on the stochastic porous medium equation abstract in this paper we discuss an initial boundary value problem and a cauchy problem for the stochastic porous medium equation our basic estimates are based on the known results due to math sbornik 67 1965 609 642 arch rational mech anal 25 1967 64 80 by the procedure developed in trans amer math soc 354 2001 1117 1135 we obtain solutions over the given probability space rather than martingale solutions we will also establish the existence of invariant measures when the space domain is bounded

optimal design of magnitude responses of rational infinite impulse response filters this correspondence considers a design of magnitude responses of optimal rational infinite impulse response iir filters the design problem is formulated as an optimization problem in which a total weighted absolute error in the passband and stopband of the filters the error function reflects a ripple square magnitude is minimized subject to the specification on this weighted absolute error function defined in the corresponding passband and stopband as well as the stability condition since the cost function is nonsmooth and nonconvex while the constraints are continuous this kind of optimization problem is a nonsmooth nonconvex continuous functional constrained problem to address this issue our previous proposed constraint transcription method is applied to transform the continuous functional constraints to equality constraints then the nonsmooth problem is approximated by a sequence of smooth problems and solved via a hybrid global optimization method the solutions obtained from these smooth problems converge to the global optimal solution of the original optimization problem hence small transition bandwidth filters can be obtained

allpass vfd filter design this correspondence proposes a general design for allpass variable fractional delay vfd digital filters with minimum weighted integral squared error subject to constraints on maximum error deviation from the desired response the resulting optimization problem is nonlinear and nonconvex with a nonlinear continuous inequality constraint stability of the designed filters are discussed an effective procedure is proposed for solving the optimization problem firstly a constraint transcription method and a smoothing technique are employed to transform the continuous inequality constraint into one equality constraint then by using the concept of a penalty function the transformed constraint is incorporated into the cost function to form a new cost function the nonlinear optimization problem subject to continuous inequality constraints is then approximated by a sequence of unconstraint optimization problems finally a global optimization method using a filled function is employed to solve the unconstraint optimization problem design example shows that a trade off can be achieved between the integral squared error and the maximum error deviation for the design of allpass vfd filters

digital filter design using root moments for sum of all pass structures from complete and partial specifications this paper is concerned with the development of digital filter design procedures for transfer functions in the form of sum of all pass in which the requirements may be partially specified specifically the requirements for a digital filter or equalizer in amplitude a spl theta or phase spl phi spl theta or possibly group delay response spl tau spl theta may be specified from measurements over a limited a set of frequencies spl theta sub 1 spl theta sub 2 the problem is to develop techniques for the design a transfer function h z satisfying these specifications and constrained to be in the form of a sum of two all pass functions the proposed solution is based on the use of root moments the companion problem concerned with the estimation of the orders of the required all pass filters is also examined and a solution proposed based on the same context of root moments

eigenfilter design of 1 d and 2 d iir digital all pass filters the eigenfilter approach is extended to design 1d and 2d iir digital all pass filters instead of the integration of the filter magnitude error the integration of phase error is minimized such that the design problem can be characterized by an eigenformulation the stability is obtained by constraining the phase specification

a technique for realizing linear phase iir filters a real time iir filter structure is presented that possesses exact phase linearity with 10 approximately 1000 times fewer general multiplies than conventional fir filters of similar performance and better magnitude characteristics than equiripple or maximally flat group delay iir filters this structure is based on a technique using local time reversal and single pass sectioned convolution methods to realized a real time recursive implementation of the noncausal transfer function h z sup 1 the time reversed section technique used to realize exactly linear phase iir filters is described the effects of finite section length on the sectional convolution are analyzed a simulation methodology is developed to address the special requirements of simulating a time reversed section filter a design example is presented with computer simulation to illustrate performance in terms of overall magnitude response and phase linearity as a function of finite section length nine example filter specifications are used to compare the performance and complexity of the time reversed section technique to those of a direct fir implementation

