1,934 research outputs found

    Operator-Valued Frames for the Heisenberg Group

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    A classical result of Duffin and Schaeffer gives conditions under which a discrete collection of characters on R\mathbb{R}, restricted to E=(1/2,1/2)E = (-1/2, 1/2), forms a Hilbert-space frame for L2(E)L^2(E). For the case of characters with period one, this is just the Poisson Summation Formula. Duffin and Schaeffer show that perturbations preserve the frame condition in this case. This paper gives analogous results for the real Heisenberg group HnH_n, where frames are replaced by operator-valued frames. The Selberg Trace Formula is used to show that perturbations of the orthogonal case continue to behave as operator-valued frames. This technique enables the construction of decompositions of elements of L2(E)L^2(E) for suitable subsets EE of HnH_n in terms of representations of HnH_n

    Coordinating Complementary Waveforms for Sidelobe Suppression

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    We present a general method for constructing radar transmit pulse trains and receive filters for which the radar point-spread function in delay and Doppler, given by the cross-ambiguity function of the transmit pulse train and the pulse train used in the receive filter, is essentially free of range sidelobes inside a Doppler interval around the zero-Doppler axis. The transmit pulse train is constructed by coordinating the transmission of a pair of Golay complementary waveforms across time according to zeros and ones in a binary sequence P. The pulse train used to filter the received signal is constructed in a similar way, in terms of sequencing the Golay waveforms, but each waveform in the pulse train is weighted by an element from another sequence Q. We show that a spectrum jointly determined by P and Q sequences controls the size of the range sidelobes of the cross-ambiguity function and by properly choosing P and Q we can clear out the range sidelobes inside a Doppler interval around the zero- Doppler axis. The joint design of P and Q enables a tradeoff between the order of the spectral null for range sidelobe suppression and the signal-to-noise ratio at the receiver output. We establish this trade-off and derive a necessary and sufficient condition for the construction of P and Q sequences that produce a null of a desired order

    Submodularity and Optimality of Fusion Rules in Balanced Binary Relay Trees

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    We study the distributed detection problem in a balanced binary relay tree, where the leaves of the tree are sensors generating binary messages. The root of the tree is a fusion center that makes the overall decision. Every other node in the tree is a fusion node that fuses two binary messages from its child nodes into a new binary message and sends it to the parent node at the next level. We assume that the fusion nodes at the same level use the same fusion rule. We call a string of fusion rules used at different levels a fusion strategy. We consider the problem of finding a fusion strategy that maximizes the reduction in the total error probability between the sensors and the fusion center. We formulate this problem as a deterministic dynamic program and express the solution in terms of Bellman's equations. We introduce the notion of stringsubmodularity and show that the reduction in the total error probability is a stringsubmodular function. Consequentially, we show that the greedy strategy, which only maximizes the level-wise reduction in the total error probability, is within a factor of the optimal strategy in terms of reduction in the total error probability

    Detection Performance in Balanced Binary Relay Trees with Node and Link Failures

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    We study the distributed detection problem in the context of a balanced binary relay tree, where the leaves of the tree correspond to NN identical and independent sensors generating binary messages. The root of the tree is a fusion center making an overall decision. Every other node is a relay node that aggregates the messages received from its child nodes into a new message and sends it up toward the fusion center. We derive upper and lower bounds for the total error probability PNP_N as explicit functions of NN in the case where nodes and links fail with certain probabilities. These characterize the asymptotic decay rate of the total error probability as NN goes to infinity. Naturally, this decay rate is not larger than that in the non-failure case, which is N\sqrt N. However, we derive an explicit necessary and sufficient condition on the decay rate of the local failure probabilities pkp_k (combination of node and link failure probabilities at each level) such that the decay rate of the total error probability in the failure case is the same as that of the non-failure case. More precisely, we show that logPN1=Θ(N)\log P_N^{-1}=\Theta(\sqrt N) if and only if logpk1=Ω(2k/2)\log p_k^{-1}=\Omega(2^{k/2})

    On the use of artificial neural networks for the analysis of survival data

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    Artificial neural networks are a powerful tool for analyzing data sets where there are complicated nonlinear interactions between the measured inputs and the quantity to be predicted. We show that the results obtained when neural networks are applied to survival data depend critically on the treatment of censoring in the data. When the censoring is modeled correctly, neural networks are a robust model independent technique for the analysis of very large sets of survival data

    Waveform libraries: Measures of effectiveness for radar scheduling

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    Our goal was to provide an overview of a circle of emerging ideas in the area of waveform scheduling for active radar. Principled scheduling of waveforms in radar and other active sensing modalities is motivated by the nonexistence of any single waveform that is ideal for all situations encountered in typical operational scenarios. This raises the possibility of achieving operationally significant performance gains through closed-loop waveform scheduling. In principle, the waveform transmitted in each epoch should be optimized with respect to a metric of desired performance using all information available from prior measurements in conjunction with models of scenario dynamics. In practice, the operational tempo of the system may preclude such on-the-fly waveform design, though further research into fast adaption of waveforms could possibly attenuate such obstacles in the future. The focus in this article has been on the use of predesigned libraries of waveforms from which the scheduler can select in lieu of undertaking a real-time design. Despite promising results, such as the performance gains shown in the tracking example presented here, many challenges remain to be addressed to bring the power of waveform scheduling to the level of maturity needed to manifest major impact as a standard component of civilian and military radar systems.Douglas Cochran, Sofia Suvorova, Stephen D. Howard and Bill Mora
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