System based Code Evaluation Criteria for CDM Applications in Sensor

and Data Transmission Systems

Peter Stapf

, Marek G¨otten

1,2

, Andreas Ahrens

1 b

and Steffen Lochmann

1 c

Bereich Elektrotechnik und Informatik, Hochschule Wismar,

University of Technology Business and Design, Philipp-M¨uller-Straße 14, 23966 Wismar, Germany

Escuela T´ecnica Superior de Ingener´ıa y Sistemas de Telecomunicaci´on, Universidad Polit´ecnica de Madrid,

Technical University of Madrid, Crtra de Valenica, km 7, Madrid, Spain

Keywords:

Code-division Multiplex, Fibre Bragg Gratings, Optical Sensors, Binary Sequences.

Abstract:

To increase the multiplexing capability of code-division multiplexing (CDM) applied in optical sensor net-

works, a system based code evaluation is required. This contribution analyses evaluation criteria for sequences

applied in CDM systems. A comparison of an optical sensor application and a single user data transmission

system is presented. While a detection signal-to-noise ratio and the bit error rate are used to evaluate data

transmission systems, the proposed optical sensor application uses a modiﬁed signal-to-multiuser-interference

ratio (mSMUI). The main difference exists in the handling of interference. In contrary to data transmission,

the mSMUI requires a separation of positive and negative interferences. Both applications are simulated for

different binary sequences. While the Legendre sequence with a length of 503 chips achieves the over all best

results for the optical sensor application, the single user data transmission simulation shows no signiﬁcant

sequence inﬂuence.

1 INTRODUCTION

Optical sensors, such as ﬁbre-Bragg gratings (FBGs),

gained increased recognition due to their preferable

attributes (Jelbuldina et al., 2018), (Presti et al.,

2019). They are light, small, are immune to an elec-

tromagnetic environment and can be used together

with already known multiplexing techniques (Rajan,

2015). These can include wavelength-division multi-

plexing (WDM), code-division multiplexing (CDM),

time-division multiplexing (TDM), frequency shifted

interferometry (FSI) and optical frequency domain

refractometry (OFDR) (G¨otten et al., 2020), (Wang

et al., 2012), (Ou et al., 2017), (Kaplan et al., 2019).

Especially CDM increases the amount of FBGs that

can be evaluated in one sensor system to multiple

thousand sensors within a single ﬁbre (G¨otten et al.,

2020). This allows detailed health monitoring cover-

ing a whole smart structure with dense spatial resolu-

tion (Braghin et al., 2013), (Nawrot et al., 2017).

With the introduction of code-division multiple

access (CDMA) in a third generation communica-

https://orcid.org/0000-0001-5032-5776

https://orcid.org/0000-0002-7664-9450

https://orcid.org/0000-0002-0938-2186

tion system the common correlation receiver and the

RAKE receiver are very popular receiver designs

(Lim et al., 2006). The correlation receiver optimizes

the signal-to-noise ratio (SNR), if only white noise

superimposes the signals at the receiver input. The

Rake receiver is a special receiver for radio channels

with multipath propagation and in systems with direct

sequence spreading. It consists of several correlators

according to the number of propagation paths (Price

and Green, 1958).

This work focusses on the evaluation of quality

criteria with regard to their applicability in system op-

timization. In data transmission the detection SNR, as

the argument of the complement error function, has

proven to be a suitable criterion when optimizing data

transmission systems. The transferability to sensor-

based systems has not yet been investigated. Section

2 explains the concept of CDM in optical sensor sys-

tems. A comparison of system based code evaluation

criteria is shown in Section 3. In Section 4 a simula-

tion of a CDM interrogated optical sensor system is

introduced. A single user data transmission simula-

tion is proposed and analysed in Section 5. Both sim-

ulations examine selected binary sequences, includ-

ing Gold-, Legendre-, M- and random sequences. A

Stapf, P., Götten, M., Ahrens, A. and Lochmann, S.

System based Code Evaluation Criteria for CDM Applications in Sensor and Data Transmission Systems.

DOI: 10.5220/0010247900780085

In Proceedings of the 10th International Conference on Sensor Networks (SENSORNETS 2021), pages 78-85

ISBN: 978-989-758-489-3

conclusion is given in Section 6.

