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Locally Optimum Detection of Signals in Multiplicative and First-Order Markov Additive Noises Jumi Lee, Member, IEEE, Iickho Song, Senior Member, IEEE, Hyoungmoon Kwon, Member, IEEE, and Hong Jik Kim, Member, IEEE
Abstract—In most previously reported studies on locally optimum detection of signals, independent observations have been assumed in various noise environments. The use of an independent observation model may cause a considerable performance degradation in detection applications of modern high data-rate communication systems exhibiting dependence among interference components. In this paper, the detection of weak known and random signals is addressed in observations corrupted by multiplicative and first-order Markov additive noises. The asymptotic and finite sample-size performance of several detectors are obtained and compared: it is confirmed that the dependence of noise components need to be taken into account to maintain detection performance appropriately. Index Terms—Known signal, locally optimum detector, Markov noise, multiplicative noise, nonadditive components, random signal, signal detection, weak signal.
I. INTRODUCTION T is generally assumed that the sampled noise components are statistically independent [1] in most signal detection problems. The independent noise assumption, however, is frequently not satisfied in discrete-time signal detection applications, especially as the sampling rate becomes high. As a consequence, a detector optimized for independent noise is often not guaranteed to be optimum in a number of practical signal detection systems operating in dependent environment [2]–[4], which becomes more crucial in modern high data-rate communication systems. A number of dependent observation models taking into account the dependence among noise components have been proposed and investigated to address such a situation.
I
Manuscript received July 24, 2006; revised July 7, 2007. This study was supported by the National Research Laboratory (NRL) Program of Korea Science and Engineering Foundation (KOSEF), Ministry of Science and Technology (MOST), under Grant R0A-2005-000-10005-0, for which the authors would like to express their thanks. The material in this paper was presented in part at the 64th IEEE Vehicular Technology Conference, Montreal, QC, Canada, September 2006. J. Lee is with Samsung Electronics Co., LTD., Suwon 443-742, Korea (e-mail:
[email protected]). I. Song and H. Kwon are with the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305-701, Korea (e-mail:
[email protected]; kwon@ Sejong.kaist.ac.kr). H. J. Kim is with the Network Laboratory, LG Electronics, Anyang 431-749, Korea (e-mail:
[email protected]). Communicated by A. Høst-Madsen, Associate Editor for Detection and Estimation. Digital Object Identifier 10.1109/TIT.2007.911254
Typical examples of dependent observation models proposed and investigated are those described by the -mixing, -dependent, and transformation noise [5]–[10]. Among the classes of general dependence model is the first-order Markov model: it has been reported that many signals and noises in telecommunication systems are of this type [11], [12]. In the meantime, the problem of designing detectors for weak signals in different types of noise has been addressed in [8], [9], [13]–[17], for instance. In a number of the studies, the asymptotic and finite sample-size performance of the locally optimum (LO) detector with statistically independent samples have been explored. Because of the simple detector structure in non-Gaussian noise environment and of the almost optimum performance even at large signal strengths in many cases, the LO detectors have been extensively investigated in the literature. It is noteworthy that, with the recent increasing interest in developing low power communication systems, the importance of weak signal detection keeps growing steadily. In this paper, based on the rationale of the necessity of dependent observation model and LO detection, LO detection of known and random signals is addressed in dependent environment described by multiplicative and first-order Markov additive noises. Interestingly, the class of multiplicative noise is known to be useful [9], [17] in modeling multipath propagation, a key aspect in mobile communication systems. The organization of this paper is as follows. The observation model is described in detail in Section II, and the test statistics of the LO detector are derived in Section III. Preliminaries to performance comparison are depicted and performance characteristics of the LO detectors are analyzed and discussed in Section IV. Finally, this paper is concluded with a summary in Section V. II. THE OBSERVATION MODEL The (purely) additive noise model, in which noise is added simply to a signal to generate observations, has been one of the most common models in many areas of signal processing such as signal detection, estimation, filtering, and restoration [9]. The main reason for the prevalence of the additive noise model is that the additive noise model is relatively easy to treat mathematically, yielding physically appealing structures of signal processors in various applications. However, observation models with nonadditive noise components should be employed in several different types of situations. In a radio system, for instance, the effect of delayed signals from
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multipath may be modeled by employing nonadditive (e.g., multiplicative) as well as purely additive noise terms, which can be modeled as a Markov process [8], [9]. are corAssume that the discrete-time observations and rupted by the multiplicative noise components in a signal detection scenario. additive noise components can be described by Then, (1)
where the only unknown is under the alternative hypothesis and
(5) is the joint pdf of the observation vector . In (5), , , and . Clearly, in (5) yields the pdf under the null hypothesis ; that is, we have
denote the known signals, In (1), is the signal strength, and is the number of observations in a sample. We assume form a zero-mean firstthat the additive noise components order Markov process and are independent of the multiplica. The multiplicative noise compotive noise components may or may not be a first-order Markov process. nents The observation model (1) can be regarded as a special case of the generalized observation model considered in [14], except that Markov environment is in addition taken into account in (1). While the observation model (1) is useful for many types of fading scenario, it is unfortunately not suitable for sonar/ radar detection problems [18], [19], where chirp signals are employed and processed and the fading components are usually represented with multiplication in frequency domain (and consequently, convolution in time domain). and are The probability density functions (pdfs) of denoted by and , respectively. Let us denote by and the joint pdfs of and , respectively, where and . In addition, for the Markov process , the transigiven is denoted by tion or conditional pdf of , where is used to de, for with and note . We assume that the pdfs and are smooth enough and satisfy regularity conditions [13] that 1) they are Lebesgue integrable, 2) their first-order derivatives exist, and 3) the derivatives are absolutely bounded by integrable functions. These regularity conditions allow us to justify some mathematical operations such as interchanges in the order of differentiation with respect to a parameter and integration of a function. For example, we have
Having a basis in the generalized Neyman–Pearson lemma [20], an LO detector has maximum slope for the detector power among all the detectors function (detection probability) at having its false-alarm probability. The power of an LO detector is guaranteed under mild conditions to be no smaller than that , where of other detectors over some nonnull interval depends on the specific detector [13]. The LO detector test versus can generally be obtained as the statistic for ratio
(2)
(8)
where denotes the pdf of conditioned on . Our detection problem can be formulated as a statistical hypothesis testing problem of choosing between the null hypothesis (3)
(6)
III. LOCALLY OPTIMUM (LO) TEST STATISTICS
(7) where is the smallest natural number for which (7) produces of . a nonzero function A. When the Means Components are Not All Zero
of Multiplicative Noise
Evaluating with (5) as in [21, App. A.1], we can easily derive from (7) the test statistic
when of the LO detector for the detection of known signals of the multiplicative noise at least one of the means components is nonzero under the model (1), where . The test statistic (8) is the same as that used for the LO Bayesian detection problem in [8]. denotes the mean of the transformed In (8), signal with , and
and the alternative hypothesis (4)
(9)
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are the LO nonlinearities defining the structure of the LO dedenotes the partial derivatector, where the operator tive with respect to a variable (10) and, for or
(11)
It is observed in (8) that the test statistic depends on the statistical characteristics of the multiplicative noise compothrough the means of the transformed signals nents through and on those of the additive noise components the LO nonlinearities . The degree of the dependence of on the statistical characteristics of the multiplicative noise components is clearly different from that on the statistical characteristics of the additive noise components. Specifically, only the means of the multiplicative noise components, but not any other statistic, of the multiplicative noise compo. On the other nents are employed in the test statistic exploits the distribution of the additive noise hand, components as a whole, not particular statistics of the additive noise components. In other words, when the means of the multiplicative noise components are not all zero, the additive noise components influence more generally on the test statistic of the LO detector than the multiplicative noise components. Apparently, the LO test statistic under the additive noise model [7] can by rebe obtained as a special case of the test statistic with in (8). placing Now, when the additive noise components are infor and dependent, we have , and consequently, the test becomes statistic (12) which is the same as the test statistic for known signal detection is under independent additive noise circumstance [9] when replaced by . In (12) (13) is the LO nonlinearity defined in [9] for the independent and identically distributed (i.i.d.) noise. One of the key differences between the two test statistics and is that, while is an instant, is memoryless linear combination where the parameter multiplied by a function of the observation only, is a ‘windowed’ linear combination requiring memory where is multiplied by a function of the three observations obtained at (time or space) index with a sliding window of width three. The difference is of course a natural consequence of the independence and Markovian character of the additive noise components assumed for the and observation models under which the test statistics are derived, respectively.
