asymptotic performance of mmse receivers for large systems using random matrix theory

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 11, NOVEMBER 2007 4173

Asymptotic Performance of MMSE Receivers forLarge Systems Using Random Matrix Theory

Ying-Chang Liang, Senior Member, IEEE, Guangming Pan, and Z. D. Bai

Abstract—Random matrix theory is used to derive the limitand asymptotic distribution of signal-to-interference-plus-noiseratio (SIR) for a class of suboptimal minimum mean-square-error(MMSE) receivers applied to large random systems with un-equal-power users. We prove that the limiting SIR convergesto a deterministic value when K and N go to infinity withlimK=N = y being a positive constant, where K is the number ofusers and N is the number of degrees of freedom. We also provethat the SIR of each particular user is asymptotically Gaussianfor large N and derive the closed-form expressions of the variancefor the SIR variable under real-spreading and complex-spreadingchannel environments. It is revealed that for a given (K;N)pair, under certain mild conditions, the variance of the SIR forcomplex-spreading channels is half of that for the correspondingreal-spreading channels. Since the suboptimal MMSE receiverbecomes optimal for the case when the users are equally powered,our results show that the conjecture made by Tse and Zeitouni forthe complex-spreading case is not affirmative. We also derive theasymptotic distribution for SIR in decibels which provides betterdescription when N is small. Numerical results and computersimulations are provided to evaluate the accuracy of variouslimiting and asymptotic results obtained in this paper.

Index Terms—Code-division multiple access (CDMA), large sys-tems, minimum mean-square-error (MMSE), multiple-input mul-tiple-output (MIMO), random matrix theory, random spreading.

I. INTRODUCTION

RANDOM matrix theory (RMT) has found wide applica-tions in wireless communications [3], [12], [14]. Using

limiting spectral properties of large random matrices, it wasshown that for frequency-flat, synchronous code-division mul-tiple access (CDMA) uplink with random spreading codes,the output signal-to-interference-plus-noise ratios (SIRs) usingwell-known linear receivers such as matched filter, decorrelator,and minimum mean-square-error (MMSE) receiver, converge todeterministic values for large systems, i.e., when both spreadinggain and number of users go to infinity, and their ratio goesto a deterministic constant. Another scenario of interest is the

Manuscript received May 26, 2005; revised April 10, 2007. The work ofZ. D. Bai was supported in part by CNSF 1057-1020 and NUS under GrantR-155-000-056-112.

Y.-C. Liang is with the Institute for Infocomm Research, Singapore 119613(e-mail: [email protected]).

G. Pan is with the Department of Statistics and Finance, University of Scienceand Technology of China, Hefei, China (e-mail: [email protected]).

Z. D. Bai is with the KLASMOE, School of Mathematics and Statistics,Northeast Normal University, Changchun, 130024, China (e-mail: [email protected]).

Communicated by X. Wang, Associate Editor for Detection and Estimation.Color versions of Figures 5 and 6 in this paper are available online at http://

ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2007.907497

multiple-input multiple-output (MIMO) antenna systems. For arich multipath environment, the channel responses between thetransmitters and the receivers are independent and identicallydistributed (i.i.d.), thus the propagation channel can be modeledas a random matrix. Inspired by the fundamental discoveries,in recent years, researchers have exploited the applications ofRMT in various aspects of wireless communications, including,e.g., i) limiting capacity and asymptotic capacity distributionfor random MIMO channels; ii) limiting SIR analysis forlinearly precoded systems, such as the multicarrier CDMAusing linear receivers [2]; iii) limiting SIR analysis for randomchannels with interference cancellation receivers [7], [8], andiv) limiting SIR analysis for coded multiuser systems [17].Other applications include the design of receivers, such as thereduced-rank MMSE receiver [9], the asymptotic normalitystudy for multiple-access interference (MAI) [16], and linearreceiver output [4]. An excellent overview of applications ofRMT in wireless communications is given by Tulino and Verdúin [11].

