IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 8,...

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 8, AUGUST 2009 3651

The Effect of Finite Rate Feedback onCDMA Signature Optimization and

MIMO Beamforming Vector SelectionWei Dai, Member, IEEE, Youjian (Eugene) Liu, Member, IEEE, and Brian Rider

Abstract—We analyze the effect of finite rate feedback oncode-division multiple-access (CDMA) signature optimization andmultiple-input multiple-output (MIMO) beamforming vector se-lection. In CDMA signature optimization, for a particular user, thereceiver selects a signature vector from a codebook to best avoidinterference from other users, and then feeds the correspondingindex back to the specified user. For MIMO beamforming vectorselection, the receiver chooses a beamforming vector from a givencodebook to maximize the instantaneous information rate, andfeeds back the corresponding index to the transmitter. Thesetwo problems are dual: both can be modeled as selecting a unitnorm vector from a finite size codebook to “match” a randomlygenerated Gaussian matrix. Assuming that the feedback link israte limited, our main result is an exact asymptotic performanceformula where the length of the signature/beamforming vector, thedimensions of interference/channel matrix, and the feedback rateapproach infinity with constant ratios. The proof rests on the largedeviations of the underlying random matrix ensemble. Further,we show that random codebooks generated from the isotropicdistribution are asymptotically optimal not only on average, butalso in probability.

Index Terms—Beamforming, code-division multiple access(CDMA), finite rate feedback, large deviations, multiple-inputmultiple-output (MIMO), random matrix theory, signatureoptimization.

I. INTRODUCTION

I N a direct-sequence code-division multiple-access(DS-CDMA) system, the performance is mainly lim-

ited by interference among users. We assume that the receiver(base station) has perfect information of all users’ signature. Fora particular user, the receiver selects a signature to minimize theinterference from other users, and then feeds the correspondingindex to the specified user through a feedback link. Dually,consider a multiple-input multiple-output (MIMO) system with

Manuscript received December 31, 2006; revised January 21, 2009. Currentversion published July 15, 2009. The work was supported in part by the NSFby Grants CCF-0728955, ECCS-0725915, DMS-0505680, and DMS-0645756,and by Thomson Inc. The material in this paper was presented in part at theConference on Information Sciences and Systems (CISS), Princeton, NJ, March2006.

W. Dai is with the Department of Electrical and Computer Engineering, Uni-versity of Illinois, Urbana, IL 61801 USA (e-mail: [email protected]).

Y. (E.) Liu is with the Department of Electrical and Computer Engi-neering, University of Colorado, Boulder, CO 80309-0425 USA (e-mail:[email protected]).

B. Rider is with the Mathematics Department, University of Colorado,Boulder, CO 80309-0425 USA (e-mail: [email protected]).

Communicated by H. Boche, Associate Editor for Communications.Digital Object Identifier 10.1109/TIT.2009.2023717

beamforming vector selection. Take the Rayleigh block fadingchannel model, and assume that the receiver knows the channelstate matrix perfectly. To aid the transmitter, the receiverchooses a beamforming vector from the codebook to maximizethe instantaneous information rate, and then feeds back thecorresponding index to the transmitter. In both scenarios, weconsider a finite feedback rate up to bits. Ideally, if thefeedback rate is unlimited, the transmitter is able to obtaininterference/channel information with arbitrary accuracy, butthis is not practically feasible and it is essential to real systemsto understand the effect of finite rate feedback.

