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IEEE COMMUNICATIONS LETTERS, VOL. 2, NO. 8, AUGUST 1998
217
A Linearly Constrained Constant Modulus Approach to Blind Adaptive Multiuser Interference Suppression Joaqu´ın M´ıguez and Luis Castedo Abstract—This letter presents a linearly constrained constant modulus approach for the blind suppression of multiuser interferences in direct-sequence code division multiple access systems. The method performs the same as minimum mean square error receivers and outperforms existing blind approaches because it only requires a rough estimate of the desired user code and timing. Index Terms—Blind filtering, CDMA, linearly constrained constant modulus, multiple access interference.
I. INTRODUCTION
M
ULTIPLE-ACCESS interference (MAI) caused by code nonorthogonality constitutes the main limitation of direct-sequence code division multiple access (DS CDMA) systems. Different techniques have been proposed to adaptively suppress MAI (see [1], and references therein, for an overview). Minimum mean square error (MMSE) receivers [2] can be used but its adaptive implementation requires the transmission of training sequences. Alternative blind implementations based on the linearly constrained minimum variance (LCMV) criterion have been proposed [3] but they are extremely sensitive to inaccuracies in the acquisition of the desired user timing and spreading code. In this letter we investigate a new blind approach to MAI suppression based on the constant modulus (CM) criterion. This criterion cannot be directly applied to our problem because it can capture an interference instead of the desired user [4]. We will show, however, that imposing the same constraint as in LCMV the capture problem is eliminated. The resulting receiver, termed linearly constrained CM (LCCM), turns out to be very robust to code and timing inaccuracies because the aim of the constraint is not to protect the desired signal from being cancelled by the adaptive algorithm, but to avoid the capture of an interference.
Fig. 1. Block diagram of a DS CDMA linear receiver for the demodulation of a single user.
where and are the -th user transmitted symbols and is the chip pulse waveform, is the received amplitude, is the symbol period and is the chip period, additive white Gaussian noise (AWGN). Fig. 1 plots the linear receiver for the demodulation of a single user in DS CDMA systems. The received signal is passed through a chip-matched filter and a -tapped delay line followed by FIR filter with coefficients a bit rate sampler. The output of the chip-matched filter is (2) . Rewriting (2) in vector form
where
(3) ,
where and .. .
II. SIGNAL MODEL Let us consider a synchronous baseband DS CDMA system users. Each user is assigned a unique code sequence with . The received signal is
..
.
.. .
(4)
Correspondingly, the receiver output is (5)
(1) Manuscript received January 6, 1998. The associate editor coordinating the review of this letter and approving it for publication was Dr. B. R. Vojcic. This work has been supported by CICYT under Grant TIC96-0500-C10-02 and by Xunta de Galicia under Grant XUGA 10502A96. The authors are with the Departamento de Electronica e Sistemas, ´ Universidade da Corona, 15.071 A Corona, Spain (e-mail: [email protected]). Publisher Item Identifier S 1089-7798(98)06961-0.
and should provide an estimate of the symbols transmitted by the desired user. III. LCCM RECEIVERS In the LCCM receiver, the filter coefficients are selected according to the following optimization problem:
1089–7798/98$10.00 1998 IEEE
(6)
218
IEEE COMMUNICATIONS LETTERS, VOL. 2, NO. 8, AUGUST 1998
where is the desired user code vector. A similar optimization problem has been previously proposed in the context of array processing [5]. In order to show the ability of our approach to remove MAI, let us consider a situation in which the noise is negligible and, therefore
expression for the Hessian is
.. .
.. .
..
.
.. .
(14)
(7) represents the user where and amplitudes. Due to the constraint the receiver output can be rewritten as (8) , and where Substituting (8) into (6) and taking into account that is a full rank matrix, the optimization problem (6) reduces to
where
is the identity matrix. For the desired solution, the Hessian is (15)
which is clearly definite positive. For the remaining stationary points it is straightforward to show that the first diagonal element of the Hessian is (16)
(9) is statistically independent of the interferences and Since and (9) simplifies to verifies
is not definite positive. Therefore, and, as a consequence, is the solution to (6). It is important to note that, when the noise is negligible, this is also the solution achieved by the decorrelation receiver [6], the MMSE receiver [2] and the LCMV [3] receiver.
