vector sampling expansion the vector sampling expansion vse is an extension of papoulis 1977 generalized sampling expansion gse to the vector case in vse n bandlimited signals all with the same bandwidth b are passed through a multi input multi output mimo linear time invariant system that generates m m spl ges n output signals the goal is to reconstruct the input signals from the samples of the output signals at a total sampling rate of n times the nyquist rate where the nyquist rate is b spl pi samples per second we find necessary and sufficient conditions for this reconstruction a surprising necessary condition for the case where all output signals are uniformly sampled at the same rate n m times the nyquist rate is that the expansion factor m n must be an integer this condition is no longer necessary when each output signal is sampled at a different rate or sampled nonuniformly this work also includes a noise sensitivity analysis of vse systems we define the noise amplification factor which allows a quantitative comparison between vse systems and determine the optimal vse systems

generalized sampling expansions with multiple sampling rates for lowpass and bandpass signals in the fractional fourier transform domain the objective of generalized sampling expansion gse is the reconstruction of an unknown continuously defined function f left t right from samples of the responses from m linear time invariant lti systems that are each sampled using the 1 m th nyquist rate in this paper we investigate the gse for lowpass and bandpass signals with multiple sampling rates in the fractional fourier transform frft domain first we propose an improvement of papoulis gse which has multiple sampling rates in the frft domain based on the proposed gse we derive the periodic nonuniform sampling scheme and the derivative interpolation method by designing different fractional filters and selecting specific sampling rates in addition the papoulis gse and the previous gse associated with frft are shown to be special instances of our results second we address the problem of the gse of fractional bandpass signals a new gse for fractional bandpass signals with equal sampling rates is derived we show that the restriction of an even number of channels in the gse for fractional bandpass signals is unnecessary and perfect signal reconstruction is possible for any arbitrary number of channels further we develop the gse for a fractional bandpass signal with multiple sampling rates lastly we discuss the application of the proposed method in the context of single image super resolution reconstruction based on gse illustrations and simulations are presented to verify the validity and effectiveness of the proposed results

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

the zak transform and sampling theorems for wavelet subspaces the zak transform is used for generalizing a sampling theorem of g waiter see ieee trans informat theory vol 38 p 881 884 1992 for wavelet subspaces cardinal series based on signal samples f a n n in z with a possibly unequal to 0 waiter s case are considered the condition number of the sampling operator and worst case aliasing errors are expressed in terms of zak transforms of scaling function and wavelet this shows that the stability of the resulting interpolation formula depends critically on a

efficient reconstruction of band limited sequences from nonuniformly decimated versions by use of polyphase filter banks an efficient polyphase structure for the reconstruction of a band limited sequence from a nonuniformly decimated version is developed theoretically the reconstruction involves the implementation of a bank of multilevel filters and it is shown that how all these reconstruction filters can be obtained at the cost of one mth band low pass filter and a constant matrix multiplier the resulting structure is therefore more general than previous schemes in addition the method offers a direct means of controlling the overall reconstruction distortion t z by appropriate design of a low pass prototype filter p z extension of these results to multiband band limited signals and to the case of nonconsecutive nonuniform subsampling are also summarized along with generalizations to the multidimensional case design examples are included to demonstrate the theory and the complexity of the new method is seen to be much lower than earlier ones

generalized sampling stability and performance analysis generalized sampling provides a general mechanism for recovering an unknown input function f x spl epsiv spl hscr from the samples of the responses of m linear shift invariant systems sampled at 1 mth the reconstruction rate the system can be designed to perform a projection of f x onto the reconstruction subspace v spl phi span spl phi x k sub k spl epsiv z for example the family of bandlimited signals with spl phi x sinc x this implies that the reconstruction will be perfect when the input signal is included in v spl phi the traditional framework of papoulis 1977 generalized sampling theory otherwise one recovers a signal approximation f x spl epsiv v spl phi that is consistent with f x in the sense that it produces the same measurements to characterize the stability of the algorithm we prove that the dual synthesis functions that appear in the generalized sampling reconstruction formula constitute a riesz basis of v spl phi and we use the corresponding riesz bounds to define the condition number of the system we then use these results to analyze the stability of various instances of interlaced and derivative sampling next we consider the issue of performance which becomes pertinent once we have extended the applicability of the method to arbitrary input functions that is when spl hscr is considerably larger than v spl phi and the reconstruction is no longer exact we show that the generalized sampling solution is essentially equivalent to the optimal minimum error approximation we then perform a detailed analysis for the case in which the analysis filters are in l sub 2 and determine all relevant bound constants explicitly finally we use an interlaced sampling example to illustrate these various calculations

implementation of jacobi rotations for accurate singular value computation in floating point arithmetic in this paper we con if begin displaymath 1 m r value decomposition svd a u sigma v tau of a a 1 a 2 in r m times 2 accurately in floating point arithmetic it is shown how the rotation angacobi rotation v the right singular vector matrix and how to compute av u sigma even if the floating point representation of v is the identity matrix in the case ns a 1 gg ns a 2 underflow can produce the identity matrix as the floating point value of v even for a 1 a 2 that are far from being mutually orthogonal this can cause loss of accuracy and failure of convergence of the floating point implementation of the jacobi method for computing the svd the modified jacobi method recommended in this paper can be implemented as a reliable and highly accurate procedure for computing the svd of general real matrices whenever the exact singular values do not exceed the underflow or overflow limits

