Partial zero-forcing adaptive mmse receiver for DS ... - IEEE Xplore
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 51, NO. 5, SEPTEMBER 2002
Partial Zero-Forcing Adaptive MMSE Receiver for DS-CDMA Uplink in Multicell Environments Ju Ho Lee and Hyung-Myung Kim, Senior Member, IEEE
Abstract—Recently, adaptive multi-user detection techniques for interference suppression in direct-sequence code-division multiple-access (DS-CDMA) systems have gained much attention since they do not require any information on interfering users. In the uplink of DS-CDMA systems, however, the base station receiver typically knows the spreading waveforms of the users within its cell but does not know those of the users in other cells. In this paper, we propose a partial zero-forcing adaptive minimum mean squared error (MMSE) receiver for the DS-CDMA uplink utilizing the spreading waveforms known at the base station as well as training data. The proposed receiver first removes the intracell interference using a linear filter based on the knowledge of the spreading waveforms of the interfering users within the cell. Then the intercell interference remaining in the output of the linear filter is mitigated by adaptive MMSE detection. To speed up the convergence of the adaptive filter weights without loss of the steady-state performance, we develop a modified least mean square (LMS) algorithm based on the canonical representation of the filter weights. It is shown through analysis and simulation results that the proposed receiver improves the convergence speed and the steady-state performance. Index Terms—Adaptive minimum mean squared error (MMSE) detection, direct-sequence code-division multiple-access (DS-CDMA) uplink, intercell interference, multicell.
I. INTRODUCTION
T
O improve the capacity and performance of directsequence code-division multiple-access (DS-CDMA) systems, there has been a great deal of interest in reducing the multiple-access interference (MAI) through the use of multi-user detectors [1]. Among various multi-user detectors proposed so far, the adaptive minimum mean squared error (MMSE) receiver [2]–[4] has gained much attention since it requires no side information on interfering users other than the code timing of the user of interest and training data. The adaptive MMSE receiver suppresses the MAI by minimizing the mean squared error (MSE) adaptively. A major drawback of the adaptive MMSE receiver is the long training duration needed to get satisfactory steady-state performance. A blind adaptive multi-user detection method proposed in [5] can elim-
Manuscript received July 25, 2000; revised July 12, 2001. This work was supported by the Korea Research Foundation under Grant KRF-1998-001-E00775. J. H. Lee is with the Global Standards and Strategy Team, Telecommunication R&D Center, Samsung Electronics Co., Suwon-Si, Gyeonggi-do 442-742, Korea. H.-M. Kim is with the Communications Signal Processing Laboratory (CSPLAB), Division of Electrical Engineering, Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology (KAIST), Taejon 305-701, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TVT.2002.800633
inate the need for training and requires the receiver to know the spreading waveform and timing of the user of interest. MAI suppression is based on the constrained minimization of the mean output energy. More recently, a subspace approach to the blind adaptive multi-user detection has been proposed in [6]. Although the blind adaptive receivers do not need any training, they do not resolve the problem of slow convergence speed. The adaptive multi-user detection techniques mentioned above are very attractive for the downlink of DS-CDMA systems, where a mobile receiver does not have the information on interfering users. In the DS-CDMA uplink, however, the base station receiver typically knows the spreading waveforms of part of the users, e.g., the users within its cell, but does not know those of the users in other cells. It is natural to expect that the performance of the adaptive receivers can be improved by utilizing the knowledge of the spreading waveforms of the in-cell users. The adaptive successive interference canceler (SIC) employs adaptive MMSE detection in combination with successive interference cancellation [7]. However, its performance improvement is limited by the cumulative noise resulting from imperfect cancellation. Some blind multi-user detection methods exploiting the information on the in-cell users’ spreading waveforms were also proposed [8], [9]. The partitioned linear interference canceler (PLIC) [8] is based on the generalized sidelobe canceler [10], which is a popular technique for array signal processing algorithms. In [9], several group-blind multi-user detection algorithms were developed based on the subspace approach to the blind multi-user detection. The algorithms proposed in [8] and [9] provide better performance than the conventional blind algorithms, but the convergence speed of those algorithms may need to be improved. In this paper, we propose a partial zero-forcing adaptive MMSE receiver for the DS-CDMA uplink, which improves the steady-state performance and the convergence speed by exploiting the in-cell users’ spreading waveforms known at the base station as well as training. The intracell interference is first removed by a linear filter, constructed utilizing the knowledge of the spreading waveforms of the interfering users within the cell. The intercell interference remaining in the output of the linear filter is then suppressed by the adaptive MMSE detection. To improve the convergence speed without loss of the steady-state performance such as bit error rate (BER) and signal-to-interference-plus-noise ratio (SINR), the adaptive MMSE detection is implemented by a modified least mean square (LMS) algorithm proposed based on the canonical representation of the adaptive filter weights. The performance of the proposed receiver is evaluated by analysis and simulation.
0018-9545/02$17.00 © 2002 IEEE
LEE AND KIM: PARTIAL ZERO-FORCING ADAPTIVE MMSE RECEIVER FOR DS-CDMA UPLINK
The paper is organized as follows. In Section II, the system model is presented. In Section III, the proposed receiver is described and its performance analysis is presented. Section IV presents numerical results. Finally, we draw conclusions in Section V. II. SYSTEM MODEL Consider a synchronous DS-CDMA system with The signal received at the base station is given by
users.
