1

On Bounds of Spectral Efficiency of Optimally
Beamformed NLOS Millimeter Wave Links

arXiv:1708.04257v2 [] 23 Nov 2017

Rakesh R T, Student Member, IEEE,

Debarati Sen, Member, IEEE, Goutam Das

Abstract—Beamforming is an indispensable feature for millimeter wave (mmWave) wireless communications in order to
compensate for the severe path loss incurred due to high
frequency operation. In this paper, we introduce a novel framework to evaluate the spectral efficiency (SE) of non-line-ofsight (NLOS) mmWave links with optimal analog beamforming.
Optimality here implies the joint selection of antenna beams at
the transmitter and receiver which simultaneously maximize the
received power. We develop a mathematical framework based
on the extended Saleh-Valenzuela channel model to embody the
impact of optimal analog beamforming into the performance
metrics for NLOS mmWave links. Practical mmWave channels
are characterized by sparsity in terms of number of multi-path
components; we exploit this feature to derive upper and lower
bounds on SE of beamformed directional links. Simulation results
reveal that the proposed approach is fairly accurate to model
beamformed links in most practical operating scenarios. We also
study the impact of overhead due to antenna beam training on
the throughput (TP) of a link and obtain an approximate solution
for optimal antenna half power beamwidth which maximizes TP.
Index Terms—MmWave Communication, Directional Antenna,
Optimal Analog Beamforming, Spectral Efficiency.

I. I NTRODUCTION

R

ECENT advances in technology have paved the way
for emergence of wideband millimeter wave (mmWave)
communications providing a viable option to meet the future demand for multi-Gbps data rates [1]. However, high
frequency mmWave transmission incurs significantly large
path loss during signal propagation, and thereby limits the
transmission range. To overcome this bottleneck, directional
antennas with beamforming capability are employed for signal
transmission and/or reception [2]. The objective of beamforming protocol is to steer the antenna beams at the transmitter
and receiver nodes of a link such that the transmission rate
is maximized [2]. This is achieved by optimizing the signalto-noise ratio (SNR) or signal-to-interference plus noise ratio
(SINR) [3] at the receiver.
Beamforming protocols essentially enable spatial filtering
of multi-path signal components based on the defined optimality criteria [3]. The quality and reliability of the link
therefore depends on the beamformed directional channel and
in this context, statistical modeling of beamformed directional
channels is essential to accurately obtain mmWave network
performance metrics such as coverage probability, spectral
efficiency (SE) etc. The schemes proposed in [4], [5] which
Copyright ©2015 IEEE. Personal use of this material is permitted. However,
permission to use this material for any other purposes must be obtained from
the IEEE by sending a request to pubs-permissions@ieee.org.
The authors are with G. S. Sanyal School of Telecommunications, Indian
Institute of Technology, Kharagpur, W.B -India . E-mail: [rakeshrt, debarati,
gdas]@gssst.iitkgp.ernet.in

evaluate the performance of mmWave networks with analog
beamforming [3] simply model the beamformed directional
channel by a random gain component assuming that the
channel is frequency flat. This is similar to the model used
for conventional sub-6 GHz systems where channel gain is
obtained as the product of a Rayleigh or Nakagami-m random
variable which accounts for small scale fading effect, and
a path loss term that models the large scale fading effect.
Similarly, a recent work on coverage analysis for mmWave
line-of-sight (LOS) links with analog beamforming [6] approximates the beamformed directional channel by a random
gain component based on the uniformly random single path
(UR-SP) assumption. However, in non-LOS (NLOS) mmWave
channels the power content of multi-path components [7] are
comparable, and thus the modeling approaches considered
for beamformed directional channels in existing literature are
not applicable. Therefore, a new mathematical framework is
required which embodies the impact of optimal beamforming
for performance study of NLOS mmWave links.
In this paper, we develop a mathematical framework to
statistically model NLOS mmWave links with optimal analog
beamforming in order to evaluate the SE of noise limited
NLOS mmWave links. We assume that the optimal transmitterreceiver antenna beam pair is chosen from a set of nonoverlapping antenna beams spanning the 3600 azimuth space
such that the received signal power is maximized. The omnidirectional propagation characteristics of the channel is represented by the extended Saleh-Valenzuela (S-V) spatial channel
model [7]–[9]. We utilize this model to derive lower and upper
bounds on SE of optimally beamformed mmWave links. We
further note that SE of a noise limited link can be enhanced
by using high resolution antenna beams albeit at the cost
of significant training overhead due to the associated analog
beamforming protocol [10]. The trade-off between training
overhead and throughput (TP) for indoor mmWave networks
is investigated through simulations in [10]. We propose a
mathematical framework to quantify TP as a function of SE
and training overhead. Moreover, the analysis also helps to
determine requirements for the design of antenna beamforming protocols in terms of an optimal antenna half power
beamwidth (HPBW) which maximizes TP and specifies the
feasible region of operation in terms of antenna HPBW so
that links are able to identify optimal antenna beam pairs. The
paper has three main contributions: (i) we introduce a novel
modeling approach to study the statistical behavior of optimal
analog beamforming in NLOS mmWave links, (ii) we obtain
tractable lower and upper bounds on SE of a NLOS mmWave
link utilizing the sparsity in practical mmWave channels, and
(iii) we provide a design insight for mmWave communication

