Blind identification and synchronization of unequal power users in CDMA systems based on eigenvalues variations in slow flat fading channels

Int. J. Electron. Commun. (AEÜ) 64 (2010) 697 – 709 www.elsevier.de/aeue

Blind identification and synchronization of unequal power users in CDMA systems based on eigenvalues variations in slow flat fading channels Siavash Ghavami∗ , Vahid Tabataba Vakili, Bahman Abolhassani Department of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran Received 22 May 2008; accepted 28 April 2009

Abstract In this paper a new blind number of users estimation and sequences synchronization method in multi-user code division multiple access (CDMA) context with unequal power signals has been proposed. Instead of the traditional approach, based on the FROBENIUS square norm behavior (FSNB) and successive detection using serial interference canceler, which was proposed previously in Ghavami and Vakili [Blind SNR estimation on WCDMA systems with unequal power signals and without any prior knowledge. In: IEEE-IST2007; July 2007. p. 1–5; Joint blind users identification and synchronization in non-cooperative CDMA systems in slow flat fading channels. In: ICEE2008; May 2008], we develop a new blind method for joint identification and synchronization. This new method is based on the adaptive threshold and eigenvalues variations (EV) in terms of processing window shifts, number of active users estimation is performed using an adaptive threshold. Theoretical analysis shows that the EV-based criterion avoids the successive detection and non-overlapping synchronized peaks and hence improves the synchronization performance, and proves that it is a powerful tool for blind synchronization especially in unequal power scenario, which we face in eavesdropping case. We show that the improvement is mainly due to the suppression of delay and error propagation that occur with the successive detection in the previous method [Ghavami S, Vakili VT. Blind SNR estimation on WCDMA systems with unequal power signals and without any prior knowledge. In: IEEE-IST2007; July 2007. p. 1–5; Ghavami S, Vakili VT. Joint blind users identification and synchronization in non-cooperative CDMA systems in slow flat fading channels. In: ICEE2008; May 2008]. Simulation results confirm the performance of the identification and synchronization process using this new criterion, allows achieving very good performance at the receiver side in terms of chip error rate (CER) and bit error rate (BER), even at very low signal to noise ratios (SNRs) in unequal power scenarios. 䉷 2009 Elsevier GmbH. All rights reserved. Keywords: Blind identification; Blind synchronization; CDMA; Unequal power; Eigenvalue; Adaptive threshold

1. Introduction Many blind schemes and algorithms have been devised to either improve the performance or reduce the complexity of a code division multiple access (CDMA) receiver in a multi-user context. Some prior knowledge of user in uplink or downlink scenarios, e.g. the signature waveform [3], the ∗ Corresponding author.

E-mail addresses: [email protected] (S. Ghavami), [email protected] (V.T. Vakili), [email protected] (B. Abolhassani). 1434-8411/$ - see front matter 䉷 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2009.04.009

processing gain, the chip rate [4], is always assumed, but their parameters depend on the technique employed and applications may be unknown. Here, we propose a blind multiuser synchronization scheme with no prior knowledge about the transmitter and we assume there is no power control and the transmitted power of the signals may be different. Typically, it is the case of blind signal interception in the CDMA systems, military field or for spectrum surveillance, and generally non-cooperative scenarios. Spread spectrum signals have been used in the military domain for a long time for secure communications [3].

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Nowadays their field of application includes civilian transmissions, especially CDMA transmissions [4]. In [5], maximum likelihood (ML) estimator, approximative maximum likelihood (AML) and multiple signal classification (MUSIC) algorithms are used for propagation delay estimation in multi-user asynchronous DS-CDMA systems. In this approach, ML estimator needs to test all possible transmitted bit sequences in order to perform a true ML estimation of the unknown parameter. AML algorithm finds estimations of all the delays simultaneously, but the AML cost function is highly nonlinear with many local minima and is therefore sensitive to incorrect initialization. Both ML and AML must know spreading sequences of all users for simultaneous delay estimation of active users. The MUSIC estimates delays one by one and requires knowledge of the spreading sequences of the users whose delays are to be estimated. Moreover, [6] proposes a subspace-based channel estimation for CDMA communication systems, it dose not need to know the spreading sequences of other users, but knowledge of desired user’s spreading sequence is necessary. Thanks to the properties of the pseudo-random sequences used, the CDMA technique allows to solve the problem of the increasing number of users in the same frequency band. Moreover, these signals are difficult to detect, especially in a non-cooperative context (e.g. spectrum surveillance), because they are often below the noise level, due to low signal to noise ratios (SNRs). Several blind approaches (i.e. the process of recovering data from multiple simultaneously transmitting users without access to training sequences) have been addressed in the literature [7,8]. In a different approach, many semi-blind multi-user schemes, that exploit some known channel properties, have been proposed [9,10]. But, all these methods are not blind in the sense used in our non-cooperative context. Indeed, they require some prior knowledge about parameters of users. This knowledge is not available in a eavesdropping receiver. Furthermore, in [11] an iterative blind synchronization algorithm based on Frobenius norm is proposed for synchronizing unequal power signals in multi-user scenario. Moreover, a blind estimation method without prior knowledge about spreading sequence of transmitted signal has been proposed in [12] for data detection of direct sequence spread spectrum signals in multipath channels. This paper investigates only single users scenario, and dose not consider multi-user scenario. Also, eavesdropping of synchronous CDMA systems based on expectation maximization has been proposed in [13] and same authors perform blind detection of synchronous CDMA systems in non-Gaussian channels in [14]. In [2] a method for synchronization and estimation of number of active user has been presented when difference between powers of received signals is low. Also in [1,11] successive or iterative methods for blind sequence synchronization and estimation of number of active users have been developed. This approach needs estimation of the spreading sequence of synchronized user, de-spreading it, bit stream estimation, re-spreading and eliminating it from received

