Achievable Rate with Correlated Hardware Impairments in Large Intelligent Surfaces
Achievable Rate with Correlated Hardware
Impairments in Large Intelligent Surfaces
Juan Vidal Alegrı́a1, Fredrik Rusek1,2
1Department of Electrical and Information Technology, Lund University, Sweden
2Sony Europe, Lund, Sweden
{juan.vidal alegria, fredrik.rusek}@eit.lth.se
Abstract—Large intelligent surface (LIS) is a new technology
yet to be developed. However, this technology is likely to be
implemented using cheap components so as to facilitate its
deployment. Therefore, we can expect a performance degradation
due to hardware impairments. In this paper, we present a model
for these hardware impairments in a receiving LIS and we
analyze their effect on the achievable rate using a simplified
receiver structure. We study the effect of the correlation structure
of these impairments by modeling this correlation as a function
of the distance between the considered points of the LIS. We
present numerical results, as well as closed form formulas, in
order to get a general intuition on the problem.
Index Terms—Large Intelligent Surface (LIS), Massive MIMO,
hardware impairments, achievable rate, Hankel transform, mis-
matched receiver.
I. INTRODUCTION
Large Intelligent Surface (LIS) [1] is gaining momentum as
a natural evolution of massive MIMO [2], [3]. LIS is already
being considered as a key technology towards the following
generations of mobile wireless communications [4], [5].
In the available literature, the concept of LIS is treated
in different ways. Some papers consider a LIS as a passive
reflecting structure able to redirect the incoming signal in an
intelligent way [6]–[8]. However, we believe that the main
advantages of LIS come from considering electromagnetically
active surfaces operating as base stations [1], [9]–[11]. There-
fore, in this paper we consider an active LIS which is able to
transmit and receive signals.
Another practical issue regarding LIS is if it is treated as a
continuous surface [1], [12] or as a discrete grid of antennas
[10]. The former can be seen as a theoretical model far from
being implementable in reality. Nevertheless, in [1] it is shown
that these two models are actually equivalent if the grid of
antennas is sufficiently dense. Therefore, in this paper we
interchange both models upon convenience. In fact, the results
of this paper remark the equivalence between both models.
The future deployment of LIS throughout large walls poten-
tially relies on the possibility of implementing it using cheap
components, which may lead to hardware impairments. These
impairments can degrade the expected performance of LIS.
The effect of these impairments in multi-antenna systems has
been widely studied in the available literature [13]–[20]. In
[13]–[15], [18] the hardware impairments are modeled as ad-
ditive Gaussian noise terms. In [15], however, a multiplicative
Gaussian noise model is also presented, which is similar to
the one considered in [17] for the reciprocity calibration in
massive MIMO. In [16] an analysis of the capacity degradation
in a LIS is studied using an uncorrelated multiplicative noise
term with variance dependent on the position on the LIS. In
this paper we model the hardware impairments at a LIS base
station using a multiplicative Gaussian noise model similar
to [15], [16], but where the noise has a specific correlation
structure dependent on the distance between the points of the
LIS considered. Note that, as specific as it may seem, our
model can apply to a wide range of receivers in which the
first and second order statistics have a fixed structure and are
the only statistics considered for receiving (the higher order
terms are ignored by the receiver).
We consider the use of a simplified receiver structure easy
to implement in a realistic scenario, but with a potential
performance loss with respect to an optimum receiver. This
receiver has an achievable rate associated to a lower bound
on the mutual information of the input output relation. The
information bounds and achievable rates with mismatched
receivers have been studied in [21]–[23]. We consider these
works and apply their insights to our specific problem.
The rest of the paper is organized as follows. The system
model is presented in section II. In section III we derive
the achievable information rate for the mismatched receiver
structure under study. In section IV we analyze the effect
of the hardware imperfections on the previous information
rate. Numerical results are presented in section V. Section VI
present the conclusions derived from this work.
II. SYSTEM MODEL
We consider the transmission from a single antenna user
located at (x0, y0, z0), with z0 > 0, to a LIS extended
throughout the xy-plane and centered at x = y = z = 0.
