- Email: [email protected]
CDMA system
Journal of Systems Engineering and Electronics Vol. 18, No. 3, 2007, pp.454-461
Multi-channel blind deconvolution algorithm for multiple-input multiple-output DS/CDMA system Cheng Hao, Guo Wei & Jiang Yi National Key Lab of Communication, Univ. of Electronic Science and Technology, Chengdu 610054, P. R. China (Received November 7, 2006)
Abstract: Direct sequence spread spectrum transmission can be realized at low SNR, and has low probability of detection. It is aly problem how to obtain the original users’ signal in a non-cooperative context. In practicality, the DS/CDMA sources received are linear convolute mixing. A more complex multichannel blind deconvolution MBD algorithm is required to achieve better source separation. An improved MBD algorithm for separating linear convolved mixtures of signals in CDMA system is proposed. This algorithm is based on minimizing the average squared cross-output-channel-correlation. The mixture coefficients are totally unknown, while some knowledge about temporal model exists. Results show that the proposed algorithm can bring about the exactness and low computational complexity.
Keywords: DS/CDMA signal, non-cooperative, MBD, stochastic gradient algorithms for MBD.
1. Introduction Spread spectrum signals have been used for secure communications for several decades. Nowadays, they are also widely used outside the military domain, especially in code division multiple access (CDMA) systems. They are mainly used for transmitting at low power without interference due to jamming to others users or multi-path propagation. The spread spectrum techniques are useful for secure transmission, because the receiver has to know the sequence used by the transmitter to recover the transmitted data using a correlator. Direct-sequence spread spectrum transmitters (DSSS) use a periodical pseudorandom sequence to modulate the baseband signal before transmission. In the context of spectrum surveillance, the pseudorandom sequence used by the transmitter is unknown (as well as other transmitter parameters such as duration of the sequence, symbol frequency and carrier frequency). Hence, in this context, a DSSS transmission is very difficult to detect and demod-
ulate, because it is often below the noise level.1 Of purpose the is to automatically determine the spreading sequence in multi-user CDMA system, whereas the receiver has no knowledge of the transmitter’s pseudo-noise (PN) code. We also present the technique of direct sequence spread spectrum (DS-SS) and explain the difficulty in recovering the data in an unfriendly context. A more complex BSS algorithm is required to achieve better source separation. We propose a method for estimating the pseudo-random sequence without prior knowledge about the transmitter. Only the duration of the pseudorandom sequence is assumed to have been estimated. (This can be done by the method proposed in Ref.[2]). This method is based on minimizing the average squared cross output channel correlation criterion[3]. There are many ways to construct a numerical algorithm based on the above criterion for blind source separation as equation Eq. (12). We derive here a new stochastic gradient algorithm, which has these characteristics. For the optimization of the demixing matrices D. Experimental
* This project was supported by the National Natural Science Foundation of China (60472052,10577007) and the Specialized Research Fund for National Key Lab of Communication (51434020105ZS04).
Multi-channel blind deconvolution algorithm for multiple-input multiple-output DS/CDMA system
455
results are given to illustrate the performances of the method and show that a good estimation can be obtained.
2. Models for MIMO blind deconvolution Let us consider a multichannel, linear time-invariant (LTI), discrete time dynamical system described in the most general form as[4] x(k) =
∞
Hp s(k − p)
(1)
p=−∞
where, Hp is an m × n-dimensional matrix of mixing coefficients at timelag p (called the impulse response at timelag p) and s(k ) is an n-dimensional vector of source signals with mutually independent components. It should be noted that the causality in time domain is satisfied only when Hp = 0 for all p < 0. The goal of the multichannel blind deconvolution is to estimate the source signals using sensor signals x (k ) only and certain knowledge of the source signal distributions and statistics. In the most general case, we attempt to estimate the sources by employing another multichannel, LTI, discrete-time, stable dynamical system (Figs. 1(a) and (b)) described as y(k) =
∞
Wp x(k − p)
(2)
Fig. 1
Illustration of the multichannel deconvolution models
or apply a non-causal (doubly-finite) feedforward multichannel filter L W (z) = Wp x−p (5) p=−K
p=−∞
where, y (k ) = [y1 (k ), y2 (k ),· · ·,yn (k )] .k is an ndimensional vector of the outputs and Wp is an n × m dimensional coefficient matrix at time lag p. We use the operator form notation. H(z) =
∞
Hp z
−p
p=−∞
W (z) =
∞
Wp z −p
(3)
p=−∞
In practical applications, we have to implement the blind deconvolution problem with a finite impulse response (FIR) multi-channel filter with matrix transfer function L W (z) = Wp x−p (4) p=0
where K and L are two given positive integers. The global transfer function is defined by G(z) = W (z)H(z)
(6)
In order to ensure that the mixing system is recoverable, we put the following constraints on the convolute, mixing systems. The filter H (z ) is stable, i.e., its impulse response satisfies the absolute summability condition ∞ p=−∞
||Hp ||22 < ∞
(7)
where || · || denotes the Euclidean norm. The filter matrix transfer function H (z ) is a full rank on the unit circle (|z | = 1), that is, it has no zero on the unit circle.