signal. This process has high computational complexity and imposes large delay time for detection process and prevents using of the successive method in real time applications. Moreover, in this method performance and accuracy of sequence synchronization in each step depend on that of the pervious steps. In [15] detection and synchronization of CDMA systems for equal power users with two active users have been considered. Multi-rate CDMA synchronization for equal power users and more number of active users have been considered in [16]. Moreover, in [17] a method is presented for synchronizing a multi-user system based on behavior of the maximum eigenvalues of received signal’s estimated covariance matrix. But, in our paper, we will show, this method cannot synchronize signals of users in the unequal power scenario without any prior knowledge about both desired and interfering users’ spreading sequences. In this paper we suggest a method for joint sequence synchronization and estimation of number of active users with unequal transmitted signals powers, when SNR of the received signals are negative in dB. The proposed method supports the synchronization and number of active users estimation when difference between power of received signals is high (advantage over [2]). Synchronization for users is performed simultaneously and there is no need to successive or iterative detection (advantage over [1,11]). Scenario of unequal power users and low SNR signals occurs in eavesdropping of CDMA systems. Normally power control algorithms equalize transmit power of users relative to the base station place, however eavesdropping receiver for practical considerations is placed in a distance relative to the base station. Therefore powers of received signals are not equal. On the other hand no prior knowledge about spreading sequence of received signals causes received signal has low SNR (negative in dB) in receiving signals. This article is organized as follows, Section 2 will introduce the system model and assumptions made. In Section 3 an adaptive threshold for discrimination of the desired signals eigenvalues from those of the noise will be introduced. In Section 4 synchronization based on FROBENIUS square norm behavior (FSNB) and successive detection will be reviewed briefly. Section 5 introduces and analyzes the new approach for joint sequence synchronization and user identification, which is based on the variations of the eigenvalues. The simulation results will be detailed in Section 6 and our conclusions will be drawn in Section 7.

2. System model The received signal in eavesdropping receiver is a combination of mobile signals (uplink scenario) and base station signal (downlink scenario). This assumption has been considered as the worst cases, which both of the synchronized and asynchronized users exist in the environment. Although in CDMA systems transmission in up-link scenario and down-link scenario have different frequency band,

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therefore they cannot interfere with each other; but in military communication, spectrum surveillance and generally non-cooperative schemes this may happen. The combined signal to be received in eavesdropping receiver is given by r (t) =

K +∞ a −1 

Ak dk [ j]h k (t − j T s − k )

k=0 j=−∞

+

K +∞ s −1 

Ak dk [ j]h k (t − j T s ) + n(t)

(1)

k=0 j=−∞

where Ak , dk [ j], h k (t), Ts and k are the received amplitude, the jth data symbol, the convolution of channel response and spreading sequence waveform , symbol duration and delay time of the kth user, respectively, in uplink scenario. Ak , dk [ j], h k (t) are the amplitude, jth data symbol and convolution of channel response and spreading sequence waveform of base station signals, respectively, in down link scenario. K a and K s are the number of asynchronous users and synchronous users, respectively, and finally n(t) is additive white Gaussian noise. The h k (t) is expressed by h k (t) = N −1/2

N −1 

ck [m] p(t − mT c )

(2)

m=0

where N is the processing gain, Tc = Ts /N is the chip time, p(t) is the convolution of the chip waveform, with channel filter (which represents the channel echoes) and receiver filter impulse response, with unit energy, and ck [m] is the magnitude of the mth chip with |ck [m]| = 1. Data symbols of different users have independent identically distributions (i.i.d.). Also channel model is assumed slow flat Rayleigh fading, and thus, channel coefficients are likely constant in duration of the processing window. According to maximum doppler frequency that can be tolerated by typical wireless communication systems, slow fading assumption is logical. (In the computer simulation, the length of the processing window is assumed 200 symbols, the duration of processing window is 0.315 ms, for the bit duration of 1.575 s, if the maximum doppler frequency becomes less than 1/(10 × 0.315 ms)=317.47, then the assumption of constant channel coefficients, in one processing window will be valid). In Eavesdropping scenarios, received signal is a combination of synchronous and asynchronous signals. Some users are synchronous relative to others such as down-link scenario. Some users are asynchronous relative to other such as uplink scenario. Covariance matrix of the received signal can be calculated as follows:  K −1 s  2 k {vk (vk0 )∗ } R = n k=0

+

K a +K s −1

⎞ k {(1 − k )vk0 (vk0 )∗ + k vk−1 (vk−1 )∗ } + I⎠

k=K s

(3)

