lossy computing of correlated sources with fractional sampling this paper considers the problem of lossy compression for the computation of a function of two correlated sources both of which are observed at the encoder due to presence of observation costs the encoder is allowed to observe only subsets of the samples from both sources with a fraction of such sample pairs possibly overlapping the rate distortion function is characterized for memoryless sources and then specialized to gaussian and binary sources for selected functions and with quadratic and hamming distortion metrics respectively the optimal measurement overlap fraction is shown to depend on the function to be computed by the decoder on the source statistics including the correlation and on the link rate special cases are discussed in which the optimal overlap fraction is the maximum or minimum possible value given the sampling budget illustrating non trivial performance trade offs in the design of the sampling strategy finally the analysis is extended to the multi hop set up with jointly gaussian sources where each encoder can observe only one of the sources

optimal phase transitions in compressed sensing compressed sensing deals with efficient recovery of analog signals from linear encodings this paper presents a statistical study of compressed sensing by modeling the input signal as an i i d process with known distribution three classes of encoders are considered namely optimal nonlinear optimal linear and random linear encoders focusing on optimal decoders we investigate the fundamental tradeoff between measurement rate and reconstruction fidelity gauged by error probability and noise sensitivity in the absence and presence of measurement noise respectively the optimal phase transition threshold is determined as a functional of the input distribution and compared to suboptimal thresholds achieved by popular reconstruction algorithms in particular we show that gaussian sensing matrices incur no penalty on the phase transition threshold with respect to optimal nonlinear encoding our results also provide a rigorous justification of previous results based on replica heuristics in the weak noise regime

stabilization of the semilinear wave equation with viscous damping that in order to achieve local stability for a certain class of nonlinear perturba tions it is sufficient to stabilize the linear part of the system the same conclusion can be obtained in the case of ode in banach spaces when i the nonlinear perturbation is represented by the action of the bounded operator or ii the underlying semigroup is analytic see h d l the problem 1 1 however does not belong to either of these categories as the underlying dynamics are of hyperbolic type and the nonlinear perturbation

c matrix transform an approximation to the discrete cosine transform dct called the c matrix transform cmt has been developed by jones et al 3 for n 8 this is extended to n 16 and 32 and its performance is compared with the dct based on some standard criteria cmt is computationally simpler as it involves only integer arithmetic it has potential in signal processing applications because of its closeness to the dct

sampling and distortion tradeoffs for bandlimited periodic signals in this paper the optimal sampling strategies uniform or nonuniform and distortion tradeoffs for gaussian bandlimited periodic signals with additive white gaussian noise are studied our emphasis is on characterizing the optimal sampling locations as well as the optimal presampling filter to minimize the reconstruction distortion we first show that to achieve the optimal distortion no presampling filter is necessary for any arbitrary sampling rate then we provide a complete characterization of optimal distortion for low and high sampling rates with respect to the signal bandwidth we also provide bounds on the reconstruction distortion for rates in the intermediate region it is shown that nonuniform sampling outperforms uniform sampling for low sampling rates in addition the optimal nonuniform sampling set is robust with respect to missing sampling values on the other hand for the sampling rates above the nyquist rate the uniform sampling strategy is optimal

stabilization of hyperbolic and parabolic systems with nonlinearly perturbed boundary conditions abstract sufficient conditions for the local exponential stabilizability of abstract systems described by an arbitrary co semigroup with nonlinear unbounded perturbations are formulated applications to parabolic and in particular hyperbolic problems with nonlinearity appearing in the boundary conditions are given