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Cheng Hao, Guo Wei & Jiang Yi
3. DS/CDMA data model’s notations and hypotheses Based on the model described in Section 2, the goal of this paper is to recover the PN sequence sk when both b (user’s data symbol) and a (user’s fading coefficient) are unknown and no train samples are available. The following assumptions will be valld throughout the rest of the paper. AS1: The carrier frequency f0 and the chip period Tc have been estimated[5] . AS2: The symbol duration Ts has beenestimated.[3] In the AS1, the method based on cyclostationarity analysis can estimate the chip period Tc and carrier frequency f0 even at low SNR. If the estimation of Tc and f0 are available, we can process the signals in the baseband and the sampling period can be set to Tc . In Section 5, we will demonstrate what is available. In the AS2, the symbol duration Ts can be estimated using fluctuations of correlation even at –10 dB (that is low enough for our method in the paper). Based on AS2, we can adopt the DS-CDMA data model as follows. The received CDMA signal can be written in the form[6−7]
r(t) =
L M K m=1 k=1 l=1
data samples rm = [r(mC)r(mC + 1) . . . . . . r(mC + C − 1)]T (9) Equation (9) can further be written with respect to the current symbol vector and the immediately preceding one: rm (t) =
K k=1
[bk,m−1
L l=1
al gkl ]+
K k=1
[bk,m
L
al gkl ]+nm
l=1
(10) where nm denotes the noise vector, gkl is the ‘early’ parts of the code vectors and gkl is the ‘late’ parts of the code vectors. gkl = [sk [C − dl + 1] · · · sk [C] 0 · · · 0]T gkl = [ 0 · · · 0 sk [1] · · · sk [C − dl ] ]T
(11)
In this paper, we assume that the codes gkl + gkl are unknown. The approach exploits directly the binary form of the sources. The proposed algorithms yield good results in test examples with CDMA data. Due to the neural structure of the algorithms, they can be used for tracking purposes when the parameters are slowly changing. Hence the Eq. (10) could be written as follows rm = G0 bm + G1 bm−1 + nm = 1
Gi bm−i + nm
(12)
i=0
aklm bkm sk (t − mT − dkl ) + n(t) (8)
In Eq. (8), M symbols are sent to K users and received via L path. aklm is the amplitude (fading) coefficient of the l th path of the kth user during the mth symbol duration, bkm is the kth user’s mth data symbol ,sk is the user’s chip sequence. sk (t )∈{–1,+1}, t ∈[0,T ], sk (t )=0, t ∈[0,T / ], and dkl is the delay of th th the k user’s l path. The delays are different for different users, and each delay is assumed to change sufficiently slowly for most of the time. n(t) denotes noise. The chip sequence length is C, and N is the number of bits. The signal is then sampled by chip rate. After sampling, the samples are collected into vectors of suitable dimension. This depends on the delay spread. We collect C-length column vectors from subsequent discrete
where G0 and G1 are C × K mixing matrices corresponding to the original and the one-time unit delayed symbols. The column vectors of G0 and G1 are as follows L L al g1l · · · al gKl ] G0 = [ l=1
l=1
L L G1 = [ al g1l · · · al gKl ] l=1
(13)
l=1
We notice from Eq. (12) that our signal model can be regarded as a linear mixture of delayed and convolved sources, in the particular case when the maximum delay is one-time unit. If we have known neither the mixing matrices nor symbol sequences, we are in the framework of blind source separation. Equation (12) is in the general form of the BSS problem. The goal of the blind source separation is to determine a de-mixing matrix of filter D(n), which, when
Multi-channel blind deconvolution algorithm for multiple-input multiple-output DS/CDMA system applied to the received sensor data, recovers (separates) the individual sources up to an unknown permutation and unknown channel gains. y(n) =
I
D(i)r(n − i)
4. Adaptive algorithm This paper uses the same basic insight, but proposes a new criterion for exploiting it, which leads to a particularly simple and convenient algorithm. We propose to minimize the following criterion ⎧ ⎫ ⎪ ⎪ M M M ⎨ ⎬ 2 2 2 J =E (15) ryi yj + νi (ryi yi − δi ) ⎪ ⎪ ⎩ i=1 j=1 ⎭ i=1 j=i
where h(n) ∗ [yi (n)yj (n)] h(k)
∇(n) =
(17)
∂J(n) ∂D(n)
∂
∂dpq (n) ⎪ ⎩ i=1
j=1 j=i
ry2i yj (n)
+
M i=1
νi [ryi yj (n) −
(19)
The above matrix expression for the stochastic gradient update yields an efficient and straightforward computation once the short-time correlations are available. We now derive efficient recursive updates for these components for a convenient form of the averaging filter. For computational efficiency, we select a first-order IIR averaging filter with impulse response h(n) = an u(k),
0
(20)
where, u (k) is the unit step function and 0 < a < 1. With this form, the elements of Ryy can easily be updated recursively according to
1 p, i M ] ryp xi (n) = aryp xi (n − 1) + (1 − a)yp (n)xi (n) 1 p, i M
(21)
This completes the following simple recursive algorithm for non-stationary blind source separation. (1.) (2.) (3.) (18). (4.) (5.)