In this paper, we are concerned with the limiting and asymp-totic performance of linear receivers for large random systemswith unequal-power users. One typical example is the CDMAuplink where each user has different received power at the basestation due to multipath fading or shadowing and imperfectpower control. Another example is the MIMO systems in whichsoft interference cancellation (SIC)-based MMSE receiver [15]is applied. While each user may have identical average receivedpowers, since the soft estimates of the transmitted symbols mayvary from one user to another, for the purpose of detecting oneparticular user, the powers of the interferers after SIC will usuallynot be identical. We consider two classes of MMSE receivers: i)the optimal MMSE receiver which requires the knowledge of theinstantaneous power profile for all users, and ii) the suboptimalMMSE receiver which treats the instantaneous powers of allusers as being equal. While the optimal MMSE receiver yieldsthe highest SIR among all linear receivers, in practice, it could bedifficult to obtain the instantaneous power profile. Furthermore,for SIC-based MMSE receiver, if the power profile changes,the MMSE weighting vector has to be recalculated. Therefore,the suboptimal MMSE receiver is of great importance from theviewpoint of practical applicability.

In this paper, we derive the limiting SIRs and asymptotic SIRdistributions for a class of suboptimal MMSE receivers. Weprove that the limiting SIR converges to a deterministic valuewhen the number of users and number of degrees of freedomgo to infinity, with their ratio being a positive constant. Wealso prove that the SIR of each particular user is asymptoticallyGaussian when the number of degrees of freedom is relatively

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4174 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 11, NOVEMBER 2007

large, and derive the closed-form expressions for the variance ofthe SIR variable under real-spreading and complex-spreadingchannel environments. Based on the limiting SIR results ob-tained for the general suboptimal MMSE receivers, we derivethe optimal noise variance with which the suboptimal MMSEreceiver provides the highest limiting SIR within this categoryof the suboptimal MMSE receivers. We also quantify the mis-match effect when the noise power is overestimated or underes-timated in designing the suboptimal MMSE receiver. Further-more, based on the asymptotic distribution of SIR derived inthis paper and using the Taylor series expansion, we also es-tablish the asymptotic normality of SIR in decibels, denoted asSIRdB, and derive the closed-form expressions for the mean andvariance of this distribution. This derivation is useful as it pro-vides better description on the distribution especially when thenumber of degrees of freedom is relatively small.

In [13], Tse and Zeitouni have proven the asymptotic nor-mality of the output SIR for the optimal MMSE receiver forlarge systems when the users are equally powered. They havealso derived rigorously the closed-form expression for the vari-ance of SIR distribution using the central limit theorem for thereal-spreading channel scenario, and provided a conjecture forthe expression of variance for the complex-spreading channelscenario. Since the system model considered in this paper in-cludes the equal-power case as a special case, based on the re-sults obtained in this paper, under certain mild conditions, itis recognized that the asymptotic SIR variance for the com-plex-spreading channel scenario is just half of that for the cor-responding real-spreading channel scenario. Our derived resultsthus show that the conjecture made in [13] is not affirmative.

Unequal-power large systems are considered in [17] wherethe limiting SIR of the mismatched MMSE receiver has beenderived. While the suboptimal MMSE receiver considered inthis paper can be considered as a special case of the mismatchedMMSE receiver, the asymptotic SIR distribution of the receiverhas not been studied in [17]. In [18], the authors use generalizedGamma distribution to describe the output SIR of the optimalMMSE receiver. This is derived by decoupling the output SIRof the receiver into two independent parts: one part related to thezero-forcing (ZF) receiver and the other related to the differencebetween the SIRs for the MMSE and ZF receivers. The unequal-power systems have also been studied in [4] using the randommatrix method for various linear polynomial receivers, and in[6] using the replica method. The work in [6] extends the resultsof [5] where equal-power systems are considered.

This paper is organized as follows. Section II presents thesystem model and reviews the optimal MMSE receiver. Sec-tion III summarizes the main results derived in this paper for thesuboptimal MMSE receiver, including the limiting SIR, optimalnoise variance selection, asymptotic normality of the SIR, andthe closed-form expressions for the variances of the asymptoticdistributions for both real-spreading and complex-spreadingchannel cases. We also establish the asymptotic normalityof SIRdB in this section. Numerical results and simulationcomparisons are given in Section IV. Finally, conclusions aredrawn in Section V.

In this paper, we use the following notations: for trans-pose; for conjugate and transpose; for expectation;

for trace; for variance; for the spectral normof a matrix or the Euclidean norm of a vector; for adiagonal matrix with diagonal elements being vector .