This paper is the first to rigorously obtain exact asymptoticperformance formulas for both problems in the regime wherethe signature/beamforming vector, the dimensions of interfer-ence/channel matrix, and the feedback rate approach infinitywith constant ratios. The same set-ups have been consideredpreviously in [1]–[3], in which a one-sided bound was presented(this was a lower bound on the CDMA performance, and anupper bound in the case of MIMO). Our approach is fundamen-tally different. Identifying the underlying problem as a largedeviation question for the connected random matrix ensembleprovides a unified framework to handle both CDMA andMIMO simultaneously.1 Further, while [3] discusses the factthat random codebooks are asymptotically optimal on average(their mean performance is the best achievable performance),here we prove the stronger result that random codebooks areasymptotically optimal in probability.2

The paper is organized as follows. After describing the systemmodels in more detail, Section III presents various needed factsfrom random matrix theory. Section IV contains the main re-sults. The basic convergence result is Theorem 1, which in turnis based on a random codebook version, Theorem 2, along witha separate argument that any given codebook will not asymp-totically outperform its random counterpart. This section con-cludes with the almost sure optimality, Theorem 4. Once again,all this is based on a large deviation principle for the spectrumof a Wishart type random matrix. That proof is found in theappendices.

1The analysis in [3] is based on extreme order statistics, applied to the case of� � � independent and identically distributed (i.i.d.) random variables with afixed distribution. The laws of the underlying random variables for the problemsat hand however depend on � in an essential way; attempting a proof throughi.i.d. order statistics results in needless complications.

2We add that, based on our conference presentation in [4], the authors of [3]have gone on to refine their own estimates in [5].

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3652 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 8, AUGUST 2009

II. SYSTEM MODEL

A. CDMA Signature Optimization

In a sampled discrete-time symbol-synchronous DS-CDMAsystem with users, the received vector can be written

where and are the transmitted symbol andthe signature vector for user , respectively, andis the additive white Gaussian noise vector with zero mean andcovariance matrix . Throughout this paper, we assume thatthe transmitted symbols ’s are independent and subject to unitpower constraints, i.e., , . The signaturevectors satisfy ; their length is oftenreferred to as processing gain in [3], [6], and [7].

We focus on matched filter receivers, in which the interfer-ence seen by user 1 is given by

where contains signature vectors of all otherusers. The information rate of user 1 is, therefore

(1)

Remark 1: This differs from the rate achieved by MMSEreceivers [6]–[8]. However, our methods carry over to ana-lyzing an MMSE receiver. Each appearance of , in say(6) below, is replaced by , and the proof maybe followed verbatim except for the few obvious (and trivial)modifications.

The signature optimization problem is as follows. Assumingthat the receiver has perfect knowledge of the ’s, it guidesa particular user, say user 1, to avoid the others’ interference.Here, a codebook of signature vectors is declared to both thereceiver and user 1. Given the other signatures , thereceiver selects

(2)

The corresponding index is then fed back to user 1 through afinite rate feedback link, with rate up to bits. The finitefeedback rate imposes a constraint on the size of the codebook

. Therefore, the average interference for user 1 isgiven by

(3)

Note that the signature vector is chosen to minimize the inter-ference to user 1 only; we are not considering a joint feedbackstrategy.

B. MIMO Beamforming Vector Selection

The signal model for a MIMO system with beamformingvector selection is

where is the received signal vector, isthe channel state matrix, is the beamforming vectorsatisfying is the transmitted signal with unit powerconstraint is the white Gaussian noisevector with mean zero and covariance . The dimensions

and are the numbers of antennas at the transmitter andreceiver.

In the above setting, beamforming vector selection proceedsas follows. We assume the Rayleigh block fading channelmodel: the channel matrix is comprised of independent iden-tically distributed (i.i.d.) complex Gaussian entries with meanzero and mean-square one remains constantwithin a fading block, and is independent from one block toanother block. In each fading block, the receiver estimates thechannel state via pilot signaling. We assume that this estimateis perfect. Now suppose that there is a codebook containing

candidate beamforming vectors declared to both trans-mitter and receiver. The receiver selects a beamforming vectorfrom the codebook to maximize the instantaneous informationrate

(4)

and then feeds the corresponding index back to the transmitterthrough a feedback link (with bits per fading block). Fur-thermore, we assume that the channel estimation and feedbackhappen at the beginning of each fading block. The introduceddelay, compared to the length of fading block, can be neglected.Feedback models of this type have been widely studied in [2],[9]–[12]. The average received signal power is

and the average information rate is given by

(5)

This equality is due to the factfor any matrices and for which both sides are defined.