699

where k is defined as k = (2sig T )/(2n Ts ). Parameters k

of Ts , T and 2sig = 2sig × 2k are symbol duration, samk t pling period, the power of kth received signal, respectively. vk0 and vk−1 are normalized eigenvectors of the covariance matrix of received signal which are obtained using singular value decomposition (SVD) (each asynchronous user produces two eigenvalues which are 0k and −1 k and two eigenvectors which are vk0 and vk−1 , and each synchronous user produces one eigenvalue which is 0k and one eigenvectors which is vk0 ). Furthermore 2sig and 2k are transmitted signal t power and channel coefficient related to kth user, respectively. Also k is delay of kth asynchronous user. In downlink scenario channel coefficients of users are the same. In the following sections Ts = 1 will be assumed for simplifying analytical computation. Eigenvalues of received signal covariance matrix are obtained as follows ⎧ 0 k = 2n (k + 1), k = 0, 1, . . . , K s − 1 ⎪ ⎪ ⎪ 0 ⎪ ⎨  = 2 ( k + 1), k = K s , . . . , K − 1 n k k (4) −1 ⎪ k = 2n (k (1 − k ) + 1), k = K s , . . . , K − 1 ⎪ ⎪ ⎪ ⎩ k = 2n , k = K , . . . , M − 1 where K = K s + K a and the numbers of signal eigenvalues is K s + 2K a . So 2k1 + · · · + 2km + K s is equal to v1 , where v1 is dimension of the signal’s subspace in covariance matrix. We assume that ki synchronous users exist, which are asynchronous with respect to other users and the beginning of the processing window. Therefore if the symbol beginning of the synchronized users is matched with beginning of the processing window, the following equations meet simultaneously: ⎧ 2k1 + 2k2 + · · · + 2km + K s = v1 ⎪ ⎪ ⎪ ⎪ ⎨ k1 + 2k2 + · · · + 2km + 2K s = v2 (5) .. ⎪ ⎪ . ⎪ ⎪ ⎩ 2k1 + 2k2 + · · · + km + 2K s = vm where vi is the dimension of signal’s subspace of the covariance matrix. By solving these equations we are able to determine the number of active users k1 , . . . , km , K s .

3. Adaptive threshold for discriminating signal eigenvalues from noise eigenvalues A classic approach for determining subspace dimension is based on information theoretic criteria e.g minimum description length (MDL) and Akaike information criterion (AIC) which proposed in [18] and performance of these methods has been studied in [19], although the accuracy of these methods are high but the computational complexity of these methods are very high. Since in this section a low complexity algorithm based on an adaptive threshold is introduced for discriminating signal eigenvalues from those of noise with acceptable performance in low SNR regime. This

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adaptive threshold update, based on delay situation relative to the beginning of the processing window. The following criteria are applied for discriminating eigenvalues of signals from eigenvalues of noise |i+1 − i | > T H R ⇒ i+1 : signal’ s eigenvalue

(6)

where i is ith normalized eigenvalue and T H R is defined as follows M−2 T H R = mean M−2 j=0 (| j+1 −  j |) + std j=0 (| j+1 −  j |)

(7) where mean(| · |) and std(| · |) are mean and standard deviation of absolute variation of eigenvalue, respectively. Since the eigenvalues of noise vary continuously, when the difference of two successive normalized eigenvalues is greater than THR, nor mali zed (i + 1) shows signal’s eigenvalue. Parameter  is regulator of adaptive threshold. In [2] this threshold level is used for synchronization but  is assumed constant. Amplitude of eigenvalues depends on delays of different user signals, since changing delays relative to each other, causes variation of eigenvalue amplitude. Therefore, a constant threshold cannot distinguish signal’s eigenvalues from those of noise, and using constant threshold, some signal’s eigenvalues are lost. Therefore,  is determined based on the situation of delays relative to the beginning of the processing window. The initial value of l=1 is selected as unity and in each step will be updated as l+k+1 = l+k + 1. The following algorithm determines the threshold adaptively: 1. l=1 = 1. 2. If the number of eigenvalues which are greater than the selected threshold for l , l+1 , . . . , l+4 remains constant, ignore step 3 and go to step 4, where l+k+1 = l+k + 1 for 0 ⱕ k ⱕ 4, otherwise go to step 3. 3. If (6) is not satisfied for any value of i, go to step 5, otherwise increase l and return to step 2. 4. Parameter l is selected as regulator of the threshold and go to step 6. 5. There is not any signal, go to step 6. 6. End. Fig. 1 shows the flow chart for determining of adaptive threshold. The selected l depends on the delays and the number of users. The criterion (6) is considered clearly in simulation results. By selecting l , since the eigenvalues are arranged in ascending order, eigenvalues with greater indexes than i are related to signal.

Fig. 1. Flow chart for determining of the adaptive threshold.

power of received signal plus the noise power; therefore it is independent of synchronization time [1,16]: M  k=0

k =

K s −1

k +

k=0

= 2n

 K −1 

K −1 

(0k + −1 k )+

k=K s

The FSNB algorithm [16] is based on the maximization of the covariance matrix of received signals eigenvalues. According to Eq. (4), the sum of eigenvalues is equal to the

2n

k=K



k + M

(8)

k=0

According to (8), an appropriate criterion for synchronization is the summation over square eigenvalues. Therefore the FROBENIUS square norm of the estimated covariance matrix of the received signal is used for avoiding calculation of covariance matrix eigenvalues. Since the square of eigenvalues summation is equal to the square norm of the covariance matrix, FROBENIUS square norm can be written as follows [1,16]:  K −1  2 2 4 R = n (2k + k ) + M k=0

4. FSNB algorithm and successive detection

M−1 

+ 24n

 K −1 

2k (−k

+ 2k )

(9)

k=0

where K = K a + K s . In order to maximize (9) in terms of k , we can simplify the optimization with concentration on

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the variable part of the above equation [1,16] F(0 , 1 , . . . ,  K −1 ) =

K −1 

−1.25

2k (2k − k )

−1.35

In (4) the relative delay of the users is constant, and we can derive the following equations:

m − n = Tdm,n (a) |m − n | = (11) n − m = Ts − Tdm,n (b)

−1.45

−1.4

k=0

22k Tdl,k −

k=0

+

m−1 

K −1 

k=K −1

22k (Ts −Tdl,k )−

k=m

2k l

k=0

+

+ Tdl,k ]

−1.7 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Delay [sec]

Fig. 2. Theoretical FROBENIUS square norm for six active users with unequal power signals (scenario 1).

