CA2277252A1 - Semi-blind multi-user detectors for cdma - Google Patents
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Abstract
We consider the problem of multi-user detection for CDMA systems where the codes of some users are known while others are unknown, semi-blind multi-user detection. An example is at the base station of a cellular communication system with interference from both in-cell users, with known codes, and out-of cell users, with unknown codes. In this paper we develop a number of semi-blind multi-user detectors using a subspace approach. One class of detectors developed has a computationally efficient structure of a subspace estimation common to all known users, followed by a simple processing specific to each user. The performance of the detectors is compared to that of the purely blind detector and the non-blind detector, and the semi-blind detectors are seen to have the best performance.
Description
Semi-Blind Multi-User Detectors for CDMA
Anders Hrast-Madsen FIELD OF THE INVENTION
This invention relates to mufti-user detection in CDMA, spread spectrum, mufti-user communications.
BACKGROUND OF THE INVENTION
Mufti-user detection is a method to improve detection performance and capacity of multiple access spread spectrum (or CDMA) systems. Mufti-user detection was introduced by Verdu in [ 1 ], where 1o the optimal mufti-user detector was derived. The optimal detector has an exponential complexity in the number of users, and less complex (linear) mufti-user detectors were therefore derived in an number of papers, in particular the decorrelating detector [2] and the minimum mean square error (MMSE) detector [3]. For a review of mufti-user detection and further references, see, e.g., [4]. All references in square brackets are listed at the end of the background section of this disclosure.
~s The early works on mufti-user detection assumed that the codes of all users were known at the receiver, and made a simultaneous detection of all users (therefrom the name mufti-user detection or joint detection). If the detection for example is at the base station of a mobile communication system, this seems realistic, as the base station needs to perform detection for all users. On the other hand, it is unrealistic that a mobile station should know the codes of all other users in a cell, and 2o therefore it is desirable to consider mufti-user detectors that need to know only the code of the desired user, blind mufti-user detection. The blind MMSE detector was introduced in a number of papers [6,7], and recently it was also shown that the decorrelating detector can be implemented blindly [13,14].
Although a base station knows all codes of the users within a cell, it typically will not know 25 the codes of interfering users from surrounding cells. Even if a base stations could obtain this knowledge from surrounding base stations, it would be a waste of resources if it were to also perform detection for these users just to cancel interference (including synchronization etc.). This is a serious problem to mufti-user detection, since typically 1/3 of the interference could be from other cells, intercell interference [8,9]. Thus, also at the base station blind detectors could be relevant. On the other hand, blind detectors do not use neither the fact that the codes of in-cell users are known at the base station, nor that these other users also have to be detected.
REFERENCES
[1] Verdil, S.: "Minimum Probability of Error for Asynchronous Gaussian Multiple-Access Channel," IEEE Trans. Info. Theo., Vol. IT-32, No. 1, pp. 85-96, Jan. 1986.
Anders Hrast-Madsen FIELD OF THE INVENTION
This invention relates to mufti-user detection in CDMA, spread spectrum, mufti-user communications.
BACKGROUND OF THE INVENTION
Mufti-user detection is a method to improve detection performance and capacity of multiple access spread spectrum (or CDMA) systems. Mufti-user detection was introduced by Verdu in [ 1 ], where 1o the optimal mufti-user detector was derived. The optimal detector has an exponential complexity in the number of users, and less complex (linear) mufti-user detectors were therefore derived in an number of papers, in particular the decorrelating detector [2] and the minimum mean square error (MMSE) detector [3]. For a review of mufti-user detection and further references, see, e.g., [4]. All references in square brackets are listed at the end of the background section of this disclosure.
~s The early works on mufti-user detection assumed that the codes of all users were known at the receiver, and made a simultaneous detection of all users (therefrom the name mufti-user detection or joint detection). If the detection for example is at the base station of a mobile communication system, this seems realistic, as the base station needs to perform detection for all users. On the other hand, it is unrealistic that a mobile station should know the codes of all other users in a cell, and 2o therefore it is desirable to consider mufti-user detectors that need to know only the code of the desired user, blind mufti-user detection. The blind MMSE detector was introduced in a number of papers [6,7], and recently it was also shown that the decorrelating detector can be implemented blindly [13,14].
Although a base station knows all codes of the users within a cell, it typically will not know 25 the codes of interfering users from surrounding cells. Even if a base stations could obtain this knowledge from surrounding base stations, it would be a waste of resources if it were to also perform detection for these users just to cancel interference (including synchronization etc.). This is a serious problem to mufti-user detection, since typically 1/3 of the interference could be from other cells, intercell interference [8,9]. Thus, also at the base station blind detectors could be relevant. On the other hand, blind detectors do not use neither the fact that the codes of in-cell users are known at the base station, nor that these other users also have to be detected.
REFERENCES
[1] Verdil, S.: "Minimum Probability of Error for Asynchronous Gaussian Multiple-Access Channel," IEEE Trans. Info. Theo., Vol. IT-32, No. 1, pp. 85-96, Jan. 1986.
[2] Lupas, R. and Verdu, S.: "Linear Multi-User Detectors for Synchronous Code-Division Multiple-Access Channels," IEEE Trans. Info. Theory, vol. 35, no. l, pp. 123-36, Jan. 1989.
[3] Xie, Z. Short, R. T. and Rushforth, C. K.: "A Family of Suboptimum Detectors for Coherent Multi-User Communications," IEEE J. on Selected Areas in Commun., vol. 8, no.
4, pp. 683-to 690, May 1990.
[4] Shimon, M.: "Multi-User Detection for DS-CDMA Communications," IEEE Comm.
Mag., pp.
124-136, Oct. 1996.
[S] Madhow, U. and Honig, M.L.: "MMSE Interference Suppression for Direct-Sequence Spread-Spectrum CDMA," IEEE Trans. Comm., Vol. 42, No. 12, pp. 3188-3188, Dec. 1994.
[6] Honig, M., Madhow, U. and Verdu, S: "Blind Adaptive Multiuser Detection,"
IEEE Trans.
Info. Theo., Vol. 41, no.4, pp. 944-960, July 1995.
[7] Tsatsanis, M.K.: "Inverse Filtering Criteria for CDMA Systems," IEEE
Trans. on. Signal Processing, vol. 45, no. 1, pp. 102-112, January 1997.
[8] Duel-Hallen, A., Holtzman, J. and Zvonar, Z.: "Multiuser Detection for CDMA Systems," IEEE
2o Personal Communications, pp. 46-57, April 1995.
[9] Viterbi, A.J.:"The Orthogonal-Random Wave form Dichotomy for Digital Mobile Communications," IEEE Personal Communications, pp. 18-24, First Quarter 1994.
[ 10] Poor, H. V. and Verdu, S.: "Probability of Error in MMSE Multi-user Detection," IEEE Trans.
Info. Theo., Vol. 43, no. 3, pp. 858-881, May 1997.
[11] Hest-Madsen, A. and Cho, K.S.: "MMSE/PIC Multi-User Detection for DS/CDMA
Systems with Inter- and Intra-Cell Interference," accepted for publication in IEEE
Trans.
Communications.
[12] Anton-Haro, C. and Fonollosa, J.R.: "Interference Cancellation for UMTS
using Adaptive Antennas," ACTS '97 (Aalborg, Denmark), pp. 320-325.
[ 13] Wang, X. and Poor, H. V.: "Blind Equalization and Multiuser Detection in Dispersive CDMA
Channels," IEEE Trans. Communications, Vol. 46, No. 1, pp. 91-103, January 1998.
[ 14] Wang, X. and Poor, H. V.: "Blind multiuser detection: A subspace approach," IEEE Trans.
Inform. Theory, to be published.
[15] Sharf, L.L.: "Statistical Signal Processing - Detection, Estimation and Time Series Analysis,"
Addison-Wesley (New York), 1991.
( 16] P. Comon, and G.H. Golub, "Tracking a few extreme singular values and vectors in signal processing," Proc. IEEE, 78(8), 1327-1343, Aug. 1990.
[17] R.A. Iltis and L. Mailaender: "Multiuser Detection of Quasisynchronous CDMA Signals Using to Linear Decorrelators," IEEE Trans. Comm., Vol. 44, No. 11, pp. 1561-1571, November 1996.
[18] H. Ge: "The LMMSE Estimate-Based Multiuser Detector: Performance Analyses and Adaptive Implementation," ICASSP'97 (Munich, Germany, April 1998).
[19] Schodorf, J.B. and Williams, D.B.: "A Constrained Optimization Approach to Multiuser Detection," IEEE Trans. Sig. Pro., Vol. 45, No. 1, pp. 258 -262, January 1997.
is [20] Schodorf, J.B. and Williams, D.B.: "Array Processing Techniques for Multiuser Detection,"
IEEE Trans. Comm., Vol. 45, No. 11, pp. 1375 -1378, November 1997.
[21 ] Host-Madsen, A.: "Semi-Blind Decorrelating Multi-User Detectors for CDMA: Subspace Methods," presented at IEEE PIMRC '98, Boston, Sept. 1998.
2o SUMMARY OF THE INVENTION
This invention provides multi-user detectors that can cancel interference from both known and unknown users, while utilizing the information about known users and the fact that detection has to be done for all known users. A blind mufti-user detector basically in some way has to estimate the codes of interfering users, and by using the known codes the estimation accuracy can be improved.
25 On the other hand, since several users have to be detected jointly, it is also advantageous, considering computational complexity, if some of the processing can be common to all users. In this patent document we propose detectors that satisfy these two criteria.
We call this class of detectors semi-blind mufti-user detectors. We use the term semi-blind to mean that the detector has partial information: some, but not all, users' codes are known, as opposed 3o to blind detector of [6], which knows only the code of the desired user.
In the preferred embodiment, we consider linear detectors, i.e., the decorrelating and the MMSE detector. For non-linear approaches see [11] and [12]. We use the subspace approach of Wang and Poor [14]. We consider synchronous systems because of their conceptual simplicity.