If we model the LIS as a continuous surface, the received
narrowband signal at any point of the LIS (x, y, 0) for a
transmitted symbol s is given by
r(x, y) = h(x, y)s+ g(x, y)h(x, y)s+ n(x, y), (1)
where h(x, y) is the channel between the user and the cor-
responding point of the LIS, n(x, y) is a zero-mean white
complex Gaussian noise realization with spectral density N0,
and g(x, y) is a degradation term associated to the hardware
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impairments throughout the LIS. Note that, even though we
are considering a continuous model in space, we use a discrete
model for the temporal dimension (for further understanding
refer to [1]). We define g(x, y) as a realization of a stationary
zero-mean complex Gaussian process with autocorrelation
function c(∆x,∆y). Assuming perfect LOS propagation con-
ditions, we have [1]
h(x, y) =
1
2
√
z0
πd3
exp
(
−
j2πd
λ
)
, (2)
where λ is the wavelength, and d is the distance between the
LIS point (x, y, 0) and the user
d =
√
z20 + (x− x0)2 + (y − y0)2. (3)
For a sufficiently large surface we can assume, without
much loss of generality, that the user is located at the central
perpendicular line (CPL) of the LIS, i.e., x0 = y0 = 0. In this
case, h(x, y) has radial symmetry, and thus can be expressed
as a function of ρ =
√
x2 + y2 as
h(ρ) =
1
2
√
z0
π(z20 + ρ
2)
3
2
exp
(
−
j2π(z20 + ρ
2)
1
2
λ
)
. (4)
It is also reasonable to say that the autocorrelation function
of the hardware imperfections between two points of the
LIS, (x, y, 0) and (x+∆x, y+∆y, 0), is equally dependent
on the distance in y and x. Therefore, c(∆x,∆y) has also
radial symmetry, and can be expressed as a function of
τ =
√
∆2x + ∆
2
y
c(∆x,∆y) = c(τ). (5)
A. Space-Sampling of the LIS
If we now consider a space-sampled version of the LIS,
with a total of M sufficiently dense samples using a square
lattice for simplicity, (1) corresponds now to
r = hs+ diag(g)hs+ n, (6)
where h is the M × 1 channel vector corresponding to the
sampled and vectorized version of h(x, y), n is the M × 1
complex white Gaussian noise vector with sample variance
σn, and g is an M × 1 vector corresponding to the sampled
and vectorized version of g(x, y). Note that, assuming that
n(x, y) is white over a huge but still limited bandwidth,
σn is approximately independent of the sampling rate for a
reasonable range. We can express g as
g = C
1
2 z, (7)
where z is a complex standard normal random M × 1 vector,
and C
1
2 is a matrix such that C = C
1
2C
H
2 is the M ×M
correlation matrix of the hardware impairments associated to
the sampled version of c(x, y). So as to simplify notation, let
us define
C̃
1
2
= diag(h)C
1
2 . (8)
Using the fact that diag(g)h = diag(h)g, (6) can be refor-
mulated as
r = hs+ C̃
1
2
zs+ n. (9)
B. Information Bounds for Mismatched Receiver Structures
We can express the mutual information of the input-output
relation defined in (9) as
I(R;S) = −ER[log(pR(r))] + ER,S [log(pR|S(r|s))], (10)
where pR|S(r|s) is the conditional PDF of the received signal
given the transmitted symbol
pR|S(r|s) =
exp
(
(r − hs)H
(
|s|2C̃
)−1
(r − hs)
)
πM det
(
|s|2C̃
) , (11)
and pR(r) is the PDF of the received signal obtained through
averaging the conditional PDF over the transmitted symbol
constellation. The uppercase notation in this case corresponds
to considering the random vector and scalar, R and S, instead
of a realization of them, r and s, respectively (the bold
uppercase notation is also used for matrices).
I(R;S) corresponds to the highest rate that can be trans-
mitted through (9) for a given transmit constellation, S.
