In Analog-to-digital (A/D) conversion, signal decimation has been proven to greatly improve the efficiency of data storage while maintaining high accuracy. When one couples signal decimation with the Σ∆ quantization scheme, the reconstruction error decays exponentially with respect to the bit-rate. In this study, similar results are proven for finite unitarily generated frames. We introduce a process called alternative decimation on finite frames that is compatible with first and second order Σ∆ quantization. In both cases, alternative decimation results in exponential error decay with respect to the bit usage.
Abstract-This paper investigates adaptive blind equalization for multiple-input and multiple-output (MIMO) channels and its application to blind separation of multiple signals received by antenna arrays in communication systems. The performance analysis of the CMA equalizer used in MIMO channels is first presented. Our analysis results indicate that for the MIMO FIR channels satisfying certain conditions, the MIMO-CMA FIR equalizer is able to remove the IS1 and co-channel interference regardless of the initial setting of the blind equalizer. To recover all input signals simultaneously, a novel MIMO channel blind equalization algorithm is then developed. The global convergence of the new algorithm for MIMO channels is proved. Hence, the new blind equalization algorithm for MIMO channels can be applied to separate and equalize the signals received by antenna arrays in communication systems. Finally, Computer simulations are presented to confirm our analysis and illustrate the performance of the new algorithm.
In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there exists a variety of results which show that a wide range of functions can be approximated with sometimes surprising accuracy by these outputs. For example, it is known that the set of functions that can be approximated with exponential accuracy (in terms of the number of parameters used) includes, on one hand, very smooth functions such as polynomials and analytic functions (see e.g. [9,11,13]) and, on the other hand, very rough functions such as the Weierstrass function (see e.g. [10,4]), which is nowhere differentiable. In this paper, we add to the latter class of rough functions by showing that it also includes refinable functions. Namely, we show that refinable functions are approximated by the outputs of deep ReLU networks with a fixed width and increasing depth with accuracy exponential in terms of their number of parameters. Our results apply to functions used in the standard construction of wavelets as well as to functions constructed via subdivision algorithms in Computer Aided Geometric Design.
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