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B. When the Means Components are All Zero
of Multiplicative Noise
In obtaining the LO test statistic, the first derivative is known to be sufficient in most cases [9] when the signals are of ‘known’ types as is apparent in the developments of Section III-A: note can be regarded known signals that the transformed signals of the multiplicative noise compowhen the means nents are not all zero. However, the test statistic derived as (8) vanishes when the means of the multiplicative noise components are all zero. This implies we need to derive the test in (7). statistic of the LO detector with Specifically, from (6), (7), and the results obtained in , we have Appendix A for
(14) as the LO detector test statistic when the multiplicative noise components are all zero-mean, where
(15) (16) and (17) denotes the correlation between the transformed signals and . As in the case of the test statistic , the test statistic , obtained when the means of the multiplicative noise components are all zero, depends on the statistical characterand istics of both the multiplicative noise components additive noise components . Yet, unlike the test statistic requiring only the first moments of the multiplicative noise components and first derivatives of the transition pdfs of the additive noise components, the test statistic requires the second-order characteristics of the multiplicative noise components and the first- and second-order derivatives of the transition pdfs of the additive noise components in deutilizes the tecting signals. Clearly, the test statistic strengths of the signals and statistical characteristics of the through the correlation multiplicative noise components , and the statistical dependence between adjacent observations resulting from the dependence of the additive noise . components through the nonlinearities uses three consecutive In addition, the test statistic observations from simultaneously. The first and second sumand ) in (14) implies that mations (the terms containing the test statistic employs a sliding window of size three
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for the linear combination of the correlations representing the influence of the multiplicative noise components representing the influand the LO nonlinearities ence of the additive noise components. The Markovian character of the additive noise components additionally affects the test through the third summation (the term constatistic ) in (14), where the nonlinearity evaluated with taining the two consecutive observations and is multiplied by of the correlation between the two conthe value and . In short, the third secutive components term tells us that, even if the multiplicative noise components are not Markov, they exhibit a first-order Markovian character to a certain degree, consequently affecting the LO test statistic additionally when the additive noise components are first-order Markov. are an If we assume that the additive noise components becomes independent random process, the test statistic
(18) since
for , where
,
, and
and multiplicative If the additive noise components are both independent processes, we get noise components
(21) , , or , which becomes the test from statistic for zero-mean independent random signals in additive is replaced by the variindependent noise [9] when of the -th component of the zero-mean random ance signals. Obviously, among the test statistics , , , and , the test statistic demands the least amount of the statistical information on the multiplicative and additive noise components in the detection structure. C. Locally Optimum Detection of Random Signals We have so far been concerned with the LO detection under are known signal components. It is the model (1), where interesting and straightforward to see that the results presented in Sections III-A and -B under the model (1) may be applied in conveniently obtaining the LO detector test statistics for the detection of random signals. Assume the observation model (22)
(19) Because the test statistic is obtained for independent additive noise components, it does not require knowledge on the transition distribution of the additive noise components, but still depends on the correlations of the multiplicative noise components. It is straightforward to see that the test statistic is essentially the same as the test statistic for random signals in behaving as independent noise environment [9], with random signals. are an inWhen the multiplicative noise components dependent random process, on the other hand, we obtain
are random signals possessing some probability diswhere tribution, and and are the same as in the model (1). Formally, the LO test statistics under the random-signal model (22) can be derived using (7): alternatively, the results we have derived in Sections III-A and -B can be directly used as simple and convenient shortcuts to the derivations, allowing us to save a lot of manipulations. Specifically, we can obtain the LO test statistics for random signals under the model (22) from the test statistics and by replacing some parameters appropriately. For example, we have (23)
(20) when , where from (14) since is the variance of the multiplicative noise . Although is obtained for the “zero-mean” case, the test statistic not having a double summation term, its structure differs from and obtained also for the the structures of is rather closer to zero-mean case. The structure of obtained for the “nonzero mean” case: the test that of statistic detects signals basically by regarding the variof the transformed signals as known signals and ances employing nonlinearities obtained by not only the first but also the second derivatives of the transition pdf of the additive noise components.
