IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, …aria/papers/MehanaNosratinia2014.pdfIEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 2, FEBRUARY 2014 1019 Diversity of MIMO

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 2, FEBRUARY 2014 1019

Diversity of MIMO Linear PrecodingAhmed Hesham Mehana, Member, IEEE, and Aria Nosratinia, Fellow, IEEE

Abstract— This paper studies multiple-input multiple-outputlinear precoding in the high-signal-to-noise-ratio regime underflat fading. The diversity at all fixed rates is analyzed for anumber of linear precoders. The diversity-multiplexing tradeoffs(DMTs) are also obtained, discovering that for many linearprecoders the DMT gives no direct insight into the intricatebehavior of fixed-rate diversity. The zero-forcing (ZF), regular-ized ZF, matched filtering, and Wiener filtering precoders areanalyzed. It is shown that regularized ZF (RZF) or matched filter(MF) suffers from error floors for all positive multiplexing gains.However, in the fixed rate regime, RZF and MF precoding achievefull diversity for spectral efficiencies up to a certain thresholdand zero diversity at rates above it. When the regularizationparameter in the RZF is optimized in the minimum mean squareerror sense, the structure is known as the Wiener precoder, whichin the fixed-rate regime is shown to have diversity that dependsnot only on the number of antennas, but also on the spectralefficiency. The diversity in the presence of both precoding andequalization is also analyzed.

Index Terms— MIMO, precoder, equalization, MMSE, zeroforcing, diversity.

I. INTRODUCTION

PRECODING is a preprocessing technique that exploitschannel-state information at the transmitter (CSIT) to

match the transmission to the instantaneous channel condi-tions [1]–[4]. In particular, linear precoding is a simple andefficient method that can reduce the complexity of the MIMOreceiver; it can also be optimal in certain situations involvingpartial CSIT [5], [6].

Linear precoders include zero-forcing (ZF), matched fil-tering (MF), Wiener filtering, and regularized zero-forcing(RZF). The ZF precoding schemes were extensively stud-ied in multiuser systems as the ZF decouples the multiuserchannel into independent single-user channels and has beenshown to achieve a large portion of dirty paper codingcapacity [7]. ZF precoding often involves channel inversion,using the pseudo-inverse of the channel or other generalizedinverses [4]. Matched filter (MF) precoding [8], similarly to

Manuscript received December 22, 2011, revised November 8, 2012and August 29, 2013, accepted September 7, 2013. Date of publicationNovember 7, 2013; date of current version January 15, 2014. This work waspresented in part at the 2012 IEEE International Symposium on InformationTheory. This work was supported by the National Science Foundation underGrant CCF-1219065.

A. H. Mehana was with the University of Texas at Dallas, Dallas,Richardson, TX 75080 USA. He is now with the Department of Electronicsand Electrical Communications Engineering, Cairo University, Cairo 12316,Egypt (e-mail: [email protected]).

A. Nosratinia is with the Department of Electrical Engineering, TheUniversity of Texas at Dallas, Richardson, TX 75080 USA (e-mail:[email protected]).

Communicated by A. M. Tulino, Associate Editor for Communications.Color versions of one or more of the figures in this paper are available

online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2013.2289860

the MF receiver, is interference limited at high SNR but itoutperforms the ZF precoder at low SNR [4]. The regularizedZF precoder, as the name implies, introduces a regularizationparameter in channel inversion. If the regularization parameteris inversely proportional to SNR, the RZF of [9] is identicalto the Wiener filter precoding [4]. Peel et al. [9] introduced avector perturbation technique to reduce the transmit power ofthe RZF method, showing that in this way RZF can operatenear channel capacity.

This paper analyzes the diversity of MIMO linear precoding,with or without linear receivers, under flat fading. We showthat in a M ×N MIMO channel with M ≥ N , the ZF precoderhas diversity M − N + 1. We show that Wiener precodersproduce a diversity that is a function of spectral efficiency aswell as the number of transmit and receive antennas. At verylow rates, the Wiener precoder enjoys diversity M N , while atvery high rates it achieves diversity M − N + 1. These resultsare reminiscent of MIMO linear equalizers [10], even thoughin general the behavior of equalizers (operating on the receiveside) can be distinct from precoders (operating on the transmitside) and the analysis does not directly carry from one to theother. We also show that MIMO systems with RZF and MFprecoders (together with optimal receivers) exhibit a new kindof rate-dependent diversity that has not to date been observedor reported, i.e., they either have full diversity or zero diversity(error floor) depending on the operating spectral efficiency R.

We also calculate the DMT for the precoders mentionedabove. The fact that DMT and the diversity under fixed-rate regime require separate analyses has been established forMIMO linear equalizers [10], [11]. We find a similar phenom-enon in MIMO precoding: various fixed rates (spectral effi-ciencies) result in distinctly different diversities, whereas DMTanalysis assigns only a single value of diversity to all fixedrates (all fixed rates correspond to multiplexing gain zero).

Remark 1: It may be tempting to substitute r = 0 in theDMT expression d(r) in an attempt to produce the diversity atmultiplexing gain of zero d(0), but in fact there is no solid rela-tionship between d(r) and d(0). DMT calculations, as outlinedby Zheng and Tse [12], depend critically on the positivity of r .For example the proof of [12, Lemma 5] depends criticallyon r being strictly positive. More importantly, the asymptoticoutage calculations in [12, p. 1079] implicitly use r > 0 andresult in the outage region:

A = {α :∑

i

(1 − αi )+ < r}

where αi are the exponential order of the channel eigenvalues,i.e., λi = ρ−αi . If we set r = 0 this expression impliesthat the outage region is always empty, which is clearly nottrue.

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1020 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 60, NO. 2, FEBRUARY 2014

Fig. 1. MIMO with linear precoder.

Fig. 2. MIMO with linear precoder with receive-side equalization.

Thus, the DMT as calculated by the standard methodsof [12] does not extend to r = 0. The DMT d(r) is sometimesright-continuous at zero, including e.g. the examples in [12],but continuity at r = 0 does not always hold. There aresituations where d(0), the diversity at multiplexing gain zero,is not even uniquely defined, instead diversity takes multiplevalues at r = 0 as a function of rate R. This fact has beenobserved and analyzed, e.g., in [10], [11], [13]. The workin the present paper also produces several examples of thisphenomenon.

For the convenience of the reader, we now present a catalogof the results obtained in this paper. The number of transmitand receive antenna is M and N respectively, with M � N ,the diversity is denoted with d , spectral efficiency (rate)with R, and multiplexing gain with r . The type of systemis shown with a superscript, including zero-forcing precoding(ZFP), regularized-ZF (RZF), matched filter precoding (MFP),Wiener filter precding (WFP), and MMSE receiver. This paperdiscovers the following precoder diversities in the fixed-rateregime:

d Z F P = M − N + 1.

d R Z F ={

M N if R < Rth

0 otherwise

d M F P ={

M N if R < Rth

0 otherwise

dW F P = �N2− RN �2 + (M − N)�N2− R

N �where Rth = N log N

N−1 .This paper establishes the following DMTs for MIMO

precoders

d Z F P(r) = (M − N + 1)(1 − r

N), r ∈ (0, N]

d R Z F (r) = 0, r ∈ (0, N]d M F P(r) = 0, r ∈ (0, N]dW F P(r) = (M − N + 1)(1 − r

N), r ∈ (0, N]

This paper also calculates diversity in the presence of botha linear precoder and a linear receiver. We use the notationd A−B to denote a system with a precoder A and receiver B .

dW F P−Z F = M − N + 1.

d R Z F−Z F = 1

2(M − N + 1).

d M F P−Z F = 1

2(M − N + 1).

dW F P−M M S E = �N2− RN �2 + (M − N)�N2− R

N �.d R Z F−M M S E = d M F P−M M S E

={

12�N2− R

N �2 + (M − N)�N2− RN � if R > Rth

M N otherwise

where Rth = N log NN−1 . Note the fractional diversities, which

are uncommon.This paper is organized as follows. Section II describes

the system model. Section III provides outage analysis ofmany precoded MIMO systems. Section IV provides the DMTanalysis. The case of joint linear transmit and receive filters isdiscussed in Section V. Section VI provides simulations thatilluminate our results.

II. SYSTEM MODEL

A MIMO system with linear precoding is depicted inFigure 1. This system uses the linear precoder to manage theinterference between the streams in a MIMO system to avoid arequirement of optimal joint decoding in the receiver, which iscostly. We consider a flat fading channel H ∈ C

N×M , whereM and N are the number of transmit and receive antennas,respectively. While M � N when using linear precodingalone, we have N � M or M � N when using precodingtogether with receive-side linear equalization depending onwhether the precoder is designed for the equalized channelor the equalizer is designed for the precoded channel (seeFigure 2). The input-output system model for flat fading

MEHANA AND NOSRATINIA: DIVERSITY OF MIMO LINEAR PRECODING 1021

MIMO precoded channel with M transmit and N receiveantennas is given by

y = HTx + n

where T ∈ CM×B is the precoder matrix. Subsequently, wewill consider the joint effect of precoding and equalization,where the system model will be

y = WHTx + Wn (1)

where W ∈ CB×N is the receiver side equalizer. The numberof information symbols is B � min(M, N), the transmittedvector is x ∈ C B×1, and n ∈ C N×1 is the Gaussian noisevector. The vectors x and n are assumed independent.

We aim to characterize the diversity gain, d(R,M, N), asa function of the spectral efficiency R (bits/sec/Hz) and thenumber of transmit and receive antennas. This requires aPairwise Error Probability (PEP) analysis which is not directlytractable. Instead, we find the exponential order of outageprobability and then demonstrate that outage and PEP exhibitidentical exponential orders.

The objective of linear precoding/equalization is to trans-form the MIMO channel into min(M, N) parallel channelsthat can be described by

yk = √γkxk + nk, k = 1, . . . , B (2)

where γk is the SINR at the k-th receiver output andB = min(M, N), and nk are the decision point noise coef-ficients. Following the notation of [14], we define the outage-type quantities

Pout (R, N,M) � P(I (x; y) < R) (3)

dout(R, N,M) � − limρ→∞

log Pout (R,M, N)

logρ(4)

where ρ is the transmitted equivalent SNR, and I (x; y)=∑

I (xk; yk), the summation of the input-output mutual infor-mation of the individual streams.