However, although the theoretical development and the simulations assume that all users are synchronous, the formulas developed can be applied without change to systems where the unknown users are not synchronous. The results can therefore be directly applied to systems where the known users are synchronous, e.g., the downlink in a cellular system, and the up-link in some systems. With some additional development, the results can also be applied to a quasi-synchronous CDMA system [17], where the users within a cell are (almost) synchronous, but the interference from other cells is 1o asynchronous.
BRIEF DESCRIPTION OF THE FIGURES
There will now be described preferred embodiments of the invention in which like reference characters denote like elements, for purposes of illustration only, and without intending to limit the scope of the invention, and in which:
Fig. 1 is a schematic of a conventional detector for use in multi-user detection;
Fig. 2 is a schematic of a conventional detector for use in non-blind multi-user detection;
Fig. 3 is a schematic of a conventional detector for use in blind multi-user detection;
Fig. 4 is a schematic showing a semi-blind multi-user detector according to the invention;
2o Fig. 5 is a schematic showing a prior art system model for synchronous CDMA;
Fig. 6 illustrates how a received signal contains interference from other users, which may be detected according to the conventional linear detector and general linear detector provided on the left hand side of the figure;
Fig. 7 illustrates operation of a conventional decorrelating detector, according to the equations shown on the left hand side;
Fig. 8 is a schematic showing a semi-blind decorrelating detector according to the invention;
Figs. 9A-9D show simulation results for:
A) detector convergence with 7 known users and 4 unknown users (the curve marked 1 is for the algorithms calculated using ( 17) and the one marked 2 for the algorithm ( 16));
3o B) simulation results of detector convergence for 7 known and 10 unknown users;
C) simulation results of detector convergence for 7 known and 10 unknown users, with subspace dimension set to 4; and D) bit error rate versus SNR for 7 known and 4 unknown users, for which, in each detector, the lower curve is the median and the upper curve is the 90 percentile.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
In this patent document, "comprising" means "including". In addition, a reference to an element by the indefinite article "a" does not exclude the possibility that more than one of the element is present. Immaterial modifications may be made to the invention described here without departing from the essence of the invention.
I. AN IMPLEMENTATION OF THE INVENTION
The principles of operation of the invention are illustrated primarily by mathematical equations and algorithms. These equations and algorithms are implemented in a computer, whether general purpose or designed for the purpose, to yield the practical implementation of multi-user detection. The apparatus used is illustrated with block diagrams accompanied by the equations since these fully define the apparatus. The design of the apparatus, apart from being configured to carry out the invention, is conventional. Upon programming in accordance with the algorithms and equations set out in this patent document, the apparatus becomes a novel apparatus. For the purpose of describing the invention, the general configuration of the apparatus will first be described, 2o followed by the algorithms and equations implemented by the invention.
Referring to Fig. l, a conventional detector is shown in which a received signal r(t) is applied to a matched filter bank 10, and the resultant applied to a multi-user detector 12. The matched filter bank 10 is achieved by correlation of the received signal with the codes sI{t) using conventional correlators 14, followed by integration over the symbol length in integrators 16, and sampling at the symbol interval at sampler 18. The multi-user detector 12 is carried out by k decision sub-systems 20. Fig. 2 shows the same matched filter bank 10 followed by the multi-user detector 12, which accomplish cancellation of intracell interference. Referring to Fig. 3, a conventional blind multi-user detector is shown which uses plural conventional adaptive ICs to detect the received signal and cancel both intracell and intercell interference.
[4] Shimon, M.: "Multi-User Detection for DS-CDMA Communications," IEEE Comm.
Mag., pp.
124-136, Oct. 1996.
[S] Madhow, U. and Honig, M.L.: "MMSE Interference Suppression for Direct-Sequence Spread-Spectrum CDMA," IEEE Trans. Comm., Vol. 42, No. 12, pp. 3188-3188, Dec. 1994.
[6] Honig, M., Madhow, U. and Verdu, S: "Blind Adaptive Multiuser Detection,"
IEEE Trans.
Info. Theo., Vol. 41, no.4, pp. 944-960, July 1995.
[7] Tsatsanis, M.K.: "Inverse Filtering Criteria for CDMA Systems," IEEE
Trans. on. Signal Processing, vol. 45, no. 1, pp. 102-112, January 1997.
[8] Duel-Hallen, A., Holtzman, J. and Zvonar, Z.: "Multiuser Detection for CDMA Systems," IEEE
2o Personal Communications, pp. 46-57, April 1995.
[9] Viterbi, A.J.:"The Orthogonal-Random Wave form Dichotomy for Digital Mobile Communications," IEEE Personal Communications, pp. 18-24, First Quarter 1994.
[ 10] Poor, H. V. and Verdu, S.: "Probability of Error in MMSE Multi-user Detection," IEEE Trans.
Info. Theo., Vol. 43, no. 3, pp. 858-881, May 1997.
[11] Hest-Madsen, A. and Cho, K.S.: "MMSE/PIC Multi-User Detection for DS/CDMA
Systems with Inter- and Intra-Cell Interference," accepted for publication in IEEE
Trans.
Communications.
[12] Anton-Haro, C. and Fonollosa, J.R.: "Interference Cancellation for UMTS
using Adaptive Antennas," ACTS '97 (Aalborg, Denmark), pp. 320-325.
[ 13] Wang, X. and Poor, H. V.: "Blind Equalization and Multiuser Detection in Dispersive CDMA
Channels," IEEE Trans. Communications, Vol. 46, No. 1, pp. 91-103, January 1998.
[ 14] Wang, X. and Poor, H. V.: "Blind multiuser detection: A subspace approach," IEEE Trans.
Inform. Theory, to be published.
[15] Sharf, L.L.: "Statistical Signal Processing - Detection, Estimation and Time Series Analysis,"
Addison-Wesley (New York), 1991.
( 16] P. Comon, and G.H. Golub, "Tracking a few extreme singular values and vectors in signal processing," Proc. IEEE, 78(8), 1327-1343, Aug. 1990.
[17] R.A. Iltis and L. Mailaender: "Multiuser Detection of Quasisynchronous CDMA Signals Using to Linear Decorrelators," IEEE Trans. Comm., Vol. 44, No. 11, pp. 1561-1571, November 1996.
[18] H. Ge: "The LMMSE Estimate-Based Multiuser Detector: Performance Analyses and Adaptive Implementation," ICASSP'97 (Munich, Germany, April 1998).
[19] Schodorf, J.B. and Williams, D.B.: "A Constrained Optimization Approach to Multiuser Detection," IEEE Trans. Sig. Pro., Vol. 45, No. 1, pp. 258 -262, January 1997.
is [20] Schodorf, J.B. and Williams, D.B.: "Array Processing Techniques for Multiuser Detection,"
IEEE Trans. Comm., Vol. 45, No. 11, pp. 1375 -1378, November 1997.
[21 ] Host-Madsen, A.: "Semi-Blind Decorrelating Multi-User Detectors for CDMA: Subspace Methods," presented at IEEE PIMRC '98, Boston, Sept. 1998.
2o SUMMARY OF THE INVENTION
This invention provides multi-user detectors that can cancel interference from both known and unknown users, while utilizing the information about known users and the fact that detection has to be done for all known users. A blind mufti-user detector basically in some way has to estimate the codes of interfering users, and by using the known codes the estimation accuracy can be improved.
25 On the other hand, since several users have to be detected jointly, it is also advantageous, considering computational complexity, if some of the processing can be common to all users. In this patent document we propose detectors that satisfy these two criteria.
We call this class of detectors semi-blind mufti-user detectors. We use the term semi-blind to mean that the detector has partial information: some, but not all, users' codes are known, as opposed 3o to blind detector of [6], which knows only the code of the desired user.
In the preferred embodiment, we consider linear detectors, i.e., the decorrelating and the MMSE detector. For non-linear approaches see [11] and [12]. We use the subspace approach of Wang and Poor [14]. We consider synchronous systems because of their conceptual simplicity.
However, although the theoretical development and the simulations assume that all users are synchronous, the formulas developed can be applied without change to systems where the unknown users are not synchronous. The results can therefore be directly applied to systems where the known users are synchronous, e.g., the downlink in a cellular system, and the up-link in some systems. With some additional development, the results can also be applied to a quasi-synchronous CDMA system [17], where the users within a cell are (almost) synchronous, but the interference from other cells is 1o asynchronous.
BRIEF DESCRIPTION OF THE FIGURES
There will now be described preferred embodiments of the invention in which like reference characters denote like elements, for purposes of illustration only, and without intending to limit the scope of the invention, and in which:
Fig. 1 is a schematic of a conventional detector for use in multi-user detection;
Fig. 2 is a schematic of a conventional detector for use in non-blind multi-user detection;
Fig. 3 is a schematic of a conventional detector for use in blind multi-user detection;
Fig. 4 is a schematic showing a semi-blind multi-user detector according to the invention;
2o Fig. 5 is a schematic showing a prior art system model for synchronous CDMA;
Fig. 6 illustrates how a received signal contains interference from other users, which may be detected according to the conventional linear detector and general linear detector provided on the left hand side of the figure;
Fig. 7 illustrates operation of a conventional decorrelating detector, according to the equations shown on the left hand side;
Fig. 8 is a schematic showing a semi-blind decorrelating detector according to the invention;
Figs. 9A-9D show simulation results for:
A) detector convergence with 7 known users and 4 unknown users (the curve marked 1 is for the algorithms calculated using ( 17) and the one marked 2 for the algorithm ( 16));
3o B) simulation results of detector convergence for 7 known and 10 unknown users;
C) simulation results of detector convergence for 7 known and 10 unknown users, with subspace dimension set to 4; and D) bit error rate versus SNR for 7 known and 4 unknown users, for which, in each detector, the lower curve is the median and the upper curve is the 90 percentile.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
In this patent document, "comprising" means "including". In addition, a reference to an element by the indefinite article "a" does not exclude the possibility that more than one of the element is present. Immaterial modifications may be made to the invention described here without departing from the essence of the invention.