2 SENSOR SYSTEM CONCEPT

A serial ﬁbre optical sensor network can be interro-

gated as depicted in Figure 1. A broadband light

source provides light for the FBGs to reﬂect. A mod-

ulator applies a sequence consisting of ’1s’ and ’0s’.

The ’1s’ and ’0s’ represent light turned on and off.

After the reﬂection at the FBGs, the light is split into

two paths, where each reﬂected sequence is again

modulated. The ﬁrst (direct) path uses the same se-

quence, while the second (inverted) path modulates

the inverted sequence. The modulators are driven

with a speciﬁc time delay, which deﬁnes the interro-

gated FBG. A spectrometer collects the light after

the corresponding modulation step. The inverted path

spectrum is than subtracted from the direct path spec-

trum which corresponds to sequence inversed key-

ing (SIK). This results in a difference spectrum. The

corresponding peak wavelength of the chosen FBG

can be detected and evaluated. The set up can be fur-

ther expanded by using WDM. With the inclusion of

multiple different Bragg wavelengths, the number of

sensors can be increased even further (G¨otten et al.,

2020).

While CDM and WDM can be used in optical

sensor networks, they originate from data transmis-

sion applications, such as cellular and global position-

ing system (GPS) systems (Lim et al., 2006). WDM

is the optical equivalent to frequency-division mul-

tiplex (FDM) in data transmission systems. CDM

spreads a single transmitted symbol, applying a code

sequence. On the receiver side correlation is used to

regain the transmitted symbol out of a signal distorted

by multiaccess-interference (MAI). With this proce-

dure multiple data transmissions can take place over

a single channel (Price and Green, 1958).

Light Source

Modulator

Modulator Modulator

Spectrometer Spectrometer

−

Differential Spectrum

Code

∆t

INV

Figure 1: Setup of CDM Interrogation System (G¨otten

et al., 2020).

-30 -20 -10 0 10 20 30

(τ) →

τ →

(a) M-Sequence

-30 -20 -10 0 10 20 30

(τ) →

τ →

(b) Legendre Sequence

-30 -20 -10 0 10 20 30

(τ) →

τ →

-30 -20 -10 0 10 20 30

(τ) →

τ →

(d) Random Sequence

Figure 2: Cyclical autocorrelation function (ACF) obtained

applying SIK of different sequence types with a length of

31 chips.

One of the important aspects for CDM is the se-

lected sequence. Since the cross-correlation func-

tion (CCF) between sequences, for example for a set

of Gold-sequences, is minimal, data can be received

asynchronously (Gold, 1967). For the optical sensor

application a single code is used to take advantage

of the different optical path lengths for different FBG

sensors. Together with an advantageous autocorrela-

tion function (ACF) of an applied sequence, this can

be used to read out a speciﬁc sensor. Due to the differ-

ent arrival times of the reﬂected sequences, the time

delay for the second set of modulators is crucial. The

reﬂected sequence of the interrogated FBG needs to

be synchronous to the modulators, which corresponds

to the ACF peak. Hence, all interfering sensors are at-

tenuated with the value of the side lobes.

Thus, the side lobes of the ACFs of each sensor

for the optical sensor application, as well as the side

lobes of the ACFsof each data transmission have to be

investigated, as they can inﬂuence the measurement.

System based Code Evaluation Criteria for CDM Applications in Sensor and Data Transmission Systems

Therefore, different sequences shown in Figure (ﬁg-

ure 2) are evaluated.

Since binary Legendre and M-sequences provide

side lobes of -1 in their cyclical bipolar-bipolar ACF,

they are chosen for these particular CDM applica-

tions (Boehmer, 1967). By nature, incoherent light

can only be turned on and off. The unipolar ver-

sion of these sequences needs to be used. Apply-

ing SIK, their cyclical unipolar-bipolar ACF is cal-

culated, which leads to side lobes with the value zero,

as depicted in Figures 2a and 2b. Gold sequences ﬁnd

usage in CDM systems because of their good cross

correlation behaviour within a set of Gold sequences.

Their cyclical unipolar-bipolar ACF is shown in Fig-

ure 2c. Additionally, random sequences with their in-

herent good orthogonality are investigated, as well.

They rely on a statistical process, that has best cor-

relation behaviour only to its identical sequence, as

seen in Figure 2d. Gold and random sequences do not

provide side lobes with all zero values.

