IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 3, MARCH 2013 1059

Per-Antenna Constant Envelope Precoding forLarge Multi-User MIMO Systems

Saif Khan Mohammed, Member, IEEE, and Erik G. Larsson, Senior Member, IEEE

Abstract—We consider the multi-user MIMO broadcast chan-nel with M single-antenna users and N transmit antennas underthe constraint that each antenna emits signals having constantenvelope (CE). The motivation for this is that CE signals facilitatethe use of power-efficient RF power amplifiers. Analytical andnumerical results show that, under certain mild conditions onthe channel gains, for a fixed M , an array gain is achievableeven under the stringent per-antenna CE constraint. Essentially,for a fixed M , at sufficiently large N the total transmittedpower can be reduced with increasing N while maintaining afixed information rate to each user. Simulations for the i.i.d.Rayleigh fading channel show that the total transmit power canbe reduced linearly with increasing N (i.e., an O(N) array gain).We also propose a precoding scheme which finds near-optimalCE signals to be transmitted, and has O(MN) complexity. Also,in terms of the total transmit power required to achieve a fixeddesired information sum-rate, despite the stringent per-antennaCE constraint, the proposed CE precoding scheme performs closeto the sum-capacity achieving scheme for an average-only totaltransmit power constrained channel.

Index Terms—Multi-user, constant envelope, per-antenna,large MIMO, GBC.

I. INTRODUCTION

WE consider a Gaussian Broadcast Channel (GBC),wherein a base station (BS) having N antennas com-

municates with M single-antenna users in the downlink. Largeantenna arrays at the BS has been of recent interest, due totheir remarkable ability to suppress multi-user interference(MUI) with very simple precoding techniques [1]. Specifi-cally, under an average only total transmit power constraint(APC), for a fixed M , a simple matched-filter precoder hasbeen shown to achieve total MUI suppression in the limitas N → ∞ [2]. Additionally, due to the inherent arraypower gain property1, large antenna arrays are also beingconsidered as an enabler for reducing power consumptionin wireless communications, especially since the operational

Manuscript received December 5, 2011; revised June 8 and September 17,2012. The associate editor coordinating the review of this paper and approvingit for publication was A. Ghrayeb.

The authors are with the Communication Systems Division, Dept. ofElectrical Engineering (ISY), Linkoping University, Linkoping, Sweden (e-mail: {saif, erik.larsson}@isy.liu.se).

This work was supported by the Swedish Foundation for Strategic Research(SSF) and ELLIIT. E. G. Larsson was a Royal Swedish Academy of Sciences(KVA) Research Fellow supported by a grant from the Knut and Alice Wallen-berg Foundation. The work of Saif Khan Mohammed was partly supported bythe Center for Industrial Information Technology at ISY, Linkoping University(CENIIT). Parts of the results in this paper were presented at IEEE ICASSP2012 [15]. Also, the simpler special case of M = 1 (i.e., single-user) hasbeen studied by us in much greater detail in [16].

Digital Object Identifier 10.1109/TCOMM.2013.012913.1108271 Under an APC constraint, for a fixed M and a fixed desired information

sum-rate, the required total transmit power decreases with increasing N [3].

power consumption at BS is becoming a matter of world-wideconcern [4], [5].

Despite the benefits of large antenna arrays at the BS,practically building them would require cheap and power-efficient RF power amplifiers (PA’s). In conventional BS,power-inefficient PA’s account for about 40-50 percent ofthe total operational power consumption [5]. With currenttechnology, power-efficient RF components are generally non-linear. The type of transmitted signal that facilitates theuse of most power-efficient/non-linear RF components, is aconstant envelope (CE) signal. In this paper, we thereforeconsider a GBC, where the amplitude of the signal trans-mitted from each BS antenna is constant and independentof the channel realization. We only consider the discrete-time complex baseband equivalent channel model, where weaim to restrict the discrete-time per-antenna channel input tohave no amplitude variations. Compared to precoding methodswhich result in large amplitude-variations in the discrete-timechannel input, the CE precoding method proposed in thispaper is expected to result in continuous-time transmit signalswhich have a significantly improved peak-to-average-power-ratio (PAPR). However, this does not necessarily mean thatthe proposed precoding method will result in continuous-timetransmit signals having a perfectly constant envelope. Gener-ation of perfectly constant envelope continuous-time transmitsignals constitutes future work for us. One possible methodto generate almost constant-envelope continuous-time signalscould be to constrain the phase variation between consecutiveconstant amplitude baseband symbols of the discrete-timechannel input.

Since the per-antenna CE constraint is much more restrictivethan APC, in this paper we investigate as to whether MUIsuppression and array power gain can still be achieved underthe stringent per-antenna CE constraint. To the best of ourknowledge, there is no reported work which addresses thisquestion. Most reported work on per-antenna communicationconsider an average-only or a peak-only power constraint (see[6], [7] and references therein). In this paper, firstly, we deriveexpressions for the MUI at each user under the per-antenna CEconstraint, and then propose a low-complexity CE precodingscheme with the objective of minimizing the MUI energy ateach user. For a given vector of information symbols to becommunicated to the users, the proposed precoding schemechooses per-antenna CE transmit signals in such a way thatthe MUI energy at each user is small (i.e., of the same orderor less than the variance of the additive white Gaussian noise).Throughout the paper, we assume that such large antennasystems will not operate in a regime where the MUI energy is

0090-6778/13$31.00 c© 2013 IEEE

1060 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 3, MARCH 2013

significantly larger than the AWGN variance, since it is highlypower-inefficient to do so [8].

Secondly, under certain mild channel conditions (includingi.i.d. fading), using a novel probabilistic approach, we ana-lytically show that, MUI suppression can be achieved evenunder the stringent per-antenna CE constraint. Specifically,for a fixed M and fixed user information symbol alphabets,an arbitrarily low MUI energy can be guaranteed at each user,by choosing a sufficiently large N . Our analysis further revealsthat, with a fixed M and increasing N , the total transmittedpower can be reduced while maintaining a constant signal-to-interference-and-noise-ratio (SINR) level at each user.

Thirdly, through simulation, we confirm our analyticalobservations for the i.i.d. Rayleigh fading channel. For theproposed CE precoder, we numerically compute an achievableergodic information sum-rate, and observe that, for a fixedM and a fixed desired ergodic sum-rate, the required totaltransmit power reduces linearly with increasing N (i.e., anO(N) array power gain is achieved under the per-antenna CEconstraint). We also observe that, to achieve a given desiredergodic information sum-rate, compared to the optimal GBCsum-capacity achieving scheme under APC, the extra totaltransmit power required by the proposed CE precoding schemeis small (roughly 2.0 dB for sufficiently large N ).

Notation: C and R denote the set of complex and realnumbers. |x|, x∗ and arg(x) denote the absolute value,complex conjugate and argument of x ∈ C respectively.‖h‖2 Δ

=∑

i |hi|2 denotes the squared Euclidean-norm ofh = (h1, · · · , hN ) ∈ CN . E[·] denotes the expectationoperator. Abbreviations: r.v. (random variable), bpcu (bits-per-channel-use), p.d.f. (probability density function).

II. SYSTEM MODEL

Let the complex channel gain between the i-th BS antennaand the k-th user be denoted by hk,i. The vector of channelgains from the BS antennas to the k-th user is denoted byhk = (hk,1, hk,2, · · · , hk,N )T . H ∈ CM×N is the channelgain matrix with hk,i as its (k, i)-th entry. Let xi denote thecomplex symbol transmitted from the i-th BS antenna. Further,let PT denote the average total power transmitted from all theBS antennas. Under APC, we must have E[

∑Ni=1 |xi|2] =

PT , whereas under the per-antenna CE constraint we have|xi|2 = PT /N , i = 1, 2, · · · , N which is clearly a morestringent constraint compared to APC. Further, due to theper-antenna CE constraint, it is clear that xi is of the formxi =

√PT /Nejθi , where θi is the phase of xi. Under CE

transmission, the symbol received by the k-th user is thereforegiven by

yk =

√PT

N

N∑i=1

hk,iejθi + wk , k = 1, 2, . . . ,M (1)

where wk ∼ CN (0, σ2) is the AWGN noise at the k-threceiver. For the sake of notation, let Θ = (θ1, · · · , θN )T

denote the vector of transmitted phase angles. Let u =(√E1u1, · · · ,

√EMuM )T be the vector of scaled information

symbols, with uk ∈ Uk denoting the information symbolto be communicated to the k-th user. Here Uk denotes the

unit average energy information alphabet of the k-th user.Ek, k = 1, 2, . . . ,M denotes the information symbol energyfor each user. Also, let U Δ

=√E1U1×

√E2U2×· · ·×√

EMUM .Subsequently, in this paper, we are interested in scenarioswhere M is fixed and N is allowed to increase. Also, through-out this paper, for a fixed M , the alphabets U1, · · · ,UM arealso fixed and do not change with increasing N .

We stress that CE transmission is entirely different fromequal gain transmission (EGT). We explain this differencefor the simple single-user scenario (M = 1). In EGT a unitaverage energy complex information symbol u is communi-cated to the user by transmitting xi = wi u from the i-thtransmit antenna (with |w1| = · · · = |wN | = √

PT /N ), andtherefore the amplitude of the signal transmitted from eachantenna is not constant but varies with the amplitude of u(|xi| =

√PT /N |u|). In contrast, the CE precoding method

proposed in this paper (Section III-B) transmits a constantamplitude signal from each antenna (i.e.,

√PT /Nejθi from

the i-th antenna), where the transmit phase angles θ1, · · · , θNare chosen in such a way that the noise-free received signalis a known constant times the desired information symbol u.