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

perfect reconstruction versus mmse filter banks in source coding classically the filter banks fbs used in source coding schemes have been chosen to possess the perfect reconstruction pr property or to be maximally selective quadrature mirror filters qmfs this paper puts this choice back into question and solves the problem of minimizing the reconstruction distortion which in the most general case is the sum of two terms a first one due to the non pr property of the fb and the other being due to signal quantization in the subbands the resulting filter banks are called minimum mean square error mmse filter banks several quantization noise models are considered first under the classical white noise assumption the optimal positive bit rate allocation in any filter bank possibly nonorthogonal is expressed analytically and an efficient optimization method of the mmse filter banks is derived then it is shown that while in a pr fb the improvement brought by an accurate noise model over the classical white noise one is noticeable it is not the case for the mmse fb the optimization of the synthesis filters is also performed for two measures of the bit rate the classical one which is defined for uniform scalar quantization and the order one entropy measure finally the comparison of rate distortion curves where the distortion is minimized for a given bit rate budget enables us to quantify the snr improvement brought by mmse solutions

accuracy of two svd algorithms for 2 2 triangular matrices a new algorithm for the accurate computation of the singular value decomposition of 2x2 triangular matrices is proposed the algorithm is based on voevodin formulas sharp accuracy bounds are derived by using a subtle error analysis which tracks the signs of the errors of intermediate quantities and does not neglect the non linear parts of the errors the analysis is fine tuned for the case of almost diagonal matrices the same analysis is also used to analyze the errors for the xlasv2 computational routine of lapack the error estimates of the new algorithm compare favorably to those of the lapack routine

exact interpolation and iterative subdivision schemes we examine the circumstances under which a discrete time signal can be exactly interpolated given only every mth sample after pointing out the connection between designing an m fold interpolator and the construction of an m channel perfect reconstruction filter bank we derive necessary and sufficient conditions on the signal under which exact interpolation is possible bandlimited signals are one obvious example but numerous others exist we examine these and show how the interpolators may be constructed a main application is to iterative interpolation schemes used for the efficient generation of smooth curves we show that conventional bandlimited interpolators are not suitable in this context a better criterion is to use interpolators that are exact for polynomial functions we demonstrate that these interpolators converge when iterated and show how these may be designed for any polynomial degree n and any interpolation factor m this makes it possible to design interpolators for iterative schemes to make best use of the resolution available in a given display medium

rewiring filterbanks for local fourier analysis theory and practice this paper describes a series of new results outlining equivalences between certain rewirings of filterbank system block diagrams and the corresponding actions of convolution modulation and downsampling operators this gives rise to a general framework of reverse order and convolution subband structures in filterbank transforms which we show to be well suited to the analysis of filterbank coefficients arising from subsampled or multiplexed signals these results thus provide a means to understand time localized aliasing and modulation properties of such signals and their subband representations notions that are notably absent from the global viewpoint afforded by fourier analysis as well as signal recovery from sampled sequences based on their filterbank characterizations the utility of filterbank rewirings is demonstrated by the closed form analysis of signals subject to degradations such as missing data spatially or temporally multiplexed data acquisition or signal dependent noise the likes of which are often encountered in practical signal processing applications

filter design for mimo sampling and reconstruction we address the problem of finite impulse response fir filter design for uniform multiple input multiple output mimo sampling this scheme encompasses papoulis generalized sampling and several nonuniform sampling schemes as special cases the input signals are modeled as either continuous time or discrete time multiband input signals with different band structures we present conditions on the channel and the sampling rate that allow perfect inversion of the channel additionally we provide a stronger set of conditions under which the reconstruction filters can be chosen to have frequency responses that are continuous we also provide conditions for the existence of fir perfect reconstruction filters and when such do not exist we address the optimal approximation of the ideal filters using fir filters and a minmax l sub 2 end to end distortion criterion the design problem is then reduced to a standard semi infinite linear program an example design of fir reconstruction filters is given

classical sampling theorems in the context of multirate and polyphase digital filter bank structures the recovery of a signal from so called generalized samples is a problem of designing appropriate linear filters called reconstruction or synthesis filters this relationship is reviewed and explored novel theorems for the subsampling of sequences are derived by direct use of the digital filter bank framework these results are related to the theory of perfect reconstruction in maximally decimated digital filter bank systems one of the theorems pertains to the subsampling of a sequence and its first few differences and its subsequent stable reconstruction at finite cost with no error the reconstruction filters turn out to be multiplierless and of the fir finite impulse response type these ideas are extended to the case of two dimensional signals by use of a kronecker formalism the subsampling of bandlimited sequences is also considered a sequence x n with a fourier transform vanishes for mod omega mod or l pi m where l and m are integers with l

u invariant sampling extrapolation and causal interpolation from generalized samples causal processing of a signal s samples is crucial in on line applications such as audio rate conversion compression tracking and more this paper addresses the problems of predicting future samples and causally interpolating deterministic signals we treat a rich variety of sampling mechanisms encountered in practice namely in which each sampling function is obtained by applying a unitary operator on its predecessor examples include pointwise sampling at the output of an antialiasing filter and magnetic resonance imaging mri which correspond respectively to the translation and modulation operators from an abstract hilbert space viewpoint such sequences of functions were studied extensively in the context of stationary random processes we thus utilize powerful tools from this discipline although our problems are deterministic by nature in particular we provide necessary and sufficient conditions on the sampling mechanism such that perfect prediction is possible for cases where perfect prediction is impossible we derive the predictor minimizing the prediction error we also derive a causal interpolation method that best approximates the commonly used noncausal solution finally we study when causal processing of the samples of a signal can be performed in a stable manner

