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Oversampled A/D Conversion and Error-Rate Dependence of Non-Bandlimited Signals with Finite Rate of Innovation

Ivana Jovanovic
2006
Article

Résumé

We study the problem of A/D conversion and error-rate dependence of a class of non-bandlimited signals which have a finite rate of innovation, particularly, a continuous periodic stream of Diracs, characterized by a finite set of time positions and weights. Previous research has only considered sampling of this type of signals, ignoring the presence of quantization, which is necessary for any practical application. We first define the concept of consistent reconstruction for these signals and introduce the operations of both: a) oversampling in frequency, determined by the bandwidth of the low pass filtering used in the signal acquisition, and b) oversampling in time, determined by the number of samples in time taken from the filtered signal. Accuracy in a consistent reconstruction is achieved by enforcing the reconstructed signal to satisfy three sets of constrains, defined by: the low-pass filtering operation, the quantization operation itself and the signal space of continuous periodic streams of Diracs. We provide two schemes to reconstruct the signal. For the first one, we prove that the mean squared error (MSE) of the time positions is of the order of O(1/R_t^2R_f^3), where R_t and R_f are the oversampling ratios in time and in frequency, respectively. For the second scheme, which has a higher complexity, it is experimentally observed that the MSE of the time positions is of the order of O(1/R_t^2R_f^5). Our experimental results show a clear advantage of consistent reconstruction over non-consistent reconstruction. Regarding the rate, we consider a threshold crossing based scheme where, as opposed to previous research, both oversampling in time and also in frequency influence the coding rate. We compare the error-rate dependence behavior that is obtained from both increasing the oversampling in time and in frequency, on the one hand, and on the other hand, from decreasing the quantization stepsize.

Official source

https://infoscience.epfl.ch/record/34279?ln=fr

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Ivana Jovanovic
2006
Article

Résumé

Official source

https://infoscience.epfl.ch/record/34279?ln=fr

À propos de ce résultat

Concepts associés (18)

Concepts associés

Chargement

Publications associées (11)

Publications associées

Chargement

Oversampled A/D Conversion of Non-Bandlimited Signals with Finite Rate of Innovation

Ivana Jovanovic

We consider the problem of A/D conversion for non-bandlimitedsignals that have a finite rate of innovation, in particular, theclass of continuous periodic stream of Diracs, characterized by aset of time positions and weights. Previous research has onlyconsidered the sampling of these signals, ignoring quantization,which is necessary for any practical application (e.g. UWB,CDMA). In order to achieve accuracy under quantization, weintroduce two types of oversampling, namely, oversampling infrequency and oversampling in time. High accuracy is achieved byenforcing the reconstruction to satisfy either three convex setsof constraints related to: 1) sampling kernel, 2) quantization and3) periodic streams of Diracs which is then said to provide {\itstrong} consistency or only the first two, providing {\it weak}consistency. We propose three reconstruction algorithms, thefirst two achieving {\it weak} consistency and the third oneachieving {\it strong} consistency. For these three algorithms,respectively, the experimental MSE performance for time positionsdecreases as

O(1/{R_t^2 R_f^3})

O(1/{R_t^2 R_f^4})

and

O(1/{R_t^2 R_f^5})

, where

R_t

and

R_f

are the oversamplingratios in time and in frequency, respectively. It is also provedtheoretically that our reconstruction algorithms satisfying {\itweak} consistency achieve an MSE performance of at least

O(1/{R_t^2 R_f^3})

2004

Chargement

Error-Rate Dependence of Non-Bandlimited Signals with Finite Rate of Innovation

Ivana Jovanovic

We consider the rate-distortion problem for non-bandlimited signals that have a finite rate of innovation, in particular, the class of continuous periodic stream of Diracs, characterized by a set of time positions and weights. Previous research has only considered the sampling of these signals, ignoring quantization, which is necessary for any practical application (e.g. UWB, CDMA). In order to achieve accuracy under quantization, we introduce two types of oversampling, namely, oversampling in frequency and oversampling in time. The reconstruction accuracy is measured by the MSE of the time positions. High accuracy is achieved by enforcing the reconstruction to satisfy either three convex sets of constraints related to: 1) sampling kernel, 2) quantization and 3) periodic streams of Diracs, which is then said to provide strong consistency or only the first two, providing weak consistency. We propose reconstruction algorithms for both weak and strong consistency. Regarding the rate, we also consider a threshold crossing based scheme, which is more efficient than the PCM encoding. We compare the rate- distortion behavior that is obtained from both increasing the oversampling in time and in frequency, on the one hand, and, on the other hand, from decreasing the quantization stepsize.

