and introducing a change of variable we see that the cutoff value ust satisfy = ( )! (n + )! e n L n () 2 d = (4) where is as defined in (7). In the following subsection, we show that for >, (4) has a unique solution. A. Uniqueness of the Cutoff Value Intuitively one would expect (4) to have a unique solution. In fact, by studying the properties of (4) we ay show that this indeed holds true. For convenience, let us define the integrand in (4) to be f n; (; z) = z e n L n () 2 : Next, define the function F (z) as ( )! F (z) = f (n + n; (; z) d : (4) )! = z Note that (4) is then equivalent to the case of F (z) =. Differentiating (4) with respect to z gives F (z) = ( )! (n + )! z 2 = z 2 e n L n () 2 d (42) and we iediately notice that, since the integrand in (42) is positive F (z) < ; for z>: (43) Siilarly, one can also show that F (z) > for z>. Next, either relying on the noralization property of a pdf or by explicitly recalling the integral equation [22, eq ] we have that li z! e n z = L n () 2 d (n + )! ; for n : (44) ( )! Using [22, eq ], for n >, we also have that li z! = e n z (n )(n + ) (n +)[( )!] 2 2 d dh F L n () 2 d n 2 ; n ; n +; 4h (+h) ( h)( + h) n h= for n > (45) ;
7 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER where F (a; b; c; x) is the hypergeoetric function defined as ,  F (a; b; c; x) = = (a) (b) x (c)! where the hypergeoetric coefficient (a) is defined as the product (a) = a(a +)(a + ) (46) with (a) =. Applying a transforation forula for a hypergeoetric function [22, eq ] to (45), we have for n > li z! z e n L n () 2 d (n + )! = ; for n >: (47) (n )( )! Substitution of (44) and (47) in (4) shows that, for n > li z! Siilarly, for n = li z! F (z) = F (z) =+; for n >: (48) = ( )! (n + )! li z! z E(z) =+; for n = (49) where E is Euler s constant and in the last step we have used the fact that li z log(z) =: z! Also, fro (4), it is easily seen that li F (z) =; for n : (5) z!+ Thus, fro (43) and (48) (5) it follows that for z>, the function F (z) has a unique zero for all n. Fro (7), then we see that for any > there exists a unique cutoff value for any n which satisfies (4), as we expected. B. Evaluation of the Cutoff Value for Multiple-Antenna Systes Substituting the polynoial representation into (4) we obtain = ( )! L a () = (n + )! p= p= q= () p + a p () p+q p!q! 2 n + q where we have defined the integral G p; q() to be G p; q () = e n+p+q d; p p! n + p (5) G p; q () = (52) for p + q =; ;...; 2( ): (53) Next, we consider the two cases of n > and n = separately in order to obtain an explicit solution to (52). ) Case : n >: Note that, when n >, for p + q = ;...; 2(), we have that n+p+q and n+p+q >. Then, we easily have that G p; q(z) = (n + p + q +;) (n + p + q; ); for p + q =; ;...; 2( ) and n > (54) where (a; x) is the copleentary incoplete Gaa function and we have also ade use of the integral identity e n d = n!e n j= j j! ; for n which can be verified straightforwardly via repeated application of integration by parts. Substituting (54) into (52) we obtain a closed-for equation that can be solved for a unique z (which is nown to exist by the previous section), in general, via nuerical root finding. However, as we did in the single receiver antenna case, we ay also obtain an approxiate solution for the cutoff by investigating sall behavior of (52). In fact, following a siilar procedure as in the case of a single receiver antenna, we ay show that B + B ; for (55) where we have defined the constants B and B 2 to be the sus ( )! () p+q B = (n + )! p!q! and B 2 = = 2 n + p 2 (n + p + q)! = ( )! (n + )! 2 n + p 2 (n + p + q )!: p= q= n + q p= q= () p+q p!q! n + q Note that for n> and =, B =and B 2 =, and thus, n (55) reduces to (8), as one would expect. 2) Case 2: n = : When n =, for p + q = ;...; 2( ), we still have that n + p + q. However, in this case n + p + q.forn =, (52) reduces to = p= q= () p+q p!q! p q and, siilarly, the integral G p; q () in (53) becoes G p; q () = Then, we can easily show that G p; q () = e p+q d; G p;q() = for p + q =; ;...; 2( ): (;) E (); if p + q = (56) (p+q+;) (p + q; ); if p + q>. On substituting (56) into (52), again we ay obtain a closed-for equation in that can be solved for a unique solution. It is also easily verified that this general equation reduces to the corresponding equation given in  for the case of N R = N T =.