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and null
.1 The projected signal does not contain the components of the subspace . spanned by can be supThe intercell interference remaining in pressed by adaptive MMSE detection. The output of the and the bit decision is made by receiver is . The fact that null implies that should also be in null . Therefore, can be expressed . Then the receiver output can be rewritten as as (6)
(1) the th bit transmitted where denotes the bit duration, the bit energy of the th user, and the by the th user, additive white Gaussian noise with power spectral density of . The spreading waveform of the th user can be expressed as (2) is the th element of the signature where the processing gain, sequence for the th user, the chip duration, and the rectangular chip waveform and the amplitude normalized so with the duration of . The th sample at the output of the that chip-matched filter is (3) . Then the vector of the where samples during the th bit interval is
received
since . The weight vector is chosen to minimize the MSE
where
(7) To reflect explicitly the reduction in degrees of freedom asso, ciated with the projection of the received signal onto null the proposed receiver is implemented in the form of the partial zero-forcing filter followed by the adaptive MMSE detection, as expressed in (6) and (7).2 The closed-form optimal solution to (7) is given by (8) and . The minimum MSE is then given by
where
(9) If there is no information on the interfering users’ spreading waveforms, the proposed receiver trivially becomes the conventional MMSE receiver. When the spreading waveforms of , all all the interfering users are known, i.e., the interfering signals are completely removed by the partial onto zero-forcing filter or, equivalently, by the projection of . And, corresponding to is, after some null simple manipulations, given by
(4) (10) where with sumed that
is the noise vector and . It is asare linearly independent.
. is the first user’s which is the matched filter for . Thus, the prospreading waveform projected onto null posed receiver becomes the decorrelator of [12] if the spreading waveforms of all the interfering users are known.
III. PARTIAL ZERO-FORCING ADAPTIVE MMSE RECEIVER In this section, we describe the partial zero-forcing adaptive MMSE receiver proposed for the DS-CDMA uplink. The first user is assumed to be the desired user. It is also assumed are the in-cell users, i.e., the that users are assumed to be spreading waveforms of users known at the base station. The intracell interference can be removed by projecting the via the projection received signal onto the subspace null matrix given by (5) is the mawhere trix whose columns form an orthonormal basis of null
A. Implementation of the Adaptive MMSE Detection To speed up the convergence speed of the adaptive filter weights without loss of steady-state performance, we now develop an implementation technique for adaptive MMSE detection based on the canonical representation of the adaptive filter weights. Unlike the (training based) conventional adaptive MMSE detection of [2]–[4], the proposed adaptive implementation employs the information on desired users’ spreading waveform as well as a training sequence. 1V can be obtained through the singular value decomposition (SVD) of C. The number of computations needed for the SVD of C is approximately
N
4
(K
0
0
1) + 22(K 1) if the R-SVD algorithm is used [11]. that V s = 0 for the interfering users k = 2; . . . ; K within the cell. Hence, V is called the partial zero-forcing filter. 2Note
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 51, NO. 5, SEPTEMBER 2002
B. Performance Analysis Let the weight-error vector in the LMS adaptation of be defined as (17) from Then, by subtracting the optimum weight vector both sides of (16), the LMS adaptation in (16) can be rewritten as Fig. 1.
Receiver structure.
The weight vector as follows [5]:
(18)
may be represented in canonical form (11)
and is orthogonal where , i.e., . Let be the to matrix whose columns form an orthonormal basis of the .3 Since null can be rewritten subspace null as (12) consists The canonical representation in (12) reveals that , which will be called the anchor of the fixed component , which is orthogonal to the and the adjustable component anchor. Then the optimization in (7) becomes the minimization of the MSE with respect to
. Comwhere paring (18) with the standard LMS update described in [10], we observe that the convergence speed and the steady-state performance of the proposed receiver can be characterized in terms of the eigenvalues of the correlation matrix of the input signal for the . adjustable weight vector The convergence speed of the LMS algorithm is governed by the eigenvalue spread of the correlation matrix of tap-input signal [10]: large eigenvalue spread slows down the converof an Hergence speed. The eigenvalue spread is defined as .4 mitian matrix The correlation matrix of the tap-input signal for the conven. Note that tional adaptive MMSE reciever is , where . Since the columns of are , it is easy to show orthonormal, i.e.,
(13) (19) The closed-form optimal solution to (13) is given by (14) The corresponding minimum MSE is given by
(15) in (15) may be different from in (9) because of in (14) results in the same BER the anchor. However, in (8) (see the discussions on the canonical and SINR as adaptively, the wellrepresentation given in [5]). To find known LMS algorithm is used (16) and . The structure of the receiver described so far is shown in Fig. 1. Due to the elimination of the intracell interference prior to the adaptive MMSE detection and due to the canonical representation-based LMS algorithm described above, the proposed receiver achieves superior transient and steady-state performance compared to the conventional adaptive MMSE receiver, as shown in the next section.
where the step size
X
s
3 can be obtained through the SVD of . The number of computations needed for the SVD of is approximately 4( K + 1) + 22 if the R-SVD algorithm is used [11].
s
N0
which indicates that the proposed receiver converges more rapidly than the conventional adaptive MMSE receiver by invoking the Poincaré separation theorem. HerPoincaré Separation Theorem [13]: Let be an matrix whose columns mitian matrix, be an . If the eigenvalues of and are orthonormal, and are arranged in increasing order, then we have (20) By using the Poincaré separation theorem, it is also easy to show (21) which implies that the proposed adaptive implementation with the anchor [from (13)] provides faster convergence speed than the direct adaptive implementation based on (7). It is well known that the LMS algorithm produces a steadythat is in excess of the minimum MSE . state MSE The relative excess MSE at the steady state is defined as (22) and provides a measure of how close the steady-state performance is to the optimal performance [10].
A
A
2
A A
4The eigenvalues ( ); . . . ; ( ) of an L L Hermitian matrix are assumed to be in increasing order ( ( ) = ( ) ( )= ( )).
