1

Multiplexing and diversity gains in

noncoherent massive MIMO systems

Mainak Chowdhury, Alexandros Manolakos, Andrea Goldsmith

Abstract

We consider a noncoherent energy-detection based uplink and downlink with a large antenna array at

the base station, performing symbol-by-symbol detection, ON-OFF signaling and single-user detection.

A ray tracing propagation model is assumed with knowledge at the base station and transmitters of only

the ray arrival angles and amplitudes. We identify the sources of performance degradation, quantify

notions of diversity gain (related to the detection error performance) and multiplexing gain (related

to the number of users simultaneously supported), and present numerical results to demonstrate these

gains. Our results indicate that in this noncoherent system, increasing the number of antenna elements

can support multiple users, with a vanishing probability of detection error, as long as the number of

users is below a certain threshold which increases with the number of antennas. This contrasts with

the fact that in a rich scattering propagation environment, uncoded noncoherent systems performing

symbol-by-symbol detection cannot support more than one user with a vanishing probability of error.

I. INTRODUCTION

Large antenna arrays allow fine angular resolution, so that radio signals can be resolved

spatially. This property is currently exploited in several wireless systems including phased array

radar and astronomical imaging [2]–[4]. In this work, we explore the implications of this fine

angular resolution for the design of a noncoherent uncoded wireless communication system, i.e.,

a system performing symbol-by-symbol detection without knowledge of the instantaneous fading

realization.

The first and the third authors are with the Department of Electrical Engineering at Stanford University, and the second

author is with Qualcomm Inc., San Diego. This work was presented in part at Asilomar Conference on Signals, Systems and

Computers, 2015 [1]. This work was supported by a 3Com Corporation, an Alcatel-Lucent Stanford Graduate Fellowship, the

NSF Center for Science of Information (CSoI): NSF-CCF-0939370 and an A.G. Leventis Foundation Scholarship.

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The rich scattering assumption [5] for MIMO systems, which results in identically and

independently distributed (i.i.d.) fading across antenna elements, is widely used in communication

system analysis due to its simplicity. This model has been used to develop theoretical performance

limits, insights into practical designs, and channel models for performance analysis, especially

for communication systems that use the instantaneous channel state information at the receiver,

i.e., coherent communication systems. There has been significant interest recently in massive

MIMO systems since, when accurate channel state information is available at the base station,

even simple low complexity architectures yield large multiplexing gains in rich scattering.

In other words, given a quality of service constraint, such systems support large numbers of

simultaneously transmitting users/data streams.

On the contrary, under rich scattering, in a system that does not have information about the

instantaneous fading channel state (in particular, the fast changing phase of the fading realization)

at the transmitter or the receiver, the gains are much less, possibly none. However, the rich

scattering model is a mathematical idealization that becomes less accurate as the number of

users and/or antennas in the system grows. For example, in [6], the authors invoke the laws

of physics and show that there is a certain limit to the number of scattering elements in any

environment. Moreover, it has been empirically observed [7] that there are correlation structures

that arise once the number of antennas is large enough. This suggests that when the number of

spatial dimensions in a communication node (either transmitter or receiver) is large enough, many

of the underlying assumptions about the propagation environment and their implications need

to be rethought. Motivated by these potential gains, in this work we analyze the performance

of noncoherent massive MIMO systems outside the idealized rich scattering model. Propagation

models with correlated fading in MIMO systems have been previously studied in the context

of coherent systems [8]–[11]. Channel state information acquisition in these systems can entail

a large overhead, especially considering the number of antenna elements involved. Keeping in

mind the difficulty of channel state acquisition in large antenna arrays, analog beamforming

architectures or hybrid architectures with a combination of both analog and digital beamforming

have been considered [12], [13]. These results suggest that massive MIMO architectures with

a reduced number of digital frontends/RF chains and carefully designed precoders yield many

benefits from massive MIMO (such as highly directional beams), while also being feasible

to build. As discussed later in this paper, a noncoherent precoder (independent of the channel

realization) arises naturally out of our assumptions and analyses with the ray tracing propagation

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model, so our results can also be used to inform precoder design.

Compared to coherent and partially coherent systems, noncoherent MIMO systems have

not been as widely studied and the results to date are mostly for point-to-point channels.

There have been works about the capacity of point-to-point noncoherent massive SIMO systems

with independent Rayleigh fading [14] where it was shown that even without knowledge of

the instantaneous channel gains, the capacity increases with the number of antennas. In the

early 2000s, the structure of capacity-achieving schemes for a general block Rayleigh fading

noncoherent MIMO channel was derived in [15]. In [16], Grassman manifold signaling was

investigated as a capacity-achieving scheme for the point-to-point noncoherent channel in the high

SNR regime. Capacity regions associated with multiuser noncoherent channels have been studied

in several regimes, including that of Rayleigh fading, high SNR regimes. Results [17]–[19]

suggest that single user transmission strategies (with time sharing) may be optimal under a variety

of power constraints and fading models. As we will see in this paper, propagation characteristics

do affect significantly the expected performance for symbol-to-symbol decoding in multiuser

noncoherent massive MIMO systems. In particular, time-sharing-based single user transmission

may not be optimal under a widely accepted class of propagation models.

In this work we assume propagation is based on a ray tracing model with a finite number

of multipath components. Note that similar models have been defined in 3GPP for LTE [20].

Contrary to results from a rich scattering model, we show how the nature of fading with large

antenna arrays can actually provide multiplexing gains even without instantaneous information

about the channel phase.

In particular, we consider a base station with a large uniform linear antenna (ULA) array

serving several single antenna users. The base station and users do not have knowledge of the

instantaneous phases associated with the propagation environment, hence have to use noncoherent

schemes which do not depend on the channel phase. To quantify the gains in the limit of a large

number of antennas, we define notions of outage, diversity and multiplexing to determine the

number of users that may be supported under a certain constraint on the symbol error rate under

symbol-by-symbol decoding. As discussed later in the text, the term outage is used to refer to

multiple rays either from a single user or from multiple users interfering with one another. Note

that an outage event may correspond to different rays being in phase and reinforcing each other

and in this sense, it is different from its usage in the context of a point-to-point fading channel

where it is used to refer to rays being out of phase and hence causing a weak channel gain [5].

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Multiplexing gain is related to the number of users simultaneously supported. This is different

from its usage in the context of a point-to-point MIMO channel [5] where it refers to the number

of simultaneously transmitted independent spatial data streams. We also define the diversity gain

to characterize the behavior of the error probability in non-outage with an increasing number

of antennas. We show, through these metrics, that the finer angular resolution afforded by large

antenna arrays not only makes it possible to drive the error probability down to zero with only

slowly varying information about the channel, but also to support multiple users/independent

data streams.