optimal design of fir filters with the complex chebyshev error criteria we present an improved algorithm for an established filter design problem the design of an fir filter that best approximates in the complex chebyshev sense a desired complex valued frequency response the algorithm is a variant of the simplex algorithm of linear programming which has an interpretation as an implicit multiple exchange it is iterative robust and exhibits good convergence speed global optimum convergence is guaranteed both complex and real valued impulse responses can be designed with it the design of complex coefficient filters is new an example is given for each case the design of noncausal filters is new in addition to these new applications we conjecture that this new algorithm may have important advantages over existing techniques with respect to the maximum filter length possible speed and stability of convergence accuracy and memory requirements the ability to design long filters is among the more significant improvements over previous work filters of length 1000 have been designed with the new method

allpass filter design and applications the problem of allpass filter design for phase approximation and equalization in the chebyshev sense is solved by using a generalized remez algorithm convergence to the unique optimum is guaranteed and is achieved rapidly in the actual implementation the well known numerical problems for higher degree filters are analyzed and solved by a simple approach the algorithm yields a solution to a variety of filter design problems such as the design of filters with a desired phase response e g a delay element the design of phase equalizers or the design of recursive filters with magnitude prescriptions using parallel allpass filters for the latter the algorithm can be modified to allow arbitrary tolerance schemes for the magnitude response

design of fir digital phase networks the problem of the minimax design of fir digital filters with prescribed phase characteristics and unit magnitude is a nonlinear optimization problem in this paper it is approximated by a linear programming problem and it is shown that the solution of this linear program is optimal to first order that is if 0 and e 0 are optimal deviations of magnitude and phase characteristics then the actual deviations obtained from the linear program solution satisfy delta leq delta 0 epsilon min 0 max 2 and epsilon leq epsilon 0 1 delta 0 1 delta 0 numerical examples are given including design results for full band m term chirp filters which like linear phase filters can be implemented with m 1 2 multiplications per point

parallel factor analysis in sensor array processing this paper links multiple invariance sensor array processing mi sap to parallel factor parafac analysis which is a tool rooted in psychometrics and chemometrics parafac is a common name for low rank decomposition of three and higher way arrays this link facilitates the derivation of powerful identifiability results for mi sap shows that the uniqueness of single and multiple invariance esprit stems from uniqueness of low rank decomposition of three way arrays and allows tapping on the available expertise for fitting the parafac model the results are applicable to both data domain and subspace mi sap formulations the paper also includes a constructive uniqueness proof for a special parafac model

iir discrete time hilbert transformers a procedure for obtaining iir hilbert transformers from generalized halfband filters 5 is described the solution is shown to be suited for specifications where the care band for the hilbert transformers is centered at 0 5 even when this condition is not satisfied the proposed hilbert transformers obtained by overdesign may often be preferable for implementation to those obtained from a general design procedure 6 analytic solutions for noncausal hilbert transformers and 90 phase shifters can be derived from special classes of classical low pass filters the design procedure described by gold et al 6 is shown to be equivalent to applying a transformation on hilbert transformers derived from elliptic low pass filters numerical approximation can be used for designing iir hilbert transformers with a right sided impulse response this gives us a single causal iir filter instead of a pair of 90 phase shifters whose output is the approximate hilbert transform of the delayed version of the input

wideband spectrum sensing based on sub nyquist sampling in this paper we consider the problem of locating multiple active spectrum subbands in a wide range of frequency bands a major challenge associated with such wideband spectrum sensing is that it is either infeasible or too expensive to perform nyquist sampling on the wideband signal in this paper we propose a sensing scheme based on a sub nyquist sampling method called multicoset sampling which is similar to the polyphase implementation of the nyquist sampling but requires less a d converters in contrast to the traditional sub nquist approaches where the wideband signal is first reconstructed from the sub nyquist samples we develop a method that directly estimates the power spectrum of the wideband signal of interest using the sub nyquist samples by exploiting its statistical properties we also characterize the statistical distribution of the proposed power spectrum estimator based on which we obtain a constant false alarm energy detector for the frequency bins simulation results are provided to demonstrate the effectiveness of the proposed multiband spectrum sensing method based on sub nyquist sampling

