300 research outputs found
On the Minimax Capacity Loss under Sub-Nyquist Universal Sampling
This paper investigates the information rate loss in analog channels when the
sampler is designed to operate independent of the instantaneous channel
occupancy. Specifically, a multiband linear time-invariant Gaussian channel
under universal sub-Nyquist sampling is considered. The entire channel
bandwidth is divided into subbands of equal bandwidth. At each time only
constant-gain subbands are active, where the instantaneous subband
occupancy is not known at the receiver and the sampler. We study the
information loss through a capacity loss metric, that is, the capacity gap
caused by the lack of instantaneous subband occupancy information. We
characterize the minimax capacity loss for the entire sub-Nyquist rate regime,
provided that the number of subbands and the SNR are both large. The
minimax limits depend almost solely on the band sparsity factor and the
undersampling factor, modulo some residual terms that vanish as and SNR
grow. Our results highlight the power of randomized sampling methods (i.e. the
samplers that consist of random periodic modulation and low-pass filters),
which are able to approach the minimax capacity loss with exponentially high
probability.Comment: accepted to IEEE Transactions on Information Theory. It has been
presented in part at the IEEE International Symposium on Information Theory
(ISIT) 201
Channel Capacity under Sub-Nyquist Nonuniform Sampling
This paper investigates the effect of sub-Nyquist sampling upon the capacity
of an analog channel. The channel is assumed to be a linear time-invariant
Gaussian channel, where perfect channel knowledge is available at both the
transmitter and the receiver. We consider a general class of right-invertible
time-preserving sampling methods which include irregular nonuniform sampling,
and characterize in closed form the channel capacity achievable by this class
of sampling methods, under a sampling rate and power constraint. Our results
indicate that the optimal sampling structures extract out the set of
frequencies that exhibits the highest signal-to-noise ratio among all spectral
sets of measure equal to the sampling rate. This can be attained through
filterbank sampling with uniform sampling at each branch with possibly
different rates, or through a single branch of modulation and filtering
followed by uniform sampling. These results reveal that for a large class of
channels, employing irregular nonuniform sampling sets, while typically
complicated to realize, does not provide capacity gain over uniform sampling
sets with appropriate preprocessing. Our findings demonstrate that aliasing or
scrambling of spectral components does not provide capacity gain, which is in
contrast to the benefits obtained from random mixing in spectrum-blind
compressive sampling schemes.Comment: accepted to IEEE Transactions on Information Theory, 201
Distortion-Rate Function of Sub-Nyquist Sampled Gaussian Sources
The amount of information lost in sub-Nyquist sampling of a continuous-time
Gaussian stationary process is quantified. We consider a combined source coding
and sub-Nyquist reconstruction problem in which the input to the encoder is a
noisy sub-Nyquist sampled version of the analog source. We first derive an
expression for the mean squared error in the reconstruction of the process from
a noisy and information rate-limited version of its samples. This expression is
a function of the sampling frequency and the average number of bits describing
each sample. It is given as the sum of two terms: Minimum mean square error in
estimating the source from its noisy but otherwise fully observed sub-Nyquist
samples, and a second term obtained by reverse waterfilling over an average of
spectral densities associated with the polyphase components of the source. We
extend this result to multi-branch uniform sampling, where the samples are
available through a set of parallel channels with a uniform sampler and a
pre-sampling filter in each branch. Further optimization to reduce distortion
is then performed over the pre-sampling filters, and an optimal set of
pre-sampling filters associated with the statistics of the input signal and the
sampling frequency is found. This results in an expression for the minimal
possible distortion achievable under any analog to digital conversion scheme
involving uniform sampling and linear filtering. These results thus unify the
Shannon-Whittaker-Kotelnikov sampling theorem and Shannon rate-distortion
theory for Gaussian sources.Comment: Accepted for publication at the IEEE transactions on information
theor
Waveform Design for 5G and beyond Systems
5G traffic has very diverse requirements with respect to data rate, delay, and reliability. The concept of using multiple OFDM numerologies adopted in the 5G NR standard will likely meet these multiple requirements to some extent. However, the traffic is radically accruing different characteristics and requirements when compared with the initial stage of 5G, which focused mainly on high-speed multimedia data applications. For instance, applications such as vehicular communications and robotics control require a highly reliable and ultra-low delay. In addition, various emerging M2M applications have sparse traffic with a small amount of data to be delivered. The state-of-the-art OFDM technique has some limitations when addressing the aforementioned requirements at the same time. Meanwhile, numerous waveform alternatives, such as FBMC, GFDM, and UFMC, have been explored. They also have their own pros and cons due to their intrinsic waveform properties. Hence, it is the opportune moment to come up with modification/variations/combinations to the aforementioned techniques or a new waveform design for 5G systems and beyond. The aim of this Special Issue is to provide the latest research and advances in the field of waveform design for 5G systems and beyond
A Modulo-Based Architecture for Analog-to-Digital Conversion
Systems that capture and process analog signals must first acquire them
through an analog-to-digital converter. While subsequent digital processing can
remove statistical correlations present in the acquired data, the dynamic range
of the converter is typically scaled to match that of the input analog signal.
