Ping-pong effects in linear parallel interference ... - IEEE Xplore
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 2, NO. 2, MARCH 2003
357
Ping-Pong Effects in Linear Parallel Interference Cancellation for CDMA Lars K. Rasmussen, Senior Member, IEEE, and Ian J. Oppermann, Member, IEEE
Abstract—In this paper, the convergence behavior of linear parallel interference cancellation is investigated. Especially the so-called ping-pong effect, where the bit-error rate performance is found to oscillate between two different convergence patterns is studied in detail. This effect is shown to be a direct consequence of the extreme eigenvalues of the correlation matrix, allowing for an analytical approach. Intervals for the dominating eigenvalues within which ping-pong effects can occur are specified and illustrated by examples. It is shown that the decision statistic for traditional parallel cancellation will always exhibit oscillating behavior with either short or long codes. Relaxation factors, leading to weighted cancellation, are shown to be effective for alleviating oscillations and ping-pong effects at the expense of convergence rate. Asymptotic analysis for large systems is applied to uncover the convergence behavior for long code systems. Index Terms—Code-division multiple access (CDMA), interference cancellation, multiuser detection.
I. INTRODUCTION
M
OBILE communication systems based on code-division multiple access (CDMA) is inherently subject to multiple-access interference (MAI), as it is virtually impossible to maintain orthogonal spreading codes in a mobile environment. MAI limits the capacity of the conventional detector [1], as well as leading to strict power control requirements to alleviate the near–far problem. The linear maximum-likelihood (ML) detector, denoted the decorrelating detector [2] has been suggested as a possible multiuser detector. In an asynchronous system, however, approximating techniques must be used as the required matrix inversion for the linear detector is generally too complex and introduces intolerable detection delays [3]. It has been shown in various papers that linear interference cancellation is a direct implementation of iterative techniques for matrix inversion [4]–[7]. Such techniques can efficiently
Manuscript received April 10, 2001; revised October 3, 2001; accepted October 13, 2001. The editor coordinating the review of this paper and approving it for publication is R. Murch. This work was supported in part by the Southern Poro Communications (Annandale, Australia) and the Swedish Research Council for Engineering Sciences under Grant 217-1997-538. The work of L. K. Rasmussen was supported by the Telecommunication Theory Group (headed by Prof. T. M. Aulin) at Chalmers University of Technology during his research visit at Southern Poro Communications. This paper was presented in part at the IEEE Global Conference on Communication, San Antonio, TX, November 2001. L. K. Rasmussen was with the Department of Computer Engineering, Chalmers University of Technology, Gothenburg, Sweden. He is now with the Institute of Telecommunications Research, University of South Australia, Mawson Lakes SA 5095, Australia (e-mail: [email protected]). I. J. Oppermann is with the Center for Wireless Communications (CWCOulu), University of Oulu, Oulu, Finland (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2003.809122
implement a decorrelator or an minimum mean squared error (MMSE) detector, even for asynchronous systems [3]. Due to the minimal detection delay for parallel interference cancellation (PIC), it is especially attractive. This corresponds to the classic Jacobi iteration which unfortunately is not guaranteed to converge [8]. The computational complexity for direct matrix inversion is , where is the size of the matrix. In comof the order of parison, the computational complexity of iterative techniques is , where is the number of iterations.1 of the order of The required number of iterations for convergence may be large if the eigenvalue spread is high, however, for well-behaved systems the number of iterations may be significantly smaller than . This is in fact the case for most encountered CDMA systems as will be illustrated through numerical examples. A phenomenon commonly observed for linear PIC is the so-called ping-pong effect where the bit-error rate (BER) tends to alternate between two-state processes as a function of the iteration index. For odd and even iteration index, respectively, the BER follows different convergence patterns where the envelope for the odd case is usually better than the envelope for the even case. Only few attempts of explaining this behavior have been offered. In [5], a bias of the decision statistic is claimed to cause the problems while similarities to over and under-damped iterations are indicated in [6] and considered further in [9], and [10]. Here, we will investigate the approach suggested in [9] and [10] in more detail. This approach is inspired by convergence analysis for adaptive filters [11]. In this paper, we first formulate an analytical description of linear PIC based on iterative methods. In the description, we include a relaxation factor which corresponds to a fixed weight in a weighted (partial) cancellation scheme [4]–[7]. The behavior of the decision statistic is investigated based on the eigenvalues of the corresponding iteration matrix. These eigenvalues are closely related to the eigenvalues of the correlation matrix. It is found that the decision statistic in most cases exhibits an oscillating behavior around some fixed point. The oscillating behavior is caused by negative eigenvalues of the iteration matrix which in turn originates from the extreme eigenvalues of the correlation matrix. The alternating polarity leads to a positive and a negative convergence envelope for the decision statistic, describing odd and even iterations, respectively. This oscillating behavior is then related to ping-pong effects. It is shown that large correlator outputs are more likely to lead to alternating decisions, explaining why odd iterations usually have better BER performance than even iterations. A weight factor is suggested which can completely alleviate oscillating 1These
estimates are based on flop counts in [8].
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behavior at the expense of convergence rate. The results are applied to large, long-code systems where the relation between oscillating behavior and convergence rate is investigated. The paper is organized as follows. In Section II, the system model is specified. In Section III, an analytical description of weighted, linear PIC (WLPIC) is given based on an eigenvalue decomposition of the correlation matrix. In this section, the oscillating behavior of the decision statistic is studied in detail. The ping-pong effect is then related to the oscillating behavior in Section IV, while a large system analysis for long code schemes is presented in Section V. Numerical examples are given in Section VI and concluding remarks are found in Section VII.
