Signal Estimation with Random Parameter Matrices and

Time-correlated Measurement Noises

R. Caballero-

Aguila

1 a

, A. Hermoso-Carazo

2 b

and J. Linares-P

erez

2 c

Departamento de Estad

ıstica e I.O., Universidad de Ja

en, Campus Las Lagunillas s/n, 23071 Ja

en, Spain

Departamento de Estad

ıstica e I.O., Universidad de Granada, Campus Fuentenueva s/n, 18071 Granada, Spain

Keywords:

Least-squares Estimation Algorithms, Filtering, Fixed-point Smoothing, Covariance Information, Random

Parameter Matrices, Time-correlated Noise.

Abstract:

This paper is concerned with the least-squares linear estimation problem for a class of discrete-time networked

systems whose measurements are perturbed by random parameter matrices and time-correlated additive noise,

without requiring a full knowledge of the state-space model generating the signal process, but only information

about its mean and covariance functions. Assuming that the measurement additive noise is the output of a

known linear system driven by white noise, the time-differencing method is used to remove this time-correlated

noise and recursive algorithms for the linear ﬁltering and ﬁxed-point smoothing estimators are obtained by an

innovation approach. These estimators are optimal in the least-squares sense and, consequently, their accuracy

is evaluated by the estimation error covariance matrices, for which recursive formulas are also deduced. The

proposed algorithms are easily implementable, as it is shown in the computer simulation example, where they

are applied to estimate a signal from measured outputs which, besides including time-correlated additive noise,

are affected by the missing measurement phenomenon and multiplicative noise (random uncertainties that can

be covered by the current model with random parameter matrices). The computer simulations also illustrate

the behaviour of the ﬁltering estimators for different values of the missing measurement probability.

1 INTRODUCTION

The signal estimation problem in networked systems

is usually based on measurements that are perturbed

not only by additive noises, but also by other stochas-

tic disturbances from multiple sources, which might

be caused, for example, by the degradation of the

measuring devices or the presence of multiplicative

noises. Such random disturbances are usually inher-

ent to the network itself (network-induced uncertain-

ties) and they must be taken into account in the design

of both the observation models and the estimation al-

gorithms, so as to get proper estimations. For this rea-

son, the study of the estimation problem in this kind

of systems with one or several network-induced un-

certainties has become a hot research topic over the

last years (see e.g. (Gao and Chen, 2014), (Chen

et al., 2015), (Tian et al., 2016), (Caballero-

Aguila

et al., 2017), (Zhao et al., 2018), (Liu et al., 2018) and

https://orcid.org/0000-0001-7659-7649

https://orcid.org/0000-0001-8120-2162

https://orcid.org/0000-0002-6853-555X

(Yang et al., 2019)).

The introduction of random parameter matrices in

the mathematical model of the measured outputs pro-

vides a global frame to deal with some of the most

common network-induced uncertainties (multiplica-

tive noise, sensor gain degradation or missing mea-

surements, among others). This fact is boosting the

rise of several estimation algorithms in networked

systems with random parameter matrices under dif-

ferent assumptions about the noises and the processes

involved (see e.g. (Yang et al., 2016), (Sun et al.,

2017), (Wang and Zhou, 2017), (Caballero-

Aguila

et al., 2018), (Han et al., 2018) and (Caballero-

Aguila

et al., 2019)).

Another relevant issue when addressing the esti-

mation problem is the presence of non-white additive

noises perturbing the sensor measurements. Conven-

tional estimation algorithms usually provide accurate

estimations when the additive noise is either white or

correlated on a ﬁnite-time interval. However, we of-

ten come across more general situations involving se-

quentially time-correlated measurement noise, which

is usually the output of a linear system with white

Caballero-Águila, R., Hermoso-Carazo, A. and Linares-Pérez, J.

Signal Estimation with Random Parameter Matrices and Time-correlated Measurement Noises.

DOI: 10.5220/0007807804810487

In Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2019), pages 481-487

ISBN: 978-989-758-380-3

481

noise. The design of recursive estimation algorithms

in this class of systems with time-correlated noise

is addressed, for instance, in (Liu, 2016), (Li et al.,

2017) or (Liu et al., 2017) and the time-differencing

approach is conventionally applied to deal with this

kind of noise correlation.

