According to Theorem1, the (non-convex) optimization problem (15) can be rewritten in the more tractable form
maxp Rtot(p)−ηEEPtot(p)
s.t.C1, C2 (16)
where the definition ofηEEis as shown in Algorithm 2. It is the ratio of the data capacity to the corresponding total power consumption in each iteration of the main loop. The above proposed approach based on KKT condition for solving the EE optimization problem in (17) can be summarized in Algorithm 2.
Algorithm 2. Iterative Power Allocation Algorithm 1: Initialize theauk,n using suboptimal Algorithm 1;
2: Initializepuk,n using equal power allocation;
3: Initialize the maximum number of iterationsLmaxand the maximum toleranceδ;
4: Set current maximum value of energy efficiency ηEE = RPtottot and iteration index l= 0;
5: repeat
6: Obtain the allocation policies of power ˆpuk,n in the current iteration according to (34);
7: Calculate the value of ˆRtotand ˆPtotby solving (7) and (10);
8: if Rˆtot−ηEEPˆtot≺δthen 9: Convergence=1
10: obtainp∗k,nu = ˆpuk,n andEEopt=RPˆˆtot
tot. 11: else
12: Convergence=0 13: setηEE=RPˆˆtot
tot andl=l+ 1.
14: end if
15: Update Lagrangian multipliers ofλ,βby solving (35);
16: untilConvergence or certain stopping criteria is met
Letωuk,n=auk,npuk,n,∀u∈ U, n∈ N, k∈ K; then we can rewrite the SINR of useruin SCBS kon VRBnas
γk,nu = 1 + ω
uk,nguk,n K
l=kbl,k
s∈Ulωl,ns gl,k,ns +
s∈{Uk|gsk,n>gu k,n}ω
k,ns guk,n+σ2 (17)
To satisfy the series of constraints, the Lagrange function of the problem (17) can be expressed as
F(λ, β, p) = maxL(λ, β, p)
=Rtot(p)−ηEEPtot(p) + U
u=1λu
K
k=1
N
n=1rk,nu −Ru
+ K k=1βk
Pk−
u∈Uk
N n=1puk,n
= U u=1
⎡
⎢⎢
⎣B
N +λuK
k=1
N n=1
log2(1 + ω
uk,nguk,n K
l=kbl,k
s∈Ulωsl,ngsl,k,n+ s∈{Uk|gsk,n>gu
k,n}ω
sk,nguk,n+σ2
)
⎤
⎥⎥
⎦
−K
k=1
(ηEE+βk) U u=1
N n=1ωk,nu
− U
u=1λuRu−ηEE
K k=1PC+
K k=1βkPk
.
(18) whereλ,βare the Lagrange multiplier vectors for the constraints in (17). Taking the first order derivation ofF(λ, β) with respect toωk,nu , we can get the optimal power allocation as
puk,n∗ = ωak,nuu∗
k,n =ln 2(NBη+λu
EE+βk)−
K l=kbl,k
s∈Ulωsl,ngl,k,ns +
s∈{Uk|gsk,n>gu k,n}ω
sk,ngk,nu +σ2
guk,n
. (19) Based on the subgradient method [16], the master dual problem in (17) can be solved by
λlu+1=
λlu−εlλ K
k=1
N
n=1ruk,n−Ru +
,∀u∈ U βkl+1= βkl −εlβ
Pk−
u∈Uk
N n=1puk,n
+
,∀k∈ K
(20)
5 Simulation Results and Discussions
In this section, simulation results are given to evaluate the performance of the proposed algorithms. For the simulation, the number of SCBS is K = 5. The maximum transmit power and circuit power consumption of each SCBS is set as 3 W attand 0.5 W attrespectively. The maximum of users can be allocated to each VRBnof SCBSkisDn = 2. The QoS requirement of each user isRu= 3 bps/Hz. The number of VRB is depend on the number of users and they are nearly full matched in the SDN-NOMA system.
In Fig.1, the performance of EE is evaluated versus the number of users with differentDn which is maximum number of matched users of each VRB. It is shown that, with the increase of users, the value of EE decreases. And for the same value of user number, the larger value ofDnleads to larger value of energy efficient. This is because a largerDn leads to more selection of users and bigger bandwidth of each VRB. And it is shown that the energy efficient in NOMA is better than the average EE in OFDMA.