where is a unit diagonal matrix, is strictly lower left triangular, is strictly upper right triangular, and which is in fact the decorrelator solution. A standard method for matrix inversion is the Jacobi iteration [8]. It is defined as follows: (2) . Equation (2) is in Here, is the iteration index, and fact identical to linear parallel interference cancellation where are used to determine updated estimates from iteration estimates in iteration [5], [13]. This iteration method is however not guaranteed to converge. The concept of over relaxation can be introduced to ensure convergence
II. SYSTEM MODEL
(3)
In this section, we describe the baseband uplink model of the CDMA communication system considered throughout this paper. The uplink model is based on a -user synchronous CDMA system with single-path channels and the presence of additive white Gaussian noise (AWGN) with zero mean and . variance A specific user in this multiuser CDMA system transmits a which is modulated binary information symbol by a length spreading sequence where each element is in . Binary data and chip formats are assumed for clarity. The continuous-time received signal is down converted to baseband, passed through a filter matched to the chip pulse-shape and sampled. The received baseband signal after chip-matched filtering is then
where is the matrix of received spreading codes, is the data symbol vector, and is a vector of independent, identically distributed Gaussian random variables of zero mean and variance . Here, we have assumed no random phase rotation and perfect power control for clarity and notational simplicity. The results do also apply to the more general cases of no or arbitrary power control, as well as for asynchronous and fading multipath systems with random phase rotations. The general model is discussed in more detail in [12]. A minimal set of sufficient is obtained through correlation with the statistics of length received spreading code of the desired user and corresponds algebraically to
Here, is the relaxation coefficient, typically less than one. The Jacobi over-relaxation iteration is identical to weighted linear PIC (WLPIC) [5], [13]. Other iterations can be applied which also correspond to cancellation structures [5]–[7], [3], [14]. In this paper, however, we limit our focus to linear PIC where the ping-pong effect is most pronounced. Based on (3), we can derive a nonrecursive description of WLPIC
(4) is defined above. Here, is known as the where iteration matrix of (3) which leads to (4) as described in general in [8]. We can further rewrite (4) to arrive at
where . This is the polynomial expansion description of WLPIC developed in [16]–[18]. To gain further insight, an eigenvalue decomposition of the correlation matrix is introduced
where where is the matched filter output vector, is the correlation matrix, and is the zero mean Gaussian noise output vector with . covariance matrix are the eigenvalues and corresponding eigenvectors. We can then get the following expression:
III. ANALYTICAL DESCRIPTION It has previously been shown that linear interference cancellation corresponds to iterative methods for solving a set of linear equations [4]. Based on the above system model, the set of equations to be solved is (1)
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Here, , are the eigenvalues of the iteration matrix. Without loss of generality, we consider user 1 in more detail
(5) is element of eigenvector . We can separate this Here, into the desired signal, the MAI and the noise (desired signal)
0
Fig. 1. Intervals for convergence/divergence for 1 and , respectively. Intervals where oscillating behavior will occur are also included.
the desired fixed point is , the solution to the equation system in (1). The second term in (5) then describes the residual error which can either converge to zero or diverge. To investigate the potential oscillating behavior more closely, we therefore condirectly. sider the error term Considering the recursive form in (3), we get
(MAI) (8) Again, considering user 1 (noise)
(6)
refers to element of vector . Here, the notation modes of convergence, It is clear from (6) that there are one for each distinct eigenvalue. In particular, convergence is assured when all modes converge, i.e., (9) for all
. The dominating mode determines the convergence rate. The fastest convergence rate is achieved when the two extreme modes are identical (7) These results are well-known and have previously been presented in [4], [7], [3], [13]. Based on (6), it is observed that if for one or more values of , the decision statistic will experience an alternating behavior between two state processes determined by the corresponding modes of convergence. Depending on the polarity of the set of convergence modes , we can expect to encounter oscillating behavior. Intervals for the dominating modes and eigenvalues for observing oscillaor , the iteration tions are shown in Fig. 1. If , oscillation will occur during will not converge. For divergence while no oscillation will be observed in the diverwas possible. However, this mode cannot be gence if and are both positive. In case , reached since , the scheme will converge smoothly, while for the scheme converges, exhibiting some level of oscillating behavior. This classification is also used in adaptive filter theory [11]. Using the optimal relaxation factor in (7), we see that one and one extreme mode will be in the interval . In this case, oscillations will be in the interval are always encountered in the decision statistic. dimensional vector which A fixed point is defined as a solves the original equation system. As previously discussed,
Similar to the decision statistic, the residual error obviously modes of convergence, each representing a geoexhibits converges as the weighted sum of metric series. Hence, . The time required for exponentials of the form of its initial value is given by each series to reach
However, the overall time constant cannot be expressed in a simple form. Nevertheless, a lower bound is attained by the slowest rate of convergence given by the eigenvalue leading to . An upper bound is found the maximum magnitude of by the equivalent eigenvalue leading to the smallest magnitude [19]. of has maximum magnitude, then we In case have a smooth exponential convergence as shown in Fig. 2. is the dominating mode, then However, when we have alternating polarity with each iteration. The mode still converges exponentially, however, only the absolute value converges smoothly. The term follows a positive envelope for odd iteration indices and a negative envelope for even iteration indices, respectively. This is illustrated in Fig. 2. This converging oscillation can cause the polarity of the decision statistic to alternate and, thus, lead to BER ping-pong. IV. PING-PONG EFFECTS The question remains however, when this oscillating behavior in the decision statistic leads to ping-pong effects in the BER.
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Fig. 2. Illustration of possible convergence envelopes for a positive and a negative dominating term.
(a)
(b)
statistic alternates in polarity between odd and even iterations. In example 1, we see that the fixed point is comfortably clear of the decision boundary. In example 2, however, the fixed point is closer to the decision boundary and the magnitude of the oscillation is larger, both undesirable features. Noise enhancement of the decorrelating solution can bring the fixed point close to the decision boundary and a large starting decision statistic relative to the fixed point leads to large oscillation magnitudes. The starting decision statistic is equal to the matched filter output. Since the conventional detector usually has a BER of less than 0.5, it is more likely that large matched filter outputs corresponds to correct decisions. It follows that even iterations experience more ping-pong errors than odd iterations. This is observed in numerical examples as demonstrated later. Large decision statistics normally considered desirable are, therefore, the cause of potentially undesirable ping-pong effects in WLPIC. The relaxation coefficient can be used to scale the effective eigenvalues such that ping-pong is avoided. The coefficient for all . The largest should be selected such that value to guarantee no ping-pong is then
Fig. 3. Two examples of converging error terms and decision statistics. Example 1 is well-behaved while example 2 exhibits ping-pong effects.