Compute output according to Eq. (14). Update short-time correlations using Eq. (20). Compute separation filter gradient using Eq. Update separation matrix using Eq. (16). Go back to step 1.
5. Simulation
∂J(n) ∇pq (n) = = ∂dpq (n)
⎧ ⎪ M M ⎨
2λ[ryp yp (n) − δp2 ]ryp xq (n, l)
ryp yi (n) = aryp yi (n − 1) + (1 − a)yp (n)yi (n)]
h(k) is a low-pass averaging filter for computing a short-term estimate of the cross-correlation of output channels yi and yj at time n and lag l. There are many ways to construct a numerical algorithm based on the above criterion for blind nonstationary source separation as equation Eq. (12). We derive here a new stochastic gradient algorithm, which has these characteristics. For the optimization of the demixing matrices, D (n)’s, a stochastic gradient update takes the form
where
ryp yi (n)ryi xq (n, l)+
(16)
k
D(n + 1) = D(n) − µ∇(n)
M i=1 i=p
(14)
i=0
ryi yj (n) =
∇pq (n, l) = 4
457
5.1 Simulation parameter setup δi2 ]2
⎫ ⎪ ⎬ ⎪ ⎭
(18) where p and q are the row and column indice gradient matrices. Note the use of the instantaneous value at time n of the error function in Eq. (15) gradient computation. The (p,q)th element of the instantaneous gradient matrix can easily be shown as
The average squared cross-output-channel-correlation minimizing algorithm is demonstrated here via computer simulation, for a DS/CDMA signal received in the presence of various levels of white Gaussian background noise. In the experiment, we considered a 2-user CDMA system. The source signal was generated by the computer and the signal parameters were set as follows: carrier frequency f0 =5 000, chip period Tc =1/1 000, symbol period Ts =31/1 000(PN
458
Cheng Hao, Guo Wei & Jiang Yi
code length C=31), users, number K=2, path L=2, SNR=3(the SNR is higher than that of AS1 and AS2). In evaluating the algorithms, two parameter matrices G0 and G1 are chosen (randomly) as convoluting mixing matrices. The algorithm is robrst to unknown signal delay .Both sources are zero-mean and have unit variances. A Gaussian vector noise process is added to the observation with noise covariance σ 2 I. The signal-to-noise ratio (SNR) is then defined as SNR = 10 lg
1 (dB) σ2
we have solved the problem of assumption 1. We can process the aboriginal signals in the baseband and set the sampling period to Tc . Using the fluctuations of correlation method, we obtain the Ts =0.031 1 s (aberrancy near upon 0.3%). So the PN code length can be known as C = Ts / T c =31. The assumption 2 is also solved. For the purpose of observation, we only display the last 800 points in the figure. Figure 3 shows the 2 users’ CDMA signal sources with AWGN channel (SNR=3).
In this section, we evaluate the performance of the adaptive algorithm by some heuristic arguments and simulation results. 5.2 Results analysis As the CDMA signal exhibits cyclostationarity, we utilize the spectral correlation method to get the carrier frequency f0 and the chip period Tc . Figure 2 shows the measured spectral correlation function magnitude for the 2-user CDMA system at the f=0 section. From Fig. 2, the first peak value coordinate appears at the α = ±1×104, according to the definition of cycliccorrelation, we can get the carrier frequency at the f0 = α/2 , the carrier frequency f0 has been estimated.
Fig. 2
Fig. 3
2 users’CDMA signal sources with AWGn channel
Figure 4 shows the users’ data after convolute mixing. Two unknown GOLD sequences are used as the spreading codes. The output data after separation between 7 200∼8 000 is shown in Fig. 4. The unitary output binary estimated data after separation between 7 200∼8 000 is shown in Fig. 5.
Measured spectral correlation function magnitude for the 2-user CDMA system at the f =0 section
The second peak value appears at the both sides of each main peak, and the coordinate is ±9 000(that is not distinct in the figure) and ±11 000 s, so we can obtain chip period Tc = 1/1 000 s. Above all,
Fig. 4
Users’data after convolute mixing
Multi-channel blind deconvolution algorithm for multiple-input multiple-output DS/CDMA system
459
points, is of near to 0. The convergence rate is accord ant with the adaptation of the coefficient in the demixing matrix D as Fig. 7.
Fig. 5
Output data after separation
Comparing the Figs. 3 and 5, the output binary estimated data after separation is similar to the source signal. In this experiment all the symbols were correctly estimated, which shows the data are received correctly. According to the assumption 2, we estimate the PN code length, which becomes very simple to seek the 2 users’ PN code. We can adopt traditional slippage correlation method or matrix eigenanalysis techniques to resolve the problem. In this paper, we use the slippage correlation method and get the 2 users’ PN code as Fig. 6.
