1

Correction Factor for Analysis of MIMO Wireless
Networks With Highly Directional Beamforming

arXiv:1710.07369v2 [] 31 Jan 2018

Mandar N. Kulkarni, Eugene Visotsky, Jeffrey G. Andrews

Abstract—In this letter, we reconsider a popular simplified
received signal power model with single stream beamforming
employed by the transmitter and the receiver in the regime when
the beams have high gain and narrow beamwidth. We define the
correction factor as the ratio of the average actual received signal
power divided by the average received signal power using the
popular simplified model. We analytically quantify this factor
for LOS and NLOS service and interfering links under some
assumptions. The analysis along with simulations using a 3GPP
compliant new radio (NR) channel model confirm the importance
of incorporating the correction factor in coverage analysis of
wireless networks that utilize the popular simplified received
power model.

I. I NTRODUCING T HE C ORRECTION FACTOR
In system level analysis for computing coverage and rate
performance of wireless networks on R2 a popular model
to compute the received signal power at X ∈ R2 from a
transmitter (serving/interfering) at Y ∈ R2 is as follows [1]–
[4].
Pr = Pt `(||X − Y ||)hGt (θ)Gr (φ),
(1)
where Pt is the transmit power, `(.) is the path loss, h is
the small scale fading, Gt (θ) is the transmit antenna gain and
Gr (φ) is the receive antenna gain. If at all blockage effects
are explicitly incorporated in the analysis by differentiating
line of sight (LOS) and non-LOS (NLOS) links, then only
h and ` are modeled differently for LOS and NLOS [3].
The antenna patterns Gt (.) and Gr (.) are considered to have
the same distribution for LOS/NLOS links. In this work,
we will show the importance of incorporating an additional
blockage dependent factor in the received signal power when
the antenna patterns have very narrow beamwidths and large
gains – for example, an antenna pattern having 36 dB gain
and 12o half power beamwidth in azimuth. Our analytical
model shows that if there are large number of antennas at the
transmitter and receiver, which employ analog beamforming,
then the additional factor (called as the correction factor) is
much less than 1 for NLOS service links but is close to 1 for
LOS service links, and equal to 1 for NLOS/LOS interfering
links. Such a factor cannot be incorporated by modifying
either h or `(.) for analyzing signal to interference plus noise
ratio (SINR) in highly directional MIMO wireless networks,
especially cellular networks, and an example to explain this is
given in the Appendix.
Email:
{mandar.kulkarni@, jandrews@ece.}utexas.edu, eugene.
visotsky@nokia − bell − labs.com. M. Kulkarni and J. Andrews are
with the UT Austin, TX. E. Visotsky is with Nokia Bell Labs, IL. Last
revised on February 2, 2018.

Most prior analyses of MIMO wireless networks computing
coverage and rate performance with highly directional single
user beamforming incorporates a received signal power similar
to (1) and do not model a channel with LOS/NLOS dependent
rank [2]–[9]1 , which gives rise to the needed correction factor
as we will show in this work. The purpose of this letter is to
make the growing research community using received power
model similar to (1) aware of the significance of how different
rank of the MIMO channel for LOS and NLOS can affect
the effective antenna gain and thus the design insights. We
will formally define effective antenna gain in this work. Also
we propose a quick way to preserve the existing analyses
by multiplication of a LOS/NLOS dependent constant for
service links but not the interfering links. The example in
Appendix is indicative of how this can be done. The correction
factor is especially important for analysis of millimeter wave
(mmWave) cellular networks, wherein inclusion of blockage
effects is crucial and the beamforming is highly directional [3].
All prior works which studied different system design issues
in these networks like [5]–[9] use the received power model in
(1) without incorporating the correction factor. In Section V,
we discuss key implications on system design resulting from
incorporation of such a factor.
The analysis in this work is for analog beamforming implementation done at the transmitter and the receiver under
consideration. Our analysis along with the simulation results
considering a more detailed wideband 3GPP channel model
suffice to motivate the inaccuracy of the popular model in (1)
when the transmit and receive beams are narrow and with large
gains. However, more detailed analysis is needed in the future
to estimate the correction factor more accurately.
II. S YSTEM M ODEL
We concentrate on a single transmitter-receiver pair in a
wireless network. Nt and Nr denote the number of transmit
and receive antennas. If the link is NLOS, the narrowband
channel between the transmitter-receiver is given by [11], [12]
s
η
`(d) X
HNLOS = κ
γi ar (φi )a∗t (θi ),
(2)
η i=1
where `(d) is the path gain (assumed deterministic function of
d for simplicity), η is the number of paths (assumed constant),
d is the transmission distance in meters and γi is the
 small