C. Unified Formulation

The average interference introduced in (3) and the averagereceived power involved in (5) are difficult to quantify as such.However, when and approach infinity with constantratios, a precise analysis is possible. In particular, let

be a codebook (a discrete subset of the unit


DAI et al.: FINITE RATE FEEDBACK ON CDMA SIGNATURE OPTIMIZATION 3653

sphere in ), and let be a random Gaussian matrixwith i.i.d. entries. Define

(6)

(7)

As with with and, we show that and converge, and also compute

their limits (Section IV). Given the limiting and ,the asymptotic behavior of the average information rate will becharacterized in Section IV.

Remark 2: We have assumed that has i.i.d. en-tries for a unified formulation while the matrixin CDMA signature optimization is composed of independentand isotropically distributed columns. Importantly, the asymp-totic spectral statistics of and are the same as

[13, Lemma 1], and so the limit of providesthe asymptotic average interference for a fixed user in CDMA.

III. PRELIMINARIES

Henceforth we will use to denote the joint proba-bility measure of the random variables where the sub-script emphasizes the dimensionality. For example, let

be the standard Gaussian random matrix,be the corresponding singular value decomposition, and

be the corresponding singular value vector. Thenis the probability measure of is the

induced joint probability measure of and , and is theinduced marginal probability measure of .

A. Asymptotic Random Matrix Theory

The performance calculation is based on the asymptotic spec-tral distribution of the matrix . Let be theeigenvalues of and define their the empirical distribu-tion function

As with

(8)

weakly almost surely, where and. The right-hand side (RHS) is known

as the Marcenko-Pastur law; a standard reference is [14]. Forlater it is useful to define

ifif

and

The asymptotic properties of the minimum and maximumeigenvalues will also figure into our analysis. For any finite ,set

and

Proposition 1: Let with .1) and almost surely.2) All moments of and converge.3) For all measurable sets such that ,

we have .Proof: The almost sure convergence goes back to [15],

[16]. The convergence of moments is implied by the tail estimate

see [17] which contains like estimates for the deviations of. The last claim of this proposition follows from these

facts and the Cauchy-Schwarz inequality

Last, consider a linear spectral statistic

Our results will hinge on the following concentration property.

Proposition 2: Let have bounded Lipschitznorm on a neighborhood of . Then

(9)

almost surely.Proof: By Proposition 1(1), we may work on the event

for small enough so that the restricted eigenvalues lie withinthe neighborhood . There, we may replace by a whichequals on and is otherwise extended to be both boundedand globally Lipschitz, with Lipschitz norm . (Note

.) To such a , [18, Corollary 1.8] applies

So the claim would follow by Borel–Cantelli granted

and that is implied by (8).

B. Isotropic Distribution

We will make use the isotropic distribution on the Stiefelmanifold . This isthe unique probability measure such that for any measurable



set and for all and[19], [20, Secs. 2 and 3], and

may be realized as the push forward of the Haar measure on, which is invariant on .

IV. MAIN RESULTS

For all , define

otherwise

(10)

and

(11)

for any .The map satisfies the basic properties of a good

rate function: it vanishes at , is decreasing/increasingto the left/right of this point, and also tends to infinity as

or . (Proposition 5 in Appendix A contains detailedstatements and proofs of these points.) It is also this case whichfigures into our basic convergence result for and .

Theorem 1: Let and approach infinity such thatand . Then, there exist unique

and for whichand

Remark 3: An analysis of shows thatand as , while and

as . This is consistent with intuition:and representing either no information, or perfect

information.

