jamming separation

7/27/2019 Jamming Separation

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Wireless Pers CommunDOI 10.1007/s11277-013-1286-6

Noisy Blind Signal-jamming Separation Algorithm

Based on VBICA

Yuling Duan Hang Zhang

Springer Science+Business Media New York 2013

Abstract Aiming at the blind signal-jamming separation (BSJS) in wireless communication

environment, we propose a noisy BSJS based on Variational Bayesian Independent Com-

ponent Analysis algorithm to separate the communication signal from jamming signals and

noises. This algorithm takes the KullbackLeibler divergence between the true post distri-

butions of source signals and the approximate ones as objective function, models sources

using mixture of Gaussians, and updates parameters of the model using variational-Bayesian

learning method, so as to make the estimated approximate posterior distributions close tothe true ones and recover source communication signals finally. The simulation results show

that the proposed algorithm is effective for the BSJS in noisy environment.

Keywords Variational Bayesian Noisy mixture MOG Blind signal-jamming separation Blind source separation

1 Introduction

Blind source separation (BSS) technique has become a new study field in the modern sig-nal processing. Its purpose is to recover the independent sources from the observed mixed

signals using the statistical characteristics when the sources and the channel parameters are

all unknown [1]. Presently, the technique has been used in a good many fields, such as

biomedicine, speech recognition, machine malfunction elimination and so on [24]. In this

paper, we use it to separate communication signal from observed signals which are mixed by

jamming signals and noises, and achieving the purpose of anti-jamming. We call it as blind

signal-jamming separation (BSJS).

Y. DuanNanjing Artillery Academy, Nanjing, China

e-mail: [email protected]

H. Zhang (B)

Institute of Communications Engineering,

PLA University of Science and Technology, Nanjing, China

e-mail: [email protected]

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Y. Duan, H. Zhang

Unknown

Mixing

matrix

Observation

signals

)(1 tx

)(2 tx

)(txM

Sour

cesignals

A

)(2 ts

)(tsN

)(1 ts

Fig. 1 Noise-free mixing model of observed signals in BSS system

Unknown

Mixing

matrix

Observation

signals

)(1 tx

)(2 tx

)(txM

A

)(1 tn

)(2 tn

)(tnM

)(2 tJ

)(tJN

)(1 tsSourcesignals

Fig. 2 Noisy mixing model of observed signals in BSJS system

1.1 Conventional BSS Model

Conventional BSS is processing in noise-free environment. Figure 1 describes the noise-free

mixing model of observed signals in BSS.

In Fig. 1, the observed signals can be described as

X(t) = AS(t) (1)

where X(t) = [x1(t),x2(t) , . . . ,xM(t)]T is a set of observed signals, t = 1, 2, . . . , L ,

L denotes the number of the sampling point. A is a M N linear mixing matrix. S(t) =[s1(t), s2(t) , . . . , sN(t)]

T is a set of source signal vector. In (1), X(t) is known only, and the

A and S(t) are both unknown.

BSS algorithms try to find a matrix W which is called as separation matrix, make the

output signals y = WX are the best estimation of source signals. In a word, the conventionalBSS algorithm is to solve the inverse transform of the mixing matrix A for getting the best

estimation of the source signals S(t) from the observed mixing signals X(t).

1.2 Noisy BSJS Model

But in actual environment, noise cannt be neglected. Especially in BSJS system, noise and

jamming are simultaneous. The noisy mixing model of the observed signals in BSJS system

is shown in Fig. 2. By this time, the mixing process is no longer invertible because of the

additive noise, and conventional BSS algorithms are difficult to get the inverse transform of

the mixing matrix directly, so they also cannt separate source communication signal from

jamming and noises effectively.

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Noisy BSJS Algorithm Based on VBICA

In Fig. 2, the observed signals can be described as

X(t) = AS(t) + n(t) (2)

where, S(t) = [s1(t), J2(t) , . . . , JN(t)]T is a set of source signal vector from N trans-

mitters, s1(t) is communication signal, Ji (t), i = 2, . . . , N are jamming signals. n(t) =[n1(t), n2(t) , . . . , nM(t)]

T is noise vector. And we assume that the source signals and addi-

tive noise components are mutually statistically independent. In this paper, we only talks

about non-underdetermined case, namely the mixing matrix A is full rank. So it needs that

the number of the observation signal is greater than or equal to the number of the source

signal. There, we use M receive antennas, and M N.Its clearly that noisy mixture blind source separation is more significant to study.

