Periodic Reporting for period 1 - QuaSiProc (Quantitative Analysis for Modern Signal Processing)
Reporting period: 2022-09-01 to 2024-08-31
Signal processing is integral to a myriad of applications that impact our daily lives. Despite its ubiquity, many of its applications still rely on methods that lack robust theoretical foundations. This gap is particularly evident in emerging sampling schemes, which are often validated through heuristic approaches rather than formal mathematical frameworks. Addressing this issue, the QuaSiProc project aims to establish a solid theoretical basis and quantitative analysis for various innovative sampling techniques. The main objectives are:
1) Advancing on the theory of dynamical sampling.
Dynamical sampling is concerned with cost-effective sampling methods and is particularly relevant in contexts where measuring devices are expensive and sampling schemes with high acquisition densities are impractical. These considerations are highly important, for instance, in the areas of environmental monitoring and health care. The main goal is to reduce the number of sensors required for a certain task by compensating with oversampling in time domain, and to quantify the corresponding trade-off. A common mathematical model postulates that the signal in question evolves over time through the action of a known evolution operator. The project aims to provide a characterization of evolutionary systems modeled through the action of two different evolution operators. In addition, the project seeks to establish a relationship to the theory of mobile sampling by identifying conditions for stable reconstruction on a sampling scheme where measurements are collected over continuous trajectories and signals evolve in time during the measuring process.
2) Integrating shift-invariant spaces into modern sampling schemes.
While signals are classically modeled as bandlimited functions (that is, functions with compact Fourier support), many real-world applications call for more flexible settings where the signals are not exactly bandlimited but approximately so. The project considers shift invariant spaces, a common alternative signal model that better fits many realistic scenarios. The contribution of the project is twofold.
2a) Classical sampling schemes for measuring continuous-time signals are based on synchronous behavior: the signal’s amplitude is measured at predetermined –uniform or irregular– instants, governed by a global clock. In many practical scenarios, conventional analog-to-digital converters can lead to deficient implementations due to their size and power consumption. The critical need for compact and energy-efficient measuring devices has driven the research of asynchronous, event-driven sampling methods; those which are not ruled by a clock-pattern but rather capture the times where significant events of a signal occur. The integrate-and-fire sampler consists of an integrator followed by a comparator, simulating the simplified behavior of a neuron; the action potentials (spikes) are generated when the accumulated stimulus (integrator) surpasses a certain threshold (comparator). The project aims to quantify the performance of the leaky integrate-and-fire samplers for signals in shift-invariant spaces.
2b) The project aims to design sampling strategies for multi-variate signals belonging to a shift-invariant space using samples taken on a periodic random set. The goal is to provide a simple strategy to reconstruction which is accompanied with explicit and possibly very favorable stability margins.
3) In multidimensional settings, finding a stable sampling set for Paley-Wiener signals remains a very challenging task. Moreover, the focus of the literature is often on existential claims rather than in the quantitative aspects, a deficiency that discourages numerical applications. Indeed, many reconstruction algorithms depend on the estimation of the stability bounds; the tighter these are, the faster and more reliable the corresponding reconstruction is. This project aims to contribute to the quantification of the stability margins for sampling and interpolation adapted to multi-spectral signals.
1) Advancing on the theory of dynamical sampling.
Dynamical sampling is concerned with cost-effective sampling methods and is particularly relevant in contexts where measuring devices are expensive and sampling schemes with high acquisition densities are impractical. These considerations are highly important, for instance, in the areas of environmental monitoring and health care. The main goal is to reduce the number of sensors required for a certain task by compensating with oversampling in time domain, and to quantify the corresponding trade-off. A common mathematical model postulates that the signal in question evolves over time through the action of a known evolution operator. The project aims to provide a characterization of evolutionary systems modeled through the action of two different evolution operators. In addition, the project seeks to establish a relationship to the theory of mobile sampling by identifying conditions for stable reconstruction on a sampling scheme where measurements are collected over continuous trajectories and signals evolve in time during the measuring process.
2) Integrating shift-invariant spaces into modern sampling schemes.
While signals are classically modeled as bandlimited functions (that is, functions with compact Fourier support), many real-world applications call for more flexible settings where the signals are not exactly bandlimited but approximately so. The project considers shift invariant spaces, a common alternative signal model that better fits many realistic scenarios. The contribution of the project is twofold.
2a) Classical sampling schemes for measuring continuous-time signals are based on synchronous behavior: the signal’s amplitude is measured at predetermined –uniform or irregular– instants, governed by a global clock. In many practical scenarios, conventional analog-to-digital converters can lead to deficient implementations due to their size and power consumption. The critical need for compact and energy-efficient measuring devices has driven the research of asynchronous, event-driven sampling methods; those which are not ruled by a clock-pattern but rather capture the times where significant events of a signal occur. The integrate-and-fire sampler consists of an integrator followed by a comparator, simulating the simplified behavior of a neuron; the action potentials (spikes) are generated when the accumulated stimulus (integrator) surpasses a certain threshold (comparator). The project aims to quantify the performance of the leaky integrate-and-fire samplers for signals in shift-invariant spaces.