Bearing these considerations in mind, this paper

addresses the least-squares linear ﬁltering and ﬁxed-

point smoothing problems of discrete-time stochastic

signals under the following conditions:

Covariance Information. The state-space model gen-

erating the signal process is not necessarily known

and only information about its mean and covari-

ance functions are required in its stead.

Random Parameter Matrices. The measured outputs

can be corrupted by random uncertainties, which

are modelled in a comprehensive way by random

parameter matrices.

Time-correlated Measurement Noises. The additive

noise in the measurements is described by a

discrete-time linear model perturbed by white

noise.

The use of the time-differencing approach allows us

to transform the original measured data into an equiv-

alent set of transformed measurements, that are per-

turbed by white noise, and the problem is then re-

duced to ﬁnd the optimal estimators based on this new

set of observations. For this purpose, the innovation

approach is used and recursive algorithms are derived

for the ﬁltering and ﬁxed-point smoothing problems.

The remainder of this paper is divided into four

sections. The measurement model and the problem

formulation are described in Section 2, where the

time-differencing approach is also detailed. Section

3 discusses the derivation of the ﬁltering and ﬁxed-

point smoothing algorithms, based on the innovation

technique. Section 4 illustrates the application of such

algorithms by a numerical simulation example, where

the performance of the estimators is assessed in terms

of their error variances. Finally, some concluding re-

marks are included in Section 5.

2 PROBLEM FORMULATION

The goal of this paper is to ﬁnd a recursive algorithm

for the least-squares (LS) linear ﬁltering and ﬁxed-

point smoothing estimators of a discrete-time random

signal from noisy measurements which are perturbed

by random parameter matrices and time-correlated

additive noise vectors. The estimation problem will

be addressed using covariance information; that is, we

will assume that the evolution model of the signal to

be estimated is unknown and only information about

the mean and covariance functions of the signal and

the processes involved in the observation model are

available.

More precisely, consider the following observa-

tion model with random parameter matrices:

= H

+ v

, k ≥ 1, (1)

where, at each sampling time k, x

∈ R

is the signal

vector to be estimated, z

∈ R

is the measurement

vector at time k. The following assumptions on the

processes involved in (1) are required:

(i) {x

}

k≥1

, the signal process, has zero mean and

its covariance function is factorized as





= Λ

, s ≤ k,

where, for k ≥ 1, Λ

,Ψ

∈ R

×n

are known ma-

trices.

(ii) {H

}

k≥1

is a sequence of independent random

parameter matrices, whose entries h

k,pq

, for

p = 1, ...,n

and q = 1, . . . , n

, have known

means, E[h

k,pq

], and second-order moments,

E[h

k,pq

k,p

], for p, p

= 1,...,n

and q,q

1,...,n

(iii) {v

}

k≥1

is a zero-mean time-correlated noise

process such that

= A

k−1

+ β

k−1

, k ≥ 1, (2)

where {A

}

k≥0

are known matrices, {β

}

k≥0

is a

zero-mean white noise with covariance matrices

E[β

] = B

, and v

is a random vector with

zero mean and covariance matrix V

(iv) The random vector v

and the processes

}

k≥1

, {H

}

k≥1

and {β

}

k≥0

are mutually in-

dependent.

2.1 Time-differencing Approach

To address the LS estimation problem from the mea-

sured outputs (1) perturbed by time-correlated addi-

tive noise, such measurements are transformed with

the aim of removing the correlated noise. For that

purpose, the time-differencing approach (Liu, 2016)

is used to deﬁne a new set of measurements, whose

additive noise is not time-correlated, according to the

following model:

= z

− A

k−1

, k ≥ 2; y

= z

. (3)

Then, if we substitute (1) into the above equation and

we use (2) for v

, we obtain

= H

− A

k−1

+ β

k−1

, k ≥ 2;

= z

(4)

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

482

Remark 1. Note that, since y

, for h = 2,. . . , L, is

obtained from z

h−1

and z

, the sets {y

,...,y

} and

,...,z

} are equivalent in the sense that they can

be obtained one from the other by linear transforma-

tions. Consequently, LS linear estimator of x

based

on the original measurements {z

,...,z

} given in (1)

is just equal to that based on the new measurements

,...,y

} given in (4). So, in order to address this

estimation problem, the ﬁrst and second-order statis-

tical properties of the process {y

}

k≥1

are necessary.