We obtain these desirable properties via a simple energy-detection based noncoherent transmis-

sion and detection scheme. We find that with this scheme, in both the uplink and the downlink,

we can simultaneously support all users with a vanishing probability of error, as long as the

number of users is smaller than a certain threshold which increases with the number of antennas

at the base station. By defining diversity gain as the dominating exponent in the non-outage

error probability, we derive the diversity gain associated with non-outage transmissions and find

that it is not affected by interference from multiple users. Since our analysis does not use the

instantaneous phase of the channel gains, we describe our system as being noncoherent.

The rest of the paper is organized as follows. In Section II we describe our system model and

our assumptions about the propagation model, transmission, and decoding schemes. We describe

our performance metrics in Section III followed by detailed results in Section IV. We finally

present extensions of these results to related settings in Section V, numerical results in Section

VI, and our conclusions in Section VII.

II. SYSTEM MODEL

In this section, we describe the propagation models considered in this paper, followed by a

description of the transmission and detection schemes.

A. Notation

We use boldface fonts (a) to refer to vectors and ak to refer to the kth element of the vector

a. We use o(g(N)) to refer to any function f(·) such that limN→∞ f(N)/g(N)→ 0. Similarly

we use Θ(g(N)) for any f(N) satisfying c1g(N) < f(N) < c2g(N) for some 0 < c1 < c2. We

say f(N) = ω(g(N)) if g(N) = o(f(N)). We say that f(N) = O(g(N)) if f(N) = Θ(g(N))

or f(N) = ω(g(N)). We use the word beamspace to refer to the transform domain obtained by

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taking a spatial Fourier transform across antenna elements at the base station. The beamspace

angle ψ corresponding to an angle of arrival θ is given by ψ = 2π sin(θ)/λ, where λ is the

wavelength. Note that both these angles are taken modulo 2π. We use |S| to refer to the cardinality

of a set S.

B. Propagation characteristics

Our propagation model is the ray tracing model (or spatial channel model with a finite number

of multipath components [21]) similar to models defined in 3GPP for LTE [20]. We assume that

the absolute values of the attenuation and the instantaneous angles of arrival for each of the rays

are known perfectly (please refer to Section V-A and Section VI-A for an in-depth discussion of

this assumption). For simplicity of analysis, the gains are assumed to be equal and the angles of

arrival of transmissions from a particular user are assumed to be identically and independently

distributed (i.i.d.) uniformly in an interval around a central angle specific to the particular user.

The central angle is also assumed to be i.i.d. with the distribution being uniform in [0, 2π].

An important assumption in our analysis is that the receiver antenna array is in the far field

and that all the elements of the antenna array are colocated. An important corollary of assuming

colocation, is that all the receiver antennas “see” the same planar wavefront. This assumption

has been used to justify the particular form for the received signal that we describe next. Note,

in particular, that this assumption precludes modeling the nonstationarity across large antenna

arrays [22]. While operating at low carrier frequencies may necessitate explicit modeling of this

nonstationarity, the use of high carrier frequencies and the resulting small form factors of large

antenna arrays makes the colocation and the stationarity assumption closer to reality. Moreover,

for simplicity of presentation, in this work, we consider only the azimuth direction, assuming

that all transmitters and receivers are at the same elevation. Extensions of this theory to scenarios

with both elevation and azimuth dimensions follow very similar lines and are deferred to future

work. The received signal in the uplink is given by

y[t] =∑b∈B

∑p∈Lb

Gb,pejφb,p[t]a(θb,p)sb[t− τb,p] + ν[t].

In the above, y[t] ∈ Cn is the received signal at the multi-antenna receiver at (discrete) time

t and ν[t] is the complex Gaussian noise, i.i.d. across antennas, with power 1 at each element.

B refers to the set of all beams (this is the same as the number of users). Lb is the set of all

multipath components associated with a beam b ∈ B. We use P to refer to the set of all paths,

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i.e., P = {(b, p) : b ∈ B, p ∈ Lb}. Gb,p is the path attenuation coefficient, φb,p is the instantaneous

phase and τb,p is the delay associated with the pth multipath component of the bth beam. Note

that due to the colocation assumption made earlier, the absolute phase φb,p is the same across all

antennas for a particular path. Physically, the absolute phase captures the phase changes that a

particular wavefront experiences either due to propagation or due to reflection/scattering before

reaching the receiver array. sb is the symbol transmitted at discrete time t by user (or beam)

b. For the simplicity of presentation, we assume in the rest of the paper that Gb,p = 1√L, and

that |Lb| = L for all b. We also assume henceforth that τb,p is zero. As discussed in Section V

later in this manuscript, the analysis when τb,p are unequal is very similar to the case when they

are all equal to zero. Under the latter assumption, the expression for the received signal in the

uplink (the subscript u stands for uplink) simplifies to

y[t] =∑b∈B

hu,b[t]sb[t] + ν[t] with

hu,b[t] ,∑p∈Lb

Gb,pejφb,p[t]a(θb,p),

(1)

where φb,ps (assumed to be unknown at the receiver) are uniformly distributed in [0, 2π] and θb,ps

are independent and identically distributed, with the distribution being uniform in [µi− c, µi+ c]

for some constant c and for µi uniformly distributed in [0, 2π]. This constant c is a function of

the propagation environment and the beamwidth at the single user transmitter. The validity of the

distribution of θb,p for a certain b depends on the wavelength and the scattering/reflecting surfaces

involved. In situations where this assumption is valid, the physical origins of the distribution of

the angles of arrival for a particular beam are in the randomness in the local radii of curvatures

of the local scattering surfaces that the rays in each beam hit (please refer to Kirchoff’s or

Kirchhoff’s approximation in e.g., [23]). a(·) is a one dimensional channel response vector

satisfying a(θ)m = ej2πmsin θλ , for m ∈ {0, . . . , N −1}, where the subscript m refers to the phase

shift in the received path at the mth receiver antenna and the inter-element spacing is assumed

to be 1 length unit. The uplink propagation model is depicted pictorially in Fig. 1.

We also consider downlink transmissions in our analysis (Fig. 2). Since there is an excess

of antennas at the downlink, the beamwidths can be much narrower. However, there does exist

multiuser interference from the sidelobes associated with the transmissions to different users

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...

User 1

User 2

θ1,1

Base station

PathBeam

Fig. 1: System model in the uplink for |B| = 2, |Lb| = 2.

...