2004

Chargement

Euclidean distance matrices (EDMs) are central players in many diverse fields including psychometrics, NMR spectroscopy, machine learning and sensor networks. However, they are not often exploited in signal processing. In this thesis, we analyze attributes of EDMs and derive new key properties of them. These analyses allow us to propose algorithms to approximate EDMs and provide analytic bounds on the performance of our methods. We use these techniques to suggest new solutions for several practical problems in signal processing. Together with these properties, algorithms and applications, EDMs can thus be considered as a fundamental toolbox to be used in signal processing. In more detail, we start by introducing the structure and properties of EDMs. In particular, we focus on their rank property; the rank of an EDM is at most the dimension of the set of points generating it plus 2. Using this property, we introduce the use of low rank matrix completion methods for approximating and completing noisy and partially revealed EDMs. We apply this algorithm to the problem of sensor position calibration in ultrasound tomography devices. By adapting the matrix completion framework, in addition to proposing a self calibration process for these devices, we also provide analytic bounds for the calibration error. We then study the problem of sensor localization using distance information by minimizing a non-linear cost function known as the s-stress function in the multidimensional scaling (MDS) community. We derive key properties of this cost function that can be used to reduce the search domain for finding its global minimum. We provide an efficient, low cost and distributed algorithm for minimizing this cost function for incomplete networks and noisy measurements. In randomized experiments, the proposed method converges to the global minimum of the s-stress in more than 99% of the cases. We also address the open problem of existence of non-global minimizers of the s-stress and reduce this problem to a hypothesis. If the hypothesis is true then the cost function has only global minimizers, otherwise, it has non-global minimizers. Using the rank property of EDMs and the proposed minimization algorithm for approximating them, we address an interesting and practical problem in acoustics. We show that using five microphones and one loudspeaker, we can hear the shape of a room. We reformulate this problem as finding the locations of the image sources of the loudspeaker with respect to the walls. We propose an algorithm to find these positions only using first-order echoes. We prove that the reconstruction of the room is almost surely unique. We further introduce a new algorithm for locating a microphone inside a known room using only one loudspeaker. Our experimental evaluations conducted on the EPFL campus and also in the Lausanne cathedral, confirm the robustness and accuracy of the proposed methods. By integrating further properties of EDMs into the matrix completion framework, we propose a new method for calibrating microphone arrays in a diffuse noise field. We use a specific characterization of diffuse noise fields to relate the coherence of recorded signals by two microphones to their mutual distance. As this model is not reliable for large distances between microphones, we use matrix completion coupled with other properties of EDMs to estimate these distances and calibrate the microphone array. Evaluation of our algorithm using real data measurements demonstrates, for the first time, the possibility of accurately calibrating large ad-hoc microphone arrays in a diffuse noise field. The last part of the thesis addresses a central problem in signal processing; the design of discrete-time filters (equivalently window functions) that are compact both in time and frequency. By properly adapting the definitions of compactness in the continuous time to discrete time, we formulate the search for maximally compact sequences as solving a semi-definite program. We show that the spectra of maximally compact sequences are a special class of Mathieu’s cosine functions. Using the asymptotic behavior of these functions, we provide a tight bound for the time-frequency spread of discrete-time sequences. Our analysis shows that the Heisenberg uncertainty bound on the time-frequency spread of sequences is not tight and the lower bound depends on the frequency spread, unlike in the continuous time case.

EPFL2013

Chargement

Oversampled A/D Conversion of Non-Bandlimited Signals with Finite Rate of Innovation

Ivana Jovanovic

O(1/{R_t^2 R_f^3})

O(1/{R_t^2 R_f^4})

and

O(1/{R_t^2 R_f^5})

, where

R_t

and

R_f

O(1/{R_t^2 R_f^3})

2004

EPFL2013