1.3 Previous Work and Our Work

Presently, some noisy BSS algorithms have been proposed. One of which is a kind of BSS

algorithm based on denoise technique, such as Wang et al. [5] presented a concurrent time-

frequency and noise-pretreatment methodology for blind source separation in an additive

white Gaussiannoise channel, which is based on waveletdenoise andempirical mode decom-

position (EMD) denoise. This kind of algorithm could eliminate some noise although, their

separation performance mainly depends on the denoise techniques and increase the cost of

system in a certain extent. The other kind of noisy BSS algorithm is a fast ICA for noisy data

using Gaussian moments algorithm proposed by Hyvarinen [6]. But, this algorithm needs

known noise precision and just aims at Gaussian noise. Besides, these above-mentioned

noisy BSS algorithms are almost based on ICA theory. But ICA theory has many limits.For example, it requires the most only one source signal is gaussian signal, and the mixing

matrix is full column rank. Most importantly, ICA theory ignores noise in mixing process. So

these algorithms also cannt solve actual BSJS problem effectively. Fortunately, Choudrey

[7] proposed an independent component analysis based on Variational Bayesian (VBICA),

which overcomes these limits of the conventional ICA theory.

Considering the noise-robustness of VBICA, and combining with the model of commu-

nication signal and the observed signals in noisy BSJS system, we propose a BSJS algorithm

based on VBICA to separate source communication signal s1(t) from jamming signals and

noises. Firstly, we model the source signals, noise and the mixing process of the observed

signals, then get the approximate posterior distributions of the source signals by variationalBayesian learning. Finally, the source communication signal is recovered. And in order to

verify this algorithm, we do a large number of simulations to separate communication sig-

nal such as BPSK, 16QAM from jamming signals and noises, and have a good separation

performance.

The rest of this paper is organized as follows. In Sect. 2, we infer the VBICA algorithm

and the modeling process of the parameters in BSJS detailedly. In the next section, the noisy

BSJS algorithm based on VBICA is proposed and a series of simulation results are provided.

At the end of the paper, the remarkable conclusions are presented.

2 The VBICA Algorithm

VBICA algorithm is a kind of BSS algorithm based on model estimation which combines

the Bayesian theory with the ICA theory. Firstly, it uses prior probability density functions

(PDFs) to model the variables in Fig. 2 including source signals, mixing matrix and noise,

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Y. Duan, H. Zhang

Apriori Knowledge of

Communication

System

Sources Model Noise Model

1x

2x

Nx

Mixing Model

Parameters

Learning

Sources Model

Parameters

LearningRecoversources

s

1

J

2

J

1

MJ

Transfer Parameters

Transfer Data

Mixing

Model

ObservedSignals

Sourcesapproximate

posterior

probabilitydensity

Noise Model

Parameters

Learning

Recovered

signals

Fig. 3 Simplified description of VBICA algorithm

constructs a parameter model (called as Generated Model) based on the probability theory,

which describes the generative process of the observed signals. Then, VB learning method is

used to update the parameters so that the parameter model can reflect the generative process

as exactly as possible. Finally, we get posterior PDFs of the variables and recover sourcesignals, especially the source communication signal. The algorithm can be simplified to

Fig. 3.

In Fig. 3, the PDF of observed signals X(t) expresses as

p(X(t)|S(t), A,) = | det(/2 )|1/2 exp[ED ] (3)

where ED = (X(t) AS(t))T(X(t) AS(t))/2. denotes diagonal precision matrix of

noise.