III. MUI ANALYSIS AND THE PROPOSED CE PRECODER

For any given information symbol vector u to be commu-nicated, with Θ as the transmitted phase angle vector, using(1) the received signal at the k-th user can be expressed as

yk =√PT

√Ekuk +

√PT sk + wk ,

skΔ=

(∑Ni=1 hk,ie

jθi

√N

−√Ekuk

)(2)

where√PT sk is the MUI term at the k-th user. In this section

we aim to get a better understanding of the MUI energy levelat each user, for any general CE precoding scheme wherethe signal transmitted from each BS antenna has constantenvelope. Towards this end, we firstly study the range of valuestaken by the noise-free received signal at the users (scaleddown by

√PT ). This range of values is given by the set

M(H)Δ={v = (v1, · · · , vM ) ∈ C

M∣∣ vk =

∑Ni=1 hk,ie

jθi

√N

,

θi ∈ [−π, π) , i = 1, . . . , N}

(3)

For any vector v = (v1, v2, · · · , vM )T ∈ M(H), from (3) itfollows that there exists a Θv = (θv1 , · · · , θvN )T such that vk =∑N

i=1 hk,iejθvi√

N, k = 1, 2, . . . ,M . This sum can be expressed

as a sum of N/M terms (without loss of generality let usassume that N/M is integral only for the argument presentedhere)

vk =

N/M∑q=1

vqk ,

vqkΔ=

( qM∑r=(q−1)M+1

hk,rejθv

r

)/√N , q = 1, . . . ,

N

M.

(4)

MOHAMMED and LARSSON: PER-ANTENNA CONSTANT ENVELOPE PRECODING FOR LARGE MULTI-USER MIMO SYSTEMS 1061

From (4) it follows that M(H) can be expressed as a direct-sum of N/M sets, i.e.

M(H) = M(H(1)

)⊕M(H(2)

)⊕ · · · ⊕M(H(N/M)

)M(

H(q)) Δ={v = (v1, · · · , vM ) ∈ C

M∣∣

vk =

∑Mi=1 hk,(q−1)M+i e

jθi

√N

, θi ∈ [−π, π)}

, q = 1, . . . , N/M (5)

where H(q) is the sub-matrix of H containing only thecolumns numbered (q − 1)M + 1, (q − 1)M + 2, · · · , qM .M(

H(q)) ⊂ CM is the dynamic range of the received

noise-free signals when only the M BS antennas numbered(q−1)M+1, (q−1)M+2, · · · , qM are used and the remainingN −M antennas are inactive. If the statistical distribution ofthe channel gain vector from a BS antenna to all the users isidentical for all the BS antennas (as in i.i.d. channels), then,on an average the sets M(

H(q)), q = 1, . . . , N/M would all

have similar topological properties. Since, M(H) is a direct-sum of N/M topologically similar sets, it is expected thatfor a fixed M , on an average the region M(H) expands withincreasing N . Specifically, for a fixed M and increasing N ,the maximum Euclidean length of any vector in M(H) growsas O(

√N), since M(H) is a direct-sum of O(N) topolog-

ically similar sets (M(H(q)) , q = 1, 2, . . . , N/M ) with themaximum Euclidean length of any vector in M(

H(q))

beingO(1/

√N) (note that in the definition of M(

H(q))

in (5), eachcomponent of any vector v ∈ M(

H(q))

is scaled down by√N ). Also, for a fixed M and increasing N , since M(H)

is a direct-sum of N/M similar sets, it is expected that theset M(H) becomes increasingly dense (i.e., the number ofelements of M(H) in a fixed volume in CM is expected toincrease with increasing N ). The above discussion leads us tothe following results in Section III-A and III-C.

A. Diminishing MUI with increasing N , for fixed M and fixedEk(k = 1, . . . ,M)

For a fixed M and fixed Ek, the information alphabets andthe information symbol energies are fixed. However, sinceincreasing N (with fixed M ) is expected to enlarge the setM(H) and make it increasingly denser, it is highly probablethat at sufficiently large N , for any fixed information symbolvector u = (

√E1u1, · · · ,

√EMuM )T ∈ U there exists a

vector v ∈ M(H) such that v is very close to u in termsof Euclidean distance. This then implies that, with increasingN and fixed M , for any u ∈ U there exists a transmit phaseangle vector Θ such that the sum of the MUI energy for allusers is small compared to the AWGN variance at the receiver.Hence, for a fixed M and fixed Ek , it is expected that the MUIenergy for each user decreases with increasing N .

This is in fact true, as we prove it formally for channelssatisfying the following mild conditions. Specifically for afixed M , we consider a sequence of channel gain matrices

{HN}∞N=M satisfying

limN→∞

|h(N)k

Hh(N)l |

N= 0 , k = l (As.1)

limN→∞

∑Ni=1 |h(N)

k1,i| |h(N)

l1,i| |h(N)

k2,i| |h(N)

l2,i|

N2= 0 , (As.2)

limN→∞

‖h(N)k ‖2N

= ck , (As.3)

k, l, k1, l1, k2, l2 ∈ (1, 2, . . . ,M) (6)

where ck are positive constants, h(N)k denotes the k-th row

of HN and h(N)k,i denotes the i-th component of h

(N)k . From

the law of large numbers, it follows that i.i.d. channelssatisfy these conditions with probability one [13]. Physicalmeasurements of the channel characteristics with large antennaarrays at the BS have revealed closeness to the i.i.d. fadingmodel, as long as the BS antennas are sufficiently spaced apart(usually half of the carrier wavelength) [14], [1].

Theorem 1: For a fixed M and increasing N , consider asequence of channel gain matrices {HN}∞N=M satisfying themild conditions in (6). For any given fixed finite alphabetU (fixed Ek, k = 1, . . . ,M ) and any given Δ > 0, thereexist a corresponding integer N({HN},U ,Δ) such that withN ≥ N({HN},U ,Δ) and HN as the channel gain matrix,for any u ∈ U to be communicated, there exist a phaseangle vector Θu

N(Δ) = (θu1 (Δ), · · · , θuN(Δ))T which whentransmitted, results in the MUI energy at each user being upperbounded by 2Δ2, i.e.∣∣∣∑N

i=1 h(N)k,i e

jθui (Δ)

√N

−√Ekuk

∣∣∣2 ≤ 2Δ2 , k = 1, . . . ,M. (7)

Proof – The proof relies on technical results in Theorem 3(stated and proved in Appendix A) and Theorem 2 (stated andproved below). All these results assume a fixed M (numberof user terminals) and increasing N (number of BS antennas).These results are stated for a fixed sequence of channelmatrices {HN}∞N=M , fixed information alphabets U1, · · · ,UM

and fixed information symbol energy E1, · · · , EM . Further,the sequence of channel matrices {HN}∞N=M is assumed tosatisfy the conditions in (6) and the information alphabetsare assumed to be finite/discrete. The proofs use a novelprobabilistic approach, treating the transmitted phase anglesas random variables. We now present the proof of Theorem1.

Let us consider a probability space with the transmittedphase angles θi, i = 1, 2, . . . , N being i.i.d. r.v’s uniformlydistributed in [−π , π). For a given sequence of channelmatrices {HN}, we define a corresponding sequence of r.v’s

{zN}, with zNΔ= (zI

(N)

1 , zQ(N)

1 , . . . , zI(N)

M , zQ(N)

M ) ∈ R2M ,where we have

zI(N)

kΔ= Re

(∑Ni=1 h

(N)k,i e

jθi

√N

)zQ

(N)

kΔ= Im

(∑Ni=1 h

(N)k,i e

jθi

√N

),

k = 1, . . . ,M. (8)

From Theorem 3 it follows that, for any channel sequence{HN} satisfying the conditions in (6), as N → ∞ (with fixed

1062 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 3, MARCH 2013

BΔ(u)

Δ=

{b = (bI1, b

Q1 , · · · , bIM , bQM )T ∈ R

2M∣∣∣ |bIk −

√Eku

Ik| ≤ Δ , |bQk −

√Eku

Qk | ≤ Δ , k = 1, 2, . . . ,M

}(9)

M ), the corresponding sequence of r.v’s {zN} converges indistribution to a 2M -dimensional real Gaussian random vec-tor X = (XI

1 , XQ1 , · · · , XI

M , XQM )T with independent zero-

mean components and var(XIk ) = var(XQ

k ) = ck/2 , k =1, 2, . . . ,M . For a given u = (

√E1u1, · · · ,

√EMuM )T ∈ U ,

and Δ > 0, we next consider the box BΔ(u) defined in (9)

(at the top of this page), where uIk

Δ= Re(uk) , u

Qk

Δ= Im(uk).