Remark 2. The assumptions on the model guarantee

that the process {y

}

k≥1

given in (4) has zero mean

and its covariance matrices Σ

≡ E





, k ≥ 1, sat-

isfy

= Σ

k,k

− Σ

k,k−1

k−1

− A

k−1

k−1,k

k−1

k−1,k−1

k−1

+ B

, k ≥ 2;

= Σ

+ A

+ B

(5)

where Σ

k,s

≡ E





is given by

k,s

= E[H

] =



, s < k,

E[H

], s = k,

(6)

with H

≡ E[H

], k ≥ 1, and the (p, p

)-th entries of

the above matrices E[H

] are calculated by







∑

q=1

∑

E[h

k,pq

k,p

](Λ

)

, p, p

= 1,. . . , n

3 LS LINEAR ESTIMATION

ALGORITHMS

Given the set of measurements {y

,...,y

} in (4), the

aim in this Section is to design recursive algorithms

for the optimal estimators

k/L

within the class of lin-

ear estimators of the signal x

, based on these mea-

surements, using the LS optimality criterion. Specif-

ically, recursive algorithms for the ﬁlter,

k/k

, and

smoother,

k/k+N

, at the ﬁxed point k for any N ≥ 1,

will be obtained.

3.1 Preliminary Results

Since the LS linear estimator of the signal,

k/L

, is

the orthogonal projection of the signal x

over the lin-

ear space spanned by the observations

{

,...,y

}

which generally are non-orthogonal vectors, to ob-

tain the estimation algorithms we will use a inno-

vation approach (Kailath et al., 2000). According

to such approach, the observation process {y

}

k≥1

transformed into an equivalent one, named innova-

tion process, of orthogonal vectors {µ

}

k≥1

, deﬁned

by µ

= y

−

k/k−1

, where

k/k−1

is the orthogonal

projection of y

onto the linear space generated by

{

,...,µ

k−1

}

, with

1/0

= E[y

] = 0.

Therefore, the LS linear estimator,

k/L

, of

any random vector α

based on the observations

,...,y

}, can be calculated as a linear combina-

tion of the corresponding innovations, {µ

,...,µ

Namely, denoting Π

= E[µ

], the LS linear esti-

mator

k/L

is expressed as the following linear com-

bination of the innovations:

k/L

∑

h=1

E[α

]Π

−1

. (7)

3.1.1 Observation Predictor

Starting from expression (4) of the observations y

and taking into account that H

k−1

is correlated with

the innovation µ

k−1

, to obtain the predictor

k/k−1

the

observations y

are rewritten as follows:

= H

− A

k−1

, k ≥ 2, (8)

where W

= β

− A



− H



, k ≥ 1.

Then, according to the projection theory, we have:

k/k−1

= H

k/k−1

− A

k−1

k−1/k−1

Now, to obtain the estimator

k/k

we use the gen-

eral expression (7). Since H

is independent of

,...,µ

k−1

, it is easy to see that E[W

] = 0, for

h < k, hence, denoting W

≡ E





, from (7) we

have that

k/k

= W

−1

, k ≥ 1. So, the observa-

tion predictor

k/k−1

satisfy:

k/k−1

= H

k/k−1

− A

k−1

k−1/k−1

k−1

−1

k−1

, k ≥ 2.

(9)

Now, we derive an expression for W

= E





Since E[W

] = 0, for h < k, it is clear that W





. Using (8) for y

, we have that E









, and, from the deﬁnition of W

, we ob-

tain that W

= E





is calculated by

= −A



k,k

− H



, k ≥ 1, (10)

where Σ

k,k

is given in (6).

3.2 LS Linear Filtering Algorithm

The starting points to derive the proposed LS linear

recursive algorithm for the ﬁltering estimators,

k/k

are the general expression (7) for the estimators as

linear combination of the innovations, along with ex-

pression (9) for the one-stage observation predictor.

Also, a recursive formula for the ﬁltering error co-

variance matrices, Σ

k/k

≡ E



−

k/k

)(x

−

k/k

)



is obtained in this Section. These covariance matrices

are used to assess the accuracy of the ﬁltering estima-

tors

k/k

when the LS criterion is used.

