In many quantization problems, the distortion function is given by the Euclidean metric to measure the distance of a source sample to any given reproduction point of the quantizer. We will in this work regard distortion functions, which are additively and multiplicatively weighted for each reproduction point resulting in a heterogeneous quantization problem, as used for example in deployment problems of sensor networks. Whereas, normally in such problems, the average distortion is minimized for given weights (parameters), we will optimize the quantization problem over all weights, i.e., we tune or control the distortion functions in our favor. For a uniform source distribution in one-dimension, we derive the unique minimizer, given as the uniform scalar quantizer with an optimal common weight. By numerical simulations, we demonstrate that this result extends to two-dimensions where asymptotically the parameter optimized quantizer is the hexagonal lattice with common weights. As an application, we will determine the optimal deployment of unmanned aerial vehicles (UAVs) to provide a wireless communication to ground terminals under a minimal communication power cost. Here, the optimal weights relate to the optimal flight heights of the UAVs.
The success of the compressed sensing paradigm has shown that a substantial reduction in sampling and storage complexity can be achieved in certain linear and non-adaptive estimation problems. It is therefore an advisable strategy for noncoherent information retrieval in, for example, sporadic blind and semi-blind communication and sampling problems. But, the conventional model is not practical here since the compressible signals have to be estimated from samples taken solely on the output of an un-calibrated system which is unknown during measurement but often compressible. Conventionally, one has either to operate at suboptimal sampling rates or the recovery performance substantially suffers from the dominance of model mismatch. In this work we discuss such type of estimation problems and we focus on bilinear inverse problems. We link this problem to the recovery of low-rank and sparse matrices and establish stable low-dimensional embeddings of the uncalibrated receive signals whereby addressing also efficient communication-oriented methods like universal random demodulation. Exemplary, we investigate in more detail sparse convolutions serving as a basic communication channel model. In using some recent results from additive combinatorics we show that such type of signals can be efficiently low-rate sampled by semi-blind methods. Finally, we present a further application of these results in the field of phase retrieval from intensity Fourier measurements.
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