h infty optimal fractional delay filters fractional delay filters are digital filters to delay discrete time signals by a fraction of the sampling period since the delay is fractional the intersample behavior of the original analog signal becomes crucial in contrast to the conventional designs based on the shannon sampling theorem with the band limiting hypothesis the present paper proposes a new approach based on the modern sampled data h optimization that aims at restoring the intersample behavior beyond the nyquist frequency by using the lifting transform or continuous time blocking the design problem is equivalently reduced to a discrete time h optimization which can be effectively solved by numerical computation softwares moreover a closed form solution is obtained under an assumption on the original analog signals design examples are given to illustrate the advantage of the proposed method

weighted least squares design of recursive allpass filters a method for the design of allpass filters is described in this method an error reflecting the difference between the desired phase response and the phase response of the practical allpass filter is formulated in a quadratic form the coefficients are obtained by solving a system of linear equations involving the sum of a toeplitz and an hankel matrix

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

design of iir digital allpass filters based on eigenvalue problem a new method is proposed for designing iir digital allpass filters with an equiripple phase response that can be proven to be optimal in the chebyshev sense the proposed procedure is based on the formulation of an eigenvalue problem by using the remez exchange algorithm since there exists more than one eigenvalue in the general eigenvalue problem we introduce a new and very simple selection rule for the eigenvalue to be searched for where the rational interpolation is performed if and only if the real maximum eigenvalue is chosen therefore the solution of the rational interpolation problem can be gotten by computing only one eigenvector corresponding to the real maximum eigenvalue and the optimal filter coefficients are easily obtained through a few iterations without any initial guess of the solution the design algorithm proposed not only retains the speed inherent in the remez exchange algorithm but also simplifies the interpolation step because it has been reduced to the computation of the real maximum eigenvalue two examples are designed to demonstrate the effectiveness of this method

complex chebyshev approximation for fir digital filters an adaptive approach to fir finite impulse response filter design is applied to a fully general complex specification showing it to be capable of giving an optimum chebyshev solution it is thought to be one of the first practical approaches to achieve this in the case of a complex function design examples include a bandpass filter with nearly linear phase and all pass filters with general shaping phase comparisons are drawn with equivalent results from other algorithms

velocity filtering of acoustic well logging waveforms a velocity filter that is based on a particularly convenient and effective combination of an up down filter and slanting time shift operations is presented the up down filter is based on a cascade of spatial and temporal hilbert transform filters the hilbert transform filters and the time shift operators are finite duration impulse response fir filters designed according to the minimax criterion the velocity filter is applied to acoustic well logging waveforms to isolate the initial p s and stoneley waves from their reflected and converted waves at bed boundaries to achieve this objective the waveforms are time shifted to align the primary wave of interest before the velocity filter is applied examples demonstrate the effectiveness of this procedure in separating the primary waves from their reflected and converted waves

design of recursive group delay filters by autoregressive modeling a new method of design of recursive group delay filters is presented in this paper the method uses autoregressive modeling to derive parameters of the filter the design is based on the following line of argument the desired group delay characteristics are to be realized through a stable all pass filter the transfer function of an all pass filter is completely determined by the coefficients of either the numerator or the denominator polynomial for a stable all pass filter the denominator polynomial must be a minimum phase polynomial for a minimum phase polynomial the magnitude function and the group delay function are related through cepstral coefficients therefore from the given group delay specification the cepstral coefficients corresponding to the pole part inverse of the denominator polynomial of the desired filter are first determined the magnitude spectrum corresponding to these cepstral coefficients is approximated using autoregressive modeling a wide variety of group delay filters can be realized by the proposed method depending on the nature of approximation used and the accuracy desired the design procedure is illustrated through examples