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

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

a potential well theory for the wave equation with a nonlinear boundary condition soit d un sous ensemble connexe ouvert borne de r n a frontiere de lipschitz d soit d l union de deux sous varietes disjointes a n 1 dimensions et et leur confluence de lipschitz on considere le probleme aux valeurs limites et initiales 2 u t 2 u dans d o t u x o u x u x o t v x dans d u x t o sur o t u n f u x t sur o t

sampling rate distortion consider a discrete memoryless multiple source with m components of which k leq m possibly different sources are sampled at each time instant and jointly compressed in order to reconstruct all the m sources under a given distortion criterion a new notion of sampling rate distortion function is introduced and is characterized first for the case of fixed set sampling next for independent random sampling performed without knowledge of the source outputs it is shown that the sampling rate distortion function is the same regardless of whether or not the decoder is informed of the sequence of sampled sets furthermore memoryless random sampling is considered with the sampler depending on the source outputs and with an informed decoder it is shown that deterministic sampling characterized by a conditional point mass is optimal and suffices to achieve the sampling rate distortion function for memoryless random sampling with an uninformed decoder an upper bound for the sampling rate distortion function is seen to possess a similar property of conditional point mass optimality it is shown by example that memoryless sampling with an informed decoder can outperform strictly any independent random sampler and that memoryless sampling can do strictly better with an informed decoder than without

construction of group transforms subject to several performance criteria a transform l is defined in this paper and studied subject to the two opposing performance criteria computational effectiveness and accuracy of approximation of a given unitary transform measured in distance related to the hilbert sehmidt norm the matrices which represent the l transform constitute a special class of group matrices which were introduced by frobenius in 1895 these n n matrices depend upon certain free parameters whose choice determines their behavior with respect to the two criteria hence part of these parameters can be chosen to affect the computational performance of l whereas the other part is used to minimize its distance from a given unitary transform the computational complexity of l is eased by making l sparse rather than a product of sparse matrices as in the fft algorithm it is shown that if only one of these two criteria is used then either all the parameters can be used to reduce the computational complexity of l to that of the fast haar transform in nontrivial cases or else all of these parameters can be used to achieve asymptotically distance zero from a given unitary transform if both criteria are used then the more parameters that are used for one of them the less parameters we have to deal with for the other one and vice versa an inverse proportional dependence various examples are given

high rate interpolation of random signals from nonideal samples we address the problem of reconstructing a random signal from samples of its filtered version using a given interpolation kernel in order to reduce the mean squared error mse when using a nonoptimal kernel we propose a high rate interpolation scheme in which the interpolation grid is finer than the sampling grid a digital correction system that processes the samples prior to their multiplication with the shifts of the interpolation kernel is developed this system is constructed such that the reconstructed signal is the linear minimum mse lmmse estimate of the original signal given its samples an analytic expression for the mse as a function of the interpolation rate is provided which leads to an explicit condition such that the optimal mse is achieved with the given nonoptimal kernel simulations confirm the reduction in mse with respect to a system with equal sampling and reconstruction rates

on the linear minimum mean squared error estimation of an undersampled wide sense stationary random process we consider the problem of linearly estimating in the sense of minimum mean squared error a wide sense stationary process in noise given uniformly spaced samples where the sampling interval is such that significant aliasing occurs we derive the corresponding aliased wiener filter and provide a technique for determining a closed form for the necessary power spectral density functions we conclude with an example where both signal and noise are modeled using a second order innovations representation

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 per unit time given 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 they describe the minimal amount of excess distortion in the reconstruction due to lossy compression of the samples and provide 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 where each sample is quantized by a scalar quantizer using a fixed number of bits

entropy of highly correlated quantized data this paper considers the entropy of highly correlated quantized samples two results are shown the first concerns sampling and identically scalar quantizing a stationary continuous time random process over a finite interval it is shown that if the process crosses a quantization threshold with positive probability then the joint entropy of the quantized samples tends to infinity as the sampling rate goes to infinity the second result provides an upper bound to the rate at which the joint entropy tends to infinity in the case of an infinite level uniform threshold scalar quantizer and a stationary gaussian random process specifically an asymptotic formula for the conditional entropy of one quantized sample conditioned on the previous quantized sample is derived at high sampling rates these results indicate a sharp contrast between the large encoding rate in bits sec required by a lossy source code consisting of a fixed scalar quantizer and an ideal sampling rate adapted lossless code and the bounded encoding rate required by an ideal lossy source code operating at the same distortion