A
A
A 111
LEE AND KIM: PARTIAL ZERO-FORCING ADAPTIVE MMSE RECEIVER FOR DS-CDMA UPLINK
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The relative excess MSE at the steady state for the conventional adaptive MMSE receiver is given by [10] (23) provided that the step size satisfies the convergence condition . In the same way, if that , then the proposed receiver has (24) Each term in the summation of (23) [or (24)] is positive and (or ) provided that the converincreasing with gence condition is satisfied. Thus, the Poincaré separation theorem yields (25) implying that the proposed receiver provides the steady-state performance closer to its own optimal performance than the conventional adaptive MMSE receiver does. To complete the evaluation of the steady-state performance of the proposed receiver, the optimal performance of the two receivers will be compared. Since the anchor is used in the adaptive implementation of the proposed receiver, it is not suitable to compare the minimum MSEs. Thus, the output SINRs corresponding to the optimal solutions of the two receivers are considered. With the filter weight applied to the received signal , the output SINR is given by SINR
(26)
The SINR of the conventional adaptive MMSE receiver corresponding to the Wiener solution given in [3] is SINR
(27)
In the case of the proposed receiver, the SINR corresponding to in (8) and is the optimal solution can be calculated using given by SINR
(28)
may be smaller than SINR since the reSINR ceived signal is filtered by the partial zero-forcing filter before the MMSE detection. However, it has been observed that the decrease in the optimal SINR is negligible or insignificant comprovided that the well-designed signapared with SINR ture sequences such as m- and Gold sequences are used. Combining this observation with (25) implies that the steady-state performance of the proposed receiver is better than that of the conventional adaptive MMSE receiver for the same step size. It can be also confirmed from the following simulation results.
K=
Fig. 2. Comparison of the relative excess MSE. 10; K = 5; E =N = 10 dB, E =E = 10 dB for k = 2; . . . ; K . The step size used in the LMS adaptation is = 1=E fkr(i)k g for each scheme.
IV. NUMERICAL RESULTS We consider a synchronous system with users and . Gold sequences of period are the processing gain used as the signature sequences. All of the results in this section are for the first user. Fig. 2 plots the relative excess MSEs of the conventional adaptive MMSE and the proposed receivers versus the number in-cell users. In case of the proof training bits with posed receiver, the relative excess MSE of the direct adaptive implementation without anchor, which is based on (7), is also plotted to assess the benefit of the anchor in the adaptive implementation [see (12) and (13)]. Each curve corresponds to the ensemble average of 500 independent experiments. The bit energy of each user is chosen as follows: dB and dB for . The step size used in the LMS adaptation for each receiver was set to (power of the received signal), i.e., to . The proposed receiver converges more rapidly than the conventional adaptive MMSE receiver. The relative excess MSE at the steady state for the proposed receiver is significantly smaller than that for the conventional adaptive MMSE receiver. Comparing the proposed receiver with the direct adaptive implementation from (7), it is seen that the use of the anchor improves the convergence speed significantly with a similar steady-state performance. In Fig. 3, the output SINRs corresponding to the optimal solutions for the conventional adaptive MMSE and the proposed dB and . The receivers are compared for difference in the ideal SINRs of the two receivers is negligible or insignificant compared with the ideal SINR of the conventional adaptive MMSE receiver. The results in Figs. 2 and 3 imply that the proposed receiver achieves faster convergence speed and better steady-state performance than the conventional adaptive MMSE receiver as illustrated in Fig. 4, which represents the output SINR versus the number of iterations. For comparison, Fig. 4 also
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 51, NO. 5, SEPTEMBER 2002
Fig. 3. SINRs corresponding to the optimal solutions for the conventional 10; K = 5; E =N = adaptive MMSE and the proposed receivers. 10 dB.
K=
Fig. 4. SINR comparison. The hybrid group-blind detector and the PLIC are blind algorithms. The others are training-based algorithms. K = 10; K = 5; E =N = 10 dB. E =E = 2(k 1) dB for k = 2; . . . ; K and E =E = 8 dB for k = K + 1; . . . ; K . The step size used in the LMS adaptation employed for all of the detectors except the hybrid group-blind detector is = . The forgetting factor used in the PASTd algorithm employed 1=E r(i) for the hybrid group-blind detector is = 0:985.
0
fk
kg
shows the results for the adaptive SIC [7], the PLIC [8], and the hybrid group-blind detector [9]. The number of in-cell . The SINR curves were obtained users is chosen as using 500 independent experiments with the bit energy chosen dB; dB for as follows: ; and dB for . used in the LMS adaptation employed for The step size all of the detectors except the hybrid group-blind detector was set to (power of the received signal). The adaptive SIC regenerates the signals transmitted by the in-cell interfering with stronger power and successively cancels users them from the received signal. In the case of the hybrid
Fig. 5. Steady-state BER of the proposed receiver versus the number of in-cell ; E =N = 10 dB, E =E = 10 dB users K . K = 10; = 1=E r(i) for k = 2; . . . ; K .
fk
kg
group-blind detector, the initial estimate of the signal subspace is obtained by applying eigenvalue decomposition to the sample autocorrelation matrix computed using the first 50 received signal vectors. The projection approximation subspace tracking (PASTd) algorithm [6] is then employed for tracking the signal . It is seen from subspace with forgetting factor Fig. 4 that the proposed receiver well outperforms the others. The performance improvement of the adaptive SIC is limited by the cumulative noise resulting from imperfect cancellation of interference although it outperforms the conventional adaptive MMSE receiver. The proposed receiver, the PLIC, and the hybrid group-blind detector utilize information on the in-cell users’ spreading waveforms to decorrelate signals of the in-cell users. However, performance of the proposed receiver is superior to that of the two blind algorithms since the proposed receiver employs training as well as the information on the in-cell users’ spreading waveforms. Finally, Fig. 5 shows the steady-state BER of the prowith posed receiver versus the number of in-cell users (power of the received signal), dB, and dB for . The steady-state BER was calculated by the Gaussian approximation [14] of the MAI-plus-noise after the convergence of the filter weights and was averaged over 500 independent experiments. The BER of the decorrelator was calculated assuming that the spreading waveforms of all the users are known. For the comparison, the steady-state BER of the conventional adaptive MMSE receiver is also shown. The steady-state BER of the proposed receiver increases. When approaches the BER of the decorrelator as , the ideal BERs of the conventional adaptive MMSE and the proposed receivers are the same. However, it appears in Fig. 5 that the steady-state BER of the proposed receiver with is slightly better than that of the conventional adaptive MMSE receiver. This difference in the steady-state BER is due to the canonical representation-based LMS algorithm used for the adaptive implementation of the proposed receiver.