2

systems by obtaining an approximate solution for optimal
antenna HPBW which maximizes TP for a given analog
beamforming protocol under a set of channel parameters.
II. S YSTEM M ODEL
As shown in Fig. 1, we consider a system model consisting
of an outdoor mmWave link with the transmitter and receiver
nodes separated by a distance d. The nodes are assumed
to be equipped with directional antennas with beamforming
capability. We further assume that direct LOS connectivity
between the transmitter and receiver is blocked and hence
the beamformed link is established through NLOS multi-path
components (Fig. 1). We approximate the antenna radiation
pattern by a sectored model [4] with zero side lobe gain.
Let θ3dB,t and θ3dB,r (in degrees) denote antenna HPBW
of the transmitter and receiver, respectively. The transmitter
and receiver main lobe antenna gain values can approximately
360
360
be calculated as Gm,t = θ3dB,t
and Gm,r = θ3dB,r
[4],
respectively. We further assume that the nodes select the
optimal antenna beam pair that maximizes the SNR at the
receiver node (out of Mt and Mr number of non-overlapping
beams at the transmitter and the receiver nodes, respectively).

ϴl

B=Mr Mt

ϴref
Blockage
d

θ3dB,t

A

Tx

Фl

θ3dB,r

Фref
Rx

C

Mt beams

Mr beams

Fig. 1: A typical node deployment scenario

We adopt a frequency flat equivalent of the extended S-V
channel model for our analysis. The total power received from
L multi-path components can thus be expressed as,
L
X
P = PT cd−α
|hl |2 GT (Θref − Θl )GR (Φref − Φl ), (1)
l=1

where PT represents the transmit power, c denotes the intercept point from the path loss formula, and α denotes the
path loss exponent. |hl | is the small scale fading amplitude
which is generally modeled as a Rayleigh or Rice random
variable [7], [8]. GT (.) and GR (.) represent the antenna gain
of the transmitter and receiver antennas respectively with
corresponding antenna pointing angles Θref and Φref . Θref
and Φref are defined as the angle between the maximal gain
direction of the antenna main lobe and the line segments Tx-A
and Rx-C at the transmitter and receiver, respectively, as shown
in Fig. 1. Θl and Φl are the angle of departure (AOD) and
angle of arrival (AOA) of the l-th multi-path component. The
number of multi-path components denoted by L is a random
variable with its average value denoted by λ0 [7]. Assuming
a sectored radiation pattern model and unit transmit power,
the signal power received by the i-th P
antenna beam pair can
2
be obtained from (1) as Pi = cd−α l∈Li |hl | Gm,t Gm,r ,