Since the sources S(t) = [s1(t), J2(t), , JN(t)]T are mutually independent, so the

distribution ofS(t) can be written as

p(S(t)) =

Ni=1

p(si (t)) (4)

Theoretically, we could recover the sources by calculating the posterior distributions of the

sources given the observed signals and the model,

p (S(t)|X(t), ) =p (X(t)|S(t), ) p (S(t)|)

p (X(t)|)(5)

where p(S(t)|) is the prior distribution of source signal model, p(X(t)|) is the marginprobability ofX(t) under the model .

2.1 Objective Function

In this algorithm, an appropriate objective function is needed as convergence condition dur-

ing the learning processing. There, we adopt negative-free-energy as the objective function.

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The detailed deduction is given as follows. The logarithm marginal likelihood of observed

signals X is

log p(X) = logp(X, W)

p(W|X)(6)

where W is an aggregate of allvariables andparameters. p(W |X ) is the true posteriordistrib-ution.As the logarithm marginal likelihood ofX does not depend on W, it can be re-writtenas

log p(X) =

p(W) log

p(W)p(X, W)

p(W)p(W|X)d W

=

p(W) log

p(X, W)

p(W)d W +

p(W) log

p(W)

p(W|X)d W (7)

= F[X] + K L[p(W) p(W |X ) ]

where p(W) is approximate posterior distribution of p(W |X ); and

F[X] = log p(X, W)p(W) + H[p(W)] (8)

K L[p(W)||p(W|X)] =

p(W) log

p(W)

p(W|X)d W (9)

F[X] is called as negative-free-energy [7] . In (8), the first term is expectation of the jointpossibility density between sources and observed signals under p(W). And H[p(W)] is theentropy of p(W). K L[p(W)||p(W|X)] is the KullbackLeibler (KL) divergence whichmeasures the difference between two probability densities, and it is strictly non-negative. So,

log p(X) F[X] (10)

with equality when p(W) equals p(W|X ).Therefore, maximizing the F[X] means to minimize the KL divergence between the

approximate posterior PDF and true one. That is to say, F[X] is higher, the estimated data ismore closing to the true one. So the objective function of this algorithm is to maximize the

F[X].

2.2 Modeling

2.2.1 Source Model

In this paper, mixture of Gaussians (MOG) model is chosen to model the source signals. Sta-

tistically, MOG model could toward any probabilitydensity distribution as exactlyas possible

by choosing appropriate number of Gaussian components and appropriate parameters. So

S(t) is modeled as N MOGs with mi Gaussian components.

p(S(t)| ) =N

i =1

miqi =1

p(qi = qi |i )p(si (t)|qi , i,qi , i,qi ) (11)

=N

i =1

miqi =1

i,qi N(si (t); i,qi , i,qi )

where qi represents the chosen Gaussian component in the i th source. The mixing weight

i,qi = p(qi = qi |i ) is the prior probability of the Gaussian component qi . N () meansGaussiandistribution,i,qi , i,qi representthemeanandprecisionof theGaussiancomponent

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Y. Duan, H. Zhang

qi respectively. The complete parameter set of the i th source is i =

i , i , i

, and

i , i , i are mi dimension vectors respectively. So the parameter set of the sources vectoris = {1, 2, . . . , N}.

The prior probability over the source model parameters is

p( ) = p()p()p() (12)

wherep()over the mixing weight is Dirichlet

p() =

Ni =1

D(i , i0) (13)

p() over the means is product of Gaussians

p() =

N

i=1

mi

qi =1

N(i,qi , mi0, i0) (14)

p() over the precisions is product of Gammas

p() =N

i =1

miqi =1

G(i,qi ; bi0, ci0) (15)

2.2.2 Noise Model

The noise n(t) is assumed to be Gaussian, with zero mean and diagonal precision matrix ,

the PDF of which is

p(n(t)) = N(n(t)|0,; ) (16)

The prior probability p() over the diagonal precision matrix is also product of Gammas

p() =

Mj=1

G(j ; bj , cj ) (17)

2.2.3 Mixing Matrix Model

The prior PDF p(A) over the mixing matrix is product of Gaussians

p(A) =

Ni =1

Mj =1

N(Aj i |0, j i ) (18)

where j i denotes the variance of the (j, i ) mixing coefficient in mixing matrix.

In (13)(17), , m, , b, c are called as hyper-parameters.