3 CODE EVALUATION CRITERIA

As the concept of CDM originates in data transmis-

sion applications, the applicable evaluation criteria

can be considered. The ﬁrst criterion, the detection

SNR ρ denotes the squared half worst case vertical

eye opening after the correlation U

divided by the

noise power P

applied to the channel

ρ =

−U

Int

)

, (1)

where U

Int

represents the sum of all interferences. For

an additive white gaussian noise (AWGN) channel,

is equal to the signal amplitude U

. Assuming

a time-dispersive channel, intersymbol-interference

(ISI) diminish the half vertical eye opening. Consid-

ering multiple users in a CDM system, MAI, which

is included in U

Int

, is subtracted, too. In Figure 3 the

relation between combined interferences U

Int

, ρ and

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

Int

→

Detection SNR →

Bit Error Rate →

Figure 3: Relation between combined interferences U

Int

detection SNR ρ and bit error rate (BER) P

, assuming

= 0.1V

and U

= 1V.

bit error rate (BER) P

is displayed. An optimum

of ρ = 10 and P

= 7.8e

−4

is depicted for no inter-

ferences. If U

Int

matches U

, ρ reaches zero, conse-

quently P

is at it’s maximum 0.5.

The bit error rate (BER) P

, which is the second

criterion, states the number of falsely transmitted bits,

that equal the symbols, the bit errors, divided by the

number of all transmitted bits. It is one of the basic

criteria to evaluate data transmission systems. For a

two level transmission system it can also be estimated

by applying the complementary error function (erfc)

to the previously deﬁned detection SNR ρ

erfc





. (2)

Hence, the detection SNR and the BER are suitable

criteria for a CDM transmission system.

A corresponding criterion can be considered for

the sensor application, derived from ρ. It is based

on the spectral behaviour of a sensor system and de-

picted in Figure 4. Referring to the high amount of

FBGs in an optical sensor network mentioned in Sec-

tion 1, it is inevitable that multiple sensors operate at

the same wavelength, which are distributed in differ-

ent sections. CDM distinguishes between these sec-

tions. The interrogated sensor is referred to as sig-

nal, while the rest is considered as positive or nega-

tive multiuser-interference (MUI), depending on the

applied code. To evaluate the measurand, the wave-

length shift of a sensor is analysed. Consequently,

the peak wavelength of a sensor can shift according

to the applied strain or temperature. This applies to

each single sensor in each section k in the network, so

that the interfering sensors can be spectrally shifted,

too. Out of all combinations, the worst-case is consid-

ered for the criterion. When all positive interference

MUI

is spectrally shifted and all negative interfer-

ence MUI

−

overlaps with the signal from the interro-

gated sensor s

, the signal peak is diminished and in

competition with the positive interference. The ratio

between the remaining signal peak and the positive

λ λ

MUI

−

MUI

−

Signal

Figure 4: Graphic representation of mSMUI, where P

the spectral optical power density depending on the wave-

length λ.

SENSORNETS 2021 - 10th International Conference on Sensor Networks

interference is deﬁned as a modiﬁed signal-to-MUI

ratio mSMUI

for section k. The spectral noise den-

sity N

can be added to the ratio as well, so that the

equation results in

mSMUI

− MUI

−

MUI

+ N

. (3)

Comparing the detection SNR with the mSMUI it can

be seen, that the signal U

for data transmission and

for the sensor application are diminished by U

Int

or MUI respectively. Due to the possibility of spec-

tral shifting and the need for a worst case considera-

tion, the MUI in the sensor network has to be split into

positive and negative parts. In data transmission sys-

tems this separation is not required, as interferences

affect the detection SNR only at the point of deci-

sion at the receiver side. Therefore, interferences can

be constructive or destructive for the half vertical eye

opening.

4 SENSOR SYSTEM

SIMULATION

Based on the previously, in Section 2, explained

CDM-WDM system, a simulation is programmed. It

is used to provide information about the usability of

applied sequences. The simulation is situated on the

chip-level, meaning the single logical ’1’s and ’0’s

of the binary sequence. This automatically includes

a rectangular chip shaping that is applied in the

mentioned interrogation system. The optical correla-

tion of this system comprises a multiplication step,

realized by an optical modulator and a summation

step, realized by a spectrometer. This setup needs to

be transferred into a mathematical description. The

size of the sum, how many chips are summed up, is

deﬁned by the integration time of the spectrometer

and the duration of each chip. At the beginning of

the integration time, the ﬁrst modulator starts and

the light needs to travel through the whole sensor

network before it arrives at the second modulator.