Fig. 7
Adaptation of the coefficient in the demixing matrix D
Fig. 8
BER for convoluted mixture mixture CDMA signal separation
Fig. 6
2 Users’ PN code
Figure 7 plots the adaptation of the coefficient in the demixing matrix D with the α=0.9. Experimental bit-error-rate (BER) results are shown in Fig. 8. 5.3 The influence of δ, ν and α Figure 9 shows the risk function J fluctuatis with the sample serial n. We know from Fig. 9, of after about 3 000 points recursion operation , the risk function J is convergent. Then the risk function J, after twice fluctuation, of recursion operation after about 6 000
Fig. 9
Fluctuation of risk function J
We also find the influence on demixing matrices about convergence factor v and the power factor δ. Power factor δ has dual effects on the demixing matrices. First, when the value of δ is larger, the convergence rate is higher but unstable. From a contrasting point of view, the convergence rate is lower but stabile. Second, the least value of δ affects the risk function
460
Cheng Hao, Guo Wei & Jiang Yi
convergence limit. The value of power factor δ is coherent with the risk function convergence limit. We test 10 000 samples 3 times under the same condition, and get the risk function J over time, as show in figure 10. We take logarithm over the y axis to make it clear to observe. The shape and energy factor’s value indicate the influence which energy factor has on the algorithm. Convergence factor v also has the similar influence to energy factor, as show in Fig.11.
Fig. 12
Influence of filter parameter ‘a’ on J
6. Conclusions
Fig. 10
Fig. 11
Fluctuation of risk function J
Influence of power factor V on J
In the algorithm, we choose the low-pass filter h(n) = an u(n), 0 < a < 1 The aim is to reduce the algorithm’s complexity and average the demixing output data. Under the same condition, we get the risk function J over , as show in Fig. 12. The conclusion has been obtained that the value of α is larger, the convergence rate is lower but stable and α is smaller; the convergence rate is higher but unstable.
A technique for BSS of DS-CDMA signals is developed and demonstrated. The technique, referred to here as the average squared cross-output-channel-correlation minimizing algorithm. We propose to minimize the following criterion function J to estimate the spreading code with the correct timing and despread the underlying message sequence, without knowledge about the content of the code or message sequences. The technique is applicable to arbitrary spreading codes and message sequences, and is effective in the environments with white background noise, as long as the DSSS signal has a constant-modulus spreading code and a modulation-on-symbol structure or nearly incommensurate code-repetition and message rates. These results show that the average squared crossoutput-channel-correlation minimizing algorithm is a promising alternative to the existing BSS algorithms for DS/CDMA signals. This algorithm is also applicable to signals with long code, such as commercial communication signals, which allows the technique to be implemented with reasonable computational complexity and over reasonably short reception intervals. This method is also particularly applicable estimation in downlink signal processing, because the codes of the interfering users are unknown. From the previous study it is clear that the algorithm to CDMA separation problem has a significant performance. It is effective to separate the signals and remove the multipath channel effect.
References [1] Uzzafer M. Single correlator receiver for wideband multi-
Multi-channel blind deconvolution algorithm for multiple-input multiple-output DS/CDMA system path channel. Proceedings of the IEEE 6th Circuits and Systems Symposium, 2004: 421–424. [2] Burel G. Detection of spread spectrum transmissions using
461
[8] Imamura H, Okello J. Blind source separation of PSK and amplitude modulated signals. Intelligent Signal Processing and Communication Systems, 2004: 343–346.
fluctuations of correlation estimators. ISPACS ’2000, 2000. [3] Krongold B S, Jones D L .Blind source separation of non-
Cheng Hao was born in 1976 and now is a Ph.
stationary convolutively mixed signals. Statistical Signal
D. student in National Key Lab of Communication, University of Electronic Science and Technology of
and Array Processing, 2000: 53–57. [4] Andrzej Cichocki, Shunichi Amari, John Wiley. Adaptive blind signal and image processing. Chi Chester, 2003. [5] Barua s, Hashimoto T. Suboptimal MMSE multiuser re-
China. His research interests include intelligent signal processing, OFDM and iterative decoding.E-mail: [email protected]
ceiver design for DS-CDMA multirate system using spectral correlations. Information Theory Proceedings, 2003: 446–449. [6] Waheed K, Salem F.M. Blind information-theoretic multiUser detection algorithms for DS-CDMA and WCDMA downlink systems. Neural Networks, IEEE Transactions,
Guo Wei was born in 1964. He is a professor and doctoral tutor with UESTC. His research interests include OFDM, MIMO, Turbo codes, LDPC codes, Ad Hoc and UWB wireless communication.