suppression of adjacent channel cochannel and intersymbol interference by equalizers and linear combiners we describe the ability of a linear equalizer combiner or decision feedback equalizer to suppress all received adjacent channel intersymbol and cochannel interference the emphasis is on values among transmitter bandwidth receiver bandwidth carrier spacing and antenna diversity which provide the best opportunities for interference suppression through analyses of the number of degrees of freedom and constraints in generalized zero forcing equalizers and partial comparisons to calculations of equalizer minimum mean square performance four results are obtained first with one antenna and a linear equalizer arbitrarily large receiver bandwidths allow for marginal improvements in spectral efficiency through decreased carrier spacing because the carrier spacing cannot be reduced to a value below the symbol rate without incurring unsuppressible interference second large receiver bandwidths assist multiple antennas in improving the spectral efficiency in that carrier spacing values may go below the symbol rate even in the presence of cochannel interference third the use of equalizers and linear combiners together with large receiver bandwidths allows large transmitter bandwidths to be used fourth for cochannel interference and intersymbol interference the number of interferers that may be suppressible by a generalized zero forcing linear equalizer combiner increases linearly with the product of the number of antennas and the truncated integer ratio of the total bandwidth to the symbol rate

matched sampling systems relation to wavelets and implementation using prcc filter banks this paper deals with two types of sampling systems namely the interpolation and approximation sampling systems closed form expressions are derived for the frequency responses of the filters used in these systems that are matched to the input process in the mean squared sense closed form expressions are also derived for the mean squared error between the input and the reconstructed processes for these matched sampling systems using these expressions it is shown that the meyer scaling function and wavelet or functions derived from these arise naturally in the context of subsampled bandlimited processes to implement these systems the perfect reconstruction circular convolution prcc filter bank is proposed as a framework for the frequency sampled implementation of these systems examples of matched interpolation and approximation sampling systems are provided and their performance is compared with some standard interpolators to demonstrate their efficacy

accurate computation of the product induced singular value decomposition with applications we present a new algorithm for floating point computation of the singular value decomposition svd of the product b tau c where b and c are full row rank matrices the algorithm replaces the pair b c with an equivalent pair b c and then it uses the jacobi svd algorithm to compute the svd of the explicitly computed matrix b tau c in this way each nonzero singular value sigma is approximated with some sigma delta sigma where the relative error delta sigma sigma is up to a factor of the dimensions of order boldmath varepsilon unboldmath min delta in cal d kappa 2 delta b min delta in cal d kappa 2 delta c where cal d denotes the set of diagonal nonsingular matrices kappa 2 cdot denotes the spectral condition number and boldmath varepsilon unboldmath is the roundoff unit of floating point arithmetic the new algorithm is applied to the eigenvalue problem hm x lambda x with symmetric positive definite h and m it is shown that each eigenvalue lambda is computed with high relative accuracy and that the relative error delta lambda lambda of the computed approximation lambda delta lambda is up to a factor of the dimension of order boldmath varepsilon unboldmath min delta in cal d kappa 2 delta h delta min delta in cal d kappa 2 delta m delta the new algorithm can also be used for accurate svd computation of a single matrix g that admits an accurate factorization g b tau c

sampling of deterministic signals and systems in this paper we analyze sampling approximations of stable linear time invariant systems for input signals in the paley wiener space pw 1 the goal is to approximate a transform tf by using only the samples of the signal f which do not necessarily have to be equidistant we completely characterize the stable linear time invariant lti systems t and sampling patterns for which the approximation process converges to tf for all signals in pw 1 furthermore we prove that for every complete interpolating sequence there exist a stable lti system and a signal in pw 1 for which the approximation error increases unboundedly as the number of samples that are used for the approximation is increased this shows that a more flexible choice of the sampling points instead of equidistant sampling where such a divergence behavior is known to exist is not sufficient to overcome the divergence thus sampling based signal processing has fundamental limits the calculations further indicate that oversampling cannot resolve the convergence problems

generalized average sampling in shift invariant spaces average sampling is motivated by realistic needs e g physical limitation of acquisition devices or characteristics of sampling procedure as an extension of the average sampling we study generalized average sampling in shift invariant spaces with frame generators in which averages are taken from suitable channeled version of signals illustrative examples are given

vector sampling expansions in shift invariant subspaces multi input multi output mimo sampling scheme which is motivated by applications in multi channel deconvolution and multi source separation has been investigated in many aspects common for most of results on mimo systems is that the input signals are supposed to be band limited in this paper we study vector sampling expansions on general finitely generated shift invariant subspaces necessary and sufficient conditions for a vector sampling theorem to hold are given we also give several examples to illustrate the main result

a two channel system approximation for bandlimited functions the approximation of stable linear time invariant lti systems is studied for the paley wiener space mathcal pw pi 1 of bandlimited functions with absolutely integrable fourier transform for pointwise sampling it is known that there exist stable lti systems and functions such that the approximation process diverges regardless of the oversampling factor recently it was shown that the divergence can be overcome by using more general measurement functionals that are based on a complete orthonormal system however this approach requires the approximation process to have an increased bandwidth in this paper a two channel approximation process is presented that is uniformly convergent for all stable lti systems and all functions in mathcal pw pi 1 an advantage of the two channel approach compared with the one channel approach is the reduction of the approximation bandwidth which can be exactly the same as the input function bandwidth

sampling theorems for uniform and periodic nonuniform mimo sampling of multiband signals we examine a multiple input multiple output mimo sampling scheme for a linear time invariant continuous time mimo channel the input signals are modeled as multiband signals with different spectral supports and the channel outputs are sampled on either uniform or periodic nonuniform sampling sets with possibly different but commensurate intervals on the different outputs this scheme encompasses papoulis generalized sampling and several nonuniform sampling schemes as special cases we derive necessary and sufficient conditions on the channel and the sampling rate that allow stable perfect reconstruction of the inputs or equivalently perfect inversion of the channel from an implementation viewpoint we note that it is desirable that the reconstruction filters have continuous frequency responses we derive necessary and sufficient conditions that guarantee this continuity property the frequency responses of the reconstruction filters are specified as solutions to a system of linear equations finally we demonstrate that perfect reconstruction may be possible even when the channel outputs are sampled at an average rate that does not allow the reconstruction of any output from its samples alone in certain instances this average rate can achieve the recently presented fundamental bounds on mimo sampling density