where Li denotes the set of multi-path components which
are located inside the antenna main lobes of the transmitter
and receiver corresponding to the i-th antenna beam pair.
We assume that the cardinality of the set Li (card(Li )) is
a Poisson random variable with average number of multi-path
components λd = λ0 /B, where B denotes the total number
of available transmitter-receiver beam pairs (B = Mt Mr ). It
should be noted that in practice the average received signal
power varies with antenna beam orientation angle [7], and
therefore λd as well as α are functions of antenna beam
orientation angle. As of now due to lack of availability
of empirical data to capture this variation, we assume λd
and α to be constant [7]–[9] which incidentally also lends
mathematical tractability for analysis. It may also be noted
that λd and α could be obtained by making use of the
analytical model reported in our prior work [11]. However,
this modeling approach is presently out of scope of this paper.
In this paper, the small scale fading gain |hl | is assumed to
be Nakagami-m distributed with
equal to 1/λ0 ,
hP mean power
i
L
2
≈
1, where E[.]
|h
|
which ensures that EL,|hl |2
l
l=1
denotes the expectation operator. The received power corresponding to the i-th antenna beam pair can thus be
h expressed
i
P
2
2
−α −1
as, Pi = cd λ0
|g
|
G
G
with
E
|g
|
= 1.
l
m,t
m,r
l
l∈Li
In practice mmWave multi-path components are sparse in
time as well as the angular dimension [8]. Consequently,
the probability of receiving multiple propagation components
inside the antenna main lobe is negligible and therefore with
most of the practical antenna radiation patterns, card(Li ) ≤
1, ∀i. Therefore, the presence of the multi-path component
inside a pair of antenna beams can be modeled by a Bernoulli
random variable with success probability p. The value of p can
be computed as, p = 1 − exp(−λd ). Based on this approximation, the received power corresponding to i-th antenna beam
2
−α
pair is simplified as Pi = Πi (p) |g| Gm,t Gm,r λ−1
,
0 cd
where Πi (.) denotes the Bernoulli random variable corresponding to the i-th antenna beam pair with success probability
p, i.e., Πi (p) = 1 with probability p; Πi (p) = 0 with
probability 1−p. The optimal transmitter and receiver antenna
beams (thick lined sectors in Fig. 1) are jointly selected based
on the maximum received signal power criteria, and therefore
the optimal received signal power is calculated as,
Popt = max (P1 , P2 , ..., PB ) .
(2)
III. C ALCULATION

OF

S PECTRAL E FFICIENCY

In this section, we first derive an upper bound on SE
of an optimally beamformed mmWave NLOS link (included
in Section III-A) by assuming Nakagami-m fading for each
multi-path component. In addition, we also present the upper
and lower bounds on SE in Section III-A. Finally, we obtain
an expression for link throughput in Section III-B which
determines the fraction of SE useful for communication after
accounting for the antenna beam training overhead.
A. SE under extended S-V channel model with the assumption
of Nakagami-m distributed |g| .

The variability in power received by each antenna beam pair
2
is essentially due to the parameters |g| and Πi (p). Therefore,
power maximization in (2) is equivalent to the calculation

3

of normalized received signal power corresponding
to the

 ′ ′
′
′
optimal antenna beam pair, i.e., Popt = max P1 , P2 , ..., PB ,
′

2

where Pi = Πi (p) |g| , i ∈ {1, ..., B}. In this section, we
first evaluate the cumulative distribution function (CDF) of
′
′
Popt . We note that Pi for i-th antenna beam pair is a mixed
random variable, since |g|2 and Πi (p) are continuous and
′
discrete random variables, respectively. Accordingly, Pi is
a continuous random variable if Πi (p) = 1; and a discrete
random variable if Πi (p) = 0. This condition also implies
′
that Popt is a continuous random variable if ∃i, where
Πi (p) = 1, ∀i ∈ {1, ..., B}. Therefore, we proceed with the
′
derivation for the CDF of Popt in two exclusive parts; the
first of which deals with the continuous case (∃i, where
Πi (p) = 1, ∀i ∈ {1, ..., B}) and the second part deals with
the discrete case (Πi (p) = 0, ∀i ∈ {1, ..., B}) only. Hence,
′
the CDF of Popt with ∃i, where Πi (p) = 1, ∀i ∈ {1, ..., B} is,
h ′
′
′
′
FPopt
(P ∗ ) =P rob (P1 ≤ P ∗ ) ∩ (P2 ≤ P ∗ )... ∩ (PB ≤ P ∗ )
i
|∃i, where Πi (p) = 1, ∀i ∈ {1, ..., B} ,
(3)