II. CHANNEL MODEL AND MMSE RECEIVER

Let us consider the following system model:

(1)

where is the number of users, is the transmitted symbol ofuser , is the channel vector of user ,is the additive white Gaussian noise vector, and isthe received signal vector. The following assumptions are madethroughout this paper.

(AS1) The channel vectors ’s can be represented as

(2)

for , where ’s are i.i.d. variables withzero mean and unit variance, i.e., for all

’s and ’s.

(AS2) The transmitted symbols ’s are i.i.d. variables withzero mean and average power , for

.

(AS3) The noise vector is i.i.d., zero mean, circularly sym-metric complex Gaussian and with covariance matrix

.

Remark 2.1: For frequency-flat CDMA uplink with per-fect power control, and denote the transmitted symboland spreading sequence of user , respectively. Thus, and

denote the number of active users and processing gain,respectively. For third-generation wideband CDMA, the up-link spreading codes use random codes, thus the propagationchannel can be modeled as a random matrix channel. Thechannel model (1) can also be used to represent MIMO antennasystems, where and denote, respectively, the transmittedsymbol through the th transmit antenna, and the channelresponses from the th transmit antenna to all receive antennas.In this case, and represent the transmit antenna numberand receive antenna number, respectively. When there are richlocal scatters around the transmitter and receiver sides, canbe modeled as an i.i.d. vector, thus channel (1) becomes arandom matrix channel.

Remark 2.2: The channel coefficients can be real or complex.For complex-spreading case, we assume that the real and imag-inary parts of each channel coefficient are independent, zeromean and with equal variance.

Remark 2.3: The use of normalization factor in (2)implies that the signal-to-noise ratio (SNR), , is de-fined as the average received SNR for user over all degrees offreedom.

Rewrite (1) as

(3)

LIANG et al.: ASYMPTOTIC PERFORMANCE OF MMSE RECEIVERS FOR LARGE SYSTEMS 4175

where and . We are theninterested in recovering the transmitted symbolsfrom the received signal vector .

For linear receivers, the equalization output of user is givenby

(4)

where is the weighting vector for user . Using (1),can be represented as

(5)

Thus, the equalization output consists of two components: i) the

desired signal component and ii) the interference-plus-

noise component .

Denote , and write as the variance of the inter-ference-plus-noise component. One measure for quantifying thereceiver performance is the output SIR of the equalizer, whichis given by

SIR (6)

Notice that

(7)

where , and. We derive

SIR (8)

Define the mean-square-error (MSE) function for user as

The MMSE receiver, which minimizes the MSE function, is represented as

(9)

where . Let and

. Using matrix inversion lemma and

noticing that , we have

(10)

The MMSE receiver also maximizes the output SIR which isgiven by

SIR (11)

III. MAIN RESULTS

The MMSE receiver shown in (9) is referred to as the optimalMMSE receiver which produces the highest output SIR amongall linear receivers. The optimal MMSE receiver however re-quires the instantaneous power profile of the transmitted sym-bols. If the power profile changes, the MMSE weighting vectorhas to be recalculated. A practical solution is to treat all trans-mitted symbols as being equally powered when designing thereceiver. This receiver is called suboptimal MMSE receiver inthis work. The weighting vector of suboptimal MMSE receiverfor user is then given by

(12)

Note that if and , thenthe weighting vector in (12) is actually a scaler version of theweighting vector given in (10). For unequal-power systems, theselection of parameter will be discussed later.

With the suboptimal receiver, from (6), the output SIR foruser can be written as (13) shown at the bottom of the page,where ,

with and . Note that, and denotes the ratio between the desired

user’s power to the average value of the interference powers. Inthe sequel, we will derive the limiting SIR and asymptotic SIRdistribution for the suboptimal MMSE receiver. Without loss ofgenerality, we only consider the detection of user 1, and assumethat . If , the limiting SIR will be scaled by afactor of , while the variance of the asymptotic distribution ofthe SIR will be scaled by a factor of .

A. Limiting SIR Analysis

In this subsection, we derive the limiting SIR for the subop-timal MMSE receiver when and in sucha way that approaches a positive constant . To simplify the

notation, we denote . Before stating the maintheorems for the suboptimal MMSE receiver, we briefly state thelimiting SIR results for equal-power systems where .

Proposition 3.1 [12], [14]: Suppose that1) for all and , ’s are i.i.d. random variables with

and ; and2) ( is a constant) as .