User 1

User 2

Base station

Fig. 2: System model in the downlink for |B| = 2 (interference from sidelobes not shown).

simultaneously. The system model is thus:

yb[t] =∑b̃∈B

Gb̃,p(̃b)ejφ[t]ap(̃b)(θb,p(b))sb̃[t] + νb[t]

,∑b̃∈B

hd,̃b[t]sb̃[t] + νb[t],(2)

where ap(·) is the beamforming gain of path p at spatial direction (·) and is given by a factor

f(·) depending on the distance in the beamspace. The exact expression for ap(b)(θ) is given by

ap(b)(θ) = f(2π(sin(θb,p(b))− sin(θ))/λ), (3)

where

f(x) =1√N

N−1∑k=0

ejkx.

Properties of this factor f(·) are described in more detail in Appendix A. Note also that for each

beam b, the base station transmitter can choose any path p ∈ Lb, denoted by p(b) to send the

information symbols along. In our analysis, we assume that the transmitter chooses any one of

the L available paths to transmit information to user b.

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C. Transmission and detection schemes

We describe these schemes separately for the uplink and the downlink. For both uplink and

downlink, the transmitter uses ON-OFF keying and the detector uses a threshold detector with

the threshold amplitude being half of√power, where power is the instantaneous received

power corresponding to the ON state.

• Uplink: The users use equiprobable ON-OFF keying with an average power P (i.e., with

peak power 2P ). For each user b, the following statistics are computed to decode the

transmission at the receiver:

Sb =

{ybeamspace,ψ =

1√N

N−1∑k=0

yke−jkψ : ψ = 2π sin(θb,p)/λ for p ∈ Lb

}.

Note that these are sufficient statistics for the detection of sb, and one cannot do better

by considering other statistics with single-user decoding. For each beam b, the beamspace

decoder subsequently either declares an outage, based on its knowledge of θb,p, or performs

single user detection using a threshold detector on the amplitude

∑t∈Sb |t||Sb|

, (4)

with the threshold amplitude being a/2 if the received amplitude at the detector correspond-

ing to peak power transmission is a. Note that we do not analyze the uplink performance

with joint decoding of multiple users in this manuscript.

• Downlink: We assume ON-OFF keying in the beamspace, i.e., the transmitter chooses any

path corresponding to the user and uses equiprobable ON-OFF signaling along that path. It

uses a total average power of |B|P, equally distributed among the |B| users. To transmit along

a path, the transmitter computes a phase shift which is a known function of the direction

corresponding to the path, which is assumed to be known/tracked perfectly. Subsequently it

transmits the same signal, phase shifted across the different antennas. To transmit to multiple

users simultaneously, the transmitter sends in a linear combination of the signals for each

path (equivalently users, since in the downlink it has been assumed that the transmitter

chooses only one path for each user).

The receiver performs amplitude level detection with the threshold detector.

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Outage: Any two multipath components are less than a certain threshold distance ∆0

in the beamspace. We also define outage for a beam b as one where there is a

single multipath component within a distance ∆0 of any one of the multipath

components for a single beam b.

Inter-path

interference (IPI):

The sidelobes of multipath components with unknown phases far away in the

beamspace introduce inter-path interference.

Additive noise: Additive white Gaussian noise at the receiver.

TABLE I: Definitions of conditions behind performance degradation for a finite N

III. METRICS AND SUMMARY OF OUR RESULTS

We analyze uncoded communication and detection by considering all the paths corresponding

to a particular beam in the beamspace, under the assumption that the realization of the random

phase φ associated with any path is not known at the detector. In this setup, an error event occurs

due to the conditions defined in Table I.

We observe that in the limit of an infinite N , all the multipath components are resolvable, but

for any finite N, there is always some limit to the resolvability of different multipath components.

For each of the conditions identified in Table I, we identify the error events and the corresponding

metrics used to quantify it.

• Outage: For a given N , we fix a distance ∆0 in the beamspace and define an outage event

as the event that the distance between two paths in the beamspace is less than ∆0. In

Fig. 3, outage occurs if there happens to be a path in any of the red regions around the

paths. Computing the probability of the outage event is related to the birthday problem

[24]. We present results describing under what conditions on ∆0 and |P|, the probability of

outage vanishes with increasing N. In the lemmas below we consider two ways of defining

outage; one considering the minimum separation across all rays, and the other considering

the distances only from the rays for the single user/beam whose transmission is being

decoded.

Lemma 1 (Outage). Let N → ∞. For a separation ∆0(N), the probability of outage

vanishes like Θ (|P|2∆0) if |P|2∆0 = o (1) , and approaches a non zero constant otherwise.

Proof. Let us consider a probability distribution on a subset of the interval [0, 2π] where

the density function is positive and bounded above and below by positive constants M and

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0 2πBeamspace angle

Fig. 3: Paths in beamspace. The red segments are of length 2∆0. An outage event occurs if the

minimum separation between paths in the beamspace is smaller than ∆0.

m respectively. Note that this includes a large class of continuous distributions, including

the distribution induced on the beamspace angle ψ = 2π sin(θ)/λ by a uniform distribution

on θ (both modulo 2π), assumed in the system model.

Let pk be the probability of non-outage after k draws, i.e., it is the probability that the

minimum distance among the k random draws from the bounded interval is greater than ∆.

Then pk satisfies the following:

pk(1− 2k∆0M) ≤ pk+1 ≤ pk (1− 2k∆0m) .

This expression follows from the independence of angles of arrival for different paths. By

induction, we have|P|−1∏t=1

(1− 2tM∆0) ≤ p|P| ≤|P|−1∏t=1

(1− 2tm∆0). (5)

We note that the convergence behavior of p|P| is determined by 2∑|P|−1

t=1 t∆0 = (|P| −1)|P|m∆0. We consider two different cases below:

– |P|2∆0 = o(1): In this case, both sides of (5), i.e., the probability of non-outage

converges to 1. This may be proved by noticing that|P|−1∏t=1

(1− 2t∆0) = 1−Θ(|P|2∆0) +

|P|−1∑t=2

Θ(|P|2t∆0t) = 1− o(1)→ 1.

The limiting probability of outage in this case is thus zero.

– |P|2∆0 = O(1): In this case, the right hand side of (5) converges to a constant strictly

less than 1. Thus the limiting probability of outage in this case is bounded below and

does not vanish to zero.

Requiring that every user be separated by ∆0 is too strict of a criterion for outage if only

transmissions for a single beam b need to be decoded. Hence we also define an outage event

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for a single beam b as one where there is at least one ray within a distance ∆0 of any ray

corresponding to beam b. We have the following lemma about this outage event:

Lemma 2 (Outage for beam b). Let N →∞. For a separation ∆0(N), the probability of

outage for a beam b vanishes like Θ (|P|∆0) if |P|∆0 = o (1) , and approaches a non zero

constant otherwise.