scale fading on path i (random variable such that E |γi |2 =
1 Except our prior work in [10] which was the first attempt to do so, to the
best of our knowledge.

2



1) and κ is a normalizing constant such that E ||HNLOS ||2F =
Nt Nr `(d).
Assuming a uniform linear array at the receiver,
the array response vector ar is given as ar (φi ) =
 −jφ −2jφ
T
i
1e i e
. . . e−(Nr −1)φi , where j is square root
of −1. Similarly, one can define at by replacing Nr with
Nt . Note that φi and θi are spatial angles of arrival and
departure (AOA/AOD). It is assumed that these AOAs and
AODs are continuous random variables and no assumption on
their distribution is made.
If the link is LOS, the narrowband channel is given by [12]
HLOS

r

p
= `(d)

s
κ

KR
ar (φ0 )a∗t (θ0 )+
KR + 1
1

η
X

η(KR + 1)

i=1

!
γi ar (φi )a∗t (θi )

, (3)

where KR is the Rician K-factor. AOA and AOD given by
φ0 and θ0 are constants corresponding to the direct LOS
path between the receiver and the transmitter. Rest of the
AOA/AOD are continuous random variables. η and κ could
have different LOS-specific values here, as compared to (2).
Assuming w is the combiner employed by the receiver and f
is the precoder employed by the transmitter, the received signal
power model is given as Prmulti = ||w∗ Hf ||2 , where H is the
Nr × Nt channel which could be either HLOS or HNLOS .
We constrain w and f to be chosen of the form N1r ar (.)
and N1t at (.), respectively, which is basically employing analog
beamforming using phase shifters at both the receiver and
the transmitter. If the transmitter-receiver pair form a desired
communication link, w and f are chosen so as to maximize
Prmulti . If the transmitter-receiver pair form an interfering link,
then w and f can be arbitrary.
Most analytical studies to compute coverage and rate performance cannot afford to use the received signal power
model defined above for tractability. As mentioned in Section I, a simplified model similar to (1) is generally used.
Now we will define a generative model for such a simplified model. We
[13].
p define a keyhole channel as 2follows

Hkeyhole =
`(d)γar (φ)at (θ), where E |γ|
= 1 and
{θ, φ} could have arbitrary distribution. Now, Prkeyhole is
defined as ||w∗ Hkeyhole f ||2 . If the transmitter-receiver pair
is a desired signal link, w = N1r ar (φ) and f = N1t at (θ)
to maximize Prkeyhole and thus, Prkeyhole = |γ|2 `(d)Nt Nr .
If the transmitter-receiver pair is an interfering link with
w = N1r ar (φ0 ) and f = N1t at (θ0 ) for some arbitrary φ0
and θ0 , then Prkeyhole = `(d)|γ|2 Gr (φ, φ0 )Gt (θ0 , θ), where
Gr (φ, φ0 ) = || √1N a∗r (φ)ar (φ0 )||2 . Similarly Gt can be writr
ten replacing subscript r with t and φ with θ. Unlike the
desired signal power case, here φ0 and θ0 are not chosen
to maximize ||w∗ Hkeyhole f ||2 but can be random angles
distributed according to some
distribution.
 continuous



We wish to compare E Prmulti with E Prkeyhole . This
comparison will highlight how important it is to consider rank
> 1 channels for LOS and NLOS in terms mean received
signal power since the keyhole channel is always rank 1. In

order to quantify this comparison, we define a correction factor
as follows.
Definition 1. The proposed correction factor to estimate the
received signal
power
link is defined

 on a serving/interfering

as Υ = E Prmulti /E Prkeyhole .