Then, with probability , the following SIR:

(14)

converges to the deterministic value

(15)

SIR (13)


where is the limiting spectral distribution function of therandom matrix .

According to [12], [14], and [31], is the Stieltjes transformof the Marchenko–Pastur law at , that is, the positive solu-tion of the equation

(16)

Equation (16) is also equivalent to the following equation whichwill be used frequently in the sequel:

(17)

From (16), one can readily derive that has a densityfunction

where and . Also, has apoint mass at the origin if .

We now provide the limiting theorem for the output SIR ofthe suboptimal MMSE receiver.

Theorem 3.1: Suppose the following conditions hold.1) For all and , ’s are i.i.d. random variables with

, , and ;2) as ; and3) the powers of all users are i.i.d. with a fixed distribution

and are uniformly bounded, that is, there exists a con-stant such that .

Then the SIR in (13), denoted by , converges almostsurely to a deterministic constant given by

(18)

where

(19)

and

(20)

Proof: The proof is given in Appendix A.

Theorem 3.1 tells us that the limiting SIR is independentof the actual power distribution if the suboptimal MMSE re-ceiver assumes equal powers for all users. The parameterin (20) can be thought of as a mismatch factor which defines

the difference between the true noise variance and selectednoise variance in designing the receiver. For equal-powersystems, the weighting vector in (12) becomes optimal if wechoose . In this case, and . This resultis consistent with the limiting result derived in [12] using therandom matrix method, and those derived in [5] and [6] usingthe replica method.

Differentiating in (18) with respect to the variable ,and setting , we get

(21)

From (16), the derivative of over can be calculated asfollows:

(22)

Applying (22) into the first term of (21), we observe that theextreme value of is taken only at point . From(22) it is also noticed that is negative, thus the maximumlimiting SIR is achieved when is chosen as . Based onthese arguments, we have the following theorem.

Theorem 3.2: Under the conditions of Theorem 3.1, the lim-iting SIR is maximized when .

B. Asymptotic Distribution of the SIR

We next establish the asymptotic normality of for whichthe following lemmas are needed.

Lemma 3.1: Under the assumptions of Theorem 3.1, we have

a.s. (23)

and

a.s. (24)

where denotes the th diagonal element of matrix , anda.s. denotes “almost surely.”

Proof: See Appendix B.

Lemma 3.2: Under the assumptions of Theorem 3.1, withprobability , we get equation (25)–(26) at the bottom of thepage and (27) at the top of the following page.

(25)

(26)


(27)

Proof: Lemma 3.2 comes immediately from Lemma 3.1.

Lemma 3.3: In addition to the conditions of Theorem 3.1, weassume that the empirical distribution of tends toa limiting distribution . Then, as

(28)

where

(29)

with for .Proof: See Appendix C.

Lemma 3.4: Suppose the assumptions of Theorem 3.1 hold,we then have

and

(30)

Proof: See Appendix D.

Before proceeding further, we introduce the following nota-tions:

where denotes the distribution function by substitutingfor in .

Lemma 3.5: In addition to the assumptions of Theorem 3.1,assume that

(31)

where denotes the empirical distribution function of. We then get

(32)

and

(33)

Proof: See Appendix E.

Theorem 3.3: Further to the conditions of Theorem 3.1, as-sume that the empirical moment of the users’ powers tends to alimit , i.e.,

(34)

Then, as

(35)

where denotes convergence in distribution, anddenotes the Gaussian distribution with zero mean and variance

. The variance can be calculated as follows.i) If is real, then

(36)

ii) If is complex and , then the variancebecomes

(37)

where

(38)

(39)

Proof: See Appendix F.

Remark 3.1: In Theorem 3.3, we consider the asymptotic dis-tribution of for a given (limited) size , in-stead of that of . This is because, in theory, therate of instantaneous load converging to may be arbitrarilyslow, thus may blow up. Please note thatis the limit of with being replaced by the instantaneousload , and the subscript in is used to denote the de-pendence on the instantaneous load only.

Remark 3.2: For equal-power systems, , thus if wechoose , we have


For the real-spreading channel case, the above results are consis-tent with the results derived in [13]. For the complex-spreadingchannel case, the variance expression of SIR distribution is dif-ferent from the conjecture made in [13]. In fact, by extendingthe covariance results from real-spreading channels to complex-spreading channels, besides the term should beused to replace the term , the term in (36) shouldalso be replaced with the term .

