1 Joural of Commucatos Vol. 9, No., October 4 Imroved ult-chael Bld De-Covoluto Algorthm for ear Covolved xture DS/CDA Sgals Hao Cheg, Na Yu, ad Ju u ...

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Improved Multi-Channel Blind De-Convolution Algorithm for Linear Convolved Mixture DS/CDMA Signals Hao Cheng1, Na Yu1, and Jun Liu2 1

2

Telecommunication Engineering Laboratory, Chengdu University National Key Lab. of Communication, University of Electronic Science and Technology, Chengdu, China Email: [email protected]; [email protected]; [email protected] present the technique of DSSS and explain the difficulty to recover the data in an unfriendly context. The algorithm is particularly important in a non-cooperative condition of the military field. This algorithm can decode the local spread spectrum sequence, so as to monitor, interference, and decipher the enemy signal. A more complex multi-channel blind de-convolution algorithm is required to achieve better source separation [2]. We propose a method to estimate the pseudo-random sequence without prior knowledge about the transmitter. Only the duration of the pseudorandom sequence is assumed to have been estimated. (This can be done by the method proposed in [3]). This method is based on minimizing the average squared cross output channel correlation criterion. There are many ways to construct a numerical algorithm based on the above criterion for blind source separation. We derive here a new stochastic gradient algorithm, which has these characteristics for the optimization of the de-mixing matrix D . Experimental results are given to illustrate the performances of the method and show that a good estimation can be obtained. This paper is organized as follows. In Section II, the fundamental models and assumptions of MIMO blind deconvolution is introduced. In Section III, we give the DS/CDMA data model and separation criterion. Then, the proposed approach is described in Section IV. Finally, experimental results are provided to illustrate the approach in Section V and a conclusion is drawn.

Abstract—In this paper, an improved multi-channel blind deconvolution algorithm for code division multiple access signals is proposed. Through minimizing the average squared crossoutput-channel-correlation, the proposed scheme achieves a better source separation performance for the linear convolved mixture system. Unlike most of the existing linear convolved mixture separation algorithms in the literature, the improved algorithm can obtain the original users’ signal even in low signal-to-noise ratio in a non-cooperation scenario. Simulation results show that the proposed optimal scheme not only achieves superior performance on the bit error rate to those of the existing ones, but also provides a convergent solution under different channel conditions. Index Terms—CDMA signal, non-cooperation, blind deconvolution, gradient algorithms

I.

INTRODUCTION

Spread spectrum signals have been used for the secure communications for several decades. Nowadays, they are also widely used outside the military domain, especially in Code Division Multiple Access (CDMA) system. They are mainly used to transmit at a low power without interference due to jamming to others users or to multipath propagation. The CDMA techniques are useful for the secure communication while the receiver has to know the sequence used by the transmitter and recover the transmitted data using a correlator [1]. Direct sequence spread spectrum (DSSS) transmitters use a periodical pseudo-random sequence to modulate the base-band signal before transmission. In the context of spectrum surveillance, the pseudo-random sequence used by the transmitter is unknown as well as other transmitter parameters such as duration of the sequence, symbol frequency and carrier frequency. Hence, in this context, a DSSS signal is very difficult to detect and demodulate, because it is often below the noise level. In this paper, our motivation is to determine the spreading sequence automatically in a multi-user CDMA system, whereas the receiver has no knowledge of the transmitter’s pseudo-noise (PN) sequence. We also

II. MODELS FOR MIMO BLIND DE-CONVOLUTION [4] A. Models for MIMO Convolution A multi-channel linear time invariant (LTI), discrete time dynamical system can be described in the most general form as:

x( k )

(1)

p

where H p is an n m dimensional matrix of mixing coefficients at time-lag p (called the impulse response at time-lag p ), and s(k - p) is an n-dimensional vector of source signals with mutually independent components. It should be noted that the causality in time domain is satisfied only when H p 0 for all p 0 . x(k ) is the received signal (sensor signals) after convolution mixed, and x(k ) [ x1 (k ), x2 (k ), L, xn (k )]T is an n dimensional vector of the outputs.

