IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 2, NO. 3, JUNE 2013 275

Diversity of MMSE Receivers in MIMO Multiple Access ChannelsAhmed Hesham Mehana, Student Member, IEEE, and Aria Nosratinia, Fellow, IEEE

Abstract—This letter analyzes the performance of MMSE re-ceivers in multiple input multiple output (MIMO) multiple accesschannel (MAC). The Diversity-Multiplexing Tradeoff (DMT) ofthe MIMO MAC under MMSE reception easily follows from thesingle-user DMT, indicating that MMSE receivers are largelysuboptimal for MIMO MAC. The main result of this letter is thecalculation of the diversity of the MIMO MAC in the fixed-rate regime, i.e., when users transmit at non-zero rates thatare not functions of SNR. In this regime, the MIMO MACMMSE receivers exhibit interesting diversity behaviors whosecharacterization is completed by the contributions of this letter.

Index Terms—MIMO, MMSE, multiple access channel, diver-sity.

I. INTRODUCTION

L INEAR receivers are widely used for their low complex-ity compared with maximum likelihood (ML) receivers.

MMSE receivers are adopted in some of the emerging stan-dards, e.g. IEEE 802.11n and 802.16e. This work investigatesthese receivers in the MIMO multiple access channel (MAC).

This letter starts by extending the Diversity-MultiplexingTradeoff (DMT) analysis of MMSE receivers from single-user to the MAC case, using the invariance of the single-userresults to coding the antenna streams jointly or separately.We then analyze the MIMO MAC MMSE receivers in thefixed rate regime. Recent developments show [1]–[3] that amore delicate analysis is called for in the fixed-rate regimethat requires tools and techniques beyond the DMT analysis.Indeed, a lower bound obtained in [4] showed the MIMOMAC MMSE can exhibit an intricate behavior. A contributionof this letter is to calculate an upper bound on diversity thatis tight against the lower bound of [4] and thus establishes thefixed-rate diversity.

A brief review of the relevant literature is as follows. Theknown MMSE diversity results, including those mentionedbelow, have largely addressed the single-user scenario. Theperformance of MMSE receiver in terms of reliability goesback to [5] where outage analysis was performed for MMSESIMO diversity combiner in a Rayleigh fading channel withmultiple interferers. Onggosanusi et al. [6] studied MMSE andzero-forcing (ZF) MIMO receivers. Hedayat and Nosratinia [1]considered the outage probability in the fixed-rate regimeunder joint and separate spatial encoding, but for MMSEthey obtained results only in the extremes of very high andvery low rates. Kumar et al. [2] provided a DMT analysisfor the system of [1] and observed that the the DMT doesnot predict the diversity of MMSE receivers at lower rates.

Manuscript received November 27, 2012. The associate editor coordinatingthe review of this letter and approving it for publication was K. Wong

The authors are with the the University of Texas at Dallas, Richardson, TX75083-0688 USA (e-mail: {axm081100, aria}@utdallas.edu).

Digital Object Identifier 10.1109/WCL.2013.13.120873

Hesham and Nosratinia [3] provide an exact characterizationfor the diversity in the fixed rate-regime.

II. SYSTEM MODEL

The input-output system model represents a MIMO MACwith K users, M transmit antennas per user, and N receiveantennas (cf. Figure 1). The MIMO channel in this workexperiences flat fading, so that the system model is given by

y =K∑i=1

Hixi + n = Hex+ n (1)

where Hi ∈ CN×M is the channel matrix for User i, with en-tries that are independent and identically distributed complexGaussian. The vector transmitted by User i is xi ∈ C M×1.The transmissions of all users are aggregated into the vec-tor x = [xT

1 xT2 . . .xT

K ]T , and the corresponding equivalentchannel matrix is He = [H1H2 . . .HK ]. n ∈ C N×1 is theGaussian noise vector. The vectors x and n are assumedindependent from each other and from the channel gains. It isassumed that perfect channel state information is available atthe receiver (CSIR) but not at the transmitter.

We aim to characterize the diversity gain, d(R,M,N),as a function of the spectral efficiency R (bits/sec/Hz) andthe number of transmit and receive antennas. This requiresa pairwise error probability (PEP) analysis which is notdirectly tractable. Thankfully outage and PEP exhibit identicalexponential orders in our case, a fact whose proof followssimilarly to [3] and is omitted in this letter for brevity.

