Capacity Degradation with Modeling Hardware
Impairment in Large Intelligent Surface
Sha Hu, Fredrik Rusek, and Ove Edfors
Department of Electrical and Information Technology,
Lund University, Lund, Sweden
{firstname.lastname}@eit.lth.se.
Abstract—In this paper, we consider capacity degradations
stemming from potential hardware impairments (HWI) of newly
proposed Large Intelligent Surface (LIS) systems. Without HWI,
the utility of surface-area (the first-order derivative of the capacity
with respect to surface-area) is shown to be proportional to the
inverse of it. With HWI, the capacity as well as the utility
of surface-area are both degraded, due to a higher effective
noise level caused by the HWI. After first modeling the HWI
in a general form, we derive the effective noise density and the
decrement of utility in closed-forms. With those the impacts of
increasing the surface-area can be clearly seen. One interesting
but also natural outcome is that both the capacity and utility
can be decreased when increasing the surface-area in the cases
with severe HWI. The turning points where the capacity and the
utility start to decrease with HWI can be evaluated from the
derived formulas for them. Further, we also consider distributed
implementations of a LIS system by splitting it into multiple
small LIS-Units, where the impacts of HWI can be significantly
suppressed due to a smaller surface-area of each unit.
I. INTRODUCTION
Large Intelligent Surface (LIS) is a newly proposed wireless
communication system [1], [2] that is beyond massive MIMO
[3]–[5] and breaks the traditional antenna-array concept. As
envisioned in [1], [2], a LIS allows for an unprecedented
focusing of energy in three-dimensional space, remote sensing
with extreme precision, and unprecedented data-transmissions,
which fulfills visions for the 5G communication systems [6]
and the concept of Internet of Things [7]. The abundant signal
dimensions of the received signal with a LIS system can also
facilitate potential applications using artificial intelligence [8].
In [1], fundamental limits on the number of independent
signal dimensions are derived under the assumption of a single
deployed LIS with infinite surface-area. The results reveal that
with matched-filter (MF) processing at the LIS, the inter-
user interference of two users is close to a sinc-function.
Consequently, as long as the distance between two users are
larger than half the wavelength, the inter-user interference is
negligible [9]. In [2], fundamental limits on positioning with
LIS are also derived, and the results show that the Cramér-Rao
lower-bound (CRLB) for positioning can decrease in the third-
order of the surface-area of the LIS. These nice properties show
the potential of LIS in future wireless communication systems.
Implementing the LIS, on the other hand, is challenging and
brings many new research questions. One potential issue is the
hardware impairments (HWI) such as Tx-RF impairments [10],
[11], analog imperfectness and quantization errors [12], [13],
nonlinearity of the power amplifies [14], time and frequency
synchronization errors [15], etc. These HWI are commonly
encountered in current communication systems, but with LIS
it gets more severe since the surface-area of LIS is typically
large (for instances, using facades of buildings or long walls in
airports as LIS). Hence, the HWI can degrade the capabilities
of the LIS in practical implementations.
In this paper, we consider the HWI in analyzing the capacity
of LIS, and target at understanding of interplays among the
surface-area, the HWI, and the degradation of capacity and util-
ity. We model the HWI in the form of errors where the center
of a LIS is used as a reference point in hardware designs, and
the impairments are caused by the distance from the considered
point to the center on the LIS. Such a modeling implies that
the farer from the center, the larger HWI will present in the
signal processing unit (SPU), which collects the received signal
reached at the LIS. The HWI are modeled by a non-negative
function with using the distance r of the consider point to the
center of the LIS as a variable. With such a principle, we carry
out analytical analysis of the capacity degradation with the
HWI. We also measure the utility degradation of the surface-
area, i.e., the slope of capacity in relation to the surface-area
A, which can be used as references when implementing a LIS
system. That is, when the utility is below a certain threshold
or even negative, it is not cost effective to further increase the
surface-area.