The outage probabilities of MIMO systems under jointspatial encoding is respectively given by [11], [13]

Pout � P

( B∑

k=1

log(1 + γk) � R

)(5)

We shall perform outage analysis for different pre-coders/equalizers as the first step towards deriving the diversityfunction. We then provide lower and upper bounds on errorprobability via outage probabilities. This two-step approachwas first proposed in [12] due to the intractability of the directPEP analysis for many MIMO architectures.

We denote the exponential equality of two functions f (ρ)and g(ρ) as f (ρ)

.= g(ρ) when

limρ→∞

log f (ρ)

log(ρ)= limρ→∞

log g(ρ)

log(ρ)

The exponential inequalities � and � are defined in a similarmanner. In the following, we shall need to specify variousupper and lower bounds or approximations of the SINRγ , which will give rise to a number of variables such as

γ , γ , and γ . Following a well-used notation, we denotef (x) = �(g(x)) when there exist two positive constants c1, c2such that c1g(x) ≤ f (x) ≤ c2g(x) for sufficiently large x .

III. PRECODING DIVERSITY

In this section we analyze a linearly precoded MIMO systemwhere M ≥ N and the number of data streams B is equal toN . For the purposes of the developments in this section, thereis no receive-side equalization.

A. Zero-Forcing Precoding

The ZF precoder completely eliminates the interference atthe receiver. ZF precoding is well studied in the literature viaperformance measures such as throughput and fairness undera total (or per antenna) power constraint [15, and referencestherein].

1) Design Method I: One approach to design the ZF pre-coder is to solve the following problem [4]

T = arg minT E[||Tx||22

]

subject to HT = I (6)

The resulting ZF transmit filter is given by

T = βHH (HHH )−1 ∈ CM×N (7)

where β is a scaling factor to satisfy the transmit powerconstraint, that is [4]

β2tr(TTH )

� ρ (8)

where we assume that the noise power is one and that theinformation streams are independent. From (8), the receivedSINR per stream is thus given by

γ Z F Pk = ρ

tr((HHH )−1). (9)

Using (5), the outage probability is given by

Pout = P

(N log

(1 + ρ

tr((HHH )−1)

)� R

)(10)

A direct evaluation of (10) is not easily tractable sincethe diagonal elements of (HHH )−1 are distributed accordingto the inverse-chi-square distribution [11], [16]. We insteadbound (10) from below and above and show that the twobounds match asymptotically.

Let {λk} be the eigenvalues of HHH . Equation (10) can bewritten as

Pout = P

(N log

(1 + ρ

∑Nk=1

1λk

)� R

)

which can be bounded as

Pout � P

(N log(1 + ρ

Nλmin) � R

)

= P

(λmin � N(2

RN − 1)Rρ−1

)(11)

= P(λmin � ρ−1) (12)

.= ρ−(M−N+1). (13)

The transition from (12) to (13) is proved in Appendix A.


We now proceed with a lower bound on outage. The outageprobability in (10) can be bounded:

Pout = P

(N log(1 + ρ

tr(HHH )−1 ) � R

)

� P

(N log(1 + ρ

(HHH )−1kk

) � R

)

= P(z � ρ−1) (14)

where we have made a change of variable z = 1(HHH )−1

kk.

The random variable z in (14) is distributed according to thechi-square distribution with 2(M − N + 1) degree of freedom,i.e. z ∼ χ2

2(M−N+1) [16]. Thus the bound in (14) can beevaluated [11] yielding:

Pout � ρ−(M−N+1). (15)

From (13) and (15), we conclude that the diversity of MIMOsystem using the ZF precoder given by (6) and joint spatialencoding is

d Z F P = M − N + 1. (16)

B. Zero-Forcing Precoding: Design Method II

Notice that the ZF precoder design in (6) minimizes thetransmitted power. Another approach for ZF precoding designallocates unequal power levels across the transmit antennas tooptimize some performance measure. For instance, considerthe optimization problem [15]

maxpk ,T

∑

k

log(1 + γ Z F Pk )

subject to HT = diag{√p1, . . . ,√

pN }E||Tx||2 � ρ (17)

where pk is the transmit power for stream k. The optimalsolution for (17) (assuming independent transmit signaling)has the following form [15, Theorem 1]:

T = HH (HHH )−1diag{√p1, . . . ,√

pN } (18)

where pk are the solution to:

maxpk

∑

k

log(1 + γ Z F Pk )

subject to∑

k

pk[(HHH )−1]

kk � ρ (19)

Due to the logarithmic form of the cost function, the solutionhas the familiar form of waterfilling. It is well-known thatwater-filling may drive some pk to zero. Depending on thevalue of ρ and realization of HHH , it may also happen thatall optimal pk are positive. The set of realizations of HHH

that satisfy this condition are collected into an event that wedenote P . Conditioned on P it is easy to verify that the optimalsolution is given by:

pk = ρ + ∑Nk=1(HHH )−1

kk

N(HHH )−1kk

− 1 k = 1, . . . , N (20)

The outage probability can then be evaluated as follows

Pout = P

(log

N∑

k=1

(pk + 1) < R

)

= P

(log

N∑

k=1

(pk + 1) < R

∣∣∣∣P)

P(P)

+P

(log

N∑

k=1

(pk + 1) < R

∣∣∣∣P)

P(P)

(21)

We will now calculate P(P)

. Using (20), we have

P(P) = P

(N(HHH )−1

kk −N∑

k=1

(HHH )−1kk > ρ

)

� P

(N(HHH )−1

kk > ρ

)

� P

(Nλmax(HHH )−1 > ρ

)

= P

(λmin(HHH ) < Nρ−1

)

.= ρ−(M−N+1). (22)

where (22) is proved in Appendix A.We now bound other terms of (21).

P

( N∑

k=1

log(ρ + ∑N

k=1(HHH )−1kk

M(HHH )−1kk

) � R

)

� P

( N∑

k=1

log(ρ

M(HHH )−1kk

) � R

)(23)

= P

( N∑

k=1

log(M(HHH )−1

kk

ρ) � −R

)(24)

� P

(N log

N∑

k=1

((HHH )−1

kk

ρ) � −R

)(25)

= P

( N∑

k=1

1

ρλk� 2− R

N

)

� P

(1

ρλmin� 1

N2− R

N

)

.= P(λmin � ρ−1) (26)

.= ρ−(M−N+1), (27)

where (23) holds by discarding the positive element∑Nk=1(HHH )−1

kk . Equation (25) follows from Jensen’s inequal-ity. Thus

Pout = P

(log

N∑

k=1

(pk + 1) < R

∣∣∣∣P)

P(P)

+P

(log

N∑

k=1

(pk + 1) < R

∣∣∣∣P)

P(P)

� P

(log

N∑

k=1

(pk + 1) < R

∣∣∣∣P)

+ P(P)

.= ρ−(M−N+1). (28)

where we have used (22) and (27) to obtain (28).


A lower bound on the outage probability can be given asfollows.

Pout = P

(log

N∑

k=1

(pk + 1) < R

∣∣∣∣P)

P(P)

+P

(log

N∑

k=1

(pk + 1) < R

∣∣∣∣P)

P(P)

� P

(log

N∑

k=1

(pk + 1) < R

∣∣∣∣P)

P(P)

.= P

(log

N∑

k=1

(pk + 1) < R

∣∣∣∣P). (29)

where (29) follows since P(P) = 1 − P(P) .= 1. Thus theoutage probability can be bounded as follows

Pout � P

( N∑

k=1

log(ρ + ∑N

k=1(HHH )−1kk

M(HHH )−1kk

) � R

)

� P

( N∑

k=1

log(ρ + N

λmin

M(HHH )−1kk

) � R

)

� P

(N log

1

M N

N∑

k=1

ρ + 1λmin

(HHH )−1kk

� R

). (30)

The singular value decomposition of H and the correspond-ing eigen decomposition of HHH are given by

H = UVH

HHH = UUH

where U ∈ CN×N and V ∈ CM×M are unitary matri-ces, ∈ RN×M is a rectangular matrix with non-negativereal diagonal elements and zero off-diagonal elements, and = T ∈ RN×N is a diagonal matrix whose diagonalelements are the eigenvalues of HHH . Let uk be the k-thcolumn of UH . We have

(HHH )−1kk = uH

k −1uk =

N∑

l=1

|ukl |2λl

� |uk1|2λ1

(31)

where ukl is the (k, l) entry of the matrix U, and λ1 is theminimum eigenvalue.

The bound in (30) can be rewritten

Pout � P

(N log

1

M N

N∑

k=1

1∑N

l=1|ukl |2

(ρ+ 1λ1)λl

� R

)

� P

(N log

1

M N

N∑

k=1

1|uk1|2

(ρ+ 1λ1 )λ1

� R

)(32)

� P

(N log

1

M N

N∑

k=1

1|uk1|2(ρλ1+1)

� R

)(33)

� P

((ρλ1 + 1)

1

N

N∑

k=1

1

|uk1|2 � M2RN

)(34)

The quantity in the left hand side of (34) is similar to [13,Eq.(18)], thus the analysis of [13] applies and we obtain

Pout � P

(λmin � ρ−1

)= ρ−(M−N+1). (35)

Thus, the MIMO ZF precoding with unequal power alloca-tion (19) achieves diversity order M − N + 1.

Recall that the diversity is defined based on the decodingerror probability, not outage. In Appendix B we provide thepairwise error probability (PEP) analysis for the zero-forcingand regularized zero-forcing precoded systems and show thatthe outage and error probabilities exhibit the same diversity.