Using ideas from [7], we also obtain fairly explicit formulasfor and .3

Corollary 1: Set

for

and

for

Then

if andotherwise

(12)

3This computation is also considered in [5], but from a different vantage point.

and

ifotherwise

(13)

Granted the existence and uniqueness of and , the proofof Theorem 1 takes the following course.

First, by computing the average performance of randomcodebooks, we obtain upper and lower bounds onand , respectively. Let be a random code-book of i.i.d. isotropically distributed unit-norm vectors.That is, , where

and are i.i.d. for alland . Let also

and

Then we have the following.

Theorem 2: As with and

and

Clearly, and.

Moving to the lower bound on and the upperbound on , recall the singular value decomposition,

where and is thediagonal matrix of eigenvalues . It is well knownthat is isotropically distributed and independent of [20],[21], [22, p. 37]. Now for a codebook , let

(14)

and

(15)

denote the conditional averages given . It is clear

and



Theorem 3: For with and

and

both with probability one. In turn

and

Here and below, the almost sure convergence may be con-sidered to take place on the probability space generated bya doubly-infinite array of ’s, on which all ’s live.Hence, Theorems 2 and 3 combine to yield Theorem 1. Asa byproduct, Theorem 2 implies that random codebooks areasymptotically optimal on average. In fact, they are also asymp-totically optimal in probability.

Theorem 4: Let again withand . Then, for any

and

Remark 4: While Theorem 4 has obvious appeal, randomcodebooks are not so practical to implement due to their built-inhigh computational complexity. The number of algebraic op-erations required to evaluate the and functions in (2)and (4) increases exponentially with the feedback rate ,and therefore increases exponentially with in our asymptoticregime. To reduce the computational load, pseudorandomcodebooks or codebooks with certain structures are preferredin practice.

We close with the following.

Corollary 2: The asymptotic average information rate of theCDMA system (1) is given by

(16)

while that of the MIMO system (5) satisfies

(17)

The proofs of Theorem 2–4 occupy the nextSections IV-A–IV.C. The key step is a large deviationprinciple established in Theorem 5 in Appendix B. Last,Corollaries 1 and 2 are proved in Appendices C and D,respectively.

A. Average Performance of Random Codebooks

As considerations for and follow thesame line, we only give the details for the limit behavior of

. For that, the bounds

(18)

and

(19)

are established separately.We start by expressing in a more convenient form.

Recalling the decomposition , write

The last equality follows from the fact that and have thesame law for any given unitary .

Note that are i.i.d. exponentialrandom variables of mean one; we will henceforth denote

. Also observe that, for , the vari-ables are conditionally independentgiven . On account of this, we omit the subscript and definethe corresponding conditional probability measure

(20)

in terms of which

Thus



and

(21)

This last expression turns attention to the large deviations of. In Theorem 5 we will show: for with

and for all

(22)

almost surely (on the probability space described after Theorem3, the left-hand side (LHS) is a function of the sequence of ’s).With this we may return to (18) and (19).

1) Proof of (18): Set aside any small for which. Since [Proposition 5(4)],

there exists a such that and

. Define

By (22) and the almost sure convergence of and(Proposition 1), . On the event

with large enough

and so also

Hence, again for large enough

Now take , then to conclude

Substituting into (21) and recalling that(Proposition 1) yields (18).

2) Proof of (19): Similar to the above, choose smallenough such that . Now using[again Proposition 5(4)], there exists a with

and . Define theevent

Then , and on it holds

So, when is sufficiently large and again on

Next write

(23)

in which denotes the complement of . The first termis less than



for large enough. More simply, the second term in (23) isbounded by which goes to zero asby Proposition 1. Letting and then produces

and substitution into (21) completes the proof.

B. Uniform Bounds for Arbitrary Codebooks

Here we prove Theorem 3. Recall the basic quantities ofinterest (14) and (15), along with the conditional distribution

is defined in (20). We first prepare two lemmas.