where P rob(.) represents probability of the given event.
′
′
identically
Since P1 ,.., PB are independent

 distributed,
 ′ and 
′
′
P rob P1 ≤ P ∗ =...=P rob PB ≤ P ∗ =P rob P ≤ P ∗ .
Hence, (3) can be simplified based on Bayes’ rule as,
PB B 
′
B−i i
p P rob(P ≤ P ∗ )i
i=1 i (1 − p)
∗
′
(P ) =
FPopt
(4)
B
1 − (1 − p)
′
Without loss of generality, we calculate P rob(Pi ≤ P ∗ ) using
the probability density function (PDF) of Gamma random
m m−1 −mx
. The CDF of
variable X defined as fX (x) = m xΓ(m)e
′
Pi ∃i, where Πi (p) = 1, ∀i ∈ {1, ..., B} is calculated as,
Z P ∗ m m−1 −mx
′
m x
e
dx
γ(m, mP ∗ )
P rob(Pi ≤ P ∗ ) =
=
(5)
Γ(m)
Γ(m)
0
where γ(x, y) denotes the lower incomplete Gamma function
′
with parameters x and y. Substituting P rob(Pi ≤ P ∗ ) in (4)
results in,

h
iB
B
p γ(m,mP ∗ )
−1
1 + 1−p Γ(m)
(1 − p)
′
(6)
FPopt
(P ∗ ) =
1 − (1 − p)B
We note that (6) is intractable owing to the incomplete Gamma
function. For further simplification for the computation
of
∗
)
SE, we explore the possibility to approximate γ(m,mP
and
Γ(m)
iB
h
p γ(m,mP ∗ )
. Since P ∗ varies from 0 to ∞, only
1 + 1−p
Γ(m)
iB
h
p γ(m,mP ∗ )
loose approximations are possible for 1 + 1−p
Γ(m)
which can aid the evaluation of SE. Also, due to the possibly
large values for B (for example, antenna HPBWs of 330 and
150 at transmitter and receiver corresponds to B = 121 and
B = 625, respectively), any approximation for the incomplete
′
Gamma function may lead to significant error in FPopt
(P ∗ ).
The only option is to minimize the error,
and therefore we
∗
∗ m
)
apply a tighter approximation, γ(m,mP
≤
1 − e−aP
Γ(m)
−1

′
with a = mΓ(m + 1) m [4]. The bound on FPopt
(P ∗ ) is
therefore achieved by introducing this approximation in (6).

′

Further, the discrete probability component of the CDF of Popt
is determined by the condition Πi (p) =0, ∀i ∈ {1, ..., B}.
Hence, P rob Πi (p) = 0, ∀i ∈ {1, ..., B} = (1 − p)B . The
′
′
PDF of Popt , fPopt
(P ∗ ) is obtained as,

m−1
mapB(1 − p)B−1 
−aP ∗
1
−
e
1 − (1 − p)B

m B−1
∗
p 
−aP ∗
1−e
e−aP (7)
× 1+
1−p
SE is calculated using the following formula,
i
h 
′
SE = E ln 1 + ρPopt


=(1 − p)B ln(1) + 1 − (1 − p)B
Z ∞
′
×
ln(1 + ρP ∗ )fPopt
(P ∗ ) dP ∗ , (8)
′
fPopt
(P ∗ ) ≤

0
−α
λ−1
0 Gm,t Gm,r cd
σ2

where ρ =
with σ 2 denotes noise power.
The upper bound on SE is obtained by substituting (7) in (8),
Z ∞


∗ m−1
SE ≤
ln(1 + ρP ∗ )mapB(1 − p)B−1 1 − e−aP
0

 B−1
∗ m
∗
p 
× 1+
1 − e−aP
e−aP dP ∗ .
(9)
1−p
The integration in (9) can be evaluated by replacing m with
m
b = ⌊m⌋. Hence, (9) is modified into,

 
i
mi−1
bX 
mi
b −1
B
p
(1 − p)
b
ami
b
SE ≤
j
i
1−p
j=0
i=1
Z ∞
∗
× (−1)j
ln(1 + ρP ∗ )e−ba(1+j)P dP ∗
B
X