8 274 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER 23 Fig. 4. Approxiation to the optial cutoff value versus SNR (in decibels). = 4. As we did earlier for the case of n >, we ay obtain an approxiate solution to the cutoff that satisfies (52), which will eliinate the need to perfor nuerical root finding. However, due to the singularity of the exponential integral function E () near zero, this becoes ore involved than the previous case. Still, after soe anipulations, we ay show that a reasonable approxiation for large is a2 + 2 a 2 2 4a a 3 2a (57) where we have defined the constants a ;a 2 ; and a 3 as a = 2 +D 3 a 2 = D 2 + [ +log() +2 E] a 3 = + D and where E is the Euler s constant and D ;D 2;D 3 are the following sus: () p+q D = p!q! p q = p= q= p+q6= 2 (p + q)!; () p+q D 2 = p!q! p q = p= q= p+q6= 2 (p + q )!; () p+q D 3 = p!q! p q = p= q= p+q6= 2 p + q (p + q )!: 2 Fig. 4 shows the typical behavior of the cutoff value for a ultiple-antenna syste with =4, along with the cutoff approxiations derived above. Note that, in the case of n =, the cutoff approxiation deviates considerably fro the true cutoff for sall values of. However, as we will see in the next section, even these cutoff values will be effective in approxiating the true capacity. It is clear that for all the other cases, derived approxiations to the cutoff value do closely estiate the true cutoff value for reasonably high SNRs and large n. Fro Fig. 4, we ay also observe that still lies in the range, and specifically! as!. C. Evaluation of Capacity Substituting (32) into (3), we obtain the capacity of the ultipleantenna syste C = log f () d; (58) where is the cutoff transission value corresponding to any unordered eigenvalue derived in the previous section, and f () is the pdf of any scaled, unordered eigenvalue given in (39). Using the explicit for of the pdf (39) and the representation of associated Laguerre polynoial given in (5), we can write (58) as C = ()! (n+)! = p= q= () p+q p!q! 2 n+ n+ J n+p+q+() (59) p q where J p (), for p = ; 2;...; is the integral defined in (6). The integral J p() can be evaluated in closed for and was given in  as p J p () =(p )! E () + j= P j() j : (6)
9 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER Fig. 5. Capacity of the ultiple-antenna syste with optial adaptive transission versus SNR (in decibels). = 4 (N = in the receiver CSI only syste). Substituting (6) into (59) we obtain the capacity of ultiple-antenna syste ( )! () p+q C =log 2 (e) (n + )! p!q! = p= q= n + n + 2 (n + p + q)! p q n+p+q P j (z) 2 E (z) + bits/channel use: (6) j j= It is easy to verify that for =, (6) reduces to (9) obtained previously for a syste with ultiple transit antennas and a single receiver antenna, as required. Fig. 5 plots the capacity of a ultiple-antenna syste for = 4 with different values of n versus the SNR. Shown on the sae figure is the capacity of the corresponding ultiple-antenna syste with only receiver CSI obtained in . While the capacity of a ultiple-antenna syste with CSI at both transitter and receiver is invariant to which end of the lin has the larger nuber of antennas, this is not the case with only receiver CSI. Thus, Fig. 5 specifically corresponds to the case when the receiver CSI syste has N T = and N R = n. Again, it is clear fro Fig. 5 that large capacity iproveents can be achieved with adaptive power and rate allocation when CSI is available at both ends of the syste as copared to the case when only receiver CSI is available. In Fig. 6, we have shown the capacity of the sae syste as that considered in Fig. 5, but this tie coparing it with a receiver-csi-only syste with N T = n and N R =. In this case, the receiver-csi-only syste has a lower capacity than in the previous case thereby resulting in a larger capacity gap copared to the adaptive transission syste. However, the capacity of the adaptive transission schee is invariant under the swapping of the transitter and receiver antennas and also is larger than either of the cases with only receiver CSI. Further, coparing these results with the capacity plots for N R =given earlier, we see that large capacity gains are available when ultiple antennas are used at both ends of the counications lin. In Fig. 7, we have shown the capacity evaluated with both the exact cutoff value and the approxiate cutoff value given by either (55) or (57) for a syste with =4. This figure shows that the derived approxiate cutoff values are indeed reasonable when the SNR is sufficiently