eigenfilter design of higher order digital differentiators an effective approach is proposed for designing higher order differentiators by the eigenfilter method by minimizing a quadratic measure of the error in the frequency band an eigenvector of an appropriate matrix is computed to get the filter coefficients this method is not only simple and fast but also optimal in the least squares sense comparison to the mcclellan parks algorithm for minimax equiripple filters shows that both are optimal in the sense of different minimum norms of the error function but much better performance is obtained with the proposed approach in most of the frequency band except in the narrowband region near the cutoff edge

eigenfilter approach for the design of allpass filters approximating a given phase response we apply the eigenfilter method to design an allpass filter that approximates a given phase response in the least squares ls sense as it is not possible to express the exact ls phase error as a quadratic form suitable for eigenfilter formulation alternative error measures that approximate the ideal ls error are proposed for each of these new formulations the allpass coefficients are obtained as the elements of the eigenvector corresponding to the minimum eigenvalue of a real symmetric and positive definite matrix we propose a fast converging iterative technique to approximate the ideal ls phase error solution by employing an iterative weighting technique the phase error can he made approximately equiripple the design methods are illustrated with various practical examples and the results are compared to allpass filters designs reported in the literature

compressive wideband power spectrum estimation in several applications such as wideband spectrum sensing for cognitive radio only the power spectrum a k a the power spectral density is of interest and there is no need to recover the original signal itself in addition high rate analog to digital converters adcs are too power hungry for direct wideband spectrum sensing these two facts have motivated us to investigate compressive wideband power spectrum sensing which consists of a compressive sampling procedure and a reconstruction method that is able to recover the unknown power spectrum of a wide sense stationary signal from the obtained sub nyquist rate samples it is different from spectrum blind sampling sbs which aims at reconstructing the original signal instead of the power spectrum in this paper a solution is first presented based on a periodic sampling procedure and a simple least squares reconstruction method we evaluate the reconstruction process both in the time and frequency domain then we examine two possible implementations for the compressive sampling procedure namely complex gaussian sampling and multicoset sampling although we mainly focus on the latter a new type of multicoset sampling is introduced based on the so called minimal sparse ruler problem next we analyze the statistical properties of the estimated power spectrum the computation of the mean and the covariance of the estimates allows us to calculate the analytical normalized mean squared error nmse of the estimated power spectrum further when the received signal is assumed to contain only circular complex zero mean gaussian i i d noise the computed mean and covariance can be used to derive a suitable detection threshold simulation results underline the promising performance of our proposed approach note that all benefits of our method arise without putting any sparsity constraints on the power spectrum

a new method for digital all pass filter design a new method for the design of digital all pass filters using chebyshev criterion is introduced it is based on a phase approximation algorithm for polynomial transfer functions the algorithm exploits a scheme of iteratively linearizing the nonlinear constraints in a nonlinear programming and converges theoretically the design method has the advantage that it finds the best uniform phase approximation to an arbitrarily specified phase response without any initial guess of the solution design examples of orders up to 80 obtained on an ibm pc xt personal computer are given to show the practicability of the method

dynamics of stochastic zakharov equations the asymptotic behavior of zakharov equations driven by random force is studied the force which is smooth enough and homogeneous in space and white noise in time acts on both equations of the zakharov system independently by some a priori estimates we prove the existence of a solution in energy spaces e1 and e2 via the galerkin approximation this solution is defined on the given probability space rather than a martingale solution then a global random attractor is constructed in energy space e2 equipped with weak topology further the existence of a stationary measure is proved in energy space e2 with usual topology

design of fir filters in the complex domain the design of fir digital filters with a complex valued desired frequency response using the chebyshev error is investigated the complex approximation problem is converted into a real approximation problem which is nearly equivalent to the complex problem a standard linear programming algorithm for the chebyshev solution of overdetermined equations is used to solve the real approximation problem additional constraints are introduced which allow weighting of the phase and or group delay of the approximation digital filters are designed which have nearly constant group delay in the passbands the desired constant group delay which gives the minimum chebyshev error is found to be smaller than that of a linear phase filter of the same length these filters in addition to having a smaller approximately constant group delay have better magnitude characteristics than exactly linear phase filters with the same length the filters have nearly equiripple magnitude and group delay