The box BΔ(u) contains all those vectors in R2M whose

component-wise displacement from u is upper bounded by Δ.Using the fact that zN converges in distribution to a Gaussianr.v. with R2M as its range space, in Theorem 2 it is shownthat, for any Δ > 0, there exist an integer N({HN},U ,Δ),such that for all N ≥ N({HN},U ,Δ)

Prob(zN ∈ BΔ(u)) > 0 , ∀u ∈ U . (10)

Since the probability that zN lies in the box BΔ(u) is strictly

positive for all u ∈ U , from the definitions of BΔ(u) in (9)

and zN in (8) it follows that, for any u ∈ U there exist a phaseangle vector Θu

N (Δ) = (θu1 (Δ), · · · , θuN (Δ))T such that

∣∣∣Re(∑N

i=1 h(N)k,i e

jθui (Δ)

√N

)−√Eku

Ik

∣∣∣ ≤ Δ ,

∣∣∣Im(∑Ni=1 h

(N)k,i e

jθui (Δ)

√N

)−√Eku

Qk

∣∣∣ ≤ Δ (11)

for all k = 1, 2, · · · ,M , which then implies (7). �Since Theorem 1 is valid for any Δ > 0 and (7) holds

for all N ≥ N({HN},U ,Δ), we can satisfy (7) for anyarbitrarily small Δ > 0 by having N ≥ N({HN},U ,Δ)i.e., a sufficiently large N . Hence, the MUI energy at eachuser can be guaranteed to be arbitrarily small by havinga sufficiently large N . Theorem 1 therefore motivates us topropose precoding techniques which can achieve small MUIenergy levels.

An essential part of the proof for Theorem 1 was thepositivity of the box event probability Prob(zN ∈ BΔ(u)),when N is sufficiently large. In the following theorem, weformally state and prove the positivity of the box eventprobability.

Theorem 2: For a given channel sequence {HN}∞N=M

satisfying (6) and a given fixed finite alphabet set U , for anyΔ > 0, there exist a corresponding integer N({HN},U ,Δ),such that for all N ≥ N({HN},U ,Δ) (with fixed M )

Prob(zN ∈ BΔ(u)) > 0 , ∀u ∈ U . (12)

where BΔ(u) is defined in (9).

Proof – We consider the probability that a n-dimensionalreal r.v. X = (X1, X2, · · · , Xn) lies in a n-dimensionalbox centered at α = (α1, . . . , αn) ∈ Rn and denoted byC(Δ,α) =

{(x1, x2, · · · , xn) ∈ Rn |αk − Δ ≤ xk ≤

αk + Δ , k = 1, 2, . . . , n}

. For notational convenience, werefer to αk+Δ and αk−Δ as the corresponding “upper” and

“lower” limits for the k-th coordinate. The probability that Xlies in the box C(Δ,α) is given by the expansion

Prob(X ∈ C(Δ,α)) =n∑

k=0

(−1)kTk(Δ,α) (13)

where Tk(Δ,α) is the probability that the r.v.(X1, X2, · · · , Xn) belongs to a sub-region of{(x1, · · · , xn) ∈ Rn | xl ≤ αl + Δ , l = 1, 2, . . . , n

},

where exactly k coordinates are less than their corresponding“lower” limit and the remaining n − k coordinates are lessthan their corresponding “upper” limit. Specifically, Tk(Δ,α)is given by2

Tk(Δ,α) =

n∑i1=1

n∑i2=i1+1

· · ·n∑

ik=ik−1+1

Prob(Xr ≤ αr −Δ

∀r ∈ {i1, i2, · · · , ik} ,

Xr ≤ αr +Δ ∀r /∈ {i1, i2, · · · , ik})

(14)

Using the expansion in (13), the probability of the box event{zN ∈ B

Δ(u)}

can be expressed as

Prob(zN ∈ BΔ(u)

)=

Prob((√Eku

Ik −Δ) ≤ zIk

(N) ≤ (√Eku

Ik +Δ) ,

(√Eku

Qk −Δ) ≤ zQk

(N) ≤ (√Eku

Qk +Δ) , k = 1, 2, . . . ,M

)

=

2M∑k=0

(−1)k2M∑i1=1

2M∑i2=i1+1

· · ·2M∑

ik=ik−1+1

Prob(z(N)l ≤ √

Elul −Δ

∀l ∈ {i1, i2, · · · , ik} ,

z(N)l ≤ √

Elul +Δ ∀l /∈ {i1, i2, · · · , ik})

(15)

where z(N)l is the l-th component of zN (i.e., z(N)

l = zQ(N)

l/2

for even l, and z(N)l = zI

(N)

(l+1)/2 for odd l) and ul is the l-th

component of the vector (uI1, u

Q1 , u

I2, u

Q2 , · · · , uI

M , uQM )T . For

notational convenience we define

T(N)

(k, i1, i2, · · · , ik,u,Δ)Δ=

Prob(z(N)l ≤ √

Elul −Δ ∀l ∈ {i1, i2, · · · , ik} ,

z(N)l ≤ √

Elul +Δ ∀l /∈ {i1, i2, · · · , ik})

1 ≤ i1 < i2 < · · · < ik ≤ 2M , 0 ≤ k ≤ 2M. (16)

Let Y = (Y1, Y2, · · · , Y2M ) denote a multivariate 2M -dimensional real Gaussian r.v. with independent zero meancomponents and var(Y2k−1) = var(Y2k) = ck/2 , k =1, 2, . . . ,M . From Theorem 3 (Appendix A) it followsthat the c.d.f. of zN converges to the c.d.f. of Yas N → ∞. This convergence in distribution impliesthat, for any given arbitrary δ > 0, for each term

2 As an example, for n = 2, we have Prob(α1 − Δ ≤ X1 ≤ α1 +

Δ , α2 − Δ ≤ X2 ≤ α2 + Δ)

= T0(Δ,α) − T1(Δ,α) + T2(Δ,α),

where T0(Δ,α)Δ= Prob(X1 ≤ α1 + Δ , X2 ≤ α2 + Δ), T2(Δ,α)

Δ=

Prob(X1 ≤ α1 − Δ , X2 ≤ α2 − Δ), and T1(Δ,α)Δ= Prob(X1 ≤

α1 +Δ , X2 ≤ α2 −Δ) + Prob(X1 ≤ α1 −Δ , X2 ≤ α2 +Δ).

MOHAMMED and LARSSON: PER-ANTENNA CONSTANT ENVELOPE PRECODING FOR LARGE MULTI-USER MIMO SYSTEMS 1063

∣∣∣T (N)

(k, i1, i2, · · · , ik,u,Δ) − Prob(Yl ≤

√Elul −Δ ∀l ∈ {i1, i2, · · · , ik} ,

Yl ≤√Elul +Δ ∀l /∈ {i1, i2, · · · , ik}

) ∣∣∣ ≤ δ. (17)

g({HN},u,Δ, δ

) Δ= max

k=0,1,··· ,2Mmax

1≤i1<i2<···<ik≤2MN(k, i1, i2, · · · , ik, δ,u,Δ) (18)

∣∣∣Prob(zN ∈ B

Δ(u))− Prob

(Y ∈ B

Δ(u))∣∣∣

=

∣∣∣∣∣2M∑k=0

2M∑i1=1

2M∑i2=i1+1

...

2M∑ik=ik−1+1

(−1)k

{T

(N)

(k, i1, i2, · · · , ik,u,Δ) − Prob(Yl ≤

√Elul −Δ ∀l ∈ {i1, i2, · · · , ik} ,

Yl ≤√Elul +Δ ∀l /∈ {i1, i2, · · · , ik}

)}∣∣∣∣∣≤

2M∑k=0

2M∑i1=1

2M∑i2=i1+1

...

2M∑ik=ik−1+1

∣∣∣∣∣{T

(N)

(k, i1, i2, · · · , ik,u,Δ) − Prob(Yl ≤

√Elul −Δ ∀l ∈ {i1, i2, · · · , ik} ,

Yl ≤√Elul +Δ ∀l /∈ {i1, i2, · · · , ik}

)}∣∣∣∣∣≤ ∑2M

k=0

∑2Mi1=1

∑2Mi2=i1+1 · · ·

∑2Mik=ik−1+1 δ = 22Mδ. (19)

T(N)

(k, i1, i2, · · · , ik,u,Δ), there exists a corresponding pos-itive integer N(k, i1, i2, · · · , ik, δ,u,Δ) such that (17) issatisfied for all N ≥ N(k, i1, i2, · · · , ik, δ,u,Δ). We thenchoose a positive integer g

({HN},u,Δ, δ)

given by (18).Combining (15), (16) and (17), for all N ≥ g

({HN},u,Δ, δ)

we have (19). Since the range space (support) of Y is theentire space R2M , it follows that Prob

(Y ∈ B

Δ(u))> 0

(i.e., strictly positive) for any Δ > 0 and all u ∈ U . For thegiven information symbol vector u and Δ > 0, we choose acorresponding δ given by

δ(u,Δ)Δ=

1

2

Prob(Y ∈ BΔ(u)

)22M

> 0 (20)

From (19) and (20) it now follows that, for all N >g({HN},u,Δ, δ(u,Δ)

)we have

∣∣∣Prob(zN ∈ BΔ (u)

)− Prob(Y ∈ BΔ(u)

)∣∣∣ ≤ 22Mδ(u,Δ)

=Prob

(Y ∈ BΔ(u)

)2

(21)

which then implies that

Prob(zN ∈ BΔ(u)

) ≥ Prob(Y ∈ BΔ(u)

)2

> 0 (22)

i.e., Prob(zN ∈ B

Δ(u))

is strictly positive for N >g({HN},u,Δ, δ(u,Δ)