The performance of iteration schemes have previously been investigated based on the mean squared error (MSE), e.g., [13], [14]. Following the approach in [13], the MSE of the WLPIC designed to converge toward the decorrelating detector can be determined as
(10) This is also the value that provides the fastest convergence given that no oscillating behavior is allowed. It should be noted that . Only oscillating behavior can virtually not be avoided for , i.e., an orthogonal system, then oscillations are when not encountered. V. LONG CODE SYSTEMS
It is clear that there is no oscillation in the MSE, regardless of the eigenvalues. This is illustrated by the results in [14]. It is not possible to derive a similar expression for the BER. The error rate can only be determined as shown in [13] and realizations of the [15] as a sum of -functions over all . For large corresponding MAI in each decision statistic systems, however, it is possible to derive an analytical expression which is asymptotically accurate. In [23], it is shown that the decision statistic at iteration for user is asymptotically stays Gaussian distributed as goes to infinity and fixed. As a consequence, the BER can be expressed as a single -function based on moments of the eigenvalues. This is used to obtain the numerical results for asymptotically large systems in Section VI. Considering (9), the potential oscillating behavior of the error term is focused around zero. Two examples are depicted , system at 10 dB. in Fig. 3(a). This is for a with . In The dominating eigenvalue is both cases, we observe oscillation, however, in example 2, the starting value (matched filter output) is more than twice as large as for example 1, leading to excessive oscillation. Fig. 3(b) depicts the decision statistics for the same examples. The fixed and , points in the two cases are respectively. From Fig. 3(b), we observe that the oscillations in example 2 does in fact lead to ping-pong as the decision
All of the above results relate to eigenvalues of the correlation matrix. For long codes, however, a new set of spreading codes is selected for each symbol interval, making it seemingly necessary to retune the relaxation coefficient at each interval. Fortunately, for large systems where and are approaching is finite, the limiting distribution infinity such that of the eigenvalues is deterministic and has finite support [20]. In particular, the limits for the maximum and minimum eigenvalues are and as [21], [22]. As shown in [7], it follows that a large converges only if the load is limited by system with . Furthermore, it was found in [7] that fastest convergence is achieved when (11) In addition to these results, a long code system will exhibit the following behavior: (OD) (OC) (SC)
RASMUSSEN AND OPPERMANN: PING-PONG EFFECTS IN LINEAR PARALLEL INTERFERENCE CANCELLATION FOR CDMA
Fig. 4. Limiting curves describing the combinations of and for which oscillating divergence, oscillating convergence and smooth convergence, respectively, is encountered. Also, the curve for the relaxation factor for fastest convergence is included.
where OD, OC, and SC denote oscillating divergence, oscillating convergence, and smooth convergence, respectively. These limits are illustrated in Fig. 4 together with the optimal relaxation factor of (11). We observe that the optimal factor curve lies within the region of oscillating convergence, resulting in potential BER ping-pong in all cases. The obvious question is now, how much is the convergence rate affected by avoiding oscillating behavior. Consider again the error term from (8)
where error, we get
. Looking at the squared magnitude of the
where denotes the extreme eigenvalue that leads to the max. This approach was first sugimum magnitude of gested in [15]. It can be shown that for large, long-code systems , then . These results have previously been derived in [7], following a different approach. The approach presented here, confirms the results in [7]. We adopt the terminology of [7] and denote the factor as the convergence factor. For the combinations of and describing the three curves in Fig. 4, the upper bounds on convergence rate are shown in Fig. 5. It is clear that there is no point
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Fig. 5. Upper bounds on convergence rate for the two limiting cases of convergence behavior. Also, the case of the optimal relaxation factor is included.
in being close to the limit for convergence as the convergence rate will be slow. It is observed that a relaxation factor chosen to avoid oscillating behavior leads to a small loss of convergence rate. It is, thus, possible to avoid oscillations at some cost of convergence rate. The cost is quantified further in the following section for some numerical examples. VI. NUMERICAL EXAMPLES Here, we consider two examples with , , , at an and an asymptotic case where dB. The first example is a short-code system with and . The relaxation factor is , and , respectively, acchosen to be cording to (7) and (10). The simulated BER is shown in Fig. 6 as a function of the iteration index. Included for reference is the decorrelator performance, the fixed point performance for the it, we have oscillating divergence eration. It is clear that for , we observe the fastest convergence, as expected. For however, we still have a slight oscillating behavior in the BER. , we have smooth convergence as the negaFinally, for tive eigenvalues of the iteration matrix have been eliminated. In this example, the loss of convergence speed is minor as only one as opposed to . extra iteration is required for In Fig. 7, a long-code example is presented. Here, . Again, , and are chosen for illustration. The same behavior as for the short-code example is observed. In this case, however, the loss of convergence speed is more significant. Here, we need ten as opposed to six iterations for iterations for to reach the decorrelator performance. The loss of convergence . speed diminishes as the system gets larger This is demonstrated in Fig. 8, where the asymptotic results presented in [23] is used to derive numerical results for , . This is based on an analytically justified Gaussian approximation which is asymptotically accurate in going to infinity with the load fixed. the limit of
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Fig. 6. Short code example with K = 16, N = 32 at E =N = 8 dB. The extreme eigenvalues are = 2:541 and = 0:1308. Included are the cases of = 1, = and = . The decorrelator performance is included for reference.
!1
Fig. 8. Long code example with = 0:5 and K; N at E =N = 8 dB. Included are the cases of = 1, = and = . The decorrelator performance is included for reference.
At intermediate iterations, the term
becomes a reasonable estimate of the MMSE or , thus providing better performance than the decorrelator. The iterative detector can in fact also be design for implementing the MMSE detector and in this case the structure converges smoothly to the MMSE performance without overshooting [7], [13]. VII. CONCLUDING REMARKS
Fig. 7. Long code example with K = 16, N = 32 at E =N = 8 dB. Included are the cases of = 1, = and = . The decorrelator performance is included for reference.
The analytical expression is based on the moments of the eigenvalues [23]. In all the numerical examples, we observe an overshooting convergence behavior. The performance within some interval of iterations is better than the decorrelator performance before converging smoothly from below. This is a commonly observed phenomenon which is due to the fact that intermediate solution vectors are closer to the MMSE solution or the single user solution than the decorrelator solution. Considering (5), we have
(12)
In this paper, we have presented an analytical description of weighted, linear PIC. The description is based on an eigenvalue decomposition of the correlation matrix. The description is used to investigate convergence characteristics, with special attention toward oscillating behavior leading to ping-pong effects in the BER. It is found that negative eigenvalues of the iteration matrix create oscillations in the decision statistic centered around the fixed point described by the decorrelator output. Large correlator outputs, combined with fixed points close to the decision boundary are, thus, exposed as the main causes leading to ping-pong effects. Intervals for oscillating divergence, oscillating convergence and smooth convergence are specified. A relaxation factor, eliminating oscillating behavior at a minimal cost of convergence rate, is suggested. The loss of convergence rate can be significant for relatively small systems. However, as for a fixed , the loss in convergence rate diminishes. REFERENCES [1] S. Moshavi, “Multiuser detection for DS-CDMA communications,” IEEE Commun. Mag., vol. 34, pp. 132–136, Oct. 1996. [2] R. Lupas and S. Verdú, “Near–far resistance of multiuser detectors in asynchronous channels,” IEEE Trans. Commun., vol. 38, pp. 496–508, Apr. 1990.