large. Moreover, they confir the earlier rear that although the cutoff estiate given in (57) deviates fro the true cutoff ore than that of (55), the approxiation (57) nevertheless results in a reasonable capacity estiate. Fig. 7 shows that the capacity coputed with approxiate cutoff values tend to get closer and closer to the exact capacity either as SNR becoes large or the axiu of the nuber of antennas n grows. Finally, Fig. 8 plots the capacity versus the iniu nuber of antennas at one of the ends of the syste against a fixed but large axiu nuber of antennas n at the other end. Fig. 8 corresponds to n =8. As observed in the case when CSI is available only at the receiver, studied in  and , fro Fig. 8 we see that again the capacity is alost linear in the iniu nuber of antennas. In Fig. 8, we have also included the capacity approxiations coputed with the estiated cutoff values. Note that these capacity approxiations are in good agreeent with the exact capacities for the values of SNR and nuber of antennas considered.
10 276 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER 23 Fig. 6. Capacity of the ultiple-antenna syste with optial adaptive transission versus SNR (in decibels). = 4 (N = in the receiver CSI only syste). Fig. 7. Multiple-antenna syste capacity evaluated with both exact and approxiate cutoff values versus SNR (in decibels). =4.
11 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER Fig. 8. Multiple-antenna syste capacity evaluated with both exact and approxiate cutoff values versus iniu nuber of antennas (). n = 8. D. Outage Probability As reared earlier, the large capacity gains possible with the adaptive transission schee derived above, copared to the capacity of a syste with CSI available only at the receiver, coe at the price of channel outage. This is so because the optial adaptive power and rate allocation schee would not be transitting at all if all the observed i s were less than the cutoff value, thus resulting in channel outage. Hence, in order to put the extraordinary capacity gains offered by the power and rate adaptation schees in perspective, it is necessary to tae into account the associated outage probability values. In what follows, we provide a siple upper bound for this outage. We denote the largest eigenvalue of the Wishart distributed atrix W as ax and the outage probability of a ultiple antenna syste by P n; out. Then, it is easily seen that P n; out = f () d (62) where f () is the pdf of the largest eigenvalue of the Wishart atrix W. An upper bound for this pdf in the case of real Gaussian rando variables was derived in  for the case of = n. Following , we ay generalize this upper bound for any and n and show that in the case of coplex Wishart atrices f () (n)() n+2 e : (63) Fro (63) and (62), we have the following upper bound for the outage probability of the ultiple-antenna syste: P n; out [(n + ) (n + ; )] (n)() p : (64) Note that, for =, this upper bound for the outage probability in fact gives the exact value of the outage. This is clear by observing that for =the right-hand side of (63) reduces to the exact pdf for this case, given by (5). Fig. 9 plots this upper bound for the outage probability as a function of SNR for =2. As one would expect, the outage probability decreases with increasing SNR values. Also, it is clear fro this plot that the outage probability bound decreases rapidly when the axiu nuber of antennas n increases for a fixed. Unfortunately, though, the above bound becoes very loose when the SNR is low and the axiu nuber of antennas are large. Especially, in soe of these cases the right-hand side of (63) ay becoe larger than unity rendering it copletely useless. In order to circuvent this shortcoing, we ay derive another bound which is always less than or equal to unity. Note that this bound is valid only for the case of = n. In order to derive this bound, we denote the sallest eigenvalue of the Wishart atrix W by in. It is shown in  that when = n the pdf f () of in is given by f () = e. Since, P ; out P( in <), we have that P ; out e p 2 : (65) Cobining (64) and (65) we have that for = n P ; out in fp ;p 2g : (66) Fig. shows this upper bound for = n ultiple-antenna syste outage probability versus the SNR for different values of. The conclusion one can draw by observing these plots is that eploying ultiple antennas at both ends of the counication lin and adapting power and rate not only provides large capacity gains but also helps in decreasing the outage probability considerably.