40 60 80 100 120 140 1.5
2 2.5 3 3.5 4 4.5
Number of users
Energy Efficiency of System (bits/Joule)
OFDMA Dn=2 Dn=3
Fig. 1.Energy efficient performance versus user number with differentDn.
6 Conclusions
We investigated the dynamic resource allocation in downlink SDN-NOMA net- works. We developed a framework in NOMA network by means of SDN technol- ogy. We considered the energy efficient of the network as optimization function.
We proposed a suboptimal VRB assignment algorithm based on the two-side matching method. By considering minimum QoS requirement and maximum power constraint, we formulated the power allocation as a mixed integer pro- gramming problem as the considered problem was transformed into an equivalent problem with a tractable iterative solution. The mathematical analysis and sim- ulation results demonstrated that the effectiveness of the proposed algorithms.
Acknowledgements. This work is supported by the National Natural Science Foun- dation of China (61471025, 61771044), the Young Elite Scientist Sponsorship Pro- gram by CAST (2016QNRC001), Research Foundation of Ministry of Education of China & China Mobile (MCM2018-1-8), Beijing Municipal Natural Science Founda- tion (L172025), and the Fundamental Research Funds for the Central Universities (FRF-GF-17-A6, etc.).
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Blind Extraction of Image Information in Positive De fi nite System
Xinwu Chen, Yaqin Xie, Erfu Wang(&), and Danyang Qin Key Laboratory of Electronics Engineering College, Heilongjiang University,
Xuefu Road 74, Harbin 150080, China
Cxw808@qq.com, 648427372@qq.com, efwang_612@163.com, qindanyang@hlju.edu.cn
Abstract. According to the renewable and the noise-like characteristic of chaotic signal, the effective frequency band and the energy band of image information are used to carry out the chaotic masking transmission, so as to achieve the security transmission of image information. Chen chaos is selected as the carrier to carry out the chaotic masking transmission for the image information of the two time-frequency aliasing under the positive definite transmission model, and the blind extraction algorithm is used to restore it in the receiver. The influence of additive noise source on the extraction effect is analyzed, and a secure transmission method of image information under chaotic masking is proposed in this paper.
Keywords: Image informationPositive definite system Masking transmissionBlind extractionChen chaos
1 Introduction
As one of the popular multimedia forms in today’s society, digital image has been widely used in politics, economy, national defense and education. In some relatively special areas, such as military, commercial, digital image has a high confidentiality requirements [1,2]. Since 1990, many researchers have made many kinds of image encryption algorithms by using the spatio-temporal property and visual perception of images [3]. Banerjee and Barrera use something similar to chaos to achieve encryption by improving and transforming chaos [4, 5]. Chaotic signal is the description of complex and irregular motion in a deterministic system. The chaotic masking tech- nology uses the chaotic signals with statistical characteristics to hide the useful signals, so that the useful signal and the chaotic signal can be superimposed to achieve the communication security effect [6–8]. Blind source separation (BSS) is a subject developed in the middle and late 80s of last century, which can recover the target source signal from the observed aliasing signal, just using the statistical characteristics of the source signal, even the input signal and channel parameters are unknown. In recent years, blind source separation (BSS) technology has been widely used in wireless communications, biomedical engineering, speech processing and image
©ICST Institute for Computer Sciences, Social Informatics and Telecommunications Engineering 2018 X. Gu et al. (Eds.): MLICOM 2017, Part II, LNICST 227, pp. 428–436, 2018.
https://doi.org/10.1007/978-3-319-73447-7_47
processing, etc. [9,10]. On the basis of this, we combine the image information and the chaotic signal, and propose a method to cover the image information by using the chaotic signal and extract the image information through the blind source separation technique in the determined model, so as to achieve the effect of the secure communication.
2 The Mathematical Model of Blind Source Separation and Chaos Signal
2.1 The Mathematical Model
Positive definite hybrid system model is the system model which assuming that the source are independent and the number isn, and at the receiving end using nreceive antennas to receivensignals. The mathematical model of the positive definite hybrid blind source separation system as shown in Fig.1.