2005, 16(4): 937–948. [7] Eikhamy S E, Lotfy M, Badawy A S. On the blind multi-
Jiang Yi was born in 1978. He received his M.S.
user detection of DS/CDMA signals using the indepen-
Degree from UESTC in 2004, and is now working as a communication engineer.
dent component analysis (ICA). Radio Science Conference, 2003: 34–39.
Multi-channel blind deconvolution algorithm for multiple-input multiple-output DS/CDMA system Cheng Hao, Guo Wei & Jiang Yi National Key Lab of Communication, Univ. of Electronic Science and Technology, Chengdu 610054, P. R. China (Received November 7, 2006)
Abstract: Direct sequence spread spectrum transmission can be realized at low SNR, and has low probability of detection. It is aly problem how to obtain the original users’ signal in a non-cooperative context. In practicality, the DS/CDMA sources received are linear convolute mixing. A more complex multichannel blind deconvolution MBD algorithm is required to achieve better source separation. An improved MBD algorithm for separating linear convolved mixtures of signals in CDMA system is proposed. This algorithm is based on minimizing the average squared cross-output-channel-correlation. The mixture coefficients are totally unknown, while some knowledge about temporal model exists. Results show that the proposed algorithm can bring about the exactness and low computational complexity.
Keywords: DS/CDMA signal, non-cooperative, MBD, stochastic gradient algorithms for MBD.
1. Introduction Spread spectrum signals have been used for secure communications for several decades. Nowadays, they are also widely used outside the military domain, especially in code division multiple access (CDMA) systems. They are mainly used for transmitting at low power without interference due to jamming to others users or multi-path propagation. The spread spectrum techniques are useful for secure transmission, because the receiver has to know the sequence used by the transmitter to recover the transmitted data using a correlator. Direct-sequence spread spectrum transmitters (DSSS) use a periodical pseudorandom sequence to modulate the baseband signal before transmission. In the context of spectrum surveillance, the pseudorandom sequence used by the transmitter is unknown (as well as other transmitter parameters such as duration of the sequence, symbol frequency and carrier frequency). Hence, in this context, a DSSS transmission is very difficult to detect and demod-
ulate, because it is often below the noise level.1 Of purpose the is to automatically determine the spreading sequence in multi-user CDMA system, whereas the receiver has no knowledge of the transmitter’s pseudo-noise (PN) code. We also present the technique of direct sequence spread spectrum (DS-SS) and explain the difficulty in recovering the data in an unfriendly context. A more complex BSS algorithm is required to achieve better source separation. We propose a method for estimating the pseudo-random sequence without prior knowledge about the transmitter. Only the duration of the pseudorandom sequence is assumed to have been estimated. (This can be done by the method proposed in Ref.[2]). This method is based on minimizing the average squared cross output channel correlation criterion[3]. There are many ways to construct a numerical algorithm based on the above criterion for blind source separation as equation Eq. (12). We derive here a new stochastic gradient algorithm, which has these characteristics. For the optimization of the demixing matrices D. Experimental
* This project was supported by the National Natural Science Foundation of China (60472052,10577007) and the Specialized Research Fund for National Key Lab of Communication (51434020105ZS04).
Multi-channel blind deconvolution algorithm for multiple-input multiple-output DS/CDMA system
455
results are given to illustrate the performances of the method and show that a good estimation can be obtained.
2. Models for MIMO blind deconvolution Let us consider a multichannel, linear time-invariant (LTI), discrete time dynamical system described in the most general form as[4] x(k) =
∞
Hp s(k − p)
(1)
p=−∞
where, Hp is an m × n-dimensional matrix of mixing coefficients at timelag p (called the impulse response at timelag p) and s(k ) is an n-dimensional vector of source signals with mutually independent components. It should be noted that the causality in time domain is satisfied only when Hp = 0 for all p < 0. The goal of the multichannel blind deconvolution is to estimate the source signals using sensor signals x (k ) only and certain knowledge of the source signal distributions and statistics. In the most general case, we attempt to estimate the sources by employing another multichannel, LTI, discrete-time, stable dynamical system (Figs. 1(a) and (b)) described as y(k) =
∞
Wp x(k − p)
(2)
Fig. 1
Illustration of the multichannel deconvolution models
or apply a non-causal (doubly-finite) feedforward multichannel filter L W (z) = Wp x−p (5) p=−K
p=−∞
where, y (k ) = [y1 (k ), y2 (k ),· · ·,yn (k )] .k is an ndimensional vector of the outputs and Wp is an n × m dimensional coefficient matrix at time lag p. We use the operator form notation. H(z) =
∞
Hp z
−p
p=−∞
W (z) =
∞
Wp z −p
(3)
p=−∞
In practical applications, we have to implement the blind deconvolution problem with a finite impulse response (FIR) multi-channel filter with matrix transfer function L W (z) = Wp x−p (4) p=0
where K and L are two given positive integers. The global transfer function is defined by G(z) = W (z)H(z)
(6)
In order to ensure that the mixing system is recoverable, we put the following constraints on the convolute, mixing systems. The filter H (z ) is stable, i.e., its impulse response satisfies the absolute summability condition ∞ p=−∞
||Hp ||22 < ∞
(7)
where || · || denotes the Euclidean norm. The filter matrix transfer function H (z ) is a full rank on the unit circle (|z | = 1), that is, it has no zero on the unit circle.
