Blind Separation of DS-CDMA Signals with ICA Method
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Blind Separation of DS-CDMA Signals with ICA Method Miao Yu Nanjing Telecommunication Technology Institute, Nanjing, China Email: [email protected]
Jianzhong Chen, Lei Shen and Shiju Li Nanjing Telecommunication Technology Institute, Nanjing, China Telecommunication School/HangzhouDianzi University, Hangzhou, China Department of ISEE/Zhejiang University, Hangzhou, China [email protected], [email protected], [email protected] Abstract—The estimation of pseudo noise sequence and information sequence is of great importance in the security of DS-CDMA system, which remains a hot research problem in reconnaissance and supervision of wireless communication. In DS-CDMA system, the pseudo noise sequences of different users are uncorrelated and the information sequences of different users are statistical independent, thus independent component analysis (ICA) could be introduced to separate the DS-CDMA signals with little prior knowledge. In multipath situations, the recovered pseudo noise sequence with ICA is overlapped. For DSCDMA signals which utilize m-sequence as pseudo noise sequence, the triple correlation function (TCF) is introduced to eliminate the influence of overlap in this paper, which increases the estimation correct ratio greatly. Under the non-cooperative conditions, it is very difficult for the interceptor to aim at the very beginning of the transmitted signal. Certain offsets between the interceptor and the transmitter deteriorate the blind separation method obviously. The relationship between none convergence and the offset is found out and explained from eigen value decomposition. The validity of the proposed method and theory analysis is proved well by the simulation results at last. Index Terms—DS-CDMA, ICA, blind estimation, pseudo noise sequence, TCF, average filter, convergence
I. INTRODUCTION Direct sequence spread spectrum (DSSS) signal is the typical form of low probability of interception signals, which has been used as secure communication for several decades. Thus the estimation of information sequence and pseudo noise sequence are of great importance in electronic warfare or wireless communication supervision. However, the power spectrum density of DS-CDMA signals is so low for common technology to estimate these critical sequences. m-sequences are easy to generate by linear feedback shift registers (LFSR), which are widely used DS-CDMA system as pseudo noise sequences. The situation mManuscript received January 8, 2010; revised October 29, 2010; accepted November 1, 2010. China Postdoctoral Science Foudation, No.20100471858.
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sequences are used as pseudo noise sequences and the symbol interval is equal to the period of m-sequences is mainly considered in this paper. In 1969, Massey [1] proposed a cipher analysis method of m-sequence utilizing the logic character of LFSR. Massey's method make use of only first order statistic therefore it is sensitive to noise. In 2000, Massey algorithm was updated by P.C.Hill [2] with maximum vote criterion to find the primitive polynomial of msequence, but when information modulation exists, the method is not available any more. In 2000, G.Burel [3] and cooperators proposed the estimation method of pseudo noise based on principal component analysis (PCA). In 2004, G.Burel’s PCA method is extended in the situation of two users [4], and the performance is analyzed from the idea of eigen value decomposition (EVD) [5]. But G.Burel’s method was not available in multiuser situations. E.R.Adams proposed a method based on triple correlation function (TCF) to estimate the primitive polynomial of m-sequences [6-8]. TCF was a certain 3rd order statistic, it could resisted additive white Gaussian noise (AWGN) and the influence of multipath in some extent. But the method was only fit for pure msequences, when information modulation exists, its performance declines greatly. With the development of DS-CDMA signal processing, some efficient methods of parameter estimation for carrier frequency [9], chip interval [10], and symbol interval [11] of the DS-CDMA signals have been developed, which lies the foundation for ICA or BSS to deal with the DS-CDMA signals. In DS-CDMA system, information sequences of different users remains statistically independent and the pseudo noise sequences of different users are uncorrelated, so the DS-CDMA signals satisfy the theory model of blind source separation (BSS) [12] very well. As the typical approach of BSS, ICA could be introduced to achieve the blind separation of DS-CDMA signals in non-cooperative situations, only when few parameters have been estimated [13]. From the Nyquist sampling theorem, the sample rate is much higher than the chip rate in DS-CDMA system, so there are many sample points during the interval of one
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chip. However, in the original blind separation of DSCDMA signals with ICA, only one sample point from each chip is used, other samples are discarded wastefully. According to this, average filter is introduced to improve the performance. Besides the preciseness, convergence is also very important for an algorithm. Under the noncooperative conditions, it is very difficult for the interceptor to aim at the very beginning of the transmitted signal. Certain offsets between the interceptor and the transmitter deteriorate the convergence greatly. In this paper, the relationship between the convergence and the offset is found out which is very useful in choosing the ideal offset in practice to achieve better performance. Besides the estimation of information sequence, Shen [14] adopted the ICA method to deal with the problem of pseudo noise estimation in multipath situations, but no further results of the estimation correct rate were given out. Accordingly, a method based on ICA and TCF is proposed here to estimate of m-sequences in the complex situations of multiuser and multipath. The validity of the proposed method and theory analysis is proved by simulation results at last. This paper is organized as follows: in section 2, the mathematic model of DS-CDMA signal is established, in section 3, the blind separation of DS-CDMA signal with ICA is introduced, in section 4, the estimation of pseudo noise sequence with ICA and TCF are mainly discussed, in section 5, average filter and the convergence characteristic are mainly focused, in section 6, some important simulation experiment results are given out to testify the performance of the proposed method. II. MATHEMATIC MODEL OF DS-CDMA SIGNALS An asynchronous BPSK DS-CDMA system with multiuser transmitting over Rayleigh fading channel is mainly considered in this paper. The received signal could be represented as r (t ) = s (t ) + n (t ) (1)
n ( t ) is noise and s ( t ) is the received DS-CDMA signal, K
s ( t ) = ∑ sk ( t )
(2)
Pk
power of the kth user
delay for lth path of the kth user
akl
attenuation for the lth path of the kth user
I k ,m
mth symbol of the kth user, I k ,m ∈ {−1, +1}
d kl
hk ( t )
convolution of transmitter filter, channel response and receiver filter convolution of the pseudo noise sequence with f (t)
ck , p
pth chip of the kth user’s pseudo noise sequence, ck , p ∈ {−1, +1}
I k ,m
mth symbol of the kth user
f (t )
Considering the practical wireless channel, the delay factor, d kl , is uniformly distributed between 0 and P , P is the period of pseudo noise sequence, the attenuation, akl ,obeys Rayleigh distribution. III. BLIND SEPARATION OF DS-CDMA SIGNALS WITH ICA METHOD When received, the signal should be demodulated and band-pass filtered firstly, which is represented as r (t ) = s (t ) + n (t ) (4) K
K
Lk
M
s ( t ) = ∑ sk ( t ) = ∑∑∑ Pk ⋅ akl I k , m hk ( t − mTb − d kl Tc ) k =1
r (t )
k =1 l =1 m =1
(5) is not suit to be process by ICA directly, it
should go through two preprocessing steps, centering and whitening. During centering, the mean value is subtracted (6) r ( t ) = r ( t ) − E ⎡⎣ r ( t ) ⎤⎦ Then r ( t ) is sampled once a chip, and the samples are
shaped to M-1 vectors with the degree of 2P×1 [16]. rm = ⎡⎣ r ( t − ( m − 1) P + 1) ; r ( t − ( m − 1) P + 2 ) ; ; r ( t − ( m + 1) P ) ⎤ ⎦
T
(7)
In asynchronous situations, the offset between the interceptor and the transmitter is supposed to be D chips. Fig .1 is the sketch map of rm . The situation that d kl > D is revealed in Fig. 1(a), while d kl < D is depicted in Fig. 1(b) [17].
k
where sk ( t ) is the kth user’s DS-CDMA signal, its complex band pass representation could be modeled as[15] Lk
M
sk ( t ) = Pk ∑∑ akl I k , m hk ( t − mTb − d kl Tc ) l =1 m =1
(3)
⋅ exp ⎡⎣ 2π jf 0 ( t − d kl Tc ) ⎤⎦
Figure 1. Sketch map of rm
P
where hk (t ) = ∑ ck , p ⋅ f ( t − pTc ) , parameters in (3) is p =1
explained in Table Ⅰ, TABLE I.