Proof. The probability of non outage is bounded both above and below by Π|P|−1k=1 (1 −

m0∆0), for some constant m0, which is related to bounds on the probability density function

on the beamspace position ψ of a ray. Thus the probability of outage scales like 1 −Π|P|−1k=1 (1 − m0∆0). This vanishes iff |P|∆0 = o(1), and approaches a non zero constant

otherwise.

• IPI: If there is no outage, the random phases from different multipath components add

up constructively or destructively. We note that the magnitude of the interference that a

path causes to another path depends on the difference between the beamspace angles of

the two paths. If the difference is ∆0, the interference magnitude is scaled by a factor

|f(∆0)| = 1√N

∣∣∣ sin(N∆0/2)sin(∆0/2)

∣∣∣. Some properties of f(·) are described in Appendix A. If the

difference in beamspace is small, the resulting interference is strong.

• Additive noise: Since the spatial Fourier transform is a unitary transform, the statistics of

the additive noise at each point in the beamspace remains the same. This is the noise floor.

The combined effect of the additive noise and the additive interference at a particular point

in the beamspace is captured by the following lemma.

Lemma 3. The received amplitude along direction ψ1,1 = 2π sin(θ1,1)/λ in the beamspace

is given by

1) Uplink:

|ybeamspace,ψ1,1| = |√N/Ls1 +

∑p 6=1

√1/L exp(jφ̃1,p)f(2π(sin(θ1,1)− sin(θ1,p))/λ)s1+

∑(b,p):b 6=1 and p 6=1

√1/L exp(jφ̃b,p)f(2π(sin(θ1,1)− sin(θb,p))/λ)sb + ν̃1,1|,

(6)

where φ̃b,p = φb,p − φ1,1, and ν̃1,1 is the additive white Gaussian noise seen along

direction (1, 1).

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2) Downlink:

|ybeamspace,ψ1,1| =∣∣√N/Ls1+∑

b:b6=1

√1/L exp(jφ̃b,p(b))f(2π(sin(θ1,1)− sin(θb,p(b)))/λ)sb + ν1

∣∣, (7)

Proof. We present our proofs for the uplink and downlink separately.

1) Uplink: We note that ybeamspace,ψ1,1 = 1√N

∑N−1k=0 yke

−jkψ1,1 . Using (1), we get that

|ybeamspace,ψ1,1| =

∣∣∣∣∣∣ 1√N

N−1∑k=0

∑(b,p)

Gb,pejφb,pej2πk sin(θb,p)/λsb + νk

e−jkψ1,1

∣∣∣∣∣∣(a)=

∣∣∣∣∣∣ 1√N

N−1∑k=0

∑(b,p)

Gb,pejφ̃b,pej2πk sin(θb,p)/λsb + νke

−jφ1,1

e−jkψ1,1

∣∣∣∣∣∣(b)= |√NGb,ps1 +

∑(b,p):b 6=1 or p6=1

Gb,p exp(jφ̃b,p)f(2π sin(θb,p)/λ− ψ1,1)sb + ν̃1,1|.

(8)

In the above (a) follows by defining φ̃b,p = φ − φ1,1, (b) follows by noting that1√N

∑N−1k=0 exp(j2πk sin(θ)/λ−jkψ) = f(2π sin(θ)/λ−ψ), f(0) =

√N, and defining

ν̃b,p ,N−1∑k=0

νke−j(kψb,p+φb,p). (9)

The expression in the Lemma follows by observing that Gb,p = 1√L.

2) Downlink: This follows directly from (2).

The multiplexing gain is defined as the number of beams |B| that are discernible at the

receiver with a vanishing outage probability and a vanishing probability of error. If each beam

corresponds to L multipath components and there are |P| multipath components in total, then

the multiplexing gain |B| satisfies

|B| = |P|L.

The diversity gain is defined as the first order exponent in the error probability of non-outage

transmissions in N , i.e., diversity gain is defined as a d such that:

limN→∞

− log(Probability of error)Nd

= 1. (10)

Note that d can be a function of N and is not unique. In particular if there exists a d so that

(10) is satisfied, then d + o(d) also satisfies (10).

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In the next section, we present results about the dependence of the outage probability and

the diversity gain d on the number of users |B|.

IV. MULTIPLEXING AND DIVERSITY GAINS

A. Overview

We observe that, with just a single beam and a single ray for that beam, the diversity gain

d is independent of the interference from the other rays and is a constant depending only on

the average received power. This may be seen by considering specializations of (6) and (7) to

L = 1, |B| = 1. In particular we get that

Uplink: |ybeamspace,ψ1,1 | = |√Ns1 + ν̃1,1|,

Downlink: |ybeamspace,ψ1,1| = |√Ns1 + ν|.

(11)

Using Laplace’s principle [25], we get that the probability of detection error is dominated,

for a large N , by the probability of the event that |ν| >√N√

max. received power/2. The

corresponding diversity gain is

d = max. received power/4.

With interfering paths however (either from multiple paths from a single user or from multiple

paths from different users), the diversity gain is drastically smaller. For the simple case of two

rays with ∆0 separation between them in the beamspace, the resulting signal at beamspace

position ψ1,1 looks like the following:

Uplink: |ybeamspace,ψ1,1| =∣∣∣∣∣√N√Ls1 +

1√2

exp(jφ̃)f(∆0) + ν̃1,1

∣∣∣∣∣ ,Downlink: |ybeamspace,ψ1,1| =

∣∣∣∣∣√N√Ls1 +

1√2

exp(jφ̃)f(∆0) + ν

∣∣∣∣∣ .(12)

We can see immediately that, if ∆0 is on the order of 1/N , then f(∆0) = Θ(√N), and as

N →∞, the diversity gain is strictly less than what would have been the case if there were no

interference. This is because the probability of ∆0 being on the order of 1/N and exp(jφ̃) being

close to −1 is also Θ(1/N) which is much larger than e−Nc for any positive constant c. Thus,

whenever ∆0 = Θ(1/N), the diversity gain d is strictly less due to destructive interference from

the interfering ray.

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small separation ∆0 large separation ∆0

smaller outage

probability,

higher |B|,smaller (or

equal) d

larger outage

probability,

smaller |B|,larger (or equal)

d

Fig. 4: Dependence of system performance on the beamspace separation ∆0 for a fixed N

The above discussions suggest that if we can guarantee a minimum separation in the beamspace

between the rays, then the diversity gain can be improved substantially. In particular declaring

an outage whenever the separation of paths in the beamspace is less than a certain threshold

∆0, the diversity gain can be improved substantially, potentially to the value attainable without

any inter-path-interference or IPI. Increasing ∆0 however, comes at the cost of decreasing the

number of users that can be supported with a vanishing probability of outage. Moreover the

magnitude of IPI from multiple users also grows as the number of users increases.