RASMUSSEN AND OPPERMANN: PING-PONG EFFECTS IN LINEAR PARALLEL INTERFERENCE CANCELLATION FOR CDMA
[3] P. H. Tan and L. K. Rasmussen, “Linear interference cancellation in CDMA based on iterative techniques for linear equation systems,” IEEE Trans. Commun., vol. 48, pp. 2099–2108, Dec. 2000. [4] H. Elders-Boll, H. D. Schotten, and A. Busboom, “Efficient implementation of linear multiuser detectors for asynchronous CDMA systems by linear interference cancellation,” Eur. Trans. Telecommun., vol. 9, no. 4, pp. 427–437, Sept.–Nov. 1998. [5] R. M. Buehrer, S. P. Nicoloso, and S. Gollamudi, “Linear versus nonlinear interference cancellation,” J. Commun. and Networks, vol. 1, June 1999. [6] D. Guo, L. K. Rasmussen, and T. J. Lim, “Linear parallel interference cancellation in long-code CDMA multiuser detection,” IEEE J. Select. Areas Commun., vol. 17, pp. 2074–2081, Dec. 1999. [7] A. J. Grant and C. B. Schlegel, “Iterative implementations for linear multiuser detectors,” IEEE Trans. Commun., vol. 49, pp. 1824–1834, Oct. 2001. [8] O. Axelsson, Iterative Solution Methods. Cambridge, U.K.: Cambridge Univ. Press, 1994. [9] L. K. Rasmussen, “On ping-pong effects in linear interference cancellation for CDMA,” in Proc. IEEE Int. Symp. Spread Spectrum Tech. and Appl., Parsipanny, NJ, Sept. 2000, pp. 348–352. [10] L. K. Rasmussen and I. J. Oppermann, “Convergence behavior of linear parallel cancellation in CDMA,” in IEEE Global Commun. Conf., San Antonio, TX, Nov. 2001, pp. 3148–3152. [11] B. Farhang-Boroujeny, Adaptive Filter, Theory and Applications. New York: Wiley, 1998. [12] L. K. Rasmussen, P. D. Alexander, and T. J. Lim, “A linear model for CDMA signals received with multiple antennas over multipath fading channels,” in CDMA Techniques for 3rd Generation Mobile Systems, F. Swarts, P. van Rooyen, I. Oppermann, and M. Lötter, Eds. Norwell, MA: Kluwer, Sept. 1998, ch. 2. [13] D. Guo, L. K. Rasmussen, S. Sun, and T. J. Lim, “A matrix-algebraic approach to linear parallel interference cancellation in CDMA,” IEEE Trans. Commun., vol. 48, pp. 152–161, Jan. 2000. [14] M. J. Juntti, B. Aazhang, and J. O. Lilleberg, “Iterative implementation of linear multiuser detection for dynamic asynchronous CDMA systems,” IEEE Trans. Commun., vol. 46, pp. 503–508, Apr. 1998. [15] L. K. Rasmussen, T. J. Lim, and A.-L. Johansson, “A matrix-algebraic approach to successive interference cancellation in CDMA,” IEEE Trans. Commun., vol. 48, pp. 145–151, Jan. 2000. [16] M. Moshavi, E. G. Kanterakis, and D. L. Schilling, “Multistage linear receivers for DS-CDMA systems,” Int. J. Wireless Inform. Networks, vol. 3, pp. 1–17, 1996. [17] D. Guo, “Linear parallel interference cancellation in CDMA,” M. Eng. thesis, National Univ. Singapore, Singapore, 1998. [18] D. Guo and L. K. Rasmussen, “Linear parallel interference cancellation using fixed weighting factors for long-code CDMA,” in Proc. IEEE Int. Symp. Information Theory, Sorrento, Italy, June 2000, p. 332. [19] S. Haykin, Adaptive Filter Theory, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.
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[20] D. Jonsson, “Some limit theorems for the eigenvalues of a sample covariance matrix,” J. Multivariate Analysis, pp. 1–38, Dec. 1982. [21] Z. D. Bai and Y. Q. Yin, “Limit of the smallest eigenvalue of a large dimensional sample covariance matrix,” Ann. Probability, pp. 1275–1294, Dec. 1993. [22] Z. D. Bai, J. W. Silverstein, and Y. Q. Yin, “A note on the largest eigenvalue of a large dimensional sample covariance matrix,” J. Multivariate Analysis, pp. 166–168, 1988. [23] D. Guo, S. Verdú, and L. K. Rasmussen, “Asymptotic normality of linear multiuser detection outputs,” in Proc. IEEE Int. Symp. Information Theory, Washington, DC, Aug. 2001, p. 307.
Lars K. Rasmussen (S’92–M’93–SM’01) was born on March 8, 1965 in Copenhagen, Denmark. He received the M.Eng. degree from the Technical University of Denmark, Copenhagen, and the Ph.D. degree from Georgia Institure of Technology, Atlanta, GA, in 1989 and 1993, respectively. From 1993 to 1995, he was with the University of South Australia, Adelaide. From 1995 to 1998, he was with the CWC at the National University of Singapore, Singapore. From 1998 to 2001, he was an Associate Professor at Chalmers University of Technology, Gothenburg, Sweden. Since February 2002, he is a Professor in Telecommunications at the Institute for Telecommunications Research, University of South Australia, Adelaide, Australia.