LEE AND KIM: PARTIAL ZERO-FORCING ADAPTIVE MMSE RECEIVER FOR DS-CDMA UPLINK
V. CONCLUSION In this paper, a partial zero-forcing adaptive MMSE receiver has been proposed for the DS-CDMA uplink. The proposed receiver utilizes the in-cell users’ spreading waveforms known at the base station as well as training. The intracell interference is first removed by the partial zero-forcing filter formed by exploiting the knowledge of the spreading waveforms of the in-cell interfering users. The adaptive MMSE detection is employed for suppressing the intercell interference remaining in the output of the partial zero-forcing filter. To speed up the convergence of the adaptive filter weights without degrading the steady-state performance, the canonical representation-based LMS algorithm has also been developed. Due to the elimination of the intracell interference prior to the adaptive MMSE detection and due to the canonical representation-based LMS algorithm, the proposed receiver achieves gains both in convergence speed and in steady-state performance. This was also confirmed through the analysis and the simulation results. The performance gains are obtained at the cost of additional computations to find the partial zero-forcing filter and to form the canonical representation-based LMS algorithm. The proposed receiver assumes perfect knowledge of the spreading waveforms of the in-cell users. However, the performance of the proposed scheme may be susceptible to error in that knowledge, which could be caused by residual error in estimating timing and channel parameters. The remedy for this problem is an important issue for future investigation. A base station will not only know all spreading waveforms of the in-cell users, but also need to decode all the in-cell users. To decode all the in-cell users, the base station will need to implement the proposed receiver for each user since the proposed receiver treats all users separately. This results in a considerable , which can be comparable to the comoverhead of plexity of the adaptive filtering itself. Devising joint detection algorithms is a topic for further research. In realistic situations, the signals transmitted by the users undergo dispersive channels due to multipath propagation and arrive at the base station asynchronously. Although a simple synchronous nondispersive channel is assumed in this paper, the proposed scheme can be extended to asynchronous dispersive channels as follows: for each user, the sum of the spreading waveforms received from multiple paths is regarded as the effective spreading waveform received at the base station, as in [9]. Then, based on the knowledge of the effective spreading waveforms of in-cell users, the proposed algorithm can be modified to process the received signal in windows of finite length, as in [9], [15], and [16]. REFERENCES [1] S. Moshavi, “Multi-user detection for DS-CDMA communications,” IEEE Commun. Mag., vol. 34, pp. 124–136, Oct. 1996. [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.
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[3] S. L. Miller, “An adaptive direct-sequence code-division multiple-access receiver for multiuser interference rejection,” IEEE Trans. Commun., vol. 43, pp. 1746–1755, Feb./Mar./Apr. 1995. , “Training analysis of adaptive interference suppression for [4] direct-sequence code-division multiple-access systems,” IEEE Trans. Commun., vol. 44, pp. 488–495, Apr. 1996. [5] M. Honig, U. Madhow, and S. Verdú, “Blind adaptive multiuser detection,” IEEE Trans. Inform. Theory, vol. 41, pp. 944–960, July 1995. [6] X. Wang and H. V. Poor, “Blind multiuser detection: A subspace approach,” IEEE Trans. Inform. Theory, vol. 44, pp. 677–690, Mar. 1998. [7] Y. Cho and J. H. Lee, “Analysis of an adaptive SIC for near-far resistant DS-CDMA,” IEEE Trans. Commun., vol. 46, pp. 1429–1432, Nov. 1998. [8] J. B. Schodorf and D. B. Williams, “A constrained optimization approach to multiuser detection,” IEEE Trans. Signal Processing, vol. 45, pp. 258–262, Jan. 1997. [9] X. Wang and A. Høst-Madsen, “Group-blind multiuser detection for uplink CDMA,” IEEE J. Select. Areas Commun., vol. 17, pp. 1971–1984, Nov. 1999. [10] S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: PrenticeHall, 1996. [11] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore, MD: Johns Hopkins Univ. Press, 1991, ch. 5. [12] R. Lupas and S. Verdú, “Linear multiuser detectors for synchronous code-division multiple-access channels,” IEEE Trans. Inform. Theory, vol. 35, pp. 123–136, Jan. 1989. [13] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1993, ch. 4. [14] H. V. Poor and S. Verdú, “Probability of error in MMSE multiuser detection,” IEEE Trans. Inform. Theory, vol. 43, pp. 858–871, May 1997. [15] S. J. Baines, A. G. Burr, and T. C. Tozer, “Double window multi-user detection for asynchronous DS-CDMA,” Electron. Lett., vol. 32, no. 24, pp. 2199–2201, Nov. 1996. [16] M. J. Juntti and B. Aazhang, “Finite memory-length linear multiuser detection for asynchronous CDMA communications,” IEEE Trans. Commun., vol. 45, pp. 611–622, May 1997.
Ju Ho Lee received the B.S., M.S., and Ph.D. degrees from Korea Advanced Institute of Science and Technology (KAIST), Taejon, Korea, in 1993, 1995, and 2000, respectively, all in electrical engineering. Currently, he is a Senior Engineer of Samsung Electronics Co., Suwon-Si, Gyeonggi-do, Korea, working on standardization of mobile communications. His main interests include wireless communications and signal processing for wireless communications including multi-user detection and adaptive antenna array.
