arXiv subjects

Andrea J. Goldsmith

Publications and source records attributed to Andrea J. Goldsmith.

At least 55 records · Page 3Linked to original sources

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.

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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.

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arXiv2018

Blind Joint MIMO Channel Estimation and Decoding

We propose a method for MIMO decoding when channel state information (CSI) is unknown to both the transmitter and receiver. The proposed method requires some structure in the transmitted signal for the decoding to be effective, in particular that the underlying sources are drawn from a hypercubic space. Our proposed technique fits a minimum volume parallelepiped to the received samples. This problem can be expressed as a non-convex optimization problem that can be solved with high probability by gradient descent. Our blind decoding algorithm can be used when communicating over unknown MIMO wireless channels using either BPSK or MPAM modulation. We apply our technique to jointly estimate MIMO channel gain matrices and decode the underlying transmissions with only knowledge of the transmitted constellation and without the use of pilot symbols. Our results provide theoretical guarantees that the proposed algorithm is correct when applied to small MIMO systems. Empirical results show small sample size requirements, making this algorithm suitable for block-fading channels with coherence times typically seen in practice. Our approach has a loss of less than 3dB compared to zero-forcing with perfect CSI, imposing a similar performance penalty as space-time coding techniques without the loss of rate incurred by those techniques.

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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.

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arXiv2017

The Fluctuating Two-Ray Fading Model: Statistical Characterization and Performance Analysis

We introduce the Fluctuating Two-Ray (FTR) fading model, a new statistical channel model that consists of two fluctuating specular components with random phases plus a diffuse component. The FTR model arises as the natural generalization of the two-wave with diffuse power (TWDP) fading model; this generalization allows its two specular components to exhibit a random amplitude fluctuation. Unlike the TWDP model, all the chief probability functions of the FTR fading model (PDF, CDF and MGF) are expressed in closed-form, having a functional form similar to other state-of-the-art fading models. We also provide approximate closed-form expressions for the PDF and CDF in terms of a finite number of elementary functions, which allow for a simple evaluation of these statistics to an arbitrary level of precision. We show that the FTR fading model provides a much better fit than Rician fading for recent small-scale fading measurements in 28 GHz outdoor millimeter-wave channels. Finally, the performance of wireless communication systems over FTR fading is evaluated in terms of the bit error rate and the outage capacity, and the interplay between the FTR fading model parameters and the system performance is discussed. Monte Carlo simulations have been carried out in order to validate the obtained theoretical expressions.

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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.

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arXiv2016

Information Recovery from Pairwise Measurements

A variety of information processing tasks in practice involve recovering $n$ objects from single-shot graph-based measurements, particularly those taken over the edges of some measurement graph $\mathcal{G}$. This paper concerns the situation where each object takes value over a group of $M$ different values, and where one is interested to recover all these values based on observations of certain pairwise relations over $\mathcal{G}$. The imperfection of measurements presents two major challenges for information recovery: 1) $\textit{inaccuracy}$: a (dominant) portion $1-p$ of measurements are corrupted; 2) $\textit{incompleteness}$: a significant fraction of pairs are unobservable, i.e. $\mathcal{G}$ can be highly sparse. Under a natural random outlier model, we characterize the $\textit{minimax recovery rate}$, that is, the critical threshold of non-corruption rate $p$ below which exact information recovery is infeasible. This accommodates a very general class of pairwise relations. For various homogeneous random graph models (e.g. Erdos Renyi random graphs, random geometric graphs, small world graphs), the minimax recovery rate depends almost exclusively on the edge sparsity of the measurement graph $\mathcal{G}$ irrespective of other graphical metrics. This fundamental limit decays with the group size $M$ at a square root rate before entering a connectivity-limited regime. Under the Erdos Renyi random graph, a tractable combinatorial algorithm is proposed to approach the limit for large $M$ ($M=n^{Ω(1)}$), while order-optimal recovery is enabled by semidefinite programs in the small $M$ regime. The extended (and most updated) version of this work can be found at (http://arxiv.org/abs/1504.01369).

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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.