Manuscript received June 10, 2014; revised October 20, 2014. This project was supported by the National Natural Science Foundation of China (61271168) ; the National Natural Science Foundation of China (20578012); the Specialized Research Fund for National Key Lab of Communication (45634020105ZS92) Corresponding author email: [email protected] doi:10.12720/jcm.9.10.785-791 ©2014 Engineering and Technology Publishing

H p s( k - p )

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Journal of Communications Vol. 9, No. 10, October 2014

discrete time series) transformation in the complex frequency domain. With the symbol W(z) , we can better

v(k) y(k)

+

s(k)

+ H(z)

understand the convolution signal than Wp . x p is

x(k) W(z)

another expression of x(k - p) , there is no variable k in

W(z) ,so we use the symbol x p instead of x(k - p) . In some other applications, a non-causal (doubly-finite) feed-forward multi-channel de-mixing matrix

(a) x1(k)

W11

y1(k)

L

W( z ) Wp x p

W12

(4)

p K

where K and L are two variables of positive integers. The global transfer function G( z ) is defined by

W1m

W21

x2(k)

W22

G( z) W( z)H( z)

y2(k)

where H( z ) is the Z-transformation of H p , and H p is the channel impulse response for time domain. We can obtain form (6), H( z ) is the channel impulse response for frequency domain, can also be called channel filter.

W2m

Wn1

H( z )

Wnm

In order to ensure that the mixed signal is recoverable, we put the following constraints on the convolute and mixed signal. The channel impulse response function H( z ) is stable, i.e., its impulse response satisfies the absolute summability condition:

(b)

The goal of the multi-channel blind de-convolution is to estimate source signals using the received sensor signals x(k ) only and certain knowledge of the source signal distributions and statistics. In the most general case, we attempt to estimate the sources by employing another multi-channel, LTI, discrete-time, stable dynamical system (Fig. 1 (a) and (b)) described at time domain:

p

(6)

where z p is the expression of H p at time lag p .

yn(k)

Fig. 1. Illustration of the multi-channel deconvolution models: (a) Functional block diagram of the feed-forward model and (b) Architecture of feed-forward neural network (each synaptic weight W is an FIR or stable IIR filter

y (k )

H p z p

p

Wn2

xm(k)

(5)

Wp x(k - p)

x(k ) [ x1 (k ), x2 (k ), , xn (k )]T acts as an n dimensional input signal vector while x(k ) is an outputs in (1). In the practical applications, we have to implement the blind de-convolution method with a finite impulse response (FIR) multi-channel filter with a matrix transfer function:

where || || denotes the Euclidean norm. The channel impulse response transfer function H p is a full rank on the unit circle ( H p 1 ), that is, it has no zeros on the unit circle.

even in a low SNR. If the estimation of Tc and f 0 are available, we can process the signals in the base band and the sampling period can be set to Tc . In the Section V, we will demonstrate that is available. In the reference [3], the symbol duration Ts can be estimated using fluctuations

L

(3)

p 0

where W(z) is the Z-transformation of Wp . The Ztransformation means the time domain signal (i.e., ©2014 Engineering and Technology Publishing

(7)

p

B. DS/CDMA Data Model’s Notations and Hypotheses Base on the model of Section II, the goal of this paper is to recover the PN sequence s k when user’s data symbol and user’s fading coefficient are both unknown, and no training samples are available either. The following assumptions will be in effect throughout the rest of the paper. We assume the parameters f 0 , Tc and Ts are known in advance. The more detailed estimation algorithms are provided in the literature [3], [5], [6]. In the reference [5] and [6], the method based on cyclo-stationarity analysis can estimate the chip period Tc and carrier frequency f 0

(2)

where y(k ) [ y1 (k ), y2 (k ), , yn (k )]T is an n dimensional vector of the outputs after the novel separation algorithm, and Wp is an n×m dimensional coefficient matrix at time lag p . Wp can be called de-mixed matrix, too.