Lemma 1: With probability one

and

Here and are defined as the (almost surely) uniquevalues such that

and

Proof: We detail the lower bound for ; the upperbound for follows suit.

First note that since the eigenvalues are almost surely distinctand the ’s have continuous distribu-

tions, is a.s. strictly increasing on .Thus the uniqueness of and .

Next, for any given of size , define the distribution

is the corresponding mean. The result is implied bythe inequality

(24)

which says that a random variable having the LHS as its distri-bution stochastically dominates a random variable with distri-bution given by the RHS (both variables here may be taken tobe supported on ). In symbols

As for (24), by definition and the rest isjust the union bound

Here, denote the individual elements of and the lastequality holds since, for fixed and isotropically dis-tributed is isotropically distributed.

Lemma 2: For in the same relationship asLemma 1,

and

almost surely.Proof: This is a direct consequence of the strict mono-

tonicity of [Proposition 5(4)]. We spell things out for( being much the same).

First, by the definition of

Next, with decreasing on , for all smallenough ( necessarily), there existsa for which

. Also, by (22)

(25)

and

(26)

almost surely. Clearly, the last three displays would stand incontradiction if remained above as .

Remark 5: Along the way we learn that

(27)

which holds on account of (25) and the bound (24). We willreturn to this point during the proof of Corollary 2.

Proof of Theorem 3: We only provide details for the lowerbounds.

For the first half, choose an small enough such that. Since , there is a so

that . Also define the event



By Proposition 1, Lemma 2, and (22): .Further, on for all large enough it holds

Since was arbitrary, the claim is proved.For the second half, define for any

By the first half, . So, for all sufficientlylarge

and again we can now take .

C. Asymptotic Optimality of the Random Codebooks

At last we come to the proof of Theorem 4. As before, it isenough to focus on the case.

While the proof of Theorem 2 rests on the probabilitymeasure , we now require the measure

. These two measures are connected bythe joint measure : for any measurable set

We first show that for any

(28)

Note that . Let be such that, and let

Then by (22). Thus, for large enough

This is (28) once .For the last step prepare the following. Given , let

be such that . Notice that

satisfies

If this were not the case, we could choose a subsequenceover which . Over the same subsequence

(with corresponding ) it would hold

but that contradicts (28).Finally, on the set with large



Fig. 1. � . (a) � � �� (b) � � �.

Here, holds since for whatever and, and uses the bound

valid for sufficiently large . Theorem 4 is proved.

V. SIMULATIONS

Figs. 1 and 2 give simulation results for several CDMAsystems and MIMO systems, respectively. In both figures, the

axis is the normalized feedback rate . The axis inFig. 1 is the and that in Fig. 2 is the . The dashed

lines with x markers are for random codebooks while the solidlines with plus markers are for well designed codebooks, whichare numerically generated by the criterion of maximizingthe minimum chordal distance of the codebook. The solidlines without any markers are the asymptotic performanceby Corollary 1. Simulations show that as increaseproportionally, the performance ( and ) will getcloser to the asymptotic one. Although random codebooks arenot optimal for finite dimensional systems, as tendto infinity in the manner discussed, the difference betweenrandom codebooks and well-designed codebooks decreases.



Fig. 2. � . (a) � � �� (b) � � �.

VI. CONCLUSION

In this paper, we analyze the effect of finite rate feedback

on CDMA signature optimization and MIMO beamforming

vector selection. The main results are the exact asymptotic

performance formulas. In addition, we prove that random code-

books are asymptotically optimal not only on average but also

in probability. The proofs rest on a large deviation principle

derived over a random matrix ensemble.

APPENDIX

A. Properties of Rate Functions

Let be an exponential random variable of mean one, andlet with corresponding distribution

ifotherwise

(29)

Recall as well the Marcenko–Pastur law from (8).



Define the moment generating functions

(30)

(31)

which, as we will show, exists as a monotone limit. Also, for alland , define the rate functions

(32)

and

(33)

The next four propositions lay out some basic properties ofthe ’s and the ’s.