12 278 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER 23 Fig. 9. Upper bound for outage probability of a ultiple-antenna syste versus SNR. = 2. Fig.. Upper bound for outage probability of a ultiple-antenna syste versus SNR. = n.
13 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER V. CONCLUSION We have considered the capacity of ultiple-antenna systes in Rayleigh flat fading under the assuption that CSI is available at both ends of the syste. First, we derived the capacity of such systes in the case when only the transitter is equipped with ultiple antennas. We showed that the capacity of this syste is, in fact, the sae as a receiver-only diversity syste with axial ratio cobining. We also proposed a transission diversity schee (axial gain transission) that is atheatically equivalent to a receiver-only diversity syste with selection cobining and evaluated its capacity. Next, we derived capacity expressions for a general syste with ultiple antennas at both transitter and receiver. We showed that the optial power allocation is given by a atrix water-filling algorith. We obtained an equation that deterines the cutoff value for such systes, which can be evaluated via nuerical root-finding, and a corresponding closed-for expression for the capacity with optial power and rate adaptation. We evaluated this capacity for soe representative situations and deonstrated siilarities with the capacity of such systes when CSI is available only at the receiver end. In all these cases, the only step that required nuerical techniques in deterining the capacity is the evaluation of the cutoff value.in order to circuvent this proble, we also derived approxiations to the cutoff value for all cases considered. Nuerical results show that these approxiations yield good capacity estiates when the SNR or the nuber of antennas is sufficiently large. Fro these capacity coputations for ultiple-antenna systes with adaptive transission techniques we observe that large capacity gains are possible copared to the receiver-csi-only systes. The tradeoff for these increased capacity values is the outage probability incurred by the adaptive power and rate allocation schees. We derived siple upper bounds for this outage probability and showed that the channel outage probability ay also be decreased by increasing the nuber of antennas. REFERENCES  T. Ericson, A Gaussian channel with slow fading, IEEE Trans. Infor. Theory, vol. IT-6, pp , May 97.  L. H. Ozarow, S. Shaai (Shitz), and A. D. Wyner, Inforation theoretic considerations for cellular obile radio, IEEE Trans. Veh. Technol., vol. 43, pp , May 994.  A. J. Goldsith and P. P. Varaiya, Capacity of fading channels with channel side inforation, IEEE Trans. Infor. Theory, vol. 43, pp , Nov  H. Viswanathan, Capacity of Marov channels with receiver CSI and delayed feedbac, IEEE Trans. Infor. Theory, vol. 45, pp , May 999.  I. C. Abou-Faycal, M. D. Trott, and S. Shaai (Shitz), The capacity of discrete-tie eoryless Rayleigh-fading channels, IEEE Trans. Infor. Theory, vol. 47, pp. 29 3, May 2.  J. S. Richters, Counication over fading dispersive channels, MIT Res. Lab. Electronics, Cabridge, MA, Tech. Rep. 464, Nov. 3, 967.  S. Shaai (Shitz) and I. Bar-David, The capacity of average and pea-power-liited quadrature Gaussian channels, IEEE Trans. Infor. Theory, vol. 4, pp. 6 7, July 995.  J. G. Sith, On the inforation capacity of pea and average power constrained Gaussian channels, Ph.D. dissertation, Dept. Elec. Eng., Univ. California, Bereley, 969.  J. G. Sith, The Inforation capacity of aplitude and variance-constrained scalar Gaussian channels, Infor. Contr., vol. 8, pp , 97.  