2.3 Initialization

In VBICA, initializing variables and parameters is needed. In cooperative communication,

some prior information of communication signals is known, such as the signals modulation

mode, shape filter, symbol rate, carrier frequency and so on. By many tests time after time,

these hyper-parameters of p(), p(), p(), p(A), p() in the model are initialized as

follows:

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2x1x

10 10,b c

10 10,m

10 1,111,m

1,1

11,m

0N ,1N , NN m ,1N , NN m

1 Nq1s N NJ

1,1

1,1a

M2

,M M

b c 2 2

,b c

1

1q

1 1,b c

0 0,N Nm 0 0,N Nb c

Mx

Fig. 4 VBICA mixing system model parameters

i0 = 5, m = 0, = 1, 000, b() = 1, 000, c() = 0.001,b() = 1, 000, c() = 0.001, b() = 1, 000, c() = 0.001

This paper performs Singular Value Decomposition (SVD) on the observation signals to

initialize the mixing matrix, noise and the sources. Then, we can get the weights, means and

precise of the MOG model by k-means clustering [8] on the initialized sources.

2.4 VB Learning

After initialization, its time for VB learning. In this model, all the parameters and their

relationship are described as Fig. 4, where circles represent random variables and rectangles

represent hyper-parameters.

In VBICA model, objective function F[X] is shown in (8), where W = {A,, S, q, } isaggregate of the source variables and parameters. Maximizing the F[X] equals to maximizethe p(W). There, we use variational approximation method [9] to maximize the factors of

p(W).

p(W) = p()p(A)p(S|q)p(q)p( ) (19)

where p( ) = p()p()p().

p (X, W) = p (X|A,, S) p (S|q, , ) p (q| ) p () p (A) p () (20)

where p(X|A,, S) =L

t=1 p(x(t)|s(t), A,).Combining (19) (20) with (8),

F = log p(X|A,, S)p(A)p()p(S) + log p(S|q, , )p(S|q)p(q)p()p()

+H[p(S|q)] + log p(q | )p(q)p( ) + H[p(q)] + log p()p()

+H[p()] + log p()p() + H[p()] + log p()p()

+H[p()] + log p(A)p(A) + H[p(A)]

+ log p()p() + H[p()] (21)

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Y. Duan, H. Zhang

where,

p(q) =

N

i =1L

t=1ti,qi (22)

ti,qi = t

i,qi

q i

ti,q i

, ti,qi = i,qi pi,qi .

p(S|q) =

Ni=1

Lt=1

N

si (t); ti,qi

, ti,qi

(23)

The other posteriors PDFs have the same form as the priors. The update equations of these

parameters are detailed in Appendix A.

Updating these parameters iteratively until convergence that means the F = |Fnew

Fold

is less than a tolerance . In this paper, = 0.0001.After VB learning, we can estimate the sources by getting the posteriors PDFs of these

variables.

3 The Noisy BSJS Based on VBICA

In VBICA algorithm, the noise is modeled and learned. Comparing with other noise-free

BSS algorithms, VBICA is robust to noise. So we employ it to solve the BSJS problem in

noisy communication system, the flow chart is shown in Fig. 5, where the pre-processing

means removing means and whitening [1]. Then, we model the sources, noise and mixing

process, and initialize the prior PDFs of all the parameters in this model. Next, VB learning

method is used to update these parameters until convergence. Finally, posterior PDFs of the

variables are estimated, and we also recover the source communication signal and jamming

signal from the observed signals.

In order to verify the performance of the noisy BSJS based on VBICA algorithm, we

do a series of simulations at Matlab 7 platform. Firstly, we simulate the MOG models

validity for the source signals. Secondly, simulations that prove the separation performance

of the algorithm for BPSK signal under multi-tone jamming signal in well-determined case

( )tX

Pre

-processing

( )tZ

N

VBICA

algorithm

Y

Rec

oversources

Variables approxi

mate posterior

probability density( ), ( ), ( ),

( ), ( ), ( )

( | )

p' q p' p'

p' p' p'

p' q

A

S

s

1J

2J

1MJ

Reco

vered

signals

new old F FConvergence?