I. AN IMPLEMENTATION OF THE INVENTION
The principles of operation of the invention are illustrated primarily by mathematical equations and algorithms. These equations and algorithms are implemented in a computer, whether general purpose or designed for the purpose, to yield the practical implementation of multi-user detection. The apparatus used is illustrated with block diagrams accompanied by the equations since these fully define the apparatus. The design of the apparatus, apart from being configured to carry out the invention, is conventional. Upon programming in accordance with the algorithms and equations set out in this patent document, the apparatus becomes a novel apparatus. For the purpose of describing the invention, the general configuration of the apparatus will first be described, 2o followed by the algorithms and equations implemented by the invention.
Referring to Fig. l, a conventional detector is shown in which a received signal r(t) is applied to a matched filter bank 10, and the resultant applied to a multi-user detector 12. The matched filter bank 10 is achieved by correlation of the received signal with the codes sI{t) using conventional correlators 14, followed by integration over the symbol length in integrators 16, and sampling at the symbol interval at sampler 18. The multi-user detector 12 is carried out by k decision sub-systems 20. Fig. 2 shows the same matched filter bank 10 followed by the multi-user detector 12, which accomplish cancellation of intracell interference. Referring to Fig. 3, a conventional blind multi-user detector is shown which uses plural conventional adaptive ICs to detect the received signal and cancel both intracell and intercell interference.
5 Referring to Fig. 4, a semi-blind mufti-user detector 24 is shown which uses an adaptive IC
26 followed by a matched filter 28 and non-blind mufti-user detector 30. Some but not all codes of users are known. This system cancels both intracell and intercell interference using all information available to the user. Only one adaptive IC is used that is common to all users, which assists in reducing complexity of the system.
Referring to Fig. 5, a conventional system model is shown in which the received signal is processed in the chip matched filter 10, sampled at the chip rate at 14 and then provided to serial to parallel buffer 32.
Referring to Fig. 6, a conventional received vector rl is formed as the desired signal plus 1o interference. The detector works by finding the projection of rl on s2. In Fig. 7, a similar principle is shown for the decorrelating detector, except that the projection on wl is found.
The complete semi-blind decorrelating detector is shown in Fig. 8. Fig. 8 essentially is a practical implementation of equation 13 described in the following. Received signal r, which includes the signal from known and unknown users received at a single cell site, is processed along two paths.
The lower path forms a conventional non-blind conventional correlating mufti-user detector as shown in Fig. 2. On the upper path a blind mufti-user detector calculates the interference due to unknown, outside, users, which is then subtracted from the output from the conventional detector 10 to generate a signal from which intracell and intercell interference has been removed.
In the lower path, a matched filter 10 (Fig. 1 ) is first applied to the received signal r. The output from the matched filter 10 is provided along path 34 where (STS)-1 at 36 followed by S at 38 is applied to the signal. This effectively projects the received data on the subspace orthonormal to the subspace of the known users, which isolates the signal from users inside the cell (the known users).
The signal from users inside the cell is then subtracted from the received signal r at 39 to yield a signal that only contains the signal from outside users. The signal from outside users is then processed along the upper path 40 to find the interference from the outside users by first applying singular value decomposition 42 to the signal. SVD 42 is followed by the equation shown at 44, which projects the output from the subtractor 39 onto a basis for the subspace defined by the codes of the mobile users whose codes are unknown. This produces the interference from the outside users, justification for which is shown in the proof of Theorem 2 given at the end of this description.
3o Equation 44 uses both the received signal r provided along path 48 and the output from the matched
26 followed by a matched filter 28 and non-blind mufti-user detector 30. Some but not all codes of users are known. This system cancels both intracell and intercell interference using all information available to the user. Only one adaptive IC is used that is common to all users, which assists in reducing complexity of the system.
Referring to Fig. 5, a conventional system model is shown in which the received signal is processed in the chip matched filter 10, sampled at the chip rate at 14 and then provided to serial to parallel buffer 32.
Referring to Fig. 6, a conventional received vector rl is formed as the desired signal plus 1o interference. The detector works by finding the projection of rl on s2. In Fig. 7, a similar principle is shown for the decorrelating detector, except that the projection on wl is found.
The complete semi-blind decorrelating detector is shown in Fig. 8. Fig. 8 essentially is a practical implementation of equation 13 described in the following. Received signal r, which includes the signal from known and unknown users received at a single cell site, is processed along two paths.
The lower path forms a conventional non-blind conventional correlating mufti-user detector as shown in Fig. 2. On the upper path a blind mufti-user detector calculates the interference due to unknown, outside, users, which is then subtracted from the output from the conventional detector 10 to generate a signal from which intracell and intercell interference has been removed.
In the lower path, a matched filter 10 (Fig. 1 ) is first applied to the received signal r. The output from the matched filter 10 is provided along path 34 where (STS)-1 at 36 followed by S at 38 is applied to the signal. This effectively projects the received data on the subspace orthonormal to the subspace of the known users, which isolates the signal from users inside the cell (the known users).
The signal from users inside the cell is then subtracted from the received signal r at 39 to yield a signal that only contains the signal from outside users. The signal from outside users is then processed along the upper path 40 to find the interference from the outside users by first applying singular value decomposition 42 to the signal. SVD 42 is followed by the equation shown at 44, which projects the output from the subtractor 39 onto a basis for the subspace defined by the codes of the mobile users whose codes are unknown. This produces the interference from the outside users, justification for which is shown in the proof of Theorem 2 given at the end of this description.
3o Equation 44 uses both the received signal r provided along path 48 and the output from the matched
6 filter 10 along path 50. The interference from outside users is then provided to subtractor 46, which subtracts the interference from outside users from the output from the matched filter 10. (STS)'' is then applied at 52 to the signal to complete the detector. A decision (sgn function) is made at 54 which generates the transmitted bits +1 or -1 for each of the K users, thus completing the detector.
The details of operation of the system elements will now be described.
The details of operation of the system elements will now be described.
7 IL SYSTEM MODEL
Consider a synchronous spread spectrum communications system with the users transmitting through an additive white l:raussian noise channel. The user population consists of K users with known codes, and K
users with unknown codes. As we consider synchronous systems, it is su~cient to consider a single symbol interval [O,T], where the received signal can be written as K K _ _ ~t ) _ ~ bkAksk (~)'f ~ bkAk sk (t ) +n(r)~ E [a. T l kal k=1 where sk is the normalized code waveform of the k'th known user with support in [O,T], sk the waveform of the k'th unknown user, bk,6k are the transmitted bits (tl), Ak,Ak the amplitudes and n white Gaussian noise. We assume that the codes are given by chip coda, sx(t) _ ~ctyr(t-iT IM), where ct a {-1,+1} and yr is the chip waveform, and similarly for the unknown users. A
suffcient statistic for the received signal is therefore the output of a chip rate sampled chip matched 5lter, and we can write this statistic on vector form as x K _ __ r = ~bkAkak +~bkAk sk +n = SAb +SAb +n k=1 k=1 where S = [sl,ai,...,:K1S =~, ii,...,sx]. The correlation matrix of r is given by R=E[rrr]=SAzSr +SAZSr +azI
We will assume that all codes, of both known a~ unknown users, are linearly independent, so that ~S S~ has full rank.
As mentioned in the iatroduaron, the system model can still be used if the unknown users are asynchronous. In that case, different bits within the interval [O,TJ from the same user are considered as different 'virtual' users, and K is now the number of 'virtual' unknown users, which can be between one and two times the actual number of unknown users.
I>tZ DETECTOR STRUCTURES
A general liar detector for user i is given by b, = sgn(w; r~
For jointly detection of all users, the detector can be written b - sgn~W r r~
Consider a synchronous spread spectrum communications system with the users transmitting through an additive white l:raussian noise channel. The user population consists of K users with known codes, and K
users with unknown codes. As we consider synchronous systems, it is su~cient to consider a single symbol interval [O,T], where the received signal can be written as K K _ _ ~t ) _ ~ bkAksk (~)'f ~ bkAk sk (t ) +n(r)~ E [a. T l kal k=1 where sk is the normalized code waveform of the k'th known user with support in [O,T], sk the waveform of the k'th unknown user, bk,6k are the transmitted bits (tl), Ak,Ak the amplitudes and n white Gaussian noise. We assume that the codes are given by chip coda, sx(t) _ ~ctyr(t-iT IM), where ct a {-1,+1} and yr is the chip waveform, and similarly for the unknown users. A
suffcient statistic for the received signal is therefore the output of a chip rate sampled chip matched 5lter, and we can write this statistic on vector form as x K _ __ r = ~bkAkak +~bkAk sk +n = SAb +SAb +n k=1 k=1 where S = [sl,ai,...,:K1S =~, ii,...,sx]. The correlation matrix of r is given by R=E[rrr]=SAzSr +SAZSr +azI
We will assume that all codes, of both known a~ unknown users, are linearly independent, so that ~S S~ has full rank.
As mentioned in the iatroduaron, the system model can still be used if the unknown users are asynchronous. In that case, different bits within the interval [O,TJ from the same user are considered as different 'virtual' users, and K is now the number of 'virtual' unknown users, which can be between one and two times the actual number of unknown users.
I>tZ DETECTOR STRUCTURES
A general liar detector for user i is given by b, = sgn(w; r~
For jointly detection of all users, the detector can be written b - sgn~W r r~
8 where W = ~w ~ w Z ~ ~ ~ w x ~ . As argued in [10], any interesting detector must have w; a range~S SD, since any component of w, outside this subspace will only increase noise without reducing interference.
A. The decorrelating detector Prior to developing the semi-blind detector we will discuss general decorrelating detectors in a geometrical framework useful to understanding the decorrelating detector. A decorrelating detector for user i satisfies w; s;=1 w; sk =O,k ~i (1) w; sk = O,k =1,...,K
These conditions imply that the decorrelating detector cancels the interference from other users completely, wrr=Atb; +wi n The conditions (1) oondidons together with w, a range~S SD determine w, uniquely, since ~S Sl has full rank. We can also write w, as w, = M,s, . Although w, is unique, clearly M, need not be unique, and in the following we will discuss some different natural choices of M,. A practical solution for M, is a solution M that does not depend on i, since then wTr=s; MTr and MTr can be calculated common to all users. Especially if M is low-rank, or if it has to be estimated, as is the case when some user codes are unknown, this can yield an efficient solution.
To understand how the deoorrelating estimator can be estimated, we will at first analyze the case when all codes are known, i.e., K = 0 . Without loss of generality we can consider detection for user 1.