as the LO test statistic for random signals when the means of the products of the random signals and multiplicative noise components under the model (22) are not all in (8). Similarly, we would zero by replacing with obtain
(24)
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TABLE I THE FIRST-ORDER LO NONLINEARITIES IN THE GAUSSIAN ADDITIVE NOISE ENVIRONMENT
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TABLE II THE SECOND-ORDER LO NONLINEARITIES IN THE GAUSSIAN ADDITIVE NOISE ENVIRONMENT
as the LO test statistic for random signals when the parameters are all zero under the model (22) by replacing with in (14). In (24)
(25) is the correlation (or covariance) between the products and with and . D. LO Test Statistics for Gaussian Distribution Let us examine the LO test statistics and and the constituting LO nonlinearities , , and when the transition pdfs are specified. Consider the conditional Gaussian pdf
It is interesting to see from (27) that, contrary to a general obin servation we have made previously, the test statistic the Markov Gaussian environment does not require a window of in the Markov Gaussian environwidth three: instead, ment is just a linear correlator [13] similar to that in independent is environment when a table look-up scheme for for all , we can show, as in available. In passing, when , a gen[21, App. H], that deeral requirement for known-signal detection, where notes the expectation under the null and alternative hypotheses , respectively. for When the means of multiplicative noise components are all zero, we will similarly have
(26) (28) of given , where denotes the correlation coefficient and , and and denote, respectively, the between mean and variance of the random variable for . As shown in is well-known [22], for the conditional pdf (26), we have the conditional mean and conditional variance of when . Now, if we assume the transition pdf of is the shown in (26) with Gaussian conditional pdf for , we get Tables I and II. Using the results in , these tables, we have , and , where and with denoting the , , and : note that the additive variance of noise components are assumed to be all zero-mean at the beginning of this paper. It is noteworthy that we have when since means the additive noise components are independent in the Gaussian environment, indirectly defined in (9) is an appropriate confirming the nonlinearity generalization of defined in (13). From the results above, we have
(27)
Note that, as and are constants in the Gaussian environment, the terms containing these nonlinearities do not take any role in signal detection and thus can be deleted without affecting the detection from the test statistic performance of the LO detector: we have already completed the is a polynomial and deletion in (28). In general, when is 2, the nonlinearities the highest degree in and will be constants. IV. PERFORMANCE COMPARISON A. Preliminaries The LO test statistics depend only on the first- and secondas we have observed previously. order characteristics of Therefore, it suffices to specify the first- and second-order moto take the influence of the multiplicative noise ments of components into account in the performance evaluation of the LO detectors. We will characterize and compare the (relative) performance of detectors when the additive noise components are depicted by the identically distributed first-order Markov Gaussian (IFMG) and first-order Markov Middleton Class A (IFMM) distributions. The Middleton Class A (MCA) distribution is a physically meaningful and widely employed non-Gaussian noise model
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[23]–[25], and is a subset of the Gauss-Gauss mixture distributions, a member of the class of -contaminated distributions [13], [24], [26]. The MCA distribution has the pdf
requires fewer which will be larger than one when detector observations (has better performance) than detector , and vice versa. The parameter
(32) (29) and transition pdf
in (31) is called the efficacy of . In (32), denotes the for sample size , test statistic of the detector and denote the expected value under the alternative hypothesis and the variance under the null hypothesis of , respectively, and
(33)
(30) for where
is the overlap index, is the average intensity, is the correlation coefficient, , and for , with called the Gaussian factor. The pdf shown in (29) has mean zero and variance . The overlap index represents the average number of source emission events of man-made and natural electromagnetic interference impinging on the receiver per second multiplied by the mean duration of a typical interfering emissions [27]. The smaller , the fewer are such source emission events and their durations, the properties of the interference are dominated more by the waveform characteristics of individual events. The Gaussian factor is defined as the ratio of the intensity of the Gaussian component of the input interference in the receiver to the intensity of the non-Gaussian components. In practice, the overlap index and Gaussian factor are typically in the ranges to and to , respectively [27]. of Combining (29) and (30), we can obtain the bivariate MCA pdf, which turns out to be a linear combination of two Gaussian pdfs when both and are nonzero and finite. When the value of or approaches zero or infinity, on the other hand, the bivariate MCA pdf becomes a bivariate Gaussian pdf, an impulse, or an impulse plus a bivariate Gaussian pdf: details can be found in [21, Appendix I]. B. Performance Measures In the comparison of the asymptotic performance of detectors, which are useful for comparing detector performance when samples of sufficiently large sizes are available, closed-form results are usually possible. The notion of the asymptotic relative efficiency (ARE) is frequently employed [13] as a convenient and useful measure in comparing the asymptotic performance of detectors: as observed in [9], the asymptotic performance comparisons are particularly meaningful for the detection problems of weak signals. The asymptotic relative efficiency ARE of a detector with respect to another detector can be expressed as the ratio ARE
(31)
. Note that we have
from (7). The second approach in the comparison of detection performance is to obtain and compare the detection probabilities of the detectors with samples of finite size. The finite sample-size performance comparison clearly has more practical meaning than the asymptotic measure of performance characteristics of signal detectors since detectors in practice are used with samples of finite size. On the other hand, explicit closed-form expressions of finite sample-size performance are rather inordinately difficult, if not impossible, to obtain and evaluate even for a reasonably small sample-size in most cases. In this paper, we employ Monte-Carlo simulations to obtain and compare the finite sample-size performance of detectors in Gaussian noise environment. In the simulations, we obtain the detection probability assuming that the sample-size is and the false-alarm probability is . The thresholds satisfying the false-alarm probability of are obtained from . The detection probabilities subject to the false-alarm probability are then obtained from . Due to the computational complexity associated with the IFMM environment, the finite sample-size performance characteristics of detectors are considered only in the IFMG environment. C. Performance of Detectors When
are Not All Zero
1) Asymptotic Performance: Using the results shown in Appendix B, we can express the efficacy of the detector employing as the test statistic (34) In (34), we have used the notation (35) for simplicity, where
(36)
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TABLE III SOME VALUES OF I
IN THE
IFMG ENVIRONMENT
are generalizations of Fisher’s information function [9] for (representing the order or ) of derivative or partial derivative with respect to and (representing the time index), . In (36), the ’single-subscript’ and nonlinearities for are defined as , , and , and the ’double-subscript’ nonlinearities for have already been deis the Fisher’s information fined in (11). Note that function of the pdf . The efficacy of the detector based on the test statistic can be similarly expressed as (37) from the results shown in Appendix C, where (38) is the Let us assume an IFMG environment, where conditional pdf shown in (26) with means zero and variances one and is the standard Gaussian pdf. Then, denoting by the correlation coefficient of the additive noise components and , we get Table III. Using the results in Table III, we can show that the ARE of with respect to is
ARE (39) from (31), (34), and (37), where (40) with
(41)
and the conventions
when
and
.
Fig. 1. Exact and approximate ARE
It is noteworthy that in (39). Note that
in the IFMG environment.
, which has been used
denotes a measure of the normalized correlation between the and its shifted version repretransformed signal sequence senting the th order similarity, with and (from Schwartz inequality). It is clearly observed in (39) that, because of the first-order Markovian character we have assumed depends not only on in the observation model (1), ARE but also on the transformed signal the sign and magnitude of sequence through the parameter . If the sequence is specified, we can evaluate analytically the ARE using (39) in some cases: for example, we have ARE , , , and when , , , and , respecdenotes a sequence of tively (Fig. 1), where period and length . On the other hand, it is not a simple task to assess generally. Fortunately, it is apparent from ARE Fig. 1 that the ARE can be reasonably approximated by its is in the order of 100 finite counterpart when the length or longer. Thus, as shown in Fig. 2, we have evaluated the approximate value of the right hand side of (39) as a function and for some finite sequences genof erated randomly with . In Fig. 2, because the approximate ARE varies depending on the particular choice even when the value of is fixed, we have of averaged over 1000 sequences. In addition, we have restricted the results only for because sequences with are highly unlikely to be generated randomly. It is observed in Fig. 2 that, as a function of , the ARE possesses a maximum at a positive (negative) value of when is positive (negative). In Figs. 1 and 2, it is observed would get larger as implying that that the ARE Markovian character of the additive noise influences more as the value of increases.