the design of arbitrary fir digital filters using the eigenfilter method the design of finite impulse response fir digital filters for approximating an arbitrary function in both magnitude and phase in the least square sense is studied the design method is based on the computation of an eigenvector of an appropriate real symmetric and positive definite matrix the design of the complex coefficient filter is shown to be an extension of the design of the real coefficient filter several design examples including the constant group delay filters and digital phase all pass filters are presented comparisons to existing methods are made

perfect reconstruction formulas and bounds on aliasing error in sub nyquist nonuniform sampling of multiband signals we examine the problem of periodic nonuniform sampling of a multiband signal and its reconstruction from the samples this sampling scheme which has been studied previously has an interesting optimality property that uniform sampling lacks one can sample and reconstruct the class spl bscr spl fscr of multiband signals with spectral support spl fscr at rates arbitrarily close to the landau 1969 minimum rate equal to the lebesgue measure of spl fscr even when spl fscr does not tile r under translation using the conditions for exact reconstruction we derive an explicit reconstruction formula we compute bounds on the peak value and the energy of the aliasing error in the event that the input signal is band limited to the span of spl fscr the smallest interval containing spl fscr which is a bigger class than the valid signals spl bscr spl fscr band limited to spl fscr we also examine the performance of the reconstruction system when the input contains additive sample noise

gridless line spectrum estimation and low rank toeplitz matrix compression using structured samplers a regularization free approach this paper considers the problem of compressively sampling wide sense stationary random vectors with a low rank toeplitz covariance matrix certain families of structured deterministic samplers are shown to efficiently compress a high dimensional toeplitz matrix of size n times n producing a compressed sketch of size o sqrt r times o sqrt r the reconstruction problem can be cast as that of line spectrum estimation whereby in absence of noise toeplitz matrices of any size n can be exactly recovered from compressive sketches of size o sqrt r times o sqrt r no matter how large n is in the presence of noise and finite data the line spectrum estimation algorithm is combined with a novel denoising technique that only exploits a positive semidefinite psd toeplitz constraint to denoise the compressed sketch using a simple least squares minimization framework a major advantage of the algorithm is that it does not require any regularization parameter it also enjoys lower computational complexity owing to its ability to predict the unobserved entries of the low rank toeplitz matrix explicit bounds on the reconstruction error are established and it is shown that the psd constraint on the denoiser is sufficient to ensure stable reconstruction from a sketch of size o sqrt r times o sqrt r extensive simulations demonstrate that the proposed algorithm provides better performance over random samplers and algorithms that use nuclear norm based regularizers

existence of self similar blow up solutions for zakhrov equation in dimension two part i we consider the zakharov equation in space dimension two left begin array 20 c iu t delta u nu frac 1 c 0 2 n tt delta n delta left u right 2 end array right we prove the existence of blow up solutions stable self similar blow up solutions for this problem and we study various properties of these solutions

simultaneous design in both magnitude and group delay of iir and fir filters based on multiple criterion optimization this work considers the simultaneous design in both magnitude and group delay of digital transfer functions on the basis of multiple criterion optimization both causal iir filters and nonlinear phase fir filters are studied examples of the optimal tradeoff filters both fir filters and iir filters are presented and their characteristics are analyzed

minimax phase error design of iir digital filters with prescribed magnitude and phase responses infinite impulse response iir digital filters with prescribed magnitude and phase responses have been used in many applications to approximate the prescribed magnitude and phase responses we propose a new approach to the design of general iir filters by minimizing the maximum phase error subject to a prescribed or simultaneously minimized maximum magnitude error where the phase error and magnitude error are controlled by two elliptic constraints respectively with major and minor axes along the desired frequency response the sequential constrained least squares method and levy sanathanan koerner strategy are used to convert the nonconvex constraints into convex ones resulting in a series of convex optimization subproblems design examples and comparisons with recent methods demonstrate the flexibility and effectiveness of the proposed methods