Ian J. Oppermann (M’97) was born in Maryborough, Qld, Australia, in 1969. He received the B.Sc., B.E. (electrical), and Ph.D. degrees from the University of Sydney, Australia, in 1989, 1991, and 1997, respectively. Before commencing postgraduate studies, he was with OTC Australia (now Telstra). Since 1996, he has been with the Center for Wireless Communications (CWC-Oulu), University of Oulu, Finland. He was the Founder and Technical Director of Southern Poro Communications, Sydney, Australia, a company that designed and implemented baseband CDMA transceivers and was heavily involved in UMTS network modeling. His research interests include wideband channel modeling, CDMA receiver structures for high-data-rate/low-mobility environments, and complex valued spreading sequences for CDMA systems. He was an Editor of CDMA Techniques for Third-Generation Mobile Systems and Associate Area Editor for Communications on the 1998 CRC Dictionary of Engineering Terms. Dr. Oppermann received the Sydney University Engineering Foundation Prize for Excellence in Teaching in 1995 and 1997
357
Ping-Pong Effects in Linear Parallel Interference Cancellation for CDMA Lars K. Rasmussen, Senior Member, IEEE, and Ian J. Oppermann, Member, IEEE
Abstract—In this paper, the convergence behavior of linear parallel interference cancellation is investigated. Especially the so-called ping-pong effect, where the bit-error rate performance is found to oscillate between two different convergence patterns is studied in detail. This effect is shown to be a direct consequence of the extreme eigenvalues of the correlation matrix, allowing for an analytical approach. Intervals for the dominating eigenvalues within which ping-pong effects can occur are specified and illustrated by examples. It is shown that the decision statistic for traditional parallel cancellation will always exhibit oscillating behavior with either short or long codes. Relaxation factors, leading to weighted cancellation, are shown to be effective for alleviating oscillations and ping-pong effects at the expense of convergence rate. Asymptotic analysis for large systems is applied to uncover the convergence behavior for long code systems. Index Terms—Code-division multiple access (CDMA), interference cancellation, multiuser detection.
I. INTRODUCTION
M
OBILE communication systems based on code-division multiple access (CDMA) is inherently subject to multiple-access interference (MAI), as it is virtually impossible to maintain orthogonal spreading codes in a mobile environment. MAI limits the capacity of the conventional detector [1], as well as leading to strict power control requirements to alleviate the near–far problem. The linear maximum-likelihood (ML) detector, denoted the decorrelating detector [2] has been suggested as a possible multiuser detector. In an asynchronous system, however, approximating techniques must be used as the required matrix inversion for the linear detector is generally too complex and introduces intolerable detection delays [3]. It has been shown in various papers that linear interference cancellation is a direct implementation of iterative techniques for matrix inversion [4]–[7]. Such techniques can efficiently
Manuscript received April 10, 2001; revised October 3, 2001; accepted October 13, 2001. The editor coordinating the review of this paper and approving it for publication is R. Murch. This work was supported in part by the Southern Poro Communications (Annandale, Australia) and the Swedish Research Council for Engineering Sciences under Grant 217-1997-538. The work of L. K. Rasmussen was supported by the Telecommunication Theory Group (headed by Prof. T. M. Aulin) at Chalmers University of Technology during his research visit at Southern Poro Communications. This paper was presented in part at the IEEE Global Conference on Communication, San Antonio, TX, November 2001. L. K. Rasmussen was with the Department of Computer Engineering, Chalmers University of Technology, Gothenburg, Sweden. He is now with the Institute of Telecommunications Research, University of South Australia, Mawson Lakes SA 5095, Australia (e-mail: [email protected]). I. J. Oppermann is with the Center for Wireless Communications (CWCOulu), University of Oulu, Oulu, Finland (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2003.809122
implement a decorrelator or an minimum mean squared error (MMSE) detector, even for asynchronous systems [3]. Due to the minimal detection delay for parallel interference cancellation (PIC), it is especially attractive. This corresponds to the classic Jacobi iteration which unfortunately is not guaranteed to converge [8]. The computational complexity for direct matrix inversion is , where is the size of the matrix. In comof the order of parison, the computational complexity of iterative techniques is , where is the number of iterations.1 of the order of The required number of iterations for convergence may be large if the eigenvalue spread is high, however, for well-behaved systems the number of iterations may be significantly smaller than . This is in fact the case for most encountered CDMA systems as will be illustrated through numerical examples. A phenomenon commonly observed for linear PIC is the so-called ping-pong effect where the bit-error rate (BER) tends to alternate between two-state processes as a function of the iteration index. For odd and even iteration index, respectively, the BER follows different convergence patterns where the envelope for the odd case is usually better than the envelope for the even case. Only few attempts of explaining this behavior have been offered. In [5], a bias of the decision statistic is claimed to cause the problems while similarities to over and under-damped iterations are indicated in [6] and considered further in [9], and [10]. Here, we will investigate the approach suggested in [9] and [10] in more detail. This approach is inspired by convergence analysis for adaptive filters [11]. In this paper, we first formulate an analytical description of linear PIC based on iterative methods. In the description, we include a relaxation factor which corresponds to a fixed weight in a weighted (partial) cancellation scheme [4]–[7]. The behavior of the decision statistic is investigated based on the eigenvalues of the corresponding iteration matrix. These eigenvalues are closely related to the eigenvalues of the correlation matrix. It is found that the decision statistic in most cases exhibits an oscillating behavior around some fixed point. The oscillating behavior is caused by negative eigenvalues of the iteration matrix which in turn originates from the extreme eigenvalues of the correlation matrix. The alternating polarity leads to a positive and a negative convergence envelope for the decision statistic, describing odd and even iterations, respectively. This oscillating behavior is then related to ping-pong effects. It is shown that large correlator outputs are more likely to lead to alternating decisions, explaining why odd iterations usually have better BER performance than even iterations. A weight factor is suggested which can completely alleviate oscillating 1These
estimates are based on flop counts in [8].
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behavior at the expense of convergence rate. The results are applied to large, long-code systems where the relation between oscillating behavior and convergence rate is investigated. The paper is organized as follows. In Section II, the system model is specified. In Section III, an analytical description of weighted, linear PIC (WLPIC) is given based on an eigenvalue decomposition of the correlation matrix. In this section, the oscillating behavior of the decision statistic is studied in detail. The ping-pong effect is then related to the oscillating behavior in Section IV, while a large system analysis for long code schemes is presented in Section V. Numerical examples are given in Section VI and concluding remarks are found in Section VII.