−1.2 Synchronization Points

−1.4 −1.6 −1.8 −2 −2.2

−2.6 −2.8 0

22k [(Tdl,k

− Ts ) + Ts − Tdl,k ] 2

K −1 

k=0

22k (Ts − Tdl,k ) +

k=m

k=K −1

2k

(13)

k=0

The solution of the above equation is m−1 2  K −1 2 1 k=0 k Tdl,k − k=m k (Ts − Tdl,k ) l = − + k=K −1 2 2 2 k=0 k

(14)

 2 since ( m−1 k=0 2k ) > 0 to, Eq. (11) for s less than l is negative and for s greater than l is positive. Therefore maximum values of F are situated in either of the ends of the interval, for l = 0 and l = max[max(Ts − Tdl,k )|0 ⱕ k ⱕ m−1 , max(Tdl,k )|m−1
0.2

0.3

0.4 0.5 0.6 Delay [sec]

0.7

0.8

0.9

1

Fig. 3. Theoretical FROBENIUS square norm for six active users with unequal power signals (scenario 2).

By differentiating Eq. (15) we will have

k=K −1 m−1   dF(l ) = 42k l − 22k Tdl,k dl k=0

0.1

(12)

k=m

+

−1.6 −1.65

−2.4

2k [Td2l,k

k=0 K −1 

−1.5 −1.55

FSNB

where Tdm,n is time distance between mth and nth user, respectively, also . Maximum value of m and n is number of active users. It is assumed that l is the desired delay and delays with index 0 to m − 1 satisfy (11-a), and delays with index m to K − 1 satisfy (11-b). They will be known without loss of analysis generality due to these assumptions. We can simplify (10) from a K -variable function to a single variable function using (11),

k=K −1  2 F(l ) = 2k l2

FSNB

k=0

−

Synchronized Point

−1.3

(10)

m−1 

701

(15)

if l =max(Ts −Tdl,k )|0 ⱕ k ⱕ m−1 and k =q then l =Ts −Tdl,k which is equivalent to q =0. If l =max(Tdl,k )|m−1
and k = p then l = Tdl, p which is equivalent to  p = 0. Therefore, for each desired value of l value, maximum values of F (as defined in (12)) are the synchronized points in respect to beginning of the processing window. In practice k s are constant, and we can shift beginning of the processing window for synchronization, therefore we will have k =d f −k . According to the periodic property of k , variations relative to beginning of the processing window k =d f −k can be substituted with (mod d f −k , Ts ) where it is residual of d f − k over Ts and F can be expressed; F(d f ) =

K −1 

2k ([mod(d f − k , Ts )]2

k=0

− mod(d f − k , Ts ))

(16)

In this section Ts = 1 has been assumed for analysis. Fig. 2 shows the FROBENIUS square norm criterion for six active users. The delays of users 1, . . . , 6 are 0.1, 0.2, 0.4, 0.6, 0.8, 0.9 s, respectively, and normalized powers of users are 0.5, 0.6, 0.7, 0.8, 0.9, 1, respectively (scenario 1).

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It is obvious that the shifts related to the local maximum points are synchronized point. When k s are unequal and difference between powers is not negligible, maximum values of FSNB related to delay of weaker signals can be masked, and one peaks are appeared in FSNB criterion, where it is related to delay of strongest signal. The reason for this problem is that when eigenvalues of weaker signals are maximized by synchronization, variations of eigenvalues of stronger user are able to mask this maximum. Therefore synchronizations for weaker users are not achieved correctly especially when signal power difference is large. Fig. 3 shows the FSNB for six active users by the same delay and power of scenario 1, except power 6th is increased 3 times of the previous values (scenario 2). It is obvious that some peaks of weaker signals have been masked. For estimating the synchronization point with respect of the beginning of the processing window for each user, FSNB can be used. The synchronization process are begun with the strongest signal because when the users power are unequal, in the FSNB curve only one peak appears which is related to the strongest user, we assume the users are arranged in ascending order by their received power at the eavesdropping receiver; that is, the strongest user is indexed as K − 1 and the weakest one as 0. To estimate synchronization instant of other users, since there is no pre-information about the spreading sequence and timing of each user, we eliminate the signal of K th user from the received signal by serial interference canceler (SIC). To do so, we must estimate the spreading sequence of the synchronized user. When signal of one user is synchronized, the covariance matrix of the received signal can be written as ∗ R = vm v m + 2n ⎧ ⎫ −1 ⎨ K ⎬ × k {(1 − k )vk0 (vk0 )∗ + k vk−1 (vk−1 )∗ } + I ⎩ ⎭ k=1k
m

(17) Hence, still under the assumption of almost uncorrelated sequences. Eq. (17) highlights a maximum eigenvalue which is associated to eigenvector which contains the corresponding spreading sequence (including the effects of the overall transmission filter). For simplicity, we assume that the length of the code or the chip rate is known at the receiver, although it can be estimated using cyclostationary property of the direct sequence spread spectrum signals [20]. In our simulations, we assume that the spreading sequence is antipodal and has binary phase shift keying (BPSK) modulation. This assumption is not necessary because we can identify the type of modulation by calculating the real and imaginary part of estimated data, and comparing of real part variance with imaginary part variance [15]. Also it is assumed that Ts has been estimated in receiver using [21,22].

