arXiv subjects

Alon Kipnis

Publications and source records attributed to Alon Kipnis.

29 records · Page 2Linked to original sources

arXiv2018

Wiener-Chaos Approach to Optimal Prediction

The chaos expansion of a general non-linear function of a Gaussian stationary increment process conditioned on its past realizations is derived. This work combines Wiener chaos expansion approach to study the dynamics of a stochastic system with the classical problem of the prediction of a Gaussian process based on a realization of its past. This is done by constructing a special basis for the Fock space of the Gaussian space generated by the process, such that each basis element is either measurable or independent with respect to the given samples. This property of the basis allows us to derive the chaos expansion of a random variable conditioned on part of the sample path. We provide a general method for the construction of such basis when the underlying process is Gaussian with stationary increment. We evaluate the basis elements in the case of the fractional Brownian motion, which leads to a prediction formula for this process.

math.PR↗

arXiv2018

Fundamental Distortion Limits of Analog-to-Digital Compression

Representing a continuous-time signal by a set of samples is a classical problem in signal processing. We study this problem under the additional constraint that the samples are quantized or compressed in a lossy manner under a limited bitrate budget. To this end, we consider a combined sampling and source coding problem in which an analog stationary Gaussian signal is reconstructed from its encoded samples. These samples are obtained by a set of bounded linear functionals of the continuous-time path, with a limitation on the average number of samples obtained per unit time available in this setting. We provide a full characterization of the minimal distortion in terms of the sampling frequency, the bitrate, and the signal's spectrum. Assuming that the signal's energy is not uniformly distributed over its spectral support, we show that for each compression bitrate there exists a critical sampling frequency smaller than the Nyquist rate, such that the distortion in signal reconstruction when sampling at this frequency is minimal. Our results can be seen as an extension of the classical sampling theorem for bandlimited random processes in the sense that it describes the minimal amount of excess distortion in the reconstruction due to lossy compression of the samples, and provides the minimal sampling frequency required in order to achieve this distortion. Finally, we compare the fundamental limits in the combined source coding and sampling problem to the performance of pulse code modulation (PCM), where each sample is quantized by a scalar quantizer using a fixed number of bits.

cs.IT↗

arXiv2018

Lossy Compression of Decimated Gaussian Random Walks

We consider the problem of estimating a Gaussian random walk from a lossy compression of its decimated version. Hence, the encoder operates on the decimated random walk, and the decoder estimates the original random walk from its encoded version under a mean squared error (MSE) criterion. It is well-known that the minimal distortion in this problem is attained by an estimate-and-compress (EC) source coding strategy, in which the encoder first estimates the original random walk and then compresses this estimate subject to the bit constraint. In this work, we derive a closed-form expression for this minimal distortion as a function of the bitrate and the decimation factor. Next, we consider a compress-and-estimate (CE) source coding scheme, in which the encoder first compresses the decimated sequence subject to an MSE criterion (with respect to the decimated sequence), and the original random walk is estimated only at the decoder. We evaluate the distortion under CE in a closed form and show that there exists a nonzero gap between the distortion under the two schemes. This difference in performance illustrates the importance of having the decimation factor at the encoder.

eess.SP↗

arXiv2018

Analog-to-Digital Compression: A New Paradigm for Converting Signals to Bits

Processing, storing and communicating information that originates as an analog signal involves conversion of this information to bits. This conversion can be described by the combined effect of sampling and quantization, as illustrated in Fig. 1. The digital representation is achieved by first sampling the analog signal so as to represent it by a set of discrete-time samples and then quantizing these samples to a finite number of bits. Traditionally, these two operations are considered separately. The sampler is designed to minimize information loss due to sampling based on characteristics of the continuous-time input. The quantizer is designed to represent the samples as accurately as possible, subject to a constraint on the number of bits that can be used in the representation. The goal of this article is to revisit this paradigm by illuminating the dependency between these two operations. In particular, we explore the requirements on the sampling system subject to constraints on the available number of bits for storing, communicating or processing the analog information.

cs.IT↗

arXiv2017

Mean Estimation from Adaptive One-bit Measurements

We consider the problem of estimating the mean of a normal distribution under the following constraint: the estimator can access only a single bit from each sample from this distribution. We study the squared error risk in this estimation as a function of the number of samples and one-bit measurements $n$. We consider an adaptive estimation setting where the single-bit sent at step $n$ is a function of both the new sample and the previous $n-1$ acquired bits. For this setting, we show that no estimator can attain asymptotic mean squared error smaller than $π/(2n)+O(n^{-2})$ times the variance. In other words, one-bit restriction increases the number of samples required for a prescribed accuracy of estimation by a factor of at least $π/2$ compared to the unrestricted case. In addition, we provide an explicit estimator that attains this asymptotic error, showing that, rather surprisingly, only $π/2$ times more samples are required in order to attain estimation performance equivalent to the unrestricted case.