For the sake of simplicity, we assume a complex Gaussian
constellation with power P , i.e., S = CN (0, P ), which may
not maximize I(R;S). However, apart from the difficulty of
obtaining pR(r) in closed form for the true pR|S(r|s), a
receiver structure operating close to I(R;S) would have to
operate on the basis of pR|S(r|s). In reality, simpler receiver
structures are of interest. We can define a lower bound to
I(R;S) [23]
I(R;S) ≥ Iq
= −ER[log(q(r))] + ER,S [log(q(r|s))],
(12)
where q(r|s) is a strictly positive function, and
q(r) =
∫
S
q(r|s)pS(s)ds. (13)
Apart from being a lower bound to I(R;S), Iq provides a
lower bound to the information rate achievable by a detector
operating over (9) on the basis of q(r|s) instead of the true
conditional PDF, pR|S(r|s) [21]. In this paper we focus on a
simplified receiver structure based on
q(r|s) =
1
(πσ̃n)M
exp
(
(r − hs)H(r − hs)
σ̃n
)
, (14)
which physically means that the simplified receiver sees the
hardware impairments as a rectification over the noise power,
σ̃n, instead of considering their specific structure. Other more
advanced, but computationally more expensive, methods have
been explored, but due to lack of space they are not included
in this paper.
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III. ACHIEVABLE INFORMATION RATE FOR THE
MISMATCHED RECEIVER
If we consider a mismatched receiver based on (14), we can
obtain q(r) through (13), where pS(s) in this case is
pS(s) =
1
πP
exp
(
−
|s|2
P
)
. (15)
The integral in (13) can be then solved in closed form giving
q(r) =
exp
(
−rH
(
σ̃nIM + Phh
H
)−1
r
)
πM det(σ̃nIM + Phh
H)
, (16)
where IM denotes the identity matrix of size M . We now
plug (14) and (16) into (12), and, after taking the expectation
and simplifying, we get
Iq = log
(
1 +
P‖h‖2
σ̃n
)
+
P‖h‖2
σ̃n
− P
P‖h‖4 + PhHC̃h + σn‖h‖2
σ̃2n + σ̃nP‖h‖2
,
(17)
where ‖h‖2 = hHh corresponds to the squared Frobenius
norm of h. It can be verified that if the effect of the hardware
imperfections is not considered, i.e., setting C̃ to be the zero
matrix and σ̃n = σn, the resulting Iq corresponds to the
capacity of a SIMO channel. We desire to optimize (17) with
respect to σ̃n so that we get as close as possible to the true
I(R, S), which would also give us an optimum achievable
rate by the considered receiver structure. Therefore, we define
the optimization problem
maximize
σ̃n
Iq, (18)
which can be solved in closed form giving
σ̃n,opt = P
hHC̃h
‖h‖2
+ σn. (19)
We now get the optimum Iq by plugging (19) to (17), and
simplifying, which gives
Iq,opt = log
(
1 +
P‖h‖4
PhHC̃h + σn‖h‖2
)
. (20)
The second term inside the logarithm in (20) can be seen
as the postprocessed SINR, which corresponds to the ratio
between the signal power after matched filtering, and the
interference due to hardware impairments plus the noise power
after matched filtering.
IV. ANALYSIS OF THE CORRELATION OF THE HARDWARE
IMPAIRMENTS
We will now study the effect of the hardware impairments
on the achievable rate for the mismatched receiver defined in
the previous section. If we rewrite (20) as
Iq,opt = log
1 + 1
hHC̃h
‖h‖4 +
σn
P‖h‖2
, (21)
we can now focus specifically on the interference caused by
the hardware impairments. We can define the term
SIR−1 =
hHC̃h
‖h‖4
=
hHdiag(h)Cdiag(h)Hh
‖h‖4
(22)
which captures the negative effect of the hardware impairments
on the achievable rate for the mismatched receiver. Note that
diag(h)Hh =
[
|h1|2, |h2|2, . . . , |hM |2
]T
=
(
hHdiag(h)
)T
,
which means that only the amplitude information of the chan-
nel affects the achievable rate. At this point, considering a huge