C. Regularized Zero-Forcing Precoding

In general, direct channel inversion performs poorly dueto the singular value spread of the channel matrix [9]. Onetechnique often used is to regularize the channel inversion:

T = β HH (HHH + c I)−1 (36)

where β is a normalization factor and c is a fixed constant.Recall

y = HTx + n = βU(+ c I)−1UH x + n (37)

allowing us to decompose the received waveform at eachantenna into signal, interference, and noise terms:

yk = β

( N∑

l=1

λl

λl + c|ukl |2

)xk

+βN∑

i=1,i �=k

( N∑

l=1

λl

λl + cuklu

∗il

)xi + nk (38)

where the scaling factor β is given by β = 1√η and

η = tr[(HHH + c I)−1HHH (HHH + c I)−1]

= tr[(UUH + c I)−1UUH (UUH + c I)−1]

= tr[U(+ c I)−1(+ c I)−1UH ]

= tr[(+ c I)−2] =

N∑

l=1

λl

(λl + c )2. (39)

The received signal power is given by

PT = E||HTx||2

= E

[β2tr

(U(+ c I)−1UH xxH U(+ c I)−1UH

)]

= E

[β2tr

((+ c I)−1UH xxH U(+ c I)−1UH U

)]

= β2tr

((+ c I)−1UH E(xxH )U(+ c I)−1

)

= β2ρ

Ntr[(+ c I)−22] = β2ρ

N

N∑

l=1

λ2l

(λl + c )2. (40)

where we have used E(xxH ) = ρN I.

The SINR is evaluated by computing the signal and inter-ference powers from (38). For a given channel H, the power


of desired and interference signals at the k-th receive antennaare respectively given by

P(k)D = β2ρ

N

( N∑

l=1

λl

λl + c|ukl |2

)2

(41)

P(k)I = β2ρ

N

N∑

i=1,i �=k

∣∣∣∣N∑

l=1

λl

λl + cukl u

∗il

∣∣∣∣2

. (42)

Thus the SINR for the k-th signal stream, assuming unitnoise power, is given by

γk = P(k)D

P(k)I + 1

=β2ρN

( ∑Nl=1

λlλl+c |ukl |2

)2

β2ρN

∑Ni=1,i �=k

∣∣∣∣∑N

l=1λl

λl+c ukl u∗il

∣∣∣∣2

+ 1

(43)

Defining the exponential order of eigenvalues λl = ρ−αl ina manner similar to [12], and using the definition of η = β−2,

γk =

( ∑l

ρ−αl

ρ−αl +c|ukl |2

)2

∑i �=k

∣∣∣∣∑N

l=1ρ−αl

ρ−αl +cukl u∗

il

∣∣∣∣2

+ N ρ−1 η

=

( ∑l ρ

−αl |ukl |2)2

∑i �=k

∣∣∣∣∑N

l=1 ukl u∗ilρ

−αl

∣∣∣∣2

+ N ρ−1∑N

l=1 ρ−αl

(44)

where we have substituted for η using (39), and the asymptoticequality follows because constant c dominates ρ−αl , a fact thatalso implies η

.= ∑l ρ

−αl .Multiplying the numerator and denominator of (44) by ρ2,

we have

γk=

( ∑l ρ

1−αl |ukl |2)2

∑i �=k

∣∣∣∣∑N

l=1 uklu∗ilρ

1−αl

∣∣∣∣2

+ N∑N

l=1 ρ1−αl

. (45)

The sum in the numerator of (45) is, in the SNR exponent,equivalent to:

∑

l

ρ1−αl |ukl |2 .= ρ1−αmin∑

l

|ukl |2

= ρ1−αmin (46)

where we use the fact that∑

l |ukl |2 = 1. Similarly, for thefirst term in the denominator of (45)

∑

i �=k

∣∣∣∣N∑

l=1

ukl u∗ilρ

1−αl

∣∣∣∣2.= ρ2−2αmin

∑

i �=k

∣∣∣∣N∑

l=1

uklu∗il

∣∣∣∣2

= ρ2−2αmin∑

i �=k

wki (47)

where we define wki �∣∣∣∑N

l=1 uklu∗il

∣∣∣2. Notice that wki ≤ 1.

Using (46) and (47), the SINR in (45) is given by

γk=(ρ1−αmin

)2

ρ2−2αmin∑

i �=k wki + N∑N

l=1 ρ1−αl

. (48)

If all α� > 1 then the exponents of ρ are negative andthe denominator is dominated by its second term, which alsodominates the numerator. If at least one of the α� ≤ 1, thenthe maximum exponent which corresponds to αmin dominateseach summation. Thus we have:

γk.=

⎧⎪⎪⎨

⎪⎪⎩

ρ1−αmin α1 > 1 , . . . , αN > 1(ρ1−αmin

)2

ρ2−2αmin∑N

i=1i �=k

wki +N ρ1−αminotherwise (49)

We now concentrate on the case where there exists at leastone α� ≤ 1. We define

μmin � mink,ik �=i

wki (50)

which is obviously a random variable, therefore in this specialcase we have:

γk �(ρ1−αmin

)2

(N − 1)(ρ1−αmin

)2μmin + N ρ1−αmin

(51)

.= 1

(N − 1)μmin� γ (52)

Thus in general

γk�ν

(N − 1)μmin� γ (53)

where ν is a new random variable defined as:

ν ={κα if αk > 1 ∀k

1 otherwise(54)

where κα � ρ1−αmin .We can now bound the outage probability as follows

Pout = P

( N∑

k=1

log(1 + γk) � R

)

� P

( N∑

k=1

log(1 + γ ) � R

)

= P

(ν

(N − 1)μmin� 2R/N − 1

)

= P

(ν

μmin� ζ

)(55)

where ζ � (2R/N − 1)(N − 1).The bound in (55) can be evaluated as follows

P

(ν

μmin� ζ

)= P

(ν

μmin� ζ

∣∣ν = κα

)P(ν = κα

)

+P

(ν

μmin� ζ

∣∣ν = 1

)P(ν = 1

)

= P(κα � ζ μmin

)P(ν = κα

)

+P( 1

μmin� ζ

)P(ν = 1

). (56)


Notice that P(κα � ζ μmin

) .= 1 since κα is vanishing at highSNR and ζ and μmin are positives. We now need to computeP(ν = κα

)and P

(ν = 1

), or equivalently P

({αk > 1 ∀k

})

and its complement. We use one of the results of [10].Lemma 1: Let {λn} denotes the eigenvalues of a Wishart

matrix HHH , where H is an N ×M matrix with i.i.d Gaussianentries, and let αn = − log(λn)

log(ρ) . If 1αn denotes the number ofαn that are greater than one, then for any integer s � N wehave [10, Section III-A] 1

P(1αn = s

) .= ρ−(s2+(M−N)s). (57)

Thus setting s = N (i.e. all αn > 1) in (57) yields

P(ν = κα

) = P(1αn = N

) .= ρ−M N (58)

P(ν = 1

) .= �(1) (59)

where �(1) is a non-zero constant with respect to ρ.Evaluating (56) depends on the values of ζ which is always

real and positive. If ζ < 1 then we have

P

(ν

μmin� ζ

).= ρ−M N (60)

because P( 1μmin

� ζ) = 0 as 1/μmin > 1. On the other hand

if ζ > 1 then

P

(ν

μmin� ζ

).= ρ−M N + P

( 1

μmin� ζ

)�(1) (61)

.= �(1) (62)

since P( 1μ � ζ

)is not a function of ρ because μ is indepen-

dent of ρ. For the set of rates where ζ > 1, equation (62)implies that the outage probability in (86) is not a function ofρ and thus the diversity is zero, i.e. the system will have errorfloor. The set of rates for which ζ > 1 are

R > N log( N

N − 1

)� Rth . (63)

This concludes the calculation of a lower bound on the out-age probability. A similar approach will yield a correspondingupper bound, as follows. Let

μmax � maxk �=i

|ukl′ u∗il′ |2 (64)

A lower bound on the SINR is given as

γk � ν

(N − 1)μmax� γ . (65)

The outage probability is bounded as

Pout � P

( N∑

k=1

log(1 + γ ) � R

)= P

(ν

μmax� ζ

). (66)

We can evaluate (66) in a similar way as (56), establishingthat the outage diversity d R Z F

out = M N if the operating spectralefficiency R is less than Rth = N log ( N

N−1 ), and d R Z Fout = 0 if

R > Rth . This shows that the performance of RZF precodercan be much better than that of the conventional ZF precoder

1Note that [10] analyzes linear MIMO receiver where it is assumed N � M.It can be easily shown that the above Lemma 1 applies for the case consideredhere where M � N .

MIMO system whose diversity is M − N + 1 independent ofrate.

Recall that diversity is the SNR exponent of the probabilityof codeword error. In Appendix B, we show that the outageexponent tightly bounds the SNR exponent of the error prob-ability. Thus we have the following theorem.

Theorem 1: For an M × N MIMO system that utilizes jointspatial encoding and regularized ZF precoder given by (36),the outage diversity is d R Z F = M N if the operating spectralefficiency R is less than Rth = N log ( N

N−1 ), and d R Z F = 0if R > Rth .

Remark 2: Rth is a monotonically decreasing function of Nwith the asymptotic value limN→∞ Rth = 1

ln 2 ≈ 1.44. Overallwe have 1.44 ≤ Rth ≤ 2, leading to an easily rememberedrule of thumb that applies to all antenna configurations.Regularized ZF precoders always exhibit an error floor atspectral efficiencies above 2 b/s/Hz, and enjoy full diversityat spectral efficiencies below 1.44 b/s/Hz.

D. Matched Filter Precoding

The transmit matched filter (TxMF) is introduced in [4], [8].The TxMF maximizes the signal-to-interference ratio (SIR)at the receiver and is optimum for high signal-to-noise-ratioscenarios [4]. The TxMF is also proposed for non-cooperativecellular wireless network [17]. The TxMF is derived bymaximizing the ratio between the power of the desired signalportion in the received signal and the signal power under thetransmit power constraint, that is [4]

T = arg maxT

E(||xH y||2)

E(||n||2)

subject to: E||Tx||2 � ρ (67)

where y is the noiseless received signal y = Tx.The solution to (67) is given by

T = βHH (68)

with

β =√

1

tr(HH H). (69)

We now analyze the diversity for the MIMO system underTxMF. The received signal is given by

y = βHHH x + n = βUUH x + n.