Given the vector quantity of original signal Sẳẵs1ð ị;t s2ð ị;t ;sNð ịt T, which means the number of unknown original signal isN. In order to transmit the signal in secrecy, one of vector quantity is chosen to as a chaotic signal, then the image information is hided in the chaotic signal effectively, then realize the secret transmis- sion. A is an unknown channel hybrid matrix of order NN, which generated by system randomly.Nẳẵn1ð ị;t n2ð ị;t ;nNð ịt Tis the additive white gaussian noise of the channel. It can conclude that the vector formulation of the positive definite hybrid system observed signal is shown as follow:
Y ẳASỵN: ð1ị
The key step of positive definite hybrid system for blind source separation is to solve the separation matrix W. S0ẳs01ð ị;t s02ð ị;t ;s0Nð ịt T
is the original signal estimated from the observed signal. Through the matrixW, the target signalsS0can be
Extraction
Image informaƟon1
ChaoƟc signal
Channel Unknown
Image informaƟon2
Fig. 1. Mathematical model of positive definite blind source separation
extracted from the observed signal Y, the output of the separation system or the extracted vector expression is
S0ẳWYẳWASỵWN: ð2ị
ICA is the common method for blind signal processing. This paper adopts the algorithm for blind source separation to get the separation matrix W, due to the FastICA algorithm has good convergence, the short training time and small dependence on learning step factor.
2.2 Chaotic Signal
This paper based on chaotic signal to do the research of target signal blind extraction, so the Chen chaotic system is selected. The dynamic expression of Chen chaotic system [11] is given.
dx
dtẳaðyxị
dy
dtẳ ðcaịxxzỵcy
dzdtẳxybz 8<
: : ð3ị
Where a;b;care the system parameters. Chen chaotic system in a state of chaos whenaẳ35;bẳ3;cẳ28.x;y;zare the state variables of the system.
3 Algorithm Performance Index and Simulation Flow
3.1 Algorithm Performance Index
For the successful separation of the target signal can be evaluated by two aspects, qualitative and quantitative. For image information, qualitative analysis can visually contrast the image information before and after blind source separation so as to obtain an intuitive evaluation [12]. Quantitative analysis can evaluate the performance of the algorithm objectively through the performance evaluation function, the similarity coefficient [13] is the most commonly used evaluation criterion.
3.2 Simulation Flow
The detailed implementation steps and algorithm flow chart are given. The chaotic signal is used as the background to judge the validity and universality of the algorithm.
The algorithmflow diagram is shown in Fig. 2.
Implementation steps:
Step 1: Select gray scale pictures from the standard test picture library and convert it from a two-dimensional image to one dimensional array data, making the one-dimensional array data into binary array data.
Step 2: Simulate the unknownness of the channel, randomly generate the mixed matrix. The observed signal is obtained after the source signal have passed by the
hybrid matrix, then the observation signal is observed to see if the image infor- mation has been hidden by chaotic signals and can not be distinguished by human eyes.
Step 3: Use the FastICA algorithm to do blind source separation for the observed signal, and extract the target signal from it.
Step 4: Observe the image information before and after blind source separation through the visual system and view the similarity coefficients. If the extracted image information can be clearly recognized by the human eye, and the similarity coef- ficient is more than 0.97, the separation is considered to be successful.
Because this paper is based on the mathematical model of positive definite mixed system to realize blind source separation, the algorithm simulation process randomly generates a full rank square matrix.
4 Simulation Experiment and Performance Analysis
4.1 Blind Separation of Two Image Signals Without Noise
Select two gray scale pictures from the standard test picture library shown as Figs.3 and4256256 and convert it from a two-dimensional image to one dimensional array data, making the one-dimensional array data into binary array data. Then encapsulate it with Chen chaotic signals. A 33 matrix is generated randomly through the system,
Fig. 2. Flow chart of positive definite blind source separation algorithm
making into aliasing with encapsulated data to obtain three way observation signals.
Then convert the data of the observation signal into a decimalization date and turn it into a two-dimensional date. The image information is shown in Figs.5,6and7. And then the matrix after aliasing was separated by FastICA algorithm to can get estimated value of each source signal. We convert the resulting estimates into a decimal and two-dimensional date. The image information can be shown in Figs.8 and 9 (The image information after separating from source signal is displayed here).