456
Cheng Hao, Guo Wei & Jiang Yi
3. DS/CDMA data model’s notations and hypotheses Based on the model described in Section 2, the goal of this paper is to recover the PN sequence sk when both b (user’s data symbol) and a (user’s fading coefficient) are unknown and no train samples are available. The following assumptions will be valld throughout the rest of the paper. AS1: The carrier frequency f0 and the chip period Tc have been estimated[5] . AS2: The symbol duration Ts has beenestimated.[3] In the AS1, the method based on cyclostationarity analysis can estimate the chip period Tc and carrier frequency f0 even at low SNR. If the estimation of Tc and f0 are available, we can process the signals in the baseband and the sampling period can be set to Tc . In Section 5, we will demonstrate what is available. In the AS2, the symbol duration Ts can be estimated using fluctuations of correlation even at –10 dB (that is low enough for our method in the paper). Based on AS2, we can adopt the DS-CDMA data model as follows. The received CDMA signal can be written in the form[6−7]
r(t) =
L M K m=1 k=1 l=1
data samples rm = [r(mC)r(mC + 1) . . . . . . r(mC + C − 1)]T (9) Equation (9) can further be written with respect to the current symbol vector and the immediately preceding one: rm (t) =
K k=1
[bk,m−1
L l=1
al gkl ]+
K k=1
[bk,m
L
al gkl ]+nm
l=1
(10) where nm denotes the noise vector, gkl is the ‘early’ parts of the code vectors and gkl is the ‘late’ parts of the code vectors. gkl = [sk [C − dl + 1] · · · sk [C] 0 · · · 0]T gkl = [ 0 · · · 0 sk [1] · · · sk [C − dl ] ]T
(11)
In this paper, we assume that the codes gkl + gkl are unknown. The approach exploits directly the binary form of the sources. The proposed algorithms yield good results in test examples with CDMA data. Due to the neural structure of the algorithms, they can be used for tracking purposes when the parameters are slowly changing. Hence the Eq. (10) could be written as follows rm = G0 bm + G1 bm−1 + nm = 1
Gi bm−i + nm
(12)
i=0
aklm bkm sk (t − mT − dkl ) + n(t) (8)
In Eq. (8), M symbols are sent to K users and received via L path. aklm is the amplitude (fading) coefficient of the l th path of the kth user during the mth symbol duration, bkm is the kth user’s mth data symbol ,sk is the user’s chip sequence. sk (t )∈{–1,+1}, t ∈[0,T ], sk (t )=0, t ∈[0,T / ], and dkl is the delay of th th the k user’s l path. The delays are different for different users, and each delay is assumed to change sufficiently slowly for most of the time. n(t) denotes noise. The chip sequence length is C, and N is the number of bits. The signal is then sampled by chip rate. After sampling, the samples are collected into vectors of suitable dimension. This depends on the delay spread. We collect C-length column vectors from subsequent discrete
where G0 and G1 are C × K mixing matrices corresponding to the original and the one-time unit delayed symbols. The column vectors of G0 and G1 are as follows L L al g1l · · · al gKl ] G0 = [ l=1
l=1
L L G1 = [ al g1l · · · al gKl ] l=1
(13)
l=1
We notice from Eq. (12) that our signal model can be regarded as a linear mixture of delayed and convolved sources, in the particular case when the maximum delay is one-time unit. If we have known neither the mixing matrices nor symbol sequences, we are in the framework of blind source separation. Equation (12) is in the general form of the BSS problem. The goal of the blind source separation is to determine a de-mixing matrix of filter D(n), which, when
Multi-channel blind deconvolution algorithm for multiple-input multiple-output DS/CDMA system applied to the received sensor data, recovers (separates) the individual sources up to an unknown permutation and unknown channel gains. y(n) =
I
D(i)r(n − i)
4. Adaptive algorithm This paper uses the same basic insight, but proposes a new criterion for exploiting it, which leads to a particularly simple and convenient algorithm. We propose to minimize the following criterion ⎧ ⎫ ⎪ ⎪ M M M ⎨ ⎬ 2 2 2 J =E (15) ryi yj + νi (ryi yi − δi ) ⎪ ⎪ ⎩ i=1 j=1 ⎭ i=1 j=i
where h(n) ∗ [yi (n)yj (n)] h(k)
∇(n) =
(17)
∂J(n) ∂D(n)
∂
∂dpq (n) ⎪ ⎩ i=1
j=1 j=i
ry2i yj (n)
+
M i=1
νi [ryi yj (n) −
(19)
The above matrix expression for the stochastic gradient update yields an efficient and straightforward computation once the short-time correlations are available. We now derive efficient recursive updates for these components for a convenient form of the averaging filter. For computational efficiency, we select a first-order IIR averaging filter with impulse response h(n) = an u(k),
0
(20)
where, u (k) is the unit step function and 0 < a < 1. With this form, the elements of Ryy can easily be updated recursively according to
1 p, i M ] ryp xi (n) = aryp xi (n − 1) + (1 − a)yp (n)xi (n) 1 p, i M
(21)
This completes the following simple recursive algorithm for non-stationary blind source separation. (1.) (2.) (3.) (18). (4.) (5.)