The received signal at the k-th antenna

yk = β

( N∑

l=1

λl |ukl |2)

xk

+βN∑

i=1,i �=k

( N∑

l=1

λluklu∗il

)xi + nk (70)

The SINR at k-th receive antenna is

γk =β2 ρ

N

( ∑Nl=1 λl |ukl |2

)2

β2 ρN

∑Ni=1,i �=k

∣∣∣∣∑N

l=1 λlukl u∗il

∣∣∣∣2

+ 1


Substitute with the value of β and λl = ρ−αl

γk =

( ∑Nl=1 ρ

−αl |ukl |2)2

∑Ni=1,i �=k

∣∣∣∣∑N

l=1 ρ−αl ukl u∗

il

∣∣∣∣2

+ N ρ−1∑N

l=1 ρ−αl

(71)

Observe that (71) is the same as the SINR of the RZF precodedsystem given by (44). Hence the analysis in the present casefollows closely that of the outage lower bound of the RZFprecoder, with the following result: the system can achieve fulldiversity as long as the operating rate is less than Rth givenin (63). The pairwise error probability analysis is also similarto that of the RZF precoding system (given in Appendix B)which we omit for brevity. Thus we conclude that Theorem 1applies for the TxMF precoder.

E. Wiener Filter Precoding

The transmit Wiener filter TxWF minimizes the weightedMSE function.

{T, β} = argminT,βE(||x − β−1y||2)

subject to E(||Tx

∣∣|2) � ρ. (72)

Solving (72) yields

T = βF−1HH (73)

with

F =(

HH H + N

ρI)

β =√

1

tr(F−2HH H)(74)

where β can be interpreted as the optimum gain for thecombined precoder and channel [4].

Notice that the TxWF precoding function is similar to thatof the MMSE equalizer [18]. Indeed the SINR of both systemsare equivalent. To see this, we first compute the SINR for theprecoded H ∈ CM×N (with M � N) MIMO channel

γk =ρ βN |(T H)kk |2

ρ βN

∑Ni �=k |(T H)ki |2 + 1

(75)

=ρN |(T H)kk |2

ρN

∑Ni �=k |(T H)ki |2 + tr(F−2HH H)

(76)

where we have used the independence of the transmitted signalto compute (75).

Now consider a MIMO channel H2 = HT ∈ CN×M . TheMMSE equalizer for this channel is given by

We = (HH2 H2 + N

ρI)−1HH

2 . (77)

The received SINR for that system is given by

γ M M S Ek =

ρN |(We H2)kk |2

ρN

∑Ni �=k |(We H2)ki |2 + tr(WeWe)

. (78)

Since We H2 = TW F PH and tr(WeWe) = tr(F−2HH H),we conclude that γ M M S E

k = γW F Pk . Hence the diversity

analysis of [10], [13] for the MIMO MMSE receiver appliesfor the MIMO Wiener precoding system. It is shown in [10]that this diversity is a function of rate R and number oftransmit and receive antennas. We thus conclude the following.

Lemma 2: Consider a channel H ∈ CM×N the diversity of

the MIMO system under Wiener filter precoding is given by

dW F P = �N2− RN �2 + (M − N)�N2− R

N � (79)

where (·)+ = max(·, 0) and �·�.Remark 3: It is commonly stated that MMSE and ZF

operators “converge” at high SNR. The developments in thispaper as well as [11] serve to show that although not false, thiscomment is essentially fruitless because the performance ofMMSE and ZF at high SNR are very different. This apparentincongruity is explained in the broadest sense as follows: Eventhough the MMSE coefficients converge to ZF coefficients asρ → ∞, the high sensitivity of logarithm of errors (especiallyat low error probabilities) to coefficients is such that theconvergence of MMSE to ZF coefficients is not fast enoughfor the logarithm of respective errors to converge.

IV. DIVERSITY-MULTIPLEXING TRADEOFF IN PRECODING

For increasing sequence of SNRs, consider a correspondingsequence of codebooks C(ρ), designed at increasing rates R(ρ)and yielding average error probabilities Pe(ρ). Then define

r = limρ→∞

R(ρ)

logρ

d = − limρ→∞

log Pe(ρ)

logρ.

For each r the corresponding diversity d(r) is defined (witha slight abuse of notation) as the supremum of the diversitiesover all possible codebook sequences C(ρ).

From the viewpoint of definitions, the traditional notionof diversity can be considered a special case of the DMTby setting r = 0. However, from the viewpoint of analysis,the approximations needed in DMT calculation make use ofR(ρ) being a strictly increasing function, while for diversityanalysis R is constant (not strictly increasing function of ρ).Thus, although sometimes DMT analysis may produce resultsthat are luckily consistent with diversity analysis2 (r = 0),in other cases the DMT analysis may produce results thatare inconsistent with diversity analysis. Certain equalizers andprecoders fall into the latter category. In the following, wecalculate the DMT of the various precoders considered up tothis point.

1) ZF Precoding: Recall that two ZF precoding designshave been considered. For the ZF precoder minimizing power,given by (7), the outage upper bound in (11) can be written as

Pout � P(λmin � ρ(

rN −1)) (80)

.= ρ−(M−N+1)(1− rN ) (81)

where we substitute R = r logρ to obtain (80), and equa-tion (81) follows in a manner identical to the procedure thatled to (13).

2E.g. the point-to-point MIMO channel with ML decoding.


Similarly the outage lower bound (14) can be written as

Pout � P(z � ρ(

rN −1))

.= ρ−(M−N+1)(1− rN ). (82)

From (81) and (82) we conclude

d Z F P(r) = (M − N + 1)(1 − r

N

)+. (83)

The DMT of the ZF precoder maximizing the throughput,given by (18), is obtained in an essentially similar manner tothe above, therefore the discussion is omitted in the interestof brevity.

2) Regularized ZF Precoding: We begin by producing anoutage lower bound. To do so, we start by the bound on theSINR of each stream k obtained in (49), and further bound itby discarding some positive terms in the denominator.

γk =(ρ1−αmin

)2

∑i �=k

∣∣ukl u∗il ρ

1−αmin∣∣2 + Nρ1−αmin

�

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(ρ1−αmin

)2

ρ2(1−αmin)∣∣ukl′ u∗

2l′∣∣2+Nρ1−αmin

k = 1

(ρ1−αmin

)2

ρ2(1−αmin)∣∣ukl′ u∗

1l′∣∣2+Nρ1−αmin

k > 1

.=⎧⎨

⎩

1|ukl′ u∗

2l′ |2k = 1

1|ukl′ u∗

1l′ |2k > 1

We can now bound the outage probability

Pout = P

( N∑

k=1

log(1 + γk) � R

)

� P

( N∑

k=1

log(1 + γk) � R

)

� P

(N log

N∑

k=1

1

N(1 + γk) � R

)(84)

.= P

( N∑

k=1

1

N(1 + γk) � ρ

rN

)(85)

.= P

( N∑

k=1

γk � ρrN

)

� P

(ν

|ukl′ u∗2l′ |2

+N∑

k=2

ν

|ukl′ u∗1l′ |2

� ρrN

). (86)

where we have used Jensen’s inequality in (84).For notational convenience define

ψ= 1

|ukl′ u∗2l′ |2

+N∑

k=2

1

|ukl′ u∗1l′ |2

.

Then the bound in (86) can be evaluated as follows:

P(νψ � ρ

rN) = P

(νψ � ρ

rN∣∣ν = 0

)P(ν = 0

)

+P(νψ � ρ

rN∣∣ν = 1

)P(ν = 1

)

= P(0 � ρ

rN)P(ν = 0

) + P(ψ � ρ

rN)P(ν = 1

)

.= ρ−M N + P(ψ � ρ

rN)�(1). (87)

� ρ−M N +�(1) (88)

= �(1) (89)

where (87) follows from Lemma 1, and (88) is true as longas P

(ψ � ρ

rN) = �(1), the proof of which is relegated to

Appendix C.Since the outage lower bound (88) is not a function of ρ,

the system will always have an error floor. In other words theDMT is given by

d R Z F(r) = 0 0 < r ≤ B (90)

We saw earlier that in the fixed-rate regime RZF precodingenjoys full diversity for spectral efficiencies below a certainthreshold, but it now appears that DMT shows only zerodiversity. DMT is not capable of predicting the complexbehavior at r = 0 because the DMT framework only assignsa single value diversity to all distinct spectral efficiencies atr = 0. A similar behavior was observed and analyzed for theMMSE MIMO receiver [10], [11], [13].

3) Matched Filter Precoding: The DMT of the MIMOsystem with the TxMF precoder is the same as the DMTgiven by (90) due to the similarity in the outage analysis (seeSection III-D). We omit the details for brevity.

4) Wiener Filter Precoding: Since the received SINR of theMIMO system using TxWF precoding is the same as that ofMIMO MMSE receiver, we conclude from [13] that the DMTfor the TxWF precoding system is

dW F P(r) = (M − N + 1)(1 − r

N

)+. (91)

Similarly to the MIMO MMSE receiver [10], [13], weobserve that DMT for the MIMO system with TxWF doesnot always predict the diversity in the fixed rate regime givenby (79).

V. EQUALIZATION FOR LINEARLY PRECODED

TRANSMISSION

The objective of a precoded transmitter is to separate thedata streams at the receiver. In other words, linear precodingis a method of interference management at the transmitter.In general, precoded systems do not require interferencemanagement at the receiver, however, once a transmitter isdesigned and standardized (as precoders have been), somestandards-compliant receivers may opt to further equalize theprecoded channel (see Figure 2). This section analyzes theequalization of precoded transmissions.

When the transmit and receive filters can be designed jointlyand from scratch, singular value decomposition becomes anattractive option whose diversity has been analyzed in [19].The distinction of the systems analyzed in this section isthat the precoders can be used with or without the receivefilters, while with the SVD solution neither the transmit northe receive filters can operate without each other.

A snapshot of some of the results of this section is asfollows. It is shown that equalization at the receiver canalleviate the error floor that was observed in matched filterprecoding as well as regularized ZF precoding. It is shown thatMMSE equalization does not affect the diversity of Wienerfilter precoding, but ZF equalization does indeed affect thediversity of Wiener filter precoding in a negative way.


Recall that in the system model given in Section II we havedefined the precoder and equalizer matrices T ∈ CM×B andW ∈ CB×N , respectively, where B is the number of datastreams, with B ≤ min(M, N). In most wireless systems, theequalizer at the receiver is designed to equalize the compoundchannel (HT) composed of the precoder and the channel(rather than designing the precoder for the equalized channel(WH) although it is possible). In such case we have M � Nand we set B = N .