Note that for serving links E Prkeyhole = Nt Nr `(d)
irrespective of LOS/NLOS as per our analytical model.
Definition 2. The effective antenna gain is defined as the
actual received signal power (on serving/interfering links)
normalized by the path loss and the transmit power of the
signal.
Note that the effective antenna gain is in general a random variable. As per our analytical model, it is equal to
Prmulti /`(d). Considering our system model, wherein `(d) is
deterministic the mean effective antenna gain for a serving
link is given by ΥNt Nr , where Υ is the correction factor
for a serving link. Our proposal is that if one wants to use
a simplified received power model like in (1) for system
level analysis, wherein the impact of beamforming is captured
through a spatial gain pattern at the transmitter and receiver,
then the corrected received signal power on serving and
interfering links is obtained by multiplying Υ to the estimate
in (1). Since here a keyhole model is used to generate the
simplified received power model in (1), the corrected received
signal power is ΥPrkeyhole .
III. C OMPUTING Υ W HEN Nt , Nr A RE L ARGE
Before we state the results, we make a quick observation
based on the result in [14].
Observation 1: As Nr → ∞ and Nt → ∞, the left
singular vectors corresponding to non-zero singular values of
(2) and (3) converge to √1N ar (φi ), with i = 1 . . . , η for
r
(2) and i = 0, . . . , η for (3). Similarly, the right singular
vectors corresponding to non-zero singular values of (2) and
(3) converge to √1N at (φi ).
t
Observation 2: As Nr → ∞, a∗r (φi )ar (φj )/Nr → 1(i =
j). Similarly a∗t (θi )at (θj )/Nt → 1(i = j) as Nt → ∞.
Theorem 1. Large Nt and Nr is assumed.
If

 the
link is a  NLOS service  link, then E Prmulti
≈
Nt Nr `(d)E maxi=1,...,η |γi |2 /η.
If the link is a LOS service

link and KR  1 then E Prmulti ≈ Nt Nr `(d)KR /(KR + 1).
Proof. Optimal combiner and precoder correspond to the
singular vectors corresponding to the maximum singular value
norm of the channel matrix. Making use of Observation 1
for NLOS channel with large number of antennas, w =
√1 ar (φ1 ) and f = √1 at (θ1 ) assuming |γ1 | = maxi |γi |,
Nr
Nt
without loss of generality. Thus,
Prmulti = ||w∗ HNLOS f ||2
s
η
`(d) X
= κ
γi a∗r (φ1 )ar (φi )a∗t (θi )at (φ1 )
ηNt Nr i=1

2

.

Note that Observation 1 implies
that the non-zero singular valq
`(d)Nt Nr
ues of (2) are given by
κγi . Thus, ||HNLOS ||2F =
η

3

Pη
κ2 `(d)Nt Nr i=1 |γi |2 /η, which is computed using the fact
that square of Frobenius norm equals
sum ofsquares of singu
lar values of a matrix. Thus E ||HNLOS ||2F = Nt Nr `(d)κ2 ,
which implies that the normalizing constant κ = 1. Similarly,
κ = 1 in (3).
Since the AODs/AOAs are continuous random variables,
any two such angles are unequal with probability 1. Using
the orthogonality of the array response vectors for unequal
2
q
`(d)
=
+
0
AODs/AOAs, we get Prmulti ≈ Nr Nt γ1 ηN
t Nr
2

Nt Nr `(d) |γη1 | with probability 1. Thus, the expectation of
E[|γ1 |2 ]
Prmulti is Nt Nr `(d) η . The result is approximate as we
used asymptotic results in Observations 1 and 2 for finite
number of antennas. 