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for , where . Consequently, we have . In Figs. 5 and 6, we can make the following observations. First, the detection probability of (solid line) is higher than that of (dash-dot). It is observed in Fig. 5 that the detecand get higher as becomes tion probabilities of larger at the same in the weak signal region. An explanation for this is possible with Fig. 7, where it is clearly anticipated that the components would tend to make, for a finite set of observations, the environment more like “slow fading” as the value of gets larger (allowing the detection easier) and “fast fading” as the value of gets smaller. In addition, the outperformance of over gets larger as grows at the same signal strength in the weak signal region. D. Performance of Detectors When Fig. 2. Approximate ARE (0).
in the IFMG environment as a function of
Now, let us assume an IFMM environment, where is the pdf shown in (29) and is the conditional pdf shown and . As in the IFMG environment, in (30) with . To obtain the we have approximated the ARE with , , ARE, we need first to evaluate , and for . Noting that , , are all zero when as shown in (68)–(70) in Appendix B, we have evaluated these three generalized Fisher’s information functions for using (36). On the other hand, the analytic evaluation of for is practically impossible even with numerical methods unfortunately: thus, based on the approximation approach in [8], we have approximated the sequence with a geometrical progression in terms of by noting that is relatively easy to evaluate. Specifically, as observed in Tables IV and V for example, we have (empirically) obtained the approximations in the structure of a geometrical progression with when , respectively, for . (The approximation interestingly resembles the relationship in the IFMG environment.) These approximations have been employed in the evaluation of the ARE shown in Figs. 3 and 4. All the values of the ARE shown in Figs. 3 and 4 are larger than and become larger as gets larger. Note that the ARE is generally greater than 1 even when because in the IFMM environment does not mean that are independent. As is anticipated easily, it is observed that all the (approximate) ARE shown in Figs. 1–4 is larger than or equal to , implying that would asymptotically perform better than under the model (1), and becomes larger as gets larger. 2) Finite Sample-Size Performance: Figs. 5 and 6 show the detection probability of the two detectors and for under the model (1). We have assumed that the additive noise components are IFMG with and for . In addition, it is assumed that and the multiplicative noise components are Gaussian with and
Are All Zero
1) Asymptotic Performance: As shown in Appendix D, we have the efficacy
(42) of the detector employing the test statistic have used the notations
, where we
(43)
(44) and
(45)
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VALUES OF I
TABLE IV NUMERICALLY EVALUATED (N) AND APPROXIMATED WITH A GEOMETRIC PROGRESSION (A) WHEN A = 0:5 AND = 0:01 IN THE IFMM ENVIRONMENT
VALUES OF I
TABLE V NUMERICALLY EVALUATED (N) AND APPROXIMATED WITH A GEOMETRIC PROGRESSION (A) WHEN A = 0:5 AND = 0:1 IN THE IFMM ENVIRONMENT
Fig. 3. Approximate ARE of when A = 0:5.
in the IFMM environment for some values
Similarly, from the results shown in [21, Appendix E], the efficacy of the detector based on the test statistic can be obtained as
Fig. 4. Approximate ARE (0).
in the IFMM environment as a function of
from (32), where
(47)
(46)
(48)
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(51) and
Fig. 5. Detection probabilities of T
and T
(52)
when n = 30.
respectively: again, details can be found in [21, App. F and G]. In the IFMG environment, we have
(53)
(54) Fig. 6. Detection probabilities of T n = 30.
and T
when r
= 00:9 and when
(55) and (56)
(49) and
(50) Finally, the efficacies of the detector based on the test statistics and can be obtained as
In Figs. 8–10, we have shown ARE , ARE , and ARE obtained from (53)–(56). It is observed in Fig. 8 that ARE becomes larger as . Depending on the sign of and the known signal sequence , ARE has different values as expected. For example, if for , ARE has its maximum value when . On the contrary, if for , ARE has its maximum value when . Particularly, the ARE when and for is the same as the ARE when and for . When , we have ARE since is the same as . Similar observations can be made in Figs. 9 and 10. 2) Finite Sample-Size Performance: The detection probabilities of , , , and for are obtained and shown in Figs. 11–13 for some values of and , where the following observations can be made. First, among the various detectors, exhibits the highest detection probability,
LEE et al.: LOCALLY OPTIMUM DETECTION OF SIGNALS IN ADDITIVE NOISES
Fig. 7. Typical realizations of Markov Gaussian random process when r =
Fig. 8. ARE
229
00 9 0
in the IFMG environment.
which gets higher (at the same signal strength) as or becomes larger, in the weak signal region. Second, the detection probabilities of and ( and ) get lower as ( ) becomes larger: this is because and ( and ) do not take the effect of ( ) into account in detecting signals. Finally, it is observed that the performance of and ( and ) gets closer to that of and ( and ), respectively, as ( ) gets closer to 0. Although we have not explicitly shown here due to restrictions on space, the observations made herein, if is
: ;
;
and 0:9.
Fig. 9. ARE
interpreted as [21].
in the IFMG environment.
, are applicable to the case of
also
V. CONCLUSION In this paper, we have addressed the problem of detecting weak known and random signals in observations corrupted by multiplicative and first-order Markov additive noises. When the means of the multiplicative noise components are not all zero, is derived by evaluating the first derivathe LO detector tive of the joint pdf of the observation vector. Similarly, for the
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Fig. 10. ARE
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in the IFMG environment.
Fig. 11. Detection probabilities of T ,T 0:0 in the IFMG environment. Note that T and T , respectively, when r = 0.
,T , and T when r = and T are the same as T
zero-mean multiplicative noise environment, we have obtained by evaluating the second derivative of the the LO detector joint pdf of the observation vector. Assuming specific conditions on the multiplicative and first-order Markov additive noise , , , and components, the LO detectors have been derived from and . It is observed that the LO test statistics depend on the means or correlations, but not on any other statistic, of the multiplicative noise components. On the other hand, the LO test statistics depend not on specific statistics but on the distributions of the additive noise components as a whole. The asymptotic and finite sample-size performance of the LO detectors has then been investigated when the additive noise components are identically distributed first-order Markov Gaussian. The dependence among the multiplicative noise components and that among the additive noise components
Fig. 12. Detection probabilities of T
,T
,T
, and T
when r
=
Fig. 13. Detection probabilities of T
,T
,T
, and T
when r
=
0:6 in the IFMG environment.
0:9 in the IFMG environment.
have been found to influence the performance of the detectors with varying degrees. Specifically, for the LO detectors designed when the means of the multiplicative noise components exhibits higher are not all zero, it is observed that (1) , (2) the detection probabilities detection probability than and both become higher as the correlation of of the multiplicative noise components becomes large in the weak over signal region, and (3) the outperformance of gets larger as the absolute value of the correlation of the additive noise components grows in the weak signal region. For the LO detectors designed for zero-mean multiplicative is the noise environment, (1) the detection probability of or highest among various detectors and gets higher as becomes larger in the weak signal region and (2) the detection and ( and ) get closer probabilities of to those of and ( and ), respectively, as ( ) gets closer to 0.
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APPENDIX A EVALUATION OF THE SECOND PARTIAL DERIVATIVE
231
and
As when are all zero, we need to evaluate the second-order derivative. For convenience, let (57) Then (64) none of which vanishes when and the test statistic have dividing the sum of (62)–(64) by
are all zero. Thus, we can be obtained by .