Compute output according to Eq. (14). Update short-time correlations using Eq. (20). Compute separation filter gradient using Eq. Update separation matrix using Eq. (16). Go back to step 1.
5. Simulation
∂J(n) ∇pq (n) = = ∂dpq (n)
⎧ ⎪ M M ⎨
2λ[ryp yp (n) − δp2 ]ryp xq (n, l)
ryp yi (n) = aryp yi (n − 1) + (1 − a)yp (n)yi (n)]
h(k) is a low-pass averaging filter for computing a short-term estimate of the cross-correlation of output channels yi and yj at time n and lag l. There are many ways to construct a numerical algorithm based on the above criterion for blind nonstationary source separation as equation Eq. (12). We derive here a new stochastic gradient algorithm, which has these characteristics. For the optimization of the demixing matrices, D (n)’s, a stochastic gradient update takes the form
where
ryp yi (n)ryi xq (n, l)+
(16)
k
D(n + 1) = D(n) − µ∇(n)
M i=1 i=p
(14)
i=0
ryi yj (n) =
∇pq (n, l) = 4
457
5.1 Simulation parameter setup δi2 ]2
⎫ ⎪ ⎬ ⎪ ⎭
(18) where p and q are the row and column indice gradient matrices. Note the use of the instantaneous value at time n of the error function in Eq. (15) gradient computation. The (p,q)th element of the instantaneous gradient matrix can easily be shown as
The average squared cross-output-channel-correlation minimizing algorithm is demonstrated here via computer simulation, for a DS/CDMA signal received in the presence of various levels of white Gaussian background noise. In the experiment, we considered a 2-user CDMA system. The source signal was generated by the computer and the signal parameters were set as follows: carrier frequency f0 =5 000, chip period Tc =1/1 000, symbol period Ts =31/1 000(PN
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Cheng Hao, Guo Wei & Jiang Yi
code length C=31), users, number K=2, path L=2, SNR=3(the SNR is higher than that of AS1 and AS2). In evaluating the algorithms, two parameter matrices G0 and G1 are chosen (randomly) as convoluting mixing matrices. The algorithm is robrst to unknown signal delay .Both sources are zero-mean and have unit variances. A Gaussian vector noise process is added to the observation with noise covariance σ 2 I. The signal-to-noise ratio (SNR) is then defined as SNR = 10 lg
1 (dB) σ2
we have solved the problem of assumption 1. We can process the aboriginal signals in the baseband and set the sampling period to Tc . Using the fluctuations of correlation method, we obtain the Ts =0.031 1 s (aberrancy near upon 0.3%). So the PN code length can be known as C = Ts / T c =31. The assumption 2 is also solved. For the purpose of observation, we only display the last 800 points in the figure. Figure 3 shows the 2 users’ CDMA signal sources with AWGN channel (SNR=3).
In this section, we evaluate the performance of the adaptive algorithm by some heuristic arguments and simulation results. 5.2 Results analysis As the CDMA signal exhibits cyclostationarity, we utilize the spectral correlation method to get the carrier frequency f0 and the chip period Tc . Figure 2 shows the measured spectral correlation function magnitude for the 2-user CDMA system at the f=0 section. From Fig. 2, the first peak value coordinate appears at the α = ±1×104, according to the definition of cycliccorrelation, we can get the carrier frequency at the f0 = α/2 , the carrier frequency f0 has been estimated.
Fig. 2
Fig. 3
2 users’CDMA signal sources with AWGn channel
Figure 4 shows the users’ data after convolute mixing. Two unknown GOLD sequences are used as the spreading codes. The output data after separation between 7 200∼8 000 is shown in Fig. 4. The unitary output binary estimated data after separation between 7 200∼8 000 is shown in Fig. 5.
Measured spectral correlation function magnitude for the 2-user CDMA system at the f =0 section
The second peak value appears at the both sides of each main peak, and the coordinate is ±9 000(that is not distinct in the figure) and ±11 000 s, so we can obtain chip period Tc = 1/1 000 s. Above all,
Fig. 4
Users’data after convolute mixing
Multi-channel blind deconvolution algorithm for multiple-input multiple-output DS/CDMA system
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points, is of near to 0. The convergence rate is accord ant with the adaptation of the coefficient in the demixing matrix D as Fig. 7.