Signal Estimation with Random Parameter Matrices and Time-correlated Measurement Noises

483

3.2.1 Signal Filtering Estimators

From the general expression (7), obtaining the signal

ﬁlter,

k/k

∑

h=1

k,h

−1

, k ≥ 1, requires calcula-

ting the coefﬁcients

k,h

≡ E





= E





−E



h/h−1



, 1 ≤ h ≤ k.

Expression (4) for y

and the separable form of

the signal covariance, speciﬁed in assumption (i), lead

easily to

E[x

] = Λ



− A

h−1



, k ≥ 2;

E[x

] = Λ

(11)

From now on, the following operator will be used

for notational simplicity:

≡ H

− A

k−1

, k ≥ 2;

≡ H

(12)

and it will be applied to the matrices ϒ

= Λ

and

= Ψ

that deﬁne the signal covariance function

(see assumption (i)).

Then, from (11) and (12), it is clear that





= Λ

, 1 ≤ h ≤ k,

and, using (9) for

h/h−1

, together with (7) for

h/h−1

and

h−1/h−1

, the following expression for the ﬁlter

coefﬁcients is obtained:

k,h

= Λ

−

h−1

∑

j=1

k, j

−1



h, j

+ A

k−1

h−1

h−1, j



−X

k,h−1

−1

h−1

, 2 ≤ h ≤ k;

k,1

= Λ

Hence, if we deﬁne a function E

satisfying

= H

−

h−1

∑

j=1

−1

−E

h−1

−1

h−1

, h ≥ 2;

= H

we obtain that the coefﬁcients X

k,h

can be expressed

as follows:

k,h

= Λ

, 1 ≤ h ≤ k.

So, by deﬁning the vectors

≡

∑

h=1

−1

, k ≥ 1,

and using (7) for

k/k

, it is clear that the signal ﬁltering

estimators are given by

k/k

= Λ

, k ≥ 1.

Now, using this expression and deﬁning

≡ E





∑

h=1

−1

, k ≥ 1,

a formula for the ﬁltering error covariance matrix

k/k

is derived. Actually, from the Orthogonal Pro-

jection Lemma (OPL), we rewrite Σ

k/k

= E





−



k/k



; then, since E



k/k



= Λ

, using

assumption (i) for E





, we have

k/k

= Λ

(Ψ

− Λ

)

, k ≥ 1.

Bearing in mind the preceding results, using the

OPL to write the innovation covariance matrix as

= E[y

] − E[

k/k−1

and taking into account that E[

k/k−1

k−1

] = H

−

, the recursive ﬁltering algorithm below can be de-

duced without trouble.

3.2.2 Recursive Filtering Algorithm

Under hypotheses (i)-(iv), the LS linear ﬁlter,

k/k

and the corresponding error covariance matrix, P

k/k

are given by

k/k

= Λ

, k ≥ 1

k/k

= Λ



− Λ



, k ≥ 1,

where the vectors e

and the matrices K

= E[e

]

are recursively obtained from

= e

k−1

+ E

−1

, k ≥ 1; e

= 0,

= K

k−1

+ E

−1

, k ≥ 1; K

= 0,

and the matrices E

= E[e

] satisfy

= H

− K

k−1

− E

k−1

−1

k−1

, k ≥ 2;

= H

The innovations, µ

, are given by

= y

− H

k−1

− W

k−1

−1

k−1

, k ≥ 2;

= y

and the innovation covariance matrices, Π

, are ob-

tained by

= Σ

−

− E

)

−W

k−1

−1

k−1



k−1

+ W

k−1



, k ≥ 2;

= Σ

The vectors y

are given in (3), and the matrices Σ

and W

are calculated by expressions (5) and (15),

respectively. Finally, the notations H

and H

are

deﬁned in (12).

