Eigenvalues of the Gram matrix formed from received data frequently appear in sufficient detection statistics for multi-channel detection with Generalized Likelihood Ratio (GLRT) and Bayesian tests. In a frequently presented model for passive radar, in which the null hypothesis is that the channels contain only complex white Gaussian noise and the alternative hypothesis is that the channels contain a common rank-one signal in the mean, the GLRT statistic is the largest eigenvalue λ1of the Gram matrix formed from data, which has a Wishart distribution. Although exact expressions for the distribution of λ1are known under both hypotheses, numerically calculating values of these distribution functions presents difficulties in cases where the dimension of the data vectors is large. Following on recent work addressing this issue under the null hypothesis, this paper presents a method to calculate values of this distribution under the alternative hypothesis, allowing tractable computation of receiver operating characteristic curves.

A class of detectors for cyclostationarity is introduced. These detectors are based on the use of generalized coherence to measure correlation among two or more collections of random vectors. The generalized coherence framework allows any finite collection of pertinent samples of the cyclic auto-correlation function estimates formed from the measured signal data to be combined into the detection statistic. The performance of this approach is demonstrated and compared against other established cyclostationarity detectors in both a cognitive radio scenario and a multi-channel passive surveillance scenario.

In multi-channel detection, sufficient statistics for Generalized Likelihood Ratio and Bayesian tests are often functions of the eigenvalues of the Gram matrix formed from data vectors collected at the sensors. When the null hypothesis is that the channels contain only independent complex white Gaussian noise, the distributions of these statistics arise from the joint distribution of the eigenvalues of a complex Wishart matrix G. This paper considers the particular case of the largest eigenvalue λ1of G, which arises in passive radar detection of a rank-one signal. Although the distribution of λ1is known analytically, calculating its values numerically has been observed to present formidable difficulties. This is particularly true when the dimension of the data vectors is large, as is common in passive radar applications, making computation of accurate detection thresholds intractable. This paper presents results that significantly advance the state of the art for this problem.

This paper considers the problem of testing for mutual independence of multiple sets of complex Gaussian vectors. This problem has classical roots in statistics and has been of recent interest in the signal processing literature in connection with multi-channel signal detection. The maximal invariant statistic for this problem is described both as a collection of subspaces of the data space (i.e., points on a complex Grassmannian manifold) and as a corresponding set of complex matrices. The distribution of the maximal invariant is also derived under both hypotheses in the testing problem.

By taking advantage of active sensing using radio waves, radar sensors can offer several advantages over passive sensors. Although much attention has been given to multistatic and multiple-input-multiple-output (MIMO) radar concepts, little has been paid to understanding radar networks (i.e., multiple individual radars working in concert). In this context, we study the coverage problem of a bistatic radar (BR) sensor network, which is very challenging due to the Cassini oval sensing region of a BR and the coupling of sensing regions across different BRs. In particular, we consider the problem of deploying a network of BRs in a region to maximize the worst-case intrusion detectability, which amounts to minimizing the vulnerability of a barrier. We show that it is optimal to place BRs on the shortest barrier if it is the shortest line segment that connects the left and right boundary of the region. Based on this, we study the optimal placement of BRs on a line segment to minimize its vulnerability, which is a nonconvex optimization problem. By exploiting certain specific structural properties pertaining to the problem (particularly an important structure of detectability), we characterize the optimal placement order and the optimal placement spacing of the BR nodes, both of which present elegant balanced structures. Our findings provide valuable insights into the placement of BRs for barrier coverage. To our best knowledge, this is the first work to explore the barrier coverage of a network of BRs.

We consider the problem of fusing measurements in a sensor network, where the sensing regions overlap and data are nonnegative real numbers, possibly resulting from a count of indistinguishable discrete entities. Because of overlaps, it is generally impossible to fuse this information to arrive at an accurate value of the overall amount or count of material present in the union of the sensing regions. Here we study the computation of the range of overall values consistent with the data and provide several results. Posed as a linear programming problem, this leads to questions associated with the geometry of the sensor regions, specifically the arrangement of their nonempty intersections. We define a computational tool called the fusion polytope , based on the geometry of the sensing regions. Its properties are explored, and in particular, a topological necessary and sufficient condition for this to be in the positive orthant, a property that considerably simplifies calculations, is provided. We show that in two dimensions, inflated tiling schemes based on rectangular regions fail to satisfy this condition, whereas inflated tiling schemes based on hexagons do.