2b) The project aims to design sampling strategies for multi-variate signals belonging to a shift-invariant space using samples taken on a periodic random set. The goal is to provide a simple strategy to reconstruction which is accompanied with explicit and possibly very favorable stability margins.
3) In multidimensional settings, finding a stable sampling set for Paley-Wiener signals remains a very challenging task. Moreover, the focus of the literature is often on existential claims rather than in the quantitative aspects, a deficiency that discourages numerical applications. Indeed, many reconstruction algorithms depend on the estimation of the stability bounds; the tighter these are, the faster and more reliable the corresponding reconstruction is. This project aims to contribute to the quantification of the stability margins for sampling and interpolation adapted to multi-spectral signals.
Throughout the duration of the project the fellow worked in a collaborative manner on the three objectives above mentioned. The results were published in 3 leading journals: Applied and Computational Harmonic Analysis; Journal of Mathematical Analysis and Applications; and IEEE Transactions on Information Theory, and 1 Conference proceedings: IEEE International Conference on Sampling Theory and Applications (SampTA). The results were also disseminated by exposition in 9 different international conferences, one of those as an invited speaker. Other publications are also in preparation.
We mention some contributions of the project. More results will be announced in forthcoming publications.
The problem of dynamical sampling can be equivalently stated as the problem of determining when the orbit of a set of vectors through a bounded operator in a Hilbert space forms a frame. The operator models the evolution of the measured signals, whereas the vectors represent the position of the sensors. The efforts of the action led to solving the problem of characterizing frames formed by the iteration of two different commuting operators, thus expanding the degrees of freedom in the evolution of the signal.
The characterization was done in terms of model subspaces of the space of square integrable functions defined on the torus and having values in some Hardy space with multiplicity. In this context, model subspace means that it is the orthogonal complement of a subspace that is reducing by the bilateral shift and invariant by the unilateral shift acting locally. This opened the follow-up question of characterizing such subspaces. The conditions obtained are of the type of the ones in the theorems of Helson and Beurling-Lax-Halmos on invariance under the bilateral and unilateral shift.
Multi-band signals are relevant in wireless communications as models for physical or commercial restrictions on certain frequency intervals. The Nyquist-Shannon Theorem states that the energy of this type of signals is captured on a uniform grid with a sampling rate determined by the spectral diameter. However, if the spectrum contains a relatively small number of active frequency bands, one could expect that number to govern the sampling rate instead. The research of this action considered more complex sampling patterns, namely periodic non-uniform sets, where the position of the channels are chosen in a random manner.
Multi-band signals can be modeled by shift-invariant spaces of generating functions with distinct frequency profiles. For this broader signal model, we showed that such a sampling strategy succeeds with high probability provided that the density of the sampling pattern exceeds by a logarithmic factor the number of active frequency profiles (as opposed to the much larger spectral diameter). This simple alternative is accompanied by favorable a priori stability margins (that lead to so-called snug frames). Importantly, the obtained reconstruction guarantees hold uniformly for all signals, rather than for a subset of well-concentrated ones. At the technical level, this was achieved by methods of concentration of measure formulated on the Zak domain.
The problem of dynamical sampling can be equivalently stated as the problem of determining when the orbit of a set of vectors through a bounded operator in a Hilbert space forms a frame. The operator models the evolution of the measured signals, whereas the vectors represent the position of the sensors. The efforts of the action led to solving the problem of characterizing frames formed by the iteration of two different commuting operators, thus expanding the degrees of freedom in the evolution of the signal.
The characterization was done in terms of model subspaces of the space of square integrable functions defined on the torus and having values in some Hardy space with multiplicity. In this context, model subspace means that it is the orthogonal complement of a subspace that is reducing by the bilateral shift and invariant by the unilateral shift acting locally. This opened the follow-up question of characterizing such subspaces. The conditions obtained are of the type of the ones in the theorems of Helson and Beurling-Lax-Halmos on invariance under the bilateral and unilateral shift.
Multi-band signals are relevant in wireless communications as models for physical or commercial restrictions on certain frequency intervals. The Nyquist-Shannon Theorem states that the energy of this type of signals is captured on a uniform grid with a sampling rate determined by the spectral diameter. However, if the spectrum contains a relatively small number of active frequency bands, one could expect that number to govern the sampling rate instead. The research of this action considered more complex sampling patterns, namely periodic non-uniform sets, where the position of the channels are chosen in a random manner.
Multi-band signals can be modeled by shift-invariant spaces of generating functions with distinct frequency profiles. For this broader signal model, we showed that such a sampling strategy succeeds with high probability provided that the density of the sampling pattern exceeds by a logarithmic factor the number of active frequency profiles (as opposed to the much larger spectral diameter). This simple alternative is accompanied by favorable a priori stability margins (that lead to so-called snug frames). Importantly, the obtained reconstruction guarantees hold uniformly for all signals, rather than for a subset of well-concentrated ones. At the technical level, this was achieved by methods of concentration of measure formulated on the Zak domain.