(10) IV. ADAPTIVE ALGORITHM is the (normalized) fourth-order moment of where Moreover, assuming the interferences are also statistically , independent among them and verify , and the above optimization problem can be further reduced to
In this section we derive an adaptive algorithm to solve the constrained optimization problem (6). This problem can be converted into an unconstrained form using the generalized sidelobe canceller (GSC) [7] decomposition (17)
(11) The minima are the points where the gradient vanishes and the Hessian matrix is positive-definite. Calculating the gradient we obtain .. .
(12)
vanishes at the point It is straightforward to show that where MAI is completely removed. Neverthealso vanishes at the additional stationary points less, (13) is the number of interferent user signals present where at the output1. The next step is to examine the positive definiteness of the Hessian matrix at the above stationary points. The general 1 This
is the quiescent vector, is the where blocking matrix whose columns span the null space of (i.e., is full rank and satisfies and is the unconstrained adaptive weight vector. The GSC decomposition always satisfies the ensures that the overall weight vector As a result, (6) is equivalent to constraint regardless of
result is true if the transmitted symbols have negative kurtosis (i.e., This is always the case in practical situations.
(18) Problem (18) can be solved using the normalized stochastic gradient algorithm (19) where and
is the output error, is chosen to eliminate the error signal at each iteration, i.e., (20)
In practical situations, is where to reduce the tipically multiplied by a positive constant algorithm misadjustment noise.
M´IGUEZ AND CASTEDO: AN APPROACH TO BLIND ADAPTIVE MULTIUSER INTERFERENCE SUPPRESSION
Fig. 2. Capture probability versus for a DS CDMA channel containing 15 interferent users, each of them 6 dB stronger than the desired one. Spreading codes are length 31 Gold sequences.
219
Let us model the estimation error as a vector of Gaussian random variables with zero mean and autocorreFig. 2 plots the capture probability of a lation matrix LCCM receiver with respect to . Each point was obtained running 10 000 simulations with the same initial conditions and counting the number of times that the receiver was captured by an interference. Note that captures were only Practical code acquisition circuits do observed for not reach such a large value of estimation error variance. The channel Gaussian noise was neglected in the analysis of Section III because otherwise it becomes rather involved. Simulations reveal that, when considering the AWGN in the channel, LCCM receivers perform the same as MMSE receivers thus exhibiting a desirable balance between noise enhancement and MAI suppression. This is in accordance with recent analytical studies of the unconstrained CM cost function where it is shown that the MSE value achieved at the CM minima is equal to the MMSE value when the SNR is high [8]. Fig. 3 plots the bit error rate (BER) for different SNR values. It is assumed that there exists a small amount of For comparison purposes, code estimation error Fig. 3 also plots the BER corresponding to a MMSE receiver with training sequences and a LCMV receiver. It is clearly seen the LCCM receiver performs practically the same as the former and considerably better than the latter. Note that a small amount of constraint mismatching severely degrades the performance of the LCMV receiver. VI. CONCLUSIONS We have presented a new blind adaptive approach to MAI cancellation for DS CDMA systems, termed LCCM, in which the CM cost function is minimized subject to a linear constraint. It is analytically demonstrated that LCCM receivers exhibit an optimal performance when the channel noise is negligible and a perfect knowledge of the desired user code is available. Computer simulations illustrate that these receivers are very robust to code estimation errors and perform the same as MMSE receivers in the presence of noise. REFERENCES
Fig. 3. BER versus SNR for a DS CDMA channel with AWGN and three interferent users, each of them 6 dB stronger than the desired one. Spreading codes are length 7 Gold sequences. MMSE and LCMV receivers are implemented with least squares adaptive algorithms. Weight vectors are obtained after 150 iterations.