densely-sampled LIS, we can asymptotically approximate (22)
using a continuous infinite surface as
SIR−1≈
∫∫∫∫
R4 |h(x, y)|
2c(∆x,∆y)|h(x′, y′)|2dxdydx′dy′(∫∫
R2 |h(x, y)|
2dxdy
)2 ,
(23)
where ∆x = x−x′ and ∆y = y−y′. Note that in the sampled
version of h(x, y) and c(∆x,∆y) there is a normalization
factor related to the sampling, but in (22) it is canceled due
to the quotient. The denominator in (23) is evaluated to [1](∫ ∞
−∞
∫ ∞
−∞
|h(x, y)|2dxdy
)2
=
1
4
. (24)
We can express the correlation function as a 2D convolution
c(∆x,∆y)= c 1
2
(∆x,∆y) ∗ ∗c∗1
2
(−∆x,−∆y)
=
∫∫
R2
c 1
2
(x′′−x, y′′−x)c∗1
2
(x′′−x′, y′′−x′)dx′′dy′′,
(25)
where we have manipulated using variable change. Therefore,
we can express (23) as
SIR−1 ≈ 4
∫∫
R2
∣∣∣|h(x, y)|2 ∗ ∗c 1
2
(x, y)
∣∣∣2 dxdy. (26)
Applying the 2D Parseval’s theorem and the 2D convolution
theorem to (26) we get
SIR−1 ≈
4
(2π)2
∫∫
R2
|Hs(ω1, ω2)|
2
C(ω1, ω2)df1df2, (27)
where Hs(f1, f2) = F2D{|h(x, y)|2} and C(f1, f2) =
F2D{c(∆x,∆y)}, F2D{.} denoting the 2D Fourier transform.
It should be noted that, due to its underlying structure,
C(f1, f2) must always be real and positive. From (4) and (5)
we know that |h(x, y)|2 and c(∆x,∆y) have radial symme-
try. Given any 2D function with radial symmetry, w(ρ), its
2D Fourier transform has also radial symmetry and can be
expressed as [24, Ch. 17]
F2D{w(ρ)} = 2πH0 {w(ρ)} , (28)
where H0 denotes the Hankel transform of order zero. There-
fore, after simplifying, we get
SIR−1 ≈ (4π)2
∫ ∞
−∞
∣∣HH0s (ν)∣∣2 CH0(ν)dν, (29)
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where HH0s (ν) and C
H0(ν) are the zero-order Hankel trans-
forms of |h(ρ)|2 and c(τ), respectively. HH0s (ν) can be
obtained from (4) as
HH0s (ν) = H0
{
z0
4π(z20 + ρ
2)
3
2
}
=
exp(−z0ν)
4π
.
(30)
However, we don’t know much about CH0(ν) since we
don’t have an expression for c(τ). We know that CH0(ν) has
to be positive from (25), and, from the nature of the hardware
impairments, it is reasonable to expect that the amplitude of
c(τ) decreases with τ and converges asymptotically to 0, i.e.,
the amplitude of the correlation of the hardware impairments
between two points of the LIS decreases with the distance
between the two points until it gets to 0. Therefore, we have
evaluated 3 possible versions of c(τ) fulfilling the previous
requirements and leading to a closed form result of (29),
namely
c1(τ) =
a
√
a2 + τ2
(31)
c2(τ) =
a3√
(a2 + τ2)3
(32)
c3(τ) =
2a
τ
J1
(τ
a
)
, (33)
where J1(.) denotes the first-order Bessel function of first
kind. Note that the previous functions have a maximum of 1,
at τ = 0, for the sake of normalization, but they can be scaled
so as to consider different levels of hardware impairments
severity. We can see a as a variable with same distance
dimension as τ , since ci(τ) is dimensionless. We can also
relate a to the coherence distance of the interference due to
hardware imperfections in the LIS, i.e., the distance required
between two points of the LIS for their hardware impairments
interference to be considered independent. If we now compute
(29) for (31)-(33), we get
SIR−11 =
1
2δ + 1
(34)
SIR−12 =
1
(2δ + 1)2
(35)
SIR−13 =
1− exp(−2δ)(1 + 2δ)
2δ2
, (36)
where δ = z0/a. If we analyze the results for asymptotical
values of δ we get
lim
δ→0
SIR−1i = 1, i ∈ {1, 2, 3} (37)
lim
δ→∞
SIR−1i = 0, i ∈ {1, 2, 3}. (38)
We can infer that, for a fixed user position, the interference
due to hardware impairments is maximum when it is correlated
throughout the LIS, while the hardware impairments have no
harmful effect if the interference caused by them is indepen-
dent and identically distributed (IID) throughout the LIS.