EXPLANATION OF THE PARAMETERS
Tb
symbol interval
K
number of total users
Tc
chip interval
M
number of intercept symbols
f0
carrier frequency
Lk
number of paths for the kth user
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rm could be written as Lk Lk Lk K ⎧ ⎫ rm = ∑ ⎨ I k , m −1 ∑ akl hklB + I k , m ∑ akl hklM + I k , m +1 ∑ akl hklF ⎬ k =1 ⎩ l =1 l =1 l =1 ⎭ (8) + nm which is composed of the end part of last spreaded symbol, the complete current spreaded symbol and the front part of the next spreaded symbol. I k , m −1 is the last
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symbol, I k , m is the current symbol and I k , m +1 is the next
where y is the stochastic variable, yg is the Gaussian
symbol, nm is 2P samples of noise in one symbol.
stochastic variable whose variance is the same as y and
B kl
M kl
F kl
h , h and h is last part, complete part and front part of th
k user’s pseudo random sequence with d kl chip delay. Equation (8) could be written in a vector form as r = ∑∑ akl ⋅ H klT ⋅ I k,m + n = H ⋅ I + n (9) k
l
H ( y ) in (17) is the differential entropy of y .
Original estimation of negentropy makes use of high order statistic (HOS), which suffers from the none robustness caused by HOS inherently. So new approximations were developed as [19] p
f ( y ) ≈ ∑ ki ⎡⎣ E {G i ( y )} − E {Gi ( v )}⎤⎦
T
I k,m = ⎡⎣ I k , m −1 , I k , m , I k , m +1 ⎤⎦ , let Dkl = mod(d kl − D, P )
2
(18)
i =1
then H could be written as
Two nonlinear functions of G ( i ) are recommended in
⎧ ⎪hk ( P − Dkl + 1), hk ( P − Dkl + 2), , hk ( P ) , 0, 0, , 0 ⎪ 2 P − Dkl Dkl ⎪⎪ T H kl = ⎨0, 0, , 0, hk (1), hk (2), , hk ( P ) , 0, 0, , 0 ⎪ Dkl P − Dkl P ⎪ ⎪0, 0, , 0, hk (1), hk (2), , hk ( P − Dkl ) P − Dkl ⎪⎩ P + Dkl (10) The covariance matrix could be calculated as follow [18]: 1 M R= (11) ∑ {rm ⋅ rmT } M − 1 m =1
[19]
T kl
where ( ⋅ )T means the transpose of the vector. From the singular value decomposition of R , the subspace of the pseudo random sequence could be estimated by EVD, 0 ⎡ λS + N1 I ⎤ ⎢ 3 K ×3 K ⎥ ⎡U ST ⎤ R = [U S U N ] ⎢ (12) ⎥⎢ T ⎥ N1 I ⎢ 0 ⎥ ⎣⎢U N ⎦⎥ ⎢⎣ ( 2 P − 3 K )× ( 2 P − 3 K ) ⎥ ⎦ where U S is the signal subspace, U N is the noise subspace, λS is the signal energy arrayed in decreasing order and I is the identity matrix. When projected to the signal subspace, the correlation between different users’ signals could be removed, which is called whitening. −1/ 2 (13) X = ( λS + N1 I ) U ST ⋅ r After the Whitening preprocessing step, X is the stochastic variable of zero mean and unit variance. Let
B = ( λS + N1 I )
−1/ 2
C = ( λS + N1 I )
U ST ⋅ H
−1/ 2
U ST
(14) (15)
When (9), (14) and (15) are imported, (13) turns into X = B ⋅ I + C ⋅ n = B ⋅ I + n' (16) Equation (16) is just the mathematic model of ICA with noise, so the unmixing matrix W could be estimated by maximizing certain object functions. Among most ICA algorithms, fastICA has become the most popular algorithm for its convergence and simplicity. From our experience, fastICA based on negentropy is much robust and precise, so it is adopted in this paper. Negentropy of a stochastic variable is defined as (17) f ( y ) = H ( yg ) − H ( y )
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1 ⎧ ⎪G1 ( u ) = a log cosh ( au ) ⎨ ⎪G2 ( u ) = − exp ( −u 2 / 2 ) ⎩
(19)
where 0 < a < 1 is some suitable constant and the differential function of (19) are ⎧⎪ g1 ( u ) = tanh ( au ) (20) ⎨ 2 ⎪⎩ g 2 ( u ) = u exp ( −u / 2 ) Then fastICA based on negentropy calculates the unmixing matrix through following steps [20,21]: 1. Choose an initial (e.g. random) weight vector w
{
} {
}
2. Let w+ = E xg ( wT x ) − E g ' ( wT x ) w
(21)
3. Let w = w+ / w+
4. If not converged, go back to 2. When W = B −1 is calculated out, the original information sequence could be estimated by Iˆ = W ⋅ X ≈ I + W ⋅ n' (22) IV BLIND ESTIMATION OF PSEUDO NOISE SEQUENCE WITH ICA AND TCF When left multiplied by W and right multiplied by the matrix of pseudo noise sequences, H , (13) turns into [14]: W ⋅ ( λS + N1 I )
−1/ 2
U ST ⋅ H ⋅ H T = H
(23)
From (12), H H ≈ U S ( λS + N1 I ) U , thus the matrix of pseudo noise sequence could be estimated by −1/ 2 H = W ⋅ ⎡( λS + N1 I ) U ST ⎤ ⋅ ⎡⎣U S ( λS + N1 I ) U ST ⎤⎦ ⎣ ⎦ (24) 1/ 2 T = W ⋅ ( λS + N1 I ) U S T
T S
If there is only one path for each user, from the expression above, the pseudo noise sequence could be recovered directly. But in the multipath situation, the same user’s pseudo random sequences of different paths overlapps together, which could not be estimated directly. According to this, triple correlation function (TCF) is introduced to suppress the influence of overlap. TCF method utilizes the shift-and-add property of msequence, M0 ⊕ MD = ME (25)
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where M 0 is the original m-sequence, M D is the shift form of M 0 with D chips delay, M E is another shift form of M 0 with E chips delay. Triple correlation function is a 3rd statistic in fact, C3, x (τ 1 ,τ 2 ) = E { x ( t ) x ( t + τ 1 ) x ( t + τ 2 )} (26) for m-sequences, let τ 1 = pTc ,τ 2 = qTc , the TCF of msequence could be written as[6]: C3, x ( p, q ) = E { x ( n ) x ( n + p ) x ( n + q )} (27) let x ( n ) be M 0 , x ( n + p ) be M p , and x ( n + q ) be
q = D1 is still the maximum of C3, hkj ( p, q ) . That is, TCF
could eliminate the influence of multipath efficiently. V. AVERAGE FILTER AND CONVERGENCE ANALYSIS
A. Average Filter in the Blind Separation In the original blind separation of DS-CDMA signals, r ( t ) is sampled once in a chip as Fig. 2 (a) [23]. 1,2,3,…,N
1,2,3,…,N
… … …
1,2,3,…,N
M q , E { x ( n ) x ( n + p )} is equal to a shift form of x ( n )
with some v chips delay, supposing the shift form is M v , then TCF could be written as: C3, x ( p, q ) = E {M v ⋅ M q } = Rxx ( v, q ) (28)
(a) Interval sample
1,2,3,…,N
1,2,3,…,N
Σ/N
Σ/N
Euqation (28) is the autocorrelation function of M v and M q , according to the autocorrelation character of msequence, if and only if v = q , Rxx ( v, q ) reached its maximum, for other v ≠ q , Rxx ( v, q ) gets its minimum. The TCF of m-sequence is of three demisions, when plotted in x-y-z coordinate system, TCF is something like peaks erected above the ground, whose pattern is uniquely determined by the primitive polynomial of msequence. Therefore, from the pattern of TCF, the primitive polynomial could be uniquely determined. In multipath situation of, data recovered by ICA are the overlapped pseudo noise sequence of different paths. Let the kth estimated data is xk ( n ) = wk ( n ) + vk ( n ) , wk ( n ) is the overlap pseudo noise sequence in the kth
degree and vk ( n ) is noise in the same degree. According to the additive character of HOS and HOS value of Gaussian signal tends to be zero, TCF of xk ( n ) is Lk
C3, xk ( p, q ) = C3, wk ( p, q ) + C3,vk ( p, q ) ≈ ∑ akl3 C3, hkl ( p, q ) l =1
(29) Two paths of hki and hkj are mainly considered here, other paths could be analyzed through the same way. 1 N C3, hki ( p, q ) = ∑ aki3 hki ( n ) hki ( n + p ) hki ( n + q ) (30) N n =1 Where hki is the d ki chips delay form of the kth pseudo noise sequence
hk ,
∑ h (n) h (n + p) ki
ki
could be
considered as a shift form of hk with some D1 chips
delay. If q = D1 , C3, hki ( p, q ) reached its maximum, for other q ≠ D1 , C3, hki ( p, q ) is the minimum. For hkj , C3, hkj ( p, q ) =
1 N
N
∑a m =1
∑ h (m) h (m + p) kj
kj
3 kj kj
h
( m ) hkj ( m + p ) hkj ( m + q )
(31)
could be considered as a shift form
of hk with E1 chips delay. m = mod ( n + d ki − d kj , P ) is different from n , but p does not change, so the point © 2011 ACADEMY PUBLISHER
… … …
1,2,3,…,N
Σ/N
Σ/N
(b) Averaged in each chip Figure 2. Sketch map of the interval sample and averaged filter.