Further, as shown in [9], compared to a centralized deploy-
ment, a LIS system that comprises a number of small LIS-
Units has several advantages. Firstly, the surface-area of each
LIS-Unit can be sufficiently small which facilitates flexible
deployments and configurations. Secondly, LIS units can be
added, removed, or replaced without significantly affecting
system design. A distributed deployment of LIS system is also
beneficial to suppress the HWI, due to the fact that the surface-
area of each LIS-Unit is smaller compared to a single LIS. We
also extensively analyze the impacts of HWI in a LIS system
that comprises an array of small LIS-Units, and show that the
HWI can be significantly suppressed, with a higher cost of
implementation complexity to build such an array.
II. NARROWBAND RECEIVED SIGNAL MODEL WITH LIS
We consider a transmission from a single-antenna user1
located in a three-dimensional space to a two-dimensional LIS
1Due to the strong interference-suppressing capability of the LIS, we
consider the case of a single user. For multiple users, as they almost do not
interfere each other [9], the analysis for each user remains the same.
ar
X
iv
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09
67
2v
1
[
cs
.I
T
]
2
3
O
ct
2
01
8
SPU
Center of the LIS
Point-element on the LIS
r
Fig. 1. An illustration of a possible implementation of the LIS system, where an SPU collects the radiation signal across the entire surface. The reference point
is the center of the LIS, and possible HWI at other points increase with the distance to the center point.
deployed on the plane z=0 as depicted in Fig. 1. Expressed
in Cartesian coordinates, the center of the LIS2 is located at
(0, 0), and the user is located at (x0, y0, z0), where z0>0 and
x0, y0 are arbitrary values. Under perfect line-of-sight (LoS)
propagation, the user transmits a symbol a to the LIS where
E{|a|2}=1.
A. Received Signal Model without HWI
Denoting λ as the wavelength and following [1], [2], the
effective channel sx0, y0, z0(x, y) at position (x, y) on the LIS
can be modeled3 as
s(x, y) =
1
2
√
z0
π
η−
3
4 exp
(
−
2πj
√
η
λ
)
, (1)
where the metric
η = (x0 − x)2 + (y0 − y)2 + z20 , for (x, y) ∈ S,
and S denotes the surface spanned by the LIS with an area A.
Based on (1), the received signal at (x, y) of the LIS is
r(x, y) =
√
Ps(x, y)a+ n(x, y), (2)
where P is the transmit power, and n(x, y) is AWGN with a
spatial power spectral density N0 across the LIS. The average
received energy stemming from a single symbol a at the LIS,
according to (2), equals
P
∫∫
(x, y)∈S
|s(x, y)|2dxdy = ζP, (3)
where
ζ =
z0
4π
∫∫
(x, y)∈S
η−
3
2 dxdy. (4)
Without loss of generality and in order to simplify the capacity
analysis, we assume the user to be located on a central
2When referring to the coordinates of points on the LIS, the z-dimension
is omitted as it is zero by default.
3Here we assume a narrowband system where the transmit time from the
user to the LIS is negligible compared to its symbol period.
perpendicular line (CPL)4, that is x0 = y0 = 0. Further, we
assume a square5 LIS with −A≤x≤A and −A≤y≤A.
Then, it can be shown that ζ equals [1]
ζ =
1
π
arctan
(
A2
z0
√
2A2 + z20
)
=
1
π
arctan
(
τ2
√
2τ2 + 1
)
, (5)
where we defined a normalized length of the LIS (by the
distance z0) as
τ = A/z0, (6)
and “arctan”denotes the inverse tangent function. This is
the array gain obtained with the LIS for any finite deployed
surface-area. If the LIS is infinitely large, i.e., A=B=∞, then
it holds that ζ = 1/2, which means half of the isotropically
transmitted power from the user is received (which is the
maximal power that can be received) by the LIS.
B. Modeling the Hardware Impairments (HWI)
Hardware impairments can result from many aspects in
the transmit and receiver chains even with calibration and
compensation techniques [8], [16]. With LIS we model the
HWI (or remaining HWI after mitigations) as a Gaussian
process, represented by a variable h(r) in the received signal,
where
r =
√
x2 + y2.
With such an assumption, the received signal (2) changes to
r(x, y) =
√
P (1 + h(r)) s(x, y)a+ n(x, y)
=
√
Ps(x, y)a+ w(x, y), (7)
where the effective noise comprising HWI equals
w(x, y) =
√
Ph(r)s(x, y)a+ n(x, y).