V. NUMERICAL RESULTS
In Fig. 1 can be seen a plot of (34), (35), and (36) as
a function of δ. The theoretical results are compared with
a simulation of (22) from an implementation of a discrete
128 × 128 LIS with λ/2-spacing working at a frequency
of 5GHz, where the user is located at the CPL at a fixed
distance of 5/3λ. Note that, even though we can talk about
frequency and wavelength for specific implementations, it
is only the absolute distances what affect our results. The
mismatch between the theoretical and simulated results are
most likely due to the fact that the simulated LIS is finite,
which means that there might be windowing effects, together
with the fact that the space sampling might be insufficient for
some values of a.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-14
-12
-10
-8
-6
-4
-2
0
S
IR
-1
(d
B
)
Theoretical result for c
1
( )
Simulation result for c
1
( )
Theoretical result for c
2
( )
Simulation result for c
2
( )
Theoretical result for c
3
( )
Simulation result for c
3
( )
Fig. 1: Theoretical and simulated SIR−1 with respect to δ
for the different correlation functions defined. The simulated
results consider a 128× 128 LIS with λ/2-spacing at 5GHz,
and a user at z0 = 5/3λ.
We can see that, in order to minimize the interference
caused by the hardware impairments, we should minimize the
correlation of this interference between the points of the LIS
separated a distance as small as possible with respect to the
distance between the LIS and the user. This also means that,
the farther the user gets from the LIS the less harmful the
correlation between the hardware impairments interference, as
well as the effect of this interference itself, will be. Thus, in
the far field we expect this interference to be almost harmless.
VI. CONCLUSIONS
In this paper we have studied the achievable information
rate for a LIS with hardware impairments using a simplified
receiver structure. We have derived the expression for the
achievable rate and analyzed how it is affected by the cor-
relation model of the hardware impairments. We can conclude
that, within the model considered, the interference due to
hardware impairments is minimized by minimizing the ratio
between the coherence distance of this interference, i.e., the
maximum distance between two points of the LIS for their
hardware impairments interference to be notably correlated,
and the distance between the LIS and the user.
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REFERENCES
[1] S. Hu, F. Rusek, and O. Edfors, “Beyond massive MIMO: The potential
of data transmission with large intelligent surfaces,” IEEE Transactions
on Signal Processing, vol. 66, no. 10, pp. 2746–2758, May 2018.
[2] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta,
O. Edfors, and F. Tufvesson, “Scaling up MIMO: Opportunities and
challenges with very large arrays,” IEEE Signal Processing Magazine,
vol. 30, no. 1, pp. 40–60, Jan 2013.
[3] T. L. Marzetta, “Noncooperative cellular wireless with unlimited num-
bers of base station antennas,” IEEE Transactions on Wireless Commu-
nications, vol. 9, no. 11, pp. 3590–3600, November 2010.
[4] E. Björnson, L. Sanguinetti, H. Wymeersch, J. Hoydis, and T. L.
Marzetta, “Massive MIMO is a reality – What is next? Five promising
research directions for antenna arrays,” 2019. [Online]. Available:
http://arxiv.org/abs/1902.07678
[5] W. Saad, M. Bennis, and M. Chen, “A vision of 6G wireless systems:
Applications, trends, technologies, and open research problems,” 2019.
[Online]. Available: http://arxiv.org/abs/1902.10265
[6] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah,
and C. Yuen, “Large intelligent surfaces for energy
efficiency in wireless communication,” 2018. [Online]. Available:
http://arxiv.org/abs/1810.06934
[7] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless
network: Joint active and passive beamforming design,” in 2018 IEEE
Global Communications Conference (GLOBECOM), Dec 2018, pp. 1–6.
[8] M. D. Renzo and J. Song, “Reflection probability in wireless
networks with metasurface-coated environmental objects: An approach
based on random spatial processes,” 2019. [Online]. Available:
http://arxiv.org/abs/1901.01046
[9] S. Hu, F. Rusek, and O. Edfors, “The potential of using large antenna
arrays on intelligent surfaces,” in 2017 IEEE 85th Vehicular Technology
Conference (VTC Spring), June 2017, pp. 1–6.