math.ST↗

arXiv2017

Compress-and-Estimate Source Coding for a Vector Gaussian Source

We consider the remote vector source coding problem in which a vector Gaussian source is to be estimated from noisy linear measurements. For this problem, we derive the performance of the compress-and-estimate (CE) coding scheme and compare it to the optimal performance. In the CE coding scheme, the remote encoder compresses the noisy source observations so as to minimize the local distortion measure, independent from the joint distribution between the source and the observations. In reconstruction, the decoder estimates the original source realization from the lossy-compressed noisy observations. For the CE coding in the Gaussian vector case, we show that, if the code rate is less than a threshold, then the CE coding scheme attains the same performance as the optimal coding scheme. We also introduce lower and upper bounds for the performance gap above this threshold. In addition, an example with two observations and two sources is studied to illustrate the behavior of the performance gap.

cs.IT↗

arXiv2016

The Distortion Rate Function of Cyclostationary Gaussian Processes

A general expression for the distortion rate function (DRF) of cyclostationary Gaussian processes in terms of their spectral properties is derived. This expression can be seen as the result of orthogonalization over the different components in the polyphase decomposition of the process. We use this expression to derive, in a closed form, the DRF of several cyclostationary processes arising in practice. We first consider the DRF of a combined sampling and source coding problem. It is known that the optimal coding strategy for this problem involves source coding applied to a signal with the same structure as one resulting from pulse amplitude modulation (PAM). Since a PAM-modulated signal is cyclostationary, our DRF expression can be used to solve for the minimal distortion in the combined sampling and source coding problem. We also analyze in more detail the DRF of a source with the same structure as a PAM-modulated signal, and show that it is obtained by reverse waterfilling over an expression that depends on the energy of the pulse and the baseband process modulated to obtain the PAM signal. This result is then used to study the information content of a PAM-modulated signal as a function of its symbol time relative to the bandwidth of the underlying baseband process. In addition, we also study the DRF of sources with an amplitude-modulation structure, and show that the DRF of a narrow-band Gaussian stationary process modulated by either a deterministic or a random phase sine-wave equals the DRF of the baseband process.

cs.IT↗

arXiv2016

Optimal Rate Allocation in Mismatched Multiterminal Source Coding

We consider a multiterminal source coding problem in which a source is estimated at a central processing unit from lossy-compressed remote observations. Each lossy-encoded observation is produced by a remote sensor which obtains a noisy version of the source and compresses this observation minimizing a local distortion measure which depends only on the marginal distribution of its observation. The central node, on the other hand, has knowledge of the joint distribution of the source and all the observations and produces the source estimate which minimizes a different distortion measure between the source and its reconstruction. In this correspondence, we investigate the problem of optimally choosing the rate of each lossy-compressed remote estimate so as to minimize the distortion at the central processing unit subject to a bound on the overall communication rate between the remote sensors and the central unit. We focus, in particular, on two models of practical relevance: the case of a Gaussian source observed in additive Gaussian noise and reconstructed under quadratic distortion, and the case of a binary source observed in bit-flipping noise and reconstructed under Hamming distortion. In both scenarios we show that there exist regimes under which having more remote encoders does reduce the source distortion: in other words, having fewer, high-quality remote estimates provides a smaller distortion than having more, lower-quality estimates.

cs.IT↗

arXiv2015

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.

cs.IT↗

arXiv2015

Indirect Rate-Distortion Function of a Binary i.i.d Source

The indirect source-coding problem in which a Bernoulli process is compressed in a lossy manner from its noisy observations is considered. These noisy observations are obtained by passing the source sequence through a The indirect source-coding problem in which a Bernoulli process is compressed in a lossy manner from its noisy observations is considered. These noisy observations are obtained by passing the source sequence through a binary symmetric channel so that the channel crossover probability controls the amount of information available about the source realization at the encoder. We use classic results in rate-distortion theory to compute an expression of the rate-distortion function for this model, where the Bernoulli source is not necessarily symmetric. The indirect rate-distortion function is given in terms of a solution to a simple equation. In addition, we derive an upper bound on the indirect rate-distortion function which is given in a closed. These expressions capture precisely the expected behavior that the noisier the observations, the smaller the return from increasing bit-rate to reduce distortion.

cs.IT↗

arXiv2012

Stochastic integration for a wide class of Gaussian stationary increment processes using an extension of the S-transform

Given a Gaussian stationary increment processes with spectral density, we show that a Wick-Ito integral with respect to this process can be naturally obtained using Hida's white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Ito formula.

math.PR↗