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arXiv2016

Information Recovery from Pairwise Measurements

This paper is concerned with jointly recovering $n$ node-variables $\left\{ x_{i}\right\}_{1\leq i\leq n}$ from a collection of pairwise difference measurements. Imagine we acquire a few observations taking the form of $x_{i}-x_{j}$; the observation pattern is represented by a measurement graph $\mathcal{G}$ with an edge set $\mathcal{E}$ such that $x_{i}-x_{j}$ is observed if and only if $(i,j)\in\mathcal{E}$. To account for noisy measurements in a general manner, we model the data acquisition process by a set of channels with given input/output transition measures. Employing information-theoretic tools applied to channel decoding problems, we develop a \emph{unified} framework to characterize the fundamental recovery criterion, which accommodates general graph structures, alphabet sizes, and channel transition measures. In particular, our results isolate a family of \emph{minimum} \emph{channel divergence measures} to characterize the degree of measurement corruption, which together with the size of the minimum cut of $\mathcal{G}$ dictates the feasibility of exact information recovery. For various homogeneous graphs, the recovery condition depends almost only on the edge sparsity of the measurement graph irrespective of other graphical metrics; alternatively, the minimum sample complexity required for these graphs scales like \[ \text{minimum sample complexity }\asymp\frac{n\log n}{\mathsf{Hel}_{1/2}^{\min}} \] for certain information metric $\mathsf{Hel}_{1/2}^{\min}$ defined in the main text, as long as the alphabet size is not super-polynomial in $n$. We apply our general theory to three concrete applications, including the stochastic block model, the outlier model, and the haplotype assembly problem. Our theory leads to order-wise tight recovery conditions for all these scenarios.

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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.

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arXiv2015

Energy-based Modulation for Noncoherent Massive SIMO Systems

An uplink system with a single antenna transmitter and a single receiver with a large number of antennas is considered. We propose an energy-detection-based single-shot noncoherent communication scheme which does not use the instantaneous channel state information (CSI), but rather only the knowledge of the channel statistics. The suggested system uses a transmitter that modulates information on the power of the symbols, and a receiver which measures only the average energy across the antennas. We propose constellation designs which are asymptotically optimal with respect to symbol error rate (SER) with an increasing number of antennas, for any finite signal to noise ratio (SNR) at the receiver, under different assumptions on the availability of CSI statistics (exact channel fading distribution or the first few moments of the channel fading distribution). We also consider the case of imperfect knowledge of the channel statistics and describe in detail the case when there is a bounded uncertainty on the moments of the fading distribution. We present numerical results on the SER performance achieved by these designs in typical scenarios and find that they may outperform existing noncoherent constellations, e.g., conventional Amplitude Shift Keying (ASK), and pilot-based schemes, e.g., Pulse Amplitude Modulation (PAM). We also observe that an optimized constellation for a specific channel distribution makes it very sensitive to uncertainties in the channel statistics. In particular, constellation designs based on optimistic channel conditions could lead to significant performance degradation in terms of the achieved symbol error rates.

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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.

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arXiv2014

Diversity-Multiplexing Tradeoff for the Interference Channel with a Relay

We study the diversity-multiplexing tradeoff (DMT) for the slow fading interference channel with a relay (ICR). We derive four inner bounds on the DMT region: the first is based on the compress-and-forward (CF) relaying scheme, the second is based on the decode-and-forward (DF) relaying scheme, and the last two bounds are based on the half-duplex (HD) and full-duplex (FD) amplify-and-forward (AF) schemes. For the CF and DF schemes, we find conditions on the channel parameters and the multiplexing gains, under which the corresponding inner bound achieves the optimal DMT region. We also identify cases in which the DMT region of the ICR corresponds to that of two parallel slow fading relay channels, implying that interference does not decrease the DMT for each pair, and that a single relay can be DMT-optimal for two pairs simultaneously. For the HD-AF scheme we derive conditions on the channel coefficients under which the proposed scheme achieves the optimal DMT for the AF-based relay channel. Lastly, we identify conditions under which adding a relay strictly enlarges the DMT region relative to the interference channel without a relay.