G. J. Foschini, Layered space-tie architecture for wireless counication in a flat fading environent when using ulti-eleent antennas, Bell Labs. Tech. J., vol., no. 2, pp. 4 59, 996.  G. J. Foschini and M. J. Gans, On liits of wireless counications in a fading environent when using ultiple antennas, Wireless Personal Coun., vol. 6, pp , 998.  İ. E. Telatar, Capacity of ulti-antenna Gaussian channels, Europ. Trans. Telecoun., vol., pp , Nov  T. L. Marzetta and B. M. Hochwald, Capacity of a obile ultiple-antenna counication lin in Rayleigh flat fading, IEEE Trans. Infor. Theory, vol. 45, pp , Jan  M. S. Alouini and A. J. Goldsith, Capacity of Rayleigh fading channels under different adaptive transission and diversity-cobining techniques, IEEE Trans. Veh. Technol., vol. 48, pp. 65 8, July 999.  W. C. Jaes, Microwave Mobile Counications. Piscataway, NJ: IEEE Press, 994.  R. A. Horn and C. R. Johnson, Matrix Analysis. Cabridge, U.K.: Cabridge Univ. Press, 985.  P. Lancaster and M. Tisenetsy, The Theory of Matrices with Applications. Orlando, FL: Acadeic, 985.  A. T. Jaes, Distributions of atric variates and latent roots derived fro noral saples, Ann. Math. Statist., vol. 35, pp , June 964.  R. G. Gallager, Inforation Theory and Reliable Counication. New Yor: Wiley, 968.  A. Leon-Garcia, Probability and Rando Processes for Electrical Enginners. Reading, MA: Addison-Wesley, 994.  A. Erdelyi, Ed., Higher Transcendental Functions. New Yor: Mc- Graw-Hill, 953.  I. S. Gradshteyn and I. M. Ryzhi, Tables of Integrals, Series, and Products. New Yor: Acadeic, 965.  H. H. Goldstine and J. von Neuann, Nuerical inverting of atrices of high order II, Proc. Aer. Math. Soc., vol. 2, pp , 95.  A. Edelan, Eigenvalues and condition nubers of rando atrices, Ph.D. dissertation, Dept. Math., MIT, Cabridge, MA, 989. On the Separability of Deodulation and Decoding for Counications Over Multiple-Antenna Bloc-Fading Channels Yibo Jiang, Student Meber, IEEE, Ralf Koetter, Meber, IEEE, and Andrew C. Singer, Meber, IEEE Abstract We study the separability of deodulation and decoding for counications over ultiple-antenna bloc-fading channels when bitlinear linear dispersion (BL-LD) codes are used. We assue the channel is nown to the receiver only, and find necessary and sufficient conditions on the dispersion atrices for the separation of deodulation and decoding at the receiver without loss of optiality. Index Ters Multiple-antenna systes, space tie codes. I. INTRODUCTION We consider a ultiple-input ultiple-output (MIMO) counication setup with t transit antennas and r receive antennas. Inforationtheoretic results by Foschini and Gans  and Telatar  have spared treendous interest and effort in the design of practical channel codes for counications over ultiple-antenna channels (i.e., MIMO chan- Manuscript received October 29, 22; revised April 29, 23. This wor was supported in part by the National Science Foundation under Grants CCR , ITR , and CCR , and in part by the Motorola Center for Counications. The aterial in this correspondence was presented at the IEEE International Syposiu on Inforation theory, Yoohaa, Japan, June/July 23. The authors are with the Coordinated Science Laboratory, Departent of Electrical and Coputer Engineering, University of Illinois at Urbana-Chapaign, Urbana, IL 68 USA (e-ail: Counicated by B. M. Hochwald, Guest Editor. Digital Object Identifier.9/TIT /3$7. 23 IEEE
Journal of Machine Learning Research (2) 245 279 Subitted 2/; Published 8/ Bayes Point Machines Ralf Herbrich Microsoft Research, St George House, Guildhall Street, CB2 3NH Cabridge, United Kingdo Thore
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