B

0


i mi−1
B  
bX 
X
B
mi
b −1
p
i
i
j
1−p
i=1
j=0
a
b(1+j)


e ρ
b
a(1 + j)
× (−1)j
,
(10)
E1
b
a(1 + j)
ρ
where E1 (.) denotes exponential integral function and
−1
c .To provide further insights into the system
b
a = mΓ(
b m
b + 1) m
design, we evaluate simplified upper and lower bounds for
SE. The upper bound on SE is derived by substituting m = 1
in (7) (equivalent to the Rayleigh assumption for |g|), i.e.,


∗
∗ B−1
pB
′
e−P
1 − pe−P
fPopt
(P ∗ ) =
B
1 − (1 − p)
∗
∗
pB
≤
exp(−λ0 e−P )e−P .
(11)
B
1 − (1 − p)
The last step in (11) is obtained from p ≈ λB0 (for large B)
B−1
followed by the relation 1 − λB0 x
≤ e−λ0 x [12].
 Ap-∗
∗
−P
plying the inequality exp(−λ0 e
) ≤ 1 − 1 − e−λ0 e−P
in (11), the upper bound on SE is evaluated using (8) as,
"

 
 #
1 − e−λ0 2
1
1
2
−
. (12)
e ρ E1
SE ≤ pB e ρ E1
ρ
2
ρ
=b
am(1
b − p)B

A closed form lower bound on SE can be derived by ignoring small scale fading for individual multi-path components.
′
Based on the aforementioned simplification, Popt becomes a

4

′

discrete random variable. Specifically, Popt = 1 with probaB
B
bility 1 − (1 − p) ; 0 with probability (1 − p) . Therefore,
lower bound on SE is determined
h as,
i
B
B
SE ≥ (1 − p) log(1) + 1 − (1 − p) ln(1 + ρ)
i
h
B
(13)
= 1 − (1 − p) ln(1 + ρ).

Interestingly, the bounds expressed in (12) and (13) can be
simplified further for highly sparse mmWave channels. Such
channels are envisaged when the transmitter-receiver distance
d is fairly large; in fact it has been reported that the number
of detectable multi-path components at the receiver decreases
with transmission distance (since the power level of most
multi-path components is below noise floor due to excessive
propagation loss at mmWave frequencies) [13]. Based on the
inequality ex E1 (x) ≤ ln(1 + x1 ) [12] and small λ0 , (12) is
approximated as SE ≤ pBln(1 + ρ) ≤ λd Bln(1 + ρ) ≤
λ0 ln(1 + ρ). Similarly,
the lower bound is approximated as

SE ≥ 1 − e−λ0 ln(1 + ρ) ≥ λ0 ln(1 + ρ), which converges
with the upper bound.
B. Computation of throughput by accounting antenna beam
training overhead
Generally, SE can be enhanced by operating with large B
which essentially increases the antenna gain. However, analog
beamforming protocols require a fixed training time (with
large B, training time increases) to identify the antenna beam
pair which maximizes SE. The training overhead reduces the
opportunity of nodes to communicate due to limited residual
duration for data transmission. This overhead is expected to
be significant for an outdoor environment since the channel
changes frequently and beamforming needs to be repeatedly
performed to discover strong multi-path components. In this
section, we quantify the TP of a link by associating antenna
beam training overhead with SE, and derive an approximate
value of optimal antenna HPBW which maximizes TP. Let
To denote the antenna beam training duration and let the total
duration due to antenna beam training plus data transmission
be T . In the present context, T can be same as the coherence
time of the channel Tc . We define TP as,