The present paper develops an approach for analog-to-digital conversion that
aims at minimizing the number of bits per sample at the output of the
converter. This is attained by reducing the dynamic range of the analog signal
by performing a modulo operation on its amplitude, and then quantizing the
result. While the converter itself is universal and agnostic of the statistics
of the signal, the decoder operation on the output of the quantizer can exploit
the statistical structure in order to unwrap the modulo folding. The
performance of this method is shown to approach information theoretical limits,
as captured by the rate-distortion function, in various settings. An
architecture for modulo analog-to-digital conversion via ring oscillators is
suggested, and its merits are numerically demonstrated
Some Theory and Applications of Probability in Quantum Mechanics
This thesis investigates three distinct facets of the theory of quantum information. The first two, quantum state estimation and quantum process estimation, are closely related and deal with the question of how to estimate the classical parameters in a quantum mechanical model. The third attempts to bring quantum theory as close as possible to classical theory through the formalism of quasi-probability.
Building a large scale quantum information processor is a significant challenge. First, we require an accurate characterization of the dynamics experienced by the device to allow for the application of error correcting codes and other tools for implementing useful quantum algorithms. The necessary scaling of computational resources needed to characterize a quantum system as a function of the number of subsystems is by now a well studied problem (the scaling is generally exponential). However, irrespective of the computational resources necessary to just write-down a classical description of a quantum state, we can ask about the experimental resources necessary to obtain data (measurement complexity) and the computational resources necessary to generate such a characterization (estimation complexity). These problems are studied here and approached from two directions.
The first problem we address is that of quantum state estimation. We apply high-level decision theoretic principles (applied in classical problems such as, for example, universal data compression) to the estimation of a qubit state. We prove that quantum states are more difficult to estimate than their classical counterparts by finding optimal estimation strategies. These strategies, requiring the solution to a difficult optimization problem, are difficult to implement in practise. Fortunately, we find estimation algorithms which come close to optimal but require far fewer resources to compute. Finally, we provide a classical analog of this quantum mechanical problem which reproduces, and gives intuitive explanations for, many of its features, such as why adaptive tomography can quadratically reduce its difficulty.
The second method for practical characterization of quantum devices takes is applied to the problem of quantum process estimation. This differs from the above analysis in two ways: (1) we apply strong restrictions on knowledge of various estimation and control parameters (the former making the problem easier, the latter making it harder); and (2) we consider the problem of designing future experiments based on the outcomes of past experiments. We show in test cases that adaptive protocols can exponentially outperform their off-line counterparts. Moreover, we adapt machine learning algorithms to the problem which bring these experimental design methodologies to realm of experimental feasibility.
In the final chapter we move away from estimation problems to show formally that a classical representation of quantum theory is not tenable. This intuitive conclusion is formally borne out through the connection to quasi-probability -- where it is equivalent to the necessity of negative probability in all such representations of quantum theory. In particular, we generalize previous no-go theorems to arbitrary classical representations of quantum systems of arbitrary dimension. We also discuss recent progress in the program to identify quantum resources for subtheories of quantum theory and operational restrictions motivated by quantum computation
Learning Theory and Approximation
The main goal of this workshop – the third one of this type at the MFO – has been to blend mathematical results from statistical learning theory and approximation theory to strengthen both disciplines and use synergistic effects to work on current research questions. Learning theory aims at modeling unknown function relations and data structures from samples in an automatic manner. Approximation theory is naturally used for the advancement and closely connected to the further development of learning theory, in particular for the exploration of new useful algorithms, and for the theoretical understanding of existing methods. Conversely, the study of learning theory also gives rise to interesting theoretical problems for approximation theory such as the approximation and sparse representation of functions or the construction of rich kernel reproducing Hilbert spaces on general metric spaces. This workshop has concentrated on the following recent topics: Pitchfork bifurcation of dynamical systems arising from mathematical foundations of cell development; regularized kernel based learning in the Big Data situation; deep learning; convergence rates of learning and online learning algorithms; numerical refinement algorithms to learning; statistical robustness of regularized kernel based learning
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