W( z ) Wp x p

|| H p ||22

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Journal of Communications Vol. 9, No. 10, October 2014

where G 0 and G1 are C K dimension mixing matrices corresponding to the original and the one time unit delayed symbols. The column vectors of G 0 and G1 as follow:

of correlation even in –10dB (that is enough low to our method in the paper). We can adapt the DS-CDMA data model as follow. The received CDMA signal r (t ) can be written in the reference [7], [8]: M

K

L L L G 0 aklm g1l ' , aklm g 2l ' , , aklm g Kl ' l 1 l 1 l 1 (13) L L L G1 aklm g1l '' , aklm g 2l '' , , aklm g Kl '' l 1 l 1 l 1 We notice from (12) that our signal model can be regarded as a linear mixture of the delayed and convolved sources, in the particular case when the maximum delay is one time unit. If we have known neither the mixing matrices nor symbol sequences, then we are in the framework of blind sources separation. The equation (12) is the general form of the BSS problem.

L

r(t ) aklmbkmsk (t mT d kl ) n(t )

(8)

m 1 k 1 l 1

where M symbols are sent to K users and received via L path. aklm is the amplitude (fading) coefficient of lth path of the kth user during the mth symbol duration, bkm is the kth user’s mth data symbol. s k is the user’s chip sequence:

sk (t ) 1, 1 , t [0, T ]; sk (t ) 0, t [0, T ] d kl is the delay of kth user's lth path. The delays are different for different users, and each delay is assumed to change sufficiently slowly for most of the time. n(t ) denotes noise. The signal is then sampled by chip rate. After sampling the samples are collected into vectors of suitable dimension. This depends on the delay spread. We collect C-length column vectors from subsequent discrete data samples vector rm :

rm [r (mC) r (mC 1)

r (mC C 1)]T

III. ADAPTIVE ALGORITHM We use the same basic insight in this paper, but propose a new criterion to exploit it, which leads to a simple and convenient algorithm. The object function is defined to minimize the risk function J :

(9)

where rm is the discrete time domain expression of the

r (t ) in (8), and C is the chip sequence length. Equation (9) can further be written with respect to the current symbol vector and the immediately preceding one: K

L

K

L

k 1

l 1

k 1

l 1

rm (t ) [bk , m 1 aklm g kl ' ] [bk , m aklm g kl '' ] nm (10) where n m is a C-length column vectors and it is the expression of n(t ) in discrete time domain, and denotes the noise vector. g kl is the ‘early’ parts of the code vectors and g kl '' is the ‘late’ parts of the code vectors.

g kl ' sk [C dkl 1], sk [C dkl 2], , sk [C ] 0, ,0

T

g kl '' 0, ,0, sk [1], sk [2], , sk [C dkl 1], sk [C d kl ]

T

(11)

where contains d kl “0” in the vectors g kl and g kl '' . In this paper, we assume that the codes vectors g kl g kl are unknown. The approach exploits directly the binary form of the sources. The proposed algorithms yield good results in test examples with CDMA data. Due to the neural structure of the algorithms, they can be used for tracking purposes when the parameters are slowly changing. Hence (10) could be written as follows

M M M (14) J E ry2i y j i (ryi yi i2 )2 i 1 j 1 i 1 j i where the variable vi is the convergence coefficients, and variable δi is the power factor, yi and yj are the channel’s output vectors. ryi y j (n) is the correlation function of yi and yj. h(n) [ yi (n) y j (n)] ryi y j (n) (15) h( k ) k

where the operator " " is the convolution representation. h(k ) and h(n) are the low-pass averaging filter vectors to compute a short-term estimate of the cross-correlation of output channels yi and yj at time n . There are a lot of numerical algorithms based on the above criterion for the non-stationary source separation problem for the equation (14). We use a new stochastic gradient algorithm here. For the optimization of the demixing matrix D(n) , a stochastic gradient [9] update takes the form

D(n 1) D(n) (n)

where (n) is the differential function of J (n) , and is the convergence coefficient 0 1 . J (n) D(n) J (n) pq (n) d pq (n) ( n)

rm G 0 bk , m G1bk , m 1 n m 1

G i bk , m i n m

(12)

i 0

, i 0

©2014 Engineering and Technology Publishing

(16)

787

M M M 2 2 2 ( ) [ ( ) ] r n r n y y i yi y j i d pq (n) i 1 j 1 i j i 1 jj ii

(17)