Following the notation of [7], we define the outage-typequantities

Pout(R,N,M) � P(I(x;y) < R) (2)

dout(R,N,M) � − limρ→∞

logPout(R,M,N)

log ρ(3)

where ρ is the per-stream signal-to-noise ratio (SNR).We say that the two functions f(ρ) and g(ρ) are exponen-

tially equal, denoted by f(p).= g(p) when

limρ→∞

log f(ρ)

log(ρ)= lim

ρ→∞log g(ρ)

log(ρ)

The ordering operators �̇ and �̇ are also defined in a similarmanner. If f(ρ)

.= ρd, we say that d is the exponential order

of f(p). The basis of the logarithm is 2 throughout this letter.

III. DMT OF THE MMSE MIMO MAC

Kumar et al. [2] obtained the DMT of MMSE single-userMIMO receivers:

d(r) = (N −M + 1)(1− r

M)+. (4)

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MMSE

User 1

MUX

Base station

He

Μ

1

User Κ1

Μ

1

Ν

Fig. 1. MIMO system with linear MMSE receiver.

This result shows that MMSE receivers are largely suboptimalin the DMT sense. This result holds regardless of whether theantenna data streams are coded independently or jointly, whichallows the result to be trivially extended to a K-user MIMOMAC system with M -antenna users and MMSE receiver. TheDMT of the MIMO MAC is identical to a single-user MIMOsystem with KM transmit antennas, since coding across theantennas or independently for each antenna does not affectthe DMT. We thus obtain the following lemma, where nowthe multiplexing gain r is normalized per-user.

Lemma 1: In a MIMO MAC system with MMSE receiverconsisting of K users, M transmit antennas per user and Nreceive antennas, the DMT is given by

d(r) =(N −KM + 1

)(1− r

M

)+

. (5)

This DMT result does not predict the interesting behavior ofdiversity in the fixed-rate regime1, which is the topic of thenext section.

IV. DIVERSITY IN THE FIXED RATE REGIME

The key result of this letter is as follows:Theorem 1: In a MIMO MAC system with MMSE receiver

consisting of K users, M transmit antennas per user and Nreceive antennas with N � KM , the per user diversity isgiven by

d(R) =⌈M2−R/M

⌉2+ |N −KM |⌈M2−R/M

⌉(6)

The proof of Theorem 1 is performed via bounding theoutage probability from below and above and then showingthat these two bounds exhibit the same diversity order. It canthen be shown that the pairwise error probability follows thesame diversity as the outage probability, a fact whose proof isvery similar to [3] and is omitted here.

A. MIMO MAC MMSE Outage Upper Bound

The MIMO MAC MMSE outage upper bound was estab-lished in [4]. We reproduce the outage upper bound here forcompleteness, with a slightly different presentation.

1Please refer to [4] for more discussions about the distinction between theDMT and the diversity at r = 0.

Without loss of generality we consider the outage probabil-ity of User 1, considering that the naming and ordering of theusers are arbitrary. The rate transmitted by this user is R.

P 1out = P

( M∑k=1

log(1 + γ1k) < R

). (7)

where γ1k is the unbiased decision-point SINR for the data

stream k of the first user:

γ1k =

1

(I+ ρHeHHe)

−1kk

− 1, k = 1, · · ·M. (8)

where (·)kk denotes the diagonal element number k. Thus theoutage probability is:

P 1out = P

( M∑k=1

log(I+ ρHeHHe)

−1kk > −R

). (9)

Using Jensen’s inequality the outage probability can bebounded as

P 1out � P

(log

( M∑k=1

1

M(I+ ρHe

HHe)−1kk

)>

−R

M

)

� P

(log

(KM∑k=1

1

M(I+ ρHe

HHe)−1kk

)>

−R

M

)(10)

= P

(KM∑k=1

1

1 + ρλk> M2−

RM

)(11)

where (10) holds by adding positive terms2 to the argumentof the logarithm, which itself is a monotonically increasingfunction. Substituting λk = ρ−αk , we have the followingexponential equality

1

1 + ρλk

.=

{ραk−1 αk < 1

1 αk > 1.(12)

Thus at high SNR, each of the additive terms in (11) is eitherzero or one, therefore to characterize

∑k

11+ρλk

at high SNRwe basically count the ones. The asymptotic slope of (11) inthe special case of K = 1 was calculated in [3] using thedistribution of {αk}. This result generalizes to K > 1 in astraight forward manner, as follows.