A. ZF Equalizer

The ZF equalizer is analyzed when operating together withvarious precoders, as follows.

1) Wiener Filter Precoding: The TxWF precoder is given by

T = β

(HH H + N

ρI)−1

HH

= βHH(

HHH + N

ρIN

)−1

(92)

where (92) follows from [20, Fact 2.16.16] 3. The scalarcoefficient β is given in (74) and, similar to (39), it can bewritten as β = 1/

√η

η = tr[(+ Nρ−1 I)−2] =

N∑

l=1

λl

(λl + Nρ−1 )2

The ZF equalizer for the precoder and the channel isgiven by

WZ F = (HHH)−1

HH (93)

The composite channel H is given by

H = HT.

The received signal is given by

y = WZ F HTx + WZ F n. (94)

The filtered noise n = WZ F n is is a complex Gaussianvector with zero-mean and covariance matrix Rn given by

Rn = [HHH]−1

= [(HHH + Nρ−1 I)−1(HHH )2(HHH + Nρ−1 I)−1]−1

= [U(+ Nρ−1 I)−1UH U(+ Nρ−1 I)−1UH ]−1

= [U2(+ Nρ−1 I)−2UH ]−1

where we have used the eigen decomposition HHH = UUH .The noise variance of the output stream k is therefore

Rn(k, k) =N∑

l=1

(λl + Nρ−1

λl

)2

|ukl |2 (95)

where (95) follows in a similar manner as (31). We cancompute the signal-to-noise ratio of the ZF filter output:

γk = ρ β2

N Rn(k, k)

= ρ/N∑N

j=1λ j

(λ j+Nρ−1 )2

∑Nl=1

( λl+Nρ−1

λl

)2|ukl |2. (96)

3Let A ∈ Cn×m and B ∈ Cm×n then (In + AB)−1A = A(Im + BA)−1.This fact can be proved via Matrix Inversion Lemma.

Due to the complexity of (96) we proceed to bound theoutage from above and below. The upper bound on outage iscalculated as follows. Since |ukl | � 1,

γk � ρ/N∑N

j=1λ j

(λ j+Nρ−1 )2

∑Nl=1

( λl+Nρ−1

λl

)2(97)

= 1/N∑N

j=1ρ

1−α j

(ρ1−α j +N )2

∑Nl=1

( ρ1−αl +Nρ1−αl

)2(98)

� γ . (99)

where we have substituted λl = ρ−αl in (98). Thus the outageprobability is bounded as

Pout = P

( N∑

k=1

log(1 + γk) � R

)

� P

( N∑

k=1

log(1 + γ ) � R

)= P

(γ � 2

RN − 1

)(100)

Similarly to the analyses of earlier cases, we examine theSINR bound γ for different values of αl . Define the setB = {l | αl > 1} and the event

L = {|B| = N} (101)

we have

Pout � P(γ � 2

RN − 1

)

= P

(γ � 2

RN − 1

∣∣∣∣L)

P(L)+ P

(γ � 2

RN − 1

∣∣∣∣L)

P(L)(102)

� P

(γ � 2

RN − 1

∣∣∣∣L)

+ P

(γ � 2

RN − 1

∣∣∣∣L). (103)

To calculate the first term in (103), we evaluate γ whenαl � 1 ∀l

γ.= 1/N

∑Nj=1 ρ

1−α j∑N

l=11

ρ2(1−αl )

(104)

� 1/N∑N

l=11

ρ2(1−αl )

(105)

.= 1

Nρ2(1−αmax) = 1

Nρ2λ2

min (106)

where (104) follows because ρ1−αl + N.= N , (105) follows

because∑N

j=1 ρ1−α j �1, and (106) follows because the sum

in (105) is asymptotically dominated by the largest component.We further bound the first term in (103)

P

(γ � 2

RN − 1

∣∣L)

� P

(1

Nρ2λ2

min � 2RN

)

.= P(λmin � ρ−1) (107)

.= ρ−(M−N+1) (108)

where (107) is the same as (12), hence (108) follows.To calculate the second term in (103), we evaluate γ when

one or more αl � 1. Consider the two summations in the


denominator of (98). The first one can be asymptoticallyevaluated as

N∑

j=1

ρ1−α j

(ρ1−α j + N )2.=

∑

α j<1

1

ρ1−α j+

∑

α j>1

ρ1−α j

.={ρ−(1−αmax) |L| = N

max(ρ−1+α′, ρ1−α′′

) � ρ−(1−αmax) 1 � |L| < N(109)

where α′ = maxα j<1 α j and α′′ = minα j>1 α j and (109) fol-lows because min(ρ−1+α′

, ρ1−α′′) � ρ−(1−αmax). The second

summation in the denominator of (98) can be evaluated asfollows

N∑

l=1

(ρ1−αl + N

ρ1−αl

)2.=

∑

αl<1

1 +∑

αl>1

1

ρ2(1−αl )

.={

1 |L| = N

ρ−2(1−αmax) 1 � |L| < N(110)

We now use (109) and (110) to bound γ

γ �{ρ1−αmax = ρλmin |L| = N

ρ2−2αmax = ρ3λ3min 1 � |L| < N

� γ (111)

We thus have

Pout � P

(γ � 2

RN − 1

∣∣∣∣L)

� P

(γ � 2

RN − 1

∣∣∣∣L)

< P

(γ � 2

RN − 1

∣∣∣∣ |B| = 0

)

+P

(γ � 2

RN − 1

∣∣∣∣ 0 < |B| < N

)

.= P(λmin � ρ−1) + P

(λ3

min � ρ−3)

.= P(λmin � ρ−1) .= ρ−(M−N+1). (112)

This concludes the calculation of outage upper bound. Wenow proceed with the outage lower bound.

Define the event Q = {|akl | � ε∀ k, l} where akl is the (k, l)entry of the unitary matrix U (cf. equation (31)). Define

γ = 1/N∑N

j=1ρ

1−α j

(ρ1−α j +N )2

∑Nl=1

( ρ1−αl +Nρ1−αl

)2ε

(113)

Notice that γ > γ because |akl | � ε∀ k, l.The outage probability is bounded as

Pout = P

( N∑

k=1

log(1 + γk) � R

)

� P

( N∑

k=1

log(1 + γk) � R

∣∣∣∣Q)

P(Q)

� P

( N∑

k=1

log(1 + γ ) � R

)P(Q) (114)

= P(γ � 2

RN − 1

)P(Q) (115)

The probability P(Q) = �(1), i.e. non-zero constantwith respect to ρ. The proof is similar to the one in

[13, Appendix A] and omitted here for brevity. Wethus have

Pout � P(γ � 2

RN − 1

)

= P

(γ � 2

RN − 1

∣∣∣∣L)

P(L)+ P

(γ � 2

RN − 1

∣∣∣∣L)

P(L)

� P

(γ � 2

RN − 1

∣∣∣∣L)

P(L).= P

(γ � 2

RN − 1

∣∣∣∣L)

(116)

where (116) holds since P(L) .= �(1) as given by (59).We further bound the outage probability by bounding γ

as follows. Once again consider the two summations in thedenominator of (113). For the first summation of (113), wehave

N∑

j=1

ρ1−α j

(ρ1−α j + N )2.=

∑

α j<1

1

ρ1−α j+

∑

α j>1

ρ1−α j

.={ρ−(1−αmax) |L| = N

max(ρ−1+α′, ρ1−α′′

) � ρ1−αmax 1 � |L| < N(117)

where the bound in the second line (117) is true because∑

α j<1

1

ρ1−α j+

∑

α j>1

ρ1−α j �∑

α j>1

ρ1−α j .= ρ1−αmax

Using (110) and (117) to bound γ and substituting backin (113) gives:

γ �{ρ1−αmax = ρλmin |L| = N

ρ1−αmax = ρλmin 1 � |L| < N� ˘γ (118)

Thus the outage bound in (116) can be then evaluated aswe did for the upper bound

Pout � P

(γ � 2

RN − 1

∣∣∣∣L)

� P

(˘γ � 2

RN − 1

∣∣∣∣L)

< P

(˘γ � 2

RN − 1

∣∣∣∣|B| = 0

)P(|B| = 0

)

+P

(˘γ � 2

RN − 1

∣∣∣∣L, 0 < |B| < N

)P(|L| < N

)

.= P(λmin � ρ−1)�(1)+ P

(λmin � ρ−1)�(1) (119)

.= P(λmin � ρ−1)

.= ρ−(M−N+1). (120)

where (119) follows as a direct result of Lemma 1 (Eq. (57)).From (112) and (120), we conclude that the diversity of MIMOsystem using TxWF precoder and ZF equalizer is

dW F P−Z F = M − N + 1.

2) Regularized Zero Forcing Precoding: The ZF equalizeris given by (93) where the composite channel H = HT. Thereceived signal to noise ratio of the k-th output symbol ofthe ZF filter as

γk = ρ β2

N Rn(k, k)= ρ/N

∑Nj=1

λ j

(λ j+N )2

∑Nl=1

(λl+Nλl

)2|ukl |2. (121)


The process of obtaining lower and upper bound has manysimilarities with the developments of Section V-A.1, thereforewe omit many of the steps in the interest of brevity by referringto the previous developments.

We begin with the outage upper bound, which is developedin a manner similar to (100).