There are two natural approaches to the decorrelating detector: the projection approach and the orthogonalization approach. In the projection approach the received vector r is projected on the subspace orthogonal to the codes of the other users, whereby their interference has been removed, followed by a projection on the subspace spanned by user 1. An explicit solution can be derived as follows.
Let S~ =[s2s3..sK] (2) and lei i P~ = SySi Sy Si (3) be the projection unto the subspace spanned by S,, with Pi = I - Pl the projection unto the orthogonal subspace. Then we can write the decorrelating detector as
A. The decorrelating detector Prior to developing the semi-blind detector we will discuss general decorrelating detectors in a geometrical framework useful to understanding the decorrelating detector. A decorrelating detector for user i satisfies w; s;=1 w; sk =O,k ~i (1) w; sk = O,k =1,...,K
These conditions imply that the decorrelating detector cancels the interference from other users completely, wrr=Atb; +wi n The conditions (1) oondidons together with w, a range~S SD determine w, uniquely, since ~S Sl has full rank. We can also write w, as w, = M,s, . Although w, is unique, clearly M, need not be unique, and in the following we will discuss some different natural choices of M,. A practical solution for M, is a solution M that does not depend on i, since then wTr=s; MTr and MTr can be calculated common to all users. Especially if M is low-rank, or if it has to be estimated, as is the case when some user codes are unknown, this can yield an efficient solution.
To understand how the deoorrelating estimator can be estimated, we will at first analyze the case when all codes are known, i.e., K = 0 . Without loss of generality we can consider detection for user 1.
There are two natural approaches to the decorrelating detector: the projection approach and the orthogonalization approach. In the projection approach the received vector r is projected on the subspace orthogonal to the codes of the other users, whereby their interference has been removed, followed by a projection on the subspace spanned by user 1. An explicit solution can be derived as follows.
Let S~ =[s2s3..sK] (2) and lei i P~ = SySi Sy Si (3) be the projection unto the subspace spanned by S,, with Pi = I - Pl the projection unto the orthogonal subspace. Then we can write the decorrelating detector as
9 R'~ = s~ P~ s~ P~ si The factor s; P~ s, is simply a normalizing factor, and can be omitted if we consider BPSK, where we are only interested in the sign of the output. It is easy to see that the conditions (I) are satisfied. The disadvantage of this solution is that P; is specific to user 1, and each user will therefore use a different projection.
We will therefore consider the orthogonalization approach. The problem with the traditional matched filter solution is that the vectors st,si,...,sx are not orthogonal. However, since sl,sZ,...,sK are linearly independent, there exists some inner product in Ru with respect to which al,s2,...,sx are orthonormal. Specifically, there exists some (positive semi definite) matrix W a R"f ""'f so that sT Ws~ = 8;,~ . Since s,, sZ,...,sx are not a basis of R"~ the matrix W
is not unique, but two solutions are W =S~STsr2ST (4) and Ww = SA~ASTSA~2 ASr , which gives w, = Ws, (5) or rl y = si Wwsi Wwsi .
The conditions (1) are satisfied (seen most easily be noticing that ST WS = I
, and that scaling of vectors do not influence orthogonality). This solution is in fact equivalent to the "traditional"
decorrelating detector [2] , since S r W r = ~S r Sr ~ S r r . The solution fiuiher has the advantage that W is independent of the user oonsider~ed and that it is a low rank matrix, which makes the transformation efficient to calculate.
To estimate W (or rather W,~ we can make use of the following. The eigenvalue decomposition of R can be written as [14]
r R ~ITs LT"~ 0' Q I jjT
where Ua is an orthonormal basis of the signal subspace spanned by S and U" is a basis for the noise subspace orthogonal to Ua. The diagonal matrix Aa contains the K largest eigenvalues of R, while ~ is the minimum eigenvalue. We then have, by examining the proofs in [14]
Proposition 1 Ww = Ua(Aa -QZI) I Us Thus, as R can be estimated from the received data only, Ww can be estimated by EVD, and the orthogonalizing decorrelating detector can be implemented blindly [14].
B. The MMSE detector The minimum mean square error (MMSE) detector is defined as the detector that minimizes the mean square error (MSE):
we _ ~~~(~i -wTr Equivalently, the solution is given as the vector that satisfies Elr{b; -w~ r~=0 which results in the equation Rw; = A;s;
Thus, we can write b, = sgn(s T R-t r~
(~ (8) b=s~~rR tr~
Notice that this detector can be implemented blindly by replacing R by a time-averaged estimate [18]. Using the eigenvalue decomposition (6), this can also be written as j!i =Sgn(S; UaAs1 Ua r!
1 (9) b =s~~T UaAalUa rl which is the idea of the subspace method in [14].
Now suppose that w, is constrained to lie in some subspace ?l, let u,, u2,...,um be an orthonormal basis of ?l, and let U = ~u, u2 ~ ~ ~ u,"~. We can then write w; = Uci for some vector c;.
Inserting this to (7) gives e; =~~UrRUrtUr9a (10) if P is the projection onto ?t we can also find w; as ct = AOP~~tPsr where t denotes pseudo-inverse (which is needed here, since PRP does not have full rank).
IV. SEMI-BLIND DETECTORS
A. Semi-blind decorrelating detectors In our first approach to semifilind decorrelating detectors, we use a mixed projection/orthogonalization approach, as described in section IILA The idea is to first project the received data r on the subspace orthogonal to span(St ) , with S, given by (2). Thereby all interference from the larown users has been removed. The orthogonalization approach is then used in span(S t ) 1. The calailarions can be done as follows. Let P, be defined by (3), and define S, =
P; .C"S. st ~, A, _ ~A A, ~ and (. r Wt =StAytSt StAtr AtSt We can write the eigenvalue decomposition of Rt = P~ RPi as As 0 0 UJ
R, _ ~U, U" Uo 0 ~zI 0 U"
0 0 0 Uo Here U, is an orthonormal basis of the signal subspace, which in this case is spanned by CS s, ~, while U" is the noise subspace. The matrix Uo is an orthonormal basis of the space spanned by St. Furthermore, Aa = diag(.tt , ~li,...,~.K+t ) with .i; > 0-2 . Similarly to the blind deoorrelating detector we now have that Wt =U.~.=-~ZIx+trtU= (I1) We can then state the semi-blind deoorrelating detector as Theorem l: The deoorrelating detector is given by wt = r Wtst (l2) si Wtsi In our setting, S; is unknown. Equation (11), however, outlines a method for estimating W,, and the method can therefore be implemented without knowledge of the unknown users' codes.
The problem of the above method is that W, is specific to user 1. It can be used with advantage in the case where some users' codes are known, but still only one user's data is of interest, as for example in a mobile station. However, if several users are to be detected simultaneously, an SVD/subspace tracking has to be done for each user, and the method is ine~cient.
We will therefore develop a method that requires only one SVD common to all users. This method will be based on the orthogonalization approach. Fiat, define the projection on span(S) , P =s~Tsr~sT
with Pl = I - P . The eigenvalue decomposition of P 1 RP 1 , is then given by At 0 0 Ua P1RP1 =[Us U" Uo 0 QZI 0 U"
0 0 0 Uo where As = diag(.i,,,.iz,...,.iK ) with ~.; > a'2 , and Uo has K rows.
The idea in the semi-blind detector is first to apply the projection P1 to the received data. Thereby the signal from known all users have been eliminated. The subspace of the unknown users, projected to span(S)1, is then found. This is span~U, ~. The last step is to combine this subspace with the code of each user through some algebraic manipulations. For details, see the proof of Theorem 2.
We can now state the semi-blind decorrelating detector as follows Theorem 2: The decorrelating detector is given by b=sg~~.STSrIST~I-RIJf~AJ -v2Ir~Ua JrJ (13) Proof see Appendix A.
Notice that all quantities in the theorem can be estimated with knowledge of only the known users' codes. Notice also that although R appears explicitly, R itself does not have to be estimated. Define T(n) = ST R(n)Us (n) where R(n) etc. denote the estimated quantities over n samples. Then we can rewrite (13) as b(n)=s~~TS~ tI ST -T(~)(Ar(n)-~~I,Us (n))r~, (14) The quantity T(n) can be calculated as follows, T(n)=Sr 1 L.rlrt Us(n) n f=I
_l ~ 1 ~, ~Srrt ~U; (n)ri)T ~ (IS) ~=t ~~Trr~'i Us (n) n .-_t The implementation using the formula on the second line only roquires KK
multiplications per bit, instead of the MZ multiplications to calculate R On the other hand this formula is not directly suitable for recursive implementation, since Us (n) has to be applied to all previous samples. The formula on the third line requires KM
multiplicadons, and can be implemented recursively as long as S is time-invariant.
B. Hybrid Semi-Blind Detectors We will in this section consider detectors that are combinations of the decorrelating detector and the MMSE detector. Specifically, we will study detectors that are decornelating among the known users and MMSE
with respect to the unknown users. We will call this class of detectors hybrid semi-blind detectors. The idea is similar to ideas used in array processing: to direct nulls in the direction of known interferers, and find the MMSE solution for the remaining interference, and has in this ~ntext also been studied for mufti-user detection [19,20].
As for the decorrelating detector, we will give two solutions for the hybrid detector: one that makes a subspace calculation for each user, and one that makes a common subspace calculation. For the former case we get the following Theorem 3. The notation is the same as for Theorem 1.
Theorem 3: The hybrid semi-blind detector is given by wl = s; UsAfIUsSt UfAsIUss~ (16) where Us and A~ is given from the eigenvalue decomposition of Ri = P~ RP~l Aj 0 0 Us R, _ ~Ua U" Uo 0 ~2I 0 U
0 0 0 Ua where As =diag(.i,,.'tz,...,~.K+~) with ~,; >a~2 This detector is similar to the detector of [19], but is here stated in a subspace context.
For the case of common subspace tracking we get, with the same notation as for Theorem 2 Tfreorem 4: The hybrid semi-blind detector is given by b =sg~~SrSr~ST(I-RU,As~U; )rJ (17) Proof-. see Appendix B.