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

484

3.3 LS Linear Smoothing Algorithm

The goal in this Section is to derive a recursive algo-

rithm for the LS linear smoothing estimators,

k/k+N

at the ﬁxed point k, for any N ≥ 1, as well as, a recur-

sive formula for the smoothing error covariance ma-

trices, Σ

k/k+N

≡ E



−

k/k+N

)(x

−

k/k+N

)



3.3.1 Recursive Fixed-point Smoothing

Algorithm

Under hypotheses (i)-(iv), for each k ≥ 1, the LS li-

near ﬁxed-point smoothers,

k/k+N

, N ≥ 1, are calcu-

lated by

k/k+N

k/k+N−1

+ X

k,k+N

−1

k+N

, N ≥ 1,

(13)

with initial condition given by the ﬁlter,

k/k

The matrices X

k,k+N

≡ E



k+N



are recursively

obtained by

k,k+N



− M

k,k+N−1



k+N

−X

k,k+N−1

−1

k+N−1

, N ≥ 1;

k,k

= Λ

(14)

where M

k,k+N

≡ E[x

k+N

] satisfy the following re-

cursive formula

k,k+N

= M

k,k+N−1

+ X

k,k+N

−1

k+N

, N ≥ 1;

k,k

= Λ

(15)

The ﬁxed-point smoothing error covariance matrices,

k/k+N

, are obtained by

k/k+N

= Σ

k/k+N−1

− X

k,k+N

−1

k+N

k,k+N

, N ≥ 1,

with initial condition given by the ﬁltering error co-

variance matrix Σ

k/k

3.3.2 Smoothing Algorithm Derivation

Using the general expression (7), the smoothing esti-

mators are written as

k/k+N

k+N

∑

h=1

k,h

−1

, N ≥ 1;

hence, it is clear that, starting from the ﬁlter,

k/k

, the

ﬁxed-point smoothing estimators are recursively ob-

tained by (13).

To obtain the recursive relation (14) for X

k,k+N



k+N



−E



k+N/k+N−1



, N ≥ 1, we proceed as

follows:

− On the one hand, the assumption (i) together with

(12), yield



k+N



= Ψ

k+N

− On the other, using that

k/k−1

= H

k−1

+ W

k−1

−1

k−1

it is clear that



k+N/k+N−1



= E



k+N−1



k+N

k,k+N−1

−1

k+N−1

, N ≥ 1.

Therefore, denoting M

k,k+N

= E



k+N



, expression

(14) holds and, using the recursive relation for the

vectors e

, given in the ﬁltering algorithm, the recur-

sive expression (15) for the matrices M

k,k+N

is also

straightforward.

Finally, using (13) for the smoothers

k/k+N

, the

recursive formula for the ﬁxed-point smoothing error

covariance matrices, Σ

k/k+N

, is immediately deduced,

and the smoothing algorithm is proven. 

4 COMPUTER SIMULATION

RESULTS

This section analyzes a numerical simulation example

to illustrate the application of the recursive ﬁltering

and ﬁxed-point smoothing algorithms proposed in the

current paper.

Speciﬁcally, we consider that the signal to be es-

timated is a scalar signal {x

}

k≥1

which is generated

by the following ﬁrst-order autoregressive model

k+1

= 0.95x

+ ω

, k ≥ 1,

where {ω

}

k≥1

is a zero-mean white Gaussian noise

with constant variance Var[ω

] = 0.1, for all k. The

autocovariance function of this signal is

E[x

] = 1.025641 × 0.95

k−s

, s ≤ k

which, in accordance with the assumption (i) on

the theoretical model, can be factorized in a semi-

degenerate kernel form, taking, for example,

= 1.025641 × 0.95

, Ψ

= 0.95

−k

The measured outputs of this signal are assumed

to be described by the following model with missing

measurements and multiplicative noise:

= θ

(1 + 0.9ε

+ v

, k ≥ 1,

where





k≥1

is a sequence of independent

Bernoulli random variables with constant probabili-

ties, P



= 1



= θ. Hence 1 − θ is the probability

that the signal is absent in the measurements; that is,

the missing measurement probability. The multiplica-

tive noise,





k≥1

, is a zero-mean Gaussian white

process with unit variance.