multiple input multiple output sampling necessary density conditions we consider the problem of multiple input multiple output mimo sampling of multiband signals in this problem a set of input signals is passed through a mimo channel modeled as a known linear time invariant system the inputs are modeled as multiband signals whose spectral supports are sets of finite measure and the channel outputs are sampled on nonuniform sampling sets the aim is to reconstruct the inputs from the output samples this sampling scheme is quite general and it encompasses various others including papoulis generalized sampling and nonuniform sampling as special cases we introduce notions of joint upper and lower densities for collections of sampling sets and then derive necessary conditions on these densities for stable sampling and consistent reconstruction of the channel inputs from the sampled outputs these results generalize classical density results for stable sampling and interpolation due to landau

modeling analysis and optimum design of quantized m band filter banks this paper provides a rigorous modeling and analysis of quantization effects in m band subband codecs followed by optimal filter bank design and compensation the codec is represented by a polyphase decomposition of the analysis synthesis filter banks and an embedded nonlinear gain plus additive noise model for the pdf optimized scalar quantizers we construct an equivalent time invariant but nonlinear structure operating at the slow clock rate that allows us to compute the exact expression for the mean square quantization error in the reconstructed output this error is shown to consist of two components a distortion component and a dominant random noise component uncorrelated with the input signal we determine the optimal paraunitary and biorthogonal fir filter coefficients compensators and integer bit allocation to minimize this mse subject to the constraints of filter length average bit rate and perfect reconstruction pr in the absence of quantizers the biorthogonal filter bank results in a smaller mse but the filter coefficients are very sensitive to signal statistics and to average bit constraints by comparison the paraunitary structure is much more robust we also show that the null compensated design that eliminates the distortion component is more robust than the optimally compensated case that minimizes the total mse but only at nominal conditions both modeling and optimal design are validated by simulation in the two channel case

on explicit solutions to the bezout equation explicit constructions are given for the solution of the bezout equation for matrices whose coefficients are multivariate polynomials or entire functions of paley wiener type this procedure also provides the 1 inverse when the bezout equation has no solutions

accuracy of the kogbetliantz method for scaled diagonally dominant triangular matrices the paper proves that kogbetliantz method computes all singular values of a scaled diagonally dominant triangular matrix which can be well scaled from both sides symmetrically to high relative accuracy special attention is paid to deriving sharp accuracy bounds for one step one batch and one sweep of the method by a simple numerical test it is shown that the methods based on bidiagonalization are generally not accurate on that class of well behaved matrices

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

finite sampling in multiple generated u invariant subspaces the relevance in a sampling theory of u invariant subspaces of a hilbert space mathcal h where u denotes a unitary operator on mathcal h is nowadays a recognized fact indeed shift invariant subspaces of l 2 mathbb r become a particular example periodic extensions of finite signals also provide a remarkable example as a consequence the availability of an abstract u sampling theory becomes a useful tool to handle these problems in this paper we derive a sampling theory for finite dimensional multiple generated u invariant subspaces of a hilbert space mathcal h as the involved samples are identified as frame coefficients in a suitable euclidean space the relevant mathematical technique is that of the finite frame theory since finite frames are nothing but spanning sets of vectors the used technique naturally meets matrix analysis

principal component filter banks for optimal multiresolution analysis an important issue in multiresolution analysis is that of optimal basis selection an optimal p band perfect reconstruction filter bank prfb is derived in this paper which minimizes the approximation error in the mean square sense between the original signal and its low resolution version the resulting prfb decomposes the input signal into uncorrelated low resolution principal components with decreasing variance optimality issues are further analyzed in the special case of stationary and cyclostationary processes by exploiting the connection between discrete time filter banks and continuous wavelets an optimal multiresolution decomposition of l sub 2 r is obtained analogous results are also derived for deterministic signals some illustrative examples and simulations are presented

higher order sampling and recovering of lowpass signals a higher order sampling function applied to lowpass signals is described to recover the original signal from its periodic nonuniform samples a formulation of the sampling function is introduced based on the occurrence of local gaps in the sampled signal spectrum as a result the interpolant is greatly simplified

sampling theorem for wavelet subspaces error estimate and irregular sampling the error estimate is useful in the application of the sampling theorem for the classical shannon sampling theorem various errors are widely studied but for the sampling theorem in general wavelet subspaces only the aliasing error is studied in this paper we study three other errors truncation error amplitude error and time jitter error with the same technique a result on irregular sampling is improved

irregular sampling for spline wavelet subspaces spline wavelets spl psi sub m t are important in time frequency localization due to i spl psi sub m can be arbitrarily close to the optimal case as m is sufficiently large ii spl psi sub m has compact support and simple analytic expression which lead to effective computation although the spline wavelet subspaces are so simple walter s well known sampling theorem does not hold if the order of spline m is even moreover when irregular sampling is considered in these spaces it is hard to determine the sampling density which is a serious problem in applications in this correspondence a general sampling theorem is obtained for m spl ges 3 in the sense of iterative construction and the sampling density spl delta sub m is estimated

asymmetric multi channel sampling in shift invariant spaces abstract we develop an asymmetric multi channel sampling on a shift invariant space v with a riesz generator t in l 2 r where each channeled signal is assigned a uniform but distinct sampling rate we use fourier duality between v and l 2 0 2 to find conditions under which there is a stable asymmetric multi channel sampling formula on v