VB learning, updating

parameters in the model

iteratively, , , , ,

, , , , ,

,

, , , , ,

, , , , ,

, , , ,

i i i i i

i i i i i

i ji j j

t t t

i q i q i q i q i q

i q i q i q i q i q

i q a ji

p

m b

c m b c

Setting up VBICA model

( )p AMixing Model

( ( ))p tnNoise Model

( ( ) | )p tSSources Model

F

Setting up Objective Function,

negative-free-energy

1x

2x

Mx

ObservedSignals

Initializing mixing matrix,

noise, sources and hyper-

parameters in the model

0 ( ) ( )

( ) ( ) ( ) ( )

, , , , ,, , ,

i m b c

b c b c

Transfer Parameters

Transfer Datas

N). The

detailed simulation analyses and results are described as follows.

3.1 Simulation for MOG Modeling and Analysis

There,sources signals includingBPSK, 16QAM, broadband-noise jamming signaland multi-

tone jamming signal are to be modeled respectively. Figure 6a, b show their time-domain

waveforms and corresponding probability density distribution histograms. Every source sig-

nalismodeled usingMOGwith five Gaussiancomponents.By learningthemodel,probability

density distributions of estimated signals are described in Fig. 6c. Many tests indicate that the

number of Gaussian components in every MOG is less work on the separation performance

of the algorithm.

In Fig. 6c, these colorful broken lines represent the probability density distributions of five

Gaussiancomponents in each source model respectively, andtheblack real lines represent theprobability density distributions of theMOGswhicharemixture of five Gaussiancomponents

in proportion respectively. Comparing Fig. 6c with b, the probability density distributions of

sources and the ones of the estimated signals using MOG are almost coincident. That is to

say, MOG could model the source signals successfully.

3.2 Simulation for BPSK Under Multi-tone Jamming Signal and Analysis

In this simulation, source communication signal s1(t) is BPSK signal, symbol rate Rb =10 kbit/s, carrier frequency fc = 20 kHz. Jamming signal J1(t) is a multi-tone with carrierfrequencies fc1 = 19kHz, fc2 = 19.99kHz, fc3 = 21 kHz respectively. The jamming-signal-ratio (JSR) is 10dB. Mixture matrix A22 is generated randomly. Noise is Gaussian

noise, and the signal-noise-ratio (SNR) is 10dB. Equation (2) is used to create X(t), sampling

frequency fs = 100 kHz. After separating the X(t) using the algorithm shown in Fig. 5,the simulation results are shown in Fig. 7. We also analyses the separation performance

in different SNR cases. Figure 8 plots the bit-error-ratio (BER) of the recovered BPSK

signals using EASI algorithm [10], FICA algorithm [11], JADE algorithm [12] and the

VBICA algorithm respectively. The result is the average of 50 Monte Carlo experiments.

Furthermore, comparing the separation running time of the four algorithms, which is related

with the sampling points of the observation signal, the result is shown in Table 1.From Fig. 7, we can see that the recovered signals waveforms are almost same as the

source signal waveforms except the scale and phase ambiguity, but it doesnt matter for

receiving communication signal correctly. In order to measure the accuracy of separation,

we calculate the similitude coefficient matrix Ee, the (i, j ) element of which is defined as

i j = (yi , sj ) =

Nk=1 yi (k)sj (k)

Nk=1 y

2i (k)

Nk=1 s

2j (k)

(24)

where N is the number of source signals; yi , sj are the recovered signals and the sourcesignals respectively. The separation performance is good when there is only one element

closes to one, and the others all close to zero at every row and every column in Ee.

Now, we calculate the Ee between the recovered signals and the source signals in Fig. 7.