Depending on the optical path length, the reﬂected

sequence arrives with a different time delay. Besides

this sequence, the modulator is synchronized to,

interfering reﬂections arrive sooner and later. There-

fore, the modulator is driven with a sequence starting

with the beginning of the integration time. To create

the appropriate time delay, the sequence is rotated,

so that the last chips of the sequence are at the very

beginning. To ﬁll the integration time, the sequence is

repeated several times and at the end both modulators

stop with the actual last chip of the sequence. The

integration time is set so that no sequence is truncated

at the end. Instead, the modulators are driven with

logical ’0’s, to ﬁll the remaining integration time.

The simulation uses a matrix calculation to obtain the

autocorrelation results for each section of the system,

assuming the minimum distance of sensors operating

at the same wavelength equal to the chip duration.

The investigated sequence c

c consists of a certain

number of chips c

where n = 0,1,.. .,N − 1 and

N denotes the length of the sequence. The arriving

sequences with different time delays at the second

modulator are represented in the matrix S

(K×I)

where

K stands for the number of sections and I for the

integration time in chips. The integration time I

needs to be equal to or greater than the length of the

sequence N.

S =







... c

N−1

0 ... 0

0 ... c

N−2

N−1

... 0

0 ... 0 c

... c

N−1







(4)

The delay is implemented on the chip level by adding

zeros in front of the ﬁrst sequence. For each section

an additional zero indicates an additional time delay

matching the distance of sensors at the same wave-

length. The sequence itself is repeated to ﬁll the in-

tegration time I. No sequence is truncated. Instead,

zeros ﬁll the remaining columns of the matrix. Thus,

all sequences from different sections, meaning with

different time delays, are represented in the matrix S

The behavior of the second modulator is deﬁned in

matrix R

(K×I)

. Instead of the leading zeros for the

time delay, the matrix is ﬁlled with rotated parts of

the sequence. Therefore the last chips of a sequence

appear in front of the ﬁrst chip.

R =







... c

N−1

0 ... 0

N−1

... c

N−2

N−1

... 0

... c

N−1







(5)

This matrix realizes each synchronization point, thus

each possible time delay for the simulated network.

The integration time here is ﬁlled with zeros at the

end, as well. Each element in one line of the matrix S

needs to be multiplied with the correspondingelement

of each line of the matrix R

R. All these elements need

to be summed up to fulﬁll the correlation of both se-

quences. This happens line-wise and can be realized

by a matrix multiplication

A = S

S· R

. (6)

The matrix A

(K×K)

contains all correlation func-

tions for each section in its lines. All side lobes can

be assigned to the corresponding section and the ACF

System based Code Evaluation Criteria for CDM Applications in Sensor and Data Transmission Systems

L31

M31

M63

L67

L127

M127

L251

M255

L503

M511

Sequences →

mSMUI (in dB) →

30 sections

50 sections

62 sections

126 sections

250 sections

254 sections

502 sections

510 sections

Figure 5: Simulated mSMUI as a function of selected code sequences and network size (with reference to Table 1).

peaks can be found along the main diagonal. Hence,

the vector s

s contains all ACF peaks

s = (a

0,0

1,1

,.. .,a

K−1,K−1

), (7)

whereas the matrix M

(K×K)

M = A

A− diag(s

s) (8)

contains all interference between each section.

The interference of section j on section k is indi-

cated by the matrix element m

k, j

. Consequently, all

elements on the main diagonal are zero, since there

cannot be an interference of a section on itself. The

interference can be positive or negative, so that the

separate sums of all positive and of all negative inter-

ference for each section k are calculated by

MUI

∑

k, j

∀ j ∈ {0, 1,.. .,K − 1|m

k, j

> 0}, (9)

MUI

−

∑

k, j

∀ j ∈ {0, 1,.. .,K − 1|m

k, j

< 0}. (10)

The corresponding signal s

of the section k is an

element in vector s

s. Thus, signal, positive interfer-

ence and negative interference are calculated and can

be used for determining the criteria mSMUI

to evalu-

ate the sequence c

c. This mSMUI

is section and inte-

gration time dependent. The integration time deﬁnes

the number of repetitions of the sequence c

c and the

amount of zeros in the last columns of matrix S

S and

matrix R

R. Non-cyclical ACFs contain boundary ef-

fects at the beginning and at the end of the sequence.