APPENDIX B EFFICACY OF THE LOCALLY OPTIMUM TEST STATISTIC
(58)
The first moment of under the null hypothesis can easily be shown to be zero from
(59)
(65) (60) Therefore, the second partial derivative of
and
is
(61) In (61), we have
(66) (62)
(63)
for using tical cases, where under the null hypothesis is
in prac. The variance of
(67)
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from (75) for (68)
(76) for
,
, and
(69)
and (77) , . Since
(70)
for . Therefore, from (32), (72), and (73), the efficacy of is the detector based on the test statistic (78)
, we have
(71)
APPENDIX C EFFICACY OF THE LOCALLY OPTIMUM TEST STATISTIC It is straightforward to show that for and that
using in practical cases,
(72)
APPENDIX D EFFICACY OF THE LOCALLY OPTIMUM TEST STATISTIC Again, it is straightforward to show that the first moment of under the null hypothesis is zero using (68)–(70) and (79) for
. Now using that
Next, we have (80) , and when shown in (68)–(70) and (79), the second moment of under the null hypothesis can be evaluated as for
(73) from (33) since (74) for
as
LEE et al.: LOCALLY OPTIMUM DETECTION OF SIGNALS IN ADDITIVE NOISES
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since
of
(83) and mean.
from the fact that
is zero-
ACKNOWLEDGMENT The authors wish to express their appreciation of the invaluably constructive suggestions and helpful comments from the anonymous reviewers and Associate Editor. REFERENCES
(81) Consequently, we have the efficacy
(82)
[1] C. Rao and B. Hassibi, “Analysis of multiple-antenna wireless links at low SNR,” IEEE Trans. Inf. Theory, vol. 50, pp. 2123–2130, Sep. 2004. [2] S. Khatalin and J. P. Fonseka, “Capacity of correlated Nakagami-m fading channels with diversity combining techniques,” IEEE Trans. Veh. Technol., vol. 55, pp. 142–150, Jan. 2006. [3] Y. Chen and V. K. Dubey, “Dynamic simulation model of indoor wideband directional channels,” IEEE Trans. Veh. Technol., vol. 55, pp. 417–430, Mar. 2006. [4] I. J. Kim, S. R. Park, I. Song, J. Lee, H. Kwon, and S. Yoon, “Detection schemes for weak signals in first order moving average of impulsive noise,” IEEE Trans. Veh. Technol., vol. 56, pp. 126–133, Jan. 2007. [5] H. V. Poor, “Signal detection in the presence of weakly dependent noise,” IEEE Trans. Inf. Theory, vol. 28, pp. 735–744, Sep. 1982. [6] G. V. Moustakides and J. B. Thomas, “Min-max detection of weak signal in -mixing noise,” IEEE Trans. Inf. Theory, vol. 30, pp. 529–537, May 1984. [7] H. V. Poor and J. B. Thomas, Advances in Statistical Signal Processing. Greenwich, CT: JAI, 1993, vol. 2, Signal Detection. [8] E. Kokkinos and A. M. Maras, “Locally optimum Bayes detection in nonadditive first-order Markov noise,” IEEE Trans. Commun., vol. 47, pp. 387–396, Mar. 1999. [9] I. Song, J. Bae, and S. Y. Kim, Advanced Theory of Signal Detection. New York: Springer, 2002. [10] X. Yang, H. V. Poor, and A. P. Petropulu, “Memoryless discrete-time signal detection in long-range dependent noise,” IEEE Trans. Signal Processing, vol. 52, pp. 1607–1619, Jun. 2004. [11] P. S. Rao, D. H. Johnson, and D. D. Becker, “Generation and analysis of non-Gaussian Markov time series,” IEEE Trans. Signal Processing, vol. 40, pp. 845–856, Apr. 1992. [12] D. Middleton, An Introduction to Statistical Communication Theory, Revised, Ed. Piscataway, NJ: IEEE Press, 1996. [13] S. A. Kassam, Signal Detection in Non-Gaussian Noise. New York: Springer, 1988. [14] I. Song and S. A. Kassam, “Locally optimum detection of signals in a generalized observation model: The known signal case,” IEEE Trans. Inf. Theory, vol. 36, pp. 502–515, May 1990. [15] R. S. Blum and S. A. Kassam, “Optimum distributed detection of weak signals in dependent sensors,” IEEE Trans. Inf. Theory, vol. 38, pp. 1066–1079, May 1992. [16] A. M. Maras, “Locally optimum Bayes detection in ergodic Markov noise,” IEEE Trans. Inf. Theory, vol. 40, pp. 41–55, Jan. 1994. [17] J. Bae, I. Song, H. Morikawa, and T. Aoyama, “Nonparametric detection of known signals based on ranks in multiplicative noise,” Signal Process., vol. 60, pp. 255–261, Jul. 1997. [18] B. R. Mahafza, Radar Systems Analysis and Design Using MATLAB. Boca Raton, FL: CRC, 2000. [19] A. B. Gershman, M. Pesavento, and M. G. Amin, “Estimating parameters of multiple wideband polynomial-phase sources in sensor arrays,” IEEE Trans. Signal Processing, vol. 49, pp. 2924–2934, Dec. 2001. [20] E. L. Lehmann, Testing Statistical Hypotheses, Second ed. New York: Wiley, 1986. [21] J. Lee, “Locally Optimum Detection Under First-Order Markov Noise Environment,” Ph.D. dissertation, Korea Advanced Inst. Science, Techn., Daejeon, 2006.
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[22] C. H. Park, I. Song, and D. K. Nam, Random Processes. Seoul, Korea: Saengneung, 2004. [23] R. Prasad, A. Kegel, and A. D. Vos, “Performance of microcellular mobile radio in a cochannel interference, natural, and man-made noise environment,” IEEE Trans. Veh. Technol., vol. 42, pp. 33–40, Feb. 1993. [24] P. A. Delaney, “Signal detection in multivariate class-A interference,” IEEE Trans. Commun., vol. 43, pp. 365–373, Feb.-Apr. 1995. [25] S. Miyamato, M. Katayama, and N. Morinaga, “Performance analysis of QAM systems under class A impulsive noise environment,” IEEE Trans. Electromagn. Compat., vol. 37, pp. 260–267, May 1995.
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 1, JANUARY 2008
Peter Harremoës (M’00) was born in 1964. He received the master of science degree from the University of Copenhagen, Copenhagen, Denmark, in 1989 with mathematics as major and archaeology as minor subject. He received the Ph.D. degree from Roskilde University, Roskilde, Denmark, 1993. From 2001 to 2005, he had a postdoctoral position at the Department of Mathematic, University of Copenhagen. Since 2006, he has been Scientific Staff Member at Centrum voor Wiskunde en Informatica, Amsterdam, The Netherlands, and is working in the group Quantum Computing and Information Theoretic Learning. Dr. Harremoës is Associate Editor of Entropy.
Miroslav Husák received the Ing. (B.Sc.) and Ing. (M.Sc.) degrees the Slovak University of Technology (STU) in 2005 and 2007, respectively. His research interests include error control coding and communications theory.