Remark 3.3: Let us consider the relationship between theasymptotic SIR distributions for real-spreading and complex-spreading channel cases. For the real-spreading channel case,we denote as . For the corresponding complex-spreading channel case, is denoted as

where and are independent but both follow the samedistribution of random variable , which is with zeromean and unit variance, according to the assumptions of The-orem 3.1. It can be easily verified that

Thus, from (34) and (35), we have .

If we further assume Gaussian distribution for , then

. On the other hand, comparing thereal-spreading channel with binary phase-shift keying (BPSK)random codes and the complex-spreading channel with quater-nary phase-shift keying (QPSK) random codes, then

. Computer simulations will be given in Sec-tion IV to compare the accuracy of the results derived in thispaper and that of [13].

The accuracy of Theorem 3.3 in describing the distribution ofSIR depends on the dimension . The larger the , the moreaccurate the real distribution approaches the theoretical distribu-tion. Using Gaussian distribution to model the SIR distribution,one potential problem is that the SIR may appear to be negative,which is not feasible in practice. The negative SIR problem alsoapplies to the results of [13]. In order to predict the SIR distri-bution more accurately for a small size system, and to avoid theSIR being negative, we turn to look at the distribution of SIRin decibels, denoted as SIRdB. The following theorem directlycomes from Theorem 3.3 using the Taylor series expansion.

Theorem 3.4: Under the conditions of Theorem 3.3, as

(40)

where the variance is given in Theorem 3.3 for both real-spreading and complex-spreading channels, and is the naturallogarithm base.

IV. NUMERICAL RESULTS AND COMPUTER SIMULATIONS

In this section, numerical results and computer simulationsare presented to evaluate the accuracy of various limitingand asymptotic results derived in this paper for large randomsystems.

A. Evaluation of Theorem 3.2

For a given noise variance , the selection of noise variancein designing the suboptimal MMSE receiver will affect the lim-iting SIR. Fig. 1 illustrates the limiting SIRs versus the selectedvariable for different parameters and . The limiting SIRis maximized when for all cases. Also, it is seen thatthe SIR loss is less than 3 dB when the mismatch betweenand is within 10 dB.

B. Limit and Asymptotic Distribution of SIR

Computer simulation results are presented to evaluate theaccuracy of the theoretical SIR distribution as compared tothe simulated distribution for finite size systems. Two types ofpower distributions were simulated: equal power distributionand random exponential distribution. We also considered twotypes of distributions for the channel coefficients: the realGaussian and the complex Gaussian. For each case, 10 000Monte Carlo runs were carried out.

1) Equal Power Distribution: For equal power distri-bution, the suboptimal MMSE receiver is equivalent to theoptimal MMSE receiver. The objective here is thus to comparethe asymptotic SIR variances for real-spreading and com-plex-spreading channel cases. Fig. 2 shows the comparisonbetween Tse and Zeitouni’s formula (TZ formula) and theformula derived in this paper (LPB formula) with the simulatedvariances. Please note here the “variance of SIR (dB)” means“ (variance of SIR).” For the real-spreading case, thetheoretical results from TZ formula and LPB formula overlap.For the complex-spreading case, the formula derived in thispaper is consistent with the simulation results, while the TZformula is away from the simulated results.

2) Exponential Power Distribution: For exponential powerdistribution, the results of this paper are needed in order to pro-duce the theoretical mean and variance for the SIR variable.Fig. 3 compares the theoretical limiting SIR with the mean ofthe simulated SIRs, and Fig. 4 compares the theoretical variancewith the variance of the simulated SIRs. It is seen that the simu-lated results are consistent with the theoretical results when thesystem load is small , but with little biaswhen the system is fully loaded .