). For a given channel sequence

{HN}, a finite U and Δ > 0, we define the integer

N({HN},U ,Δ)Δ= max

u∈Ug({HN},u,Δ, δ(u,Δ)

). (23)

Combining this definition with the result in (22) proves thetheorem. �

B. Proposed CE Precoding Scheme

For reliable communication to each user, the precoder atthe BS must choose a Θ such that the MUI energy is assmall as possible for each k = 1, 2, . . . ,M . This motivatesus to consider the following non-linear least squares (NLS)problem, which for a given u to be communicated, findsthe transmit phase angles that minimize the sum of the MUIenergy for all users:

Θu = (θu1 , · · · , θuN ) = arg minθi∈[−π,π) , i=1,...,N

g(Θ,u)

g(Θ,u)Δ=

M∑k=1

∣∣∣sk∣∣∣2 =M∑k=1

∣∣∣∑Ni=1 hk,ie

jθi

√N

−√Ekuk

∣∣∣2.(24)

This NLS problem is non-convex and has multiple localminima. However, as the ratio N/M becomes large, due tothe large number of extra degrees of freedom (N − M ), thevalue of the objective function g(Θ,u) at most local minimahas been observed to be small, enabling gradient descent basedmethods to be used.3 However, due to the slow convergenceof gradient descent based methods, we propose a novel it-erative method, which has been experimentally observed toachieve similar performance but with a significantly fasterconvergence.

In the proposed iterative method to solve (24), we startwith the p = 0-th iteration, where we initialize all the

3 This observation is expected, since the strict positivity of the box eventprobability in (10) (proof of Theorem 1), implies that there are many distincttransmit phase angles Θ such that the received noise-free vector lies in a small2M -dimensional cube (box) centered at the desired information symbol vectoru, i.e., the MUI energy at each user is small for many different Θ.

1064 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 3, MARCH 2013

angles to 0. Each iteration consists of N sub-iterations. LetΘ(p,q) = (θ

(p,q)1 , · · · , θ(p,q)N )T denote the phase angle vector

after the q-th sub-iteration (q = 1, 2, . . . , N ) of the p-thiteration (subsequently we shall refer to the q-th sub-iterationof the p-th iteration as the (p, q)-th iteration). After the (p, q)-th iteration, the algorithm moves either to the (p, q + 1)-thiteration (if q < N ), or else it moves to the (p + 1, 1)-thiteration. In general, in the (p, q+1)-th iteration, the algorithmattempts to reduce the current value of the objective functioni.e., g(Θ(p,q),u) by only modifying the (q+1)-th phase angle(i.e., θ(p,q)q+1 ) while keeping the other phase angles fixed to theirvalues from the previous iteration. The new phase angles afterthe (p, q + 1)-th iteration, are therefore given by

θ(p,q+1)q+1 = arg min

Θ=

(θ(p,q)1 ,··· ,θ(p,q)q ,φ,θ

(p,q)q+2

,··· ,θ(p,q)N

)T, φ∈[−π,π)

g(Θ,u)

= π + arg

(M∑k=1

h∗k,q+1√N

[ ( 1√N

N∑i=1, �=(q+1)

hk,i ejθ

(p,q)i

)

−√Ekuk

])

θ(p,q+1)i = θ

(p,q)i , i = 1, 2, . . . , N , i �= q + 1. (25)

The algorithm is terminated after a pre-defined number ofiterations. We denote the phase angle vector after the lastiteration by Θu = (θu1 , · · · , θuN)T . Experimentally, we haveobserved that, for the i.i.d. Rayleigh fading channel, with asufficiently large N/M ratio, beyond the p = L-th iteration(where L is some constant integer), the incremental reductionin the value of the objective function is minimal. Therefore,we terminate at the L-th iteration. Since there are totally LNsub-iterations, from the phase angle update equation in (25), itfollows that the complexity of the proposed iterative algorithmis O(MN).

With Θu as the transmitted phase angle vector, the receivedsignal and the MUI term are given by

yk =√PT

√Ekuk +

√PT sk + wk ,

skΔ=

(∑Ni=1 hk,ie

jθui√

N−√Ekuk

)(26)

The received signal-to-noise-and-interference-ratio (SINR) atthe k-th user is therefore given by

γk(H, E,PT

σ2) =

Ek

Eu1,··· ,uM

[|sk|2]+ σ2

PT

(27)

where EΔ= (E1, E2, · · · , EM )T is the vector of information

symbol energies. Note that the above SINR expression is fora given channel realization H. For each user, we would beideally interested to have a low value of the MUI energyE[|sk|2], since this would imply a larger SINR.

To illustrate the result of Theorem 1, in Fig. 1, for the i.i.d.CN (0, 1) Rayleigh fading channel, with fixed informationalphabets U1 = U2 = · · · = UM = (16-QAM and Gaussian)and fixed information symbol energy Ek = 1, k = 1, . . . ,M ,we plot the ergodic (averaged over channel statistics) MUIenergy EH[|sk|2] with the proposed CE precoding scheme(using the discussed iterative method for solving (24)) as afunction of increasing N (sk is given by (26)).4 It is observed

4 We have observed that EH[|sk|2] is the same for all k = 1, . . . ,M .

10 20 30 40 50 60 70 80 90 10010

−5

10−4

10−3

10−2

10−1

100

No. of base station antennas (N)

Erg

odic

per

−use

r M

UI e

nerg

y

M = 12, 16−QAMM = 24, 16−QAMM = 12, GaussianM = 24, Gaussian

Ek = 1, k=1,2....,M

Fig. 1. Reduction in the ergodic per-user MUI energy EH

[|sk|2] withincreasing N . Fixed M , fixed U1 = · · · = UM = 16-QAM,Gaussian andfixed Ek = 1 , k = 1, 2, . . . ,M . IID CN (0, 1) Rayleigh fading.

that, for a fixed M , fixed information alphabets and fixedinformation symbol energy, the ergodic per-user MUI energydecreases with increasing N . This is observed to be true,not only for a finite/discrete 16-QAM information symbolalphabet, but also for the non-discrete Gaussian informationalphabet.

C. Increasing Ek with increasing N , for a fixed M , fixedU1, · · · ,UM and fixed desired MUI energy level

It is clear that, for a fixed M and N , increasing Ek, k =1, . . . ,M would enlarge U which could then increase MUIenergy level at each user (enlarging U might result in U /∈M(H)). However, since an increase in N (with fixed Mand Ek) results in a reduction of MUI (Theorem 1), itcan be argued that for a fixed M , with increasing N theinformation symbol energy of each user can be increased whilemaintaining a fixed MUI energy level at each user. Further,from (2), it is clear that for a fixed PT the effective SINR at thek-th user (i.e., Ek/(Eu[|sk|2] + σ2/PT )) will increase withincreasing N , since Ek can be increased while maintaininga constant MUI energy. Finally, since σ2/PT increases withdecreasing PT and the MUI energy |sk|2 is independentof PT , by appropriately decreasing PT and increasing Ek

with increasing N (fixed M ), a constant SINR level can bemaintained at each user.

This observation is based entirely on Theorem 1 (whichholds for a broad class of fading channels satisfying theconditions in (6), including i.i.d. fading channels).5 The aboveobservation implies that as long as the channel satisfies theconditions in (6), the total transmit power can be reducedwithout affecting user information rates, by using a sufficiently

5 Since Theorem 1 holds for all finite information alphabets, the aboveobservation is valid even for the special case when the information alphabetitself has constant amplitude symbols, e.g. PSK. However, with the proposedCE transmission scheme the per-antenna transmit signals have a constantenvelope irrespective of the information alphabet used, and therefore usingPSK type information alphabet offers no extra advantage in terms of the PAPRof the transmitted signals.

MOHAMMED and LARSSON: PER-ANTENNA CONSTANT ENVELOPE PRECODING FOR LARGE MULTI-USER MIMO SYSTEMS 1065

20 40 60 80 100 120 140 1600

1

2

3

4

5

6

7

8

9

No. of base station antennas (N)

E*

Ik = 0.1 , 16−QAM

Ik = 0.01 , 16−QAM

Ik = 0.1 , Gaussian

Ik = 0.01 , Gaussian

M = 12 users,

Ik : Ergodic per−user MUI energy

Fig. 2. E� vs. N for a fixed desired ergodic MUI energy level Ik (same foreach user). Fixed M = 12, fixed U1 = · · · = UM = 16-QAM,Gaussian.IID CN (0, 1) Rayleigh fading.

large antenna array at the BS (i.e., an achievable array gaingreater than one). We illustrate this through the followingexample using the proposed CE precoding scheme. Let thefixed desired ergodic MUI energy level for the k-th user bedenoted by Ik , k = 1, 2, · · · ,M . For the sake of simplicitywe consider U1 = U2 = · · · = UM . Consider

E� Δ= max

p>0∣∣ Ek=p ,EH

[Eu1,··· ,uM

[|sk|2

]]= Ik , k=1,··· ,M

p (28)

which finds the highest possible equal energy of the infor-mation symbols under the constraint that the ergodic MUIenergy level is fixed at Ik , k = 1, 2, · · · ,M . In (28), skis given by (26). In Fig. 2, for the i.i.d. Rayleigh fadingchannel, for a fixed M = 12 and a fixed U1 = · · · =UM = (16-QAM and Gaussian), we plot E� as a functionof increasing N , for two different fixed desired MUI energylevels, Ik = 0.1 and Ik = 0.01 (same Ik for each user6).From Fig. 2, it can be observed that for a fixed M and fixedU1, · · · ,UM , E� increases linearly with increasing N , whilestill maintaining a fixed MUI energy level at each user. At lowMUI energy levels, from (27) it follows that γk ≈ PTEk/σ

2.Since Ek (k = 1, 2, · · · ,M ) can be increased linearly withN (while still maintaining a low MUI level), it can be arguedthat a desired fixed SINR level can be maintained at eachuser by simply reducing PT linearly with increasing N . Thissuggests the achievability of an O(N) array power gain forthe i.i.d. Rayleigh fading channel. In the next section wederive an achievable sum-rate for the proposed CE precodingscheme, using which (in Section V), for an i.i.d. Rayleighfading channel, through simulations we show that indeed anO(N) array power gain can be achieved.