optimal sub nyquist nonuniform sampling and reconstruction for multiband signals we study the problem of optimal sub nyquist sampling for perfect reconstruction of multiband signals the signals are assumed to have a known spectral support spl fscr that does not tile under translation such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching landau s 1967 lower bound equal to the measure of spl fscr for signals with sparse spl fscr this rate can be much smaller than the nyquist rate unfortunately the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity in a previous study we derived bounds on the error due to mismodeling and sample additive noise adopting these bounds as performance measures we consider the problems of optimizing the reconstruction sections of the system choosing the optimal base sampling rate and designing the nonuniform sampling pattern we find that optimizing these parameters can improve system performance significantly furthermore uniform sampling is optimal for signals with spl fscr that tiles under translation for signals with nontiling spl fscr which are not amenable to efficient uniform sampling the results reveal increased error sensitivities with sub nyquist sampling however these can be controlled by optimal design demonstrating the potential for practical multifold reductions in sampling rate

irregular sampling theorems and series expansions of band limited functions abstract we present a new approach to the problem of irregular sampling of band limited functions that is based on the approximation and factorization of convolution operators a special case of our main result is the following theorem if r n is compact g l 1 rn a band limited function with g t 0 on and xi i i and yj j j are two discrete sets in rn which are dense enough then every band limited function f lwp 1 p our methods work a for a very general class of function spaces not only weighted lwp spaces b in all dimensions even in general locally compact abelian groups c to provide the correct behavior of the coefficients in contrast to the traditional estimates currently used by engineers furthermore the method is constructive and stable with respect to input errors such as round off errors truncation effors jitter errors or aliasing errors

statistical and computational performance of a class of generalized wiener filters corresp a class of suboptimal wiener filters is considered and their computational and statistical performances and the trade off between the two are studied and compared with those for known classes of suboptimal wiener filters a general model of a suboptimal wiener filter over a group is defined which includes as special cases the known filters based on the discrete fourier transform dft in the case of a cyclic group and the walsh hadamard transform wht in the case of a dyadic group statistical and computational performances of various group filters are investigated the cyclic and the dyadic group filters are known to be computationally the best ones among all the group filters however they are not always the best ones statistically and other not necessarily abelian group filters are studied results are compared with those for the cyclic group filters dft and the general problem of selecting the best group filter is posed that problem is solved numerically for small size signals leq 64 for the first order markov process and random sine wave corrupted by white noise for the first order markov process with the covariance matrix b s l rho s l as rho increases the use of various non abelian groups results in improved statistical performance of the filter as compared to the dft similarly for the random sine wave with covariance matrix b s l cos lambda s l as lambda decreases non abelian groups result in a better statistical performance of the filter than the dft does however that is compensated for by the increased number of computations to perform the filtering

information rates of densely sampled data distributed vector quantization and scalar quantization with transforms for gaussian sources this paper establishes rates attainable by several lossy schemes for coding a continuous parameter source to a specified mean squared error distortion based on sampling at asymptotically large rates first a densely sampled spatiotemporal stationary gaussian source is distributively encoded the berger tung bound to the distributed rate distortion function and three convergence theorems are used to obtain an upper bound expressed in terms of the source spectral density to the smallest attainable rate at asymptotically large sampling rates the bound is tighter than that recently obtained by kashyap both indicate that with ideal distributed lossy coding dense sensor networks can efficiently sense and convey a field in contrast to the negative result obtained by marco for encoders based on scalar quantization and slepian wolf distributed lossless coding the second scheme is transform coding with scalar coefficient quantization a new generalized transform coding analysis as well as the aforementioned convergence theorems is used to find the smallest attainable rate at asymptotically large sampling rates in terms of the source spectral density and the operational rate distortion function of the family of quantizers which in contrast to previous analyses need not be convex the result shows that when a transform is used scalar quantization need not cause the poor performance found by marco as a corollary the final result pursues an approach originally proposed by berger to show that the inverse water pouring formula for the rate distortion function can be attained at high sampling rates by transform coding with ideal vector quantization to encode the coefficients also established in the paper are relations between operational rate distortion and distortion rate functions for a continuous parameter source and those for the discrete parameter source that results from sampling