The zero-mean variable h(r) is Gaussian and its variance is
a non-negative function with respect to the distance r, which
4Such an ideal assumption holds when the LIS is sufficiently large or small.
For other cases, approximation techniques in [2] can be applied to carry out
the analysis based on the results obtained with the CPL case.
5Similar analysis can be carried out for other shapes of the LIS, or
alternatively, we can transfer the LIS of other shapes into a square one with
the same surface-area.
we denote as f(r) and its expressions will be discussed later.
The rationale behind (7) is that, we model the HWI at each
point on the LIS as a function of the distance to the center.
This can be due to that we use the central point of the LIS as a
reference point in hardware design, and the larger the distance
from the center, the more severe the HWI is (more difficult to
synchronize and tune the hardwares).
Taking (7) into consideration, the received power at the LIS
remains the same as (3), but the effective noise density (with
MF process6) changes to
Ñ = N0 +
z0P
4π
∫∫
(x, y)∈S
f(r)η−3dxdy
∫∫
(x, y)∈S
η−
3
2 dxdy
. (8)
The proof of (8) is given in Appendix. As the surface-area A
increases, the effective noise density may get higher, therefore
it is of interest to analyze the capacity changes with the HWI
for guiding practical implementations of LIS systems.
C. Capacity Bound with the LIS
The capacity (nat/s/Hz) corresponding to the received model
(7) equals [1]
C = log
(
1 +
ζP
Ñ
)
, (9)
where ζ is given in (5).
Definition 1. To analyze the changes of C in relation to the
surface-area, we define a utility γ (nat/s/Hz/m2) of the LIS as
γ ,
∂C
∂A
=
1
8A
∂C
∂A
. (10)
From (9), it can be shown that
γ =
P
8A(Ñ + ζP )
(
∂ζ
∂A
−
ζ
Ñ
∂Ñ
∂A
)
, (11)
where the first-order derivatives
∂ζ
∂A
=
2τ
πz0
√
2τ2 + 1 (τ2 + 1)
, (12)
and
∂Ñ
∂A
≥ 0, (13)
are yet to be discussed. In an ideal case when there is no HWI,
we have Ñ =N0 and the equality in (13) holds. In this case,
the utility γ is maximized.
To bound γ, from (11) it holds that
γ ≤
P
8A(Ñ + ζP )
∂ζ
∂A
≤
1
8Aζ
∂ζ
∂A
= γ0, (14)
6Although with HWI, the MF process is not optimal anymore since the
noise density now is not flat across the LIS, we still assume the same MF
process and use it as an approximation to the capacity due to simplicity.
10
-1
10
0
10
1
10
2
0
2
4
6
8
10
-1
10
0
10
1
10
2
10
-4
10
-2
10
0
10
2
Numerical
Fig. 2. The capacity C and utility γ of a LIS without HWI, and with settings
z0 =4, noise PSD N0 =1. The transmit power is P =20 dB (with marker
‘*’) and 30 dB (with marker ‘o’), respectively. The higher the transmit power
is, the closer the utility is to the upper-bound.
where
γ0 =
1
4z20
1
arctan
(
τ2√
2τ2+1
) 1√
2τ2 + 1 (τ2 + 1)
. (15)
As can be seen, the upper-bound in (14) is tight under two
conditions: 1) there is no HWI, and 2) N0 is negligible, i.e., in
high signal-to-noise ratio (SNR) cases. Further, for a given τ ,
the bound of utility γ0 decreases quadratically in the distance
z0. That is, when z0 increases, the length A must increase faster
than z0 in order to have the same utility of the surface-area.
D. The Decreasing Rate of the Utility
With the bound γ0 in (15), we can analyze the changes of
utility of the surface-area. Firstly, we notice that as τ →∞,
γ0 decrease to 0, which is natural since when the surface-area
is infinitely large the capacity saturates. Secondly, when the
surface-area is small, i.e., τ→0, the LIS has the highest utility
and it holds that
γ0 ≈
1
4z20
1
arctan
(
τ2√
2τ2+1
) ≈ 1
4τ2z20
=
1
A
.
Summarizing the discussions above, we can have the below
property.