We notice that the hybrid semi-blind detector (17) is very similar to the semi-blind decorrelating detector (13). The only difference is that As -6~I is replaced by A, . As for (17), R itself does not have to be estimated, as the quantities needed can be calculated using (15).
In general, the MMSE detector has superior performance to the decorrelating detector [10], and it would therefore be natural to study pure MMSE semi-blind detectors. However, the hybrid approach has a number of advantages. For the known users, the decorrelating detector does not need a~ estimated information, whereas the MMSE solution requires estimates and tracking of user powers. On the other hand, the deoornelating detector for the unknown users requires a noise estimate and is therefore rather dependent on an accurate estimate of the number of users, while the MMSE
detector does not have these shortcomings. Finally, the semi-blind hybrid detector has a compact, simple expression (17), while it seems impossible to derive the true MMSE detector by the methods used here.
One way to make ( 17) closer to a true MMSE solution, is to replace it with the following expression b=sgnC(STS+QZA-ZrIST(I-RUtAs~Us)r~ (18) The argument for (18) is heuristic. We first find the MMSE solution in the subspace spanned by the known users, ignoring all interference from unknown users. For each user, we then find the MMSE solution in the subspace spanned by this solution and iJ s .
This solution can avoid the amplification of noise that ~uld happen in ( 17), but it is not an NIIviSE solution and does not satisfy any orthogonality or IvFvISE conditions.
V. SUBSPACE TRACKIrIG
A large number of different subspace tracking methods exist, with varying convergence speed and complexity, and most of them can be applied to the current problem, but with some modifications. To illustrate the principles we have chosen to use the F2 algorithm ( 16], summarized in Tabte 1. We use a standard exponential window with It(n) _ .ZR(n -1) +r"r~
Table I: The F2 algorithm [16] for tracking a d-dimensional subspace.
Let the data vecxors be v~, v2,...
U(0) arbitrary m x d matrix, U(0)T U(0) = I
E(0) = I.
For n = 1,2,3,...
Define W(n)_~U(n-1)E(n-1) v"~
Compute the m x d and d x d matrices, U(n), E(n) from the SVD:
~U(n) u~~~n) ~~Y(n)T = W (n) For the semi-blind algorithms (16) and (12) the input to the F2 algorithm for user 1 are the vectors V n = Pl rn The subspace dimension is d = K +1. The matrix U,(n) is directly the output U(n) of the F2 algorithm, while Ad(n) = E(n)2 .
For the semi-blind detectors (17) and (13), the input to the F2 algorithm is Vn =Plrn and the subspace dimension is d = K . Again, the subspace information can be obtained directly from the output of the F2 algorithm with the matrix Us(n) equal to U(n) and Aj(n) = E(n)Z . However, also T(n) (equation (15)) has to be calculated to implement the algorithms. As stated below equation (15) this formula on the second line is not suitable for recursive implementation. We will here give some approximate formula for a recursive implementation, that can also be used when S is time-varying.
With the weighting factor included, T(n) can be rewritten as T(n)=~~° ~(Srrr~Us(~)ryr (19) The level of approximation used in evaluating (19) should correspond to the level of approximation used in the specific subspace tracking algorithm used, so that there is an agreement in convergence speed and complexity between the subspace tracking algorithm and the update formula for T(n). Thus, a general update formula cannot be given, and we will look at two examples. The first possibility is to replace U~(n) with i1,(i) inside the summation in (19), to get T(n) ~ ~ ~~f ~rrt ~Us (l)rt \r r= J~
~., r =~,T(n_1)+Prrn'lUi (n)r~J
This approximation is essentially the same as the key approximation used in deriving PASTd, and gives good performance with this algorithm. However, for a fast converging algorithm such as F2, we lave found this does not result in good convergence. Another approximation is T(n) ~ ~.T(n-1)Us (n-1)Us(n)+Srr"(i1; (n)r"~r (20) The justification of this approximation is geometrical. The quantity Us (n -i)Tr (n -1) represents the orthogonal projection of R(n-I~Sr onto the subspace spanned by Ua(n-1). To find the projection of R(n-1)Sr onto the subspace spanned by Us(n) we should reproject R(n -1)Sr . Instead we project the previous projection. If the subspace does not change too much from n to n+l, the approximation is only slight. Notice also that this approximation is similar to the key approximation in deriving F2 (equation (27) in [16]) The calculation of Us (n -1)Us (n) is complex in itself. Fortunately, this can be obtained as an intermediate result in the FZ
algorithm. Notice that since ~U(n) u~ ~U(n) u~= I (Table i), we have C~n) O~YrC~n_1) 0~_~
0 a ~
~F(n_1) U~_ _ ~U(n) u~ W(n _ ~~U(n)r U(n -1) ~~
where ~ indicates elements with values irrelevant to the result. Thus, LTs (n-1)IJs(n) can be found from the upper Left K x K
matrix of the above matrix. The total multiplication count for this approach is KK z + 2K , which compares well with the multiplication count of 3MK Z +MK for the F2 algorithm itself. However, if K
is large, the formula in the second line of equation ( 15) has less complexity.
VL SIMULATION STUDIES
We consider a system with K--7 users with known codes, all with the same power, and a variable number K of users with unknown codes. The users are assigned purely random codes of length M=31, and the SNR
is 20 dB. An ensemble of 50 different random code assignments is generated, and the median signal to inference and noise ratio (SINK) is calculated over all code choices and users, in total an ensemble of 350. The SINK for a user is calculated using a moving average filter of length 20.
In all cases we consider 8 different detectors:
single user The oom~entional single user detector. This is the lower bound on performance of any mufti-user detector.
Full The hypothetical non-blind MMSE mufti-user detector that knows all codes.
This is the upper bound of performance for any linear mufti-user detector.
non-blind The nonfilind MMSE detector that uses only the known codes and ignores the interference from the unknown users.
Direct The blind MMSE detector calculated using direct inversion of the correlation matrix (equation (8)).
Blind The blind MMSE detector calculated using (9) and using SVD for subspace calculation.
Semi-blind SVD The semi-blind detector calculated using (17) and using SVD for subspace calculation.
Semi-blind 2 The stmt-blind detector calculated using (16) and using SVD for subspace calculation.
Semi-blind F2 The semi-blind detector calculated using (17) and using the F2 algorithm for subspace calculation with T(n) calculated using (20).
Decorrelating detectors are not considered, since they in general have an inferior performance and are sensitive to estimation errors. Performance of the dacorrelating detectors can be seen in [21]. For the F2 algorithm, it was initialized with an ordinary SVD over the first 50 samples, i.e., U(0) and E(0) in Table 1 was initialized from the SVD over the first 50 samples.
Without this initialization, the F2 algorithm diverged in the beginning, until it would start converging.
QA
Figure'l4shows the convergence with K = 4 unknown users, all of the same power as the known users. It can be seen that the semi-blind detector has a performance gain over the blind detector starting at 4 dB and decreasing to 2 dB. As the number of bits increase both detectors should converge towards the full MMSE, and the performance gain go towards zem. It also seen that the performance of the semi-blind detector is better than the non-blind MMSE
already after 50 bits, whereas the blind detector requires 100 bits.
The performance of the semi-blind algorithm using F2 is almost indistinguishable from the performance using SVD, and is therefore di~eult to see in the figures. The performance of the semi-blind algorithm using (16) is slightly worse than the one using (17). From other simulations we have made, this seems to be a general behavior. Since (17) has much less complexity than (16) for simultaneous detection, ( 17) must be prefered.
The performance gain of course depends on many factors: SNR, user powers, the codes used, and the number of users.
Figure shows the performance when the number of unknown users is increased to K =10 users. Of these, 4 have the same power as the known users, while the 6 others have power -6dB. As expected, the performance gain of the semi-blind detector over the blind is less than for Figure~~ Basically, when the number of unknown users is high, the information on known users does help relatively less. However, the gain is still around 2dB.
It can also be seen that the performance of both blind and semi-blind detectors is worse than the non-blind detector in the beginning. In this case, the semi-blind detector has a further advantage to the blind detector: the non-blind part can be used alone until the blind part has converged. This cannot be done for the blind detector. An even better way is to replace (14) with the following formula b(n)=s~~TSr~(sT -X(n)T(n)Aa'(n)Uf (n))rnJ
where OSx(n)sl is a weighting function with x(Or-0 and x(aor 1. The optimal (or even a good) way of choosing x(n) is a subject for further study. However, some preliminary tests show that this gives a superior performance in the beginning.
_ K
The previous results were under the assumption that K was known or estimated without errors. In Figure'B the same data g is used as for Figure, i.e., K =10 , but the value of K used for the estimators is set to 4, i.e., equal to the number of high power users. Remarkably, the performance of both the blind and the semi-blind algorithms improve, at least for a small number of bits, and in addition, the gain of the semi-blind algorithm is increased, now again ranging between 2 and 4 dB. Two conclusions can be drawn 1. The estimators, both blind and semi-blind, are relatively insensitive to estimation errors on K .
2. The performance is better if only the largest eigem~alues are used in the subspace algorithms, rather than the full interference subspace.
Item 2 is in particular important. This can be interpreted so that it pays off to track only a few high power unknown users.
In a cellular system, this means that it is advantageous to track only the out-of-cell users on the boundary of the cell considered, while the interference from other users inside neighboring cells are considered background noise because of path loss. This means that the number of users (or, rather, eigenmodes) that need to be tracked in the semi-blind algorithms can be quite low, and low-complexity subspace tracking algorithms such as PASTd could be employed, resulting in a very low complexity system.
This conclusion is only true for a small number of bits. As the number of bits go towards infinity, the blind and semi-blind algorithm with the correct value of K should converge towards the full MMSE
detector, which is not true if the estimated value of K is too small. Thus, in a dynamic environment, the best value of K depends not only on the number of users and their power, but also on the time scale for change in the environment.