IV. ACHIEVABLE INFORMATION SUM RATE

In this section we study the ergodic information sum-rateachieved by the CE precoding scheme proposed in Section

6 Due to same channel gain distribution and information alphabet for eachuser, it is observed that the ergodic MUI energy level at each user is alsosame if the users have equal information symbol energy.

III-B. For a given channel realization H, Gaussian informa-tion alphabets7,8 U1, · · · ,UM , information symbol energiesE1, · · · , EM and total transmit power to receiver noise ratioPT /σ

2, the mutual information between yk and uk is givenby

I(yk;uk) = h(uk)− h(uk | yk)= h(uk)− h

(uk − yk√

PT

√Ek

∣∣∣ yk)≥ h(uk)− h

(uk − yk√

PT

√Ek

)(29)

where h(z) denotes the differential entropy of a continuousvalued r.v. z. The inequality in (29) follows from the fact thatconditioning of a r.v. reduces its entropy. Further, using (26)in (29) we have

I(yk;uk) ≥ h(uk)− h( sk√

Ek

+wk√

PT

√Ek

)= log2(πe)− h

( sk√Ek

+wk√

PT

√Ek

)≥ log2(πe)− log2

(πe var

[ sk√Ek

+wk√

PT

√Ek

])

≥ log2(πe)− log2

(πeE

[ ∣∣∣ sk√Ek

+wk√

PT

√Ek

∣∣∣2 ])

= log2(πe)− log2

(πe[E[|sk|2]Ek

+σ2

PTEk

])= log2

(γk(H, E,

PT

σ2))

= Rk

(H, E,

PT

σ2

)(30)

where Rk

(H, E, PT

σ2

)Δ= log2

(γk(H, E, PT

σ2 ))

is an achiev-able information rate for the k-th user, with the proposed CEprecoding scheme. In (30), we have used the fact that thedifferential entropy of a complex Gaussian circular symmetricr.v. z having variance σ2

z is log2(πeσ2z). Further, for any

complex scalar r.v. z, var[z]Δ= E[|z − E[z]|2]. The second

inequality in (30) follows from the fact that, for a complexscalar r.v., among all possible probability distributions havingthe same variance, the complex circular symmetric Gaussiandistribution is the entropy maximizer [9]. The third inequalityfollows from the fact that, for any complex scalar r.v. z,var[z] ≤ E[|z|2]. From (30) it follows that an achievable

7We restrict the discussion to Gaussian information alphabets, due to thedifficulty in analyzing the information rate achieved with discrete alphabets.This is not a concern since, through Figs. 1 and 2, we have already observedthat the two important results in Section III-A and III-C hold true for Gaussianalphabets as well.

8Gaussian information alphabets need not be optimal w.r.t. achieving themaximum sum-rate of a per-antenna CE constrained GBC. As an example, in[16], we have considered the capacity of a single-user MISO channel with per-antenna CE constraints at the transmitter. Due to the scenario in [16] beingmuch simpler compared to the multi-user scenario discussed here, in [16]we were able to show that the optimal capacity achieving complex alphabetis discrete-in-amplitude and uniform-in-phase (DAUIP) (i.e., non-Gaussian).However, since it appears that the analytical tools and techniques in [16]cannot be used to derive the optimal alphabet for the multiuser scenario, werestrict ourselves to Gaussian alphabets here.

1066 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 3, MARCH 2013

N=60 N=80 N=100 N=120 N=160 N=200 N=240 N = 320 N = 400GBC Sum Capacity Upper Bound (M = 10) -2.8 -4.0 -5.1 -5.8 -7.2 -8.2 -8.9 -10.2 -11.2

Proposed CE Precoder (M = 10) -0.8 -2.1 -3.3 -4.1 -5.5 -6.5 -7.2 -8.6 -9.6Power Gap (M = 10) 2.0 1.9 1.8 1.7 1.7 1.7 1.7 1.6 1.6

GBC Sum Capacity Upper Bound (M = 40) 3.8 2.4 1.3 0.6 -0.9 -2.0 -2.7 -4.1 -5.1Proposed CE Precoder (M = 40) 9.2 6.0 4.1 3.2 1.4 -0.1 -0.9 -2.3 -3.5

Power Gap (M = 40) 5.4 3.6 2.8 2.6 2.3 1.9 1.8 1.8 1.6

Fig. 3. Minimum PT /σ2 (DB) required to achieve a per-user ergodic rate of 2 bpcu.

0 50 100 150 200 250 300−12

−9

−6

−3

0

3

6

9

12

15

No. of Base Station Antennas (N)

Min

. req

d. P

T/σ2 (

dB)

to a

chie

ve a

per

−use

r ra

te o

f 2 b

pcu

M = 10, Proposed CE Precoder (CE)M = 10, ZF Phase−only Precoder (CE)M = 10, GBC Sum Cap. Upp. Bou. (APC)M = 40, Proposed CE Precoder (CE)M = 40, ZF Phase−only Precoder (CE)M = 40, GBC Sum Cap. Upp. Bou. (APC)

1.7 dB

Fig. 4. Required PT /σ2 vs. N , to achieve a fixed desired ergodic per-user rate = 2 bpcu. Gaussian information alphabets U1 = · · · = UM . IIDCN (0, 1) Rayleigh fading.

ergodic information sum-rate for the GBC under the per-antenna CE constraint, is given by

RCE(E,

PT

σ2

)Δ=

M∑k=1

EH

[Rk

(H, E,

PT

σ2

) ]. (31)

Subsequently, we consider the scenario where all users havethe same unit energy Gaussian information alphabet and thesame information symbol energy.9 Further optimization ofRCE

(E, PT

σ2

)over E subject to E1 = · · · = EM , results

in an achievable ergodic information sum-rate which is givenby

RCE(PT

σ2

)Δ= max

E |E1=E2=···=EM>0RCE

(E,

PT

σ2

)(32)

Since it is difficult to analyze the sum-rate expression in (32),we have studied it through exhaustive numerical simulationsfor an i.i.d. CN (0, 1) Rayleigh fading channel. In the follow-ing section, we present some important observations based onthese numerical experiments.

V. SIMULATION RESULTS ON THE ACHIEVABLE ERGODIC

INFORMATION SUM-RATE RCE(

PT

σ2

)All reported results are for the i.i.d. CN (0, 1) Rayleigh

fading channel. In Fig. 4, for a fixed M we plot the minimumPT /σ

2 required by the proposed CE precoder, to achieve an

9We impose this constraint so as to reduce the number of parametersinvolved, thereby simplifying the study of achievable rates in a multi-userGBC with per-antenna CE transmission. Nevertheless, for the i.i.d. Rayleighfading channel with each user having the same Gaussian information alphabet,it is expected that the optimal E which maximizes the ergodic sum-rate in(31), has equal components.

ergodic per-user information rate of RCE(PT /σ2)/M = 2

bpcu as a function of increasing N (Due to the same channeldistribution for each user, we have observed that the ergodicinformation rate achieved by each user is 1/M of the ergodicsum-rate). The minimum required PT /σ

2 is also tabulated inFig. 3. It is observed that, for a fixed M , at sufficiently largeN , the required PT /σ

2 reduces by roughly 3 dB for everydoubling in N . This shows that, for a fixed M , an array powergain of O(N) can indeed be achieved even under the stringentper-antenna CE constraint. For the sake of comparison, wehave also plotted a lower bound on the PT /σ

2 required toachieve a per-user ergodic rate of 2 bpcu under the APCconstraint (we have used the cooperative upper bound on theGBC sum-capacity [10]).10 We observe that, for large N anda fixed per-user desired ergodic information rate of 2 bpcu,compared to the APC only constrained GBC, the extra totaltransmit power (power gap) required under the per-antennaCE constraint is small (1.7 dB).