Using the estimated spread sequence and the estimated delay of the K th user we can de-spread the signal of this user and estimate his data as T r (t) dˆm [ j] = vm

(18)

To eliminating the estimated signal from the received signal, its amplitude should be known. Therefore we must estimate SNR of the synchronized signal. After synchronization, eigenvalue of the synchronized signal can be written as m = 0 ⇒ m = 2n (m + 1)

(19)

The above equation is simplified as m =

T 2  + 2n Ts sig

(20)

and the SNR of strongest user in dB can be obtained as    Ts n − 1 (21) S N R m = 10 log T 2n where 2n is estimated as 2n =

M−1  1 i M−p

(22)

i=K

where p is dimension of covariance matrix of signal which is equal to obtained i in Section 3. The i is obtained according to the adaptive threshold in (6), it is equal to the minimum index of signal’s eigenvalue. According to Eq. (4) k s depend on variance of both signal and noise when 0 < k < K − 1, and k s depend on variance of noise for 2K < k < M − 1. The Eq. (5) shows that the total space consists of a signal’s and noise’s subspaces, which noise’s subspace spanned by the remaining M − 2K eigenvalues. With the help of this subspace decomposition, the power of the total received signals can easily be separated into the power of the desired signal and the noise power, yielding an efficient estimate of the SNR. The estimated signal are re-spread using the estimated spreading sequence. Multiplying the estimated power of user by re-spreading signal, we are able to eliminate this signal from the received signal in processing window as follows ⎧ +∞ ⎨ ri (t) = ri−1 (t) − Ai vi  dˆi [ j], 1 ⱕ i ⱕ K − 1 (23) j=−∞ ⎩ r0 = r (t) where ri (t) is residual signal after i − 1 times signal subtraction, this procedure is performed using the SIC receiver. Synchronization, SNR estimation and spreading sequence estimation are repeated, and finally the estimated signal of user removes from received signal. Eqs. (22), (23) are performed K − 1 times for estimating the SNR of users. Finally after finishing this process we are able to estimate data, synchronization point in respect to the beginning of the processing window and the SNR of all users.

S. Ghavami et al. / Int. J. Electron. Commun. (AEÜ) 64 (2010) 697 – 709

5. The EV based algorithm for synchronization

For any shift of the processing window (e.g. d f ) where k ⱕ d f ⱕ k+1 and for each user above equation shows when 0k increases −1 k decreases and vice versa. In Eq. (25) 0 k has a maximum for d f = k . When 0k is maximized for d f = k , −1 k is minimized at this point. Number of signal eigenvalues are determined using the adaptive threshold, which defined in (6). 0k and −1 k are calculated in terms of different shifts of the processing window. Number of maximum values are 2K a and there are 2K a delays corresponding to this maximum values. K a points of this points are real synchronized points, which maximize 0k , while K a points are virtual synchronized points which maximize −1 k . For correct synchronization the virtual synchronized point must be identified and then they must be eliminated. The processing window is shifted for each estimated delay because the virtual synchronized points are identified. Then the numbers of signals eigenvalues in the estimated covariance matrix of the received shifted signal are calculated for all estimated delays using the adaptive threshold in (6). If the number of signals eigenvalues are reduced (number of signals

In this section a new synchronization method based on eigenvalues variations in terms of the processing window’s shifts is developed. Eigenvalues of the received signal covariance matrix for asynchronous scenario according to (4) and k = 0, . . . , K a − 1 is obtained as

0 k = 2n (k (1 − k ) + 1) (24) 2 −1 k = n (k k + 1). But in practice k s are constant, and we can shift the processing window, therefore k = d f − k . According to periodic property of k (relative to the beginning of the processing window), k = d f −k can be substitute by mod(d f −k , Ts ), where it is residual of d f − k over Ts . Therefore eigenvalues of signal for k = 0, . . . , K a − 1 in the asynchronous scenario can be obtained as follows

0 k = 2n (k (1 − mod(d f − k , Ts )) + 1) (25) 2 −1 k = n (k mod(d f − k , Ts ) + 1).

1.8

0

λ1

1.7

λ−1 1

1.4

1.8 λ02 λ−1 2

1.6

λ1

λ2

1.3

1.2

1.1

1 0.3

0.5

0.7

1.5

1.5

1.4

1.4

1.3

1.3

1.2

1.2

1.1

1.1

0.9

1 0

0.2

Delay [sec]

0.4

0.6

0.8

1

0

0.2

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Delay [sec]

1.8

0.8

1

4 λ06

λ05

3.5

λ−1 5

1.8

1.6

0.6

Delay [sec]

2

1.7

λ−1 6

3 1.6

0

1.4

λ5

λ4

λ6

1.5 λ4

λ−1 3

1.6

1 0.1

λ03

1.7

λ3

1.5

703

−1

λ4

2.5

1.4

1.3

2

1.2

1.2

1.5

1.1 1

1 0

0.2

0.4

0.6

Delay [sec]

0.8

1

1 0

0.2

0.4

0.6

Delay [sec]

0.8

1

0.1

0.3

0.5

0.7

Delay [sec]

Fig. 4. Variations of i0 and i−1 (theoretical EV) for i = 1, 2, . . . , 6 in terms of processing window shifts.

0.9 1

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S. Ghavami et al. / Int. J. Electron. Commun. (AEÜ) 64 (2010) 697 – 709

−0.9 MEVB

−1

FSNB

−1.1 −1.2

0.101 0.101 0.1009 0.1009 0.1008 0.1008

0

0.1

0.2

0.3

0.4 0.5 0.6 Delay[sec]

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4 0.5 0.6 Delay[sec]

0.7

0.8

0.9

1

−1.3 0.18 MEVB

−1.4 −1.5

0.16 0.14 0.12

−1.6 0

0.1

0.2

0.3

0.4 0.5 0.6 Delay [Sec]

0.7

0.8

0.9

1

Fig. 5. Analytical diagram of FROBENIUS square norm behavior in terms of shifts of processing window with equal power users.