recovery of missing samples for oversampling in shift invariant spaces abstract it is well known that any finite missing samples of a band limited signal can be recovered from the remaining known samples when the signal is oversampled at a rate higher than the minimum nyquist rate we consider a similar problem for oversampled signals in shift invariant spaces with compactly supported continuous riesz generators using a multi channel oversampling we find conditions under which finitely many missing samples can be recovered

oversampling pcm techniques and optimum noise shapers for quantizing a class of nonbandlimited signals we consider the efficient quantization of a class of nonbandlimited signals namely the class of discrete time signals that can be recovered from their decimated version the signals are modeled as the output of a single fir interpolation filter single band model or more generally as the sum of the outputs of l fir interpolation filters multiband model these nonbandlimited signals are oversampled and it is therefore reasonable to expect that we can reap the same benefits of well known efficient a d techniques that apply only to bandlimited signals we first show that we can obtain a great reduction in the quantization noise variance due to the oversampled nature of the signals we can achieve a substantial decrease in bit rate by appropriately decimating the signals and then quantizing them to further increase the effective quantizer resolution noise shaping is introduced by optimizing prefilters and postfilters around the quantizer we start with a scalar time invariant quantizer and study two important cases of linear time invariant lti filters namely the case where the postfilter is the inverse of the prefilter and the more general case where the postfilter is independent from the prefilter closed form expressions for the optimum filters and average minimum mean square error are derived in each case for both the single band and multiband models the class of noise shaping filters and quantizers is then enlarged to include linear periodically time varying lptv sub m filters and periodically time varying quantizers of period m we study two special cases in great detail

generalized sampling in shift invariant spaces with multiple stable generators abstract the aim of this paper is to derive stable generalized sampling in a shift invariant space with l stable generators this is done in the light of the theory of frames in the product hilbert space l l 2 0 1 l 2 0 1 l 2 0 1 l times the generalized samples are expressed as the frame coefficients of an appropriate function in l l 2 0 1 with respect to some particular frame in l l 2 0 1 since any multiply stable generated shift invariant space is the image of l l 2 0 1 by means of a bounded invertible operator the generalized sampling is obtained from some dual frame expansions in l l 2 0 1 an example in the setting of the hermite cubic splines is exhibited

sampling theorem and irregular sampling theorem for multiwavelet subspaces we generalize the multiwavelet sampling theorem by reproducing a kernel that is easy to use multiscaling functions that have orthogonality regularity short compact support symmetry and high approximation order satisfy the conditions in the sampling theorem which is not possible in the case of the scalar wavelet we consider the general cases of the uniform noninteger and irregular sampling a reconstruction from more general sets of points is necessary since the measurements may not be made at uniform points finally we establish a general irregular sampling theorem for multiwavelet subspaces and derive an estimate for the perturbations of uniform noninteger sampling in shift invariant spaces

exact image deconvolution from multiple fir blurs we address the problem of restoring an image from its noisy convolutions with two or more blur functions channels deconvolution from multiple blurs is in general better conditioned than from a single blur and can be performed without regularization for moderate noise levels we characterize the problem of missing data at the image boundaries and show that perfect reconstruction is impossible even in the no noise case almost surely unless there are at least three channels conversely when there are at least three channels we show that perfect reconstruction is not only possible almost surely in the absence of noise but also that it can be accomplished by finite impulse response fir filtering such fir reconstruction is vastly more efficient computationally than the least squares solution and is suitable for low noise levels even in the high noise case the estimates obtained by fir filtering provide useful starting points for iterative least squares algorithms we present results on the minimum possible sizes of such deconvolver filters we derive expressions for the mean square errors in the fir reconstructions and show that performance comparable to that of the least squares reconstruction may be obtained with relatively small deconvolver filters finally we demonstrate the fir reconstruction on synthetic and real data

minimum mean squared error reconstruction for generalized undersampling of cyclostationary processes the generalized sampling problem is considered using a filter bank model with a stochastic process framework a signal reconstruction solution minimizing the time averaged mean squared error is found using a filter bank optimization technique for cyclostationary signals the use of a stochastic model approach to find a minimized error solution rather than a perfect reconstruction solution allows consideration of sampling problems that deterministic approaches cannot solve for instance a particular sampling density is not required allowing reconstruction optimization in cases of undersampling an extension is also presented for optimal reconstruction in the presence of additive noise and interference the result is a generalized sampling model which can be used for an extensive variety of scenarios combining aspects of undersampling multiple input multiple output mimo sampling and periodic nonuniform sampling

the structure of finitely generated shift invariant spaces in l2 rd a simple characterization is given of finitely generated subspaces of l2 rd which are invariant under translation by any multi integer and is used to give conditions under which such a space has a particularly nice generating set namely a basis and more than that a basis with desirable properties such as stability orthogonality or linear independence the last property makes sense only for local spaces i e shift invariant spaces generated by finitely many compactly supported functions and special attention is paid to such spaces as an application we prove that the approximation order provided by a given local space is already provided by the shift invariant space generated by just one function with this function constructible as a finite linear combination of the finite generating set for the whole space hence compactly supported this settles a question of some 20 years standing