To
TP = 1 −
SE.
(14)
T
To evaluate TP for a practical network scenario, we consider
the Multiple Sector ID Capture (MIDC)1 scheme enabled
antenna beamforming protocol specified in the IEEE 802.11ad
standard [15]. In the recent past, several commercial products
compliant with the IEEE 802.11ad standard have been
released for outdoor communications [16], although the
standard was originally proposed for indoor communications.
We also note that the standard also allows nodes to employ
an antenna beam tracking mechanism to track the channel
variations due to mobility [15]. The change in direction of
arrival of strongest multi-path component is identified by
sending a channel estimation sequence appended to the data
1 The antenna beamforming protocols may also identify sub-optimal antenna
beam pairs as the solution. However, experimental evaluations confirm that
the MIDC based protocol obtains the optimal antenna beam pair with fairly
high probability [14].

frames. However, continuous antenna beam tracking results
in reduction of data transmission duration which eventually
degrades the TP of the link. Interestingly we observe that
antenna beamforming (though it requires more search time
compared to the antenna beam tracking mechanism) allows
data transmission for a longer duration in comparison to
the antenna beam tracking mechanism since it does not
require any prior knowledge of the channel. As such, a
judicial selection of analog beamforming and beam tracking
mechanism is required for communications in highly mobile
environments. However, an analysis based on this observation
is presently out of scope of this paper, and we take into
account analog beamforming only. To derive TP, we assume
same number of antenna

 √ and receiver
√ beams at transmitter
nodes (Mt = Mr = B) and To = 2 2 B + Nb2 Tf
[14], where Tf represents the transmission duration of the
control frame for antenna beam training and Nb = 4 [14].
Further, for analytical tractability, the lower bound on SE in
this section is considered with (1 − p)B = exp(−λ0 ) and
λ−1 cd−α
ρ = BK, where K = 0 σ2 . Based on this parameter
setting, TP in (14) is modified as,

 √
T 
f
T P = 1 − 2 2 B + Nb2
(1 − e−λ0 )ln(1 + BK).
T
(15)
The optimal value of B is determined by equating the first
derivative of TP from (15) to zero and hence we obtain,

 

c∗ K ln 1 + B
c∗ K
 p

1+B
1
c∗ + Nb2 , (16)
p
− 2 B
=
Ft
c∗
K B
2Tf
c∗ is the optimal value for B which can
where Ft = T and B
be found by numerically solving (16). However, it is possible
c∗ by
to derive an approximated closed form expression√for B
applying the simplification (1 + x)ln(1 + x) ≈ x x in (16)
which results√in,
p
c∗ + 2Ft B
c∗ + N 2 Ft − 1 = 0.
Ft K B
(17)
b

It is interesting to note that
pthe approximation in (16)
p leads
c∗ in (17). Therefore, B
c∗ is,
to a quadratic equation of B
r

√
√
1 − KNb2 Ft2 + KFt
−Ft +
p
c∗ =
√
.
(18)
B
Ft K
Correspondingly, the optimal antenna HPBW for the trans∗
∗
mitter and receiver nodes is determined as, θ3dB,t
= θ3dB,r
=
360
∗
θ3dB ≈ √ .
c∗
B

IV. P ERFORMANCE E VALUATION
Extensive Monte Carlo numerical simulations were performed to validate the analysis presented in the preceding
section. We set the simulation parameters as cd−α /σ 2 = 0.01,
Tf = 5 µs [15], and a variable number of antenna beam
pair B is selected. We consider the values of λ0 spanning
from 1 to 3.5 including the experimentally reported values
λ0 = 1.9 [7] and 3.3 [13]. Firstly, we compare the simulated
plot for SE and the plots for the analytically evaluated bounds
on SE in Fig. 2(a) for varying λo and arbitrarily chosen

5

3

2.5
B=625(θ3dB ≈ 14.50 )

2.5

2
Simulation
Upper bound from (10)
Upper bound from (12)
Lower bound from (13)

1.5
1

SE (bps/Hz)

2

B=625

SE (bps/Hz)

SE (bps/Hz)

2.5

1.5
1
Simulation
Upper bound from (10)
Lower bound from (13)
Based on [4],[5]

0.5

0.5

B=121

1

1.5

2

2.5

3

0

3.5

0

200

400

600

2
1.5

Simulation
Upper bound from (10)
Lower bound from (13)

1
0.5

B=121(θ3dB ≈ 330 )

0

800

5

6

B

λ0

(a)

7

8

KdB (dB)

(b)

(c)

Fig. 2: (a) Comparison of simulated plot for SE and the plots for the analytically evaluated bounds on SE for varying λ0 and m = 3.2 (b) Comparison of
simulated plot for SE and the plots for the analytically evaluated bounds on SE for varying B, m = 3.2 and λ0 = 1.9 (c) Comparison of simulated plot for
SE and the plots for the analytically evaluated bounds on SE for varying KdB with λ0 = 1.9.