Generally speaking, the sample rate is much higher than the chip rate in DS-CDMA system, so there are many sample points during one chip. Many other samples are discarded wastefully in the original method. So it is easy to think about averaging the samples in each chip as depicted in Fig. 2(b). The proposed average filtered method of blind separation could be depicted in Fig. 3. Object Function
Average Filter
Centering
Whitening
Unmixing Matrix
Figure 3. Flow chart of the average filtered blind separation
Average filter is introduced at the beginning of the flow. The noise suppression acquired by averaging filter is proportional to the number of samples averaged [22]. If sample rate is eight times of chip rate, the improvement of average filter is about 10 log10 8 ≈ 9 dB.
B. Relationship between Convergence characteristic and offset Under the non-cooperative conditions, it is very difficult for the interceptor to aim at the very beginning of the transmitted DS-CDMA signals. Supposing the offset between the interceptor and the transmitter is D chips, then the offset D has great influence on the convergence of the blind separation method. It is very obviously that certain offset deteriorates the convergence performance greatly. Look back on Fig. 1, the 2 P averaged chips ( rm ) from the dash line with up arrow are mainly considered here. As depicted in Fig. 1(a), D < d kl , rm are composed of the last d kl − D chips of the last spreaded symbol, the complete current spreaded symbol and the front
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P + D − d kl chips of the next spreaded symbol. In Fig.
1(b), D > d kl , so the number of chips of the last symbol, next symbol in rm is P + d kl − D and D − d kl .Because mod(d kl − D, P ) = mod( P + d kl − D, P ) and mod( D − d kl , P ) is equal to mod( P + D − d kl , P ) , let Dkl = mod(d kl − D, P ) , the 2 P averaged chips could be written in a uniform form, rm is composed of the last Dkl chips of the last spreaded symbol, complete current spreaded symbol and the front P − Dkl chips of the next spreaded symbol. From Fig .1 and (8), there are three energy values for one user, for K users, there will be 3K values in total. The diagonal of λS in (12) are just these 3K energy Lk
Lk
l =1
l =1
frequency is set to be f s = 504MHz , that is, each chip is sampled 8 times. The attenuation factor, akl , are chosen from Rayleigh distribution. and the delay parameter, d kl , obeys the uniform distribution of 0—P chips, where P = 63 is the period of the pseudo noise sequence.
A. Estimation of Information Sequence Fig. 4 shows the waveform of the original information sequence and estimated information sequence.
values. Let EkB = I k , m −1 ∑ akl hklB , EkM = I k , m ∑ akl hklM Lk
and EkF = I k , m +1 ∑ akl hklF . Among EkB , EkM and EkF , l =1
EkM is the biggest because it is corresponding to the complete spread symbol. From (11), it is reasonable to estimate EkB and EkF by the length of none zeros chips
in hklB and hklF .Thus the difference between EkB and EkF could be approximated by the difference between Dkl and P − Dkl . d kl obeys the uniform between 0 and P, so its mathematical expectation is P/2 . When 0 < D < P / 2 , 0 < Dkl < P / 2 , when P / 2 < D < P , P > Dkl > P / 2 . Let ∆ = Dkl − ( P − Dkl )
(32)
Let d kl = P / 2 , when D = 0 , Dkl and P − Dkl are both similar to P / 2 , and the difference value, ∆ , is very small. When D ≈ P / 2 , Dkl is very small and P − Dkl is much bigger, thus ∆ ≈ P . When D = P , the difference value turns back to zero again. Thus, the offset D causes energy transfer between EkB and EkF . When the difference, ∆ , is too big, one of EkB and EkF is much smaller than the other. Thus the subspace of U S corresponding to the smaller energy is much similar to the noise subspace. Only when EkB and EkF is very close, should the subspace of pseudo noise sequence be much distinguishable from the noise subspace. So the offsets which make EkB close to EkF enable the algorithm converge much easily. VI. SIMULATION RESULTS In this section, several experiments have been completed to testify the performance of the proposed method and theory analysis above. All these experiments were accomplished by MATLAB®R2009a. The simulation time is T = 2 × 10−3 second, the symbol rate Rb = 106 bps , the chip rate Rc = 63M chips/s. The sample
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Figure 4. Original and estimated information sequence
Fig. 4 (b) is one user's estimated information sequence which is acquired when SNR=-12dB. The users number and the multipath number are both set to be six, the offset between the interceptor and the transmitter is 15 chips. Compared with the transmitted information sequence in Fig. 4 (a), when judged by zero threshold, the information sequence could be recovered with a high correct rate of 96.30% in this experiment. Fig .5 is the bit error rate (BER) curve which is counted when the algorithm converges. SNR is set from 21dB to -3dB with the step of 3dB, and it is the result of 500 Monte Carlo experiments in average.
Figure 5. Bit error rate of the blind separation method
BER of original blind separation method declines slowly with the increase of SNR. While BER with average filter is much lower than original method. Seen from Fig. 5, the average filter brings great SNR improvement than original method. Partial data in Fig. 5 are shown in Table . TABLE II.
SNR ICA ICA-AF
-15dB 0.4173 0.0943
-12dB 0.3671 0.0306
BER VERSUS SNR
-9dB 0.2392 0.0066
-6dB 0.0954 0.0008
-3dB 0.0316 0
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When SNR=-6dB, the BER of original ICA method is 9.54%, the average filter method get the close BER of 9.43% when SNR=-15dB. So, the distance between these two methods is about 10 log10 ( 504 / 63) ≈9dB, which proves the correctness of the analysis above.
B. Estimation of Pseudo Noise Sequence The situation of single path is firstly considered in this subsection. The number of users is set to be K = 2 . Fig. 6 is the true value and the estimated value of the first user's pseudo noise sequence.
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The number of total users is set to be K = 4 , and the number of paths is set to be L = 3 , i.e., there are four users totally and each user's DS-CDMA signal has three paths. The delays are set to be d = [50, 57, 45; 62, 62, 38; 48, 43, 48; 59, 2, 48]; the attenuation factors are set to be A = [1.7906, 0.8170, 0.8977; 0.7126, 1.4681, 1.2755; 0.6493, 0.9200, 0.1234; 1.5560, 2.6588, 2.2564]. The asynchronous offset between the interceptor and the transmitter is still set to be D = 15 chips, and the noise is set to be SNR = -12 dB. Fig. 7(a) is the estimated data of the first user with ICA, from which the original pseudo noise sequence could not be recognized any more. Fig. 7(b) is the true pseudo noise sequence with a shift of Dkl = 30 chips. Fig. 7(c) is the overlap form of true pseudo noise sequence with the shift of 35, 42 and 30 chips respectively. Compared with Fig. 7(a), when multipaths exist, the estimated sequence from ICA is very similar to the overlap form of true pseudo noise sequence. Because of the influence of overlap, the true value could not be acquired directly. So, TCF is introduced to deal with the multipah influence of overlap. When plotted in the x-y-z coordinate system, TCF is much like peaks erecting above the ground. The TCF peaks of first user’s true pseudo noise sequence in Fig. 8.
Figure 6. Estimated pseudo noise sequence under single path
Fig. 6 (a) shows the estimated pseudo noise sequence of the first user when SNR=-12dB. The delay factor of the first user is set to be 10 chips, the offset between the transmitter and the interceptor is set to be 15 chips. From the analysis above, there will be Dkl = mod(10-15,63) = 58 chips zeros at the beginning of the estimated sequence, and the number of zeros at the end of the estimated sequence is P − Dkl = 5 chips. To be compared easily, the shift form with Dkl chips of true pseudo noise sequence is shown in Fig. 6(b). From Fig. 6(a) and Fig. 6(b), when there is only a single path for each user, the estimation sequence from ICA is the accurate duplicate of ture pseudo noise sequence. The estimation of the second user is much similar to the first user, its waveform is not illustrated here for the limit of pages. Then comes the problem of multipath. Fig. 7 reveals the waveforms of the first user's pseudo noise sequence before and after being processed by ICA.