The random generation of the hybrid matrix for this experiment is
Aẳ
0:8694 0:1014 0:2086 0:4122 0:7794 0:8096 0:1678 0:1066 0:2961 2
4
3 5
Fig. 3. Image information offirst source signal
Fig. 4. Image information of second source signal
Fig. 5. Image information of the first observation signal in noise free model
Fig. 6. Image information of the second observation signal in noise free model
According to Figs.5, 6 and 7 we can clearly find that the Image information reconstructed by observed signal could not be identified which shows that the image signals are obscured by Chen chaotic signals and cannot be recognized by human eyes.
By comparing Fig.3 with Fig.8 and Fig.4 with Fig.9, we can clearlyfind that the similarity between the two images is high and almost no difference, and the image information can be easily seen with human vision. By calculation, the similarity coefficient between Figs.3 and 8 is 0.9999, and the similarity coefficient between Figs.4and9is 0.9999. In summary, it can be concluded that the image information of the source signal is well separated, and the simulation has also achieved the desired results.
4.2 Blind Separation of Two Images with Superimposed Noise
We select two gray scale pictures from the standard test picture library, as shown in Figs.3and 4, and convert it from a two-dimensional image to one dimensional array data, then making the one-dimensional array data into binary array data. A Gaussian white noise is randomly generated by the system, and then the Gaussian white noise, Fig. 7. Image information of the third
observation signal in noise free model Fig. 8. Image information of thefirst extracting signal in noise free model
Fig. 9. Image information of the second extracting signal in noise free model
the binary array data and the Chen chaotic signal are encapsulated. A 44 matrix is randomly generated by the system and mixed with the encapsulated data to obtain four channel observation signals. The data of the observation signal is converted into decimal and then the four channel image information which made by converting one-dimensional data into two-dimensional data can be shown in Figs.10,11,12and 13. And then the matrix after aliasing was separated by FastICA algorithm, we can get the estimated value of the source signal. Then the image information shown in Figs.14 and15can be obtained by repeating the above steps.
The random generation of the hybrid matrix for this experiment is
Aẳ
0:3935 0:5669 0:8033 0:5702 0:0788 0:8792 0:0240 0:4017 0:2789 0:7586 0:7554 0:9707 0:4431 0:4590 0:4078 0:1747 2
66 4
3 77 5
Fig. 11. Image information of the second observation signal in noise model Fig. 10. Image information of the first
observation signal in noise model
Fig. 12. Image information of the third observation signal in noise model
Fig. 13. Image information of the fourth observation signal in noise model
In Figs.10,11,12 and 13observations can be found that the image information can’t be identified only by the human eye. It shows that the image signal is covered well by the Chen chaos signal, which is not recognized by the human eye. Through the comparison between Figs.3and14and Figs.4and15, it can be clearly seen that the extracted image information has a high similarity with the image information of the source signal, which shows that the content of the image information can be easily found. By calculation, the similarity coefficient between Figs. 3and14is 0.9999, and the similarity coefficient between Figs.4 and 15 is 0.9999. In summary, it can be basically concluded that the image information of the source signal is well separated, and the validity of the algorithm is also verified.
5 Conclusion
The secure communication transmission technology based on chaotic signals is widely used in various information security fields. Considering the importance of image information for secure transmission of the signal, this paper proposes that the signal of image information can be masked by Chen chaotic signal. After the channel trans- mission of determined system, the image information is extracted by blind extraction.
The simulation results show that the method can obscure the image information and can extract the image information well. Even in the case of white Gaussian noise (WGN), better experimental results can be obtained, and the validity is verified as well.
That will prepare for the secrecy and blind separation of the image information under the under-determined background.
Acknowledgments. This work was supported by the National Natural Science Foundation of China (grant 61571181), Postdoctoral Research Foundation of Heilongjiang Province (grant LBH-Q14136), and Graduate Student Innovation Research Project Fundation of Heilongjiang University (grant YJSCX2017-148HLJU).
Fig. 14. Image information of thefirst extracting signal in noise model
Fig. 15. Image information of the second extracting signal in noise model
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