Property 1. Without HWI and a small N0 (such that γ is close
to γ0), we have
γA ≈ 1 (16)
for a small surface-area A of the LIS.
We plot both the capacity and utility in Fig. 2 for cases
without HWI, where we see that the upper-bound of the utility
is tight as A or transmit power P increases. Further, Property
1 can also be clearly seen from Fig. 2, where log γ+logA ≈ 0
holds when the surface-area is small.
III. DEGRADATIONS CAUSED BY HWI
Next we discuss the degradations in capacity and utility of
the LIS systems caused by the HWI.
A. General HWI Model
Without loss of generality, we model the HWI in a general
form as
f(r) = αr2β , (17)
where α and β are non-negative, and when α=0 it holds that
f(r) = 0 which represent the case without HWI. The model
(17) is meaningful in the sense that any other form of f(r)
(with similar properties) can be approximated by its Taylor
series whose components are of the forms in (17).
B. Capacity Degradation
With the model (17), the effective noise density in (8) with
HWI equals
Ñ = N0 +
Pz0α
4π
∫∫
0≤x, y≤A
r2β
(
x2 + y2 + z20
)−3
dxdy
∫∫
0≤x, y≤A
(
x2 + y2 + z20
)− 3
2 dxdy
. (18)
Analytical analysis of (18) is possible, but the form of (18)
is rather complicated, and it is hard to obtain insights on the
impact of HWI. Alternatively, as shown in Fig. 2, we are more
interested in the case when τ is small which yields a high
utility. Therefore, we assume that A� z0, and (18) can then
be simplified into
Ñ ≈ N0 +
Pα
4πz20A
2
∫∫
0≤x, y≤A
(
x2 + y2
)β
dxdy. (19)
To further simplify the integrals, we approximate the square
shaped domain −A≤x, y≤A by a disk x2+y2≤4A2/π with
the same area. Then, (19) can be simplified into
Ñ ≈ N0 +
Pα
8z20A
2
∫ 2A√
π
0
r2β+1dr
= N0 +
4β−1PαA2β
(β + 1)z20π
β+1
. (20)
Note that, the impact of the surface-area is A2β (i.e., in the
order β of the surface-area) in (20), and the approximation is
exact when there is no HWI, i.e., α=0.
Now we can evaluate the capacity degradation with HWI. As
seen from the capacity formula (9), the capacity degradation
can be measured in terms of the received SNR losses with
HWI, which equals
σ ≈
Ñ
N0
= 1 +
4β−1PαA2β
(β + 1)z20π
β+1N0
. (21)
Under the case that β�1, it holds that
σ ≈ 1 +
PαA2β
4πz20N0
. (22)
where P/(4πz20) is the averaged power (total received power
divided by the surface-area) reached at each point on the LIS.
Further, under the extreme case β = 0, the approximation in
(22) is exact (under A�z0).
50 100 150 200 250 300 350 400
0
1
2
3
4
5
6
7
8
9
10
[
d
B
]
=0.1, =0.1
=0.5, =0.5
=0.5, =1
=1, =0.5
Fig. 3. The degradation in terms of losses of the received SNR (σ), for
different values of α and β and under the same settings as Fig.2 with P =20
dB.
In Fig. 3, we show the received SNR losses σ for different
values of α and β, where we can see that the losses are quite
significant when the surface-area increases and for large values
of α and β.
C. Utility Degradation
Based on (20), we can also compute the derivative ∂Ñ/∂A,
which equals
∂Ñ
∂A
=
β4β−
1
2PαA2β−1
(β + 1)z20π
β+1
. (23)
From (11), the degradation of the utility γ comprises two
aspects. Firstly, the slope of γ decreases as
P
8A(N0 + ζP )
−→
P
8A(Ñ + ζP )
. (24)
Secondly, except for the slope, the utility γ is also decreased
by an offset
ζ
Ñ
∂Ñ
∂A
=
ζβ4β−
1
2PαA2β−1
(β + 1)z20π
β+1N0 + 4β−1PαA2β
. (25)
Unlike the case without HWI (where the utility γ is always
positive and asymptotically decreases to 0 as the surface-area
increases), γ is negative with HWI whenever
∂ζ
∂A
<
ζ
Ñ
∂Ñ
∂A
. (26)
With equations (5), (12), and (25), the condition (26) can be
rewritten as
1
πz0
√
2τ2+1 (τ2+1)
<
ζβ4β−1Pατ2β−2z
2β−3
0
(β + 1)πβ+1N0 + 4β−1Pατ2βz
2β−2
0
.