330

0 11.4 dB

0
330

30
1.41 dB

300

-8.59 dB

60

300

-4.56 dB

-18.6 dB
-28.6 dB

270

15.4 dB
30
5.44 dB

-24.6 dB

270

1

∗
θ3dB
= 13.570

v=1 m/s
v=1.5 m/s
v=2 m/s
v=11.1 m/s

0.8
0.6

∗
θ3dB
= 19.880

0.4

∗
θ3dB
= 26.980

0.2
60

0

-14.6 dB
90

Fig. 2(b), the error between upper bound in (10) and the
simulated SE is reduced from approximately 8.7% to 4.6%
for B = 100 (θ3dB = θ3dB,t = θ3dB,r = 360 ) and B = 1000
(θ3dB = θ3dB,t = θ3dB,r = 11.40). We also observe that the
error due to the lower bound from (13) is as large as 15.6% for
B = 100. However, for higher values of B, the error is seen
to reduce to 6.8%, which indicates that the closed form lower
bound may also applicable for SE evaluation of optimally
beamformed links for higher values of B or equivalently while
operating with low resolution antenna beams.

TP (bps/Hz)

m = 3.2. The simulated SE plot is generated by averaging
the capacity evaluated for 105 realizations of channel. In nth realization of the channel, the capacity Cn is calculated
P
(Cn = log2 (1 + σopt
2 )) based on the criteria given in (2),
where power received
in i-th antenna beam pair is determined
P
2
as Pi = cd−α l∈Li |hl | Gm,t Gm,r . The variables card(Li )
and |hl | are generated randomly based on their respective
PDFs (as discussed in the System Model section). Further,
we choose two different values for B in the simulation. From
Fig. 2(a), the maximum error between the upper bound derived
in (10) and the simulated SE is found as approximately 7.2%
and 9.6% for B = 625 and B = 121, respectively. Moreover,
the plots in Fig. 2(a) also reveal that the upper bound from
(12) and the lower bound from (13) are tight bounds in lower
λ0 regime (for example, the lower and upper bounds show
error of 2.8% and 6.1%, respectively at λ0 = 1.25), which
indicate that the derived bounds on SE are fairly accurate for
highly sparse mmWave channels.

90

0

200

400

600

800

1000

B
Fig. 4: Effect of channel dynamics on TP

240

120
- Radiation pattern of patch antenna
- Sectored model of radiation pattern

210

150
180

(a)

240

120
- Radiation pattern of patch antenna
- Sectored model of radiation pattern

210

150
180

(b)

Fig. 3: Radiation pattern of different patch antennas with corresponding
sectored models for (a) θ3dB = 330 , (b) θ3dB = 14.50 .

Further, we compare the simulated plot for SE and the
plots for the analytically evaluated bounds on SE in Fig. 2(b)
for varying B with m = 3.2 and λ0 = 1.9. We exclude the plot for the SE bound obtained in (12) since the
corresponding error is significantly large at λ0 = 1.9 as
observed from Fig. 2(a). For comparison, we also illustrate
the result generated for the scenario where the directional
channel is simply modeled as a Nakagami-m (m = 2) random
variable, an assumption made in [4], [5]. As can be seen from

The plot for simulated SE assuming Rician distributed |hl |
is presented in Fig. 2(c). In the same figure we also plot the
bounds on SE (analytical) for varying Rician shape parameter,
KdB and λ0 = 1.9. Experimentally reported values of KdB [8]
were used in the simulation. The analytical plot corresponding
to the upper bound in (10) is generated for the values of
m determined from chosen KdB using (2.54) in [17]. An
omni-directional channel is generated in each iteration of the
simulation with Rician distributed multi-path amplitude for
adopted shape parameter KdB . Antenna radiation pattern is
then applied (based on discrete set of antenna pointing directions) to obtain optimally beamformed directional channel.
Two different patch antenna radiation patterns with antenna
HPBW, θ3dB = θ3dB,t = θ3dB,r = 14.50 and 330 depicted
in Fig. 3 are used for simulation. Correspondingly, we choose