In Fig. 4, we also consider another CE precoding scheme,where, for a given information symbol vector u, the precoderfirstly computes the zero-forcing (ZF) vector x = H†u,

(H† Δ= HH

(HHH

)−1

is the pseudo-inverse of H). Priorto transmission, each component of x is normalized to have amodulus equal to

√PT /N , i.e., the signal transmitted from the

i-th BS antenna is√PT /N xi/|xi|. At each user, the received

signal is scaled by a fixed constant.11 We shall hence-forthrefer to this precoder as the ZF phase-only precoder. In Fig. 4,we observe that the PT /σ

2 required by the proposed CEprecoder is always less than that required by the ZF phase-onlyprecoder. In fact, for moderate values of N/M , the proposedCE precoder requires significantly less PT /σ

2 as compared tothe ZF phase-only precoder (e.g. with N = 100,M = 40, therequired PT /σ

2 with the proposed CE precoder is roughly 3dB less than that required with the ZF phase-only precoder).At very large values of N/M , the ZF phase-only precoder hassimilar performance as the proposed CE precoder. However,in terms of complexity the ZF phase-only precoder does notnecessarily have a lower complexity than the proposed CEprecoder. This is because, the ZF phase-only precoder needsto compute the pseudo-inverse of the channel gain matrix(a M × N matrix) and also the matrix vector product ofthe pseudo-inverse times the information symbol vector u.Computing the pseudo-inverse has a complexity of O(M2N)and that for the matrix vector product is O(MN), resulting in

10 The cooperative upper bound on the GBC sum capacity gives a lowerbound on the PT /σ2 required by a GBC sum-capacity achieving scheme toachieve a given desired ergodic information sum-rate.

11This constant is chosen in such a way that the ergodic per-user informa-tion rate is maximized. It is therefore fixed for all channel realizations anddepends only upon the statistics of the channel, PT /σ2, N and M .

MOHAMMED and LARSSON: PER-ANTENNA CONSTANT ENVELOPE PRECODING FOR LARGE MULTI-USER MIMO SYSTEMS 1067

0 0.5 1 1.5 2 2.5 31

2

3

4

5

6

7

8

9

Desired per−user information rate (bpcu)

Ext

ra T

rans

mit

Pow

er R

equi

red

w.r

.t. S

um C

ap. U

pp. B

ou. (

dB)

ZF Phase−only Precoder

Proposed CE Precoder

M = 12, N = 48

1.5 dB

Fig. 5. The extra PT /σ2 (in dB) required (vertical axis) by the proposedCE precoder and by the ZF phase-only precoder, respectively, to achievethe same ergodic per-user information rate as predicted by the GBC sum-capacity cooperative upper bound (horizontal axis). Here the number of basestation antennas is N = 48 and the number of users is M = 12. All usersuse Gaussian information alphabets U1 = · · · = UM = Gaussian and allchannels are i.i.d. CN (0, 1) Rayleigh fading.

a total complexity of O(M2N). In contrast, the proposed CEprecoder does not need to compute the pseudo-inverse, andhas a complexity of O(MN) (see Section III-B).

To gain a better understanding of the power-efficiency of theconsidered CE precoders, in Fig. 5, for a fixed N = 48,M =12 we plot an upper bound on the extra PT /σ

2 required by theconsidered CE precoding schemes when compared to a GBCsum-capacity achieving scheme under APC,12 as a functionof the desired per-user ergodic information rate (note that inFig. 4, the desired per-user rate was fixed to 2 bpcu). It isobserved that, for a desired ergodic per-user information ratebelow 2 bpcu, the ZF phase-only precoder requires roughly1− 1.5 dB more transmit power as compared to the proposedCE precoder. For rates higher than 2 bpcu, this gap increasesvery rapidly (at 3 bpcu, this power gap is roughly 6 dB).In Fig. 6, we plot the results of a similar experiment butwith N = 480,M = 12 (a very large ratio of N/M ).It is observed that, the ZF phase-only precoder has similarperformance as the proposed CE precoder for per-user ergodicinformation rates below 3 bpcu. For rates higher than 3 bpcu,the performance of the ZF phase-only precoder deterioratesrapidly, just as it did in Fig. 5. In Figs. 5 and 6, we also notethat the extra total transmit power required by the proposedCE precoder (Section III-B) increases slowly w.r.t. increasingrate, and is less than 2.5 dB for a wide range of desired per-user information rates. From exhaustive experiments, we haveconcluded that, for moderate values of N/M , the proposedCE precoder is significantly more power efficient than theZF phase-only precoder, whereas for very large N/M bothprecoders have similar performance when the desired per-userergodic information rate is below a certain threshold (beyondthis threshold, the performance of the ZF phase-only precoder

12 Since we use the cooperative upper bound to predict the PT /σ2 requiredby a GBC sum-capacity achieving scheme, the reported values of the extraPT /σ2 required by the considered CE precoders are infact an upper boundon the minimum extra PT /σ2 required.

0 1 2 3 4 51

2

3

4

5

6

7

8

9

10

11

Fixed desired per−user information rate (bpcu)

Ext

ra T

rans

mit

Pow

er R

equi

red

w.r

.t. S

um C

ap. U

pp. B

ou. (

dB)

ZF Phase−only Precoder

Proposed CE Precoder

M = 12, N = 480

Fig. 6. Same as Fig. 5, but for N = 480 base station antennas.

0 50 100 150 200 250 300 350 4000.8

1

1.2

1.4

1.6

1.8

2

2.2

Number of BS antennas (N)

Erg

odic

Per

Use

r In

form

atio

n R

ate

(bpc

u)

GBC Sum Capacity Upper Bound (APC)Proposed CE PrecoderLimiting Information Rate for the Proposed CE Precoder

M = 12 users

PT = 38.4 / NInformation rate limit

for the proposed CE precoder

96 BS antennas required to achieve aninformation rate which is 95% of the limit.

Fig. 7. Ergodic per-user information rate for a fixed M = 12, with the totaltransmit power scaled down linearly with increasing N . Gaussian informationalphabets U1 = · · · = UM . IID CN (0, 1) Rayleigh fading.

deteriorates).In Fig. 4, for the proposed CE precoder, we had observed

that for a fixed M and fixed desired per-user information rate,with “sufficiently large” N , the total transmit power can bereduced linearly with increasing N . We next try to understandas to how “large” must N be. In Fig. 7, for a fixed M = 12users, we plot the achievable per-user ergodic informationrate under per-antenna CE transmission (i.e., RCE

(PT

σ2

)/M )

as a function of increasing N and PT = P0/N (i.e., welinearly decrease PT with increasing N , P0 = 38.4). It isobserved that, the per-user ergodic information rate increasesand approaches a limiting information rate as N → ∞ (shownby the dashed curve in the figure). P0 = 38.4 correspondsto a limiting per-user information rate of roughly 1.7 bpcu.This then suggests that, in the limit as N → ∞, the per-userinformation rate remains fixed as long as PT is scaled downlinearly with increasing N . A similar behaviour is observedunder APC (see the GBC sum capacity upper bound curve inthe figure). In Fig. 8, similar results have been illustrated forM = 24 users and PT = P1/N (P1 = 72.3, corresponding

1068 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 3, MARCH 2013

0 100 200 300 400 500 6000.8

1

1.2

1.4

1.6

1.8

2

Number of BS antennas (N)

Erg

odic

Per

Use

r In

form

atio

n R

ate

(bpc

u)

GBC Sum Capacity Upper Bound (APC)Limiting Information Rate for the Proposed CE PrecoderProposed CE Precoder

192 BS antennas required to achieve aninformation rate within 95% of the limit.

M = 24 users

PT = 72.3 / N

Fig. 8. Same as Fig. 7, but with a fixed M = 24 and PT = 72.3/N .

to a limiting per-user information rate of roughly 1.7 bpcu).With regards to the question on how “large” must N be, it isnow clear that N must at least be so large that the achievableper-user ergodic information rate is sufficiently close to itslimiting information rate (i.e., in the flat region of the curve).In general, for a desired closeness to the limiting informationrate, the minimum number of BS antennas required dependson M . Our numerical experiments suggest that, to achievea fixed desired ratio of the per-user ergodic information rateto the limiting information rate, a channel with a large Mrequires a large N also. As an example, for a fixed ratio of0.95 between the achievable per-user ergodic information rateand the limiting information rate, a channel with M = 12users requires a BS with at least N = 96 antennas, whereasa channel with M = 24 users requires a BS with at leastN = 192 antennas.

VI. CONCLUSION

We have considered per-antenna constant envelope (CE)transmission in the downlink of multi-user MIMO systems(GBC) employing a large number of BS antennas. Undercertain mild conditions on the channel, even with a strin-gent per-antenna CE constraint, array power gain can stillbe achieved. We have also proposed a low-complexity CEprecoding scheme. For the proposed CE precoding scheme,through exhaustive simulations for the i.i.d. Rayleigh fad-ing channel, we showed that, compared to an APC onlyconstrained GBC, the extra total transmit power requiredby the proposed CE precoder to achieve a given per-userergodic information rate is small (less than 2 dB for thescenarios of interest). Typically, a non-linear power-efficientamplifier is about 4 − 6 times more power-efficient than ahighly linear amplifier [11]. Combining this fact with the factthat per-antenna CE signals require an extra 2 dB transmitpower, we arrive at the conclusion that, for a given desiredachievable information sum-rate, with sufficiently large N , abase station having power-efficient amplifiers with CE inputswould require 10 log10(4) − 2.0 = 4.0 dB less total transmitpower compared to a base station having highly linear power-inefficient amplifiers with high PAPR inputs.

APPENDIX ACONVERGENCE (IN DISTRIBUTION) OF THE SEQUENCE{

zN

}The convergence in distribution of the sequence of random

variables{zN

}(as N → ∞ with fixed M ) is stated and

proved in Theorem 3. Its proof relies on three known resultswhich have been stated below.