A geometrical perspective is introduced that enables unification and generalization of several results regarding the distributions of quantities that arise in connection with an important class of multiple-channel detectors. Standard models on sets of normalized vectors following from joint Gaussian assumptions in this context are relaxed to the geometrically appealing model of uniform distributions on the Stiefel manifold of K-frames in N-dimensional space. In addition to bolstering geometric insight, several prior results are subsumed and strengthened by results obtained under this formulation. Additionally, a generalization of a classical theorem of W. G. Cochran is enabled by this framework.

A measure of the smallest angle between two p-dimensional subspaces of a higher-dimensional vector space is introduced. This p-coherence is seen to generalize standard measures of coherence, and its relationship to canonical correlation is noted. The use of p-coherence as a test statistic for the presence of a common p-dimensional signal subspace in data collected by two arrays of K > p sensors is proposed.

Estimating the number of linearly independent signals impinging on a collection of receivers is of increasing recent interest, particularly in connection with MIMO systems for communications and sensing. This paper derives maximum a posteriori estimators for signal rank from noisy data collected at multiple receivers. Situations in which noise variance is known and unknown are both treated. These estimators are shown to significantly outperform maximum-likelihood/BIC rank estimators.

The problem of detecting the presence of a known signal in multiple channels of additive noise, such as occurs in active radar with a single transmitter and multiple geographically distributed receivers, is addressed via coherent multiple-channel techniques. The transmitted signal replica is treated as one channel in a suitable M-channel detector with the remaining M - 1 channels comprised of data from the receivers. To accommodate this approach, an invariance result for the distribution of the eigenvalues of a Gram matrix is derived. The result implies that false alarm probabilities for any detector based on a function of the eigenvalues are not changed if one data channel contains a signal replica, provided that the other M - 1 channels only Gaussian noise and all channels are independent.

This paper considers the problem of testing for the independence among multiple (≥ 2) random vectors with each random vector representing a time series captured at one sensor. Implementing the Generalized Likelihood Ratio Test involves testing the null hypothesis that the composite covariance matrix of the channels is block-diagonal through the use of a generalized Hadamard ratio. Using the theory of linear prediction and its connection with Gram determinants, it is shown that this generalized Hadamard ratio can be written as a product of scalars which are independently drawn from a beta distribution under the null hypothesis. This result is useful from a Monte Carlo analysis standpoint in that it is much more computationally efficient to form a product of scalar beta random variables than it is to compute the determinant of complex Wishart matrices.

Bayesian and generalized likelihood ratio tests are derived for detection of a common unknown signal of known rank K in M > K independent channels of white gaussian noise. The cases of known and unknown noise variance are both treated. These derivations encompass the development of explicit expressions for an invariant measure on the grassmannian manifold of K-dimensional subspaces of complex N-dimensional space and parameterization of this manifold to enable the calculation of the necessary marginalization integrals. Performance of the detectors is compared by simulation.

Radar sensors, which actively transmit radio waves and collect RF energy scattered by objects in the environment, offer a number of advantages over purely passive sensors. An important issue in radar is that the transmitted energy may be scattered by objects that are not of interest as well as objects of interest (e.g., targets). The detection performance of radar systems is affected by such clutter as well as noise. Further, in many applications, clutter can be substantially stronger than the signals of interest. To combat the effect of clutter, a popular method is to take advantage of the Doppler frequency shift (DFS) extracted from the echo signal due to the relative motion of a target with respect to the radar. Unfortunately, a sensor coverage model that only depends on the distance to a target would fail to capture the DFS. In this paper, we set forth the concept of Doppler coverage for a network of spatially distributed radars. Specifically, a target is said to be Doppler-covered if, regardless of its direction of motion, there exists some radar in the network whose signalto-noise ratio (SNR) is sufficiently high and the DFS at that radar is sufficiently large. Based on the Doppler coverage model, we first propose an efficient method to characterize Dopplercovered regions for arbitrarily deployed radars. Then we design an algorithm for deriving the minimum radar density required to achieve Doppler coverage in a region under any polygonal deployment pattern, and further apply it to investigate the regular triangle based deployment.