Proposition 3: (Properties of the ’s):1) For all satisfies the

conditions of Proposition 2.2) For all

if

otherwise

(34)

3) is strictly convex for allis strictly convex for and

.4) For all , if is sufficiently large, there exists an

such that .Furthermore, if if .

5) For all and

.

Proposition 4: (Properties of ):1) .2) For all , if is large enough, there exists a

such that and .More specifically, when , and when

.3) For all and sufficiently large , let

and be achieved at and , respec-tively. Then

where equality holds if and only if .

Proposition 5: (Properties of ):1) .2) Let . For

For all ,

3) .4) strictly decreases on and strictly in-

creases on .5) As or .6) For any , there are unique and

such that .And last, the following.

Proposition 6: Let . For all.

1) Proof of Proposition 3:1) Let

. By dominated con-vergence we can differentiate inside the integral toproduce the bound

using the nonnegativity of the integrand in the numerator.2) The monotonicity of in is

obvious from its definition; the limit of asthen exists as an extended real number.

3) For any nondegenerate random variableis strictly convex on its domain of definition, see for in-stance, [23, Lemma 2.2.31 ]. (While the reference onlystates convexity via Hölder’s inequality, the inequalityis easily seen to be strict by the criteria for equalityin Hölder’s bound.) In the setting here this applies to

, and the additional averagingover does not effect the (strict) convexity. Also,restricted to and

and the same argu-ments hold.

4) For short, denote . So

and

(35)

by dominated convergence. When , then and

(36)

To evaluate , note that

onon

andonon



Applied to the integrand in (35) we find that

onon ,

and

onon .

Therefore

and

which, along with (36), complete the proof.5) For all and , there

exists an such that . Hence theabsolute value of the integrand in (35) is bounded above by

for sufficiently large; there being a large enough sothat for all . As is integrableunder , and since

dominated convergence yields .2) Proof of Proposition 4:

1) Quite simply, .2) This follows directly from Proposition 3(4) and the strict

convexity of [Proposition 3(3)].3) Let and be such that and

. Clearly,and . If , thenand

where is from the mean value theorem for some, and follows from the strict

convexity of : if , then and; if , then and

.3) Proof of Proposition 5:

1) This is the same as Proposition 4(1).2) We restrict attention to ; the case is

dual. First

because if . Next, forany ,

while when the is restricted to. Since is contin-

uous on , it suffices to take supremums

over .

3) By part (1) . On the other hand

and, therefore, . We con-clude .

4) If ,

where follows from restricting the range of , anduses and . An identical argumentshows if .

5) In order to prove that implies , picksequences and

Then is well defined for all

and it is enough to show that . First, whenwe have and

Since



for any , the first term above remains finite forand so tends to infinity with .

If instead , then , and

The third term is clearly under control and so nowtends to infinity with . For the other endpoint,choose and

and use the obvious complimentary argument.6) This follows from Proposition 4 (3)–(5).4) Proof of Proposition 6: First notice that, since

monotonically increases as [Prop. 3(2)], wealways have that

The work comes in showing that

(37)

Introduce the following shorthand:. Also let be the point where

which is well defined according to Proposition 4(2). Recall thatis strictly concave on . We will

treat the three cases in turn:1) is achieved at some ;2) ;3) .