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

a generalization of nonuniform bandpass sampling nth order nonuniform sampling is described for generalized bandpass signal frequency position bandwidth sampling rate frequency shift and phase shift a bandpass extension to the nyquist criterion is derived showing that restrictions on bandpass frequency position for odd orders of nonuniform sampling tend to zero as n tends to infinity bandpass interpolants based on the sinc function are derived for the generalized nth order sampled bandpass signals it is shown that for minimum nyquist rate sampling these interpolants are comprised of n bandpass filters each with independent phase the number of bandpass filters comprising the interpolant is found to decrease as the sample rate increases the advantage of describing nth order sampling as the nth replication and uniform sampling of a signal is demonstrated finally digital implementation of the nth order bandpass sampling interpolants is discussed it is established that it is not practicable to attempt to perform nonuniform bandpass sampling at the theoretical minimum rate where the interpolation is to be performed digitally

optimization of filter banks using cyclostationary spectral analysis this paper proposes a design method of optimal biorthogonal fir filter banks that minimize the time averaged mean squared error tamse when the high frequency subband signal is dropped to study filter banks from a statistical point of view cyclostationary spectral analysis is used since the output of the filter bank for a wide sense stationary input is cyclostationary first the cyclic spectral density of the output signal is derived and an expression for the tamse is presented then optimal filter banks are given by minimizing the tamse with respect to the coefficients of the filters under the biorthogonality condition by imposing the additional constraints on the coefficients the optimal biorthogonal linear phase filter bank can be obtained

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

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

sampling signals with finite rate of innovation the authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation examples of signals with a finite rate of innovation include streams of diracs e g the poisson process nonuniform splines and piecewise polynomials even though these signals are not bandlimited we show that they can be sampled uniformly at or above the rate of innovation using an appropriate kernel and then be perfectly reconstructed thus we prove sampling theorems for classes of signals and kernels that generalize the classic bandlimited and sinc kernel case in particular we show how to sample and reconstruct periodic and finite length streams of diracs nonuniform splines and piecewise polynomials using sinc and gaussian kernels for infinite length signals with finite local rate of innovation we show local sampling and reconstruction based on spline kernels the key in all constructions is to identify the innovative part of a signal e g time instants and weights of diracs using an annihilating or locator filter a device well known in spectral analysis and error correction coding this leads to standard computational procedures for solving the sampling problem which we show through experimental results applications of these new sampling results can be found in signal processing communications systems and biological systems

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

a tangent algorithm for computing the generalized singular value decomposition we present two new algorithms for floating point computation of the generalized singular values of a real pair a b of full column rank matrices and for floating point solution of the generalized eigenvalue problem hx lambda mx with symmetric positive definite matrices h and m the pair a b is replaced with an equivalent pair a b and the generalized singular values are computed as the singular values of the explicitly computed matrix f a b 1 the singular values of f are computed using the jacobi method the relative accuracy of the computed singular value approximations does not depend on column scalings of a and b that is the accuracy is nearly the same for all pairs ad 1 bd 2 with d 1 d 2 arbitrary diagonal nonsingular matrices similarly the pencil h lambda m is replaced with an equivalent pencil h lambda m and the eigenvalues of h lambda m are computed as the squares of the singular values of g l h l m 1 where l h l m are the cholesky factors of h m respectively and the matrix g is explicitly computed as the solution of a linear system of equations for the computed approximation lambda delta lambda of any exact eigenvalue lambda the relative error delta lambda lambda is of order p n mbox boldmath varepsilon max min delta in cal d kappa 2 delta h delta min delta in cal d kappa 2 delta m delta where p n is a modestly growing polynomial of the dimension of the problem mbox boldmath varepsilon is the round off unit of floating point arithmetic cal d denotes the set of diagonal nonsingular matrices and kappa 2 cdot is the spectral condition number furthermore floating point computation corresponds to an exact computation with h delta h m delta m where for all i j delta h ij sqrt h ii h jj and delta m ij sqrt m ii m jj are of order of mbox boldmath varepsilon times a modest function of n

reconstruction of nonuniformly sampled bandlimited signals by means of digital fractional delay filters this paper considers the problem of reconstructing a class of nonuniformly sampled bandlimited signals of which a special case occurs in e g time interleaved analog to digital converter adc systems due to time skew errors to this end we propose a synthesis system composed or digital fractional delay filters the overall system i e nonuniform sampling and the proposed synthesis system can be viewed as a generalization of time interleaved adc systems to which the former reduces as a special case compared with existing reconstruction techniques our method has major advantages from an implementation point of view to be precise 1 we can perform the reconstruction as well as desired in a certain sense by properly designing the digital fractional delay filters and 2 if properly implemented the fractional delay filters need not be redesigned in case the time skews are changed the price to pay for these attractive features is that we need to use a slight oversampling it should be stressed however that the oversampling factor is less than two as compared with the nyquist rate the paper includes error and quantization noise analysis the former is useful in the analysis of the quantization noise and when designing practical fractional delay filters approximating the ideal filters

new quadrature mirror filter structures this paper introduces new quadrature mirror filter qmf structures for the frequency domain analysis and synthesis of digital signals the conventional qmf technique is first extended to cover complex quadrature mirror filters cqmf in which a digital signal is split into n adjacent complex subbands where the real and imaginary parts are subsampled by 1 2n with respect to the original signal the computational complexity of qmf banks is then analyzed and a new scheme which reduces the computational complexity by about a factor of two over conventional qmf implementations is proposed finally the filter design tradeoffs are discussed and the microprogramed implementation of qmf banks is evaluated

filterbank reconstruction of bandlimited signals from nonuniform and generalized samples this paper introduces a filterbank interpretation of various sampling strategies which leads to efficient interpolation and reconstruction methods an identity which is referred to as the interpolation identity is developed and is used to obtain particularly efficient discrete time systems for interpolation of generalized samples as well as a class of nonuniform samples to uniform nyquist samples either for further processing in that form or for conversion to continuous time the interpolation identity also leads to new sampling strategies including an extension of papoulis 1977 generalized sampling expansion