Syed A. Jafar (S’ 99–M’04) received the B. Tech. degree in electrical engineering from the Indian Institute of Technology (IIT), Delhi, India, in 1997, the M.S. degree in electrical engineering from California Institute of Technology (Caltech), Pasadena, in 1999, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 2003. He was a summer intern in the Wireless Communications Group of Lucent Bell Laboratories, Holmdel, NJ, in 2001. He was an Engineer in the Satellite Networks Division of Hughes Software Systems from 1997 to 1998 and a Senior Engineer at Qualcomm Inc., San Diego, CA, in 2003. He is currently an Assistant Professor in the Department of Electrical Engineering and Computer Science at the University of California Irvine. His research interests include multiuser information theory and wireless communications. Dr. Jafar received the NSF CAREER award in 2006. He is the recipient of the 2006 UC Irvine Engineering Faculty of the Year award for excellence in teaching. He serves as the Editor for Wireless Communication Theory and CDMA for the IEEE TRANSACTIONS ON COMMUNICATIONS.
Miroslav Janoˇsov received the Ing. (B.Sc.) and Ing. (M.Sc.) degrees from the Slovak University of Technology (STU) in 2005 and 2007, respectively. His research interests include error control coding and communications theory.
Jinhua Jiang (S’06) received the B.Eng. degree in computer engineering from the National University of Singapore, Singapore, in 2004. He is currently working toward the Ph.D. degree in the Electrical and Computer Engineering Department at the National University of Singapore. His research interests include multi-user information theory and low-density parity-check codes.
Yibo Jiang was born in Ningbo, China, in 1976. He received the B. Eng. degree in information engineering and the M. Eng. degree in communications and information systems from Zhejiang University, Hangzhou, China, in 1997 and 2000, respectively. In 2005, he received the Ph.D. degree in electrical engineering from University of Illinois at Urbana-Champaign. Since July 2005, he has been working on physical layer of wireless communications at Qualcomm, San Diego, CA.
Rolf Johannesson (S’72–M’75–SM’96–F’98) was born in Hässleholm, Sweden, on July 3, 1946. He received the M.S. and Ph.D. degrees in 1970 and 1975, respectively, both from Lund University, Lund, Sweden. Since 1976, he has been with Lund University where he is now Professor of Information Theory. His scientific interests include information theory, error correcting codes, and cryptography. In addition to papers and book chapters in the area of convolutional codes and cryptography, he has authored two textbooks on switching theory and digital design and one on information theory and coauthored Fundamentals of Convolutional Coding (Piscataway, NJ: IEEE
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Press, 1999), and Understanding Information Transmission (New York: IEEE Press/Wiley, 2005). Prof. Johannesson was awarded the degree of Professor honoris causa from the Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia, in 2000. He was elected a member of the Royal Swedish Academy of Engineering Sciences in 2006.
Rie Johnson (formerly, Rie Kubota Ando) received the Ph.D. degree in computer science from Cornell University, Ithaca, NY, in 2001. She was a Research Scientist at IBM T. J. Watson Research Center, Yorktown Heights, NY, until 2007. Her research interests are in machine learning and its applications.
Hong Jik Kim was born in Suwon, Korea, in 1972. He received the B.S. degree in electronic and electrical engineering from Pohang University of Science and Technology (POSTECH), Pohang, Korea, in 1995, and the M.S.E. and Ph.D. degrees in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1997 and 2005, respectively. He was a Research Assistant with the Department of Electrical Engineering and Computer Science, KAIST, from March 1997 to August 2002. He joined the Network Laboratory, LG Electronics, Anyang, Korea, in 2002. Since 2005, he has been with the Mobile Technology Team, LG-Nortel Co. Ltd., Anyang, Korea, where he is currently a Chief Research Engineer. His current research interests include mobile communications, OFDM communications, spread-spectrum communications, and SDR modem development.
Jon-Lark Kim (S’01–A’03) was born in Taejon, Korea, on November 15, 1970. He received the B.S. degree in mathematics from POSTECH, Pohang, Korea, in 1993, the M.S. degree in mathematics from Seoul National University, Seoul, Korea, in 1997, and the Ph.D. degree in mathematics from the University of Illinois at Chicago, in 2002 under the guidance of Prof. Vera Pless. From 2002 to 2005, he was with the Department of Mathematics at the University of Nebraska-Lincoln as a Research Assistant Professor. Since 2005, he has been an Assistant Professor in the Department of Mathematics at the University of Louisville, KY. His areas of interest include algebraic coding theory with connections to combinatorics, graph theory, finite geometry, number theory, algebraic geometry, and quantum information. Dr. Kim was awarded a 2004 Kirkman Medal of the Institute of Combinatorics and its Applications. He is a member of the editorial board of the International J. of Information and Coding Theory.
Peter T. Kim was born in Seoul, Korea, in 1957. He received the B.A. degree in economics from the University of Toronto, Toronto, ON, Canada, in 1980, the M.A. degree in economics from the University of Southern California, Los Angeles, in 1982, and the Ph.D. degree in mathematics from the University of California at San Diego in 1987. He is currently a Professor in the Department of Mathematics and Statistics, University of Guelph, where he has been since 1989. His research interests include nonparametric function estimation, inverse problems, geometry, and topology.
Ralf Koetter (M 91–SM 06) received a Diploma degree in electrical engineering from the Technical University Darmstadt, Darmstadt, Germany in 1990 and the Ph.D. degree from the Department of Electrical Engineering at Linköping University, Linköping, Sweden. From 1996 to 1997, he was a Visiting Scientist at the IBM Almaden Research Laboratory in San Jose, CA. He was a Visiting Assistant Professor at the University of Illinois at Urbana-Champaign, Urbana, IL, and a Visiting Scientist at CNRS in Sophia-Antipolis, France, from 1997 to 1998. In the years 1999–2006, he was member of the faculty of the University of Illinois at Urbana-Champaign. In 2006, he joined the faculty of the Technical University of München as the Head of the Institute for Communications Engineering. His research interests include coding and information theory and their application to communication systems.
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From 1999 to 2001, Prof. Koetter he served as Associate Editor for Coding Theory and Techniques for the IEEE TRANSACTIONS ON COMMUNICATIONS. In 2003, he concluded a term as Associate Editor for Coding Theory of the IEEE TRANSACTIONS ON INFORMATION THEORY. He received an IBM Invention Achievement Award in 1997, an NSF CAREER Award in 2000, an IBM Partnership Award in 2001, and a 2006 XEROX award for faculty research. He is corecipient of the 2004 Paper Award of the IEEE Information Theory Society. Since 2003 he has been a Member of the Board of Governors of the IEEE Information Theory Society.
Ja-Yong Koo was born in Seoul, Korea, in 1961. He received the B.S. degree in statistics from the Seoul National University in 1984 and the Ph.D. degree in statistics from the University of California at Berkeley in 1988. He is currently a Professor in the Department of Statistics, Korea University, where he has been since 2004. Formerly, he was with Hallym University from 1988 to 2001 and Inha University from 2001 to 2004. His research interests include nonparametric function estimation, wavelets, inverse problems (image reconstruction and deconvolution), data mining, bioinformatics and estimation of functions on non-Euclidean spaces.