Figure 8. TCF of true pseudo noise sequence
Because there is no influence of multipath, only one maximum value in each row and each column among all TCF values, while other values are very small and close to 1/P. From the analysis above, the pattern of peaks from TCF is uniquely determined by primitive polynomial of the m-sequence, thus, it is probable to derive the primitive polynomial from the TCF values. Fig. 9 shows the TCF of the estimated data.
Figure 7. Estimated pseudo noise sequence under multipath Figure 9. TCF of estimated data
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Though some crossed items swell above the horizontal surface in Fig. 9, the peaks of the TCF are still very distinguishable. The relationship between the estimated correct rate and SNR is revealed in Fig. 10.
Figure 10. Estimation correct rate versus SNR
Comparing these two lines, the improvement of the ICA-TCF method is very obviously.
Fig. 12 reveals the relationship between the number of none convergence times and the signal to noise ratio (SNR). “ ― o” represents the total none convergence number for a SNR value, and “ ― ” represents none convergence times when P / 4 < mod( D, P) < 3P / 4 . When SNR=-18dB, the none convergence times of special offsets are almost the half of the total number. When SNR increases, special offsets occupy the majority of none convergence times. When SNR=-12dB, the number of none convergence times caused by special offset is 1663 occupying the 79.57 percentage of the total number. Define the sum of the K singular values as 2K ⎧ ⎪ S1 = ∑ λS (i ) ⎪ i = K +1 (33) ⎨ 3K ⎪S 2 = λ ( i ) ∑ S ⎪⎩ i = 2 K +1 Fig. 13 reveals the relationship between the difference of S1 and S2 with the change of offset value.
C. Convegence analysis of the Blind Sepration Suppose the offset between the interceptor and the transmitter is D chips. In the experiments, certain offsets deteriorate the convergence of the blind separation method obviously, which is exhibited in Fig. 11.
Figure 13. Difference of eigen value with D
Figure 11. Number of none convergence times versus the offset
Fig. 11 shows the statistic results of none convergence times with different offset value when SNR=-18dB, 15dB and -12dB. D changes from 1 to 252 ( 4P ), there are 100 Monte Carlo experiments for each D value. It is clear that the algorithm converges difficult when mod( D , P ) locates in the middle of the period of the pseudo noise sequence.
Figure 12. Relationship between non-converging times and SNR
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Fig. 13 is almost the same as Fig. 11, which indicates that the theory analysis above is reasonable. Offset causes the energy transfer between two small singular values of each user, which makes certain signal subspace undistinguished with the subspace of noise, resulting in the none-converging of the algorithm at last. VII. CONCLUSION In DS-CDMA system, information sequence of different users remain statistical independent, and the pseudo noise sequence of different users keeps uncorrelated each other, so ICA could be introduced to achieve the blind estimation of information sequence and pseudo noise sequence, which is of great importance in the security of DS-CDMA signals. When there is only one path for each user, the blind estimated pseudo sequence is the accurate duplicate of the original pseudo sequence. However, under the conditions of multipath, the estimated sequence of DS-CDMA signals with ICA method is the overlap form of the true pseudo noise sequence, from which the original pseudo noise sequence could not be recovered directly. TCF is introduced in this paper to deal with the influence of multipath, which increases the estimation correct rate greatly. Average filter is also introduced to suppress the noise in some extent. When the sample rate is much higher than the chip rate, a big improvement of SNR could be acquired by the
JOURNAL OF NETWORKS, VOL. 6, NO. 2, FEBRUARY 2011
average filter. Besides the preciseness, the relationship between convergence and the offset is found out and explained from the view of singular value decomposition. REFERENCES [1] J. Massey, “Shift register synthesis and BCH decoding,” IEEE Trans. on Information Theory, 1969, vol. 15(1), pp. 122-127 [2] P. Hill and M. Ridley, “Blind estimation of direct-sequence spread spectrum m-sequence chip codes,” IEEE 6th International Symposium on Spread Spectrum Techniques and Applications, vol. 1, 2000. [3] G. Burel and C. Boude, “Blind estimation of the pseudo random sequence of a direct sequence spread spectrum signal,” IEEE MILCOM2000. [4] C. Nzoza, R. Gautier, and G. Burel, “Blind synchronization and sequences identification in CDMA transmissions,” IEEE MILCOM 2004, vol. 3. [5] C. Bouder, S. Azou, and G. Burel, “Performance analysis of a spreading sequence estimator for spread spectrum transmissions,” Journal of the Franklin Institute vol. 341(2004), pp. 595-614. [6] E. R. Adams and P. C. J. Hill, “Detection of direct sequence spread spectrum signals using higher-order statistical processing,” IEEE ICASSP-97, vol. 5, 1997. [7] K. Batty and E. R. Adams, “Detection and blind identification of m-sequence codes using higher-order statistics,” Proceedings of the IEEE Signal Processing Workshop, pp. 16-20, 1999. [8] E. R. ADAMS, “Identification of pseudo-random sequences in DS/SS intercepts by higher-order statistics,” 2004. [9] D. Hill and J. Bodie, “Carrier detection of PSK signals,” IEEE Trans. on Communications, vol. 49(3), pp. 487-496, 2001. [10] J. Yan and J. Hongbing, “A cyclic-cumulant based method forDS-SS signal detection and parameter estimation,” IEEE International Symposium on Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications, vol. 2, 2005. [11] G. Burel, “Detection of spread spectrum transmissions uing fluctuations of correlation estimators,” IEEE ISPACA, 2000. [12] O. Ekici and A. Yongacoglu, Application of noisyindependent component analysis for CDMA signal separation, VTC-Fall, vol. 5, 2004,. [13] K. Raju, T. Ristaniemi, J. Karhunen, and E. Oja, “Jammer suppression in DS-CDMA arrays using independent component,” IEEE Trans. On Wireless Communications, vol. 5(1), pp. 77-82, 2006. [14] L. Shen, S. J. Li, Y. B. Wang, et., “Blind estimation of pseudo-random sequences of spread spectrum signals in multi-paths,” Journal of Zhejiang University, vol.41(11), pp. 1828-1833, 2007. [15] A. Viterbi,et., CDMA: Principles of spread spectrum communication, MA, US:Addison-Wesley Reading, 1995. [16] T. Ristaniemi, and J. Joutsensalo, “Advanced ICA-based receivers for block fading DS-CDMA channels,” PIMRC2000, vol. 1, 2000. [17] M. Yu, S. J. Li, and J. Z. Chen, “Convergence characteristics analysis on blind separation of DS-CDMA Signals with ICA method,” WCNIS2010, pp. 238-242, 2010. [18] L. Smith, “A tutorial on principal components analysis”, Cornell University, USA,2002.51:52. [19] A. Hyvarinen, “Fast and robust fixed-point algorithms for independent component analysis,” IEEE Trans. Neural Networks, vol. 10, pp. 626-634, 1999.
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[20] A. Hyvarinen and E. Oja, “Independent component analysis: algorithms and applications,” IEEE Trans. Neural Networks, vol.13, pp. 411-430, 2000. [21] A. Hyvärinen, J. Karhunen, and O. Oja, Independent component analysis, John Wiley & Sons, Inc, United States, 2001. [22] R. G. Lyons, Understanding Digital Signal Processing, 2nd ed., Prentice Hall PTR, New Jersey, USA, 2004. [23] M. Yu, J. Z. Chen, and S. J. Li “Average filtered blind separation of DS-CDMA signals with ICA method,” WCNIS2010, pp. 234-237, 2010.
Miao Yu was born in Jilin, China, on August 29, 1975. He received M.Sc. and Ph.D. degrees in information and communication engineering from Zhejiang University, Hangzhou, China, in 2005 and 2009 respectively. His research interests include spread spectrum signal processing and independent component analysis.
Jianzhong Chen was born in Taixing, China, on June 18, 1962. He received M.Sc. degree in wireless communication engineering in 1993. His research interests include wireless communication and signal processing.
Lei Shen was born in Zhoushan, China, on March 15, 1977. He received B.Sc. and Ph.D. degrees information and communication engineering from Zhejiang University, Hangzhou, China, in 2002 and 2007 respectively. His research interests include wireless comunication and signal processing. Dr. Shen is an associate professor of Hangzhou Dianzi University, China.
Shiju Li was born in Linhai, China, on February 28, 1947. He graduated from the radio science and electronic engineering department of Zhejiang University in 1969. His research interests include digital communication and mobile multimedia communication. Prof. Li is the author of more than 100 papers and 3 books.