Pout = P

( N∑

k=1

log(1 + γk) � R

)

� P

( N∑

k=1

log(1 + γ ) � R

)

= P(γ � 2

RN − 1

)(122)

where

γ = ρ/N∑N

j=1λ j

(λ j+N )2

∑Nl=1

( λl+Nλl

)2

= ρ/N∑N

j=1ρ

−α j

(ρ−α j +N )2

∑Nl=1

( ρ−αl +Nρ−αl

)2

.= ρ/N∑N

j=1 ρ−α j

∑Nl=1 ρ

2αl(123)

� ρ/N∑N

l=1 ρ2αl

.= ρ/N

ρ2αmax. (124)

Thus the outage in (122) can be bounded as

Pout � P(γ � 2

RN − 1

)

� P

(ρ/N

ρ2αmax� 2

RN − 1

)

.= P(λmin � ρ−0.5)

.= ρ− 12 (M−N+1). (125)

We now turn to the lower bound, which is obtained in thesame manner as (116):

Pout = P

( N∑

k=1

log(1 + γk) � R

)

� P

( N∑

k=1

log(1 + γ ) � R

)

= P(γ � 2

RN − 1

)(126)

where

γ = ρ/N∑N

j=1λ j

(λ j +N )2

∑Nl=1

(λl+Nλl

)2ε

= ρ/N∑N

j=1ρ

−α j

(ρ−α j +N )2

∑Nl=1

(ρ−αl +Nρ−αl

)2ε

.= ρ/N∑N

j=1 ρ−α j

∑Nl=1 ερ

2αl

� ρ/N

ρ−α j∑N

l=1 ερ2αl

for arbitrary j

.= ρ/N

ε ρ−α jρ2αmax= ρ/Nλ2

min

ε λ j� ˘γ. (127)

Let C1 = (2RN − 1) ε N , C2 = C1ξ where ξ is a fixed

positive constant (independent of ρ), we have

Pout � P(γ � 2

RN − 1

)

� P( ˘γ � 2

RN − 1

)

� P

(ρλ2

min

λ j� C1

)

� P

(ρλ2

min

λ j� C1

∣∣∣∣λ j � ξ

)P(λ j � ξ

)

� P(ρλ2

min � C2)P(λ j � ξ

)

.= P(ρλ2

min � C2). (128)

The exponential inequality (128) holds because P(λ j � ξ

) =�(1), as proved in Appendix D. We thus conclude:

d R Z F−Z F = 1

2(M − N + 1).

Remark 4: We note that the diversity of regularized zero-forcing precoder together with a zero-forcing equalizer canbe fractional. To our knowledge this is the first instance offractional diversity uncovered in the literature.

3) Matched Filter Precoding: In this case, the compositechannel is

H = HT = βHHH .

The noise correlation matrix is given by

Rn = [HHH]−1 = 1

β2 [(HHH )2]−1 = 1

β2 (U2UH )−1.

Thus

Rn(k, k) = 1

β2

B∑

l=1

1

λ2l

|ukl |2 (129)

The precoder normalization factor β = 1/√η, where η is

given by

η = tr[HHH ] =

N∑

l=1

λl

The signal to noise ratio of the k-th symbol of the ZF filter is

γk = ρ

N Rn(k, k)

= ρ/N∑N

j=1 λ j∑N

l=11λ2

l|ukl |2

. (130)

Notice that the SINR γk in (130) is similar to the SINR γk ofthe RZF precoding system with ZF equalizer given by (121).The only difference is the term λk + N which, when applyingthe transformation of λk = ρ−αk , has no effect on the diversityanalysis as detailed in the previous section. We then concludethat the diversity of the MIMO system applying MF precoderand ZF equalizer is the same as the diversity of the RZFprecoder with ZF equalizer. Thus:

d M F P−Z F = 1

2(M − N + 1). (131)


B. MMSE Equalizer

The MMSE equalizer has better performance comparedto ZF and is therefore widely popular. We investigate thediversity of MIMO systems that deploy different precodersat the transmitter and MMSE equalizer at the receiver.

1) MFTx Precoding: The MFTx precoder, TM F P , is givenby (68). The MMSE equalizer for the precoded channel isgiven by

WM M S E =[H

HH + Nρ−1I

]−1

HH (132)

where H = H TM F P = βM F PHHH and βM F P is givenby (69).

The SINR at the output of the MMSE filter is given by [18]

γk = ρ

Nhk

[I + ρ

NHkH

Hk

]−1

hk

= 1[

I + ρN HH H

]−1

kk

− 1 (133)

where Hk is a submatrix of H obtained by removing the k-thcolumn, hk .

The diversity analysis of the precoded system uses someresults from the un-precoded MMSE MIMO equalizers [10],which we quote in the following lemma.

Lemma 3: consider a quasi-static Rayleigh fading MIMOchannel H ∈ C

M×N (M � N), the outage probability of theMMSE receiver satisfies

Pout.= P

(tr(I + ρ

NHH H)−1 � N2− R

N

)(134)

= P

( N∑

k=1

1

1 + ρN λ

′k

� N2− RN

)(135)

.= ρ−d M M SE(136)

where {λ′k} are the eigenvalues of H and d M M S E is given

by (79).Substituting λ′

k = ρ−α′k , we have

1

1 + ρN λ

′k

.={ρα

′k−1 α′

k < 1

1 α′k > 1

(137)

thus the term 11+ρλ′

k/Nis either zero or one at high SNR,

and therefore to characterize the sum in (135) at high SNRwe count the number of ones, or equivalently the number ofα′

k > 1. Hence the outage probability reduces to [10]

Pout.= P

( ∑

α′k>1

1 = ⌈N2− R

N⌉). (138)

Now we apply the matched filter precoder. Similarlyto (134), the outage portability is given by

Pout.= P

(tr(I + ρ

NHH

H )−1 � N2− RN

)(139)

= P

( N∑

k=1

1

1 + ρNη λ

2k

� N2− RN

)(140)

where we have used HHH = 1

η (HHH )2 = 1ηU2UH to

obtain (140), and {λk} are the eigenvalues of the Wishartmatrix HHH . The scaling factor η = tr(HHH ) = ∑N

l=1 λl .We begin with a hypothetical precoder whose transmit

power is not normalized, i.e., η = 1. The outage probabilityof this un-normalized precoder is similar to that of the MMSEreceiver with no precoding at the transmitter, as given in (136),except that the eigenvalues are now squared. Thus similarlyto (137), we have the exponential inequality

1

1 + ρN λ

2k

.={ρ2αk−1 αk < 0.5

1 αk > 0.5.(141)

The analysis of [10] then follows and we have

d = 1

2

(�N2− R

N �2 + (M − N)�M2− RN �

). (142)

We conclude that the un-normalized matched filter pre-coding with MMSE receiver results in 50% diversity losscompared to MMSE receiver with no transmit precoding.

For the normalized precoder, we begin with the outageprobability in (140). Assume α1 � α2 · · · � αN , the sumterm in (140) is given by

N∑

k=1

1

1 + ρNη λ

2k

=N∑

k=1

η

η + ρN λ

2k

=N∑

k=1

∑l ρ

−αl

∑l ρ

−αl + ρN ρ

−2αk

.=N∑

k=1

ρ−αN

ρ−αN + ρ1−2αk. (143)

where we have used the fact that the∑

l ρ−αk is dominated

by the maximum element at high SNR. It is easy to seethat the terms of (143) are either one or zero at high SNR,depending on whether ρ−αN asymptotically dominates ρ1−2αk

or vice versa. These two cases are delineated with the thresholdαk ≶ 0.5 max(1 , αN + 1), or, considering that αN is positive,αk ≶ 0.5(αN + 1). Thus at high SNR, the outage probabilityis evaluated by counting the ones

Pout.= P

( N∑

k=1

1

1 + ρNη λ

2k

� N2− RN

)

.= P

( ∑

αk>0.5 (αN +1)

1 � N2− RN

)

.= P

( ∑

αk>0.5 (αN +1)

1 = L

)(144)

where L = ⌈N2− R

N⌉

. The conversion from inequality toequality in equation (144) follows from arguments developedin [10, Section III-A] .

Therefore, the outage probability is asymptotically evalu-ated by:

Pout.=

∫

S+P(α) dα (145)


where P(α) is the joint distribution of the ordered α1 � · · · �αN and the region of integration is defined as S+ = S∩RN+,where S is given as follows:

• If L = N , then we seek the probability thatαk > 1

2 (αN + 1) for k = 1, . . . , N , which impliesαN ∈ (1,∞). Thus the integration region can be tightlyrepresented as:

S = {αN > 1 , min

1≤k<Nαk > 0.5(αN + 1)

}

• If L < N , then we seek the joint probability thatαk >

12 (αN + 1) for k = 1, . . . , L and αk ≤ 1

2 (αN + 1)for k = L + 1, . . . , N , implying αN ∈ (0, 1). Thus theregion of integration is represented as:

S ={αN < 1 , min

1<k≤Lαk > 0.5(αN + 1) ,

maxL<k<N

αk < 0.5(αN + 1)

}

Using methods similar to [12] and [10, Eq (20) - (23)],exponential equality relations can be used to reduce theintegrand to the following:

Pout.=

∫

S+

∏

k

ρ−(2k−1+M−N)αk d(α) (146)

First we consider L = N . The probability expression isevaluated by simply taking the integral over all variablesexcept αN , and then taking an integral over αN .

Pout.=

∫ ∞

αN =1ρ−(2N−1+M−N)αN

×N−1∏

k=1

ρ−(2k−1+M−N)(0.5+0.5αN )d(α) (147)

.=N∏

k=1

ρ−(2k−1+M−N)

= ρ∑N

k=1 −(2k−1+M−N) (148)= ρ−M N . (149)

When L < N , we repeat the same integration strategy.

Pout.=

∫ 1


×N∏

l=L+1

(1 − ρ−(2l−1+M−N)(0.5+0.5αN )

)

×L∏

k=1

ρ−(2k−1+M−N)(0.5+0.5αN )d(α) (150)

.=∫ 1


×L∏

k=1

ρ−(2k−1+M−N)(0.5+0.5αN )d(α) (151)

.=L∏

k=1

ρ− 12 (2k−1+M−N)

= ρ∑L

k=1 − 12 (2k−1+M−N)

= ρ− 12 (L

2+(M−N)L) (152)

In deriving (150) and (151) we have used∫ b

a ρ−ckαk d(αk)

.=ρ−ack [10]. Equations (149) and (152) show that the systemexhibits two distinct diversity behaviors based on whetherL = �N2− R

N � < N . We can solve to find the boundary ofthe two regions R = N log N

N−1 . To summarize:

d M F P−M M S E

={

12

(�N2− RN �2 + (M − N)�M2− R

N �) R > N log NN−1

M N otherwise.