Journal of Communications Vol. 9, No. 10, October 2014

In evaluating the algorithms, two parameter matrices, G 0 and G1 are chosen (randomly) as the convoluted mixing matrices. Both sources are zero-mean and have unit variances. A Gaussian vector noise process is added to the observation with noise covariance 2 I . The Signal-to-Noise Ratio (SNR) is then defined as

where D(n) is the discrete time domain of the de-mixing matrix D and its dimension p and q are the row and column indices gradient matrix that is designed by the number of users, and J (n) is the nth step criterion function J in (14). Note the use of the instantaneous value at time n of the error function in (15) gradient computation. The ( p, q)th element of the instantaneous gradient matrix can easily be shown to be [10]

SNR 10log10

M

2

(dB)

(21)

where 2 is a covariance factor, and I is a unit matrix

pq (n, l ) 4 ry p yi (n)ryi xq (n, l ) ... i 1 i p

1

(18)

1 0 0 I 0 1 0 0 0 1

2[ry p y p (n) ]ry p xq (n, l ) 2 p

The above matrix expression for the stochastic gradient update yields a straightforward computation method once the short-time correlations are available. We now derive efficient recursive updates form for the averaging filter. For the computational efficiency, we select a first-order IIR averaging filter with impulse response

h(n) an u(k ) 0 a 1

In this section, we evaluate the performance of the adaptive algorithm by some heuristic arguments and simulation results. B. Results Analysis As the CDMA signal exhibit cyclo-stationary, we utilize the spectral correlation method in the reference [11] to get the carrier frequency f 0 and the chip period Tc . Fig. 2 shows the measured spectral correlation function magnitude for the 2-users’ CDMA system. Form Fig. 2, the first peak value coordinate appear at

(19)

where u (k ) is the unit step function, and a is a constant 0 a 1 . With this form, the elements of (1 a) ryy can easily be updated recursively according to

the frequency 1104 Hz, according the definition of cyclic-correlation, we can get the carrier frequency at f0 12 104 Hz , the carrier frequency f 0 has been

ry p yi (n) ary p yi (n 1) (1 a) y p (n) yi (n) 1 p, i M

(22)

(20)

ry p xi (n) ary p xi (n 1) (1 a) y p (n) xi (n)

estimated. The second peak value appear at the both sides of each main peak, and the coordinates is 9000 Hz (that is not distinct in Fig. 2) and 11000 Hz , so we can

1 p, i M ry p yi (n) and ry p xi (n) are the correlation functions

1 s . We can process the obtain the chip period Tc 1000

between y p yi and y p xi .

aboriginal signals in the base-band and set the sampling period to Tc . Using the fluctuations of correlation method, we obtain the Ts 0.0311 s (aberrancy near upon 0.3%). So the PN

This completes the simple recursive algorithm for non stationary blind source separation as below. Step 1: Compute output according to (14). Step 2: Update short-time correlations using (20). Step 3: Compute separation filter gradient using (18). Step 4: Update separation matrix using (16). Step 5: Go back to step 1.

code length can be known as C Ts Tc 31 .

IV. PERFOMANCE EVALUATION AND SIMULATION A. Simulation Parameter Setup The minimizing average squared cross-output-channelcorrelation algorithm is demonstrated here via computer simulation, for a DS/CDMA signal received in the presence of various levels of white Gaussian background noise. In the experiment, we considered a 2-users CDMA system. The source signal was generated by the computer and the signal parameters were set as follow: carrier frequency f0 5000 , chip period Tc 1/1000 , symbol

Fig. 2. Spectral correlation function magnitude for the 2 users’ CDMA system.

period Ts 31/1000 (PN code length C 31 ), users number K 2 , path L 2 , SNR 2 (the SNR is higher than assumption 1 and assumption 2). ©2014 Engineering and Technology Publishing

For the aim of observation, we only display 800 points in Fig. 2. 788

Journal of Communications Vol. 9, No. 10, October 2014

Fig. 3 shows the 2-users’ CDMA signal sources with AWGN channel (SNR=3).

With the comparison of the results in the Fig. 3 and Fig. 5, the estimated data after separation is similar to the sources signal. In this experiment, the all symbols were correctly estimated, which shows the data are received correctly. According the assumption before, we have estimated the PN code length that becomes very simple to seek the 2-users’ PN code. We adopt traditional slippage correlation method or matrix Eigen analysis techniques to resolve the problem. In the paper, we use the slippage correlation method, and get the 2-users’ PN code as Fig. 6.