The rest of this section investigates this interplay and characterizes the number of users that

can be supported with a vanishing outage probability and with a diversity gain equal to that

without any inter-path interference (IPI).

B. Diversity and multiplexing gains with ray tracing model

Both diversity and multiplexing gains as defined in the earlier section depend on the separation

∆0 in the beamspace used to declare outage. Generally for a fixed N , a higher ∆0 results in a

higher value of the outage probability, a lower value of the multiplexing gain (number of beams

supported) and a higher diversity gain (Fig. 4). Given that the number of users is low enough,

however, one can show that the diversity gain is not affected.

A result to that effect is described in Theorem 1. Details about these derivations are presented

in Appendix B.

15

Theorem 1. Let N → ∞ the number of users |B| = o(N/ logN), and the separation used to

declare outage in the beamspace be ∆0 = o(1). Then the diversity gain is equal to that without

any IPI, (i.e., equal to (total max. received power)/4) iff

|B|√N∆0

= o(√N).

We next present results about the multiplexing gain that can be supported with a vanishing

outage probability.

Theorem 2. Let N →∞, and the separation used to declare outage be ∆0(N). The number of

users |B| that can be supported with a vanishing (per-beam) outage probability satisfies

|B| = o(1)

∆0

.

Proof. This follows directly from Lemma 2, using which we get that

|P|∆0 = o(1), (13)

for a vanishing outage probability. Since L is finite and |B| = |P|/L, we get that |B| = o(1)/∆0.

We now present a corollary on the number of users/beams that can be supported with a

vanishing probability of outage with a diversity gain equal to that without IPI.

Corollary 1. The number of users that can be supported with a vanishing outage probability,

and a diversity gain equal to that without any IPI is given by

|B| = o(√N).

For a larger number of users, with symbol-by-symbol decoding, there is either a non vanishing

probability of outage or a diversity gain different from that without any IPI.

Proof. Follows directly by eliminating ∆0 from |B| = o(1)/∆0 and |B|/(√N∆0) = o(

√N).

The corresponding choice for ∆0 is given by

∆0 =o(1)

|B| .

16

C. Comparison with IID Rayleigh fading across antenna elements

In this section we describe results obtained using i.i.d. Rayleigh fading across antenna elements

to contrast them with results derived under the ray tracing propagation model. The i.i.d. Rayleigh

fading model corresponds to the situation where there is a large excess of multipath components

(i.e., L = O(N)), i.e., the number of multipath components goes to infinity much faster than

the number of antenna elements. In this setting, the outage probability as defined earlier in the

manuscript is always bounded away from 0, hence we consider only the achievable diversity

gain in this section. For analytical purposes, we assume that the number of rays in the multipath

approaches infinity for any finite number of antennas N. Under this model, by the multivariate

central limit theorem [26], for beam b in the uplink, we have, for a fixed N,

limL→∞

L∑i=1

1√Lejφiaθi ∼ CN (0, σ2

hI).

The randomness comes from the randomness in φi. The asymptotic variance σ2h = 1. A proof

of this result is presented below.

Proof. Let us first assume that θ is drawn from a random distribution. We observe that the

random variable h = ejφaθ is zero mean and has a covariance matrix of Cs = I. This follows

from our assumption on the (uniform) marginal distributions on both φ and θ. Note that the mth

element of aθ is given by e2πm sin(θ)/λ.

Thus the conditions of the CLT are satisfied and we see that, if sis are samples from the

distribution mentioned above then, by the multivariate central limit theorem [26], we have that

limL→∞

∑Li=1 hi√L∼ CN (0,Cs) ≡ CN (0, σ2

hI), (14)

where σ2h = 1, i.e., the average total power across all the rays reaching the receiver is 1.

The proof differs slightly from the above if the θis are assumed to be drawn randomly, but

known (instead of just being drawn randomly), but the result (14) continues to hold.

In other words if there is a large excess of the number of paths compared to the number of

antennas at the receiver, we cannot resolve the angles of arrival in the beamspace. Moreover,

since the phases φb,p for the multipath component (b, p) are not recoverable at the receiver

due to the large excess in the number of unknown phases, the multiuser interference cannot

be cancelled out. Thus the fading seen at the receiver is identical in distribution to a Rayleigh

17

fading distribution, even if the path attenuations (1/√L) and the angles of arrival θ are known.

Assuming i.i.d. Rayleigh fading which is unknown to the receiver for the rest of this section,

we mention our observations for the uplink and the downlink respectively.

1) Uplink: The output of the multi-antenna receiver follows a multivariate product distribution.

In other words

y ∼ 1

π(σ2 +∑

b σ2hs

2b)e−‖y‖

2/(σ2+∑b σ

2hs

2b),

where sb is the symbol transmitted in the bth beam. In the uplink, since the distribution of the

output depends only on the sum of the powers in all beams, the only information that may be

transmitted is through the sum of energies of the constellation points [27]. Thus, in particular,

choosing the transmit codebook such that sb ∈ {0,√

2P} for all b will yield a diversity gain of

d = Θ(1/N) as N increases in a Rayleigh fading channel with the fading realization unknown

at the receiver elements. This is because s0 =√

2P , s1 = 0 is indistinguishable from s0 = 0 and

s1 =√

2P at the receiver, so the probability of error is bounded away from zero and does not

vanish with N .

By choosing the transmit codebooks appropriately, however, one may be able to multiplex a

finite number of users/transmissions [27] in a channel which experiences i.i.d. Rayleigh fading

across antenna elements.

2) Downlink: In this case, without instantaneous channel state information, the beamforming

gains are lost subject to a sum power constraint. This is because of the fact that, without knowing

the exact channel realization at the transmitter, whatever performance is obtained for N > 1 under

an expected sum power constraint can also be obtained with a single antenna. An argument for

why this is the case is presented next. Consider uncoded communication without instantaneous

channel state information under an average transmit power constraint

E

[∑i

|si|2]

= P̃ , (15)

where si is the information transmitted along the ith transmitter element. Let XN be a constella-

tion of points in an N dimensional space, such that |XN | represents the size of the constellation.

We observe that for any fixed {si}Ni=1, the distribution of the received signal at any user is given

by

y ∼ 1

π(σ2 +∑

i σ2h|si|2)

e−‖y‖2

σ2+∑i σ

2h|si|2 .

18

The uncoded symbol-by symbol detection performance is based on decoding∑

i σ2h|si|2 from

the observed y.