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Blind Separation of DS-CDMA Signals with ICA Method Miao Yu Nanjing Telecommunication Technology Institute, Nanjing, China Email: [email protected]
Jianzhong Chen, Lei Shen and Shiju Li Nanjing Telecommunication Technology Institute, Nanjing, China Telecommunication School/HangzhouDianzi University, Hangzhou, China Department of ISEE/Zhejiang University, Hangzhou, China [email protected], [email protected], [email protected] Abstract—The estimation of pseudo noise sequence and information sequence is of great importance in the security of DS-CDMA system, which remains a hot research problem in reconnaissance and supervision of wireless communication. In DS-CDMA system, the pseudo noise sequences of different users are uncorrelated and the information sequences of different users are statistical independent, thus independent component analysis (ICA) could be introduced to separate the DS-CDMA signals with little prior knowledge. In multipath situations, the recovered pseudo noise sequence with ICA is overlapped. For DSCDMA signals which utilize m-sequence as pseudo noise sequence, the triple correlation function (TCF) is introduced to eliminate the influence of overlap in this paper, which increases the estimation correct ratio greatly. Under the non-cooperative conditions, it is very difficult for the interceptor to aim at the very beginning of the transmitted signal. Certain offsets between the interceptor and the transmitter deteriorate the blind separation method obviously. The relationship between none convergence and the offset is found out and explained from eigen value decomposition. The validity of the proposed method and theory analysis is proved well by the simulation results at last. Index Terms—DS-CDMA, ICA, blind estimation, pseudo noise sequence, TCF, average filter, convergence
I. INTRODUCTION Direct sequence spread spectrum (DSSS) signal is the typical form of low probability of interception signals, which has been used as secure communication for several decades. Thus the estimation of information sequence and pseudo noise sequence are of great importance in electronic warfare or wireless communication supervision. However, the power spectrum density of DS-CDMA signals is so low for common technology to estimate these critical sequences. m-sequences are easy to generate by linear feedback shift registers (LFSR), which are widely used DS-CDMA system as pseudo noise sequences. The situation mManuscript received January 8, 2010; revised October 29, 2010; accepted November 1, 2010. China Postdoctoral Science Foudation, No.20100471858.
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sequences are used as pseudo noise sequences and the symbol interval is equal to the period of m-sequences is mainly considered in this paper. In 1969, Massey [1] proposed a cipher analysis method of m-sequence utilizing the logic character of LFSR. Massey's method make use of only first order statistic therefore it is sensitive to noise. In 2000, Massey algorithm was updated by P.C.Hill [2] with maximum vote criterion to find the primitive polynomial of msequence, but when information modulation exists, the method is not available any more. In 2000, G.Burel [3] and cooperators proposed the estimation method of pseudo noise based on principal component analysis (PCA). In 2004, G.Burel’s PCA method is extended in the situation of two users [4], and the performance is analyzed from the idea of eigen value decomposition (EVD) [5]. But G.Burel’s method was not available in multiuser situations. E.R.Adams proposed a method based on triple correlation function (TCF) to estimate the primitive polynomial of m-sequences [6-8]. TCF was a certain 3rd order statistic, it could resisted additive white Gaussian noise (AWGN) and the influence of multipath in some extent. But the method was only fit for pure msequences, when information modulation exists, its performance declines greatly. With the development of DS-CDMA signal processing, some efficient methods of parameter estimation for carrier frequency [9], chip interval [10], and symbol interval [11] of the DS-CDMA signals have been developed, which lies the foundation for ICA or BSS to deal with the DS-CDMA signals. In DS-CDMA system, information sequences of different users remains statistically independent and the pseudo noise sequences of different users are uncorrelated, so the DS-CDMA signals satisfy the theory model of blind source separation (BSS) [12] very well. As the typical approach of BSS, ICA could be introduced to achieve the blind separation of DS-CDMA signals in non-cooperative situations, only when few parameters have been estimated [13]. From the Nyquist sampling theorem, the sample rate is much higher than the chip rate in DS-CDMA system, so there are many sample points during the interval of one
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chip. However, in the original blind separation of DSCDMA signals with ICA, only one sample point from each chip is used, other samples are discarded wastefully. According to this, average filter is introduced to improve the performance. Besides the preciseness, convergence is also very important for an algorithm. Under the noncooperative conditions, it is very difficult for the interceptor to aim at the very beginning of the transmitted signal. Certain offsets between the interceptor and the transmitter deteriorate the convergence greatly. In this paper, the relationship between the convergence and the offset is found out which is very useful in choosing the ideal offset in practice to achieve better performance. Besides the estimation of information sequence, Shen [14] adopted the ICA method to deal with the problem of pseudo noise estimation in multipath situations, but no further results of the estimation correct rate were given out. Accordingly, a method based on ICA and TCF is proposed here to estimate of m-sequences in the complex situations of multiuser and multipath. The validity of the proposed method and theory analysis is proved by simulation results at last. This paper is organized as follows: in section 2, the mathematic model of DS-CDMA signal is established, in section 3, the blind separation of DS-CDMA signal with ICA is introduced, in section 4, the estimation of pseudo noise sequence with ICA and TCF are mainly discussed, in section 5, average filter and the convergence characteristic are mainly focused, in section 6, some important simulation experiment results are given out to testify the performance of the proposed method. II. MATHEMATIC MODEL OF DS-CDMA SIGNALS An asynchronous BPSK DS-CDMA system with multiuser transmitting over Rayleigh fading channel is mainly considered in this paper. The received signal could be represented as r (t ) = s (t ) + n (t ) (1)
n ( t ) is noise and s ( t ) is the received DS-CDMA signal, K
s ( t ) = ∑ sk ( t )
(2)
Pk
power of the kth user
delay for lth path of the kth user
akl
attenuation for the lth path of the kth user
I k ,m
mth symbol of the kth user, I k ,m ∈ {−1, +1}
d kl
hk ( t )
convolution of transmitter filter, channel response and receiver filter convolution of the pseudo noise sequence with f (t)
ck , p
pth chip of the kth user’s pseudo noise sequence, ck , p ∈ {−1, +1}
I k ,m
mth symbol of the kth user
f (t )
Considering the practical wireless channel, the delay factor, d kl , is uniformly distributed between 0 and P , P is the period of pseudo noise sequence, the attenuation, akl ,obeys Rayleigh distribution. III. BLIND SEPARATION OF DS-CDMA SIGNALS WITH ICA METHOD When received, the signal should be demodulated and band-pass filtered firstly, which is represented as r (t ) = s (t ) + n (t ) (4) K
K
Lk
M
s ( t ) = ∑ sk ( t ) = ∑∑∑ Pk ⋅ akl I k , m hk ( t − mTb − d kl Tc ) k =1
r (t )
k =1 l =1 m =1
(5) is not suit to be process by ICA directly, it
should go through two preprocessing steps, centering and whitening. During centering, the mean value is subtracted (6) r ( t ) = r ( t ) − E ⎡⎣ r ( t ) ⎤⎦ Then r ( t ) is sampled once a chip, and the samples are
shaped to M-1 vectors with the degree of 2P×1 [16]. rm = ⎡⎣ r ( t − ( m − 1) P + 1) ; r ( t − ( m − 1) P + 2 ) ; ; r ( t − ( m + 1) P ) ⎤ ⎦
T
(7)
In asynchronous situations, the offset between the interceptor and the transmitter is supposed to be D chips. Fig .1 is the sketch map of rm . The situation that d kl > D is revealed in Fig. 1(a), while d kl < D is depicted in Fig. 1(b) [17].
k
where sk ( t ) is the kth user’s DS-CDMA signal, its complex band pass representation could be modeled as[15] Lk
M
sk ( t ) = Pk ∑∑ akl I k , m hk ( t − mTb − d kl Tc ) l =1 m =1
(3)
⋅ exp ⎡⎣ 2π jf 0 ( t − d kl Tc ) ⎤⎦
Figure 1. Sketch map of rm
P
where hk (t ) = ∑ ck , p ⋅ f ( t − pTc ) , parameters in (3) is p =1
explained in Table Ⅰ, TABLE I.