The additive noise, {v

}

k≥1

, is a zero-mean time-

correlated noise process such that

= 0.4v

k−1

+ β

k−1

, k ≥ 1,

Signal Estimation with Random Parameter Matrices and Time-correlated Measurement Noises

485

Time k

5 10 15 20 25 30 35 40 45 50

0.55

0.6

0.65

0.7

0.75

Filtering error variances, Σ

k/k

Smoothing error variances, Σ

k/k+1

Smoothing error variances, Σ

k/k+3

Smoothing error variances, Σ

k/k+5

Smoothing error variances, Σ

k/k+7

Figure 1: Error variance comparison of the ﬁlter and

smoothers, when θ = 0.5.

where {β

}

k≥0

is also a zero-mean Gaussian white

process with unit variance, and v

is a standard gaus-

sian random variable.

Finally, we assume that the processes involved in

the system model satisfy the independence hypothe-

ses assumptions imposed on the theoretical model.

The proposed algorithms have been implemented

in a MATLAB program to obtain the ﬁltering and

ﬁxed-point smoothing estimators, as well as the cor-

responding estimation error variances. Fifty iterations

of this program have been run to show the feasibil-

ity and effectiveness of the proposed estimation algo-

rithms. The estimation accuracy has been examined

by analyzing the error variances for different proba-

bilities, θ, of the Bernoulli variables modelling the

missing measurement phenomenon.

Performance of the Filtering and Smoothing Estima-

tors. The performance of the proposed LS linear es-

timators, measured by the estimation error variances,

has been assessed when θ = 0.5. The error variances

of both ﬁltering and smoothing estimators are dis-

played in Figure 1 which shows that the error vari-

ances corresponding to the smoothers are less than

those of the ﬁlters (and, consequently, the perfor-

mance of the smoothing estimators is better), as it

could be expected. From this ﬁgure it is also gath-

ered that the estimation accuracy of the smoothers at

each ﬁxed-point, k, becomes better as the number of

available observations increases.

Inﬂuence of the Probability θ. Analogous results to

those shown in Figure 1 are obtained for other values

of the probability θ that the signal is present in the

measurements. A global analysis of the ﬁltering er-

ror variances versus the probability

θ is presented in

Figure 2. From this ﬁgure, it is inferred that as θ de-

creases (and, consequently, the missing measurement

probability, 1 − θ, increases) the ﬁltering error vari-

Time k

5 10 15 20 25 30 35 40 45 50

Filtering error variances

0.4

0.5

0.6

0.7

0.8

0.9

θ = 0.1

θ = 0.2

θ = 0.3

θ = 0.4

θ = 0.5

θ = 0.6

θ = 0.7

θ = 0.8

θ = 0.9

Figure 2: Filtering error variances versus θ.

ance becomes greater and, hence, as expected, worse

estimations are obtained. Similar results are deduced

for the ﬁxed-point smoothers.

5 CONCLUSIONS

Based on the LS optimality criterion, optimal algo-

rithms have been designed for the linear ﬁltering and

ﬁxed-point smoothing estimators of discrete-time ran-

dom signals using measured outputs with random pa-

rameter matrices and time-correlated additive noise.

The use of random parameter matrices yields a widely

applicable measurement model, which is suitable to

cope with different network-induced uncertainties in

the sensor measurements. The additive measurement

noise is assumed to obey a dynamic linear equation

corrupted by white noise. As it is usual in these

situations, the time-differencing approach has been

adopted to deﬁne, at each sampling time, an equiv-

alent measurement, which is a linear combination of

two consecutive measurements. After this transfor-

mation, we get an equivalent set of measurements,

where the time-correlation of the noise has been elim-

inated. Since the LS estimators of the signal based on

the original set of observations is equal to that based

on the new set of transformed observations, the orig-

inal estimation problem is simpliﬁed and reduced to

that of designing recursive algorithms for the ﬁlter-

ing and ﬁxed-point smoothing estimators of the sig-

nal based on the transformed measurements. The al-

gorithm design has been carried out by an innova-

tion approach and without requiring the knowledge

of the signal evolution model, but only the ﬁrst and

second-order moments of the processes involved in

the measurement model. To conclude, some com-

puter simulations have shown how the proposed al-

ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics

486

gorithms are applicable to some common engineering

problems, involving missing measurements and mul-

tiplicative noise, which satisfy the system model un-

der consideration.

ACKNOWLEDGEMENTS

This research is supported by Ministerio de

Econom

ıa, Industria y Competitividad, Agencia

Estatal de Investigaci

on and Fondo Europeo de

Desarrollo Regional FEDER (grant no. MTM2017-

84199-P).

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