theory of optimal orthonormal subband coders the theory of the orthogonal transform coder and methods for its optimal design have been known for a long time we derive a set of necessary and sufficient conditions for the coding gain optimality of an orthonormal subband coder for given input statistics we also show how these conditions can be satisfied by the construction of a sequence of optimal compaction filters one at a time several theoretical properties of optimal compaction filters and optimal subband coders are then derived especially pertaining to behavior as the number of subbands increases significant theoretical differences between optimum subband coders transform coders and predictive coders are summarized finally conditions are presented under which optimal orthonormal subband coders yield as much coding gain as biorthogonal ones for a fixed number of subbands

reconstruction algorithms in irregular sampling a constructive solution of the irregular sampling problem for band limited functions is given we show how a band limited function can be com pletely reconstructed from any random sampling set whose density is higher than the nyquist rate and give precise estimates for the speed of convergence of this iteration method variations of this algorithm allow for irregular sampling with derivatives reconstruction of band limited functions from local averages and irregular sampling of multivariate band limited functions in the irregular sampling problem one is asked whether and how a band limited function can be completely reconstructed from its irregularly sam pled values f xi this has many applications in signal and image processing seismology meteorology medical imaging etc finding constructive solutions of this problem has received considerable attention among mathematicians and engineers the mathematical literature provides several uniqueness results 1 2 17 18 19 it is now part of the folklore that for stable sampling the sampling rate must be at least the nyquist rate 18 these results as deep as they are have had little impact for the applied sciences because they were not constructive if the sampling set is just a perturbation of the regular oversampling then a reconstruction method has been obtained in a seminal paper by duffin and schaeffer 6 see also 29 if for some l 0 a 0 and o 0 the sampling points xk k e z satisfy a xk ok a k i then the norm equivalence a ir f x 2dx with w n o this norm equivalence implies that it is possible to reconstruct through an iterative procedure the so called frame method most of the later work on constructive methods consists of variations of this method 3 21 22 26 the above conditions on the sampling set exclude random irregular sampling sets e g sets with regions of higher sampling density a partial but undesirable remedy to handle highly irregular sampling sets would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration

approximate eigenvectors as preconditioner given approximate eigenvector matrix u of a hermitian nonsingular matrix h the spectral decomposition of h can be obtained by computing h u hu and then diagonalizing h this work addresses the issue of numerical stability of the transition from h to h in finite precision arithmetic our analysis shows that the eigenvalues will be computed with small relative error if i the approximate eigenvectors are sufficiently orthonormal and ii the matrix h h 2 is of the form dad with diagonal d and well conditioned a in that case h can be efficiently and accurately diagonalized by the jacobi method if u is computed by fast eigensolver based on tridiagonalization this procedure usually gives the eigensolution with high relative accuracy and it is more efficient than accurate jacobi type methods on their own

implicit standard jacobi gives high relative accuracy we prove that the jacobi algorithm applied implicitly on a decomposition a xdx t of the symmetric matrix a where d is diagonal and x is well conditioned computes all eigenvalues of a to high relative accuracy the relative error in every eigenvalue is bounded by o epsilon kappa x where epsilon is the machine precision and kappa x equiv x 2 cdot x 1 2 is the spectral condition number of x the eigenvectors are also computed accurately in the appropriate sense we believe that this is the first algorithm to compute accurate eigenvalues of symmetric indefinite matrices that respects and preserves the symmetry of the problem and uses only orthogonal transformations

oversampling and reconstruction functions with compact support assume that a sequence of samples of a filtered version of a function in a shift invariant space is given this paper deals with the existence of a sampling formula involving these samples and having reconstruction functions with compact support this is done in the light of the generalized sampling theory by using the oversampling technique a necessary and sufficient condition is given in terms of the smith canonical form of a polynomial matrix finally we prove that the aforesaid oversampled formulas provide nice approximation schemes with respect to the uniform norm

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

kronecker products unitary matrices and signal processing applications discrete unitary transforms are extensively used in many signal processing applications and in the development of fast algorithms kronecker products have proved quite useful in this semitutorial paper we briefly review properties of kronecker products and direct sums of matrices which provide a compact notation in treating patterned matrices a generalized matrix product which inherits some useful algebraic properties from the standard kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula is then introduced the notation is intimately related to sparse matrix factorizations and examples are included illustrating the utility of the new notation in signal processing applications finally some novel characteristics of hadamard transforms and polyadic permutations are derived in the framework of kronecker products

minimum rate sampling and reconstruction of signals with arbitrary frequency support we examine the question of reconstruction of signals from periodic nonuniform samples this involves discarding samples from a uniformly sampled signal in some periodic fashion we give a characterization of the signals that can be reconstructed at exactly the minimum rate once a nonuniform sampling pattern has been fixed we give an implicit characterization of the reconstruction system and a design method by which the ideal reconstruction filters may be approximated we demonstrate that for certain spectral supports the minimum rate can be approached or achieved using reconstruction schemes of much lower complexity than those arrived at by using spectral slicing as in earlier work previous work on multiband signals have typically been those for which restrictive assumptions on the sizes and positions of the bands have been made or where the minimum rate was approached asymptotically we show that the class of multiband signals which can be reconstructed exactly is shown to be far larger than previously considered when approaching the minimum rate this freedom allows us in certain cases to have a far less complex reconstruction system