By taking advantage of active sensing using radio waves, radar sensors can offer several advantages over passive sensors. Although much recent attention has been given to multistatic and MIMO radar concepts, little has been paid to understanding the performance of radar networks (i.e., multiple individual radars working in concert). In this context, we study the optimal placement of a bistatic radar (BR) sensor network for barrier coverage. The coverage problem in a bistatic radar network (BRN) is challenging because: 1) in contrast to the disk sensing model of a traditional passive sensor, the sensing region of a BR depends on the locations of both the BR transmitter and receiver, and is characterized by a Cassini oval; 2) since a BR transmitter (or receiver) can potentially form multiple BRs with different BR transmitters (or receivers, respectively), the sensing regions of different BRs are coupled, making the coverage of a BRN highly non-trivial. This paper considers the problem of deploying a network of BRs in a region for maximizing the worst-case intrusion detectability, which amounts to minimizing the vulnerability of a barrier. We show that the shortest barrier-based placement is optimal if the shortest barrier is also the shortest line segment connecting the region's two boundaries. Based on this observation, we study the optimal placement of the BRs on a line segment for minimizing its vulnerability, which is a non-convex optimization problem. By exploiting some specific structural properties pertaining to the problem (particularly an important structure of detectability), we find the optimal placement order and the optimal placement spacing of the BR nodes, both of which exhibit elegant balanced structures. Our findings give valuable insight for the placement of BRs for barrier coverage. To our best knowledge, this is the first work to explore the coverage of a network of BRs.

We study the diffusion of information in an overlaying social-physical network. Specifically, we consider the following set-up: There is a physical information network where information spreads amongst people through conventional communication media (e.g., face-to-face communication, phone calls), and conjoint to this physical network, there are online social networks where information spreads via web sites such as Facebook, Twitter, FriendFeed, YouTube, etc. We quantify the size and the critical threshold of information epidemics in this conjoint social-physical network by assuming that information diffuses according to the SIR epidemic model. One interesting finding is that even if there is no percolation in the individual networks, percolation (i.e., information epidemics) can take place in the conjoint social-physical network. We also show, both analytically and experimentally, that the fraction of individuals who receive an item of information (started from an arbitrary node) is significantly larger in the conjoint social-physical network case, as compared to the case where the networks are disjoint. These findings reveal that conjoining the physical network with online social networks can have a dramatic impact on the speed and scale of information diffusion.

An information-geometric approach to sensor management is introduced that is based on following geodesic curves in a manifold of possible sensor configurations. This perspective arises by observing that, given a parameter estimation problem to be addressed through management of sensor assets, any particular sensor configuration corresponds to a Riemannian metric on the parameter manifold. With this perspective, managing sensors involves navigation on the space of all Riemannian metrics on the parameter manifold, which is itself a Riemannian manifold. Existing work assumes the metric on the parameter manifold is one that, in statistical terms, corresponds to a Jeffreys prior on the parameter to be estimated. It is observed that informative priors, as arise in sensor management, can also be accommodated. Given an initial sensor configuration, the trajectory along which to move in sensor configuration space to gather most information is seen to be locally defined by the geodesic structure of this manifold. Further, divergences based on Fisher and Shannon information lead to the same Riemannian metric and geodesics.

The generalized coherence (GC) estimate is a well studied statistic for detection of a common but unknown signal on several noisy channels. In this paper, it is shown that the GC detector arises naturally from a Bayesian perspective. Specifically, it is derived as a test of the hypothesis that the signals in the channels are independent Gaussian processes against the hypothesis that the processes have some arbitrary correlation. This is achieved by introducing suitable non-informative priors for the covariance matrices across the channels under the two hypotheses. Subsequently, reduced likelihoods are obtained by marginalizing the joint distribution of the data and the covariance matrix in each case. The likelihood ratio is then shown to be a monotonic function of the GC detection statistic. This derivation extends to the case of time-correlated signals, allowing comparison with the generalized likelihood ratio test (GLRT) recently proposed by Ramírez et al.

We consider a cyber-physical system consisting of two interacting networks, i.e., a cyber network overlaying a physical network. It is envisioned that these systems are more vulnerable to attacks since node failures in one network may result in (due to the interdependence) failures in the other network, causing a cascade of failures that would potentially lead to the collapse of the entire infrastructure. The robustness of interdependent systems against this sort of catastrophic failure hinges heavily on the allocation of the (interconnecting) links that connect nodes in one network to nodes in the other network. In this paper, we characterize the optimum inter-link allocation strategy against random attacks in the case where the topology of each individual network is unknown. In particular, we analyze the “regular” allocation strategy that allots exactly the same number of bidirectional internetwork links to all nodes in the system. We show, both analytically and experimentally, that this strategy yields better performance (from a network resilience perspective) compared to all possible strategies, including strategies using random allocation, unidirectional interlinks, etc.