EXPLANATION OF THE PARAMETERS
Tb
symbol interval
K
number of total users
Tc
chip interval
M
number of intercept symbols
f0
carrier frequency
Lk
number of paths for the kth user
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rm could be written as Lk Lk Lk K ⎧ ⎫ rm = ∑ ⎨ I k , m −1 ∑ akl hklB + I k , m ∑ akl hklM + I k , m +1 ∑ akl hklF ⎬ k =1 ⎩ l =1 l =1 l =1 ⎭ (8) + nm which is composed of the end part of last spreaded symbol, the complete current spreaded symbol and the front part of the next spreaded symbol. I k , m −1 is the last
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symbol, I k , m is the current symbol and I k , m +1 is the next
where y is the stochastic variable, yg is the Gaussian
symbol, nm is 2P samples of noise in one symbol.
stochastic variable whose variance is the same as y and
B kl
M kl
F kl
h , h and h is last part, complete part and front part of th
k user’s pseudo random sequence with d kl chip delay. Equation (8) could be written in a vector form as r = ∑∑ akl ⋅ H klT ⋅ I k,m + n = H ⋅ I + n (9) k
l
H ( y ) in (17) is the differential entropy of y .
Original estimation of negentropy makes use of high order statistic (HOS), which suffers from the none robustness caused by HOS inherently. So new approximations were developed as [19] p
f ( y ) ≈ ∑ ki ⎡⎣ E {G i ( y )} − E {Gi ( v )}⎤⎦
T
I k,m = ⎡⎣ I k , m −1 , I k , m , I k , m +1 ⎤⎦ , let Dkl = mod(d kl − D, P )
2
(18)
i =1
then H could be written as
Two nonlinear functions of G ( i ) are recommended in
⎧ ⎪hk ( P − Dkl + 1), hk ( P − Dkl + 2), , hk ( P ) , 0, 0, , 0 ⎪ 2 P − Dkl Dkl ⎪⎪ T H kl = ⎨0, 0, , 0, hk (1), hk (2), , hk ( P ) , 0, 0, , 0 ⎪ Dkl P − Dkl P ⎪ ⎪0, 0, , 0, hk (1), hk (2), , hk ( P − Dkl ) P − Dkl ⎪⎩ P + Dkl (10) The covariance matrix could be calculated as follow [18]: 1 M R= (11) ∑ {rm ⋅ rmT } M − 1 m =1
[19]
T kl
where ( ⋅ )T means the transpose of the vector. From the singular value decomposition of R , the subspace of the pseudo random sequence could be estimated by EVD, 0 ⎡ λS + N1 I ⎤ ⎢ 3 K ×3 K ⎥ ⎡U ST ⎤ R = [U S U N ] ⎢ (12) ⎥⎢ T ⎥ N1 I ⎢ 0 ⎥ ⎣⎢U N ⎦⎥ ⎢⎣ ( 2 P − 3 K )× ( 2 P − 3 K ) ⎥ ⎦ where U S is the signal subspace, U N is the noise subspace, λS is the signal energy arrayed in decreasing order and I is the identity matrix. When projected to the signal subspace, the correlation between different users’ signals could be removed, which is called whitening. −1/ 2 (13) X = ( λS + N1 I ) U ST ⋅ r After the Whitening preprocessing step, X is the stochastic variable of zero mean and unit variance. Let
B = ( λS + N1 I )
−1/ 2
C = ( λS + N1 I )
U ST ⋅ H
−1/ 2
U ST
(14) (15)
When (9), (14) and (15) are imported, (13) turns into X = B ⋅ I + C ⋅ n = B ⋅ I + n' (16) Equation (16) is just the mathematic model of ICA with noise, so the unmixing matrix W could be estimated by maximizing certain object functions. Among most ICA algorithms, fastICA has become the most popular algorithm for its convergence and simplicity. From our experience, fastICA based on negentropy is much robust and precise, so it is adopted in this paper. Negentropy of a stochastic variable is defined as (17) f ( y ) = H ( yg ) − H ( y )
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1 ⎧ ⎪G1 ( u ) = a log cosh ( au ) ⎨ ⎪G2 ( u ) = − exp ( −u 2 / 2 ) ⎩
(19)
where 0 < a < 1 is some suitable constant and the differential function of (19) are ⎧⎪ g1 ( u ) = tanh ( au ) (20) ⎨ 2 ⎪⎩ g 2 ( u ) = u exp ( −u / 2 ) Then fastICA based on negentropy calculates the unmixing matrix through following steps [20,21]: 1. Choose an initial (e.g. random) weight vector w
{
} {
}
2. Let w+ = E xg ( wT x ) − E g ' ( wT x ) w
(21)
3. Let w = w+ / w+
4. If not converged, go back to 2. When W = B −1 is calculated out, the original information sequence could be estimated by Iˆ = W ⋅ X ≈ I + W ⋅ n' (22) IV BLIND ESTIMATION OF PSEUDO NOISE SEQUENCE WITH ICA AND TCF When left multiplied by W and right multiplied by the matrix of pseudo noise sequences, H , (13) turns into [14]: W ⋅ ( λS + N1 I )
−1/ 2
U ST ⋅ H ⋅ H T = H
(23)
From (12), H H ≈ U S ( λS + N1 I ) U , thus the matrix of pseudo noise sequence could be estimated by −1/ 2 H = W ⋅ ⎡( λS + N1 I ) U ST ⎤ ⋅ ⎡⎣U S ( λS + N1 I ) U ST ⎤⎦ ⎣ ⎦ (24) 1/ 2 T = W ⋅ ( λS + N1 I ) U S T
T S
If there is only one path for each user, from the expression above, the pseudo noise sequence could be recovered directly. But in the multipath situation, the same user’s pseudo random sequences of different paths overlapps together, which could not be estimated directly. According to this, triple correlation function (TCF) is introduced to suppress the influence of overlap. TCF method utilizes the shift-and-add property of msequence, M0 ⊕ MD = ME (25)
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where M 0 is the original m-sequence, M D is the shift form of M 0 with D chips delay, M E is another shift form of M 0 with E chips delay. Triple correlation function is a 3rd statistic in fact, C3, x (τ 1 ,τ 2 ) = E { x ( t ) x ( t + τ 1 ) x ( t + τ 2 )} (26) for m-sequences, let τ 1 = pTc ,τ 2 = qTc , the TCF of msequence could be written as[6]: C3, x ( p, q ) = E { x ( n ) x ( n + p ) x ( n + q )} (27) let x ( n ) be M 0 , x ( n + p ) be M p , and x ( n + q ) be
q = D1 is still the maximum of C3, hkj ( p, q ) . That is, TCF
could eliminate the influence of multipath efficiently. V. AVERAGE FILTER AND CONVERGENCE ANALYSIS
A. Average Filter in the Blind Separation In the original blind separation of DS-CDMA signals, r ( t ) is sampled once in a chip as Fig. 2 (a) [23]. 1,2,3,…,N
1,2,3,…,N
… … …
1,2,3,…,N
M q , E { x ( n ) x ( n + p )} is equal to a shift form of x ( n )
with some v chips delay, supposing the shift form is M v , then TCF could be written as: C3, x ( p, q ) = E {M v ⋅ M q } = Rxx ( v, q ) (28)
(a) Interval sample
1,2,3,…,N
1,2,3,…,N
Σ/N
Σ/N
Euqation (28) is the autocorrelation function of M v and M q , according to the autocorrelation character of msequence, if and only if v = q , Rxx ( v, q ) reached its maximum, for other v ≠ q , Rxx ( v, q ) gets its minimum. The TCF of m-sequence is of three demisions, when plotted in x-y-z coordinate system, TCF is something like peaks erected above the ground, whose pattern is uniquely determined by the primitive polynomial of msequence. Therefore, from the pattern of TCF, the primitive polynomial could be uniquely determined. In multipath situation of, data recovered by ICA are the overlapped pseudo noise sequence of different paths. Let the kth estimated data is xk ( n ) = wk ( n ) + vk ( n ) , wk ( n ) is the overlap pseudo noise sequence in the kth
degree and vk ( n ) is noise in the same degree. According to the additive character of HOS and HOS value of Gaussian signal tends to be zero, TCF of xk ( n ) is Lk
C3, xk ( p, q ) = C3, wk ( p, q ) + C3,vk ( p, q ) ≈ ∑ akl3 C3, hkl ( p, q ) l =1
(29) Two paths of hki and hkj are mainly considered here, other paths could be analyzed through the same way. 1 N C3, hki ( p, q ) = ∑ aki3 hki ( n ) hki ( n + p ) hki ( n + q ) (30) N n =1 Where hki is the d ki chips delay form of the kth pseudo noise sequence
hk ,
∑ h (n) h (n + p) ki
ki
could be
considered as a shift form of hk with some D1 chips
delay. If q = D1 , C3, hki ( p, q ) reached its maximum, for other q ≠ D1 , C3, hki ( p, q ) is the minimum. For hkj , C3, hkj ( p, q ) =
1 N
N
∑a m =1
∑ h (m) h (m + p) kj
kj
3 kj kj
h
( m ) hkj ( m + p ) hkj ( m + q )
(31)
could be considered as a shift form
of hk with E1 chips delay. m = mod ( n + d ki − d kj , P ) is different from n , but p does not change, so the point © 2011 ACADEMY PUBLISHER
… … …
1,2,3,…,N
Σ/N
Σ/N
(b) Averaged in each chip Figure 2. Sketch map of the interval sample and averaged filter.