Hyung-Myung Kim (S’86–M’86–SM’99) received the B.S. degree in electronics engineering from Seoul National University, Seoul, Korea, in 1974 and the M.S. and Ph.D. degrees in electrical engineering from the University of Pittsburgh, Pittsburgh, PA, in 1982 and 1985, respectively. He is now a Professor at the Department of Electrical Engineering and Computer Science, The Korea Advanced Institute of Science and Technology (KAIST), Taejon, Korea. His research interests include digital signal/image processing, digital transmission of voice, data, and image, and multidimensional system theory. Dr. Kim was the Treasurer of the IEEE Taejon Section in 1992. He has been an editorial board member of Multidimensional Systems and Signal Processing since 1990.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 51, NO. 5, SEPTEMBER 2002
Partial Zero-Forcing Adaptive MMSE Receiver for DS-CDMA Uplink in Multicell Environments Ju Ho Lee and Hyung-Myung Kim, Senior Member, IEEE
Abstract—Recently, adaptive multi-user detection techniques for interference suppression in direct-sequence code-division multiple-access (DS-CDMA) systems have gained much attention since they do not require any information on interfering users. In the uplink of DS-CDMA systems, however, the base station receiver typically knows the spreading waveforms of the users within its cell but does not know those of the users in other cells. In this paper, we propose a partial zero-forcing adaptive minimum mean squared error (MMSE) receiver for the DS-CDMA uplink utilizing the spreading waveforms known at the base station as well as training data. The proposed receiver first removes the intracell interference using a linear filter based on the knowledge of the spreading waveforms of the interfering users within the cell. Then the intercell interference remaining in the output of the linear filter is mitigated by adaptive MMSE detection. To speed up the convergence of the adaptive filter weights without loss of the steady-state performance, we develop a modified least mean square (LMS) algorithm based on the canonical representation of the filter weights. It is shown through analysis and simulation results that the proposed receiver improves the convergence speed and the steady-state performance. Index Terms—Adaptive minimum mean squared error (MMSE) detection, direct-sequence code-division multiple-access (DS-CDMA) uplink, intercell interference, multicell.
I. INTRODUCTION
T
O improve the capacity and performance of directsequence code-division multiple-access (DS-CDMA) systems, there has been a great deal of interest in reducing the multiple-access interference (MAI) through the use of multi-user detectors [1]. Among various multi-user detectors proposed so far, the adaptive minimum mean squared error (MMSE) receiver [2]–[4] has gained much attention since it requires no side information on interfering users other than the code timing of the user of interest and training data. The adaptive MMSE receiver suppresses the MAI by minimizing the mean squared error (MSE) adaptively. A major drawback of the adaptive MMSE receiver is the long training duration needed to get satisfactory steady-state performance. A blind adaptive multi-user detection method proposed in [5] can elim-
Manuscript received July 25, 2000; revised July 12, 2001. This work was supported by the Korea Research Foundation under Grant KRF-1998-001-E00775. J. H. Lee is with the Global Standards and Strategy Team, Telecommunication R&D Center, Samsung Electronics Co., Suwon-Si, Gyeonggi-do 442-742, Korea. H.-M. Kim is with the Communications Signal Processing Laboratory (CSPLAB), Division of Electrical Engineering, Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology (KAIST), Taejon 305-701, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TVT.2002.800633
inate the need for training and requires the receiver to know the spreading waveform and timing of the user of interest. MAI suppression is based on the constrained minimization of the mean output energy. More recently, a subspace approach to the blind adaptive multi-user detection has been proposed in [6]. Although the blind adaptive receivers do not need any training, they do not resolve the problem of slow convergence speed. The adaptive multi-user detection techniques mentioned above are very attractive for the downlink of DS-CDMA systems, where a mobile receiver does not have the information on interfering users. In the DS-CDMA uplink, however, the base station receiver typically knows the spreading waveforms of part of the users, e.g., the users within its cell, but does not know those of the users in other cells. It is natural to expect that the performance of the adaptive receivers can be improved by utilizing the knowledge of the spreading waveforms of the in-cell users. The adaptive successive interference canceler (SIC) employs adaptive MMSE detection in combination with successive interference cancellation [7]. However, its performance improvement is limited by the cumulative noise resulting from imperfect cancellation. Some blind multi-user detection methods exploiting the information on the in-cell users’ spreading waveforms were also proposed [8], [9]. The partitioned linear interference canceler (PLIC) [8] is based on the generalized sidelobe canceler [10], which is a popular technique for array signal processing algorithms. In [9], several group-blind multi-user detection algorithms were developed based on the subspace approach to the blind multi-user detection. The algorithms proposed in [8] and [9] provide better performance than the conventional blind algorithms, but the convergence speed of those algorithms may need to be improved. In this paper, we propose a partial zero-forcing adaptive MMSE receiver for the DS-CDMA uplink, which improves the steady-state performance and the convergence speed by exploiting the in-cell users’ spreading waveforms known at the base station as well as training. The intracell interference is first removed by a linear filter, constructed utilizing the knowledge of the spreading waveforms of the interfering users within the cell. The intercell interference remaining in the output of the linear filter is then suppressed by the adaptive MMSE detection. To improve the convergence speed without loss of the steady-state performance such as bit error rate (BER) and signal-to-interference-plus-noise ratio (SINR), the adaptive MMSE detection is implemented by a modified least mean square (LMS) algorithm proposed based on the canonical representation of the adaptive filter weights. The performance of the proposed receiver is evaluated by analysis and simulation.
0018-9545/02$17.00 © 2002 IEEE
LEE AND KIM: PARTIAL ZERO-FORCING ADAPTIVE MMSE RECEIVER FOR DS-CDMA UPLINK
The paper is organized as follows. In Section II, the system model is presented. In Section III, the proposed receiver is described and its performance analysis is presented. Section IV presents numerical results. Finally, we draw conclusions in Section V. II. SYSTEM MODEL Consider a synchronous DS-CDMA system with The signal received at the base station is given by
users.
