Marten J. Klok’s research while affiliated with Delft University of Technology and other places

We study sample covariance matrices of the form W = 1 n CC T , where C is a k × n matrix with i.i.d. mean zero entries. This is a generalization of so-called Wishart matrices, where the entries of C are independent and identically distributed standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of W when either k is fixed and n → ∞, or kn → ∞ with kn = o(n/ log log n), in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving a.s. limits of the eigenvalues, only require finite fourth moments. Our most explicit results for k large are for the case where the entries of C are ±1 with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalue to the rate functions for independent and identically distributed standard normal entries of C. This case is of particular interest, since it is related to the problem of the decoding of a signal in a code division multiple access system arising in mobile communication systems. In this example, k plays the role of the number of users in the system, and n is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency, and the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.

We study sample covariance matrices of the form W = (1 / n ) C C T , where C is a k x n matrix with independent and identically distributed (i.i.d.) mean 0 entries. This is a generalization of the so-called Wishart matrices, where the entries of C are i.i.d. standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of W when either k is fixed and n → ∞ or k n → ∞ with k n = o ( n / log log n ), in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving almost sure limits of the eigenvalues, require only finite fourth moments. Our most explicit results for large k are for the case where the entries of C are ∓ 1 with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalues to the rate functions for i.i.d. standard normal entries of C . This case is of particular interest since it is related to the problem of decoding of a signal in a code-division multiple-access (CDMA) system arising in mobile communication systems. In this example, k is the number of users in the system and n is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency; the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.

We study sample covariance matrices of the form $W=\frac 1n C C^T$, where C is a $k\times n$ matrix with i.i.d. mean zero entries. This is a generalization of so-called Wishart matrices, where the entries of C are independent and identically distributed standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when k is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of W when either k is fixed and $n\to \infty$, or $k_n\to \infty$ with $k_n=o(n/\log\log{n})$, in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving a.s. limits of the eigenvalues, only require finite fourth moments. Our most explicit results for k large are for the case where the entries of C are $\pm1$ with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalue to the rate functions for independent and identically distributed standard normal entries of C. This case is of particular interest, since it is related to the problem of the decoding of a signal in a code division multiple access system arising in telecommunications. In this example, k plays the role of the number of users in the system, and n is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency, and the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.

The introduction of the so-called third-generation wireless communication system, also known as UMTS or IMT-2000, is a large-scale revolution in telecommunications. It uses a technique called code division multiple access (CDMA). An advanced algorithm to improve the performance of such a CDMA system is called hard-decision parallel interference cancellation and was studied by van der Hofstad and Klok for a rather basic model. We extend many of their results to a more realistic model, where different users transmit at different powers and where additive noise is present.

The third-generation (3G) mobile communication system uses a technique called code division multiple access (CDMA), in which multiple users use the same frequency and time domain. The data signals of the users are distinguished using codes. When there are many users, interference deteriorates the quality of the system. For more efficient use of resources, we wish to allow more users to transmit simultaneously, by using algorithms that utilize the structure of the CDMA system more effectively than the simple matched filter (MF) system used in the proposed 3G systems. In this paper, we investigate an advanced algorithm called hard-decision parallel interference cancellation (HD-PIC), in which estimates of the interfering signals are used to improve the quality of the signal of the desired user. We compare HD-PIC with MF in a simple case, where the only two parameters are the number of users and the length of the coding sequences. We focus on the exponential rate for the probability of a bit-error, explain the relevance of this parameter, and investigate how it scales when the number of users grows large. We also review extensions of our results, proved elsewhere, showing that in HD-PIC, more users can transmit without errors than in the MF system.