P 1out � P

(KM∑k=1

1

1 + ρλk> M2−

RM

)

2Recall that (I+ ρHeHHe) is a positive definite matrix [8].

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MEHANA and NOSRATINIA: DIVERSITY OF MMSE RECEIVERS IN MIMO MULTIPLE ACCESS CHANNELS 277

.= ρ−d(R) (13)

where

d(R) =⌈M2−R/M

⌉2+ |N −KM |⌈M2−R/M

⌉. (14)

B. MIMO MAC MMSE Outage Lower Bound

The lower bound is based on Jensen’s inequality

P 1out = P

( M∑k=1

log1

(I+ ρHeHHe)

−1kk

< R

)

� P

(M log

M∑k=1

1

M

1

(I+ ρHeHHe)

−1kk

< R

)(15)

= P

( M∑k=1

1

M

1

(I+ ρHeHHe)

−1kk

< 2RM

)

(16)

Let the eigen decomposition of HeHHe be given by

HeHHe = UHΛU where U is unitary and Λ is a diagonal

matrix that has the eigenvalues of the matrix HeHHe on its

diagonal. Let the eigenvalues of HeHHe be given by {λ�}

with λ1 � λ2 · · · � λkM . Let the vector uk be the column kof the matrix U, we have

(I+ ρHeHHe)

−1kk = uH

k (I+ ρΛ)−1uk

=

KM∑�=1

|u�k|21 + ρλ�

� Sk.

Let k̄ = argmink Sk. we can bound the sum in (15)

1

M

M∑k=1

1

(I+ ρHeHHe)

−1kk

=1

M

M∑k=1

1

Sk

� 1

mink Sk

=1

Sk̄

(17)

thus the outage bound in (15) can be further bounded

Pout � P

(Sk̄ > 2−

RM

)(18)

We now bound (18) by conditioning on the event

B �{|u�k̄|2 � a

M, � = KM −M + 1, · · · ,KM

}(19)

where a is a positive real number that is slightly smallerthan one a = 1 − ε1, and ε1 is a small positive number. Wethen have

Pout � P

(Sk̄ > 2−

RM

)

� P

(Sk̄ > 2−

RM

∣∣B)P(B)= P

(KM∑�=1

|u�k̄|21 + ρλ�

> 2−RM

∣∣∣∣B)P(B)

� P

( KM∑�=KM−M+1

|u�k̄|21 + ρλ�

> 2−RM

∣∣∣∣B)P(B) (20)

� P

(1

M

KM∑�=KM−M+1

a

1 + ρλ�> 2−

RM

)P(B)

.= P

(1

M

KM∑�=KM−M+1

a

1 + ρλ�> 2−

RM

)(21)

= P

( KM∑�=KM−M+1

1

1 + ρλ�>

M

a2−

RM

)(22)

where (20) follows by removing some of the elements of thesum corresponding to the largest eigenvalues. Eq. (21) followsbecause P(B) is finite and independent of ρ. The proof issimilar to [2, Appendix A]

Note that HeHHe is not a Wishart matrix, hence the

analysis of Section IV.A does not directly apply here. Theblock diagonal elements of He

HHe are similar and are givenby

D =ν∑

i=0

HHi Hi. (23)

The matrix HeHHe is Toeplitz and Hermitian. Moreover,

the matrix D given by (23) is a Wishart matrix3.Observe that the probability in (22) depends on the M

smallest eigenvalues. We now bound these eigenvalues withthe eigenvalues of the matrix D via the Sturmian separationtheorem [9, P.1077].

Theorem 2: (Sturmian Separation Theorem) Let {Ar, r =1, 2, . . .} be a sequence of symmetric r× r matrices such thateach Ar is a submatrix of Ar+1. Then if {λk(Ar) , k =1, . . . , r} denote the ordered eigenvalues of each matrix Ar

in descending order, we have

λk+1(Ai+1) ≤ λk(Ai) ≤ λk(Ai+1).