V. COMPUTER SIMULATIONS Section III proves that LCCM receivers do not capture interferences when the constraint is implemented using the In practice, however, the constraint exact received code provided by a code acquisition is built with an estimate and LCCM receivers may still circuit that is not equal to be captured by an interference. Computer simulations were carried out to study this phenomenon.
[1] S. Verd´u, “Adaptive multiuser detection,” in Code Division Multiple Access Communications, S. G. Glisic and P. A. Leppannen, Eds. Amsterdam, The Netherlands: Kluwer Academic, 1995. [2] U. Madhow and M. L. Honig, “MMSE interference suppression for direct-sequence spread-spectrum CDMA,” IEEE Trans. Commun., vol. 42, pp. 3178–3188, Dec. 1994. [3] M. L. Honig, U. Madhow, and S. Verd´u, “Blind adaptive multiuser detection,” IEEE Trans. Inform. Theory, vol. 41, pp. 944–960, July 1995. [4] J. R. Treichler and M. G. Larimore, “The tone capture properties of CMA-based interference suppressors,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 946–958, Aug. 1985. [5] M. J. Rude and L. J. Griffiths, “A Linearly constrained Adaptive Algorithm for Constant Moudulus Signal Processing,” in Proc. EUSIPCO’92, Barcelona, Spain, Sept. 1992, pp. 237–240. [6] R. Lupas and S. Verd´u, “Linear multiuser detectors for synchronous code-division multiple access channels,” IEEE Trans. Inform. Theory, vol. 35, pp. 123–136, Jan. 1989. [7] L. J. Griffiths and C. W. Jim, “An alternative approach to linearly constrained adaptive beamforming,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-30, pp. 2737–2746, Nov. 1982. [8] I. Fijalkow, A. Touzni, and J. R. Treichler, “Fractionally spaced equalization using CMA: Robustness to channel noise and lack of disparity,” IEEE Trans. Signal Processing, vol. 45, pp. 55–56, Jan. 1997.
217
A Linearly Constrained Constant Modulus Approach to Blind Adaptive Multiuser Interference Suppression Joaqu´ın M´ıguez and Luis Castedo Abstract—This letter presents a linearly constrained constant modulus approach for the blind suppression of multiuser interferences in direct-sequence code division multiple access systems. The method performs the same as minimum mean square error receivers and outperforms existing blind approaches because it only requires a rough estimate of the desired user code and timing. Index Terms—Blind filtering, CDMA, linearly constrained constant modulus, multiple access interference.
I. INTRODUCTION
M
ULTIPLE-ACCESS interference (MAI) caused by code nonorthogonality constitutes the main limitation of direct-sequence code division multiple access (DS CDMA) systems. Different techniques have been proposed to adaptively suppress MAI (see [1], and references therein, for an overview). Minimum mean square error (MMSE) receivers [2] can be used but its adaptive implementation requires the transmission of training sequences. Alternative blind implementations based on the linearly constrained minimum variance (LCMV) criterion have been proposed [3] but they are extremely sensitive to inaccuracies in the acquisition of the desired user timing and spreading code. In this letter we investigate a new blind approach to MAI suppression based on the constant modulus (CM) criterion. This criterion cannot be directly applied to our problem because it can capture an interference instead of the desired user [4]. We will show, however, that imposing the same constraint as in LCMV the capture problem is eliminated. The resulting receiver, termed linearly constrained CM (LCCM), turns out to be very robust to code and timing inaccuracies because the aim of the constraint is not to protect the desired signal from being cancelled by the adaptive algorithm, but to avoid the capture of an interference.
Fig. 1. Block diagram of a DS CDMA linear receiver for the demodulation of a single user.
where and are the -th user transmitted symbols and is the chip pulse waveform, is the received amplitude, is the symbol period and is the chip period, additive white Gaussian noise (AWGN). Fig. 1 plots the linear receiver for the demodulation of a single user in DS CDMA systems. The received signal is passed through a chip-matched filter and a -tapped delay line followed by FIR filter with coefficients a bit rate sampler. The output of the chip-matched filter is (2) . Rewriting (2) in vector form
where
(3) ,
where and .. .