For LOS, since
E |γi |2 = 1, by Markov inequality

P |γi |2 > ηKR < 1/ηKR . Thus, owing to KR  1 with
high probability the maximum singular value corresponds to
the direct LOS path. This implies that w = √1N ar (φ0 ) and
r
f = √1N at (θ0 ), which are singular vectors corresponding
t
to the maximum singular value as per Observation 1. Thus,
it is concluded that Prmulti ≈ ||a∗r (φ0 )HLOS at (θ0 )||2 =
2
p
`(d)Nt Nr KR /(1 + KR ) + ρ , where
ρ= p

1
η(KR + 1)

η
X

γi a∗r (φ0 )ar (φi )a∗t (θi )at (θ0 ).

i=1

Note that using Observation 2, we have ρ ≈ 0 by similar arguments as for NLOS case considering the angles
of arrival/departure
are continuous random variables. Thus,

R
E Prmulti ≈ Nt Nr `(d) KK
.
R +1
Corollary 1. Large Nt and Nr is assumed and the link under
consideration is assumed to be a service link. If γi are complex
normal
Pη random variables and independent of each other, Υ ≈
1
k=1 (1/k) if the link is NLOS. If γi are all identical to
η
complex normal γ1 , Υ ≈ η1 for NLOS link. For LOS link and
KR
KR  1, Υ ≈ 1+K
≈ 1.
R
Proof. If γi are complex normal random variables, |γi |2
are exponentially distributed with unit mean.
Also these are

independent
random
variables.
Thus,
E
max
|γi |2 =
i=1,...,η
 keyhole
Pη
1 and
=
k=1 (1/k) [15].  By Theorem

Pη E Pr
Nt Nr `(d), Υ ≈ E maxi |γi |2 /η = η1 k=1 (1/k) if γi are
complex normal random variables and independent of each
other. Similarly, the other two results are derived.
From Corollary 1, NLOS received signal power can be
significantly overestimated with the keyhole model for η = 10,
which translates to Υ = −4.6dB if γi are identically equal
to γ, and to Υ = −10dB if γi are independently but
identically distributed. Note that this is an analytical result
and that well accepted wideband models (like in [12]) will
have unequal distribution of powers amongst paths within and
across different clusters. Estimating Υ in these settings is an
avenue for further research.
Theorem 2. Let
the transmitter
and receiver beamforming
0
)
ar (φ0 )
√
vectors be a√t (θ
and
.
If
γ
are independent zero mean
i
N
N
t

r

random variables with unit variance, {θi } are identically
distributed to θ, {φi } are identically distributed to φ and
{γi } are independent
of all AOAs/AODs, then E Prmulti =

keyhole
E Pr
for NLOS interfering links.
Proof. The received signal power considering a multipath
channel in (2) is given by
Prmulti

η
`(d) X γi
√
a∗r (φ0 )ar (φi )a∗t (θi )at (θ0 )
=
η
N
N
t
r
i=1

2

.

(4)
Using independence of γi and that these are zero mean
random variables, the cross terms while expanding the norm
squared in (4) become zero and thus,
η
 multi  `(d) X


E Pr
=
E |γi |2 E [Gr (φ0 , φi )Gt (θi , θ0 )]
η i=1

=



`(d)ηE [Gr (φ0 , φ1 )Gt (θ1 , θ0 )]
= E Prkeyhole .
η

Theorem 2 indicates that a correction factor is not necessary
for NLOS interfering links, if the assumptions in the theorem
hold true. Similar result can be stated for LOS interfering links.
However, depending on the structure of the arrays, the perelement antenna gains and joint distribution of {γi , φi , θi } a
non-unity correction factor may be necessary for interfering
links. Next, we will validate the need for a correction factor
with some simulations using the 3GPP NR channel model
[12].
IV. S IMULATION R ESULT WITH 3GPP MODEL
We consider two MIMO systems with link lengths 100
meters operating at 73 GHz carrier frequency. One is LOS
and the other is NLOS. 8 × 8 uniform planar array with
half wavelength spacing is assumed at the transmitters and
the receivers. Note that considering a 8 × 8 antenna array
system is realistic for mmWave backhaul networks wherein
both ends of a communication link are base stations (BSs)
[16]. Effective antenna gain for each of these MIMO systems is
computed as Prmulti /`(d) as per Definition 2. Here, Prmulti was
computed considering the 3GPP NR channel model [12] along
with optimal precoders and combiners that maximize the SNR
and a unit transmit power. Several realizations of the 3GPP
channel were simulated for both the links. The distribution
of effective antenna gain seen by the LOS and NLOS link is
plotted in Fig. 1. As seen from Fig. 1 there is a drop of about
12 dB in NLOS median gain compared to LOS, which is very
significant. The implication of such drop in effective antenna
gain is discussed in next section. The LOS effective antenna
gain in Fig. 1 is very close to 10 log10 (64 × 64) = 36dB , as
expected, since correction factor for LOS links is negligible as
per our analysis. Surprisingly the drop in NLOS gain is equal
to −10 log10 19, wherein 19 is the mean number of NLOS
clusters in the 3GPP model. This equals our analytical estimate
of 1/η considering η = 19. A more accurate analysis explicitly
modeling different clusters with multiple rays and correlated
small scale fading is a possible future work.