1067
and null
.1 The projected signal does not contain the components of the subspace . spanned by can be supThe intercell interference remaining in pressed by adaptive MMSE detection. The output of the and the bit decision is made by receiver is . The fact that null implies that should also be in null . Therefore, can be expressed . Then the receiver output can be rewritten as as (6)
(1) the th bit transmitted where denotes the bit duration, the bit energy of the th user, and the by the th user, additive white Gaussian noise with power spectral density of . The spreading waveform of the th user can be expressed as (2) is the th element of the signature where the processing gain, sequence for the th user, the chip duration, and the rectangular chip waveform and the amplitude normalized so with the duration of . The th sample at the output of the that chip-matched filter is (3) . Then the vector of the where samples during the th bit interval is
received
since . The weight vector is chosen to minimize the MSE
where
(7) To reflect explicitly the reduction in degrees of freedom asso, ciated with the projection of the received signal onto null the proposed receiver is implemented in the form of the partial zero-forcing filter followed by the adaptive MMSE detection, as expressed in (6) and (7).2 The closed-form optimal solution to (7) is given by (8) and . The minimum MSE is then given by
where
(9) If there is no information on the interfering users’ spreading waveforms, the proposed receiver trivially becomes the conventional MMSE receiver. When the spreading waveforms of , all all the interfering users are known, i.e., the interfering signals are completely removed by the partial onto zero-forcing filter or, equivalently, by the projection of . And, corresponding to is, after some null simple manipulations, given by
(4) (10) where with sumed that
is the noise vector and . It is asare linearly independent.
. is the first user’s which is the matched filter for . Thus, the prospreading waveform projected onto null posed receiver becomes the decorrelator of [12] if the spreading waveforms of all the interfering users are known.
III. PARTIAL ZERO-FORCING ADAPTIVE MMSE RECEIVER In this section, we describe the partial zero-forcing adaptive MMSE receiver proposed for the DS-CDMA uplink. The first user is assumed to be the desired user. It is also assumed are the in-cell users, i.e., the that users are assumed to be spreading waveforms of users known at the base station. The intracell interference can be removed by projecting the via the projection received signal onto the subspace null matrix given by (5) is the mawhere trix whose columns form an orthonormal basis of null
A. Implementation of the Adaptive MMSE Detection To speed up the convergence speed of the adaptive filter weights without loss of steady-state performance, we now develop an implementation technique for adaptive MMSE detection based on the canonical representation of the adaptive filter weights. Unlike the (training based) conventional adaptive MMSE detection of [2]–[4], the proposed adaptive implementation employs the information on desired users’ spreading waveform as well as a training sequence. 1V can be obtained through the singular value decomposition (SVD) of C. The number of computations needed for the SVD of C is approximately
N
4
(K
0
0
1) + 22(K 1) if the R-SVD algorithm is used [11]. that V s = 0 for the interfering users k = 2; . . . ; K within the cell. Hence, V is called the partial zero-forcing filter. 2Note
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B. Performance Analysis Let the weight-error vector in the LMS adaptation of be defined as (17) from Then, by subtracting the optimum weight vector both sides of (16), the LMS adaptation in (16) can be rewritten as Fig. 1.
Receiver structure.
The weight vector as follows [5]:
(18)
may be represented in canonical form (11)
and is orthogonal where , i.e., . Let be the to matrix whose columns form an orthonormal basis of the .3 Since null can be rewritten subspace null as (12) consists The canonical representation in (12) reveals that , which will be called the anchor of the fixed component , which is orthogonal to the and the adjustable component anchor. Then the optimization in (7) becomes the minimization of the MSE with respect to
. Comwhere paring (18) with the standard LMS update described in [10], we observe that the convergence speed and the steady-state performance of the proposed receiver can be characterized in terms of the eigenvalues of the correlation matrix of the input signal for the . adjustable weight vector The convergence speed of the LMS algorithm is governed by the eigenvalue spread of the correlation matrix of tap-input signal [10]: large eigenvalue spread slows down the converof an Hergence speed. The eigenvalue spread is defined as .4 mitian matrix The correlation matrix of the tap-input signal for the conven. Note that tional adaptive MMSE reciever is , where . Since the columns of are , it is easy to show orthonormal, i.e.,
(13) (19) The closed-form optimal solution to (13) is given by (14) The corresponding minimum MSE is given by
(15) in (15) may be different from in (9) because of in (14) results in the same BER the anchor. However, in (8) (see the discussions on the canonical and SINR as adaptively, the wellrepresentation given in [5]). To find known LMS algorithm is used (16) and . The structure of the receiver described so far is shown in Fig. 1. Due to the elimination of the intracell interference prior to the adaptive MMSE detection and due to the canonical representation-based LMS algorithm described above, the proposed receiver achieves superior transient and steady-state performance compared to the conventional adaptive MMSE receiver, as shown in the next section.
where the step size
X
s
3 can be obtained through the SVD of . The number of computations needed for the SVD of is approximately 4( K + 1) + 22 if the R-SVD algorithm is used [11].
s
N0
which indicates that the proposed receiver converges more rapidly than the conventional adaptive MMSE receiver by invoking the Poincaré separation theorem. HerPoincaré Separation Theorem [13]: Let be an matrix whose columns mitian matrix, be an . If the eigenvalues of and are orthonormal, and are arranged in increasing order, then we have (20) By using the Poincaré separation theorem, it is also easy to show (21) which implies that the proposed adaptive implementation with the anchor [from (13)] provides faster convergence speed than the direct adaptive implementation based on (7). It is well known that the LMS algorithm produces a steadythat is in excess of the minimum MSE . state MSE The relative excess MSE at the steady state is defined as (22) and provides a measure of how close the steady-state performance is to the optimal performance [10].
A
A
2
A A
4The eigenvalues ( ); . . . ; ( ) of an L L Hermitian matrix are assumed to be in increasing order ( ( ) = ( ) ( )= ( )).