The third-generation (3G) mobile communication system uses a technique called code division multiple access (CDMA), in which multiple users use the same frequency and time domain. The data signals of the users are distinguished using codes. When there are many users, interference deteriorates the quality of the system. For more efficient use of resources, we wish to allow more users to transmit simultaneously, by using algorithms that utilize the structure of the CDMA system more effectively than the simple matched filter (MF) system used in the proposed 3G systems. In this paper, we investigate an advanced algorithm called hard-decision parallel interference cancellation (HD-PIC), in which estimates of the interfering signals are used to improve the quality of the signal of the desired user. We compare HD-PIC with MF in a simple case, where the only two parameters are the number of users and the length of the coding sequences. We focus on the exponential rate for the probability of a bit-error, explain the relevance of this parameter, and investigate how it scales when the number of users grows large. We also review extensions of our results, proved elsewhere, showing that in HD-PIC, more users can transmit without errors than in the MF system.

We analytically compute a mea-sure of performance of various linear Paral-lel Interference Cancellation (PIC) decod-ing schemes in the infinite stage limit, for moderately loaded CDMA systems with-out AWGN, or with a sufficiently small amount of AWGN. This measure is the ex-ponential rate of the BEP, which does not involve Gaussian approximations. We ob-tain these rates using large deviation the-ory for the eigenvalues of the code corre-lation matrix. We find that the decorre-lator performs best, followed by infinite-stage SD-PIC, which is found to perform better than infinite stage HD-PIC.

We study a multiuser detection system for code-division multiple access (CDMA). We show that applying multistage hard-decision parallel interference cancellation (HD-PIC) significantly improves performance compared to the matched filter system. In (multistage) HD-PIC, estimates of the interfering signals are used iteratively to improve knowledge of the desired signal. We use large deviation theory to show that the bit-error probability (BEP) is exponentially small when the number of users is fixed and the processing gain increases. We investigate the exponential rate of the BEP after several stages of HD-PIC. We propose to use the exponential rate of the BEP as a measure of performance, rather than the signal-to-noise ratio (SNR), which is often not reliable in multiuser detection models when the system is lightly loaded. We show that the exponential rate of the BEP remains fixed after a finite number of stages, resulting in an optimal hard-decision system. When the number of users becomes large, the exponential rate of the BEP converges to (log 2)/2 $1/4. We provide guidelines for the number of stages necessary to obtain this asymptotic exponential rate. We also give Chernoff bounds on the BEPs. These estimates show that the BEPs are quite small as long as k = o(n/log n) when the number of stages of HD-PIC is fixed, and even exponentially small when k = O(n) for the optimal HD-PIC system, and where k is the number of users in the system and n is the processing gain. Finally, we extend the results to the case where the number of stages depends on k in a certain manner. The above results are proved for a noiseless channel, and we argue that we expect similar results in a noisy channel as long as the two-sided spectrum of the noise decreases proportionally to n.

For a simple CDMA system, we compute the bit-error probability (BEP)with soft-decisionparallel-interference-cancellation (SD-PIC). Instead of approximatingthe signal-to-noise ratio, we use a different measure to calculateperformance. This measure is the exponential rate of the BEP, i.e., thelimit of n–1 log(BEP) = –I, for the processinggain n , where I depends only on the number of users. Weshow, using the rate as a measure, that SD-PIC improves the performance.The values of I follow as the solution of an optimization problem whichcan be calculated numerically. We use these results to derive theasymptotic behaviour of the rate for large k. We also derive results forthe second order asymptotics of the BEP. Inclusion of second orderasymptotics leads to excellent approximations.

We derive approximations for the probability of a bit error for a code division multiple access (CDMA) system with one-stage soft decision parallel interference cancellation. More precisely, we derive the exponential rates, Jk with cancellation and Ik without cancellation, of a CDMA system with k users and processing gain equal to n as n → ∞. Whereas the rates Ik follow explicitly from Cramér's theorem, the rates Jk are given in terms of an optimization problem that can be evaluated numerically. We prove that Jk > Ik for k ≥ 3, which shows that interference cancellation is effective. For the case without interference cancellation, we investigate the second order (Bahadur-Rao) asymptotics. For the case with interference cancellation, we can obtain second order asymptotics only for k = 3. Together the limits provide excellent approximations for the probability of a bit error in a wide range of interest.