(27)
That is, when the condition (27) holds, the utility γ is negative
and it is not beneficial to increase the surface-area A. Further,
when N0 approaches 0, the condition (27) changes to, after
some manipulations,
β >
τ2
√
2τ2 + 1 (τ2 + 1) arctan
(
τ2√
2τ2+1
) . (28)
10
-2
10
-1
10
0
10
1
10
2
10
3
0
0.5
1
1.5
2
2.5
3
3.5
4
w/o HWI
w. HWI, = =0.5
w. HWI, = =1
w. HWI, =1, =2
w. HWI, =2, =3
Fig. 4. The capacity degradation under the same settings as Fig.2 with P =20
dB with numerically compute the effective noise density in (18).
10
-2
10
-1
10
0
10
1
10
2
10
3
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
w/o HWI
w. HWI, = =0.5
w. HWI, = =1
w. HWI, =1, =2
w. HWI, =2, =3
Fig. 5. The utility degradation under the same settings as Fig.4.
Fist of all, we notice that with larger α and β, the value of
the r.h.s gets smaller and the condition (28) is easier to meet.
That is, a negative utility γ occurs with severe HWI. Secondly,
although the utility γ can be negative, when τ→∞, i.e., the
surface-area is infinitely large, the utility always converges to
zero as ∂C/∂A approaches zero. Thirdly, when τ→0, i.e., the
surface-area is very small, the r.h.s. of (28) equals 1, and the
condition for a negative utility happens is β>1 in high SNR
cases. While when τ→∞, the r.h.s. of (28) equals 0. That is,
for a sufficiently large LIS with HWI, the utility can always
be negative for any β > 0 (although it converges to zero with
an infinitely large surface-area as mentioned above), and the
capacity can be decreased if further increasing the surface-area.
Summarizing the discussions above, we have the below
Property 2.
Property 2. With the general HWI modeled in (17), the
capacity with the LIS decreases if the condition (27) is met.
That is, with HWI it is not always beneficial to increase the
surface-area.
We show some numerical results in Fig. 4 and 5 to illustrate
the capacity and utility with different HWI parameters. As can
be seen, the capacity and utility can both be severely decreased
with HWI. The larger the values of α and β are, the larger
the decrements with a sufficiently large surface-area are. More
interestingly, according to Property 2 and with conditions (27)
and (28), the capacity can even decrease with a larger surface-
area (with β = 2 and 3 in the plots). For the case α= 2 and
β = 3 (and P = 20 dB, N0 = 1, and z0 = 2 as in Fig. 4),
the turning point for β = 0 according to (27) is τ ≈ 0.3827,
where according to the simulation results in Fig. 5, the value
γ =0 corresponds to a surface-area 2.66 m2, or equivalently,
τ≈0.4077. Therefore, these two values are close, and the error
is due to our approximations (A�z0 and replacing the square
shape of the LIS by a disk) in (20) for deriving the analytical
expression.
IV. SPLITTING THE LIS INTO MULTIPLE SMALL
LIS-UNITS
To reduce the impacts of HWI when implementing a LIS
system, as can be seen from the discussions in Sec. III, it is
beneficial to split the LIS into an array of small LIS-Units.
For each LIS-Unit, there is a standalone SPU that collects the
signal, with a synthesis unit that combines the signal from
all SPUs. As for each LIS-Unit, the surface-area A is small,
the HWI can be reduced. Such a distributed implementation
of the LIS system increases the complexity and costs for
synchronizing different process units and synthesizing the
received signal from different units. Hence, it is of interest to
analyze the possible gains with such a distributed deployment
as depicted in Fig. 6.