on sampling theorem wavelets and wavelet transforms the classical shannon sampling theorem has resulted in many applications and generalizations from a multiresolution point of view it provides the sine scaling function in this case for a band limited signal its wavelet series transform wst coefficients below a certain resolution level can be exactly obtained from the samples with a sampling rate higher than the nyquist rate the authors study the properties of cardinal orthogonal scaling functions cosf which provide the standard sampling theorem in multiresolution spaces with scaling functions as interpolants they show that cosf with compact support have and only have one possibility which is the haar pulse they present a family of cosf with exponential decay which are generalizations of the haar function with these cosf an application is the computation of wst coefficients of a signal by the mallat 1989 algorithm they present some numerical comparisons for different scaling functions to illustrate the advantage of cosf for signals which are not in multiresolution spaces they estimate the aliasing error in the sampling theorem by using uniform samples

generalized sampling theorems in multiresolution subspaces it is well known that under very mild conditions on the scaling function multiresolution subspaces are reproducing kernel hilbert spaces rkhss this allows for the development of a sampling theory in this paper we extend the existing sampling theory for wavelet subspaces in several directions we consider periodically nonuniform sampling sampling of a function and its derivatives oversampling multiband sampling and reconstruction from local averages all these problems are treated in a unified way using the perfect reconstruction pr filter bank theory we give conditions for stable reconstructions in each of these cases sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported except in trivial cases this restriction no longer applies for the generalizations we study here due to the existence of fir pr banks in fact with nonuniform sampling oversampling and reconstruction from local averages we can guarantee compactly supported synthesizing functions moreover local averaging schemes have additional nice properties robustness to the input noise and compression capabilities we also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis after this we extend the sampling theory to random processes we require autocorrelation functions to belong to some subspace related to wavelet subspaces it turns out that we cannot recover random processes themselves unless they are bandlimited but only their power spectral density functions we consider both uniform and nonuniform sampling

u invariant sampling and reconstruction in atomic spaces with multiple generators given a u invariant sampling scheme on a hilbert space h this paper characterizes atomic subspaces a of h such that every signal x a can be reconstructed from its generalized samples acquired with the u invariant sampling scheme if signal recovery is possible a linear filter is derived which reconstructs the signal from the samples moreover the paper gives necessary and sufficient conditions on a sequence of sampling functions such that it forms a pseudoframe or pseudo riesz basis for a given atomic subspace a

a hierarchically blocked jacobi svd algorithm for single and multiple graphics processing units we present a hierarchically blocked one sided jacobi algorithm for the singular value decomposition svd targeting both single and multiple graphics processing units gpus the blocking structure reflects the levels of the gpus memory hierarchy the algorithm may outperform magma s textttdgesvd while retaining high relative accuracy to this end we developed a family of parallel pivot strategies on the gpu s shared address space but applicable also to inter gpu communication unlike common hybrid approaches our algorithm in a single gpu setting needs a cpu for the controlling purposes only while utilizing the gpu s resources to the fullest extent permitted by the hardware when required by the problem size the algorithm in principle scales to an arbitrary number of gpu nodes the scalability is demonstrated by more than twofold speedup for sufficiently large matrices on a tesla s2050 system with four gpus versus a single fermi card

quadrature mirror filter design in the time domain a new technique for designing quadrature mirror filters is described the formulation carried out in the time domain is shown to result in an optimization problem requiring minimization of a quartic multinomial an iterative solution is suggested which involves computation of the eigenvector of a matrix with a dimensionality equal to one half the number of filter taps our experiments show that convergence to the optimum tap weights is stable and the accuracy of the final solution is limited only by the accuracy of the eigenvalue eigenvector routine as in an earlier technique the user can specify the stop band frequency the relative weights of the passband ripple energy and the stopband residual energy and of course the number of filter taps

quadrature mirror filter design for an arbitrary number of equal bandwidth channels the two channel quadrature mirror filter structure of croisier and esteban may be extended to an arbitrary number of equal bandwidth channels given certain restrictions on the bandpass filters the most serious restriction is that the stopband attenuation of eacli band pass filter must be high for all frequencies outside twice the nominal 3 db bandwidth of the filter this restriction is not really a limiting factor for speech subband waveform coding since high adjacent channel attenuation is a necessity for the confinement of quantization noise a key feature of our filter structure is that the number of multiplies adds and stored coefficients required for implementation is significantly less than those needed for the conventional qmf structure given the same number of channels fortran code for a 16 channel filter structure is listed as an example of efficient implementation

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

harmonic maps and ideal fluid flows using harmonic maps we provide an approach towards obtaining explicit solutions to the incompressible two dimensional euler equations more precisely the problem of finding all solutions which in lagrangian variables describing the particle paths of the flow present a labelling by harmonic functions is reduced to solving an explicit nonlinear differential system in mathbb c n with n 3 or n 4 while the general solution is not available in explicit form structural properties of the system permit us to identify several families of explicit solutions

subband decomposition an lms based algorithm to approximate the perfect reconstruction bank in the general case an algorithm based on least mean squares lms is described given an arbitrary invertible decomposition decimation process the algorithm will find the finite impulse response reconstruction filters which best approximate the perfect reconstruction ones by allowing the reconstruction filters impulse responses to be sufficiently long the quality of the approximation can be made as good as required two examples are presented for the implementation of this algorithm one in the case of a decomposition by a filter bank of galand 1977 where the reconstruction bank is already known the other in the situation of a two subband decomposition where one of the subbands covers two thirds of the frequency space and the other covers the remaining one third

riesz bases in l2 0 1 related to sampling in shift invariant spaces abstract the fourier duality is an elegant technique to obtain sampling formulas in paley wiener spaces in this paper it is proved that there exists an analogue of the fourier duality technique in the setting of shift invariant spaces in fact any shift invariant space v with a stable generator is the range space of a bounded one to one linear operator t between l 2 0 1 and l 2 r thus regular and irregular sampling formulas in v are obtained by transforming via t expansions in l 2 0 1 with respect to some appropriate riesz bases