0.1

Fig. 7. MEVB for different shifts of the processing window in equal (a) and unequal (b) power scenario.

2

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Fig. 6. Analytical diagram of greatest eigenvalue behavior in terms of shifts of processing window with equal power users.

eigenvalues for one user are reduced from 2 to 1 by synchronization of signal relative to the beginning of the processing window) the estimated delay is delay of desired user, otherwise the user with this estimated delay will not exist in the processing window. In Fig. 4, T = 1 is assumed, delay and power of users are assumed similar to defined scenario in Fig. 3. Fig. 4a shows 01 and −1 1 in terms of different shifts of the process0 ing window, 1 has been maximized in d f = 0.1 s, while −1 1 has been minimized at this point. Fig. 4b, . . ., Fig. 4f show i0 and i−1 for i = 1, 2, . . . , 6 in terms of shifts of the processing window, respectively. Delay of maximum point in i0 is estimated delay of ith user, while at this point i−1 is minimized. In the computer simulation only the maximum value points are used. When power of users are different in received signal, mentioned process must be performed for all eigenvalues of signals (Number of eigenvalues of signal are determined using the adaptive threshold in (6)), but if power of users

6 Active User Equal Power 4 Active User Equal Power 3 Active User Unequal Power

40

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Fig. 8. Number of signal eigenvalues as a function of .

are equal, By using variation of greatest eigenvalue in terms of processing window shift, estimation of synchronization points is possible. Eq. (25) for k = 0, . . . , K a − 1 is simplified as

0 k = 2n (2 − mod(d f − k , Ts ) + 1) (26) 2 −1 k = n (mod(d f − k , Ts ) + 1) The eigenvalues of signals depend on delays and noise power only, since powers of users are equal and k s are assumed equal to 1, greatest eigenvalue have local maximums in d f = k s. In computer simulation scenario of Figs. 5 and 6 numbers of active users have been assumed 8, delay of users are 0.1, 0.11, 0.2, 0.21, 0.4, 0.6, 0.8 and 0.9 s relative to beginning of the processing window, respectively, while power of all users are assumed equal. Fig. 5 shows the FSNB criteria in terms of different shifts of the processing window. According to Fig. 5, peaks related to delays of 0.11 and 0.21 s, are masked. Fig. 6 is similar to Fig. 5, except it shows behavior of greatest eigenvalue in terms of shifts of the processing window. In Fig. 6, peaks corresponding to

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Fig. 9. Six greatest eigenvalues behavior in terms of different shifts of processing window for SNR = −11 dB for weakest signal.

delays 0.11 s and 0.21 s are obvious, which shows better delay resolution of greatest eigenvalue, which proposed in [17], in relative to the FSNB. In Fig. 7 we consider six active users; all of them are asynchronous relative to the beginning of the processing window. In Fig. 7a power of all active users are equal to 0 dB Watt and noise variance is equal to −10 dB Watt. In Fig. 7b, users power are 0, 3, 6, 6.5, 8 and 18 dB Watt and noise variance is equal to −10 dBWatt. Their delays relative to the beginning of the processing window are 0.1, 0.2, 0.4, 0.6, 0.8 and 0.9 s, respectively. The maximum eigenvalues behavior (MEVB) criterion proposed by [17] is plotted for an equal power scenario in terms of delay in Fig. 7a. It is obvious MEVB can estimate delay of users, but in aforementioned unequal power scenario, Fig. 7b shows the MEVB which proposed in [17] cannot estimate delay of users with smaller power even in noise free scenario. Although computational complexity of using eigenvalues in each time is more than using the FSNB in each time, but it has lower computational complexity in relative to the FSNB plus successive detection totally (it will be shown in simulation results by comparing dedicated CPU time of both methods), because synchronization in unequal power scenario based on the EV based method is executed simultaneously for all users, while in the FSNB and the suc-

cessive detection method synchronization are executed for each user separately, and for each user must be repeated. The EV based method has better performance in scenario of unequal power users. Furthermore in the unequal power scenario and when time interval between successive peaks is low EV method has better performance on synchronization in relative to the FSNB. Moreover, synchronization accuracy in each step on the successive detection using the FSNB depends on synchronization error of the FSNB in pervious steps, SNR and the spreading sequence estimation error of stronger users which have been estimated in pervious steps. Also re-spreading and elimination of the detected signal from the received signal do not perform ideally. These factors cause error in estimation of the weaker users delays. On the other hand, we can say that in successive detection, estimation error in each step depends on that error of the pervious steps. Finally imposed delay of successive detection prevents application of it in real time applications.

6. Simulation results In this section simulation results are presented. In all simulations, the spreading sequences for all users is Gold

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Fig. 10. Number of eigenvalues of signal after shifting the processing window with estimated delays from Fig. 8.