II. SIGNAL MODEL Let us consider a synchronous baseband DS CDMA system users. Each user is assigned a unique code sequence with . The received signal is
..
.
.. .
(4)
Correspondingly, the receiver output is (5)
(1) Manuscript received January 6, 1998. The associate editor coordinating the review of this letter and approving it for publication was Dr. B. R. Vojcic. This work has been supported by CICYT under Grant TIC96-0500-C10-02 and by Xunta de Galicia under Grant XUGA 10502A96. The authors are with the Departamento de Electronica e Sistemas, ´ Universidade da Corona, 15.071 A Corona, Spain (e-mail: [email protected]). Publisher Item Identifier S 1089-7798(98)06961-0.
and should provide an estimate of the symbols transmitted by the desired user. III. LCCM RECEIVERS In the LCCM receiver, the filter coefficients are selected according to the following optimization problem:
1089–7798/98$10.00 1998 IEEE
(6)
218
IEEE COMMUNICATIONS LETTERS, VOL. 2, NO. 8, AUGUST 1998
where is the desired user code vector. A similar optimization problem has been previously proposed in the context of array processing [5]. In order to show the ability of our approach to remove MAI, let us consider a situation in which the noise is negligible and, therefore
expression for the Hessian is
.. .
.. .
..
.
.. .
(14)
(7) represents the user where and amplitudes. Due to the constraint the receiver output can be rewritten as (8) , and where Substituting (8) into (6) and taking into account that is a full rank matrix, the optimization problem (6) reduces to
where
is the identity matrix. For the desired solution, the Hessian is (15)
which is clearly definite positive. For the remaining stationary points it is straightforward to show that the first diagonal element of the Hessian is (16)
(9) is statistically independent of the interferences and Since and (9) simplifies to verifies
is not definite positive. Therefore, and, as a consequence, is the solution to (6). It is important to note that, when the noise is negligible, this is also the solution achieved by the decorrelation receiver [6], the MMSE receiver [2] and the LCMV [3] receiver.
(10) IV. ADAPTIVE ALGORITHM is the (normalized) fourth-order moment of where Moreover, assuming the interferences are also statistically , independent among them and verify , and the above optimization problem can be further reduced to
In this section we derive an adaptive algorithm to solve the constrained optimization problem (6). This problem can be converted into an unconstrained form using the generalized sidelobe canceller (GSC) [7] decomposition (17)
(11) The minima are the points where the gradient vanishes and the Hessian matrix is positive-definite. Calculating the gradient we obtain .. .
(12)
vanishes at the point It is straightforward to show that where MAI is completely removed. Neverthealso vanishes at the additional stationary points less, (13) is the number of interferent user signals present where at the output1. The next step is to examine the positive definiteness of the Hessian matrix at the above stationary points. The general 1 This
is the quiescent vector, is the where blocking matrix whose columns span the null space of (i.e., is full rank and satisfies and is the unconstrained adaptive weight vector. The GSC decomposition always satisfies the ensures that the overall weight vector As a result, (6) is equivalent to constraint regardless of
result is true if the transmitted symbols have negative kurtosis (i.e., This is always the case in practical situations.
(18) Problem (18) can be solved using the normalized stochastic gradient algorithm (19) where and
is the output error, is chosen to eliminate the error signal at each iteration, i.e., (20)
In practical situations, is where to reduce the tipically multiplied by a positive constant algorithm misadjustment noise.
M´IGUEZ AND CASTEDO: AN APPROACH TO BLIND ADAPTIVE MULTIUSER INTERFERENCE SUPPRESSION
Fig. 2. Capture probability versus for a DS CDMA channel containing 15 interferent users, each of them 6 dB stronger than the desired one. Spreading codes are length 31 Gold sequences.
