The need of reconstructing discrete-valued sparse signals from few measurements, that is solving an undetermined system of linear equations, appears frequently in science and engineering. Those signals appear, for example, in error correcting codes as well as massive Multiple-Input Multiple-Output (MIMO) channel and wideband spectrum sensing. A particular example is given by wireless communications, where the transmitted signals are sequences of bits, i.e., with entries in f0; 1g. Whereas classical compressed sensing algorithms do not incorporate the additional knowledge of the discrete nature of the signal, classical lattice decoding approaches do not utilize sparsity constraints. In this talk, we present an approach that incorporates a discrete values prior into basis pursuit. In particular, we address finite-valued sparse signals, i.e., sparse signals with entries in a finite alphabet. We will introduce an equivalent null space characterization and show that phase transition takes place earlier than when using the classical basis pursuit approach. We will further discuss robustness of the algorithm and show that the nonnegative case is very different from the bipolar one. One of our findings is that the positioning of the zero in the alphabet - i.e., whether it is a boundary element or not - is crucial.
The classical sampling theorem, attributed to Whittaker, Shannon, Nyquist, and Kotelnikov, states that a
bandlimited function can be recovered from its samples, as long as we use a sufficiently dense sampling grid.
Here, we review the recent development of an operator sampling theory which allows for a "widening" of the
classical sampling theorem. In this realm, bandlimited functions are replaced by "bandlimited operators". that
is, by pseudodifferential operators which have bandlimited
Similar to the Nyquist sampling density condition alluded to above, we discuss sufficient and necessary
conditions on the bandlimitation of pseudodifferential operators to ensure that they can be recovered by their
action on a single distribution. In fact, we show that an operator with Kohn-Nirenberg symbol bandlimited to
a Jordan domain of measure less than one can be recovered through its action on a distribution defined on a
appropriately chosen sampling grid. Further, an operator with bandlimitation to a Jordan domain of measure
larger than one cannot be recovered through its action on any tempered distribution whatsoever, pointing towards
a fundamental difference to the classical sampling theorem where a large bandwidth could always be compensated
through a sufficiently fine sampling grid. The dichotomy depending on the size of the bandlimitation is related
to Heisenberg's uncertainty principle.
Further, we discuss an application of this theory to the channel measurement problem for Multiple-Input
Multiple-Output (MIMO) channels.
Proc. SPIE. 4119, Wavelet Applications in Signal and Image Processing VIII
KEYWORDS: Wavelets, Continuous wavelet transforms, Signal detection, Wavelet transforms, Algorithm development, Signal processing, Image processing, Iterated function systems, Multidimensional signal processing, Signal analyzers
Generalized Haar wavelets were introduced in connection with the problem of detecting specific periodic components in noisy signals. We showed that the non-normalized continuous wavelet transform of a periodic function taken with respect to a generalized Haar wavelet is periodic in time as well as in scale, and that generalized Haar wavelets are the only bounded functions with this property.
We compare three types of coherent Riesz families with respect to their perturbation stability under convolution with elements of a family of typical channel functions. This problem is of key relevance in the design of modulation signal sets for digital communication over time-invariant channels. Upper and lower bounds on the orthogonal perturbation are formulate din terms of spectral spread and temporal support of the prototype, and by the approximate design of worst case convolution kernels. Among the considered bases, the Weyl-Heisenberg structure which generates Gabor systems turns out to be optimal.
This paper deals with the analysis of time series with respect to certain known periodicities. In particular, we shall present a fast method aimed at detecting periodic behavior inherent in noise data. The method is composed of three steps: (1) Non-noisy data are analyzed through spectral and wavelet methods to extract specific periodic patterns of interest. (2) Using these patterns, we construct an optimal piecewise constant wavelet designed to detect the underlying periodicities. (3) We introduce a fast discretized version of the continuous wavelet transform, as well as waveletgram averaging techniques, to detect occurrence and period of these periodicities. The algorithm is formulated to provide real time implementation. Our procedure is generally applicable to detect locally periodic components in signals s which can be modeled as s(t) equals A(t)F(h(t)) + N(t) for t in I, where F is a periodic signal, A is a non-negative slowly varying function, and h is strictly increasing with h' slowly varying, N denotes background activity. For example, the method can be applied in the context of epileptic seizure detection. In this case, we try to detect seizure periodics in EEG and ECoG data. In the case of ECoG data, N is essentially 1/f noise. In the case of EEG data and for t in I,N includes noise due to cranial geometry and densities. In both cases N also includes standard low frequency rhythms. Periodicity detection has other applications including ocean wave prediction, cockpit motion sickness prediction, and minefield detection.