design of iir eigenfilters in the frequency domain the eigenfilter approach is an appealing way of designing digital filters mainly because of the simplicity of its implementation in this correspondence a new method of applying the eigenfilter approach to the design of infinite impulse response iir filters is described the procedure works in the frequency domain and yields the coefficients of a causal rational transfer function having an arbitrary number of poles and zeros some examples of filter design are given to show the effectiveness of the method presented

a technique for multiple criterion approximation of fir filters in magnitude and group delay in spite of the attention received by nonlinear phase nlp fir filter design the best way to solve this problem is still open to debate the formulation of nonlinear phase fir filter design in terms of simultaneous magnitude and group delay approximation addresses the two parameters of ultimate interest for applications the simultaneous minimization of these two functions one of which the group delay rational is approached on the basis of an original extension of the differential correction algorithm the proposed design tool enjoys interesting theoretical properties and works very effectively the filters obtained according to the multiple criterion optimization mco formulation are compared against filters that are optimal in the complex chebychev norm then the flexibility characteristic of mco formulations of trading off between magnitude and group delay performance is exemplified

sub nyquist cyclostationary detection for cognitive radio cognitive radio requires efficient and reliable spectrum sensing of wideband signals in order to cope with the sampling rate bottleneck new sampling methods have been proposed that sample below the nyquist rate however such techniques decrease the signal to noise ratio snr deteriorating the performance of subsequent energy detection cyclostationary detection which exploits the periodic property of communication signal statistics absent in stationary noise is a natural candidate for this setting in this paper we consider cyclic spectrum recovery from sub nyquist samples in order to achieve both efficiency and robustness to noise to that end we propose a structured compressed sensing algorithm which extends orthogonal matching pursuit to account for the structure imposed by cyclostationarity next we derive a lower bound on the sampling rate required for perfect cyclic spectrum recovery in the presence of stationary noise in particular we show that the cyclic spectrum can be reconstructed from samples obtained at 4 5 of the nyquist rate without any sparsity constraints on the signal if the signal of interest is sparse then the sampling rate may be further reduced to 8 5 of the landau rate once the cyclic spectrum is recovered we estimate the number of transmissions that compose the input signal along with their carrier frequencies and bandwidths simulations show that cyclostationary detection outperforms energy detection in low snrs in the sub nyquist regime this was already known in the nyquist regime but is even more pronounced at sub nyquist sampling rates

sub nyquist sampling for power spectrum sensing in cognitive radios a unified approach in light of the ever increasing demand for new spectral bands and the underutilization of those already allocated the concept of cognitive radio cr has emerged opportunistic users could exploit temporarily vacant bands after detecting the absence of activity of their owners one of the crucial tasks in the cr cycle is therefore spectrum sensing and detection which has to be precise and efficient yet crs typically deal with wideband signals whose nyquist rates are very high in this paper we propose to reconstruct the power spectrum of such signals from sub nyquist samples rather than the signal itself as done in previous work in order to perform detection we consider both sparse and non sparse signals as well as blind and non blind detection in the sparse case for each one of those scenarios we derive the minimal sampling rate allowing perfect reconstruction of the signal s power spectrum in a noise free environment and provide power spectrum recovery techniques that achieve those rates the analysis is performed for two different signal models considered in the literature which we refer to as the analog and digital models and shows that both lead to similar results simulations demonstrate power spectrum recovery at the minimal rate in noise free settings and the impact of several parameters on the detector performance including signal to noise ratio sensing time and sampling rate

the design of approximate hilbert transform pairs of wavelet bases several authors have demonstrated that significant improvements can be obtained in wavelet based signal processing by utilizing a pair of wavelet transforms where the wavelets form a hilbert transform pair this paper describes design procedures based on spectral factorization for the design of pairs of dyadic wavelet bases where the two wavelets form an approximate hilbert transform pair both orthogonal and biorthogonal fir solutions are presented as well as iir solutions in each case the solution depends on an allpass filter having a flat delay response the design procedure allows for an arbitrary number of vanishing wavelet moments to be specified a matlab program for the procedure is given and examples are also given to illustrate the results