Generally speaking, the sample rate is much higher than the chip rate in DS-CDMA system, so there are many sample points during one chip. Many other samples are discarded wastefully in the original method. So it is easy to think about averaging the samples in each chip as depicted in Fig. 2(b). The proposed average filtered method of blind separation could be depicted in Fig. 3. Object Function
Average Filter
Centering
Whitening
Unmixing Matrix
Figure 3. Flow chart of the average filtered blind separation
Average filter is introduced at the beginning of the flow. The noise suppression acquired by averaging filter is proportional to the number of samples averaged [22]. If sample rate is eight times of chip rate, the improvement of average filter is about 10 log10 8 ≈ 9 dB.
B. Relationship between Convergence characteristic and offset Under the non-cooperative conditions, it is very difficult for the interceptor to aim at the very beginning of the transmitted DS-CDMA signals. Supposing the offset between the interceptor and the transmitter is D chips, then the offset D has great influence on the convergence of the blind separation method. It is very obviously that certain offset deteriorates the convergence performance greatly. Look back on Fig. 1, the 2 P averaged chips ( rm ) from the dash line with up arrow are mainly considered here. As depicted in Fig. 1(a), D < d kl , rm are composed of the last d kl − D chips of the last spreaded symbol, the complete current spreaded symbol and the front
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P + D − d kl chips of the next spreaded symbol. In Fig.
1(b), D > d kl , so the number of chips of the last symbol, next symbol in rm is P + d kl − D and D − d kl .Because mod(d kl − D, P ) = mod( P + d kl − D, P ) and mod( D − d kl , P ) is equal to mod( P + D − d kl , P ) , let Dkl = mod(d kl − D, P ) , the 2 P averaged chips could be written in a uniform form, rm is composed of the last Dkl chips of the last spreaded symbol, complete current spreaded symbol and the front P − Dkl chips of the next spreaded symbol. From Fig .1 and (8), there are three energy values for one user, for K users, there will be 3K values in total. The diagonal of λS in (12) are just these 3K energy Lk
Lk
l =1
l =1
frequency is set to be f s = 504MHz , that is, each chip is sampled 8 times. The attenuation factor, akl , are chosen from Rayleigh distribution. and the delay parameter, d kl , obeys the uniform distribution of 0—P chips, where P = 63 is the period of the pseudo noise sequence.
A. Estimation of Information Sequence Fig. 4 shows the waveform of the original information sequence and estimated information sequence.
values. Let EkB = I k , m −1 ∑ akl hklB , EkM = I k , m ∑ akl hklM Lk
and EkF = I k , m +1 ∑ akl hklF . Among EkB , EkM and EkF , l =1
EkM is the biggest because it is corresponding to the complete spread symbol. From (11), it is reasonable to estimate EkB and EkF by the length of none zeros chips
in hklB and hklF .Thus the difference between EkB and EkF could be approximated by the difference between Dkl and P − Dkl . d kl obeys the uniform between 0 and P, so its mathematical expectation is P/2 . When 0 < D < P / 2 , 0 < Dkl < P / 2 , when P / 2 < D < P , P > Dkl > P / 2 . Let ∆ = Dkl − ( P − Dkl )
(32)
Let d kl = P / 2 , when D = 0 , Dkl and P − Dkl are both similar to P / 2 , and the difference value, ∆ , is very small. When D ≈ P / 2 , Dkl is very small and P − Dkl is much bigger, thus ∆ ≈ P . When D = P , the difference value turns back to zero again. Thus, the offset D causes energy transfer between EkB and EkF . When the difference, ∆ , is too big, one of EkB and EkF is much smaller than the other. Thus the subspace of U S corresponding to the smaller energy is much similar to the noise subspace. Only when EkB and EkF is very close, should the subspace of pseudo noise sequence be much distinguishable from the noise subspace. So the offsets which make EkB close to EkF enable the algorithm converge much easily. VI. SIMULATION RESULTS In this section, several experiments have been completed to testify the performance of the proposed method and theory analysis above. All these experiments were accomplished by MATLAB®R2009a. The simulation time is T = 2 × 10−3 second, the symbol rate Rb = 106 bps , the chip rate Rc = 63M chips/s. The sample
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Figure 4. Original and estimated information sequence
Fig. 4 (b) is one user's estimated information sequence which is acquired when SNR=-12dB. The users number and the multipath number are both set to be six, the offset between the interceptor and the transmitter is 15 chips. Compared with the transmitted information sequence in Fig. 4 (a), when judged by zero threshold, the information sequence could be recovered with a high correct rate of 96.30% in this experiment. Fig .5 is the bit error rate (BER) curve which is counted when the algorithm converges. SNR is set from 21dB to -3dB with the step of 3dB, and it is the result of 500 Monte Carlo experiments in average.
Figure 5. Bit error rate of the blind separation method
BER of original blind separation method declines slowly with the increase of SNR. While BER with average filter is much lower than original method. Seen from Fig. 5, the average filter brings great SNR improvement than original method. Partial data in Fig. 5 are shown in Table . TABLE II.
SNR ICA ICA-AF
-15dB 0.4173 0.0943
-12dB 0.3671 0.0306
BER VERSUS SNR
-9dB 0.2392 0.0066
-6dB 0.0954 0.0008
-3dB 0.0316 0
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When SNR=-6dB, the BER of original ICA method is 9.54%, the average filter method get the close BER of 9.43% when SNR=-15dB. So, the distance between these two methods is about 10 log10 ( 504 / 63) ≈9dB, which proves the correctness of the analysis above.
B. Estimation of Pseudo Noise Sequence The situation of single path is firstly considered in this subsection. The number of users is set to be K = 2 . Fig. 6 is the true value and the estimated value of the first user's pseudo noise sequence.
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The number of total users is set to be K = 4 , and the number of paths is set to be L = 3 , i.e., there are four users totally and each user's DS-CDMA signal has three paths. The delays are set to be d = [50, 57, 45; 62, 62, 38; 48, 43, 48; 59, 2, 48]; the attenuation factors are set to be A = [1.7906, 0.8170, 0.8977; 0.7126, 1.4681, 1.2755; 0.6493, 0.9200, 0.1234; 1.5560, 2.6588, 2.2564]. The asynchronous offset between the interceptor and the transmitter is still set to be D = 15 chips, and the noise is set to be SNR = -12 dB. Fig. 7(a) is the estimated data of the first user with ICA, from which the original pseudo noise sequence could not be recognized any more. Fig. 7(b) is the true pseudo noise sequence with a shift of Dkl = 30 chips. Fig. 7(c) is the overlap form of true pseudo noise sequence with the shift of 35, 42 and 30 chips respectively. Compared with Fig. 7(a), when multipaths exist, the estimated sequence from ICA is very similar to the overlap form of true pseudo noise sequence. Because of the influence of overlap, the true value could not be acquired directly. So, TCF is introduced to deal with the multipah influence of overlap. When plotted in the x-y-z coordinate system, TCF is much like peaks erecting above the ground. The TCF peaks of first user’s true pseudo noise sequence in Fig. 8.
Figure 6. Estimated pseudo noise sequence under single path
Fig. 6 (a) shows the estimated pseudo noise sequence of the first user when SNR=-12dB. The delay factor of the first user is set to be 10 chips, the offset between the transmitter and the interceptor is set to be 15 chips. From the analysis above, there will be Dkl = mod(10-15,63) = 58 chips zeros at the beginning of the estimated sequence, and the number of zeros at the end of the estimated sequence is P − Dkl = 5 chips. To be compared easily, the shift form with Dkl chips of true pseudo noise sequence is shown in Fig. 6(b). From Fig. 6(a) and Fig. 6(b), when there is only a single path for each user, the estimation sequence from ICA is the accurate duplicate of ture pseudo noise sequence. The estimation of the second user is much similar to the first user, its waveform is not illustrated here for the limit of pages. Then comes the problem of multipath. Fig. 7 reveals the waveforms of the first user's pseudo noise sequence before and after being processed by ICA.