A
A
A 111
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The relative excess MSE at the steady state for the conventional adaptive MMSE receiver is given by [10] (23) provided that the step size satisfies the convergence condition . In the same way, if that , then the proposed receiver has (24) Each term in the summation of (23) [or (24)] is positive and (or ) provided that the converincreasing with gence condition is satisfied. Thus, the Poincaré separation theorem yields (25) implying that the proposed receiver provides the steady-state performance closer to its own optimal performance than the conventional adaptive MMSE receiver does. To complete the evaluation of the steady-state performance of the proposed receiver, the optimal performance of the two receivers will be compared. Since the anchor is used in the adaptive implementation of the proposed receiver, it is not suitable to compare the minimum MSEs. Thus, the output SINRs corresponding to the optimal solutions of the two receivers are considered. With the filter weight applied to the received signal , the output SINR is given by SINR
(26)
The SINR of the conventional adaptive MMSE receiver corresponding to the Wiener solution given in [3] is SINR
(27)
In the case of the proposed receiver, the SINR corresponding to in (8) and is the optimal solution can be calculated using given by SINR
(28)
may be smaller than SINR since the reSINR ceived signal is filtered by the partial zero-forcing filter before the MMSE detection. However, it has been observed that the decrease in the optimal SINR is negligible or insignificant comprovided that the well-designed signapared with SINR ture sequences such as m- and Gold sequences are used. Combining this observation with (25) implies that the steady-state performance of the proposed receiver is better than that of the conventional adaptive MMSE receiver for the same step size. It can be also confirmed from the following simulation results.
K=
Fig. 2. Comparison of the relative excess MSE. 10; K = 5; E =N = 10 dB, E =E = 10 dB for k = 2; . . . ; K . The step size used in the LMS adaptation is = 1=E fkr(i)k g for each scheme.
IV. NUMERICAL RESULTS We consider a synchronous system with users and . Gold sequences of period are the processing gain used as the signature sequences. All of the results in this section are for the first user. Fig. 2 plots the relative excess MSEs of the conventional adaptive MMSE and the proposed receivers versus the number in-cell users. In case of the proof training bits with posed receiver, the relative excess MSE of the direct adaptive implementation without anchor, which is based on (7), is also plotted to assess the benefit of the anchor in the adaptive implementation [see (12) and (13)]. Each curve corresponds to the ensemble average of 500 independent experiments. The bit energy of each user is chosen as follows: dB and dB for . The step size used in the LMS adaptation for each receiver was set to (power of the received signal), i.e., to . The proposed receiver converges more rapidly than the conventional adaptive MMSE receiver. The relative excess MSE at the steady state for the proposed receiver is significantly smaller than that for the conventional adaptive MMSE receiver. Comparing the proposed receiver with the direct adaptive implementation from (7), it is seen that the use of the anchor improves the convergence speed significantly with a similar steady-state performance. In Fig. 3, the output SINRs corresponding to the optimal solutions for the conventional adaptive MMSE and the proposed dB and . The receivers are compared for difference in the ideal SINRs of the two receivers is negligible or insignificant compared with the ideal SINR of the conventional adaptive MMSE receiver. The results in Figs. 2 and 3 imply that the proposed receiver achieves faster convergence speed and better steady-state performance than the conventional adaptive MMSE receiver as illustrated in Fig. 4, which represents the output SINR versus the number of iterations. For comparison, Fig. 4 also
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Fig. 3. SINRs corresponding to the optimal solutions for the conventional 10; K = 5; E =N = adaptive MMSE and the proposed receivers. 10 dB.
K=
Fig. 4. SINR comparison. The hybrid group-blind detector and the PLIC are blind algorithms. The others are training-based algorithms. K = 10; K = 5; E =N = 10 dB. E =E = 2(k 1) dB for k = 2; . . . ; K and E =E = 8 dB for k = K + 1; . . . ; K . The step size used in the LMS adaptation employed for all of the detectors except the hybrid group-blind detector is = . The forgetting factor used in the PASTd algorithm employed 1=E r(i) for the hybrid group-blind detector is = 0:985.
0
fk
kg
shows the results for the adaptive SIC [7], the PLIC [8], and the hybrid group-blind detector [9]. The number of in-cell . The SINR curves were obtained users is chosen as using 500 independent experiments with the bit energy chosen dB; dB for as follows: ; and dB for . used in the LMS adaptation employed for The step size all of the detectors except the hybrid group-blind detector was set to (power of the received signal). The adaptive SIC regenerates the signals transmitted by the in-cell interfering with stronger power and successively cancels users them from the received signal. In the case of the hybrid
Fig. 5. Steady-state BER of the proposed receiver versus the number of in-cell ; E =N = 10 dB, E =E = 10 dB users K . K = 10; = 1=E r(i) for k = 2; . . . ; K .
fk
kg
group-blind detector, the initial estimate of the signal subspace is obtained by applying eigenvalue decomposition to the sample autocorrelation matrix computed using the first 50 received signal vectors. The projection approximation subspace tracking (PASTd) algorithm [6] is then employed for tracking the signal . It is seen from subspace with forgetting factor Fig. 4 that the proposed receiver well outperforms the others. The performance improvement of the adaptive SIC is limited by the cumulative noise resulting from imperfect cancellation of interference although it outperforms the conventional adaptive MMSE receiver. The proposed receiver, the PLIC, and the hybrid group-blind detector utilize information on the in-cell users’ spreading waveforms to decorrelate signals of the in-cell users. However, performance of the proposed receiver is superior to that of the two blind algorithms since the proposed receiver employs training as well as the information on the in-cell users’ spreading waveforms. Finally, Fig. 5 shows the steady-state BER of the prowith posed receiver versus the number of in-cell users (power of the received signal), dB, and dB for . The steady-state BER was calculated by the Gaussian approximation [14] of the MAI-plus-noise after the convergence of the filter weights and was averaged over 500 independent experiments. The BER of the decorrelator was calculated assuming that the spreading waveforms of all the users are known. For the comparison, the steady-state BER of the conventional adaptive MMSE receiver is also shown. The steady-state BER of the proposed receiver increases. When approaches the BER of the decorrelator as , the ideal BERs of the conventional adaptive MMSE and the proposed receivers are the same. However, it appears in Fig. 5 that the steady-state BER of the proposed receiver with is slightly better than that of the conventional adaptive MMSE receiver. This difference in the steady-state BER is due to the canonical representation-based LMS algorithm used for the adaptive implementation of the proposed receiver.