4

1

CDF

0.8

NLOS
LOS

10xlog10(1/19) 12 dB drop in
median NLOS antenna gain

0.6
0.4

10*log10(64x64) = 36dB

0.2
0
15

20

25

30

35

40

Effective antenna gain in dB

Fig. 1: Comparison of effective antenna gain for LOS and NLOS
links with new radio 3GPP channel model [12].

as well as for service and interfering links. For simplicity of
exposition, we will consider the correction factor to be much
less than 1 for NLOS serving links and equal to 1 for rest of the
cases, which is an outcome of our asymptotic analysis. First to
evaluate whether Y is an interferer or a serving BS – usually
the serving BS is the one with maximum received signal power
averaged over h – one has to multiply a correction factor that is
much less than 1 to the received signal power from Y to origin.
However, if it is determined that the BS does not serve the user
but is a potentially interfering BS, then the correction factor
is equal to 1 while computing the interference power from the
same BS at Y to the receiver at origin. Such a modification
in received power, which is done differently for service and
interfering links cannot be done by modifying `(.) or h.

V. I MPLICATIONS AND A PPLICABILITY OF THE W ORK
This work is applicable for MIMO wireless networks with
highly directional single stream beamforming at the transmitter
and the receiver. The analysis can also be extended for
multi-user MIMO with large number of transmit and receive
antennas. In short, whenever the underlying signal processing
of a large MIMO system is abstracted to compute the received
signal power as a product of a single input single output
(SISO) received signal power and some spatial antenna gain
patterns at the transmitter/receiver for simplified analysis, there
will be a need for incorporating the correction factor to make
sure that identical antenna gain patterns are not multiplied for
LOS and NLOS links, as well as serving and interfering links.
The implications of the work are prominent in the following
scenarios. For dense outdoor-to-outdoor cellular networks, a
user would likely associate with a LOS BS and thus the signal
to noise ratio (SNR) coverage estimates wouldn’t vary significantly, except for the tail probability when a user associates
with a NLOS BS that affects the cell edge rates. Otherwise,
we expect such a correction factor to be significant since there
is significant probability of connecting to a NLOS BS since
the SNR distribution itself will shift by Υ. We expect the
significance of such a correction factor to also be significant
in analysis of multi-hop mmWave cellular networks wherein
the fiber site deployment will be relatively sparse and thus
there will be a question as to whether a relay should go for
a NLOS direct hop to fiber base station or whether it should
relay over multiple LOS hops. Given that the correction factor
introduced in this letter doesn’t affect LOS links but strongly
affects NLOS links, LOS hops will be even more strongly
favoured over NLOS hops. Neglecting the correction factor
but using a model like (1) can lead to misleading insights.
A PPENDIX : A N E XAMPLE D EMONSTRATING T HE U SE O F
C ORRECTION FACTOR
Consider a receiving user at origin and a collection of
transmitting base stations (BSs) in R2 including a BS at
Y ∈ R2 which is NLOS with respect to the user. We want
to understand the SINR performance of that user using the
simplified received signal power in (1) that models beamforming through a spatial antenna pattern. Our proposal is to
introduce the correction factor to compute the received signal
powers. In principle, this factor is different for LOS and NLOS

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