Ee =

0.9996 0.0059

0.0014 0.9560

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Y. Duan, H. Zhang

0 5000 10000-1

0

116QAM

0 5000 10000-5

0

5Broadband-noise

0 5000 10000-1

0

1Multi-tone

0 5000 10000-0.5

0

0.5BPSK

sampling points

scale

/ V

(a)

-1 -0.5 0 0.5 10

100

200

300

16QAM

-4 -2 0 2 40

200

400

Broadband-noise

-1 -0.5 0 0.5 10

1000

2000 Multi-tone

-0.5 0 0.50

1000

2000 BPSK

(b)

-4 -2 0 2 40

0.5

1Broadband-noise

-1.5 -1 -0.5 0 0.5 1 1.50

2

4

6

8Multi-tone

-1.5 -1 -0.5 0 0.5 1 1.50

0.5

1

1.5

216QAM

-5 0 50

0.5

1

1.5

2BPSK

(c)

Fig. 6 Modeling source signals using MOG. a Time-domain waveforms of the source signals. b Probability

density distribution histograms of the source signals. c Probability density distribution of the estimated source

signals using MOG

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0 50 100-0.5

0

0.5BPSK

0 500 1000-5

0

5Multi-tone

0 500 1000-5

0

5Observation signal 1

0 500 1000-5

0

5Observation signal 2

0 50 100-2

0

2BPSK

0 500 1000-5

0

5Multi-tone

Scale

/ V

Sampling Points

Fig. 7 Separation performance using VBICA for BPSK under multi-tone jamming signal (JSR= 10dB,

SNR=10dB). The first row plots the time-domain waveforms of sources, the second row plots the onesof observed signals and the last row plots the time-domain waveforms of recovered signals

0 2 4 6 8 10 12 1410

-5

10-4

10-3

10-2

10-1

100

SNR/ dB

BER

BSJS algorithm based on VBICA

EASI algorithm

FICA algorithm

JADE algorithm

Fig. 8 The BER of the recovered BPSK signal (JSR= 10dB)

The 1st column ofEe denotes the similarity between the recovered BPSK signal and the

source BPSK signal, which is 0.9996 (see the Ee). That means the recovered signal is a good

estimation to the source signal.

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Y. Duan, H. Zhang

Table 1 The separation running

time of these algorithms in the

same case

Sampling points VBICA/s EASI/s FICA/s JADE/s

1,000 2.34 0.03 0.05 0.01

In Fig. 8, when SNR is more than 9dB, the BER of the recovered BPSK signal by the

proposed algorithm is < 103. Comparing with the other three conventional BSS algorithms,

the BER using VBICA is lower at the same SNR, namely, the noise-robustness of the VBICA

algorithm is stronger than the other three ones. Whereas, the separation running time of

VBICA is further longer (see Table 1). So we must try to reduce its computing complexity

in next work.

3.3 Simulation for 16QAM Under Broadband-Noise Jamming Signal and Analysis

In this section, source communication signal s1(t) is 16QAM signal. Shape filter is raised

cosine filter whose roll-off factor is 0.5. Rb = 1 kbit/s. fc = 10 kHz. Jamming signal J1(t)is broadband- noise whose bandwidth is 5 45 kHz, JSR is 5dB. Mixture matrix A32 isgenerated randomly. Noise is Gaussian noise, and the SNR is 10 dB. Equation (2) is used

to create X(t), fs = 100 kHz. Similarly, X(t) is separated according to Fig. 5, but the pre-processing would be replaced by quasi-whitening processing [13] in this simulation. The

simulation results are shown in Fig. 9.

Comparing Fig. 9a with c, the waveforms of the recovered signals are almost coinci-

dent with the ones of sources. To calculate the Ee between the sources and the recovered

signals,

Ee =

0.0207 0.9405

0.9739 0.0085

The similarity between the recovered 16QAM and the source 16QAM is 0.9739 (see the

Ee). That means the recovered signals is a good estimation to the sources. So the VBICA

algorithm achieved the BSJS for the 16QAM under broadband-noise in a certain SNR

case.

3.4 Simulation for 16QAM Under Multi-tone Jamming Signal and Analysis

Simulation parameters and condition are same as Sect. 3.3 except that the jamming signal

is multi-tone jamming signal, fc1 = 9kHz, fc2 = 10kHz, fc3 = 11kHz, JSR is 5 dB.Simulation results are shown in Fig. 10.