where is a unit diagonal matrix, is strictly lower left triangular, is strictly upper right triangular, and which is in fact the decorrelator solution. A standard method for matrix inversion is the Jacobi iteration [8]. It is defined as follows: (2) . Equation (2) is in Here, is the iteration index, and fact identical to linear parallel interference cancellation where are used to determine updated estimates from iteration estimates in iteration [5], [13]. This iteration method is however not guaranteed to converge. The concept of over relaxation can be introduced to ensure convergence
II. SYSTEM MODEL
(3)
In this section, we describe the baseband uplink model of the CDMA communication system considered throughout this paper. The uplink model is based on a -user synchronous CDMA system with single-path channels and the presence of additive white Gaussian noise (AWGN) with zero mean and . variance A specific user in this multiuser CDMA system transmits a which is modulated binary information symbol by a length spreading sequence where each element is in . Binary data and chip formats are assumed for clarity. The continuous-time received signal is down converted to baseband, passed through a filter matched to the chip pulse-shape and sampled. The received baseband signal after chip-matched filtering is then
where is the matrix of received spreading codes, is the data symbol vector, and is a vector of independent, identically distributed Gaussian random variables of zero mean and variance . Here, we have assumed no random phase rotation and perfect power control for clarity and notational simplicity. The results do also apply to the more general cases of no or arbitrary power control, as well as for asynchronous and fading multipath systems with random phase rotations. The general model is discussed in more detail in [12]. A minimal set of sufficient is obtained through correlation with the statistics of length received spreading code of the desired user and corresponds algebraically to
Here, is the relaxation coefficient, typically less than one. The Jacobi over-relaxation iteration is identical to weighted linear PIC (WLPIC) [5], [13]. Other iterations can be applied which also correspond to cancellation structures [5]–[7], [3], [14]. In this paper, however, we limit our focus to linear PIC where the ping-pong effect is most pronounced. Based on (3), we can derive a nonrecursive description of WLPIC
(4) is defined above. Here, is known as the where iteration matrix of (3) which leads to (4) as described in general in [8]. We can further rewrite (4) to arrive at
where . This is the polynomial expansion description of WLPIC developed in [16]–[18]. To gain further insight, an eigenvalue decomposition of the correlation matrix is introduced
where where is the matched filter output vector, is the correlation matrix, and is the zero mean Gaussian noise output vector with . covariance matrix are the eigenvalues and corresponding eigenvectors. We can then get the following expression:
III. ANALYTICAL DESCRIPTION It has previously been shown that linear interference cancellation corresponds to iterative methods for solving a set of linear equations [4]. Based on the above system model, the set of equations to be solved is (1)
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Here, , are the eigenvalues of the iteration matrix. Without loss of generality, we consider user 1 in more detail
(5) is element of eigenvector . We can separate this Here, into the desired signal, the MAI and the noise (desired signal)
0
Fig. 1. Intervals for convergence/divergence for 1 and , respectively. Intervals where oscillating behavior will occur are also included.
the desired fixed point is , the solution to the equation system in (1). The second term in (5) then describes the residual error which can either converge to zero or diverge. To investigate the potential oscillating behavior more closely, we therefore condirectly. sider the error term Considering the recursive form in (3), we get
(MAI) (8) Again, considering user 1 (noise)
(6)
refers to element of vector . Here, the notation modes of convergence, It is clear from (6) that there are one for each distinct eigenvalue. In particular, convergence is assured when all modes converge, i.e., (9) for all
. The dominating mode determines the convergence rate. The fastest convergence rate is achieved when the two extreme modes are identical (7) These results are well-known and have previously been presented in [4], [7], [3], [13]. Based on (6), it is observed that if for one or more values of , the decision statistic will experience an alternating behavior between two state processes determined by the corresponding modes of convergence. Depending on the polarity of the set of convergence modes , we can expect to encounter oscillating behavior. Intervals for the dominating modes and eigenvalues for observing oscillaor , the iteration tions are shown in Fig. 1. If , oscillation will occur during will not converge. For divergence while no oscillation will be observed in the diverwas possible. However, this mode cannot be gence if and are both positive. In case , reached since , the scheme will converge smoothly, while for the scheme converges, exhibiting some level of oscillating behavior. This classification is also used in adaptive filter theory [11]. Using the optimal relaxation factor in (7), we see that one and one extreme mode will be in the interval . In this case, oscillations will be in the interval are always encountered in the decision statistic. dimensional vector which A fixed point is defined as a solves the original equation system. As previously discussed,
Similar to the decision statistic, the residual error obviously modes of convergence, each representing a geoexhibits converges as the weighted sum of metric series. Hence, . The time required for exponentials of the form of its initial value is given by each series to reach
However, the overall time constant cannot be expressed in a simple form. Nevertheless, a lower bound is attained by the slowest rate of convergence given by the eigenvalue leading to . An upper bound is found the maximum magnitude of by the equivalent eigenvalue leading to the smallest magnitude [19]. of has maximum magnitude, then we In case have a smooth exponential convergence as shown in Fig. 2. is the dominating mode, then However, when we have alternating polarity with each iteration. The mode still converges exponentially, however, only the absolute value converges smoothly. The term follows a positive envelope for odd iteration indices and a negative envelope for even iteration indices, respectively. This is illustrated in Fig. 2. This converging oscillation can cause the polarity of the decision statistic to alternate and, thus, lead to BER ping-pong. IV. PING-PONG EFFECTS The question remains however, when this oscillating behavior in the decision statistic leads to ping-pong effects in the BER.
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Fig. 2. Illustration of possible convergence envelopes for a positive and a negative dominating term.
(a)
(b)
statistic alternates in polarity between odd and even iterations. In example 1, we see that the fixed point is comfortably clear of the decision boundary. In example 2, however, the fixed point is closer to the decision boundary and the magnitude of the oscillation is larger, both undesirable features. Noise enhancement of the decorrelating solution can bring the fixed point close to the decision boundary and a large starting decision statistic relative to the fixed point leads to large oscillation magnitudes. The starting decision statistic is equal to the matched filter output. Since the conventional detector usually has a BER of less than 0.5, it is more likely that large matched filter outputs corresponds to correct decisions. It follows that even iterations experience more ping-pong errors than odd iterations. This is observed in numerical examples as demonstrated later. Large decision statistics normally considered desirable are, therefore, the cause of potentially undesirable ping-pong effects in WLPIC. The relaxation coefficient can be used to scale the effective eigenvalues such that ping-pong is avoided. The coefficient for all . The largest should be selected such that value to guarantee no ping-pong is then
Fig. 3. Two examples of converging error terms and decision statistics. Example 1 is well-behaved while example 2 exhibits ping-pong effects.