code with length of 63. Both data and spreading sequence modulation are assumed BPSK. Chip time of 50 ns, sampling frequency of Fs = 200 MHz has been considered and the number of symbols in each window is 200 symbols. Fig. 8 shows number of eigenvalues of signals (which identified using the adaptive threshold in (6)) in terms of the  parameter. In the received signal, all users relative to the beginning of the processing window have been assumed asynchronous. Fig. 8 shows three scenarios: three active users, four active users and six active users, in all scenarios SNR of weakest signal is −11 dB. Fig. 8 shows number of signals eigenvalues for three active user with power of 0, 3 and 6 dB and delays of 0.2, 0.8 and 1.4 s relative to the beginning of the processing window, number of active users are identified 6, and  is obtained 9 according to the adaptive threshold in (6). In the same conditions but with equal powers again number of signal eigenvalues will be 6, which dose not show in Fig. 8 Number of eigenvalues of signal is obtained eight for four active users with equal power scenario and delays of them are 0.2, 0.6, 1 and 1.4 s relative to the beginning of processing window. Parameter  similar to the pervious case is obtained 9 according to the adaptive threshold in (6). Also Fig. 8 shows the number of signal eigenvalues for six active users with equal power scenario

and delays of 0.1, 0.3, 0.5, 0.7, 0.9, 1.1, 1.3 and 1.5 s relative to the beginning of the processing window are obtained 12. Furthermore parameter  is obtained 7. Now three active users scenario with unequal power which mentioned in pervious paragraph is considered. Eigenvalues of received signal, e.g. 1 , . . . and 6 (Number of them are obtained using the adaptive threshold in (6).) are plotted in terms of shifts of the processing window relative to the beginning of symbol in Fig. 9a–f, respectively. In Fig. 9a–f delays of maximum values of 1 , . . . and 6 are obtained as 1.4, 0.8, 0.64, 0.24, 0.2 and 1.12 s, respectively. It is obvious, delays of 1.4, 0.8 and 0.2 s are matched with delays of users in received signal. Three other delays are virtual delays and must be identified and eliminated. In continuing, the procedure of identification of virtual synchronous points is described. Fig. 10a shows the eigenvalues of the covariance matrix of the received signal, when the processing window is shifted 1.575.1.4 s=0.175 s (the estimated delay in Fig. 9a) relative to the initial state. Numbers of eigenvalues of signal are reduced from 6 to 5, according to using the adaptive threshold in (6). Therefore the signal with delay of 1.4 s exists in the received signal, which matches with the assumptions for simulation.

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Fig. 10b is similar to Fig. 10a except it is related to the signal with the delay of 0.8 s. It is a real delay similar to the delay of 1.4 s (because by shifting the processing window of 0.775 s number of eigenvalues of the signal reduces from 6 to 5). Fig. 10c are related to the delay of 0.64 s, it is obvious that number of eigenvalues of the signal with shift of processing window 1.575.0.64 s=0.935 s are not reduced using the adaptive threshold in (6). Therefore signal with the delay of 0.64 s dose not exist in the received signal and it is a virtual synchronization point. Furthermore, Fig. 10d and f show the signal with delay of 0.23 and 1.12 s do not exist in the received signal and according to Fig. 10e it is obvious that the delay of 0.2 s are exist in the received signal. Fig. 11 shows the number of estimated eigenvalues of the signal using (6) in terms of different values of the  for the shifted processing window using the adaptive threshold in (6), the processing window has been shifted as 1.4, 0.8, 0.64, 0.24, 0.2 and 1.12 s, which are estimated delays and obtained using EV based criterion. Fig. 10 shows that  = 9 must be selected, this value of  are selected according to the adaptive algorithm which was described in Section 3. This figure shows five eigenvalue are related to signal for the processing window with delay of 1.575.1.4 s = 0.175 s in relative to the initial state, therefore user with this delay exist in the received signal. For delays of 1.575.0.8 s=0.775, 1.575.0.64 s=0.935, 1.575.0.23 s=1.372, 1.575.0.2 s= 1.375 and 1.575.1.12 s = 0.473 s, number of eigenvalues of signals are 5, 6, 6, 5 and 6, respectively. Reduction of eigenvalues of the signal for each shifted processing window shows user with this delay exists in received signal. Unchanging of eigenvalues of the signal for each shifted processing window shows that user with this delay dose not exist in the received signal. Fig. 12 shows that equal power scenario with four active users and SNR of received signal for each user is −11 dB. This figure shows behavior of greatest eigenvalue in terms of shifts of the processing window. According to Fig. 12 the synchronization points are 1 = 0.2 s, 2 = 0.6 s, 3 = 1 s

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and 4 = 1 s. The considered scenario for Fig. 13 is similar to Fig. 12, except, this figure shows FSNB in terms of shifts of the processing window. FSNB criterion in equal power scenario dose not need to successive detection and can estimate synchronization points. Advantage of using greatest eigenvalue in relative to FSNB is better resolution in estimation of delays. A Computer with CPU Pentium(R) 4 and 3.2 GHz CPU clock pulse and 1Gbyte RAM have been used for the computer simulation. Fig. 14 shows dedicated time by CPU for simulation of the FSNB with successive detection which proposed in [1], this dedicated time for synchronization is similar to dedicated time for synchronization using FSNB with iterative detection which proposed in [11], and the EV based method with estimation of spreading sequence, it is obvious from Fig. 14, increasing number of users increase computational complexity of synchronization method using the FSNB and successive detection. While computational complexity of the EV based method are remained constant approximately, and this have lower computational

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complexity in relative to the FSNB with successive detection for number of users greater than unity. We have estimated the spreading sequence of each user in the EV based synchronization method for fair comparison between computational complexities of both methods.

7. Conclusions In this paper, a new method for synchronization and identifying number of active users has been presented in unequal power scenario of CDMA systems in slow flat Rayleigh fading channel. This method is based on behavior of eigenvalues of the received signal covariance matrix in terms of shifts of the processing window. The proposed method in comparing with the FSNB with successive detection offers better synchronization accuracy and reduced computational complexity. Also an adaptive threshold for determining number of active users is determined. Joint identification of synchronous and asynchronous users is possible using this adaptive threshold and the proposed synchronization method. The proposed method has good accuracy for identifying number of users and their synchronization in low signal to noise ratios, this feature fits with application of proposed method in non-cooperative context such as eavesdropping scenario.