The Ee between the sources and the recovered signals is

Ee = 0.0539 0.94310.9754 0.0205

Comparing Fig. 10a with c, we can see that the waveforms of the recoveredsignals are almost

coincident with the ones of sources clearly. Moreover, the similarity between the recovered

16QAM and the source 16QAM is 0.9754 (see the Ee). So, the recovered signals is a good

estimate to the sources and the VBICA algorithm achieved the BSJS for the 16QAM under

multi-tone jamming signal in a certain SNR case effectively.

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0 1000 2000 3000-1

-0.5

0

0.5

1

16QAM

0 1000 2000 3000-2

-1

0

1

2

Broadband-noise

0 2 4 6

x 104

0

10

20

30

0 1 2 3 4 5

x 104

0

1

2

3

16QAM Broadband-noise

Sampling Point

Frequency /Hz

Scale

/ V

(a)

0 500 1000 1500 2000 2500 3000-1

0

1

observation signal 1

0 500 1000 1500 2000 2500 3000-1

0

1


0 500 1000 1500 2000 2500 3000-1

0

1

Sampling Point


Scale

/ V

(b)

0 1000 2000 3000-1

0

1

Broadband-noise

0 1000 2000 3000-2

0

2

16QAM

0 1 2 3 4 5

x 104

0

0.5

1

0 2 4 6

x 104

0

100

200

16QAMWideband Noise

Sampling point

Frequency / Hz

Scale

/ V

(c)

Fig. 9 The noisy BSJS based on VBICA for 16QAM under broadband-noise jamming signal (JSR=5 dB,

SNR=10dB). a Time and frequency waveforms of sources. b Time-domain waveforms of observed signals.

c Time-domain and frequency-domain waveforms of recovered sources

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Y. Duan, H. Zhang

0 1000 2000 3000-1

-0.5

0

0.5

116QAM

0 1000 2000 3000-2

-1

0

1

2Multi-tone

0 2 4 6

x 104

0

5

10

15

0 2 4 6

x 104

0

50

100

16QAM Multi-tone

Sampling Point

Frequency/ Hz

Scale/V

(a)

0 500 1000 1500 2000 2500 3000-1

0

1


0 500 1000 1500 2000 2500 3000-1

0

1observation signal 2

0 500 1000 1500 2000 2500 3000-1

0

1

Sampling Point


Scale

/ V

(b)

0 1000 2000 3000-1

0

1Multi-tone

0 1000 2000 3000-2

0

216QAM

0 2 4 6

x 104

0

5

10

15

0 2 4 6

x 104

0

20

40

60

Multi-tone 16QAM

Sampling Point

Frequency / Hz

Scale/ V

(c)

Fig. 10 The noisy BSJS based on VBICA for 16QAM anti multi-tone jamming signal (JSR= 5 dB,

SNR=10dB). a Time and frequency waveforms of sources. b Time-domain waveforms of observed signals.

c Time-domain and frequency-domain waveforms of recovered signals

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Table 2 The separation similitude coefficient matrix Ee of these algorithms in the different SNRs case

SNR (dB) Algorithm

Ee

VBICA FICA based onGaussianmoments

Noise-robust EASI

0

0.0191 0.5087

0.8513 0.0501

0.4138 0.4653

0.462 0.4082

0.3055 0.4902

0.5762 0.3509

2

0.0007 0.5903

0.8934 0.0173

0.5134 0.4699

0.4892 0.5085

0.4443 0.5252

0.5578 0.4469

3

0.0157 0.65

0.9165 0.0245

0.5159 0.5144

0.5255 0.5543

0.2651 0.6327

0.7766 0.3115

5 0.0107 0.7370.9452 0.0201

0.9448 0.01160.0057 0.7369

0.0926 0.73210.9338 0.125

7

0.0179 0.8155

0.9614 0.0166

0.9615 0.0637

0.05 0.8118

0.0875 0.8096

0.9558 0.1076

10

0.0408 0.8791

0.9791 0.0548

0.9801 0.0435

0.0382 0.8765

0.0711 0.8645

0.9783 0.1653

In addition, we compare the separation similitude of the VBICA algorithm with other two

noisy BSS algorithms that are noise-robust EASI algorithm [14] and noisy ICA algorithm

based on Gaussian moments [6] in same simulation condition. The simulation parameters

and condition are almost same as Sect. 3.4. Considering the condition that the energies of thecommunication signal and jamming signal are almost equal, JSR is 1dB. The Ee is shown

in Table 2.