(153)

Remark 5: The outcome is interesting for its practicalimplications: An MMSE receiver working with matched-filterprecoding will suffer a significant diversity loss compared toan MMSE receiver without precoding, except for very lowrates corresponding to R < N log N

N−1 , where the combinationof MMSE receiver with matched filter precoding has exactlythe same diversity as the MMSE receiver alone.

Remark 6: Recall that R = N log NN−1 is exactly the

same threshold below which matched filter precoding (withoutreceiver-side equalization) achieves full diversity.

2) WFTx Precoding: Using the Wiener filter precoding atthe receiver results in the composite channel

H = HT = βHHH (HHH + ρ−1 NI)−1.

Using the eigen decomposition HHH = UUH , it can beshown that

HH

H = β2U(+ ρ−1 NI)−22UH (154)

Similar to the case of MF precoder with MMSE receiver,the outage probability of WF precoder with MMSE receiveris given by (cf. (139))

Pout.= P

(tr(I + ρ

NHH

H )−1 � N2− RN

)

= P

( N∑

k=1

1

1 + ρNη λk

� N2− RN

)(155)

where {λk} are the eigenvalues of HH H and η is the scalefactor. Using (154), {λk} are given by

λk = λ2k

(λk + ρ−1 N)2, k = 1, . . . , N (156)

The scale factor η is calculated as in (39)

η =N∑

l=1

λl

(λl + ρ−1 N )2.

Thus the outage probability can be written as

Pout.= P

( N∑

k=1

γk � N2− RN

)(157)

where

γk � 1

1 + ρNη λk

= ρ−1η

ρ−1η + 1N λk

= ρ−1η

ρ−1η + υk


Fig. 3. ZF and Wiener filtering precoded 2 × 2 MIMO for rates (left toright): R = 1.9, 2.5, and 3 b/s/Hz.

Fig. 4. Wiener precoded 3 × 3 MIMO. The diversities are d = 9, 4 and 1for R = 1.5, 4 and 5 b/s/Hz respectively.

where we define υk = 1N λk . We now proceed to express both

ρ−1η and υk in terms of {αk}, the exponential orders of {λk}.

ρ−1η =N∑

l=1

ρ−1λl

(ρ−1λl + N )2=

N∑

l=1

ρ1−αl

(ρ1−αl + N )2

.=∑

αl>1

ρ1−αl +∑

αl<1

ραl−1 (158)

observe that all the terms in (158) have negative exponent.Using (156),

υk = 1

N

ρ−2αk

(ρ−αk + ρ−1 N)2

= 1

N

ρ2(1−αk)

(ρ1−αk + N)2

.={

1 αk < 1

ρ2(1−αk) αk > 1.(159)

From (158) and (159), we see that when αk < 1 then υk +ρ−1η

.= υk.= 1. On the other hand, when αk > 1 then

υk + ρ−1η.= ρ2(1−αk) +

∑

αl>1

ρ1−αl +∑

αl<1

ραl−1

= ρ2(1−αk) + ρ1−αk +∑

αl>1l �=k

ρ1−αl +∑

αl<1l �=k

ραl−1

.= ρ1−αk +∑

αl>1l �=k

ρ1−αl +∑

αl<1l �=k

ραl−1 (160)

.= ρ−1η (161)

where (160) follows because αk > 1. Thus we have

γk = ρ−1η

ρ−1η + υk

.={ρ−1η αk < 1

1 αk > 1(162)

and ρ−1η has negative exponent thus vanishes at high SNR.Observe that (162) is similar to (137) which corresponds to

the case of the MMSE-only system (i.e. with no precoding).Thus substituting (162) in the outage probability (157) andrepeating the same analysis of the MMSE-only system asin [10], we conclude that the diversity of the MMSE receiverwhen using WFTx precoding is the same as the diversity ofthe MMSE receiver with no linear precoding, which is givenby (79).

3) RZF Precoding: Using the Regularized Zero Forcingprecoding at the receiver results in the composite channel

H = HT = βHHH (HHH + c I)−1.

where c is a fixed constant, β = 1/η and η is given by (39)

η =N∑

l=1

λl

(λl + c )2=

N∑

l=1

ρ−αl

(ρ−αl + c )2. (163)

Similar to (155), the outage probability of RZF precoderwith MMSE receiver is given by

Pout.= P

( N∑

k=1

γk � N2− RN

)

and

γk � η

η + ρN λk

where {λk} are the eigenvalues of HH

H given by

λk = λ2k

(λk + c)2= ρ−2αk

(ρ−αk + c)2, k = 1, . . . , N (164)

Notice that at high SNR we have

η.=

N∑

l=1

ρ−αl

c2 λk.= ρ−2αk

c2 .

Thus the SINR is given by (cf. (143))

γk.=

∑Nl=1 ρ

−αl

∑Nl=1 ρ

−αl + ρ1−2αk

.= ρ−αN

ρ−αN +ρ1−2αk,

k = 1, . . . , N


Fig. 5. MF and regularized ZF precoded 2×2 MIMO. The diversity is d = 4for R = 1.9 b/s/Hz and d = 0 otherwise.

Fig. 6. Matched filter precoding and ZF equalization. The diversity is d = 0.5for a 2 × 2 MIMO system and d = 1 for a 3 × 2 MIMO system.

which are the same terms as in (143), implying that theoutage probability of the MMSE receiver working with theregularized zero-forcing precoder is asymptotically the sameas the outage probability of the MMSE receiver working withthe matched filter precoder. This means:

d R Z F−M M S E = d M F P−M M S E

={

12

(�N2− RN �2 + (M − N)�M2− R

N �) R > N log NN−1

M N otherwise.

(165)

VI. SIMULATION RESULTS

This section produces numerical results for the outageprobabilities of ZF, regularized ZF (RZF), matched filter (MF)and Wiener precoding systems. Figure 3 shows the outageprobabilities of the ZF and Wiener-filter precoded 2×2 MIMOsystems. The diversity in the case of the ZF case is the sameas the one predicted by the DMT. In the case of Wiener

Fig. 7. Wiener filter precoding and ZF equalization. The diversity is d = 1for a 2 × 2 MIMO and d = 2 for a 3 × 2 MIMO system.

Fig. 8. Wiener filter precoding and MMSE equalization for 2 × 2 MIMO.The diversity d = 1 for R = 3, 4 b/s/Hz, and d = 4 for R = 1.5 b/s/Hz.

precoding, the diversity is the same as the one predicted by theDMT for high rate (R) values and it departs from the DMTfor low rate values. A complete diversity characterization isgiven by (79) which is similar to that of the MMSE MIMOequalizer [10]. Figure 4 shows outage probabilities for a 3×3MIMO system with Wiener precoding. The diversity for therates R = 1.5, 4, and 5 b/s/Hz is 9, 4 and 1 respectively.Figure 5 shows an error floor for the regularized ZF andmatched filtering precoded 2×2 system at high rates. Howeverwe observe that the maximum diversity is achieved for anyrate R < 2 (cf. Equation (63)). Figure 6 shows outageprobabilities for a 2 × 2 and a 3 × 3 MIMO system withmatched filter precoding and ZF equalization. The observeddiversity values are consistent with Eq. (131). Figure 7 showsoutage probabilities for a 2 × 2 and a 3 × 3 MIMO systemwith Wiener filter precoding and ZF equalization. Figure 8 andFigure 9 show outage probabilities for a 2 × 2 and a 3 × 3MIMO system, respectively, with Wiener filter precoding and


Fig. 9. Wiener filter precoding and MMSE equalization for 3×3 MIMO. Thediversity mimics that of Wiener precoding without equalization (Eq. (79)).

Fig. 10. MF precoding and MMSE equalization 2× 2 MIMO. The diversityis given by Eq. (153).

MMSE equalization. The diversity for the 3 × 3 system is thesame as the diversity of the Wiener filtering precoding-only(cf. Figure 4).

Figure 10 shows the outage probability of a 2 × 2 MIMOsystem with matched filter precoding and MMSE equalization,which is consistent with Eq. (153). We also plot the outageprobability of the MMSE MIMO equalizer (without any pre-coding) for comparison.

VII. CONCLUSION

Linear precoders provide a simple and efficient process-ing, and have been shown to be optimal in some scenarios[5]–[7]. This paper studies the high-SNR performance of linearprecoders. It is shown that the zero-forcing precoder under twocommon design approaches, maximizing the throughput andminimizing the transmit power, achieves the same DMT asthat of MIMO systems with ZF equalizer. When a regularizedZF (RZF) precoder (for a fixed regularization term that is

independent of the signal-to-noise ratio) or matched filter (MF)precoder is used, we have d(r) = 0 for all r , implying an errorfloor under all conditions. It is also shown that in the fixed rateregime RZF and MF precoding achieve full diversity up to acertain spectral efficiency, while at higher spectral efficienciesthey produce an error floor. If the regularization parameter inthe RZF is optimized in the MMSE sense, the RZF precodedMIMO system exhibits a complex rate-dependent behavior. Inparticular, the diversity of this system (also known as Wienerfilter precoding) is characterized by d(R) = �N2− R

N �2 +(M − N)�N2− R

N � where M and N are the number of transmitand receive antennas. This is the same behavior observed inlinear MMSE MIMO receivers [10]. Various results for thediversity in the presence of both precoding and equalizationhave also been obtained.

APPENDIX A

ASYMPTOTIC MARGINAL DISTRIBUTION OF SMALLEST

EIGENVALUE OF WISHART MATRIX

Define a Wishart matrix W using the Gaussian matrix H.

W ={

H HH M > N

HH H N � N .

Let m = max(M, N) and n = min(M, N). The matrix W ism×m random non-negative definite that has real, non-negativeeigenvalues with λ1 � · · · � λn � 0, where for emphasis wedenote λn = λmin. The joint density of the ordered eigenvaluesis [21]

f (λ) = K −1m,ne− ∑

i λi∏

i

λn−mi

∏

i< j

(λi − λ j )2. (166)

Define

αk = − logλk

ρ(167)

Using (166) and (167), the joint distribution of α

f (α) = K −1m,n exp

[−

n∑

i=1

ρ−αi

](logρ)n

n∏

i=1

ρ−(m−n+1)αi

×∏

i< j

|ρ−αi − ρ−α j |2

Define the event A = {αk : αk � 1}. We now compute theprobability that λmin < ρ

−1.