First of all, the received signal power for the distributed
deployment remains the same as a single LIS with the same
surface-area, which is ζP . The advantage comes from the
decrement of the noise density Ñ by splitting a single LIS
into M small LIS-Units, which from (8) is upper-bounded by
Ñ ≤ N0 + Ñs
(
A
M
)
, (29)
where Ñs(A) denotes the HWI for a LIS-Unit with a length
A and and a user located at the CPL of the LIS-Unit. The
upper-bound comes from the CPL assumption, that is, the
user is located at the CPL for all LIS-Units, which is only
approximately true when A� z0. Nevertheless, from (20) it
holds that
Ñs(A) =
4β−1PαA2β
(β + 1)z20π
β+1
. (30)
Inserting (30) back into (29) yields
Ñ ≤ N0 +
1
M2β
4β−1PαA2β
(β + 1)z20π
β+1
. (31)
That is, the HWI is decreased at the order of M2β , and under
the cases that β = 0, it is not beneficial to split the LIS into
small pieces for HWI suppression.
In Fig. 7, we show the HWI decrements by splitting a single
LIS with surface-area A=16 m2 into M small LIS-Units. As
can be seen, the HWI are significantly decreased with splitting
the LIS into 3 or 5 small units. By splitting the LIS into 11
small units, the HWI are almost negligible. Note that, the larger
the β, the smaller Ñs(A) it gets when the surface-area is small.
SPU
SPU
SPU
Synthesis
Unit
Array of SPU corresponding
to small LIS-Units
Fig. 6. Splitting a LIS into a number of small LIS-Units to suppress HWI,
where each small LIS-Unit has a standalone SPU and the HWI is limited by
its small surface-area.
0 2 4 6 8 10 12 14
M
-40
-35
-30
-25
-20
-15
-10
-5
0
5
10
w. HWI, = =0.1
w. HWI, = =0.5
w. HWI, = =1
Fig. 7. The decrements of HWI by splitting a single LIS with surface-area
A=16 m2 into M small LIS-Units.
V. SUMMARY
We have considered capacity effects of hardware impair-
ments (HWI) in large intelligent surface (LIS) systems. We
have modeled the HWI in a general form in relation to the
distance from a considered point on the LIS to its center, with
the latter one used as a reference-point in hardware designs.
Without HWI, the utility of surface-area in terms of capacity is
proportional to the inverse of the surface-area for a small LIS.
However, with HWI the capacity can be significantly decreased
and we have derived closed-form expressions for both the
capacity and the utility of surface-area. With severe HWI,
increasing surface-area can decease the capacity which should
be considered in practical implementations of the LIS system.
Further, we have also considered the case with splitting a large
LIS into an array comprising a number of small LIS-Units, and
we have shown that the HWI can be greatly suppressed with
such an approach.
APPENDIX: THE PROOF OF (8)
With applying an MF (with normalization) on (7) and
integrating over the entire surface, the received signal equals
r̃ =
1
√
ζ
∫∫
(x, y)∈S
s∗(x, y)r(x, y)dxdy =
√
Pζa+
w̃
√
ζ
,
where
w̃=
∫∫
(x, y)∈S
√
Ph(r)|s(x, y)|2dxdy+
∫∫
(x, y)∈S
s∗(x, y)n(x, y)dxdy.
Since the noise and the HWI are independent and white
Gaussian variables, it holds that
1
ζ
E(w̃w̃H) = P
∣∣∣∣∣∣∣
∫∫
(x, y)∈S
h(r)|s(x, y)|2dxdy
∣∣∣∣∣∣∣
2
+
∣∣∣∣∣∣∣
∫∫
(x, y)∈S
s∗(x, y)n(x, y)dxdy
∣∣∣∣∣∣∣
2
=
P
ζ
∫∫
(x, y)∈S
f(r)|s(x, y)|4dxdy +N0,
which is the noise density in (8), after some manipulations.
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I Introduction
未标题
未标题
II Narrowband Received Signal Model with LIS
II-A Received Signal Model without HWI
II-B Modeling the Hardware Impairments (HWI)
II-C Capacity Bound with the LIS
II-D The Decreasing Rate of the Utility
III Degradations Caused by HWI
III-A General HWI Model
III-B Capacity Degradation
III-C Utility Degradation
IV Splitting the LIS into Multiple Small LIS-Units
V Summary
References