[10] M. Jung, W. Saad, Y. R. Jang, G. Kong, and S. Choi, “Performance
analysis of large intelligence surfaces (LISs): Asymptotic data
rate and channel hardening effects,” 2018. [Online]. Available:
http://arxiv.org/abs/1810.05667
[11] S. Hu, K. Chitti, F. Rusek, and O. Edfors, “User assignment with
distributed large intelligent surface (LIS) systems,” in 2018 IEEE 29th
Annual International Symposium on Personal, Indoor and Mobile Radio
Communications (PIMRC), Sep. 2018, pp. 1–6.
[12] S. Hu, F. Rusek, and O. Edfors, “Beyond massive MIMO: The potential
of positioning with large intelligent surfaces,” IEEE Transactions on
Signal Processing, vol. 66, no. 7, pp. 1761–1774, April 2018.
[13] C. Studer, M. Wenk, and A. Burg, “MIMO transmission with residual
transmit-RF impairments,” in 2010 International ITG Workshop on
Smart Antennas (WSA), Feb 2010, pp. 189–196.
[14] E. Björnson, J. Hoydis, M. Kountouris, and M. Debbah, “Massive
MIMO systems with non-ideal hardware: Energy efficiency, estimation,
and capacity limits,” IEEE Transactions on Information Theory, vol. 60,
no. 11, pp. 7112–7139, Nov 2014.
[15] U. Gustavsson, C. Sanchz-Perez, T. Eriksson, F. Athley, G. Durisi,
P. Landin, K. Hausmair, C. Fager, and L. Svensson, “On the impact
of hardware impairments on massive MIMO,” in 2014 IEEE Globecom
Workshops (GC Wkshps), Dec 2014, pp. 294–300.
[16] S. Hu, F. Rusek, and O. Edfors, “Capacity degradation with modeling
hardware impairment in large intelligent surface,” in 2018 IEEE Global
Communications Conference (GLOBECOM), Dec 2018, pp. 1–6.
[17] J. Vieira, F. Rusek, O. Edfors, S. Malkowsky, L. Liu, and F. Tufvesson,
“Reciprocity calibration for massive MIMO: Proposal, modeling, and
validation,” IEEE Transactions on Wireless Communications, vol. 16,
no. 5, pp. 3042–3056, May 2017.
[18] S. Zarei, W. H. Gerstacker, J. Aulin, and R. Schober, “Multi-cell massive
MIMO systems with hardware impairments: Uplink-downlink duality
and downlink precoding,” IEEE Transactions on Wireless Communica-
tions, vol. 16, no. 8, pp. 5115–5130, Aug 2017.
[19] X. Xia, D. Zhang, K. Xu, W. Ma, and Y. Xu, “Hardware impairments
aware transceiver for full-duplex massive MIMO relaying,” IEEE Trans-
actions on Signal Processing, vol. 63, no. 24, pp. 6565–6580, Dec 2015.
[20] C. Shan, Y. Zhang, L. Chen, X. Chen, and W. Wang, “Performance
analysis of large scale antenna system with carrier frequency offset,
quasi-static mismatch and channel estimation error,” IEEE Access, vol. 5,
pp. 26 135–26 145, 2017.
[21] A. Ganti, A. Lapidoth, and I. E. Telatar, “Mismatched decoding re-
visited: General alphabets, channels with memory, and the wide-band
limit,” IEEE Transactions on Information Theory, vol. 46, no. 7, pp.
2315–2328, Nov 2000.
[22] F. Rusek and D. Fertonani, “Bounds on the information rate of inter-
symbol interference channels based on mismatched receivers,” IEEE
Transactions on Information Theory, vol. 58, no. 3, pp. 1470–1482,
March 2012.
[23] F. Rusek and A. Prlja, “Optimal channel shortening for MIMO and
ISI channels,” IEEE Transactions on Wireless Communications, vol. 11,
no. 2, pp. 810–818, February 2012.
[24] A. D. Poularikas, The Handbook of Formulas and Tables for Signal
Processing. Springer-Verlag Berlin Heidelberg, 1999.
563
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