For our purposes, we consider a special case of theSturmian Theorem by constructing a set of matrices

AM ,AM+1, . . . ,ALdM starting by the largest one ALdM�=

HeHHe and making all other matrices Ai to be (successively

embedded) i × i principal submatrices of HeHHe, such that

the smallest matrix is AM = DLd. Then we repeatedly apply

the first inequality in the Sturmian to get:

λMLd (AMLd) ≤ λMLd−1(AMLd−1) ≤ · · · ≤ λM (AM )

λMLd−1(AMLd) ≤ λMLd−2(AMLd−1) ≤ · · · ≤ λM−1(AM )

......

λMLd−M+1(AMLd) ≤ λMLd−M (AMLd−1)≤ · · · ≤ λ1(AM )

This implies that the smallest M eigenvalues of HeHHe

are bounded above by the M eigenvalues of D, respectively.Hence:

Pout�̇ P

( M∑k=1

1

1 + ρλk(D)>

M

a2−

RM

). (24)

3 Let W(n,∑

) denote a Wishart distribution with degree of freedomn and covariance (also called scale) matrix

∑. Any of the diagonal block

matrices Dj given by (23) follows a Wishart distribution since if B1 ∈W(n1,

∑) and B2 ∈ W(n2,

∑) then B1 +B2 ∈ W(n1 + n2,

∑).

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278 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 2, NO. 3, JUNE 2013

D is a sum of (ν + 1) central Wishart matrices each with Ndegrees of freedom and with identity covariance matrix, i.e.D ∈ W(KN, I). Therefore analysis similar to the analysisof [4, Section III-B] applies here and we get

P

( M∑k=1

1

1 + ρλk(D)>

M

a2−

RM

).= ρ−d(R). (25)

Remark 1: Using the linear MMSE receiver, the diversityof each user depends on the rate transmitted by that user, butis independent of the rate of other users. This is due to thefact that the residual interference is considered as noise.

Remark 2: For simplicity the results of this letter weredeveloped under the assumption that all users experience thesame transmit-side equivalent SNR of ρ, however, it is straightforward to show that the result applies to unequal SNR as longas ρi − ρj = O(1).

Remark 3: The results of this letter can easily be general-ized to remove the restriction N � KM . The main differenceis that some eigenvalues of He

HHe are zero when N < KM .The generalized result is:

d(R) =⌈(M2−R/M − (KM −N)+

)+⌉2+ |N −KM |⌈(M2−R/M − (KM −N)+

)+⌉(26)

V. CONCLUSION

In this letter the diversity of the MMSE MIMO receiver inmultiple access channel is calculated. The diversity in the fixedrate-regime (where the spectral efficiency R is not a functionof SNR) is fully characterized.

REFERENCES

[1] A. Hedayat and A. Nosratinia, “Outage and diversity of linear receiversin flat-fading MIMO channels,” IEEE Trans. Signal Process., vol. 55,no. 12, pp. 5868–5873, Dec. 2007.

[2] K. R. Kumar, G. Caire, and A. L. Moustakas, “Asymptotic performanceof linear receivers in MIMO fading channels,” IEEE Trans. Inf. Theory,vol. 55, no. 10, pp. 4398–4418, Oct. 2009.

[3] A. Hesham Mehana and A. Nosratinia, “Diversity of MMSE MIMOreceivers,” in Proc. 2010 IEEE ISIT.

[4] A. H. Mehana and A. Nosratinia, “Diversity of MMSE MIMO receivers,”IEEE Trans. Inf. Theory, vol. 58, no. 11, pp. 6788–6805, Nov. 2012

[5] H. Gao, P. J. Smith, and M. V. Clark, “Theoretical reliability of MMSElinear diversity combining in Rayleigh-fading additive interference chan-nels,” IEEE Trans. Commun., vol. 46, no. 5, pp. 666–672, May 1998.

[6] E. N. Onggosanusi, A. G. Dabak, T. Schmidl, and T. Muharemovic,“Capacity analysis of frequency-selective MIMO channels with sub-optimal detectors,” in Proc. 2002 IEEE ICASSP, vol. 3, pp. 2369–2372.

[7] A. Tajer and A. Nosratinia, “Diversity order in ISI channels with single-carrier frequency-domain equalizer,” IEEE Trans. Wireless Commun.,vol. 9, no. 3, pp. 1022–1032, Mar. 2010.

[8] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” EuropeanTrans. Telecommun., vol. 10, pp. 585–595, Nov./Dec. 1999.

[9] D. S. Bernstein, Matrix Mathematics: Theory, Facts and Formulas.Princeton University Press, 2009.

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