Boris D. Kudryashov was born in Leningrad, U.S.S.R. (now St.-Petersburg, Russia), on July 9, 1952. He received the Diploma degree in electrical engineering in 1974, the Ph.D. degree in technical sciences in 1978, both from the Leningrad Institute of Aircraft Instrumentation (LIAP), and the Doctor of Science degree from the Institute for Information Transmission Problems (IPPI), Moscow, in 2005. Since 1978, he has been first Assistant Professor and then Associate Professor and Professor at the State University of Aerospace Instrumentation (former LIAP), St.-Petersburg, Russia. His research interests include coding theory, information theory, and applications to speech, audio, and image coding. He has published more than 70 papers in journals and proceedings of international conferences, and has nine U.S. patents in speech, audio, and image coding. Dr. Kudryashov served as a member of Organizing Committees of ACCT International Workshops.
Hyoungmoon Kwon (S’00–M’08) was born in Seoul, Korea, in 1976. He received the B.S. degree in electronics engineering from Yonsei University, Seoul, Korea, in 2000, and the M.S.E. and Ph.D. degrees in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2002 and 2007, respectively. He has been a Teaching and Research Assistant at the Department of Electrical Engineering, KAIST, from March 2000 to February 2007. Since March 2007, he has been with the Department of Electrical Engineering and Computer Science, KAIST, as a Postdoctoral Research Fellow. His research interests include spread-spectrum systems and detection theory.
Michael Langberg received the B.Sc. degree in mathematics and computer science from Tel-Aviv University, Israel, in 1996, and the M.Sc. and Ph.D. degrees in computer science from the Weizmann Institute of Science, Israel, in 1998 and 2003, respectively. He is a Senior Lecturer in the Computer Science Division at the Open University of Israel. From 2003 to 2006, he was a Postdoctoral Scholar in the Computer Science and Electrical Engineering Departments at the California Institute of Technology, Pasadena. His research is in the fields of Theoretical Computer Science and Information Theory. His work focuses on the design and analysis of algorithms for combinatorial problems; emphasizing on algorithmic aspects of information theory, the study of approximation algorithms for NP-hard problems, and probabilistic methods in combinatorics.
Jumi Lee (S’99–M’06) was born in Seoul, Korea, in 1974. She received the B.S. degree in mathematics and electronics engineering from Ewha Womans University, Seoul, Korea, in 1998, and the M.S.E. and Ph.D. degrees in electrical engineering from Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2000 and 2006, respectively.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 1, JANUARY 2008
She was a Research Assistant at the Department of Electrical Engineering and Computer Science (EECS), KAIST, from March 1999 to August 2006. From September 2006 to February 2007, she was with the Department of EECS, KAIST, as a Postdoctoral Research Fellow. Since May 2007, she has been with the 4G System Lab., Samsung Electronics Co., LTD., Suwon, Korea, where she is currently a Senior Engineer. Her current research interests include mobile communications, detection and estimation theory, and statistical signal processing.
Yoav Levinbook (S’04–M’07) was born in 1974. He received the B.Sc. degree (magna com laude) in electrical and electronic engineering from Tel-Aviv University, Tel-Aviv, Israel, in 2000 and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Florida, Gainesville, in 2006 and 2007, respectively. His doctoral work was supported by a University of Florida alumni fellowship. Prior to his graduate studies, he was with the Motorola Semiconductor, Herzliya, Israel, and Smartlink, Netanya, Israel, as an electrical engineer. In August 2007, he joined NextWave Wireless, San Diego, CA, where he is currently working on WiMax systems.
Hans-Andrea Loeliger (S’85–M’90–F’04) received the diploma in electrical engineering in 1985 and Ph.D. degree in 1992, both from ETH Zurich, Switzerland. From 1992 to 1995, he was with Linköping University, Linköping, Sweden. From 1995 to 2000, he was with Endora Tech AG, Basel, Switzerland, of which he is a Cofounder. Since 2000, he has been a Professor at ETH Zurich. His research interests lie in the broad areas of signal processing, information theory, communications and electronics.
Alan C. H. Ling received the Ph.D. degree from the University of Waterloo, Waterloo, ON, Canada, in 1997. He is currently an Associate Professor at the University of Vermont, Burlington, VT. His research interests include combinatorial design theory, coding theory, sequence designs and connection to design theory, applications of design theory to computer science, engineering and computational biology, and statistical design of experiments. Dr. Ling received the 2001 Kirkman Medal from the Institute of Combinatorics and its Applications.
San Ling received the B.A. degree in mathematics from the University of Cambridge, Cambridge, U.K., and the Ph.D. degree in mathematics from the University of California, Berkeley. Since April 2005, he has been a Professor with the Division of Mathematical Sciences, School of Physical and Mathematical Sciences, in the Nanyang Technological University, Singapore. Prior to that, he was with the Department of Mathematics, National University of Singapore. His research fields include: arithmetic of modular curves and application of number theory to combinatorial designs, coding theory, cryptography, and sequences.
Simon Litsyn (M’94–SM’99) was born in Khar’kov, U.S.S.R., in 1957. He received the M.Sc. degree from Perm Polytechnical Institute, Perm, U.S.S.R., in 1979 and the Ph.D. degree from Leningrad Electrotechnical Institute, Leningrad, U.S.S.R., in 1982, all in electrical engineering. Since 1991, he has been with the Department of Electrical Engineering– Systems, Tel-Aviv University, Tel-Aviv, Israel, where he is a Professor. His research interests include coding theory, communications, and applications of discrete mathematics. He authored Covering Codes (Elsevier, 1997) and Peak Power Control in Multicarrier Communications (Cambridge University Press, 2007). Dr. Litsyn received the Guastallo Fellowship in 1992. During 2000-2003, he has served as an Associate Editor for Coding Theory for IEEE TRANSACTIONS ON INFORMATION THEORY.
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Oren Somekh (S’90–M’06) Received the B.Sc., M.Sc, and Ph.D. degrees in electrical engineering from the Technion–Israel Institute of Technology, Haifa, Israel, in 1989, 1991, and 2005, respectively. During 1991–1996, he served in the Israel Defense Forces (IDF) in the capacity of a Research Engineer. During 1998–2002, he was the Vice President of Research and Development and later Chief Technical Officer (CTO) of Surf Communication Solutions Ltd., Yokneam, Israel. From April 2005 to October 2006, he was a Visiting Research Fellow at the Center for Communications and Signal Processing Research, NJIT, Newark, NJ. Since November 2006, he has been a Visiting Research Fellow at the Electrical Engineering Department of Princeton University, Princeton, NJ. His research interests include information theoretical aspects of cooperative wireless networks. Dr. Somekh is a recipient of the Marie-Curie Outgoing International Fellowship (OIF).