219
Let us model the estimation error as a vector of Gaussian random variables with zero mean and autocorreFig. 2 plots the capture probability of a lation matrix LCCM receiver with respect to . Each point was obtained running 10 000 simulations with the same initial conditions and counting the number of times that the receiver was captured by an interference. Note that captures were only Practical code acquisition circuits do observed for not reach such a large value of estimation error variance. The channel Gaussian noise was neglected in the analysis of Section III because otherwise it becomes rather involved. Simulations reveal that, when considering the AWGN in the channel, LCCM receivers perform the same as MMSE receivers thus exhibiting a desirable balance between noise enhancement and MAI suppression. This is in accordance with recent analytical studies of the unconstrained CM cost function where it is shown that the MSE value achieved at the CM minima is equal to the MMSE value when the SNR is high [8]. Fig. 3 plots the bit error rate (BER) for different SNR values. It is assumed that there exists a small amount of For comparison purposes, code estimation error Fig. 3 also plots the BER corresponding to a MMSE receiver with training sequences and a LCMV receiver. It is clearly seen the LCCM receiver performs practically the same as the former and considerably better than the latter. Note that a small amount of constraint mismatching severely degrades the performance of the LCMV receiver. VI. CONCLUSIONS We have presented a new blind adaptive approach to MAI cancellation for DS CDMA systems, termed LCCM, in which the CM cost function is minimized subject to a linear constraint. It is analytically demonstrated that LCCM receivers exhibit an optimal performance when the channel noise is negligible and a perfect knowledge of the desired user code is available. Computer simulations illustrate that these receivers are very robust to code estimation errors and perform the same as MMSE receivers in the presence of noise. REFERENCES
Fig. 3. BER versus SNR for a DS CDMA channel with AWGN and three interferent users, each of them 6 dB stronger than the desired one. Spreading codes are length 7 Gold sequences. MMSE and LCMV receivers are implemented with least squares adaptive algorithms. Weight vectors are obtained after 150 iterations.
V. COMPUTER SIMULATIONS Section III proves that LCCM receivers do not capture interferences when the constraint is implemented using the In practice, however, the constraint exact received code provided by a code acquisition is built with an estimate and LCCM receivers may still circuit that is not equal to be captured by an interference. Computer simulations were carried out to study this phenomenon.
[1] S. Verd´u, “Adaptive multiuser detection,” in Code Division Multiple Access Communications, S. G. Glisic and P. A. Leppannen, Eds. Amsterdam, The Netherlands: Kluwer Academic, 1995. [2] U. Madhow and M. L. Honig, “MMSE interference suppression for direct-sequence spread-spectrum CDMA,” IEEE Trans. Commun., vol. 42, pp. 3178–3188, Dec. 1994. [3] M. L. Honig, U. Madhow, and S. Verd´u, “Blind adaptive multiuser detection,” IEEE Trans. Inform. Theory, vol. 41, pp. 944–960, July 1995. [4] J. R. Treichler and M. G. Larimore, “The tone capture properties of CMA-based interference suppressors,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 946–958, Aug. 1985. [5] M. J. Rude and L. J. Griffiths, “A Linearly constrained Adaptive Algorithm for Constant Moudulus Signal Processing,” in Proc. EUSIPCO’92, Barcelona, Spain, Sept. 1992, pp. 237–240. [6] R. Lupas and S. Verd´u, “Linear multiuser detectors for synchronous code-division multiple access channels,” IEEE Trans. Inform. Theory, vol. 35, pp. 123–136, Jan. 1989. [7] L. J. Griffiths and C. W. Jim, “An alternative approach to linearly constrained adaptive beamforming,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-30, pp. 2737–2746, Nov. 1982. [8] I. Fijalkow, A. Touzni, and J. R. Treichler, “Fractionally spaced equalization using CMA: Robustness to channel noise and lack of disparity,” IEEE Trans. Signal Processing, vol. 45, pp. 55–56, Jan. 1997.