In order to find the performance in terms of bit error rate (BER) versus SNR a difi'erent simulation scenario was used. Three detectors were considered: the non-blind MMSE detector, the blind MMSE
detector, and the semi-blind hybrid detector implemented using (17), with the latter two implemented using SVD for a fixed block size of 200. The data was generated in the following way:
the number of known users was K--7, and the number of unknown users K = 4 . An ensemble of 20 different code matrices was generated. For each code matrix, 100 different ensembles of 10,000 bits were generated. For each ensemble of 10,000 bits, the detectors were estimated over the 200 first bits, and the BER was calculated over all 10,000 bits, and averaged over the 100 different ensembles (i.e., the total number of bits were 1,000,000). Thus, for each SNR
value, 20 code matrices x 7 users = 140 ensembles were available. For these, the median and 90-percentile (i.e., the 14'~ worst) BER were calculated, shown in Figure ~. This gives a reasonable impression of performance, but does not completely reveal the performance of the detectors. By plotting each of the 140 ensembles and observing them manually, we made the following observations for high SNR (> 10 dB) ~ The semi-blind detector always had a lower BER than the blind detector ~ The blind detector was in most cases better than the non-blind detector. In some cases, the non-blind detector was somewhat better (up to 1 dB).
~ The semi-blind detector was almost always better than the non-blind detector. In a few cases the non-blind detector was better, but then only insignificantly (0.1-0.2 dB).
qD
~ In some cases the non-blind detector had a very poor performance (which is reflected in the 90-percentile in Figure ~). The reason must be that there is an unknown user with a code very close to a known user. Thus, even a few high power out-of-cell users can corrupt the performance of a non-blind multi-user detector in a cellular system.
The last observation is reason enough that the non-blind detector should not be used when there is inter~ell interference. Since out-of-cell users cannot be controlled the same way that in-cell users can, this last case could happen, and as the worst~case performance essentially determines the quality of service, this would not be acceptable.
VIL CONCLUSION
In this paper we have developed a class of new multi-user detectors, called semi-blind linear detectors. They are distinguished by the fact that they can cancel interference from both known and unknown users, like blind detectors, while simultaneously using the knowledge of known users. Simulations have shown that the semi~lind detectors have notably better performance than both the pure blind detectors and the non-blind detectors, and furthermore they have a considerably lower computational complexity.
Our simulations have also shown that the interference from even a few out-of-cell users can totally corrupt the performance of a non-blind multi-user detector. This problem can e~ciently be handled by the semi-blind detector.
APPENDIX A: PROOF OF THEOREM 2 We will approach the proof constructively. Denote in the following by w, the weight vector for the decorrelating detector for user 1, and let v, be defined by vt =Wat.W =Sr'rSl 2Sr (see also (5), and the discussion), and consider the subspace V = spanQUs vt D. Notice that U J S = 0 and Ul vt = 0 by the definition of U~ . We must have w, a ~, sinoe~ is orthogonal to all interference by the other known users. Thus, wt = Lil. vt Ft for some vector ct. As in the proof of proposition 1 of [14], we now see that we can write Ct =argmincr UT J(SAZSr +SAZSr~Ut vl~
c V1 __ T D UsSAZSrvt argmmc Cv; SA2SrU, vi ~SA2Sr +SAZSr~'t~
c 511bjeCttO wj S~ =1, With D=As _o.2lx Now notice that Pt ~ Ua Rvt = U; ~.SAZSr +SAZSr +Q2Iwt = Us SAZSry and v~ Rvt = vi AZSr +SAZSr +QZI}y =v~ AZST +SAZSr}vt +QZV;
Thus, D
c, =argmincT T vrRvP' QZVz CPt t t t Following the proof of [ 14J proposition 1, we then find that wt = 1 Mvi v~ Mvt with M=~U v D P~ ~ iJf s 1 T vTRv -~2v2 Here we apply a matrix inversion theorem (see, e.g., (15], sedion 2.9) to get Or 0 M ~UJ v~ CD ' OJ
1 D ~P~Pi D 1 -D ~P~ Us vi Rv~ -pi D-~P~ -~ZVi -(D 'Py 1 vi To calculate Mvl notice that since Ul v~ = 0 in fad only the last row of the matrix inside { } is needed, which yields Mv, =~U, v~ v~ -D- p~
v~ Rvl -p~ D Cpl -o~ZV~ ~ 11 2 _ - '_ wt -UdD ~Py vi Rv~ 'Pi D 1P~ -~ZVi Thus, wi r =xv; (I-RUsD 'Us for some positive constant ~ Combining the solutions for the different users, we obtain (13).
APPENDIX B: PROOF OF THEOREM 4 The proof follows the proof of theorem 2 closely. Thus, at first we see that we can write wl _ LUs vu For some vector c,. By (10) we now have, except for a scaling constant -i c~ - T R U, vl r y _ ~U~ ~ ( 1 ~U=
v~ v .~, _ r _ -' r wi = ~U~ y CUT JRLUr vy CUr ~s~
v, v, Similarly to the proof of theorem 2, we see that s R[U v ]= As P~
v s ~ PT vi Rv~
We can now follow the same steps as the proof of theorem 2 with D replaced by As and v; Rv, -~ZV; replaced by v; Rv, , arriving at ( 17).
We will therefore consider the orthogonalization approach. The problem with the traditional matched filter solution is that the vectors st,si,...,sx are not orthogonal. However, since sl,sZ,...,sK are linearly independent, there exists some inner product in Ru with respect to which al,s2,...,sx are orthonormal. Specifically, there exists some (positive semi definite) matrix W a R"f ""'f so that sT Ws~ = 8;,~ . Since s,, sZ,...,sx are not a basis of R"~ the matrix W
is not unique, but two solutions are W =S~STsr2ST (4) and Ww = SA~ASTSA~2 ASr , which gives w, = Ws, (5) or rl y = si Wwsi Wwsi .
The conditions (1) are satisfied (seen most easily be noticing that ST WS = I
, and that scaling of vectors do not influence orthogonality). This solution is in fact equivalent to the "traditional"
decorrelating detector [2] , since S r W r = ~S r Sr ~ S r r . The solution fiuiher has the advantage that W is independent of the user oonsider~ed and that it is a low rank matrix, which makes the transformation efficient to calculate.
To estimate W (or rather W,~ we can make use of the following. The eigenvalue decomposition of R can be written as [14]
r R ~ITs LT"~ 0' Q I jjT
where Ua is an orthonormal basis of the signal subspace spanned by S and U" is a basis for the noise subspace orthogonal to Ua. The diagonal matrix Aa contains the K largest eigenvalues of R, while ~ is the minimum eigenvalue. We then have, by examining the proofs in [14]
Proposition 1 Ww = Ua(Aa -QZI) I Us Thus, as R can be estimated from the received data only, Ww can be estimated by EVD, and the orthogonalizing decorrelating detector can be implemented blindly [14].
B. The MMSE detector The minimum mean square error (MMSE) detector is defined as the detector that minimizes the mean square error (MSE):
we _ ~~~(~i -wTr Equivalently, the solution is given as the vector that satisfies Elr{b; -w~ r~=0 which results in the equation Rw; = A;s;
Thus, we can write b, = sgn(s T R-t r~
(~ (8) b=s~~rR tr~
Notice that this detector can be implemented blindly by replacing R by a time-averaged estimate [18]. Using the eigenvalue decomposition (6), this can also be written as j!i =Sgn(S; UaAs1 Ua r!
1 (9) b =s~~T UaAalUa rl which is the idea of the subspace method in [14].
Now suppose that w, is constrained to lie in some subspace ?l, let u,, u2,...,um be an orthonormal basis of ?l, and let U = ~u, u2 ~ ~ ~ u,"~. We can then write w; = Uci for some vector c;.
Inserting this to (7) gives e; =~~UrRUrtUr9a (10) if P is the projection onto ?t we can also find w; as ct = AOP~~tPsr where t denotes pseudo-inverse (which is needed here, since PRP does not have full rank).
IV. SEMI-BLIND DETECTORS
A. Semi-blind decorrelating detectors In our first approach to semifilind decorrelating detectors, we use a mixed projection/orthogonalization approach, as described in section IILA The idea is to first project the received data r on the subspace orthogonal to span(St ) , with S, given by (2). Thereby all interference from the larown users has been removed. The orthogonalization approach is then used in span(S t ) 1. The calailarions can be done as follows. Let P, be defined by (3), and define S, =
P; .C"S. st ~, A, _ ~A A, ~ and (. r Wt =StAytSt StAtr AtSt We can write the eigenvalue decomposition of Rt = P~ RPi as As 0 0 UJ
R, _ ~U, U" Uo 0 ~zI 0 U"
0 0 0 Uo Here U, is an orthonormal basis of the signal subspace, which in this case is spanned by CS s, ~, while U" is the noise subspace. The matrix Uo is an orthonormal basis of the space spanned by St. Furthermore, Aa = diag(.tt , ~li,...,~.K+t ) with .i; > 0-2 . Similarly to the blind deoorrelating detector we now have that Wt =U.~.=-~ZIx+trtU= (I1) We can then state the semi-blind deoorrelating detector as Theorem l: The deoorrelating detector is given by wt = r Wtst (l2) si Wtsi In our setting, S; is unknown. Equation (11), however, outlines a method for estimating W,, and the method can therefore be implemented without knowledge of the unknown users' codes.
The problem of the above method is that W, is specific to user 1. It can be used with advantage in the case where some users' codes are known, but still only one user's data is of interest, as for example in a mobile station. However, if several users are to be detected simultaneously, an SVD/subspace tracking has to be done for each user, and the method is ine~cient.
We will therefore develop a method that requires only one SVD common to all users. This method will be based on the orthogonalization approach. Fiat, define the projection on span(S) , P =s~Tsr~sT
with Pl = I - P . The eigenvalue decomposition of P 1 RP 1 , is then given by At 0 0 Ua P1RP1 =[Us U" Uo 0 QZI 0 U"
0 0 0 Uo where As = diag(.i,,,.iz,...,.iK ) with ~.; > a'2 , and Uo has K rows.
The idea in the semi-blind detector is first to apply the projection P1 to the received data. Thereby the signal from known all users have been eliminated. The subspace of the unknown users, projected to span(S)1, is then found. This is span~U, ~. The last step is to combine this subspace with the code of each user through some algebraic manipulations. For details, see the proof of Theorem 2.