Result 1: (Multivariate Central Limit Theorem (CLT)) LetFn denote the joint cumulative distribution function (c.d.f.)of the k-dimensional real random variable (X

(1)n , · · · , X(k)

n ),n = 1, 2, . . . and for each real vector Λ = (λ1, λ2, · · · , λk)

T ,let FΛn be the c.d.f. of the random variable λ1X

(1)n +λ2X

(2)n +

· · ·+ λkX(k)n . A necessary and sufficient condition for Fn to

converge to a limiting distribution (as n → ∞) is that FΛn

converges to a limit for each vector Λ.Proof – For details please refer to [12] . �This result basically states that, if F is the joint c.d.f. of a

k-dimensional real random variable (X(1), X(2), · · · , X(k)),and if FΛn → FΛ for13 each vector Λ, then Fn → F asn → ∞.

Result 2: (Lyapunov-CLT) Let {Xn}, n = 1, 2, . . . be asequence of independent real-valued scalar random variables.Let E[Xn] = μn, E[(Xn − μn)

2] = σ2n, and for some fixed

ξ > 0, E[|Xn − μn|2+ξ] = βn exists for all n. Furthermorelet

BnΔ=( n∑

i=1

βi

) 12+ξ

, CnΔ=( n∑

i=1

σ2i

) 12. (33)

Then iflim

n→∞Bn

Cn= 0, (34)

the c.d.f. of Yn =∑n

i=1(Xi−μi)

Cnconverges (in the limit as

n → ∞) to the c.d.f. of a real Gaussian random variable withmean zero and unit variance.

Proof – For details please refer to [13] . �Result 3: (Slutsky’s Theorem) Let {Xn} and {Yn} be a

sequence of scalar random variables. If {Xn} converges indistribution (as n → ∞) to some random variable X , and{Yn} converges in probability to some constant c, then theproduct sequence {XnYn} converges in distribution to therandom variable cX .

Proof – For details please refer to [17] . �Theorem 3: For any channel sequence {HN} satisfying

the conditions in (6), the associated sequence of randomvectors {zN} (defined in (8)) converges (as N → ∞ withfixed M ) in distribution to a multivariate 2M -dimensionalreal Gaussian random vector X = (XI

1 , XQ1 , · · · , XI

M , XQM )T

with independent zero-mean components and var(XIk ) =

var(XQk ) = ck/2 , k = 1, 2, . . . ,M (note that ck , k =

1, 2, . . . ,M is defined in (6)).Proof – Consider a multivariate 2M -dimensional real ran-

dom variable (XI1 , X

Q1 , · · · , XI

M , XQM ), whose components

are i.i.d. real Gaussian with mean zero and var(XIk ) =

var(XQk ) = ck/2 , k = 1, 2, . . . ,M . Then, for any vector

Λ = (λI1, λ

Q1 , · · · , λI

M , λQM )T ∈ R2M , the scalar random vari-

able (λI1X

I1+λQ

1 XQ1 +· · ·+λI

MXIM+λQ

MXQM ) is real Gaussian

with mean zero and variance∑M

k=1 ck((λI

k)2

+ (λQk )

2)/2.

13 FΛ is the c.d.f. of λ1X(1) + · · ·+ λkX(k).

MOHAMMED and LARSSON: PER-ANTENNA CONSTANT ENVELOPE PRECODING FOR LARGE MULTI-USER MIMO SYSTEMS 1069

If we can show that for any arbitrary vector Λ ∈ R2M , thelimiting distribution of zTNΛ is also real Gaussian with meanzero and the same variance

∑Mk=1 ck

((λI

k)2

+(λQk )

2)/2, then

using Result 1 it will follow that the c.d.f. of zN convergesto the c.d.f. of (XI

1 , XQ1 , · · · , XI

M , XQM ) as N → ∞. This

would then complete the proof. In the following we show thatthis is indeed true.

For a given 2M -dimensional real vector Λ =(λI

1, λQ1 , · · · , λI

M , λQM )T , let

ζNΔ= zTNΛ =

M∑k=1

(λIkz

Ik(N)

+ λQk zQk

(N)). (35)

From the above definition and (8), it follows that r.v. ζN canbe expressed as14

ζN =

N∑i=1

(ai cos(θi) + bi sin(θi))

=N∑i=1

√a2i + b2i cos(θi − tan−1 bi

ai)

aiΔ=

∑Mk=1(λ

Ikh

I(N)

k,i + λQk h

Q(N)

k,i )√N

,

biΔ=

∑Mk=1(λ

Qk h

I(N)

k,i − λIkh

Q(N)

k,i )√N

(36)

where hI(N)

k,iΔ= Re(h(N)

k,i ) , hQ(N)

k,i

Δ= Im(h

(N)k,i ). We further

define

ηiΔ=

√a2i + b2i cos(θi − tan−1 bi

ai) (37)

Since, the phase angles θi, i = 1, 2, . . . , N are independentof each other, ηi, i = 1, 2, · · · , N are also independent.Therefore, ζN is nothing but the sum of N independentrandom variables. We can therefore apply the Lyapunov-CLT(Result 2) to study the convergence of the c.d.f. of ζN asN → ∞.

We firstly see that μiΔ= E[ηi] = 0 and σ2

iΔ= E[η2i ] =

(a2i + b2i )/2 since θi is uniformly distributed in [−π, π). Wenext show that the conditions of the Lyapunov-CLT ((34) inResult 2) are satisfied with ξ = 2. We see that

βiΔ= E[η4

i ]

= (a2i + b2i )

2E[cos4(θi − tan−1 bi

ai)]

=3

8(a2

i + b2i )2 (38)

exists for all i. In order that the condition in (34) is satisfied,we must show that

limN→∞

BN

CN= 0 (39)

where

BNΔ=

( N∑i=1

βi

) 14=(38

N∑i=1

(a2i + b2i )

2) 1

4,

CNΔ=

( N∑i=1

σ2i

) 12=( N∑

i=1

(a2i + b2i )/2

) 12

(40)

14Note that the randomness in zN is only due to the random variablesθi , i = 1, 2, . . . , N .

As a note, from (36) it follows that both BN and CN arestrictly positive for all N ≥ M . Since M is fixed, proving(39) is equivalent to proving that

limN→∞

B4N

C4N

= 0 (41)

Using (6) we firstly show that

limN→∞

C2N =

1

2

M∑k=1

ck((λI

k)2 + (λQ

k )2) (42)

i.e., C2N converges to a constant as N → ∞. We then show

that, again under (6),

limN→∞

B4N = 0 (43)

Equation (41) would then follow from (42) and (43). Wenext show (42). Using (40) we have 2C2

N =∑N

i=1(a2i + b2i ).

Expanding the expressions for ai and bi in∑N

i=1(a2i + b2i )

using (36), we have

2C2N =

M∑k=1

((λIk)

2 + (λQk )

2)‖h(N)

k ‖2N

+2M∑k=1

M∑l=k+1

{(λI

kλIl + λQ

k λQl )

∑Ni=1(h

I(N)

k,i hI(N)

l,i + hQ(N)

k,i hQ(N)

l,i )

N

+(λIkλ

Ql − λQ

k λIl )

∑Ni=1(h

I(N)

k,i hQ(N)

l,i − hQ(N)

k,i hI(N)

l,i )

N

}.

(45)

From As.1 and As.3 in (6) it follows that

limN→∞

∑Ni=1(h

I(N)

k,i hI(N)

l,i + hQ(N)

k,i hQ(N)

l,i )

N= 0 ,

limN→∞

∑Ni=1(h

I(N)

k,i hQ(N)

l,i − hQ(N)

k,i hI(N)

l,i )

N= 0 ,

limN→∞

‖h(N)k ‖2N

= ck. (46)

Using (46) in (45) and taking the limit as N → ∞ we get (42)(note that M is fixed). We now show (43). Before proceedingfurther, we define the complex numbers λk

Δ= (λI

k+jλQk ), k =

1, 2, . . . ,M . Expanding the expressions for ai and bi insidethe summation in B4

N (see (40)) we get (44)(top of the nextpage). From (As.2) in (6) it follows that for all k1, k2, l1, l2 ∈(1, 2, . . . ,M)

limN→∞

∑Ni=1 Re(h(N)∗

k1,ih(N)l1,i

)Re(h(N)∗k2,i

h(N)l2,i

)

N2= 0 ,

limN→∞

∑Ni=1 Re(h(N)∗

k1,ih(N)l1,i

)Im(h(N)∗k2,i

h(N)l2,i

)

N2= 0

limN→∞

∑Ni=1 Im(h

(N)∗k1,i

h(N)l1,i

)Re(h(N)∗k2,i

h(N)l2,i

)

N2= 0 ,

limN→∞

∑Ni=1 Im(h

(N)∗k1,i

h(N)l1,i

)Im(h(N)∗k2,i

h(N)l2,i

)

N2= 0

limN→∞

∑Ni=1 |h(N)

k1,i|2Re(h(N)∗

k2,ih(N)l2,i

)

N2= 0 ,

limN→∞

∑Ni=1 |h(N)

k1,i|2Im(h

(N)∗k2,i

h(N)l2,i

)

N2= 0. (47)

Substituting (47) into (44) and taking the limit, we have

limN→∞

8

3B4

N = limN→∞

{N∑i=1

( M∑k=1

|λk|2|h(N)k,i |2

N

)2}(48)