Amplitude(dB)

2 1

0

-1

-2 7200

7300

7400

7500

7600

7700

7800

7900

8000

7300

7400

7500

7600

7700

7800

7900

8000

Amplitude(dB)

2 1

0

-1

-2 7200

Sampling time(ms)

Fig. 3. 2-users’ CDMA signal sources with AWGN channel

Amplitude(dB)

2

1

0

-1

-2 7200

7300

7400

7500

7600

7700

7800

7900

8000

7300

7400

7500

7600

7700

7800

7900

8000

Fig. 6. The 2-users’ PN codes

Amplitude(dB)

2

1

0

-1

-2 7200

Sampling time(ms)

Fig. 4. The 2-users’ data after convolute mixing 2

Amplitude(dB) Amplitude(dB)

2 1

1 0

0

Fig. 7. Adaptation of the coefficient in the de-mixing matrix

-1

D

-1 -2 7200 -2 7200

10

7300

7400

7500

7600

7700

7800

7900

8000

7300

7400

7500

7600

7700

7800

7900

8000

9 8

Fig. 5.

7

2 1

6

BER (%)

Amplitude(dB) Amplitude(dB)

2

1 0

4

0 -1 -1 -2 7200 -2 7200

5

3 2

7300

7400

7500

7300

7400

7500

7600

7700

Sampling time(ms) 7600

7700

7800

7900

8000

7800

7900

8000

1 0

Sampling time(ms) The output data after separation

0

5

10

15

SNR

Fig. 4 shows the 2-users’ data after convolute mixing. Two unknown GOLDEN sequences are used as the spreading codes. The output data after separation between 7200~8000 is shown in Fig. 4. The unitary output binary estimated data after separation between 7200~8000 is shown in Fig. 5. ©2014 Engineering and Technology Publishing

Fig. 8. BER for convoluted mixture CDMA signals separation

The self-adaptive coefficients in the de-mixing matrix D are shown in Fig. 7 when the 0.9 . The experiment result of the bit-error-rate (BER) to SNR is shown in Fig. 8. 789

Journal of Communications Vol. 9, No. 10, October 2014

C. The Influence of J,, and

Second, the least value of affects the convergence limit of risk function. The value of power factor is coherent with the risk function convergence limit. We test 10000 samples for 3 times under the same condition, and get the risk function J over time, as Fig. 10. The shape and energy factor's value indicate the influence which energy factor has on the algorithm. Convergence factor v also exist the similar influence like energy factor as Fig. 11. In the algorithm, we choose the low pass filter (19). The aim is to abbreviate the algorithm complexity and average the de-mixing output data. Under the same condition, we get the risk function J over , as Fig. 12. The conclusion has been received that the value of is ager, the convergence rate is slower but stabilization and is smaller; the convergence rate is faster but unstable.

The risk function J fluctuation with the sample serial n is shown in the Fig. 9. The recursion operation after about 3000 point, the risk function J is convergent. The recursion operation after about 6000 point, the risk function J is near to 0. The convergence rate is in accordance with the adaptation of the coefficient in the de-mixing matrix D as Fig. 7.

Fig. 9. Fluctuation of risk function J

Fig. 12. Variety filter parameter influence on the

J

V. CONCLUSIONS Fig. 10. Varity power factor influence on the

J

Fig. 11. Varity power factor

J

v

influence on the

A blind source separation algorithm for the DS-CDMA signals is developed and demonstrated in this paper. By minimizing the average squared cross-output-channelcorrelation, a criterion function J is proposed to estimate the spreading code. This technology not only can be used in military for CDMA signal interception, also can be used in a commercial anti-jamming for CDMA signal. The experiment results show that the proposed algorithm is a promising alternative to the existing BSS algorithms for DS/CDMA signals. This algorithm is also applicable to signals with long code, such as commercial communication signals, which allow it to be implemented with a reasonable computational complexity and over reasonably short reception intervals. This method is also a particular applicable estimation in downlink signal processing, because the codes of the interfering users are unknown. It is capable to separate the signals and remove the multi-path channel effect.