C. Comparison of Asymptotic Distribution for SIR and SIRdB

Finally, the tail behavior is of interest for the distributions ofSIR and SIRdB. 1 000 000 Monte Carlo runs were simulated.We compare the simulated histogram, mean and 1% outageSIR or SIRdB from both theoretical and simulated results.Here the 1% outage SIR is defined as the value such that

SIR . Figs. 5 and 6 illustrate the compar-isons for complex Gaussian channels and exponential powerdistribution with and and ,respectively. Here, the “variance of SIRdB” means the variance


Fig. 1. Limiting SIRs versus parameter � for (a) y = 1 and � = 0:1 and (b) y = 0:5 and � = 0:01.

of “ (SIR).” Comparing the histograms and the 1%outage SIR and 1% outage SIRdB, it is seen that for a small

load and reasonably large , the distributionof SIR is close to Gaussian, but this approximation is less


Fig. 2. Comparison of simulated variance and theoretical variance for systems with equal power distribution and (a) real-spreading channel; (b) complex-spreadingchannel.

accurate as compared to SIRdB. When (smalland large load), the SIR distribution seems to be non-Gaussian,

however, the SIRdB still follows the Gaussian distribution moreclosely.


Fig. 3. Comparison of simulated mean and limiting SIR for systems with exponential power profile and (a) real-spreading channel; (b) complex-spreading channel.

V. CONCLUSION

In this paper, random matrix theory has been used to derivethe limiting SIR and asymptotic SIR distribution for a class

of suboptimal MMSE receivers applied to unequal-power largesystems. We have proven that the limiting SIR converges to a de-terministic value when and go to infinity with


Fig. 4. Comparison of simulated variance and theoretical variance for systems with exponential power profile and (a) real-spreading channel; (b) complex-spreading channel.

being a constant, where is the number of users and isthe number of degrees of freedom. We have also proven that

the output SIR is asymptotically Gaussian for large and de-rived the closed-form expressions of the variance for both real-


Fig. 5. Comparison of histograms for (a) SIR, (b) SIRdB when SNR = 20 dB; and mean and 1% outage of (c) SIR and (d) SIRdB under complex Gaussianchannels with K = 32 and N = 64 and exponential power distribution.

spreading and complex-spreading channels. It is noticed that thecomplex channels produce half the variance of the real chan-nels. We have also derived the asymptotic distribution of SIRdB,and obtained the closed-form expressions for its mean and vari-ance for finite size . Numerical results and computer simula-tions have evaluated the accuracy of the limiting results obtainedin this paper. Specifically, the distribution of SIRdB providesa more accurate description than the distribution of SIR, espe-cially when is small and system load is equal to .

APPENDIX APROOF OF THEOREM 3.1

We first prove a lemma which will be frequently used in thefollowing sections.

Lemma 5.1: In addition to the assumptions of Theorem 3.1,suppose that random matrices are in-dependent of for each and have a uniform

upper bound in the spectral norm, that is, , whereand are two positive constants. Then

Proof: Define if and otherwise.Let and

. Then by [25, Lemma 2] and the as-sumption of the finite fourth moment of the random variables

’s, we have

(41)

(42)

Here the second limit holds since for any large


Fig. 6. Comparison of histograms for (a) SIR, (b) SIRdB when SNR = 20 dB; and mean and 1% outage of (c) SIR and (d) SIRdB under complex Gaussianchannels with K = 16 and N = 16 and exponential power distribution.

which can be made arbitrarily small by taking large . In theabove, is the indicator function which takes value if thestatement in the bracket is true, and otherwise.

By the assumption of the lemma and using (41) and (42)

uniformly for .

Now, by [19, Lemma 2.7], for any

Using the Borel–Cantelli lemma and choosing , weobtain

The lemma then follows by noting that .

Now we turn to the proof of Theorem 3.1.


Proof: From the trivial inequality

one can easily show that the norm of the matrix

is bounded by

, where is a bound for the relative powers of allusers.

Using Lemma 5.1, we have

(43)

and

a.s. (44)

Making use of the convergence of the Marchenko–Pastur law(see [31, Theorem 2.5]), we obtain that

(45)

where and is the Marchenko–Pastur lawdefined earlier.

On the other hand, we have

and similarly , where

. Therefore, we obtain

a.s. (46)

where the convergence is uniform in .Let

and

Since , with probability , we get

(47)

Therefore, by the Vitali theorem (refer to [22, p. 168]) anddominated convergence theorem, we conclude that

(48)

In fact, it is easy to see that is

an analytic function on some bounded region of and that

where denotes some constant independent of and .By similar arguments, we may prove that

a.s. (49)

Thus, by substituting the above two limits into the numeratorand denominator of SIR , we show that, with probability

(50)

By (16), we have that

where . Substituting this into the right-hand sideof (50), (18) immediately follows. Further, (19) is the positivesolution of (16).