A recent approach to compressed sensing using deterministic sensing matrices formed from discrete frequency-modulated chirps or from Reed–Muller codes is extended to support efficient deterministic reconstruction of signals that are much less sparse than envisioned in the original work. In particular, this allows the application of this approach in imaging. The reconstruction algorithm developed for images incorporates several new elements to improve computational complexity and reconstruction fidelity in this application regime.

A two-user interference channel is considered where each transmitter has access to a part of the information intended to the other destination. A primary objective is to maximize the information rates, by exploring the cooperation between the transmitters for interference mitigation, based on the partial side information. It is clear that full cooperation between the transmitters is not possible since each transmitter has only a part of the side information. With this insight, several “layered coding” schemes, consisting of binning and superposition at different stages, are developed. These schemes are are carefully built on coding strategies for the classical interference channel and node cooperation mechanisms. In particular, two layered coding schemes, which are based on a combination of MIMO broadcast coding and the Han-Kobayashi (HK) coding, are thoroughly studied : The first one, namely layered coding with binning, makes heavy use of the Gelfand-Pinsker binning and the HK coding and the second one, namely layered superposition coding, involves superposition coding over different tiers. Rate regions corresponding to the proposed schemes are derived. Then the application of these coding schemes are illustrated for the Gaussian case and numerical results corroborate that the proposed layered coding schemes yield substantial gains at high SNR.

The effective and reliable detection of explosive compounds in complex environments is an important problem in many environment and security-related applications. This paper develops an explosive detection approach based on multi-modal sensing and sensor data fusion. A least-squares feature extraction technique is designed to isolate explosive signatures in data collected using electrochemical and polymer nanojunction sensors. The information obtained from the two sensors is then efficiently combined using a Bayesian decision fusion scheme. Results are presented for the detection of the explosive compound TNT showing the merit of the proposed approach.

An image reconstruction algorithm using compressed sensing (CS) with deterministic matrices of second-order Reed-Muller (RM) sequences is introduced. The 1D algorithm of Howard et al. using CS with RM sequences suffers significant loss in speed and accuracy when the degree of sparsity is not high, making it inviable for 2D signals. This paper describes an efficient 2D CS algorithm using RM sequences, provides medical image reconstruction examples, and compares it with the original 2DCS using noiselets. This algorithm entails several innovations that enhance its suitability for images: initial best approximation, a greedy algorithm for the nonzero locations, and a new approach in the least-squares step. These enhancements improve fidelity, execution time, and stability in the context of image reconstruction.

Nonlinear data-driven dimensionality reduction techniques have recently gained popularity due to the emergence of high dimensional data sets. The algorithmic complexity and storage requirements of these techniques, however, can make them prohibitive in resource-limited applications. It is therefore beneficial to reduce the number of exemplar samples required for performing an out-of-sample extension to a test point. In this paper, we propose a novel method for selecting a minimal set of exemplars and performing the out-of-sample extension. In the case of two-class target recognition with synthetic aperture radar (SAR) data, we compare the efficacy of the proposed approach with other approaches for selecting a subset of the available training samples. We show that the proposed algorithm outperforms the existing methods by providing low-dimensional embeddings that maintain interclass separability using fewer retained exemplars.

In this paper, we propose a method to exploit waveform agility in modern radars to improve performance in the challenging task of detecting small targets on the ocean surface in heavy clutter. The approach exploits the compound-Gaussian model for sea clutter returns to achieve clutter suppression by forming an orthogonal projection of the received signal into the clutter subspace. Waveform scheduling is then performed by incorporating the information about the clutter into the design of the next transmitted waveform. A simulation study demonstrates the effectiveness of our approach.

Continuing advances in device technologies are producing active sources that are increasingly agile in their waveform transmission capabilities. Additionally, modern receivers are generally designed to support adaptive processing. This combination presents rich opportunities for new sensing methodologies that exploit the agility of both transmitter and receiver to yield improved performance in such applications as detection, classification, and tracking. The paper outlines some of these opportunities and sketches some challenges entailed in realizing these opportunities and approaches being brought to bear on these challenges.