We also find the impacts on de-mixing matrices about convergence factor v and the power factor . The power factor exist dual effects on the de-mixing matrices. First, when the value of is lager, the convergence rate is faster but un-stabilization [12]. From a contrasting point of view, the convergence rate is slower but stable. ©2014 Engineering and Technology Publishing

REFERENCES [1] T. C. Yang and W. B. Yang, “Low signal-to-noise-ratio underwater acoustic communications using direct-sequence

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[2]

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[6]

[7] [8]

[9]

[10]

[11]

spread-spectrum signals,” in Proc. OCEANS 2007 – Europe, Aberdeen, 2007, pp. 1-6. J. M. Yang and G. K. Hong, “Adaptive multichannel linear prediction based dereverberation in time-varying room environments,” in Proc. 21th Signal Processing Conference, European Morocco, 2013 ,pp. 1 – 5. G. Burel, “Detection of Spread spectrum transmissions using fluctuations of correlation estimators,” in Proc. ISPACS’2000, 2000. Z. Koldovsky and P. Tichavsky, “Time-domain blind separation of audio sources on the basis of a complete ica decomposition of an observation space,” IEEE Trans. on Audio, Speech, and Language Processing, vol. 19, pp. 406-416, Feb. 2011. L. K. Liu, L. Hong, and J. Zhang, “An iterative carrier frequency estimate algorithm based on high-order cyclic cumulants,” in Proc. IEEE CIE International Conference on Radar, Chengdu, 2011, pp. 1817-1820. K. Waheed and F. M. Salem, “Blind information-theoretic multiuser detection algorithms for DS-CDMA and WCDMA downlink systems,” IEEE Trans. on Neural Networks, pp. 937-948, June 2005. L. H. Lim and P. Comon, “Blind multilinear identification,” IEEE Trans. on Information Theory, vol. 60 ,pp. 1260–1280, Feb. 2014. S. Saberali and H. Amindavar, “Nonlinear detector design for cdma signals in the presence of unknown interferers using maximum entropy method and comparison with SIC,” IEEE Communications Letters, pp. 1-4, Jan. 2014. F. Ding, G. G. Liu, and X. P. Liu, “Partially coupled stochastic gradient identification methods for non-uniformly sampled systems,” IEEE Trans. on Automatic Control, vol. 55, pp. 19761981, Aug. 2010. S. C. Douglas and M. Gupta, “Scaled natural gradient algorithms for instantaneous and convolutive blind source separation,” in Proc. Acoustics, Speech and Signal Processing 2007, April 2007, pp. 637-640. A. Abdi, J. A. Barger, and M. Kaveh, “A parametric model for the distribution of the angle of arrival and the associated correlation function and power spectrum at the mobile station,” IEEE Trans. on Vehicular Technology, vol. 51, pp. 425-434, May 2002.

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[12] C. M. Chen, R. Bhatia, and R. K. Sinha, “Multidimensional declustering schemes using Golden Ratio and Kronecker Sequences,” IEEE Trans. on Knowledge and Data Engineering, vol. 15, pp. 659-670, June 2003. Hao Cheng was born in Jiangsu, China in 1976. He received the B.S. degree in applied instrument technology and M.S. degree in physical electronics in 1999 and 2002, respectively, both from South East University of China, Nanjing, China, and the Ph.D. degree in communication engineering from University of Electronic Science and Technology of China (UESTC), Chengdu, China in 2007. He is currently working in the telecommunication engineering College of Chengdu University, Sichuan, China. His current research interests include wireless signal processing, spectral estimation, and array theory. Na Yu was born in Sichuan Province, China, in 1981. He received the B.S. and M.S. degree in electronic information science and technology from the Southwest Jiaotong University of China, Chengdu. She is currently working in the Telecommunication Engineering College of Chengdu University, Chengdu, China. Her research interests include research interests include information security, computer networks, and mobile application development. Jun Liu was born in Sichuan, China in 1975. He received the B.S. , M.S. and Ph.D. degree in electronic information science and technology from the University of Electronic Science and Technology of China (UESTC), Chengdu in 1997 , 2001and 2006 respectively. His research interests include His research interests include information security, image processing, and computing network.

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