We now observe that for all |XN |, this same detection performance can be obtained by just

considering a single antenna, i.e., N = 1 and then constructing the following codebook X1

based on XN . The construction is as follows: for all x ∈ XN , we define x to be such that∑i |si|2 = |x|2. A collection of all such xs is the set X1. In other words, the new alphabet is

given by

X1 =

√∑

i

|si|2 : s ∈ XN

.

We note that the codebook X1 still satisfies the average power constraint (15). Since the square of

the norm of the transmit codeword is a sufficient statistic for decoding information the detection

performance also remains the same.

Thus, utilizing multiple antennas in this case does not yield any benefits in terms of detection

performance for uncoded communication under i.i.d. Rayleigh fading, under our assumption that

the instantaneous channel state information is not available at the transmitter or the receiver.

On the other hand if there exists more structure, either due to some known statistical correlation

patterns or due to the ray tracing models assumed in this work, one can benefit from multiple

antennas even without the knowledge of the instantaneous fading channel state information [9].

Multiple transmit antennas are also beneficial if instead of symbol-by-symbol decoding, one

exploits temporal correlation (this is exploited in space time codes like Alamouti codes).

V. DISCUSSIONS AND RELATED EXTENSIONS

A. Coherence time of angles of arrival

In this section, we show how the coherence time of the channel coefficient at each received

antenna compares with the coherence time of the underlying process on the angle of arrival

processes. Together with the results in the numerical section, this serves to show that by per-

forming appropriate noncoherent transforms, channel estimation requirements can be significantly

reduced. This serves as a justification for our assumption in Section II-B about the precise

knowledge of the angle of arrival and the gain for each multipath. For simplicity of presentation,

in this section, we assume just a single beam, i.e., |B| = 1 and consider that s1 = 1, and that

the path attenuations are fixed.

19

We perform this analysis for a single beam and multipath components with the same atten-

uation. Let’s assume that the autocorrelation function of the ej2π sin(θ(t))/λ process is given by

Aθ(τ), i.e.,

E[e2π sin(θ(t))/λe−j2π sin(θ(t+τ))/λ] = Aθ(τ).

The channel coefficient seen at the uplink by the receiver antenna is given by

h =∑p∈L1

G1,pejφ1,pa(θ1,p).

Due to our assumption that the phases are uncorrelated with the angle and the fading processes,

we have that

E[hi(t)h∗i (t+ τ)] =

∑p∈L1

G21,pAφ(τ)Aθ(τ),

where hi is the fading coefficient seen by the ith antenna element, h∗i is its complex conjugate,

and Aφ(τ) is the autocorrelation function of the complex exponential of the phase process defined

by

Aφ(τ) = E[ejφ(t)e−jφ(t+τ)].

We note that φ(t + τ) − φ(t) is on the order of fDτ [5], where fD is the Doppler associated

with the channel. Typically fD ≈ vλ

, where v is the velocity and λ is the carrier wavelength,

whereas the frequency of change of the spatial parameters like θb,p is given by vX

where X is the

dimension of scattering objects (on the order of 1 meter) and is typically much larger than the

wavelengths used for radio communications (on the order of centimeters or millimeters). The

potential savings in channel estimation requirements (estimating φ versus estimating θ) with

representative numbers are explored in more detail in the numerical section VI-A.

B. Extensions to other antenna geometries

Our results are based on a specific antenna geometry, that of an uniformly linear antenna

array, which allows us to take the Fourier transforms to sample the beamspace and hence extract

the signals in particular locations in the beamspace. For more general geometric setups, we

would require generalizations of spectral estimation algorithms (like MUSIC, ESPRIT [28]) or

even more general parameter estimation algorithms (e.g., estimating both the angle of arrival

and the distance of the source from the receiver). Wherever there exists a structure, these

algorithms take particularly simple forms. For example, under the planar wavefront assumption,

20

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

......

User 1

User 2

Base station

PathBeam

Fig. 5: A 2D array

the received signal vector in the uplink lies in a low dimensional manifold, parametrized only by

the angle(s) of arrival (the number of angles depends on whether we consider two dimensional

or three dimensional propagation). The phase shifts between received signals from the same

planar wavefront can be shown to be a known function only of the angles of arrival.

To further illustrate this point, we can consider the rectangular array as depicted in Fig. 5.

Observe that one can use 2D Fourier transform algorithms to transform to the beamspace in

such a geometry. Note that the beamspace for a two dimensional array is also two dimensional,

with the width of the mainlobe corresponding to a ray along each dimension depending on the

number of antenna elements along that dimension. In general the area occupied by the main

lobe is proportional to the inverse of the number of antenna elements. Thus, results similar to

the ones derived in this paper for a uniform linear array can be obtained for two dimensional

regular arrays. The exact dependence of the multiplexing and diversity gains on the number of

antenna elements may be different, however.

C. Extension of analysis to unequal times of arrival

The number of multipath components has been assumed to be finite in the main theorems

presented in this paper and the results depend on the scaling behavior of the number of interfering

rays as |B| → ∞. Thus the assumption that all the rays do not arrive at the receiver at the same

time does not change the scaling behavior of the multiuser interference or IPI terms; hence, both

the theorems in this paper are exactly the same for unequal arrival times of the multipath rays.

In other words, while assuming that all the rays reach the receiver at the same time may change

the interference terms by a constant scaling, it does not change our derivations of the diversity

and the multiplexing gains presented in this paper which focus on the scaling behavior with N .

21

D. Complexity

The complexity for the beamspace-based uplink detection and downlink precoding is given

by Θ(|B|N logN). The complexity of uplink transmission or downlink detection is just Θ(1).

In comparison, the complexity for detection in systems employing channel estimation involves

computing the pseudo-inverse of channel matrices [29]. This has a complexity of |B|3+|B|2N for

uplink detection or downlink precoding. We thus see that for any |B| > log(N), the beamspace-

based detection is more efficient in terms of the number of operations.

E. Analog architectures for Fourier transforms

The use of DSP is prevalent in phased array systems for Fourier transform and spatial signal

processing. However, if the number of elements is large, especially under the constraints of a

real time communication systems, even such DSP operations may not be feasible. But it may still

be possible to employ analog architectures for Fourier transforms. For example, from the lens

equation we know that the projection of an image of a lens in its focal plane is a Fourier transform

of the far field [30]. The prospect of using a lens for efficient multiplexing in the beamspace,

while not used in any communication system until now, offers an intriguing possibility to open

up dimensions previously not utilized in earlier radio systems. This technique also gets around

limitations in engineering high speed RF circuits for small wavelengths.