References [1] Ghavami S, Vakili VT. Blind SNR estimation on WCDMA systems with unequal power signals and without any prior knowledge. In: IEEE-IST2007; July 2007. p. 1–5. [2] Ghavami S, Vakili VT. Joint blind users identification and synchronization in non-cooperative CDMA systems in slow flat fading channels. In: ICEE2008; May 2008. [3] Picholtz RL, Schilling DL, Milstein LB. Theory of spread spectrum communications – a tutorial. IEEE Transactions on Communications 1982;30(5):884–5.

[4] Magill DT, Natali FD, Edwards GP. Spread spectrum technology for commercial applications. IEEE Journal 1994;82(4):572–84. [5] Strom E, Parkvall S, Miller S, Ottersten B. Propagation delay estimation in asynchronous direct-sequence codedivision multiple access systems. IEEE Transactions on Communications 1996;44(1):84–93. [6] Bensley SE, Aazhang B. Use of electronic speckle pattern interferometry (ESPI) in the measurement of static and dynamic surface displacements. IEEE Transactions on Communications 1996;44(8):1009–20. [7] Wang X, Poorng H. Blind multiuser detection: a subspace approach. IEEE Transactions on Information Theory 1998;44(2):677–90. [8] Buzzi S, Lops M, Pauciullo A. Iterative cyclic subspace tracking for blind adaptive multiuser detection in multirate CDMA systems. IEEE Transactions on Vehicular Technology 2003;52(6):1463–75. [9] Wang X, Poor HV. Blind adaptive interference suppression for CDMA communication based on eigenspace tracking. In: IEEE-CISS; 1997. p. 468–73. [10] Honig M. Blind estimation of the pseudo-random sequence of a direct spread spectrum signal. In: MILCOM; November 1997. p. 836–40. [11] Koivisito T, Koivunen V. Blind despreading of short-code DSCDMA signals in asynchronous multi-user systems. Signal Processing Journal 2007;87(11):2560–8. [12] Tsatsanis MK, Giannakis GB. Blind estimation of direct sequence spread spectrum signals in multipath. IEEE Transactions on Signal Processing 1997;45(5):1241–51. [13] Yao Y, Poor HV. Eavesdropping in the synchronous CDMA channel: an EM-based approach. IEEE Transactions on Signal Processing 2001;49(8):1748–56. [14] Yao Y, Poor HV. Blind detection of synchronous CDMA in non-Gaussian channels. IEEE Transactions on Signal Processing 2004;52(1):271–9. [15] Nzea CN, Gautier R, Burel G. Blind synchronization and sequences identification in CDMA transmissions. In: MILCOM; November 2004. p. 1384–90. [16] Nzea CN, Gautier R, Burel G. Parallel blind multiuser synchronization and sequences estimation in multirate CDMA transmissions. In: IEEE-ACSSC; November 2006. p. 2157–61. [17] Nzea CN, Gautier R, Burel G. Blind multiuser identification in multirate CDMA transmissions: a new approach. In: IEEE Conference on Signals, Systems and Computers; October–November 2006. p. 2162–66. [18] Wax M, Kailath T. Detection of signals by information theoretic criteria. IEEE Transactions on Acoustic, Speech, and Signal Processing 1985;33(2):387–92. [19] Fishler E, Grosmann M, Messer H. Detection of signals by information theoretic criteria: general asymptotic performance analysis. IEEE Transactions on Signal Processing 2002;50(5):1027–36. [20] Yan J, Hongbing J. A cyclic-cumulant based method for DS-SS signal detection and parameter estimation. In: IEEE Conference on Signals, Systems and Computers; August 2005. p. 966–9. [21] Burel G, Bouder C, Berder O. Detection of direct sequence spread spectrum transmissions without prior knowledge. In: IEEE-GLOBECOME; November 2001. p. 236–9.

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[22] Ghavami S, Abolhassani B. Detection of DS-SS signals over fading channels without prior knowledge of spreading sequence by measuring signal non-Gaussianity. In: IEEEPIMRC2006; September 2005. p. 1–5. Siavash Ghavami received the B.S. and M.S. degree both in electrical engineering from Iran University of Science and Technology (IUST), Tehran, Iran, in 2006 and 2009, respectively; he is currently Ph.D. candidate. His research interests include mobile cellular systems, multi-user detection for CDMA systems, blind detection in multi-user spread spectrum systems, network information theory and cognitive radios. Vahid Tabataba Vakili received the B.S. degree from Sharif University of Technology, Tehran, Iran, in 1970, the M.S. degree from the University of Manchester, Manchester, UK, in 1973, and the Ph.D. degree from the University of Bradford, Bradford, U.K., in 1977, all in electrical engineering. In 1985, he joined the Department of

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Electrical Engineering, Iran University of Science and Technology, Tehran. In 1997, he was promoted to associate professor. He has served as the head of the Communications Engineering Department and as the head of postgraduate studies. His research interests are in the areas of mobile cellular systems, interference cancelation for CDMA systems, and space–time processing and coding.

Bahman Abolhassani received the B.S. degree from Iran University of Science and Technology, Tehran, Iran, in 1980, the M.S. and the Ph.D. degree from the University of Saskatchewan, Saskatoon, Saskatchewan, Canada, in 1995 and 2001, respectively, all in electrical engineering. In 2002, he joined as an assistant professor to the Department of Electrical Engineering, Iran University of Science and Technology, Tehran. He has served as the head of the Electrical Engineering Department. His research interests are in the areas of wireless communications, interference cancelation for CDMA systems, spread spectrum, wireless sensor networks and cognitive radios.