We can now state the semi-blind decorrelating detector as follows Theorem 2: The decorrelating detector is given by b=sg~~.STSrIST~I-RIJf~AJ -v2Ir~Ua JrJ (13) Proof see Appendix A.
Notice that all quantities in the theorem can be estimated with knowledge of only the known users' codes. Notice also that although R appears explicitly, R itself does not have to be estimated. Define T(n) = ST R(n)Us (n) where R(n) etc. denote the estimated quantities over n samples. Then we can rewrite (13) as b(n)=s~~TS~ tI ST -T(~)(Ar(n)-~~I,Us (n))r~, (14) The quantity T(n) can be calculated as follows, T(n)=Sr 1 L.rlrt Us(n) n f=I
_l ~ 1 ~, ~Srrt ~U; (n)ri)T ~ (IS) ~=t ~~Trr~'i Us (n) n .-_t The implementation using the formula on the second line only roquires KK
multiplications per bit, instead of the MZ multiplications to calculate R On the other hand this formula is not directly suitable for recursive implementation, since Us (n) has to be applied to all previous samples. The formula on the third line requires KM
multiplicadons, and can be implemented recursively as long as S is time-invariant.
B. Hybrid Semi-Blind Detectors We will in this section consider detectors that are combinations of the decorrelating detector and the MMSE detector. Specifically, we will study detectors that are decornelating among the known users and MMSE
with respect to the unknown users. We will call this class of detectors hybrid semi-blind detectors. The idea is similar to ideas used in array processing: to direct nulls in the direction of known interferers, and find the MMSE solution for the remaining interference, and has in this ~ntext also been studied for mufti-user detection [19,20].
As for the decorrelating detector, we will give two solutions for the hybrid detector: one that makes a subspace calculation for each user, and one that makes a common subspace calculation. For the former case we get the following Theorem 3. The notation is the same as for Theorem 1.
Theorem 3: The hybrid semi-blind detector is given by wl = s; UsAfIUsSt UfAsIUss~ (16) where Us and A~ is given from the eigenvalue decomposition of Ri = P~ RP~l Aj 0 0 Us R, _ ~Ua U" Uo 0 ~2I 0 U
0 0 0 Ua where As =diag(.i,,.'tz,...,~.K+~) with ~,; >a~2 This detector is similar to the detector of [19], but is here stated in a subspace context.
For the case of common subspace tracking we get, with the same notation as for Theorem 2 Tfreorem 4: The hybrid semi-blind detector is given by b =sg~~SrSr~ST(I-RU,As~U; )rJ (17) Proof-. see Appendix B.
We notice that the hybrid semi-blind detector (17) is very similar to the semi-blind decorrelating detector (13). The only difference is that As -6~I is replaced by A, . As for (17), R itself does not have to be estimated, as the quantities needed can be calculated using (15).
In general, the MMSE detector has superior performance to the decorrelating detector [10], and it would therefore be natural to study pure MMSE semi-blind detectors. However, the hybrid approach has a number of advantages. For the known users, the decorrelating detector does not need a~ estimated information, whereas the MMSE solution requires estimates and tracking of user powers. On the other hand, the deoornelating detector for the unknown users requires a noise estimate and is therefore rather dependent on an accurate estimate of the number of users, while the MMSE
detector does not have these shortcomings. Finally, the semi-blind hybrid detector has a compact, simple expression (17), while it seems impossible to derive the true MMSE detector by the methods used here.
One way to make ( 17) closer to a true MMSE solution, is to replace it with the following expression b=sgnC(STS+QZA-ZrIST(I-RUtAs~Us)r~ (18) The argument for (18) is heuristic. We first find the MMSE solution in the subspace spanned by the known users, ignoring all interference from unknown users. For each user, we then find the MMSE solution in the subspace spanned by this solution and iJ s .
This solution can avoid the amplification of noise that ~uld happen in ( 17), but it is not an NIIviSE solution and does not satisfy any orthogonality or IvFvISE conditions.
V. SUBSPACE TRACKIrIG
A large number of different subspace tracking methods exist, with varying convergence speed and complexity, and most of them can be applied to the current problem, but with some modifications. To illustrate the principles we have chosen to use the F2 algorithm ( 16], summarized in Tabte 1. We use a standard exponential window with It(n) _ .ZR(n -1) +r"r~
Table I: The F2 algorithm [16] for tracking a d-dimensional subspace.
Let the data vecxors be v~, v2,...
U(0) arbitrary m x d matrix, U(0)T U(0) = I
E(0) = I.
For n = 1,2,3,...
Define W(n)_~U(n-1)E(n-1) v"~
Compute the m x d and d x d matrices, U(n), E(n) from the SVD:
~U(n) u~~~n) ~~Y(n)T = W (n) For the semi-blind algorithms (16) and (12) the input to the F2 algorithm for user 1 are the vectors V n = Pl rn The subspace dimension is d = K +1. The matrix U,(n) is directly the output U(n) of the F2 algorithm, while Ad(n) = E(n)2 .
For the semi-blind detectors (17) and (13), the input to the F2 algorithm is Vn =Plrn and the subspace dimension is d = K . Again, the subspace information can be obtained directly from the output of the F2 algorithm with the matrix Us(n) equal to U(n) and Aj(n) = E(n)Z . However, also T(n) (equation (15)) has to be calculated to implement the algorithms. As stated below equation (15) this formula on the second line is not suitable for recursive implementation. We will here give some approximate formula for a recursive implementation, that can also be used when S is time-varying.
With the weighting factor included, T(n) can be rewritten as T(n)=~~° ~(Srrr~Us(~)ryr (19) The level of approximation used in evaluating (19) should correspond to the level of approximation used in the specific subspace tracking algorithm used, so that there is an agreement in convergence speed and complexity between the subspace tracking algorithm and the update formula for T(n). Thus, a general update formula cannot be given, and we will look at two examples. The first possibility is to replace U~(n) with i1,(i) inside the summation in (19), to get T(n) ~ ~ ~~f ~rrt ~Us (l)rt \r r= J~
~., r =~,T(n_1)+Prrn'lUi (n)r~J
This approximation is essentially the same as the key approximation used in deriving PASTd, and gives good performance with this algorithm. However, for a fast converging algorithm such as F2, we lave found this does not result in good convergence. Another approximation is T(n) ~ ~.T(n-1)Us (n-1)Us(n)+Srr"(i1; (n)r"~r (20) The justification of this approximation is geometrical. The quantity Us (n -i)Tr (n -1) represents the orthogonal projection of R(n-I~Sr onto the subspace spanned by Ua(n-1). To find the projection of R(n-1)Sr onto the subspace spanned by Us(n) we should reproject R(n -1)Sr . Instead we project the previous projection. If the subspace does not change too much from n to n+l, the approximation is only slight. Notice also that this approximation is similar to the key approximation in deriving F2 (equation (27) in [16]) The calculation of Us (n -1)Us (n) is complex in itself. Fortunately, this can be obtained as an intermediate result in the FZ
algorithm. Notice that since ~U(n) u~ ~U(n) u~= I (Table i), we have C~n) O~YrC~n_1) 0~_~
0 a ~
~F(n_1) U~_ _ ~U(n) u~ W(n _ ~~U(n)r U(n -1) ~~
where ~ indicates elements with values irrelevant to the result. Thus, LTs (n-1)IJs(n) can be found from the upper Left K x K
matrix of the above matrix. The total multiplication count for this approach is KK z + 2K , which compares well with the multiplication count of 3MK Z +MK for the F2 algorithm itself. However, if K
is large, the formula in the second line of equation ( 15) has less complexity.
VL SIMULATION STUDIES
We consider a system with K--7 users with known codes, all with the same power, and a variable number K of users with unknown codes. The users are assigned purely random codes of length M=31, and the SNR
is 20 dB. An ensemble of 50 different random code assignments is generated, and the median signal to inference and noise ratio (SINK) is calculated over all code choices and users, in total an ensemble of 350. The SINK for a user is calculated using a moving average filter of length 20.
In all cases we consider 8 different detectors:
single user The oom~entional single user detector. This is the lower bound on performance of any mufti-user detector.
Full The hypothetical non-blind MMSE mufti-user detector that knows all codes.
This is the upper bound of performance for any linear mufti-user detector.
non-blind The nonfilind MMSE detector that uses only the known codes and ignores the interference from the unknown users.
Direct The blind MMSE detector calculated using direct inversion of the correlation matrix (equation (8)).
Blind The blind MMSE detector calculated using (9) and using SVD for subspace calculation.
Semi-blind SVD The semi-blind detector calculated using (17) and using SVD for subspace calculation.
Semi-blind 2 The stmt-blind detector calculated using (16) and using SVD for subspace calculation.
Semi-blind F2 The semi-blind detector calculated using (17) and using the F2 algorithm for subspace calculation with T(n) calculated using (20).
Decorrelating detectors are not considered, since they in general have an inferior performance and are sensitive to estimation errors. Performance of the dacorrelating detectors can be seen in [21]. For the F2 algorithm, it was initialized with an ordinary SVD over the first 50 samples, i.e., U(0) and E(0) in Table 1 was initialized from the SVD over the first 50 samples.
Without this initialization, the F2 algorithm diverged in the beginning, until it would start converging.
QA
Figure'l4shows the convergence with K = 4 unknown users, all of the same power as the known users. It can be seen that the semi-blind detector has a performance gain over the blind detector starting at 4 dB and decreasing to 2 dB. As the number of bits increase both detectors should converge towards the full MMSE, and the performance gain go towards zem. It also seen that the performance of the semi-blind detector is better than the non-blind MMSE
already after 50 bits, whereas the blind detector requires 100 bits.
The performance of the semi-blind algorithm using F2 is almost indistinguishable from the performance using SVD, and is therefore di~eult to see in the figures. The performance of the semi-blind algorithm using (16) is slightly worse than the one using (17). From other simulations we have made, this seems to be a general behavior. Since (17) has much less complexity than (16) for simultaneous detection, ( 17) must be prefered.
The performance gain of course depends on many factors: SNR, user powers, the codes used, and the number of users.
Figure shows the performance when the number of unknown users is increased to K =10 users. Of these, 4 have the same power as the known users, while the 6 others have power -6dB. As expected, the performance gain of the semi-blind detector over the blind is less than for Figure~~ Basically, when the number of unknown users is high, the information on known users does help relatively less. However, the gain is still around 2dB.