In contrary, the cyclical ACF attained with SIK for M-

sequences and Legendre sequences contains zeros for

all side lobes. Meaning, for every additional repeti-

tion of the sequence, the signal s

increases while the

interference stays the same. Therefore, the worst-case

criterion mSMUI

can be improved by additional rep-

etitions. In the testbed they are limited by the integra-

tion time of the spectrometer. Hence, the simulation

can calculate the criterion for a few repetitions of the

sequence c

c and the results can be scaled to the corre-

sponding integration time. The scaling only applies

when a full repetition is added. Therefore, the simu-

lation calculates all mSMUI

(I)

for each section k and

for each integration time I until another repetition ﬁts

inside without truncating the sequence. Out of these

results the worst mSMUI

(I)

is considered for the eval-

uation of the sequence c

The simulation parameters are depicted in Table 1.

The chip duration T

equals 5 ns which corresponds to

a distance of 1 m between two sensors operating at the

same wavelength (G¨otten et al., 2020). The integra-

tion time is set from 13000 chips to 13000+ N chips,

where N indicates the length of the sequence. Thus,

all boundary effects resulting in interference are con-

sidered for the mSMUI. The actual integration time

can be calculated by multiplying the amount of chips

Table 1: Simulation parameters.

Parameter Value

Chip duration T

5ns

Scaling factor ×100

Integration time 13000 ... 13000+ N chips

Resulting I-time 6.5 ms ... 6.5ms+ N · T

Number of sections 30 . .. 510 sections

SENSORNETS 2021 - 10th International Conference on Sensor Networks

Symbol Chip Symbol

Symbol

Stream

CDM

Spreader

Channel

Chip

Equalizer

CDM

Receiver

Noise

Figure 6: Simulation of data transmission using a single user CDM system.

with T

that results in 6.5 µs for 13000 chips. Mea-

surements with the interrogator testbed show integra-

tion times of ∼6.5 ms. Therefore, a scaling factor

of ×100 is chosen to correspond to actual measure-

ments. The number of sections varies from 31 to 510.

It is chosen to exhaust the maximum coverage of the

investigated sequences which is one section less than

the length N. 50 serial WDM sections have already

been interrogated in the testbed. The analyzed se-

quences range from suitable Legendre sequences with

the lengths from 31 to 503 chips (L31 – L503) and

M-sequences ranging from 31 to 511 chips (M31 –

M511). In the case of M-sequences, all possible se-

quences with different generator polynomials are an-

alyzed. Gold and random sequences are not depicted

since they provide side lobes in their cyclical ACF

that lead to a very low mSMUI.

Figure. 5 depicts the simulation results. The criterion

is represented in the dB-scale, since possible nega-

tive ratios (when s

< MUI

−

) can be excluded and set

to 0 dB. The dependency of the maximum coverage

on the sequence length can be seen by the amount of

sections each sequence can handle. The scenario for

30 sections leads to results for all tested sequences.

50 sections are not supported by sequence lengths of

31 chips. The more sections, the longer the sequence

has to be. The number of repetitions of a sequence

does not inﬂuence the mSMUI. Sequence L31 pro-

vides an mSMUI of 53.45dB for 30 sections. Se-

quences M255 and L503 reach similar values, while

the rest is lower. Whereas the Legendre sequence L31

in the 30 sections scenario provides better results than

the M-sequence M31, no general rule for Legendre

and M-sequences can be found. For complete cover-

age and equal lengths, the Legendre sequence L31 is

superior to M31 and L127 is equal to M127. L503

seems to be an all-rounder for all simulated scenar-

ios. It has no major drawback for a small amount of

sections and is superior for all numbers of sections

until complete coverage of 502 sections. The inter-

rogation of such a network length suffers from other

inﬂuences that cannot be improved by the sequence

itself (G¨otten et al., 2020).