The performance of iteration schemes have previously been investigated based on the mean squared error (MSE), e.g., [13], [14]. Following the approach in [13], the MSE of the WLPIC designed to converge toward the decorrelating detector can be determined as
(10) This is also the value that provides the fastest convergence given that no oscillating behavior is allowed. It should be noted that . Only oscillating behavior can virtually not be avoided for , i.e., an orthogonal system, then oscillations are when not encountered. V. LONG CODE SYSTEMS
It is clear that there is no oscillation in the MSE, regardless of the eigenvalues. This is illustrated by the results in [14]. It is not possible to derive a similar expression for the BER. The error rate can only be determined as shown in [13] and realizations of the [15] as a sum of -functions over all . For large corresponding MAI in each decision statistic systems, however, it is possible to derive an analytical expression which is asymptotically accurate. In [23], it is shown that the decision statistic at iteration for user is asymptotically stays Gaussian distributed as goes to infinity and fixed. As a consequence, the BER can be expressed as a single -function based on moments of the eigenvalues. This is used to obtain the numerical results for asymptotically large systems in Section VI. Considering (9), the potential oscillating behavior of the error term is focused around zero. Two examples are depicted , system at 10 dB. in Fig. 3(a). This is for a with . In The dominating eigenvalue is both cases, we observe oscillation, however, in example 2, the starting value (matched filter output) is more than twice as large as for example 1, leading to excessive oscillation. Fig. 3(b) depicts the decision statistics for the same examples. The fixed and , points in the two cases are respectively. From Fig. 3(b), we observe that the oscillations in example 2 does in fact lead to ping-pong as the decision
All of the above results relate to eigenvalues of the correlation matrix. For long codes, however, a new set of spreading codes is selected for each symbol interval, making it seemingly necessary to retune the relaxation coefficient at each interval. Fortunately, for large systems where and are approaching is finite, the limiting distribution infinity such that of the eigenvalues is deterministic and has finite support [20]. In particular, the limits for the maximum and minimum eigenvalues are and as [21], [22]. As shown in [7], it follows that a large converges only if the load is limited by system with . Furthermore, it was found in [7] that fastest convergence is achieved when (11) In addition to these results, a long code system will exhibit the following behavior: (OD) (OC) (SC)
RASMUSSEN AND OPPERMANN: PING-PONG EFFECTS IN LINEAR PARALLEL INTERFERENCE CANCELLATION FOR CDMA
Fig. 4. Limiting curves describing the combinations of and for which oscillating divergence, oscillating convergence and smooth convergence, respectively, is encountered. Also, the curve for the relaxation factor for fastest convergence is included.
where OD, OC, and SC denote oscillating divergence, oscillating convergence, and smooth convergence, respectively. These limits are illustrated in Fig. 4 together with the optimal relaxation factor of (11). We observe that the optimal factor curve lies within the region of oscillating convergence, resulting in potential BER ping-pong in all cases. The obvious question is now, how much is the convergence rate affected by avoiding oscillating behavior. Consider again the error term from (8)
where error, we get
. Looking at the squared magnitude of the
where denotes the extreme eigenvalue that leads to the max. This approach was first sugimum magnitude of gested in [15]. It can be shown that for large, long-code systems , then . These results have previously been derived in [7], following a different approach. The approach presented here, confirms the results in [7]. We adopt the terminology of [7] and denote the factor as the convergence factor. For the combinations of and describing the three curves in Fig. 4, the upper bounds on convergence rate are shown in Fig. 5. It is clear that there is no point
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Fig. 5. Upper bounds on convergence rate for the two limiting cases of convergence behavior. Also, the case of the optimal relaxation factor is included.
in being close to the limit for convergence as the convergence rate will be slow. It is observed that a relaxation factor chosen to avoid oscillating behavior leads to a small loss of convergence rate. It is, thus, possible to avoid oscillations at some cost of convergence rate. The cost is quantified further in the following section for some numerical examples. VI. NUMERICAL EXAMPLES Here, we consider two examples with , , , at an and an asymptotic case where dB. The first example is a short-code system with and . The relaxation factor is , and , respectively, acchosen to be cording to (7) and (10). The simulated BER is shown in Fig. 6 as a function of the iteration index. Included for reference is the decorrelator performance, the fixed point performance for the it, we have oscillating divergence eration. It is clear that for , we observe the fastest convergence, as expected. For however, we still have a slight oscillating behavior in the BER. , we have smooth convergence as the negaFinally, for tive eigenvalues of the iteration matrix have been eliminated. In this example, the loss of convergence speed is minor as only one as opposed to . extra iteration is required for In Fig. 7, a long-code example is presented. Here, . Again, , and are chosen for illustration. The same behavior as for the short-code example is observed. In this case, however, the loss of convergence speed is more significant. Here, we need ten as opposed to six iterations for iterations for to reach the decorrelator performance. The loss of convergence . speed diminishes as the system gets larger This is demonstrated in Fig. 8, where the asymptotic results presented in [23] is used to derive numerical results for , . This is based on an analytically justified Gaussian approximation which is asymptotically accurate in going to infinity with the load fixed. the limit of
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Fig. 6. Short code example with K = 16, N = 32 at E =N = 8 dB. The extreme eigenvalues are = 2:541 and = 0:1308. Included are the cases of = 1, = and = . The decorrelator performance is included for reference.
!1
Fig. 8. Long code example with = 0:5 and K; N at E =N = 8 dB. Included are the cases of = 1, = and = . The decorrelator performance is included for reference.
At intermediate iterations, the term
becomes a reasonable estimate of the MMSE or , thus providing better performance than the decorrelator. The iterative detector can in fact also be design for implementing the MMSE detector and in this case the structure converges smoothly to the MMSE performance without overshooting [7], [13]. VII. CONCLUDING REMARKS
Fig. 7. Long code example with K = 16, N = 32 at E =N = 8 dB. Included are the cases of = 1, = and = . The decorrelator performance is included for reference.
The analytical expression is based on the moments of the eigenvalues [23]. In all the numerical examples, we observe an overshooting convergence behavior. The performance within some interval of iterations is better than the decorrelator performance before converging smoothly from below. This is a commonly observed phenomenon which is due to the fact that intermediate solution vectors are closer to the MMSE solution or the single user solution than the decorrelator solution. Considering (5), we have
(12)
In this paper, we have presented an analytical description of weighted, linear PIC. The description is based on an eigenvalue decomposition of the correlation matrix. The description is used to investigate convergence characteristics, with special attention toward oscillating behavior leading to ping-pong effects in the BER. It is found that negative eigenvalues of the iteration matrix create oscillations in the decision statistic centered around the fixed point described by the decorrelator output. Large correlator outputs, combined with fixed points close to the decision boundary are, thus, exposed as the main causes leading to ping-pong effects. Intervals for oscillating divergence, oscillating convergence and smooth convergence are specified. A relaxation factor, eliminating oscillating behavior at a minimal cost of convergence rate, is suggested. The loss of convergence rate can be significant for relatively small systems. However, as for a fixed , the loss in convergence rate diminishes. REFERENCES [1] S. Moshavi, “Multiuser detection for DS-CDMA communications,” IEEE Commun. Mag., vol. 34, pp. 132–136, Oct. 1996. [2] R. Lupas and S. Verdú, “Near–far resistance of multiuser detectors in asynchronous channels,” IEEE Trans. Commun., vol. 38, pp. 496–508, Apr. 1990.