P(λmin � ρ−1) = 1 − P

(λmin � ρ−1)

= 1 − P(λi � ρ−1,∀i

)

= 1 − P(ρ−αi � ρ−1,∀i

)

= 1 − P(αi � 1,∀i

)

= 1 −∫

Af (α) dα (168)

Define B = A ∩ Rn+ = {αk : 0 � αk � 1}. Following the

same analysis as in [10], [12], the integral in (168) can be


asymptotically evaluated as

∫

Af (α) dα

.=∫

B

n∏

i=1

ρ−(2i−1+m−n)αi dα

=n∏

i=1

(1 − ρ−(2i−1+m−n)) (169)

= 1 −n∑

i=1

ρ−(2i−1+m−n) + G(ρ) (170)

where (170) expands the product in (169), and G(ρ) col-lects all the higher-order terms. It can be easily seen that∑n

i=1 ρ−(2i−1+m−n) dominates G(ρ) at high SNR, thus we

have

1−n∑

i=1

ρ−(2i−1+m−n)+G(ρ).=1 −

n∑

i=1

ρ−(2i−1+m−n) (171)

Moreover, the term ρ−(m−n+1) dominates all other terms of∑ni=2 ρ

−(2i−1+m−n) at high SNR, i.e.

n∑

i=1

ρ−(2i−1+m−n) .= ρ−(m−n+1) (172)

which yields the following∫

Af (α) dα

.= 1 − ρ−(m−n+1) (173)

We now evaluate (168) using (173)

P(λn � ρ−1) = 1 −

∫

Af (α) dα

.= 1 − (1 − ρ−(m−n+1))

= ρ−(m−n+1) (174)

which concludes the proof.Remark 7: The result proven in this appendix, namely

f (λmin) ∝ λm−nmin for λmin < ε, has been used earlier in the

literature [13], [19] with a simple reference to the seminalwork of Telatar [21] but as far as we know a detailed proofhas not been available until now. Indeed a direct proof usingTelatar’s result is possible and is sketched as follows. UsingTelatar’s joint distribution of unordered eigenvalues, it is easyto see that the marginal distribution of an unordered eignevalueis in fact f (λi ) ∝ λm−n

i as λi → 0. To complete the prooffor λmin it remains to be shown that close to the origin,f (λmin) ∝ f (λi ), or equivalently P(λmin < ρ−1)

.= P(λi <ρ−1). This can be accomplished by noting {λmin < ρ−1} ⊂{λi < ρ−1)}, then showing the difference of the two eventsconstitutes a volume in the eigenvalue space that vanishessufficiently fast with ρ → ∞ so that its probability can bebounded (using boundedness of the joint distribution), andthus the SNR exponent of the two probabilities remain equal.Details of this alternative proof are omitted for brevity.

For the proof in this appendix, we have taken a differentapproach based on the exponential order of the eigenvalues,which is by now a well-established tool in diversity analysis,and seems better-suited to the tone and technique of this paper.

APPENDIX B

PAIRWISE ERROR PROBABILITY (PEP) ANALYSIS

In this section we perform PEP analysis for the zero-forcing (ZF) and the regularized ZF (RZF) precoding systems.The presented analysis can be easily extended to all otherprecoding systems. The basic strategy is to show the SNRexponent of outage probability bounds the SNR exponent ofPEP from both sides The PEP analysis follows from [10],[14], with careful attention to the system model given byEquation (1).

The lower bound immediately follows from [14, Lemma 3]by recognizing that although it was developed for SISO blockequalization, nowhere in its development does it depend onthe number of receive antennas, therefore we can directly useit for our purposes:

Perr � Pout . (175)

The upper bound on PEP for the ZF/RZF precoding systemsreceiver is developed using the union bound. Denote thechannel outage event by O and the error event by E . ThePEP is given by

Perr = P(E |O) Pout + P(E, O)

� Pout + P(E, O). (176)

In order to show that Pout dominates the right hand side of(176), it is shown in [10] that the probability P(E, O) can bebounded as follows using the union bound

P(E, O) � 2Rle− ρ/N

σ2n (k) � ρ−M N (177)

where l is the codeword length and σ 2n (k) is the variance of the

interference plus noise signal n in the k-th receive stream4. Theproof of [14] does not depend on the codeword length for bothupper and lower PEP bounds. The bound are tight and wereconfirmed by simulations for outage and error probabilities.

We now show that a similar proof holds for regularizedzero-forcing (RZF). Recall that the outage probability of theRZF can be upper bounded by (66)

Pout � P( ν

μmax� ζ

)� Pb

out (178)

We will use Pbout to further bound (176). Moreover P(E, O)

can be upper bounded by bounding the noise variance σ 2n (k)

in (177)

σ 2n (k) = PI + Pn < PT + 1 (179)

where we have used the noise power Pn = 1, and bound theinterference power by the total received power PT . We willfirst consider the case of RZF precoding since the case of ZFprecoding can be easily deduced from RZF by substituting theregularization parameter c = 0. For the RZF precoding system

4 [14] analyzes linear receivers so n is the k-th output filtered interferenceplus noise signals. By symmetry assumption all the equalizer outputs haveequal noise variance.


we use the PT given by (40) which can be simplified in a waysimilar to earlier sections

PT = β2ρ

N

N∑

l=1

λ2l

(λl + c )2

= 1∑N

l=1λl

(λl+c )2

ρ

N

N∑

l=1

λ2l

(λl + c )2

= 1∑N

l=1ρ−αl

(ρ−αl +c )2

ρ

N

N∑

l=1

ρ−2 αl

(ρ−αl + c )2

.= 1

ρ−αmin

ρ

Nρ−2 αmin

= 1

Nρ1−αmin . (180)

Using the union bound (177),

P(E, O) �{

2Rle−ραmin αmin < 1

2Rle− ρN αmin > 1

(181)

Since the exponential function dominates polynomials wehave

limρ→∞

e−ραmin

ρ−M N= 0

and

limρ→∞

e−ρ

ρ−M N = 0

which in turns gives

P(E, O) � ρ−M N . (182)

Using (178) and (182), the PEP given by (176) is boundedas

Perr � Pout + P(E, O)

� Pbout + ρ−M N

.= Pbout

= ρ−dout . (183)

therefore d � dout which concludes the proof for the RZFsystem.

For the ZF precoding system, it can be directly shown thata similar proof holds for both ZF precoding designs.

APPENDIX C

PROOF OF EQ. (88)

Recall that

ψ � 1

|u1l′u∗2l′ |2

+N∑

k=2

1

|ukl′ u∗1l′ |2

.

All terms of ψ the common factor 1|u1l′ |2 . Thus we have

ψ = ψaψb

ψa = 1

|u1l′ |2ψb =

(1

|u∗2l′ |2

+ 1

|u2l′ |2 + 1

|u3l′ |2 + 1

|u4l′ |2 + · · · + 1

|uNl′ |2).

(184)

Observe that all the terms of ψb are distinct except for thefirst two.

We now bound the probability P(ψ � ρ

rN).

P(ψ � ρ

rN)

� P(ψ � ρ

rN | ψ < c

)P(ψ < c)

� P(c � ρ

rN)

P(ψ < c).= P(ψ < c) (185)

Using ψ = ψaψb we can further bound (185)

P(ψ < c) = P(ψaψb < c)

� P(ψaψb � c

∣∣ψa < c2)

P(ψa < c2)

� P(c2ψb � c

)P(ψa < c2).

We thus have

P(ψ � ρ

rN)

� P(ψb � c′)

P(ψa < c2) (186)

and c′ = c/c2.We now evaluate the two probabilities in the right hand side

of (186). The first probability P(ψb � c′) = �(1). The proof

easily follows from [13, Appendix A] with the observationthat this proof holds even when the two first elements ofψb are the same. The second probability P(ψa < c2) isevaluated as follows. Let q = |u1l′ |2. We use the followingdistributions from [9, Appendix A]

f (q) = (N − 1)(1 − q)N−2, 0 � q � 1

then

P(ψa < c2) = P(q >1

c2)

=∫ 1

1c2

f (q) dq

= (1 − 1

c2)N−2 (187)

Observing that (187) is not a function of ρ concludes theproof.

APPENDIX D

PROOF OF P(λl � ξ

) = �(1) FOR ANY l

Using the development of Appendix A,

P(λl � ξ

)� P

(λmin � ξ

)

= P(λi � ξ,∀i

)

= P(ρ−αi � ξ,∀i

)

.= P(ρ−αi � ρ0,∀i

)(188)

= P(αi � 0,∀i

).= 1 (189)

where (188) follows since ξ is a fixed constant that does notdepend on ρ and (189) follows via steps similar to those inAppendix A.


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Ahmed Hesham Mehana (S’10–M’13) is currently with the Department ofElectronics and Electrical Communications Engineering at Cairo University.He received his B.Sc. and M.Sc. degrees from Cairo University in 2004 and2007, respectively, both in Electrical Engineering, and his Ph.D. degree inElectrical Engineering at the University of Texas at Dallas in 2012. He waswith Samsung Telecommunication America in 2013, and an intern at Researchin Motion Co. Ltd in 2010. His current interests include MIMO precoding,linear receivers, and interference management.

Aria Nosratinia (S’87–M’97–SM’04–F’10) is Erik Jonsson DistinguishedProfessor and associate head of the Electrical Engineering Department atthe University of Texas at Dallas. He received his Ph.D. in Electrical andComputer Engineering from the University of Illinois at Urbana-Champaignin 1996. He has held visiting appointments at Princeton University, RiceUniversity, and UCLA. His interests lie in the broad area of information theoryand signal processing, with applications in wireless communications. He wasthe secretary for the IEEE Information Theory Society in 2010-2011 and wasthe treasurer for ISIT 2010 in Austin, Texas. He has served as editor for theIEEE TRANSACTIONS ON INFORMATION THEORY, IEEE TRANSACTIONSON WIRELESS COMMUNICATIONS, IEEE SIGNAL PROCESSING LETTERS,IEEE TRANSACTIONS ON IMAGE PROCESSING, and IEEE WIRELESS COM-MUNICATIONS (MAGAZINE). He has been the recipient of the NationalScience Foundation career award, and is a fellow of IEEE.

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