high resolution distributed sampling of bandlimited fields with low precision sensors the problem of sampling a discrete time sequence of spatially bandlimited fields with a bounded dynamic range in a distributed communication constrained processing environment is studied a central unit having access to the data gathered by a dense network of low precision sensors is required to reconstruct the field snapshots to maximum accuracy both deterministic and stochastic field models are considered for stochastic fields results are established in the almost sure sense the feasibility of having a flexible tradeoff between the oversampling rate sensor density and the analog to digital converter adc precision while achieving an exponential accuracy in the number of bits per nyquist interval per snapshot is demonstrated this exposes an underlying conservation of bits principle the bit budget per nyquist interval per snapshot the rate can be distributed along the amplitude axis sensor precision and space sensor density in an almost arbitrary discrete valued manner while retaining the same exponential distortion rate characteristics

data compression and harmonic analysis in this paper we review some recent interactions between harmonic analysis and data compression the story goes back of course to shannon s r d theory in the case of gaussian stationary processes which says that transforming into a fourier basis followed by block coding gives an optimal lossy compression technique practical developments like transform based image compression have been inspired by this result in this paper we also discuss connections perhaps less familiar to the information theory community growing out of the field of harmonic analysis recent harmonic analysis constructions such as wavelet transforms and gabor transforms are essentially optimal transforms for transform coding in certain settings some of these transforms are under consideration for future compression standards we discuss some of the lessons of harmonic analysis in this century typically the problems and achievements of this field have involved goals that were not obviously related to practical data compression and have used a language not immediately accessible to outsiders nevertheless through an extensive generalization of what shannon called the sampling theorem harmonic analysis has succeeded in developing new forms of functional representation which turn out to have significant data compression interpretations we explain why harmonic analysis has interacted with data compression and we describe some interesting recent ideas in the field that may affect data compression in the future

a general sampling theory for nonideal acquisition devices the authors first describe the general class of approximation spaces generated by translation of a function spl psi x and provide a full characterization of their basis functions they then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle the theory puts no restrictions on the system input which can be an arbitrary finite energy signal bandlimitedness is not required in contrast to previous approaches this formulation allows for an independent specification of the sampling analysis and approximation synthesis spaces in particular when both spaces are identical the theorem provides a simple procedure for obtaining the least squares approximation of a signal they discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited and polynomial spline signal representations they also define a spectral coherence function that measures the similarity between the sampling and approximation spaces and derive a relative performance bound for the comparison with the least squares solution

rate distortion performance in coding bandlimited sources by sampling and dithered quantization the rate distortion characteristics of a scheme for encoding continuous time band limited stationary sources with a prescribed band is considered in this coding procedure the input is sampled at nyquist s rate or faster the samples undergo dithered uniform or lattice quantization using subtractive dither and the quantizer output is entropy coded the rate distortion performance and the tradeoff between the sampling rate and the quantization accuracy is investigated utilizing the observation that the coding scheme is equivalent to an additive noise channel it is shown that the mean square error of the scheme is fixed as long as the product of the sampling period and the quantizer second moment is kept constant while for a fixed distortion the coding rate generally increases when the sampling rate exceeds the nyquist rate finally as the lattice quantizer dimension becomes large the equivalent additive noise channel of the scheme tends to be white gaussian and both the rate and the distortion performance become invariant to the sampling rate

the distortion rate function of sampled wiener processes we consider the recovery of a continuous time wiener process from a quantized or a lossy compressed version of its uniform samples under limited bitrate and sampling rate we derive a closed form expression for the optimal tradeoff among sampling rate bitrate and quadratic distortion in this setting this expression is given in terms of a reverse waterfilling formula over the asymptotic spectral distribution of a sequence of finite rank operators associated with the optimal estimator of the wiener process from its samples we show that the ratio between this expression and the standard distortion rate function of the wiener process describing the optimal tradeoff between bitrate and distortion without a sampling constraint is only a function of the number of bits per sample we also consider a sub optimal lossy compression scheme in which the continuous time process is estimated from the output of an encoder that is optimal with respect to the discrete time samples we show that the latter is strictly greater than the distortion under optimal encoding but only by at most 3 we therefore conclude that near optimal performance is attained even if the encoder is unaware of the continuous time origin of the samples