Figure 8. TCF of true pseudo noise sequence
Because there is no influence of multipath, only one maximum value in each row and each column among all TCF values, while other values are very small and close to 1/P. From the analysis above, the pattern of peaks from TCF is uniquely determined by primitive polynomial of the m-sequence, thus, it is probable to derive the primitive polynomial from the TCF values. Fig. 9 shows the TCF of the estimated data.
Figure 7. Estimated pseudo noise sequence under multipath Figure 9. TCF of estimated data
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Though some crossed items swell above the horizontal surface in Fig. 9, the peaks of the TCF are still very distinguishable. The relationship between the estimated correct rate and SNR is revealed in Fig. 10.
Figure 10. Estimation correct rate versus SNR
Comparing these two lines, the improvement of the ICA-TCF method is very obviously.
Fig. 12 reveals the relationship between the number of none convergence times and the signal to noise ratio (SNR). “ ― o” represents the total none convergence number for a SNR value, and “ ― ” represents none convergence times when P / 4 < mod( D, P) < 3P / 4 . When SNR=-18dB, the none convergence times of special offsets are almost the half of the total number. When SNR increases, special offsets occupy the majority of none convergence times. When SNR=-12dB, the number of none convergence times caused by special offset is 1663 occupying the 79.57 percentage of the total number. Define the sum of the K singular values as 2K ⎧ ⎪ S1 = ∑ λS (i ) ⎪ i = K +1 (33) ⎨ 3K ⎪S 2 = λ ( i ) ∑ S ⎪⎩ i = 2 K +1 Fig. 13 reveals the relationship between the difference of S1 and S2 with the change of offset value.
C. Convegence analysis of the Blind Sepration Suppose the offset between the interceptor and the transmitter is D chips. In the experiments, certain offsets deteriorate the convergence of the blind separation method obviously, which is exhibited in Fig. 11.
Figure 13. Difference of eigen value with D
Figure 11. Number of none convergence times versus the offset
Fig. 11 shows the statistic results of none convergence times with different offset value when SNR=-18dB, 15dB and -12dB. D changes from 1 to 252 ( 4P ), there are 100 Monte Carlo experiments for each D value. It is clear that the algorithm converges difficult when mod( D , P ) locates in the middle of the period of the pseudo noise sequence.
Figure 12. Relationship between non-converging times and SNR
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Fig. 13 is almost the same as Fig. 11, which indicates that the theory analysis above is reasonable. Offset causes the energy transfer between two small singular values of each user, which makes certain signal subspace undistinguished with the subspace of noise, resulting in the none-converging of the algorithm at last. VII. CONCLUSION In DS-CDMA system, information sequence of different users remain statistical independent, and the pseudo noise sequence of different users keeps uncorrelated each other, so ICA could be introduced to achieve the blind estimation of information sequence and pseudo noise sequence, which is of great importance in the security of DS-CDMA signals. When there is only one path for each user, the blind estimated pseudo sequence is the accurate duplicate of the original pseudo sequence. However, under the conditions of multipath, the estimated sequence of DS-CDMA signals with ICA method is the overlap form of the true pseudo noise sequence, from which the original pseudo noise sequence could not be recovered directly. TCF is introduced in this paper to deal with the influence of multipath, which increases the estimation correct rate greatly. Average filter is also introduced to suppress the noise in some extent. When the sample rate is much higher than the chip rate, a big improvement of SNR could be acquired by the
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average filter. Besides the preciseness, the relationship between convergence and the offset is found out and explained from the view of singular value decomposition. REFERENCES [1] J. Massey, “Shift register synthesis and BCH decoding,” IEEE Trans. on Information Theory, 1969, vol. 15(1), pp. 122-127 [2] P. Hill and M. Ridley, “Blind estimation of direct-sequence spread spectrum m-sequence chip codes,” IEEE 6th International Symposium on Spread Spectrum Techniques and Applications, vol. 1, 2000. [3] G. Burel and C. Boude, “Blind estimation of the pseudo random sequence of a direct sequence spread spectrum signal,” IEEE MILCOM2000. [4] C. Nzoza, R. Gautier, and G. Burel, “Blind synchronization and sequences identification in CDMA transmissions,” IEEE MILCOM 2004, vol. 3. [5] C. Bouder, S. Azou, and G. Burel, “Performance analysis of a spreading sequence estimator for spread spectrum transmissions,” Journal of the Franklin Institute vol. 341(2004), pp. 595-614. [6] E. R. Adams and P. C. J. Hill, “Detection of direct sequence spread spectrum signals using higher-order statistical processing,” IEEE ICASSP-97, vol. 5, 1997. [7] K. Batty and E. R. Adams, “Detection and blind identification of m-sequence codes using higher-order statistics,” Proceedings of the IEEE Signal Processing Workshop, pp. 16-20, 1999. [8] E. R. ADAMS, “Identification of pseudo-random sequences in DS/SS intercepts by higher-order statistics,” 2004. [9] D. Hill and J. Bodie, “Carrier detection of PSK signals,” IEEE Trans. on Communications, vol. 49(3), pp. 487-496, 2001. [10] J. Yan and J. Hongbing, “A cyclic-cumulant based method forDS-SS signal detection and parameter estimation,” IEEE International Symposium on Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications, vol. 2, 2005. [11] G. Burel, “Detection of spread spectrum transmissions uing fluctuations of correlation estimators,” IEEE ISPACA, 2000. [12] O. Ekici and A. Yongacoglu, Application of noisyindependent component analysis for CDMA signal separation, VTC-Fall, vol. 5, 2004,. [13] K. Raju, T. Ristaniemi, J. Karhunen, and E. Oja, “Jammer suppression in DS-CDMA arrays using independent component,” IEEE Trans. On Wireless Communications, vol. 5(1), pp. 77-82, 2006. [14] L. Shen, S. J. Li, Y. B. Wang, et., “Blind estimation of pseudo-random sequences of spread spectrum signals in multi-paths,” Journal of Zhejiang University, vol.41(11), pp. 1828-1833, 2007. [15] A. Viterbi,et., CDMA: Principles of spread spectrum communication, MA, US:Addison-Wesley Reading, 1995. [16] T. Ristaniemi, and J. Joutsensalo, “Advanced ICA-based receivers for block fading DS-CDMA channels,” PIMRC2000, vol. 1, 2000. [17] M. Yu, S. J. Li, and J. Z. Chen, “Convergence characteristics analysis on blind separation of DS-CDMA Signals with ICA method,” WCNIS2010, pp. 238-242, 2010. [18] L. Smith, “A tutorial on principal components analysis”, Cornell University, USA,2002.51:52. [19] A. Hyvarinen, “Fast and robust fixed-point algorithms for independent component analysis,” IEEE Trans. Neural Networks, vol. 10, pp. 626-634, 1999.
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[20] A. Hyvarinen and E. Oja, “Independent component analysis: algorithms and applications,” IEEE Trans. Neural Networks, vol.13, pp. 411-430, 2000. [21] A. Hyvärinen, J. Karhunen, and O. Oja, Independent component analysis, John Wiley & Sons, Inc, United States, 2001. [22] R. G. Lyons, Understanding Digital Signal Processing, 2nd ed., Prentice Hall PTR, New Jersey, USA, 2004. [23] M. Yu, J. Z. Chen, and S. J. Li “Average filtered blind separation of DS-CDMA signals with ICA method,” WCNIS2010, pp. 234-237, 2010.
Miao Yu was born in Jilin, China, on August 29, 1975. He received M.Sc. and Ph.D. degrees in information and communication engineering from Zhejiang University, Hangzhou, China, in 2005 and 2009 respectively. His research interests include spread spectrum signal processing and independent component analysis.
Jianzhong Chen was born in Taixing, China, on June 18, 1962. He received M.Sc. degree in wireless communication engineering in 1993. His research interests include wireless communication and signal processing.
Lei Shen was born in Zhoushan, China, on March 15, 1977. He received B.Sc. and Ph.D. degrees information and communication engineering from Zhejiang University, Hangzhou, China, in 2002 and 2007 respectively. His research interests include wireless comunication and signal processing. Dr. Shen is an associate professor of Hangzhou Dianzi University, China.
Shiju Li was born in Linhai, China, on February 28, 1947. He graduated from the radio science and electronic engineering department of Zhejiang University in 1969. His research interests include digital communication and mobile multimedia communication. Prof. Li is the author of more than 100 papers and 3 books.