5 DATA TRANSMISSION

SYSTEM SIMULATION

The simulation of data transmission describes a sin-

gle user CDM system, since ISI is the equivalent to

the MUI in the sensor system and therefore, a bet-

ter comparability is achieved. Included are a binary

generator, a CDM spreader and receiver, a pair of

root-raised cosine ﬁlters, a chip level equalizer and a

frequency selective channel with noise inﬂuence. An

overview is provided in Figure 6. The binary gener-

ator provides a bipolar ’1’, ’-1’ symbol stream. The

CDMA spreader increases the required bandwidth, by

spreading each transmitted data symbol with a spe-

ciﬁc sequence. The CDM receiver works by corre-

lating the received signal with the original sequence.

Channel coefﬁcients and noise can be individually de-

ﬁned. The equalizer works on chip level (Darwood

et al., 2001), (Elders-Boll, 2001). After the CDM re-

ceiver, evaluation mechanisms are used to assess the

received signal. These mechanisms include a decider,

which converts the received signal back into symbols,

and a comparison of received and transmitted symbol

stream.

The channel impulse response g

(t) is given with

(t) =g

· δ(t)

· δ(t − T

)

· δ(t − 1.5T

). (11)

The channel coefﬁcients are selected with g

0.8578, g

= 0.4289 and g

= 0.2831, so that the

channel impulse response is power neutral.

The comparedcodes include M-sequences, Legendre-

sequences, Gold-sequences and random sequences

with the length of 31 and 127 chips. Furthermore,

an example of M-sequences with a length of 15 and

63 chips are tested. The noise is given through the

energy per bit to noise power spectral density ratio

10 · lg(

) with 10 dB. As this simulation is de-

scribing a single user system, the MAI, mentioned in

Section 3, is set to zero. Every simulation was done

with and without the application of a chip level equal-

izer.

System based Code Evaluation Criteria for CDM Applications in Sensor and Data Transmission Systems

M15

L31

M31

G31

R31

M63

L127

M127

G127

R127

-2

-1

Sequences →

Bit Error Rate →

without equalizer, est.

without equalizer, sim.

with equalizer, est.

with equalizer, sim.

Figure 7: Simulated BER as a function of selected code sequences with an SNR 10· log

(

) of 10 dB.

The results are shown in Figure 7. For the simula-

tions without equalizer it is shown, that the calculated

values are about half of the simulated values. This

is explained by taking into account, that the calcula-

tion assumes the worst case half vertical eye opening,

while the simulation includes all possible half vertical

eye openings. This difference changes, for the simu-

lations with equalizer, as it sets all half vertical eye

openings near to the value of 1V. It can be seen, that

over all sequences the BER P

for the respective set

ups and calculations is similar.

The channel coefﬁcient g

is multiplied by the cor-

responding side lobe of the ACF, which affects the

half vertical eye opening. This explains the differ-

ences in results for estimated values for G127 and

R127 without equalizer. Simulated results are subject

to statistical processes, such as symbol stream gener-

ation and noise application. With the inclusion of the

equalizer into the estimation, the values are the same

for all sequences up to the length 63. The half verti-

cal eye opening for these sequences is set to 1 V. As

the half vertical eye opening drops slightly for the se-

quences with a length of 127, down to 0.98V for M-,

Legendre- and Gold sequences and 0.985V for the

random sequence, P

is slightly worse for the same

amount of equalizer coefﬁcients.

6 CONCLUSION

In this work three system based code evaluation crite-

ria are introduced. The main difference between data

transmission and sensor applications is the handling

of interferences. For data transmission applications,

the worst half vertical eye opening caused by all in-

terferences is considered to estimate the criteria. The

worst case criteria for the sensor application is not a

superposition of all inﬂuences, but a separate eval-

uation of positive and negative MUI. Different se-

quences result in different mSMUI and the sequence

length deﬁnes the maximum number of sections, that

can be interrogated. While the ACF obtained by SIK

of Legendre- and M-sequences provides zero values

for all side lobes, Gold- and random sequences show

non-zero values and are therefore not applicable. The

Legendre sequence with a length of 503 chips shows

an overall best result. In contrary, in the proposed sin-

gle user CDM system the spreading sequence shows

no major inﬂuence.

ACKNOWLEDGEMENTS

This work has been funded by the German Ministry

of Education and Research (No. 13FH030PX8).

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