In Table 2, the similitude coefficient with read in Ee represents the similarity between the

recovered 16QAM and the source 16QAM. We can see that the separation performances of

the three algorithms are better as the SNR is higher. Concerning the anti-noise performance,

the Ees of VBICA have smaller changes than those of other two algorithms when SNR

changes. That is to say, the anti-noise performance of VBICA is more excellent. Besides,

when the SNR is less than 5 dB, the separation predominance of VBICA is more obvious.

So VBICA is more effective to solve BSJS problem in lower SNR cases.

4 Conclusions

Aiming at the blind signal-jamming separation problem in noisy communication system,

this paper applies the Variational Bayesian ICA theory, uses MOG to model the sources,

and proposes a noisy BSJS based on VBICA algorithm. Simulation results prove that the

algorithm is effective to solve the BSJS problem in a certain SNR case for BPSK signal

under multi-tone jamming signal and 16QAM signal under broadband-noise or multi-tone

jamming signal. Besides, its anti-noise performance exceeds conventional BSS algorithmsand the other two noisy BSS algorithms. But, some improvements need to be done about its

computing complexity.

Acknowledgments The authors would like to thank the anonymous reviewers for their constructive com-

ments and suggestions. This work is supported in part by Natural Science Foundation of China under Grant

61001106 and National Program on Key Basic Research Project of China under Grant 2009CB320400.

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Y. Duan, H. Zhang

Appendix A: The Update Equations of the Parameters in VBICA Model

In p(S|q),

ti,qi =1

ti,qi

i,qi i,qi +

Mj =1

j

aj i

xj (t)

xj,k=i (t)

(25)

= ti,qi =i,qi

+

Mj =1

j

a2j i

(26)

In p(q),

ti,qi = i,qi pi,qi (27)

ti,qi =ti,qiq i

ti,q i

(28)

i,qi = exp

i,qi

q i

i,q i

(29)

pi,qi = i,qi

ti,qi

12exp

12

ti,qi t2

i,qi i,qi

2i,qi

(30)

i,qi = bi,qi exp

ci,qi

(31)

In (5) and (7),() is Digamma function.In p(),

mi,qi =1

i,qii0mi0 + i,qi

L

t=1

t

i,qisi (t)|q t

i (32)

i,qi = i0 +i,qi

Lt=1

ti,qi (33)

In p(),

bi,qi = 1bi0

+1

2

i,qi1

(34)

ci,qi = ci0 +1

2

Lt=1

ti,qi (35)

i,qi =

Lt=1

ti,qi

s2

i(t)|q ti

2

i,qi

si (t)|q

ti

+2

i,qi

(36)

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In p(A),

maj i =

j

j i

L

t=1si (t)

xj (t)

xj,k=i (t)

(37)

j i = j i +j

Lt=1

s2i (t)

(38)

si (t) =

miqi =1

p

q ti = qi

si (t)|qti

(39)

s2i (t)

=

miqi =1

p

q ti = qi

s2i (t)|qti

(40)

p

qti = qi

=

ti,qi (41)

si (t)|qti

= ti,qi (42)

s2i (t)|qti

=

ti,qi

2+

1

ti,qi

(43)

In p(),

bj =

1

bj+

1

2

L

t=1

xj (t) xj (t)2

1

(44)

cj = cj +L

2(45)

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Author Biographies

Yuling Duan received the master degree from Institute of Communi-

cations Engineering, University of Science and Technology, China in

2012. Now, she works as a teacher at Nanjing Artillery College. Ms.

Duans research interests are in the areas of signal processing and esti-

mation theory, and their applications to communication systems.

Hang Zhang has been a professor and Ph.D. supervisor in Institute of

Communications Engineering, University of Science and Technology,

China. Her research interests are in the areas of communication system,

satellite communication technologies and signal processing, especially,

blind signal processing technique.

jamming separation

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