APPENDIX BPROOF OF LEMMA 3.1

Proof: Using the formula

where and are nonsingular and . Then

it follows that for any

(51)


where

and

Here we would remind the readers to note the difference be-tween and : the former is the complex conjugate transposeof the row vector while the latter is the column vector of the ma-trix .

By applying [25, Lemma 2], we have

(52)

almost surely and uniformly for . Employing Lemma 5.1,it follows that

(53)

It is obvious that

(54)

uniformly for . Then (23) follows from (51)–(54)and (17).

Applying (23) and the Vitali theorem, one can get

(55)

uniformly for .Let , the element is only in the

th position. Then, using the arguments in proving (46), we have

(56)

Hence, we conclude that

(57)

uniformly for .Thus, we complete the proof of (24) by using the fact that

and .

APPENDIX CPROOF OF LEMMA 3.3

Proof: The following identity will be frequently used:

Write , , and define

and . We de-

compose as

(58)

where

Applying Lemma 5.1, one can easily prove that

a.s.

where


(61)

Therefore

(59)

where .To find the limit of the first term on the right-hand side of

(58), we use again Lemma 5.1 and obtain

(60)

To complete the proof of this lemma, we employ the Vitalitheorem twice (shown in (61) at the top of the page.

APPENDIX DPROOF OF LEMMA 3.4

Proof: Let for denote conditional expectationwith respect to the -field generated by and letdenote expectation.

We shall frequently use the inequality

(62)

if is independent of and is bounded by a nonrandomconstant. In fact, the inequality is an easy consequence of [19,Lemma 3.3].

We use the following martingale decomposition:

(63)

where


Now, it suffices to evaluate the variance of above items. To esti-mate , using (62) we get the equation at the top of thepage, where we have used the fact that ,

and is the bound for ’s. This proves that isbounded by a constant.

Since

which shows that is bounded by a constant.To verify that is bounded by a constant, we need

only to note that

and . Similarly, we may verify thatand are bounded.

As for estimating the variance of , we only need to

repeat the previous steps for . Therefore

APPENDIX EPROOF OF LEMMA 3.5

Proof: For any function analytic in a region con-taining as inner points

where is a contour enclosing

and which was defined in (19). Applying theresult in [19, Sec. 4], we have

a limit

(64)

Then the conclusion (32) follows from (64) with.

Similarly, one can prove that for any positive integer

a limit (65)

which is true when is replaced by .By the Cauchy–Schwartz inequality and Lemma 3.4, for

, we have

Now, let and. By the identity and

the fact that and , we have

Applying (65) to the above estimate, we complete the proofof (33).

APPENDIX FPROOF OF THEOREM 3.3

Proof: In the sequel, we will use the notationif for any there exists a positive number so that

and the notation if

. Since the norms of the matrices

involved are bounded, we have


and

Further, by Theorem 3.1 and Lemma 3.1

(66)

Thus, to complete the proof of the theorem, we only need toprove that

and

(67)

(68)

The conclusion (67) follows from Lemmas 3.4 and 3.5. Toprove (68), we consider the real and complex cases separately.

Write the left-hand side of (68) as

where

It is easy to verify that

(69)

In the real-spreading case, the second term is the same as thefirst term. For the complex-spreading case, may take anyvalue in . If it is not zero, may not havea limit. For this reason, we need an additional assumption toeliminate this term, that is, .

Our strategy is the following: given all , by the resultin an accepted paper [30], the conditional distribution of

tends to the normal distributionwith mean zero and a variance which is independent of ,and Theorem 3.3 then follows by the Fubini theorem.

Before concluding this appendix, we shall derive the limitsof the first and the third terms in (69) and thus the asymptoticvariance of . Using the Vitali theorem and (48), we have

(70)

and furthermore

(71)

where

Using Lemmas 3.2 and 3.3

(72)

and

(73)


Thus, we obtain

(74)

It is easy to see that if and , and thesum of the first three terms and that of the last six terms in (74)are equal to and , respectively. Thus, we have

and

Since these equalities are independent of and , they aretrue in the general case. Thus

(75)

Next, we need to determine the limit of the third term in (69),that is, the limit of .

By Lemma 3.1, we have

Therefore, we finally have

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asymptotic performance of mmse receivers for large systems using random matrix theory

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