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[3] P. H. Tan and L. K. Rasmussen, “Linear interference cancellation in CDMA based on iterative techniques for linear equation systems,” IEEE Trans. Commun., vol. 48, pp. 2099–2108, Dec. 2000. [4] H. Elders-Boll, H. D. Schotten, and A. Busboom, “Efficient implementation of linear multiuser detectors for asynchronous CDMA systems by linear interference cancellation,” Eur. Trans. Telecommun., vol. 9, no. 4, pp. 427–437, Sept.–Nov. 1998. [5] R. M. Buehrer, S. P. Nicoloso, and S. Gollamudi, “Linear versus nonlinear interference cancellation,” J. Commun. and Networks, vol. 1, June 1999. [6] D. Guo, L. K. Rasmussen, and T. J. Lim, “Linear parallel interference cancellation in long-code CDMA multiuser detection,” IEEE J. Select. Areas Commun., vol. 17, pp. 2074–2081, Dec. 1999. [7] A. J. Grant and C. B. Schlegel, “Iterative implementations for linear multiuser detectors,” IEEE Trans. Commun., vol. 49, pp. 1824–1834, Oct. 2001. [8] O. Axelsson, Iterative Solution Methods. Cambridge, U.K.: Cambridge Univ. Press, 1994. [9] L. K. Rasmussen, “On ping-pong effects in linear interference cancellation for CDMA,” in Proc. IEEE Int. Symp. Spread Spectrum Tech. and Appl., Parsipanny, NJ, Sept. 2000, pp. 348–352. [10] L. K. Rasmussen and I. J. Oppermann, “Convergence behavior of linear parallel cancellation in CDMA,” in IEEE Global Commun. Conf., San Antonio, TX, Nov. 2001, pp. 3148–3152. [11] B. Farhang-Boroujeny, Adaptive Filter, Theory and Applications. New York: Wiley, 1998. [12] L. K. Rasmussen, P. D. Alexander, and T. J. Lim, “A linear model for CDMA signals received with multiple antennas over multipath fading channels,” in CDMA Techniques for 3rd Generation Mobile Systems, F. Swarts, P. van Rooyen, I. Oppermann, and M. Lötter, Eds. Norwell, MA: Kluwer, Sept. 1998, ch. 2. [13] D. Guo, L. K. Rasmussen, S. Sun, and T. J. Lim, “A matrix-algebraic approach to linear parallel interference cancellation in CDMA,” IEEE Trans. Commun., vol. 48, pp. 152–161, Jan. 2000. [14] M. J. Juntti, B. Aazhang, and J. O. Lilleberg, “Iterative implementation of linear multiuser detection for dynamic asynchronous CDMA systems,” IEEE Trans. Commun., vol. 46, pp. 503–508, Apr. 1998. [15] L. K. Rasmussen, T. J. Lim, and A.-L. Johansson, “A matrix-algebraic approach to successive interference cancellation in CDMA,” IEEE Trans. Commun., vol. 48, pp. 145–151, Jan. 2000. [16] M. Moshavi, E. G. Kanterakis, and D. L. Schilling, “Multistage linear receivers for DS-CDMA systems,” Int. J. Wireless Inform. Networks, vol. 3, pp. 1–17, 1996. [17] D. Guo, “Linear parallel interference cancellation in CDMA,” M. Eng. thesis, National Univ. Singapore, Singapore, 1998. [18] D. Guo and L. K. Rasmussen, “Linear parallel interference cancellation using fixed weighting factors for long-code CDMA,” in Proc. IEEE Int. Symp. Information Theory, Sorrento, Italy, June 2000, p. 332. [19] S. Haykin, Adaptive Filter Theory, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.
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[20] D. Jonsson, “Some limit theorems for the eigenvalues of a sample covariance matrix,” J. Multivariate Analysis, pp. 1–38, Dec. 1982. [21] Z. D. Bai and Y. Q. Yin, “Limit of the smallest eigenvalue of a large dimensional sample covariance matrix,” Ann. Probability, pp. 1275–1294, Dec. 1993. [22] Z. D. Bai, J. W. Silverstein, and Y. Q. Yin, “A note on the largest eigenvalue of a large dimensional sample covariance matrix,” J. Multivariate Analysis, pp. 166–168, 1988. [23] D. Guo, S. Verdú, and L. K. Rasmussen, “Asymptotic normality of linear multiuser detection outputs,” in Proc. IEEE Int. Symp. Information Theory, Washington, DC, Aug. 2001, p. 307.
Lars K. Rasmussen (S’92–M’93–SM’01) was born on March 8, 1965 in Copenhagen, Denmark. He received the M.Eng. degree from the Technical University of Denmark, Copenhagen, and the Ph.D. degree from Georgia Institure of Technology, Atlanta, GA, in 1989 and 1993, respectively. From 1993 to 1995, he was with the University of South Australia, Adelaide. From 1995 to 1998, he was with the CWC at the National University of Singapore, Singapore. From 1998 to 2001, he was an Associate Professor at Chalmers University of Technology, Gothenburg, Sweden. Since February 2002, he is a Professor in Telecommunications at the Institute for Telecommunications Research, University of South Australia, Adelaide, Australia.
Ian J. Oppermann (M’97) was born in Maryborough, Qld, Australia, in 1969. He received the B.Sc., B.E. (electrical), and Ph.D. degrees from the University of Sydney, Australia, in 1989, 1991, and 1997, respectively. Before commencing postgraduate studies, he was with OTC Australia (now Telstra). Since 1996, he has been with the Center for Wireless Communications (CWC-Oulu), University of Oulu, Finland. He was the Founder and Technical Director of Southern Poro Communications, Sydney, Australia, a company that designed and implemented baseband CDMA transceivers and was heavily involved in UMTS network modeling. His research interests include wideband channel modeling, CDMA receiver structures for high-data-rate/low-mobility environments, and complex valued spreading sequences for CDMA systems. He was an Editor of CDMA Techniques for Third-Generation Mobile Systems and Associate Area Editor for Communications on the 1998 CRC Dictionary of Engineering Terms. Dr. Oppermann received the Sydney University Engineering Foundation Prize for Excellence in Teaching in 1995 and 1997