Time-warped signal spaces have received attention in the literature. Among the topics of particular interest are sampling of time-warped signals and signal analysis using warped analysis functions, including wavelets. This correspondence introduces a reproducing kernel (RK) structure for time-warped signal spaces that unifies multiple perspectives on sampling in such spaces.

In situations where the presence of a signal is to be detected in several noisy channels, often one channel will have higher signal-to-noise ratio (SNR) than the others. When the SNR on one channel is sufficiently high that the signal can be extracted from that channel, it may be possible to use the extracted signal to aid in detecting the presence of the signal on the other channels. The matching pursuit time-frequency method with matched signal dictionaries is used to extract a chirp signal from a noisy channel. The extracted signal is used in one channel of a generalized coherence (GC) detector with the goal of detecting the presence of the signal on other, even noisier, channels. This approach is compared via simulation to a GC detector that does not pre-process the highest SNR channel to extract the signal. Detector performance is shown to be significantly enhanced by matching pursuit signal extraction prior to coherence estimation.

Reconstruction of a signal from its spectral phase or magnitude is in general an ill-posed problem. Various conditions restricting the class of signals under consideration have been shown to be sufficient to regularize the problem so that a unique or essentially unique signal corresponds to any given spectral magnitude or spectral phase function. This paper shows that a finite discrete-time signal is characterized by its spectral magnitude (or phase) and the spectral magnitude (or phase) of an ancillary signal obtained by windowing the original signal.

Given a signal space of functions on the real line, a time-warped signal space consists of all signals that can be formed by composition of signals in the original space with an invertible real-valued function. Clark's (1985) theorem shows that signals formed by warping bandlimited signals admit formulae for reconstruction from samples. This paper considers time warping of more general signal spaces in which Kramer's (1959) generalized sampling theorem applies and observes that such spaces admit sampling and reconstruction formulae. This observation motivates the question of whether Kramer's theorem applies directly to the warped space, which is answered affirmatively by introduction of a suitable reproducing kernel Hilbert space structure. This result generalizes one of Zeevi (1993), who pointed out that Clark's theorem is a consequence of Kramer's.

The use of the generalized coherence estimate as a statistic for detection of a common signal in multiple independent channels of additive gaussian noise has been studied in several papers. This work has relied on simulations to evaluate detector performance because the distribution of the generalized coherence estimate with signal present is unknown. This paper derives an asymptotic expression for the non-null distribution of the estimate as the length of the sample sequences approaches infinity, develops an asymptotic performance analysis based on this distribution, and compares the receiver operating characteristics derived from this theoretical approach to those obtained using simulations with large sample sequence lengths.

This paper considers a class of detection/localization problems in which the detector offers multiple operating modes. The modes differ in their detection performance and geographical coverage: "focused" modes offer higher detection performance but less coverage area than "broad search" modes. It is assumed that a signal source is to be detected and localized using a sequence of tests, each possibly employing a different mode. The goal is to determine a strategy for mode selection in the sequence of tests that will yield optimal payoff in terms of a pre-established criterion. A mathematical model capturing the key characteristics of this situation is proposed and used to develop optimal mode selection strategies.

This paper is concerned with a class of dynamic estimation problems in which the estimator has the ability to dynamically select, from among a temporally evolving set of possibilities, the source of the data on which the estimate will be based. After motivating and formulating this class of "attentive estimation" problems in some generality, the paper focuses on the special case in which the state of a linear discrete-time dynamical system driven by gaussian noise is to be estimated using linear measurements corrupted by additive gaussian noise. This differs from the standard Kalman filtering problem in that the measurement map at each time step is selectable from a pre-determined set of such maps. When the system dynamics and noise statistics are known, the problem admits a "sensor scheduling" solution i.e., a criterion for measurement selection that can be used to determine an optimal sequence of output functions in an open-loop fashion prior to the onset of estimation. When the noise statistics or other parameters are unknown, however, closed-loop adaptive strategies for measurement selection can improve estimator performance.

A nonideal two-dimensional optical system, as encountered in digital holographic data storage applications, can modify the intensity of transmitted digital data through beam shaping, focal surface distortion, and moire patterns. Such changes in intensity can have significant adverse effects on digital data recovery at the receiver (e.g., a CCD camera). Current research seeks to detect and correct classes of such distortion so that recovery methods can be applied to the received data. This paper discusses methods used to predict the locations of bit errors in the recovered data. Prediction information may be used as weighting information in the recovery algorithm and in the design of channel codes. Furthermore, the higher the level of distortion that can be tolerated in the system, the lower the cost of the corresponding lenses, making the system more tractable for commercialization.