LEE AND KIM: PARTIAL ZERO-FORCING ADAPTIVE MMSE RECEIVER FOR DS-CDMA UPLINK
V. CONCLUSION In this paper, a partial zero-forcing adaptive MMSE receiver has been proposed for the DS-CDMA uplink. The proposed receiver utilizes the in-cell users’ spreading waveforms known at the base station as well as training. The intracell interference is first removed by the partial zero-forcing filter formed by exploiting the knowledge of the spreading waveforms of the in-cell interfering users. The adaptive MMSE detection is employed for suppressing the intercell interference remaining in the output of the partial zero-forcing filter. To speed up the convergence of the adaptive filter weights without degrading the steady-state performance, the canonical representation-based LMS algorithm has also been developed. Due to the elimination of the intracell interference prior to the adaptive MMSE detection and due to the canonical representation-based LMS algorithm, the proposed receiver achieves gains both in convergence speed and in steady-state performance. This was also confirmed through the analysis and the simulation results. The performance gains are obtained at the cost of additional computations to find the partial zero-forcing filter and to form the canonical representation-based LMS algorithm. The proposed receiver assumes perfect knowledge of the spreading waveforms of the in-cell users. However, the performance of the proposed scheme may be susceptible to error in that knowledge, which could be caused by residual error in estimating timing and channel parameters. The remedy for this problem is an important issue for future investigation. A base station will not only know all spreading waveforms of the in-cell users, but also need to decode all the in-cell users. To decode all the in-cell users, the base station will need to implement the proposed receiver for each user since the proposed receiver treats all users separately. This results in a considerable , which can be comparable to the comoverhead of plexity of the adaptive filtering itself. Devising joint detection algorithms is a topic for further research. In realistic situations, the signals transmitted by the users undergo dispersive channels due to multipath propagation and arrive at the base station asynchronously. Although a simple synchronous nondispersive channel is assumed in this paper, the proposed scheme can be extended to asynchronous dispersive channels as follows: for each user, the sum of the spreading waveforms received from multiple paths is regarded as the effective spreading waveform received at the base station, as in [9]. Then, based on the knowledge of the effective spreading waveforms of in-cell users, the proposed algorithm can be modified to process the received signal in windows of finite length, as in [9], [15], and [16]. REFERENCES [1] S. Moshavi, “Multi-user detection for DS-CDMA communications,” IEEE Commun. Mag., vol. 34, pp. 124–136, Oct. 1996. [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.
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[3] S. L. Miller, “An adaptive direct-sequence code-division multiple-access receiver for multiuser interference rejection,” IEEE Trans. Commun., vol. 43, pp. 1746–1755, Feb./Mar./Apr. 1995. , “Training analysis of adaptive interference suppression for [4] direct-sequence code-division multiple-access systems,” IEEE Trans. Commun., vol. 44, pp. 488–495, Apr. 1996. [5] M. Honig, U. Madhow, and S. Verdú, “Blind adaptive multiuser detection,” IEEE Trans. Inform. Theory, vol. 41, pp. 944–960, July 1995. [6] X. Wang and H. V. Poor, “Blind multiuser detection: A subspace approach,” IEEE Trans. Inform. Theory, vol. 44, pp. 677–690, Mar. 1998. [7] Y. Cho and J. H. Lee, “Analysis of an adaptive SIC for near-far resistant DS-CDMA,” IEEE Trans. Commun., vol. 46, pp. 1429–1432, Nov. 1998. [8] J. B. Schodorf and D. B. Williams, “A constrained optimization approach to multiuser detection,” IEEE Trans. Signal Processing, vol. 45, pp. 258–262, Jan. 1997. [9] X. Wang and A. Høst-Madsen, “Group-blind multiuser detection for uplink CDMA,” IEEE J. Select. Areas Commun., vol. 17, pp. 1971–1984, Nov. 1999. [10] S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: PrenticeHall, 1996. [11] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore, MD: Johns Hopkins Univ. Press, 1991, ch. 5. [12] R. Lupas and S. Verdú, “Linear multiuser detectors for synchronous code-division multiple-access channels,” IEEE Trans. Inform. Theory, vol. 35, pp. 123–136, Jan. 1989. [13] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1993, ch. 4. [14] H. V. Poor and S. Verdú, “Probability of error in MMSE multiuser detection,” IEEE Trans. Inform. Theory, vol. 43, pp. 858–871, May 1997. [15] S. J. Baines, A. G. Burr, and T. C. Tozer, “Double window multi-user detection for asynchronous DS-CDMA,” Electron. Lett., vol. 32, no. 24, pp. 2199–2201, Nov. 1996. [16] M. J. Juntti and B. Aazhang, “Finite memory-length linear multiuser detection for asynchronous CDMA communications,” IEEE Trans. Commun., vol. 45, pp. 611–622, May 1997.
Ju Ho Lee received the B.S., M.S., and Ph.D. degrees from Korea Advanced Institute of Science and Technology (KAIST), Taejon, Korea, in 1993, 1995, and 2000, respectively, all in electrical engineering. Currently, he is a Senior Engineer of Samsung Electronics Co., Suwon-Si, Gyeonggi-do, Korea, working on standardization of mobile communications. His main interests include wireless communications and signal processing for wireless communications including multi-user detection and adaptive antenna array.
Hyung-Myung Kim (S’86–M’86–SM’99) received the B.S. degree in electronics engineering from Seoul National University, Seoul, Korea, in 1974 and the M.S. and Ph.D. degrees in electrical engineering from the University of Pittsburgh, Pittsburgh, PA, in 1982 and 1985, respectively. He is now a Professor at the Department of Electrical Engineering and Computer Science, The Korea Advanced Institute of Science and Technology (KAIST), Taejon, Korea. His research interests include digital signal/image processing, digital transmission of voice, data, and image, and multidimensional system theory. Dr. Kim was the Treasurer of the IEEE Taejon Section in 1992. He has been an editorial board member of Multidimensional Systems and Signal Processing since 1990.