Repeated use will be made of the following elementary fact. Ifis strictly concave on an open interval , then

(38)

for all .For the first case start with the relation . Since

is strictly increasing on (Proposition 3(3)),there exists an such that

and for all . Note that

[Proposition 3(5)] and is strictly increasing. Whenis sufficiently large,

(39)

for all . Let be achieved at .Then

(40)

for sufficiently large . At the same time, since

and is monotonically increasing in , we have

(41)

for sufficiently large . As a result, there exists an such thatfor all

Letting , produces (37) in this case.For the second case notice that the strict concavity of

implies that strictly increases as increases, and

for all . So, there exists a constant such that

It also can be verified that is finite:follows from the fact that and again strictconcavity; the finiteness is from the uniform upper bound

for all . Now take any and letbe achieved at . Note that

and . So, when is sufficiently large, wehave

(42)

and

(43)

On the other hand, since



and , we have

(44)

for sufficiently large . At the same time, since

and

(45)

As a result, there is an so that for

Again take .The third case is dual to the second case, and so running

through a similar set of arguments will complete the proof.

B. Large Deviations

Here we prove our fundamental large deviation principle for.

Theorem 5: Let with . For any,

(46)

almost surely. Similarly, for any

(47)

almost surely.Proof: The proof uses the strategy behind the Gartner–Ellis

Theorem (see [23, Sec. 2.3]). We only report the details for, as the goes through similarly.

As is typical, the upper bound follows from Chebyshev’s in-equality. Take an

almost surely. The last equality rests on the fact that, given ,for all satisfies theconditions of Proposition 2 (with an perhaps depending crit-

ically on ). Continuing, after taking supremums over in thedesignated range

again almost surely, where we have used Proposition 5(2).For the lower bound we bring in the truncated random vari-

ables . Plainly

Assume that we knew that

(48)

Then, since by Proposition 6 and, the proof of Theorem 5 would be com-

plete. We will actually prove

(49)

almost surely, which readily implies (48).The whole point of the introduced truncation is to enable the

standard change of measure argument. Let be such that

(50)

or equivalently, . Such a exists according toProposition 4(2). Noting that definethe new probability measure

Next let



Then

(51)

Now, by Propositions 2 and 3(1), we have

almost surely. Thus, the first two terms of (51) combine to pro-duce the desired lower bound of (recall the choiceof from (50)). Hence, it remains to show that

(52)

almost surely.For (52), writing

shows it is enough that the second term on the right is boundedaway from one. We will employ the upper bound to prove thatin fact

(53)

with probability one.Indeed, we know that

(54)

almost surely. Also, setting

and

(55)

inherits the strict convexity of (Proposition3(3)). Further, since for large enough

ififif

for in the same range. Therefore, is achievedat an and

where follows from (55), from the fact that, and by Proposition 4(3).

Similarly, is achieved at an and. So, by the same Chebyshev argument used in

our upper bound

(56)

and

(57)

almost surely. That is, (53) is verified and the proof of the The-orem is complete.

C. Proof of Corollary 1

We first record the fact, established by Verdu [7].

Lemma 3: [See [7, eq. (9) and (90)]]. With as in (8) andall

(58)

and

(59)

in which

Next, we identify the at which the supremum definingis achieved.



Proposition 7: For , let satisfy. Then

ifif andotherwise

(60)

Proof: Since is strictlyconcave on [Proposition 3(3)], can befound by differentiation. There are obviously three possibilities

1) If on , then .

2) If on , then .

3) Otherwise, satisfies .Begin by observing that

(61)

When , the above is particularly simple

(62)

Otherwise, when , we have

(63)

where in the last line have used (58).Consider first , and examine at the

boundary points and . Notice that,

for the second point, it must be assumed that soand hence . Using the definition of it can be verifiedthat

which is strictly positive if and only if , and

which is strictly negative if and only if whichenforces . Looking back to (63), we have accounted forthe first two cases.

Next we identify the situation where there existsfor which . Still as-

suming , (63) implies when we have

Letting , this may be written out as in

Since , we can clear the denominators in the above, iso-late the radical, and square both sides to produce the quadraticequation

The roots here are and . As the former has

been ruled out in advance we conclude that ,or

(64)

Now return to the case that . Recall the threepossibilities for set out at the beginning of the proof. Thesecond possibility (that is at the right endpoint) can beexcluded immediately. Therefore, . Further,we can show . Suppose to the contrary that

. Then . By our assumption

and so . On the other hand, the factthat and

implies that according to (62), and that con-tradicts that . In summary, we have shown that

and . It must then be that

. From (62) we then learn thatand therefore and .