This paper describes the design and implementation of a simple Internet-based "virtual systems laboratory" teaching tool. The tool is intended to provide a mechanism for assignment of system characterization/identification exercises in courses where students have access to Internet electronic mail, but not necessarily to a common computing environment or more sophisticated Internet resources such as Mosaic or Netscape. Specific exercises developed for use with this software in classes at Arizona State University are described.

This paper presents a detector for cyclostationary signals that uses an estimate of spectral autocoherence as a detection statistic. The detector's probability of false alarm is analytically derived and verified with computer simulations. Receiver operating characteristic curves are determined from the simulation data and the analytical expression for the probability of false alarm. A detector for polycyclic signals based on generalized coherence estimation is also described but is not analyzed.

Motivated by their utility in coding of digital communication signals, numerous examples of orthonormal wavelet sets that are generated from a bandlimited mother wavelet have been constructed. In most of these examples, orthogonality of the replicates at different scales is achieved by ensuring that the frequency domain supports at every scale are disjoint. Given the mother wavelet for such a set, this paper describes how it can be extended to yield a new bandlimited mother wavelet which generates an orthonormal set and whose dilated replicates overlap in the frequency domain. The utility of these wavelets to improve robustness of wavelet-coded communication signals with respect to demodulation by an unintended receiver is also discussed. The possibility of developing a new class of symbols by unitary warping of the frequency axis is also considered, but is shown not to be viable. If used as symbols in a wavelet-based multiple access communication system, these wavelets should improve robustness with respect to demodulation.

Distortion due to complete loss or perturbation of phase or magnitude information is considered for energy normalized images. With certain statistical assumptions on the image structure, phase information is shown to be more crucial.

The probability distribution function of the three-channel generalized coherence estimate is shown not to depend on the statistical behavior of the data on one channel, provided the other two channels contain white Gaussian noise and all channels are independent. A technique for multiple-channel matched filtering that is made viable by this invariance result is also discussed.>

The performance of nonparametric multiple-channel detectors based, respectively on the multiple coherence estimate and the generalized coherence estimate in detecting broadband signals in simulated deep-ocean noise are evaluated and compared. Probabilities of detection estimated from simulations are plotted against probabilities of false alarm predicted theoretically from the distribution functions of the detection statistics under the assumption that the channels contain independent white noise. The actual false-alarm performance for both detection statistics in the presence of simulated ocean noise is seen to be significantly different from what is predicted theoretically for white noise. The common practice of establishing detection thresholds for coherence detectors by using theoretically predicted false alarm probabilities is thus shown to be of limited value when colored noise is involved.>

The sampling and reconstruction problem for time-warped bandlimited signals is examined, and its relationship to the more general sampling problem for nonbandlimited signals is discussed. In particular, time-warped bandlimited signals in L/sup 2/ are considered, covering approximation of L/sup 2/ signals, representational ambiguity, and approximation with nonaffine warpings.>

The properties and performance of the generalized coherence (GC) estimate as a statistic for detecting the presence of a common but unknown signal on several noisy channels are examined. The H/sub 0/ distribution function of the GC estimate derived in earlier work is applied to develop detection thresholds corresponding to a range of false alarm probabilities for various numbers of channels and data sequence lengths. Results of simulations to determine the performance of a GC-based detector as a function of the signal-to-noise ratios on the data channels are presented.>

Normalization of signals is used as a means of dynamic range regulation in many applications, but the inherent nonlinear nature of normalization and its interaction with other commonly used tools of signal processing are often ignored. The author presents some of the issues involved in signal normalization and examines the role of normalization in the context of spectrum analysis. In particular, the question of the relative significance of spectral phase and magnitude is considered for normalized signals.>

The authors introduce a test for detecting the presence of a common signal on M noisy channels that generalizes a popular technique using the magnitude-squared coherence (MSC) estimate for detecting a common signal on two channels. The basis of this test is the generalized coherence estimate, a statistic involving the normalized M*M Gram matrix determinant, which reduces to the MSC estimate for M=2. The distribution of the generalized coherence estimate is derived under the assumptions of independent noise and no common signal on the M channels.>