F. Improving detection performance

Our results consider only single user detection and symbol by symbol decoding. Multiuser

detection in general has the potential to improve detection performance and thereby support more

users, by helping in the estimation of phase and removal of IPI. Coding across time also has

similar benefits in terms of reducing IPI. The definition of a relevant outage event or an outage

metric also affects the non-outage performance. Characterization of the performance benefits

from an exhaustive investigation of these ideas in terms of outage and detection error is part of

our future work.

VI. NUMERICAL RESULTS

For this section, unless mentioned otherwise, we consider L = 4, i.e., each beam has 4

multipath components. The wavelength λ = 0.05 units (note that the inter-element spacing at

the base station antennas is chosen to be 1 unit). The distance unit is taken to be 10 cm, so

22

that the wavelength corresponds to the mmWave bands (60 GHz). The phases φb,p are chosen

uniformly at every symbol time, and the gains are fixed. c, the angle spread parameter in the

generation of the angles of arrival, is fixed to be π/3. The average transmit power is assumed

to be P = 2.

A. Channel properties as N →∞

Fig. 6 shows the effect of increasing the number of antennas. Note that as the number of

antennas increases, the resolution in the beamspace becomes finer and finer. We see that for 256

antennas, the slow changing parameters of the propagation environment are better identifiable

and can be tracked better compared to the small scale Rayleigh fading seen in the beamspace

when the number of antennas is 8.

Fig. 6: The blue lines correspond to the actual noisy beamspace pattern seen at the receiver. The

red lines correspond to the beamspace pattern that would have resulted if there were no noise

and if all the users were to be transmitting 1. Note how the positions of incoming data symbols

stay relatively constant in the beamspace regardless of the noise and the multiuser interference.

We now demonstrate how having a fine angular resolution affects the coherence time of the

measured channel parameters and how this fine angular resolution can be exploited to reduce the

channel estimation requirements. We fix a velocity of v = 24.8 mph, a typical residential driving

speed, and consider that the slow changing parameters (gain and the angles of arrival/departure)

change over time scales determined by the size of scatterers divided by the velocity; whereas

the phase φ changes on the order of the carrier wavelengths divided by the velocity. For our

simulations we asssume fixed gains and that Aθ(τ) = e−|τ |, whereas for phases we assume that

23

0.00 0.02 0.04 0.06 0.08 0.10

Time (s)

0

1

2

3

4

5

6

Ang

leof

arriv

al

0.00 0.02 0.04 0.06 0.08 0.10

Time (s)

0

1

2

3

4

5

6

Inst

anta

neou

sph

ase

Fig. 7: Angle in beamspace 2π sin(θ1,1)/λ and instantaneous phase φ1,1 for beam 1, path 1

(both in radians). Note the more rapid variations of the instantaneous phase as compared to the

beamspace angle.

0.00 0.02 0.04 0.06 0.08 0.10

Time (s)

0

1

2

3

4

5

Am

plitu

deof

chan

nelc

oeffi

cien

t

Channel coefficient amplitude evolution in time

Element 1Element 2

0.00 0.02 0.04 0.06 0.08 0.10

Time (s)

−4

−3

−2

−1

0

1

2

3

4

Pha

seof

chan

nelc

oeffi

cien

t

Channel coefficient phase evolution in time

Element 1Element 2

Fig. 8: Instantaneous amplitudes and phases of channel coefficients in the uplink at antenna

elements 1 and 2 of the base station. The phase is in radians.

Aφ = e−50|τ |. We use appropriate Brownian motion processes to generate processes with these

characteristics. For |B| = 6, |Lb| = 4, we generate processes θb,p(t), φb,p(t), plot them in Fig. 7

and plot some of the resulting channel coefficients seen at the receiver antenna array according

to Equation (1) in Fig. 8. Note that the individual channel coefficients change much faster than

the slow changing angle of arrival parameters.

24

B. Plots showing outage and detection error performance

Since we do not use joint decoding in the uplink or in the downlink, we are not able to cancel

the effects of IPI (inter-path interference) along the direction of the beam. We now present the

probability of error curves in nonoutage. We present simulation results on the uplink only. In

the simulations we use the per beam outage condition (for detecting beam 1) and a threshold

value of ∆0 = N−0.1/|B| to declare outage. This choice gives a vanishing probability of outage

as N →∞, but for any finite value for N, the probability of outage is finite.

Fig. 9a shows the BER performance of the single user detection scheme in non outage for

beam 1 with different values for the total number of beams |B|. Note that the only source of

performance degradation in the curve in Fig. 9a corresponding to |B| = 1 is additive noise and

the interference from other multipath components from the same beam located more than ∆0

distance apart in beamspace. We note that as the number of antennas increases the probability

of error in single user detection goes down. The shape of the plot (convex upwards) for |B| = 1

however suggests that d as defined in (10) decreases with increasing N . If this were not to be

the case, the shape of the curve would have been convex downwards (or concave). Thus, even

though the BER performance increases with increasing N, we see that diversity gain is affected

as N increases. The exact nature of this dependence depends both on the choice of the ∆0 used

to declare outage and the finite value of N considered and is deferred to future work. If all

multipath components were to be absent, the diversity gain would have been constant and equal

to the limiting value as N becomes larger and larger (shown in the analysis).

Fig. 9a also shows the detection performance whenever there is no outage for different numbers

of users |B|. We consider two cases, one where |B| is below the threshold N0.3 = o(√N)

identified in the analysis, and where |B| is above the threshold (= N0.6). The separation for

declaring outage for user 1 is taken to be ∆0 = N−0.1/|B|. Although the BER curves for both

are convex downwards (on the loglog axis) with increasing N , the shape of the curves reveal

that for the values of N considered, the lower order terms in the non outage detection error

characterization are significant and deserve further investigation for the values of N considered.

As suggested by the proofs of Theorems 1 and 2, in particular the expressions in (18) and (17),

the exact constants and form of the lower order terms depend on the particular assumptions

about the statistics of the angle of arrival, the path gains and the instantaneous phases.

Next we compare how many beams we can support in the uplink for a certain number of

25

101 10210−5

10−4

10−3

10−2

10−1

N

BER

|B| = 1

|B| = dN0.3e|B| = dN0.6e

(a) Error prob. for a single user (loglog axis)

103 104

101

102

N

Number

ofbeamssupported

(b) Number of beams versus N

Fig. 9: Performance metrics for beamspace detection.

antennas subject to a particular error probability bound. We compute detection error performance

at position θ1,1 = 0. For each number of base station antennas N , we plot the number of users

B such that the average probability of error is less than 10−2. The resulting estimates are plotted

in Fig. 9b. The plots support the fact that the number of supported users grows roughly as√N .