1070 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 3, MARCH 2013

8

3B4

N =N∑i=1

(a2i + b2i )

2

=N∑i=1

{M∑k=1

|λk|2|h(N)k,i |2

N+ 2

M∑k=1

M∑l=k+1

(Re(λ∗

kλl)Re(h(N)∗k,i h

(N)l,i ) + Im(λ∗

kλl)Im(h(N)∗k,i h

(N)l,i )

)N

}2

=

{N∑i=1

( M∑k=1

|λk|2|h(N)k,i |2

N

)2}+ 4

[M∑

k1=1

M∑k2=1

M∑l2=k2+1

(|λk1 |2Re(λ∗

k2λl2)

∑Ni=1 |h(N)

k1,i|2Re(h(N)∗

k2,ih(N)l2,i

)

N2

+|λk1 |2Im(λ∗k2λl2)

∑Ni=1 |h(N)

k1,i|2Im(h

(N)∗k2,i

h(N)l2,i

)

N2

)]

+4

M∑k1=1

M∑k2=1

M∑l1=k1+1

M∑l2=k2+1

{Re(λ∗

k1λl1)Re(λ∗

k2λl2)

∑Ni=1 Re(h(N)∗

k1,ih(N)l1,i

)Re(h(N)∗k2,i

h(N)l2,i

)

N2+

Re(λ∗k1λl1)Im(λ∗

k2λl2)

∑Ni=1 Re(h(N)∗

k1,ih(N)l1,i

)Im(h(N)∗k2,i

h(N)l2,i

)

N2+

Im(λ∗k1λl1)Re(λ∗

k2λl2)

∑Ni=1 Im(h

(N)∗k1,i

h(N)l1,i

)Re(h(N)∗k2,i

h(N)l2,i

)

N2+

Im(λ∗k1λl1)Im(λ∗

k2λl2)

∑Ni=1 Im(h

(N)∗k1,i

h(N)l1,i

)Im(h(N)∗k2,i

h(N)l2,i

)

N2

}. (44)

Further,

limN→∞

{N∑i=1

( M∑k=1

|λk|2|h(N)k,i |2

N

)2}=

M∑k1=1

M∑k2=1

(|λk1 |2|λk2 |2 lim

N→∞

(∑Ni=1 |h(N)

k1,i|2|h(N)

k2,i|2

N2

))(49)

From (As.2) in (6) it follows that

limN→∞(∑N

i=1 |h(N)k1,i|2|h

(N)k2,i|2

N2

)= 0 and therefore using

this result in (49) and (48) we get (43). From (42) it followsthat C4

N converges to a positive constant as N → ∞. Hencewe have now shown (41), and therefore the Lyapunov-CLTconditions for the convergence of the c.d.f. of the randomvariable ζN are indeed satisfied.

Therefore invoking Result 2 (Lyapunov-CLT), it fol-lows that the c.d.f. of ζN/CN converges to the c.d.f.of a zero mean real Gaussian random variable with unitvariance. Further, since CN converges to the constant√

12

∑Mk=1 ck

((λI

k)2 + (λQ

k )2)

(see (42)), using Result 3 (Slut-sky’s Theorem) it follows that the c.d.f. of ζN converges tothe c.d.f. of a zero mean real Gaussian random variable withvariance 1

2

∑Mk=1 ck

((λI

k)2 + (λQ

k )2). �

REFERENCES

[1] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, O. Edfors, F. Tufvesson,and T. L. Marzetta, “Scaling up MIMO: opportunities and challengeswith very large arrays,” IEEE Signal Process. Mag., vol. 30, pp. 40-60,Jan. 2013.

[2] T. L. Marzetta, “Non-cooperative cellular wireless with unlimited num-bers of base station antennas,” IEEE. Trans. Wireless Commun., vol. 9,no. 11, pp. 3590–3600, Nov. 2010.

[3] D. N. C. Tse, Fundamentals of Wireless Communications. CambridgeUniversity Press, 2005.

[4] Greentouch Consortium. Available: http://www.eweekeurope.co.uk/news/greentouch-shows-low-power-wireless-19719.

[5] V. Mancuso and S. Alouf, “Reducing costs and pollution in cellularnetworks,” IEEE Commun. Mag., pp. 63–71, Aug. 2011.

[6] W. Yu and T. Lan, “Transmitter optimization for the multi-antennadownlink with per antenna power constraints,” IEEE Trans. SignalProcess., pp. 2646–2660, vol. 55, June 2007.

[7] K. Kemai, R. Yates, G. Foschini, and R. Valenzuela, “Optimum zero-forcing beamforming with per-antenna power constraints,” in Proc. 2007IEEE International Symp. Inf. Theory, pp. 101–105, 2007.

[8] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Energy and spectral effi-ciency of very large multiuser MIMO systems,” IEEE Trans. Commun.,to appear. arXiv:1112.3810v2[cs.IT].

[9] T. M. Cover and J. A. Thomas, Elements of Information Theory. JohnWiley and Sons, 1991.

[10] S. Vishwanath, N. Jindal, and A. Goldsmith, “Duality, achievable ratesand sum-rate capacity of Gaussian MIMO broadcast channels,” IEEETrans. Inf. Theory, vol. 49, no. 10, pp. 2658–2668, Oct. 2003.

[11] S. C. Cripps, RF Power Amplifiers for Wireless Communications. ArtechPublishing House, 1999.

[12] V. S. Varadarajan, A Useful Convergence Theorem. Sankhya, 20, 221–222, 1958.

[13] P. Billingsley, Probability and Measure, 3rd edition. John Wiley andSons, 1995.

[14] S. Payami and F. Tufvesson, “Channel measurements and analysis forvery large array systems at 2.6 GHz,” in Proc. 2012 European Conf.Antennas Propagation.

[15] S. K. Mohammed and E. G. Larsson, “Constant envelope precoding forpower-efficient downlink wireless communication in multi-user MIMOsystems using large antenna arrays,” in Proc. 2012 IEEE ICASSP.

[16] S. K. Mohammed and E. G. Larsson, “Single-user beamforming in large-scale MISO systems with per-antenna constant-envelope constraints: theDoughnut channel,” IEEE Trans. Wireless Commun., vol. 11, pp. 3992–4005, Nov. 2012.

[17] A. K. Basu, Measure Theory and Probability. Prentice Hall of India,1999.

Saif Khan Mohammed (S’08-M’11) received theB.Tech degree in Computer Science and Engineeringfrom the Indian Institute of Technology (I.I.T.), NewDelhi, India, in 1998 and the Ph.D. degree from theElectrical and Communication Engineering Depart-ment, Indian Institute of Science, Bangalore, India,in 2010. Currently, he is an Assistant Professor atthe Communication Systems Division (Commsys)in the Electrical Engineering Department (ISY) atLinkoping University, Sweden. From 2010 to 2011,he was a Postdoctoral Researcher at Commsys. He

has previously worked as a Systems and Algorithm designer in the WirelessSystems Group at Texas Instruments, Bangalore (India) (2003 - 2007). From2000 to 2003, he worked with Ishoni Networks, Inc., Santa Clara, CA (USA),as a Senior Chip Architecture Engineer. From 1998 to 2000, he was a ASICDesign Engineer with Philips, Inc., Bangalore.

MOHAMMED and LARSSON: PER-ANTENNA CONSTANT ENVELOPE PRECODING FOR LARGE MULTI-USER MIMO SYSTEMS 1071

His main research interests include wireless communication using largeantenna arrays, coding and signal processing for wireless communicationsystems, and statistical signal processing. He is a member of the IEEE, theIEEE Communication Society, the IEEE Signal Processing Society and theIEEE Information Theory Society. He is also a Technical Program Committeemember for the IEEE International Conference on Communications (ICC’2013), the IEEE Vehicular Technology Conference (VTC) in Spring 2013,and the IEEE Swedish Communication Theory Workshop (Swe-CTW) in fall2012. Dr. Mohammed was awarded the Young Indian Researcher Fellowshipby the Italian Ministry of University and Research (MIUR) for the year 2009-10. He has also been awarded the CENIIT research grant for the years 2012and 2013.

Erik Larsson received his Ph.D. degree from Up-psala University, Sweden, in 2002. Since 2007, heis Professor and Head of the Division for Commu-nication Systems in the Department of ElectricalEngineering (ISY) at Linkoping University (LiU)in Linkoping, Sweden. He has previously been As-sociate Professor (Docent) at the Royal Instituteof Technology (KTH) in Stockholm, Sweden, andAssistant Professor at the University of Florida andthe George Washington University, USA.

His main professional interests are within theareas of wireless communications and signal processing. He has publishedsome 80 journal papers on these topics, he is co-author of the textbookSpace-Time Block Coding for Wireless Communications (Cambridge Univ.Press, 2003) and he holds 10 patents on wireless technology.

He is Associate Editor for the IEEE TRANSACTIONS ON COMMUNICA-TIONS and he has previously been Associate Editor for several other IEEEjournals. He is a member of the IEEE Signal Processing Society SPCOMtechnical committee. He is active in conference organization, most recentlyas the Technical Chair of the Asilomar Conference on Signals, Systemsand Computers 2012 and Technical Program co-chair of the InternationalSymposium on Turbo Codes and Iterative Information Processing 2012. Hereceived the IEEE Signal Processing Magazine 2012 Best Column Award.