But this agrees with the claimed formula (64) [and so the finalcase in (60)] when setting . The proof is finished.

We are now ready to pick up the proof of Corollary 1, againwe will treat separately the cases identified by (60).

For the first case:

(65)

in which the last equality follows from (59). Therefore

At the same time, the assumption implies that



according to (65). This yields the first part of (13).For the second case: and

where we work on the principle branch of the logarithm andagain the last step uses (59). Therefore

and

producing the first part of (12).For the last case of (60)

once more using (59). This establishes the second parts of (12)and (13), completing the proof of the full statement.

D. Proof of Corollary 2

In both cases we set without any loss of generality.We first prove (16). Since is convex on

by Jensen’s inequality. Hence

(66)

as we know that

On the other hand, given any there exists sufficientlylarge so that

where is defined in Section IV-B. The firstinequality follows from splitting the integration domain into twoparts:

and

The second inequality uses (25). This yields

(67)

and taking after the fact gives the complementary upperbound for (16).

For MIMO systems, the upper bound on the average informa-tion rate is proved by Jensen’s inequality

(68)

since is increasing.The MIMO lower bound will be proved by random coding

argument. In order to show

it is sufficient to prove that



or, what is the same

(69)

Next, for sufficiently small, there is a such that

(70)

[Proposition 5(4)]. Set

(71)

and write

(72)

By Theorem 5

In fact

for some constant . To see this, note from the proof ofTheorem 5 that for all large enough is contained in theevent that both and are within some of theirlimits, and the appropriate (Lipschitz) linear statistics is alsoclose to its (almost sure) limit. The probability of the comple-ment of the latter event is exponentially small in (see the proofof Proposition 2). Thus, the second term in (72) tends to zero.

As for the first term in (72): for large , on the set , wehave

where is from the independence of is fromthe definition of in (71), recalls ,

and uses (70). Therefore, for sufficiently large , on the set

(73)

Substituting (73) into (72) and taking completes the proof.

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Wei Dai (S’01–M’08) received the M.S. and Ph.D. degrees in electrical andcomputer engineering from the University of Colorado at Boulder in 2004 and2007, respectively.

He is currently a Postdoctoral Researcher with the Department of Electricaland Computer Engineering, University of Illinois at Urbana-Champaign. His re-search interests include compressive sensing, bioinformatics, communicationstheory, information theory, and random matrix theory.

Youjian (Eugene) Liu (M’04) received the M.S. and Ph.D. degrees in elec-trical engineering from the Ohio State University, Columbus, in 2001 and 1998,respectively.

Since August 2002, he has been an Assistant Professor with the Departmentof Electrical and Computer Engineering, University of Colorado at Boulder.From January 2001 to August 2002, he worked on 3G mobile communicationsystems as a Member of Technical Staff with Wireless Advanced TechnologyLaboratory, Lucent Technologies, Bell Labs Innovations, Whippany, NJ. Hisresearch interests include MIMO communications, coding theory, and informa-tion theory.

Dr. Liu is a recipient of the 2005 Junior Faculty Development Award at theUniversity of Colorado.

Brian Rider received the Ph.D. degree in mathematics from the Courant Insti-tute, New York University, in 2000.

After a Lady Davis Fellowship at the Technion–Israel Institute of Technology,Haifa, he had Postdoctoral positions with Duke University, Durham, NC, andMSRI. Since 2004, he has been an Assistant Professor of Mathematics with theUniversity of Colorado at Boulder. His research interests include random matrixtheory and spectral properties of random Schroedinger operators.

Dr. Rider is a recipient of a 2007 NSF CAREER Grant, as well as a 2008Rollo Davidson Prize.


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