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arXiv2014

Backing off from Infinity: Performance Bounds via Concentration of Spectral Measure for Random MIMO Channels

The performance analysis of random vector channels, particularly multiple-input-multiple-output (MIMO) channels, has largely been established in the asymptotic regime of large channel dimensions, due to the analytical intractability of characterizing the exact distribution of the objective performance metrics. This paper exposes a new non-asymptotic framework that allows the characterization of many canonical MIMO system performance metrics to within a narrow interval under moderate-to-large channel dimensionality, provided that these metrics can be expressed as a separable function of the singular values of the matrix. The effectiveness of our framework is illustrated through two canonical examples. Specifically, we characterize the mutual information and power offset of random MIMO channels, as well as the minimum mean squared estimation error of MIMO channel inputs from the channel outputs. Our results lead to simple, informative, and reasonably accurate control of various performance metrics in the finite-dimensional regime, as corroborated by the numerical simulations. Our analysis framework is established via the concentration of spectral measure phenomenon for random matrices uncovered by Guionnet and Zeitouni, which arises in a variety of random matrix ensembles irrespective of the precise distributions of the matrix entries.

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arXiv2014

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.

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arXiv2014

An Algorithm for Exact Super-resolution and Phase Retrieval

We explore a fundamental problem of super-resolving a signal of interest from a few measurements of its low-pass magnitudes. We propose a 2-stage tractable algorithm that, in the absence of noise, admits perfect super-resolution of an $r$-sparse signal from $2r^2-2r+2$ low-pass magnitude measurements. The spike locations of the signal can assume any value over a continuous disk, without increasing the required sample size. The proposed algorithm first employs a conventional super-resolution algorithm (e.g. the matrix pencil approach) to recover unlabeled sets of signal correlation coefficients, and then applies a simple sorting algorithm to disentangle and retrieve the true parameters in a deterministic manner. Our approach can be adapted to multi-dimensional spike models and random Fourier sampling by replacing its first step with other harmonic retrieval algorithms.

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arXiv2013

The One-Bit Null Space Learning Algorithm and its Convergence

This paper proposes a new algorithm for MIMO cognitive radio Secondary Users (SU) to learn the null space of the interference channel to the Primary User (PU) without burdening the PU with any knowledge or explicit cooperation with the SU. The knowledge of this null space enables the SU to transmit in the same band simultaneously with the PU by utilizing separate spatial dimensions than the PU. Specifically, the SU transmits in the null space of the interference channel to the PU. We present a new algorithm, called the One-Bit Null Space Learning Algorithm (OBNSLA), in which the SU learns the PU's null space by observing a binary function that indicates whether the interference it inflicts on the PU has increased or decreased in comparison to the SU's previous transmitted signal. This function is obtained by listening to the PU transmitted signal or control channel and extracting information from it about whether the PU's Signal to Interference plus Noise power Ratio (SINR) has increased or decreased. In addition to introducing the OBNSLA, this paper provides a thorough convergence analysis of this algorithm. The OBNSLA is shown to have a linear convergence rate and an asymptotic quadratic convergence rate. Finally, we derive bounds on the interference that the SU inflicts on the PU as a function of a parameter determined by the SU. This lets the SU control the maximum level of interference, which enables it to protect the PU completely blindly with minimum complexity. The asymptotic analysis and the derived bounds also apply to the recently proposed Blind Null Space Learning Algorithm.

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arXiv2013

Shannon Meets Nyquist: Capacity of Sampled Gaussian Channels

We explore two fundamental questions at the intersection of sampling theory and information theory: how channel capacity is affected by sampling below the channel's Nyquist rate, and what sub-Nyquist sampling strategy should be employed to maximize capacity. In particular, we derive the capacity of sampled analog channels for three prevalent sampling strategies: sampling with filtering, sampling with filter banks, and sampling with modulation and filter banks. These sampling mechanisms subsume most nonuniform sampling techniques applied in practice. Our analyses illuminate interesting connections between under-sampled channels and multiple-input multiple-output channels. The optimal sampling structures are shown to extract out the frequencies with the highest SNR from each aliased frequency set, while suppressing aliasing and out-of-band noise. We also highlight connections between undersampled channel capacity and minimum mean-squared error (MSE) estimation from sampled data. In particular, we show that the filters maximizing capacity and the ones minimizing MSE are equivalent under both filtering and filter-bank sampling strategies. These results demonstrate the effect upon channel capacity of sub-Nyquist sampling techniques, and characterize the tradeoff between information rate and sampling rate.

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