Fig. 5
Output data after separation
Comparing the Figs. 3 and 5, the output binary estimated data after separation is similar to the source signal. In this experiment all the symbols were correctly estimated, which shows the data are received correctly. According to the assumption 2, we estimate the PN code length, which becomes very simple to seek the 2 users’ PN code. We can adopt traditional slippage correlation method or matrix eigenanalysis techniques to resolve the problem. In this paper, we use the slippage correlation method and get the 2 users’ PN code as Fig. 6.
Fig. 7
Adaptation of the coefficient in the demixing matrix D
Fig. 8
BER for convoluted mixture mixture CDMA signal separation
Fig. 6
2 Users’ PN code
Figure 7 plots the adaptation of the coefficient in the demixing matrix D with the α=0.9. Experimental bit-error-rate (BER) results are shown in Fig. 8. 5.3 The influence of δ, ν and α Figure 9 shows the risk function J fluctuatis with the sample serial n. We know from Fig. 9, of after about 3 000 points recursion operation , the risk function J is convergent. Then the risk function J, after twice fluctuation, of recursion operation after about 6 000
Fig. 9
Fluctuation of risk function J
We also find the influence on demixing matrices about convergence factor v and the power factor δ. Power factor δ has dual effects on the demixing matrices. First, when the value of δ is larger, the convergence rate is higher but unstable. From a contrasting point of view, the convergence rate is lower but stabile. Second, the least value of δ affects the risk function
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Cheng Hao, Guo Wei & Jiang Yi
convergence limit. The value of power factor δ is coherent with the risk function convergence limit. We test 10 000 samples 3 times under the same condition, and get the risk function J over time, as show in figure 10. We take logarithm over the y axis to make it clear to observe. The shape and energy factor’s value indicate the influence which energy factor has on the algorithm. Convergence factor v also has the similar influence to energy factor, as show in Fig.11.
Fig. 12
Influence of filter parameter ‘a’ on J
6. Conclusions
Fig. 10
Fig. 11
Fluctuation of risk function J
Influence of power factor V on J
In the algorithm, we choose the low-pass filter h(n) = an u(n), 0 < a < 1 The aim is to reduce the algorithm’s complexity and average the demixing output data. Under the same condition, we get the risk function J over , as show in Fig. 12. The conclusion has been obtained that the value of α is larger, the convergence rate is lower but stable and α is smaller; the convergence rate is higher but unstable.
A technique for BSS of DS-CDMA signals is developed and demonstrated. The technique, referred to here as the average squared cross-output-channel-correlation minimizing algorithm. We propose to minimize the following criterion function J to estimate the spreading code with the correct timing and despread the underlying message sequence, without knowledge about the content of the code or message sequences. The technique is applicable to arbitrary spreading codes and message sequences, and is effective in the environments with white background noise, as long as the DSSS signal has a constant-modulus spreading code and a modulation-on-symbol structure or nearly incommensurate code-repetition and message rates. These results show that the average squared crossoutput-channel-correlation minimizing algorithm is a promising alternative to the existing BSS algorithms for DS/CDMA signals. This algorithm is also applicable to signals with long code, such as commercial communication signals, which allows the technique to be implemented with reasonable computational complexity and over reasonably short reception intervals. This method is also particularly applicable estimation in downlink signal processing, because the codes of the interfering users are unknown. From the previous study it is clear that the algorithm to CDMA separation problem has a significant performance. It is effective to separate the signals and remove the multipath channel effect.
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Multi-channel blind deconvolution algorithm for multiple-input multiple-output DS/CDMA system path channel. Proceedings of the IEEE 6th Circuits and Systems Symposium, 2004: 421–424. [2] Burel G. Detection of spread spectrum transmissions using
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Cheng Hao was born in 1976 and now is a Ph.
stationary convolutively mixed signals. Statistical Signal
D. student in National Key Lab of Communication, University of Electronic Science and Technology of
and Array Processing, 2000: 53–57. [4] Andrzej Cichocki, Shunichi Amari, John Wiley. Adaptive blind signal and image processing. Chi Chester, 2003. [5] Barua s, Hashimoto T. Suboptimal MMSE multiuser re-
China. His research interests include intelligent signal processing, OFDM and iterative decoding.E-mail: [email protected]
ceiver design for DS-CDMA multirate system using spectral correlations. Information Theory Proceedings, 2003: 446–449. [6] Waheed K, Salem F.M. Blind information-theoretic multiUser detection algorithms for DS-CDMA and WCDMA downlink systems. Neural Networks, IEEE Transactions,
Guo Wei was born in 1964. He is a professor and doctoral tutor with UESTC. His research interests include OFDM, MIMO, Turbo codes, LDPC codes, Ad Hoc and UWB wireless communication.
2005, 16(4): 937–948. [7] Eikhamy S E, Lotfy M, Badawy A S. On the blind multi-
Jiang Yi was born in 1978. He received his M.S.
user detection of DS/CDMA signals using the indepen-
Degree from UESTC in 2004, and is now working as a communication engineer.
dent component analysis (ICA). Radio Science Conference, 2003: 34–39.