It can also be seen that the performance of both blind and semi-blind detectors is worse than the non-blind detector in the beginning. In this case, the semi-blind detector has a further advantage to the blind detector: the non-blind part can be used alone until the blind part has converged. This cannot be done for the blind detector. An even better way is to replace (14) with the following formula b(n)=s~~TSr~(sT -X(n)T(n)Aa'(n)Uf (n))rnJ
where OSx(n)sl is a weighting function with x(Or-0 and x(aor 1. The optimal (or even a good) way of choosing x(n) is a subject for further study. However, some preliminary tests show that this gives a superior performance in the beginning.
_ K
The previous results were under the assumption that K was known or estimated without errors. In Figure'B the same data g is used as for Figure, i.e., K =10 , but the value of K used for the estimators is set to 4, i.e., equal to the number of high power users. Remarkably, the performance of both the blind and the semi-blind algorithms improve, at least for a small number of bits, and in addition, the gain of the semi-blind algorithm is increased, now again ranging between 2 and 4 dB. Two conclusions can be drawn 1. The estimators, both blind and semi-blind, are relatively insensitive to estimation errors on K .
2. The performance is better if only the largest eigem~alues are used in the subspace algorithms, rather than the full interference subspace.
Item 2 is in particular important. This can be interpreted so that it pays off to track only a few high power unknown users.
In a cellular system, this means that it is advantageous to track only the out-of-cell users on the boundary of the cell considered, while the interference from other users inside neighboring cells are considered background noise because of path loss. This means that the number of users (or, rather, eigenmodes) that need to be tracked in the semi-blind algorithms can be quite low, and low-complexity subspace tracking algorithms such as PASTd could be employed, resulting in a very low complexity system.
This conclusion is only true for a small number of bits. As the number of bits go towards infinity, the blind and semi-blind algorithm with the correct value of K should converge towards the full MMSE
detector, which is not true if the estimated value of K is too small. Thus, in a dynamic environment, the best value of K depends not only on the number of users and their power, but also on the time scale for change in the environment.
In order to find the performance in terms of bit error rate (BER) versus SNR a difi'erent simulation scenario was used. Three detectors were considered: the non-blind MMSE detector, the blind MMSE
detector, and the semi-blind hybrid detector implemented using (17), with the latter two implemented using SVD for a fixed block size of 200. The data was generated in the following way:
the number of known users was K--7, and the number of unknown users K = 4 . An ensemble of 20 different code matrices was generated. For each code matrix, 100 different ensembles of 10,000 bits were generated. For each ensemble of 10,000 bits, the detectors were estimated over the 200 first bits, and the BER was calculated over all 10,000 bits, and averaged over the 100 different ensembles (i.e., the total number of bits were 1,000,000). Thus, for each SNR
value, 20 code matrices x 7 users = 140 ensembles were available. For these, the median and 90-percentile (i.e., the 14'~ worst) BER were calculated, shown in Figure ~. This gives a reasonable impression of performance, but does not completely reveal the performance of the detectors. By plotting each of the 140 ensembles and observing them manually, we made the following observations for high SNR (> 10 dB) ~ The semi-blind detector always had a lower BER than the blind detector ~ The blind detector was in most cases better than the non-blind detector. In some cases, the non-blind detector was somewhat better (up to 1 dB).
~ The semi-blind detector was almost always better than the non-blind detector. In a few cases the non-blind detector was better, but then only insignificantly (0.1-0.2 dB).
qD
~ In some cases the non-blind detector had a very poor performance (which is reflected in the 90-percentile in Figure ~). The reason must be that there is an unknown user with a code very close to a known user. Thus, even a few high power out-of-cell users can corrupt the performance of a non-blind multi-user detector in a cellular system.
The last observation is reason enough that the non-blind detector should not be used when there is inter~ell interference. Since out-of-cell users cannot be controlled the same way that in-cell users can, this last case could happen, and as the worst~case performance essentially determines the quality of service, this would not be acceptable.
VIL CONCLUSION
In this paper we have developed a class of new multi-user detectors, called semi-blind linear detectors. They are distinguished by the fact that they can cancel interference from both known and unknown users, like blind detectors, while simultaneously using the knowledge of known users. Simulations have shown that the semi~lind detectors have notably better performance than both the pure blind detectors and the non-blind detectors, and furthermore they have a considerably lower computational complexity.
Our simulations have also shown that the interference from even a few out-of-cell users can totally corrupt the performance of a non-blind multi-user detector. This problem can e~ciently be handled by the semi-blind detector.
APPENDIX A: PROOF OF THEOREM 2 We will approach the proof constructively. Denote in the following by w, the weight vector for the decorrelating detector for user 1, and let v, be defined by vt =Wat.W =Sr'rSl 2Sr (see also (5), and the discussion), and consider the subspace V = spanQUs vt D. Notice that U J S = 0 and Ul vt = 0 by the definition of U~ . We must have w, a ~, sinoe~ is orthogonal to all interference by the other known users. Thus, wt = Lil. vt Ft for some vector ct. As in the proof of proposition 1 of [14], we now see that we can write Ct =argmincr UT J(SAZSr +SAZSr~Ut vl~
c V1 __ T D UsSAZSrvt argmmc Cv; SA2SrU, vi ~SA2Sr +SAZSr~'t~
c 511bjeCttO wj S~ =1, With D=As _o.2lx Now notice that Pt ~ Ua Rvt = U; ~.SAZSr +SAZSr +Q2Iwt = Us SAZSry and v~ Rvt = vi AZSr +SAZSr +QZI}y =v~ AZST +SAZSr}vt +QZV;
Thus, D
c, =argmincT T vrRvP' QZVz CPt t t t Following the proof of [ 14J proposition 1, we then find that wt = 1 Mvi v~ Mvt with M=~U v D P~ ~ iJf s 1 T vTRv -~2v2 Here we apply a matrix inversion theorem (see, e.g., (15], sedion 2.9) to get Or 0 M ~UJ v~ CD ' OJ
1 D ~P~Pi D 1 -D ~P~ Us vi Rv~ -pi D-~P~ -~ZVi -(D 'Py 1 vi To calculate Mvl notice that since Ul v~ = 0 in fad only the last row of the matrix inside { } is needed, which yields Mv, =~U, v~ v~ -D- p~
v~ Rvl -p~ D Cpl -o~ZV~ ~ 11 2 _ - '_ wt -UdD ~Py vi Rv~ 'Pi D 1P~ -~ZVi Thus, wi r =xv; (I-RUsD 'Us for some positive constant ~ Combining the solutions for the different users, we obtain (13).
APPENDIX B: PROOF OF THEOREM 4 The proof follows the proof of theorem 2 closely. Thus, at first we see that we can write wl _ LUs vu For some vector c,. By (10) we now have, except for a scaling constant -i c~ - T R U, vl r y _ ~U~ ~ ( 1 ~U=
v~ v .~, _ r _ -' r wi = ~U~ y CUT JRLUr vy CUr ~s~
v, v, Similarly to the proof of theorem 2, we see that s R[U v ]= As P~
v s ~ PT vi Rv~
We can now follow the same steps as the proof of theorem 2 with D replaced by As and v; Rv, -~ZV; replaced by v; Rv, , arriving at ( 17).
Claims (3)
1. A method of reducing interference in a CDMA communications system, in which a code is assigned to each mobile station in the CDMA communications system, the method comprising the steps of:
A) receiving signals at a base station from a plurality of mobile stations, where codes of at least two and not all of the mobile stations are known;
B) processing the received signals to reduce interference between signals for mobile stations the codes of which are known; and C) processing the received signals to reduce interference between signals from mobile stations whose codes are unknown.
A) receiving signals at a base station from a plurality of mobile stations, where codes of at least two and not all of the mobile stations are known;
B) processing the received signals to reduce interference between signals for mobile stations the codes of which are known; and C) processing the received signals to reduce interference between signals from mobile stations whose codes are unknown.
2. The method of claim 1 in which step C comprises estimating a basis for the subspace defined by the codes of the mobile stations whose codes are known.
3. The method of claim 1 in which step B comprises estimating a basis for the subspace defined by the codes of the mobile stations whose codes are known.
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US7116271B2 (en) * | 2004-09-23 | 2006-10-03 | Interdigital Technology Corporation | Blind signal separation using spreading codes |
| WO2007037634A2 (en) * | 2005-09-28 | 2007-04-05 | Lg Electronics Inc. | A method and apparatus for mitigating multiuser access interference |
| CN116318690A (en) * | 2023-05-25 | 2023-06-23 | 山东大学 | Safe outsourcing calculation method and system based on matrix singular value decomposition |
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Cited By (9)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US7116271B2 (en) * | 2004-09-23 | 2006-10-03 | Interdigital Technology Corporation | Blind signal separation using spreading codes |
| WO2006034425A3 (en) * | 2004-09-23 | 2006-12-07 | Interdigital Tech Corp | Blind signal separation using spreading codes |
| EP1792197A2 (en) * | 2004-09-23 | 2007-06-06 | Interdigital Technology Corporation | Blind signal separation using spreading codes |
| EP1792197A4 (en) * | 2004-09-23 | 2007-09-19 | Interdigital Tech Corp | Blind signal separation using spreading codes |
| KR100847502B1 (en) | 2004-09-23 | 2008-07-22 | 인터디지탈 테크날러지 코포레이션 | Blind signal separation using spreading codes |
| WO2007037634A2 (en) * | 2005-09-28 | 2007-04-05 | Lg Electronics Inc. | A method and apparatus for mitigating multiuser access interference |
| WO2007037634A3 (en) * | 2005-09-28 | 2008-05-02 | Lg Electronics Inc | A method and apparatus for mitigating multiuser access interference |
| CN116318690A (en) * | 2023-05-25 | 2023-06-23 | 山东大学 | Safe outsourcing calculation method and system based on matrix singular value decomposition |
| CN116318690B (en) * | 2023-05-25 | 2023-08-15 | 山东大学 | Safe outsourcing calculation method and system based on matrix singular value decomposition |
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