VII. CONCLUSIONS

We have considered a ray tracing propagation model with a finite number of multipath compo-

nents and an uncoded noncoherent communication scheme which does not require knowledge of

the instantaneous phase of the channel gains. We assume perfect knowledge of the amplitude of

the gains and the angles of arrival for each of the multipath components. Under this assumption

we find that, in the limit of a large number of antennas N , we can support up to o(√N) users both

in the uplink and the downlink with a vanishing outage and error probability. This is in sharp

contrast to noncoherent communications in a rich scattering environment where multiplexing

gains are nonexistent. We define the diversity gain as being equal to the dominating exponent

in the error probability associated with an equiprobable on-off transmission scheme. Under this

definition we derive the diversity gains associated with non-outage communications and find that

they are equal to that without any multiuser interference if the number of users is less than a

certain threshold which increases with the number of antennas.

26

Our results indicate that the spatial resolution inherent in large antenna arrays can not only

help us resolve each of the multipath components in beamspace, it can also let us use noncoher-

ent methods to design and construct simple energy-detection based transmission and detection

schemes. Considering that only spatial parameters (angle of arrival and the gains) need to be

tracked, the potential overhead of parameter extraction for the channel is also much smaller than if

the instantaneous phase also needs to be tracked. The gains are not limited to noncoherent systems

only, keeping track of the instantaneous phase along each multipath component is also made

easier by the improved resolution — this has been exploited in coherent architectures in [31],

[32], and can potentially be exploited in precoder design in hybrid beamforming architectures.

ACKNOWLEDGEMENTS

This work was supported by a 3Com Corporation Stanford Graduate Fellowship, the Electron-

ics and Telecommunications Research Institute, an Alcatel-Lucent Stanford Graduate Fellowship,

the NSF Center for Science of Information (CSoI): NSF-CCF-0939370 and an A.G. Leventis

Foundation Scholarship. We would also like to thank Prof. Robert Heath, Prof. Claude Oestges

and Prof. Angel Lozano for pointing out relevant prior art. Finally, we would like to thank

the editor and the anonymous reviewers, especially Reviewer 1, whose constructive criticisms

contributed greatly to the readability of the manuscript.

APPENDIX A

In this appendix, we show some properties of f(x) which is defined as

f(x) ,1√N

N−1∑k=0

ejkx =1√N

1− ejNx1− ejx =

1√Nej(N−1)x/2 sin(Nx/2)

sin(x/2). (16)

−2 0 2

0

1

2

3

x

|f(x

)|

27

Figure: |f(x)| for N = 10.

We list some properties of f(x) for any fixed N .

• We have |f(x)| = 1√N

∣∣∣ sin(Nx/2)sin(x/2)

∣∣∣. This may be shown by summing the geometric series in

(16).

• As x→ 0, f(x)→√N. Also if x = Θ(1/N), or x = o(1/N),

f(x) = Θ(√N),

as N →∞.• f(x) = 0 iff x = 2nπ for n belonging to the set of non zero integers Z\{0}. This suggests

that the “width” of the mainlobe is given by 4πN

.

• For x ∈ [2πN, o(1)], |f(x)| / |1/(

√N sin(x/2))| ≈ 2√

Nx2.

APPENDIX B

In this appendix, we prove Theorem 1. We saw in Section IV-A that detection error probability

for a single user is on the order of exp(−cN) for some constant c greater than 0. Thus any

event which is more likely than that asymptotically will affect the detection error probability.

We argue that IPI, when the number of users grows with the number of antennas N, affects the

detection error probability. In particular, for |B| = o(N/ logN), we observe that the probability

that all paths are concentrated in an interval of length ∆ = Θ(∆0) at a distance ∆0 away from

a particular ray in the beamspace is given by

(∆)|B| � exp(−cN),

for a large enough N, since |B| = o(N/ logN). Also, since the probability of exp(jφ̃) being in

a neighborhood of constant length around −1 is constant, we have that

Prob({| exp(jφ̃b,p) + 1| ≤ ε for all (b, p) 6= (1, 1)}) ≥ exp(−c0|B|)� exp(−cN),

for some constant c0. Thus the magnitude of the IPI terms scales as

|B|f(∆0), (17)

with probability at least exp (−c0|B|)� exp(−cN). Since f(∆0) = Θ

(1√N∆0

2

)for ∆0 = o(1),

we get that the amplitude of the IPI term is at most

Θ

( |B|∆0

√N

).

REFERENCES 28

Since the received signal amplitude is Θ(√N), for the IPI to not change the diversity gain to a

value different from what would have been obtained without any interference, we need to have

|B|∆0

√N

= o(√N).

Hence the result in the theorem.

We also present a short derivation of the diversity gain without any IPI for the system model

presented in the paper for completeness. We present results for the uplink, with the downlink

result following very similar lines.

• Uplink: The decision statistic in (4) without any IPI (or any IPI of magnitude o(√N)) can

be written as

r =

∑p∈L1 |ybeamspace,ψ1,p|

L=

∑p∈L1 |

√N/Ls1 + o(

√N) + ν̃1,p|

L.

When s1 =√

2P (full power) we have r =√

2NP/L. Also when s1 = 0 (no power),

we have r = 0. Thus the threshold amplitude is given by rthres =√NP/(2L). We note

that for ∆0 = ω(1/N), as N becomes larger and larger, we have that ν̃1,p ∼ CN (0, 1),

and E[ν̃1,p1 ν̃∗1,p2

] =∑N−1

k=01N

exp(jk(ψ1,p1 − ψ1,p2)) → 0, as N → ∞ if p1 6= p2, and

|ψ1,p1 − ψ1,p2| ≥ ∆0 = ω(1/N). Thus, for a large enough N, {ν1,p}Lp=1 can be considered

to be independent and identically distributed. We observe that, for a large N, the most

probable reason for error is that

|ν̃1,p| >√NP/(2L)− o(

√N). (18)

for all p. This follows from Laplace’s principle and the observation that

Prob(|ν̃1,p| > f(N)) is dominated by exp

(−

L−1∑k=0

f 2(N)

)for any f(N), which satisfies f(N)→∞, as N →∞. This argument can be made precise

by observing the exponent of the pdf of all noise levels that can lead to an error event

(r ≶ rthres accordingly for ON or OFF transmissions respectively) and using Laplace’s

principle [25]. In any case, the probability of this event is given by ΠLp=1e

−NP/(2L)+o(N) =

exp(−NP/2+o(N)). The total maximum received power is given by 2P. Thus the diversity

gain is

max. received power/4.

• Downlink: Follows very similar arguments.

REFERENCES 29

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