6

B = 625 and B = 121 for the analytical plots assuming
that each node employs the same antenna HPBW (14.40
and 32.730, respectively) for communication. As shown in
Fig. 2(c), a maximum error of 8.2% and 18.5% is observed
for B = 625 and B = 121, respectively as a result of the
joint impact of the simplification of antenna radiation pattern
with sectored model and the approximations adopted for the
evaluation of the bound. We also note that the maximum error
between the simulated SE and the lower bound on SE is only
10.2% for B = 121.
Finally in Fig. 4, we plot simulated TP of a link with SE
determined using the simulation procedure used for Fig. 2(a)
and Fig. 2(b). We use T = Tc for the simulation. To the
best of our knowledge, an investigation pertaining to channel
dynamics in outdoor millimeter wave environment is unavailable in literature. Therefore, Tc is analytically determined by
assuming that the channel dynamics is only due to the motion
of transmitter or receiver node alone (related mathematical
expressions are available in [18]). In Fig. 4, we plot TP
for four scenarios. For three scenarios, one of the nodes is
assumed to be carried by a moving person and the in the last
scenario, the node is assumed to be located in a moving vehicle
(accordingly, velocity of the node varies from v = 1 m/s to
v = 11.1 m/s (= 3.6 km/hr − 40 km/hr)). As shown in Fig. 4,
TP initially increases and then reduces owing to the increasing
training overhead due to antenna beamforming. Moreover,
when velocity of the node increases, the maximum achievable
TP reduces. As evident from Fig. 4, there also exists a range of
B for which TP of the link becomes negative (corresponding
values of TP are truncated to zero in Fig. 4). We note that it
is not possible to complete the beamforming procedure within
the duration of Tc , a fact also evident from (15). Consequently,
for the nodes employing fixed antenna radiation pattern, beamforming protocol may have to choose a sub-optimal antenna
beam pairs so that those nodes are able to commence data
transmission before the channel starts changing significantly.
This effect is severe for highly mobile environments as shown
by the plot corresponding to v = 11.1 m/s, which reveals
that the identification of optimal antenna beam pairs is not
possible no matter what is the value of B. We also calculate
∗
∗
≈ √360 . The
using the formula θ3dB
approximated θ3dB
c∗
B

∗
calculated values are θ3dB
≈ 13.160, 18.320 , and 23.930,
respectively for v = 1, 1.5, and 2 m/s and the corresponding
∗
simulated values are θ3dB
= 13.570, 19.880 , and 26.980.
Thus, the analytical framework presented in Section III-B can
serve as a design tool for beamforming protocols for outdoor
mmWave communications.

V. C ONCLUSION
In this paper, we present a novel methodology to incorporate
optimal analog beamforming into the framework for evaluation
of bounds on SE of NLOS mmWave links. We establish that
the simplistic assumption of Rayleigh or Nakagami-m probability distribution for the beamformed directional channel gain
is inadequate to characterize NLOS mmWave beamformed
channels. In addition, we also investigate the effect of antenna
beam training overhead on throughput of a link, and identify
the necessary conditions for its maximization. The evaluation

of throughput based on an standard antenna beamforming
protocol reveals that for a highly mobile environment, it may
not be possible to identify optimal antenna beam pairs which
maximize SE, and as such the nodes may end up operating
with sub-optimal antenna beams.
As future work, it would be interesting to explore the
scenario where average number multi-path components per
antenna beam and the path loss exponent is considered to be
a function of the antenna beam orientation angle.
ACKNOWLEDGMENT
The author would like to thank to Mr. Shajahan Kutty for
proof reading this manuscript, and also express gratitude to
Dr. Saswati Ghosh for providing simulated antenna radiation
patterns for analysis.
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