Iickho Song (S’80–M’87–SM’96) was born in Seoul, Korea, in 1960. He received the B.S.E. (magna cum laude) and M.S.E. degrees in electronics engineering from Seoul National University in 1982 and 1984, respectively, and the M.S.E. and Ph.D. degrees in electrical engineering from the University of Pennsylvania, Philadelphia, in 1985 and 1987, respectively. He was a Member of Technical Staff at Bell Communications Research in 1987. In 1988, he joined the Department of Electrical Engineering, Korea Advanced Science and Technology (KAIST), where he is now a Professor. He has coauthored Advanced Theory of Signal Detection (New York: Springer, 2002) and Random Processes (in Korean; Saengneung, 2004), and has published a number of papers on signal detection, statistical signal processing, and mobile communications. His research interests include detection and estimation theory, statistical communication theory and signal processing, and mobile communications. Dr. Song served as the Treasurer of the IEEE Korea Section in 1989. He has received many awards, including the Young Scientists Award presented by the President of the Republic of Korea in 2000 and the Achievement Award from the Institution of Engineering and Technology in 2006.
Umberto Spagnolini (SM’03) received the Dott.Ing. Elettronica degree (cum laude) from Politecnico di Milano, Milan, Italy, in 1988. Since 1988, he has been with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, where he is Full Professor in Telecommunications. His general interests are in the area of statistical signal processing. The specific areas of interest include: distributed estimation, synchronization, and space–time processing for wireless communication systems, estimation and tracking of timevarying parameters, signal processing and wavefield interpolation applied to UWB radar, geophysics, and remote sensing. He is the cofounder of WiSyTech (Wireless System Technology), a spinoff company of Politecnico di Milano on Software Defined Radio. Dr. Spagnolini served (1999–2006) as an Associate Editor for the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING.
Chi Wan Sung received the B.Eng., M.Phil., and Ph.D. degrees in information engineering from the Chinese University of Hong Kong in 1993, 1995, and 1998, respectively. From 1998 to 1999, he worked in the Chinese University of Hong Kong as an Assistant Professor. He joined City University of Hong Kong in 2000, and is now an Assistant Professor of the Department of Electronic Engineering. His current research interests include multiuser information theory, wireless networking, and distributed algorithms.
Raghava N. Swamy received the B.Tech. degree in electrical engineering from the Indian Institute of Technology, Madras, India, in 2006. He is currently a graduate student in the Electrical and Computer Engineering Department at the University of California, San Diego, La Jolla, CA. His research interests include wireless communication and signal processing. M. Swamy was awarded the National Talent Search Scholarship in 2000, a gold medal in the Indian National Physics Olympiad in 2002, and the Electrical and Computer Engineering Department fellowship at the University of California, San Diego, in 2006.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 1, JANUARY 2008
Xusheng Tian received the B.S. degree from Southeast University, Nanjing, China, in 1991, the M.S. degree from Tsinghua University, Beijing, China, in 1994, and the Ph.D. degree from Rensselaer Polytechnic Institute, Troy, NY, in 2002, all in electrical engineering. He was the Principal Engineer at Premonitia, Inc., in 2002. From 2003 to 2004, he was a Visiting Assistant Professor of electrical and computer engineering at University of Miami, Coral Gables, FL. Currently, he works in the area of quantitative finance at Bloomberg LP, New York, NY. His research interests include stochastic modeling, video transmission over packet networks, computer communication networks with a focus on measurementbased network traffic modeling and network management, and resource management of wireless networks. Igor Vajda (M’90–SM’91–F’01) was born in Martin, Czechoslovakia. He graduated in mathematics from the Czech Technical University, Prague, 1965. He received the Ph.D. degree in probability and statistics from the Charles University, Prague, in 1968, and the Dr.Sci. degree in mathematical informatics from the same university in 1990. He has served as a Research Assistant in 1965, in 1966 he became a Researcher, and since 1990, has been a principal researcher at the Institute of Information Theory and Automation, Czech Academy of Sciences, Prague. During 1966–2003, he was for short periods of time a Visiting Scientist or a Visiting Professor at the Institute of Information Transmission Theory, Moscow, U.S.S.R.; University of Rostock, Germany; Complutense University of Madrid, Spain; University of Maryland, Baltimore; Catholic University of Leuven, Belgium; Technical University of Budapest, Hungary; University of Montpellier II, France; and M. Hernandez University of Alicante, Spain. Recently, he also served as a Vice-Chairman of the Board of doctoral studies in mathematical engineering at the Czech Technical University in Prague. His research interests include information measures and statistical distances, information-theoretic statistical inference, statistical inference about random processes, and statistical decisions. Dr. Vajda is an Associate Editor of Kybernetika, Revista Matematica Complutense, Test, and Applications of Mathematics. In 1988, he was awarded the Prize of the Czechoslovak Academy of Sciences for his part in a research on speech coding. The American Journal of Mathematical and Management Sciences awarded him in 1997 the Jacob Wolfowitz Prize for a joint paper with L. Györfi and E. C. van der Meulen. Since 1990, he has co-organized the Prague Conferences on Information Theory, Statistical Decision Functions, and Random Processes and in 2002, he co-organized the European Meeting of Statisticians in Prague. A. J. Han Vinck (M’77–SM’91– F’03) received the Ph.D. degree in electrical engineering from the University of Eindhoven, Eindhoven, The Netherlands, in 1980. Since 1990, he has been a Full Professor in Digital Communications at the University of Essen, Essen, Germany. From 1991 to 1993, from 1998 to 2000, and from 2006 until now, he has served as the Director of the Institute for Experimental Mathematics in Essen. From 1997 to 1999, he was also the Director of the Post-Graduate School on Networking, “CINEMA”. From 2000 to 2004, he was the Chairman for the communication division of the Institute for Critical Infrastructures, CRIS. He was an Adjoint Professor in 2003 and a Guest Profession in 2004 at the Sun Yat-Sen University, Kaohsiung, Taiwan. His interests includes Information and Communication theory, Coding and Network aspects in digital communications. Prof. Vinck he organized the IEEE Information Theory Workshop in Veldhoven, the Netherlands, in 1990. From 1995 to 1998, he was founding Chairman of the IEEE German Chapter on Information Theory. From 1997 to 2006, he served on the Board of Governors of the IEEE Information Theory Society. In 1997, he acted as Co-Chairman for the 1997 IEEE Information Theory Symposium in Ulm, Germany. In 1999, he was the Program Chairman for the IEEE IT Workshop in Kruger Park, South Africa. From 1999 to 2000, he served as Chairman of the Benelux Information and Communication Theory Society. From 2001 to 2002, he served as Member-at-Large in the Meetings and Services Committee for the IEEE. In 2003, he was elected President of the IEEE Information theory Society. IEEE elected him as a fellow for his “Contributions to Coding Techniques”. He is the initiator of the Japan-Benelux (now Asia-Europe) Workshops on Information Theory and the International Winter School on Coding, Cryptography and Information theory. Since 1997, he has been, and still supports, the organization of the series of conferences on Power Line Communications and its Applications. In 2006, he received the IEEE ISPLC2006 Achievement award in Olando, FL, for his “Contributions to Power Line Communications”. He is Co-Founder and President of the Shannon and the Gauss foundations. These foundations stimulate research and help young scientists in the field of Information Theory and Digital Communications.
