Construction of Sampling and Interpolating Sequences for Multi-Band Signals. The Two-Band Case (original) (raw)
Sampling, interpolation and Riesz bases in small Fock spaces
We give a complete description of Riesz bases of reproducing kernels in small Fock spaces. This characterization is in the spirit of the well known Kadets-Ingham 1/4 theorem for Paley-Wiener spaces. Contrarily to the situation in Paley-Wiener spaces, a link can be established between Riesz bases in the Hilbert case and corresponding complete interpolating sequences in small Fock spaces with associated uniform norm. These results allow to show that if a sequence has a density stricly different from the critical one then either it can be completed or reduced to a complete interpolating sequence. In particular, this allows to give necessary and sufficient conditions for interpolation or sampling in terms of densities.
Sampling and reconstruction of operators
We study the recovery of operators with bandlimited Kohn-Nirenberg symbol from the action of such operators on a weighted impulse train, a procedure we refer to as operator sampling. Kailath, and later Kozek and the authors have shown that operator sampling is possible if the symbol of the operator is bandlimited to a set with area less than one. In this paper we develop explicit reconstruction formulas for operator sampling that generalize reconstruction formulas for bandlimited functions. We give necessary and sufficient conditions on the sampling rate that depend on size and geometry of the bandlimiting set. Moreover, we show that under mild geometric conditions, classes of operators bandlimited to an unknown set of area less than one-half permit sampling and reconstruction. A similar result considering unknown sets of area less than one was independently achieved by Heckel and Boelcskei. Operators with bandlimited symbols have been used to model doubly dispersive communication channels with slowly-time-varying impulse response. The results in this paper are rooted in work by Bello and Kailath in the 1960s.
Sampling and recovery of bandlimited functions and applications to signal processing
Advanced Courses of Mathematical Analysis IV - Proceedings of the Fourth International School – In Memory of Professor Antonio Aizpuru Tomás, 2011
Bandlimited functions, i.e square integrable functions on R d , d ∈ N, whose Fourier transforms have bounded support, are widely used to represent signals. One problem which arises, is to find stable recovery formulae, based on evaluations of these functions at given sample points. We start with the case of equally distributed sampling points and present a method of Daubechies and DeVore to approximate bandlimited functions by quantized data. In the case that the sampling points are not equally distributed this method will fail. We are suggesting to provide a solution to this problem in the case of scattered sample points by first approximating bandlimited functions using linear combinations of shifted Gaussians. In order to be able to do so we prove the following interpolation result. Let (x j : j ∈ Z) ⊂ R be a Rieszbasis sequence. For λ > 0 and f ∈ P W , the space of square-integrable functions on R, whose Fourier transforms vanish outside of [−1, 1], there is a unique sequence (a j) ∈ 2 (Z), so that the function I λ (f)(x) := a j e −λ x−xj 2 2 , x ∈ R is continuous, square integrable, and satisfies the interpolatory conditions I λ (f)(x k) = f (x k), for all k ∈ Z. It is shown that I λ (f) converges to f in L 2 (R d) and uniformly on R, as λ → 0 + .
Riesz bases and exact controllability of C0-groups with one-dimensional input operators
Systems & Control Letters, 2004
This paper considers linear inÿnite dimensional systems with C0-group generators and one-dimensional admissible input operators. The exact controllability and Riesz basis generation property are discussed. The corresponding results of Jacob and Zwart (Advances in Mathematical Systems Theory, Birkh auser, Boston, MA, 2000) under the assumption of algebraic simplicity for eigenvalues of the generator are generalized to the case in which the eigenvalues are allowed to be algebraically multiple but with a uniform bound on the multiplicity. (B.-Z. Guo). deÿnes a bounded linear operator from L 2 (0; t) to H for some (and hence all) t ¿ 0. The input function u is assumed to be in L 2 loc (0; ∞). Under these conditions, for any x 0 ∈ H and u ∈ L 2 loc (0; ∞), (1) admits a unique solution given by
Discrete Sampling and Interpolation: Universal Sampling Sets for Discrete Bandlimited Spaces
IEEE Transactions on Information Theory, 2000
We study the problem of interpolating all values of a discrete signal f of length N when d < N values are known, especially in the case when the Fourier transform of the signal is zero outside some prescribed index set J ; these comprise the (generalized) bandlimited spaces B J. The sampling pattern for f is specified by an index set I, and is said to be a universal sampling set if samples in the locations I can be used to interpolate signals from B J for any J. When N is a prime power we give several characterizations of universal sampling sets, some structure theorems for such sets, an algorithm for their construction, and a formula that counts them. There are also natural applications to additive uncertainty principles.
A generalized sampling theory without band-limiting constraints
IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 1998
We consider the problem of the reconstruction of a continuous-time function f (x) ∈ H from the samples of the responses of m linear shift-invariant systems sampled at 1/m the reconstruction rate. We extend Papoulis' generalized sampling theory in two important respects. First, our class of admissible input signals (typ. H = L 2 ) is considerably larger than the subspace of bandlimited functions. Second, we use a more general specification of the reconstruction subspace V (ϕ), so that the output of the system can take the form of a bandlimited function, a spline, or a wavelet expansion. Since we have enlarged the class of admissible input functions, we have to give up Shannon and Papoulis' principle of an exact reconstruction. Instead, we seek an approximationf ∈ V (ϕ) that is consistent in the sense that it produces exactly the same measurements as the input of the system. This leads to a generalization of Papoulis' sampling theorem and a practical reconstruction algorithm that takes the form of a multivariate filter. In particular, we show that the corresponding system acts as a projector from H onto V (ϕ). We then propose two complementary polyphase and modulation domain interpretations of our solution. The polyphase representation leads to a simple understanding of our reconstruction algorithm in terms of a perfect reconstruction filterbank. The modulation analysis, on the other hand, is useful in providing the connection with Papoulis' earlier results for the bandlimited case. Finally, we illustrate the general applicability of our theory by presenting new examples of interlaced and derivative sampling using splines.
Interpolation of subspaces and applications to exponential bases
2000
We give precise conditions under which the real interpolation space [Y 0 , X 1 ] θ,p coincides with a closed subspace of [X 0 , X 1 ] θ,p when Y 0 is a closed subspace of codimension one. We then apply this result to nonharmonic Fourier series in Sobolev spaces H s (−π, π) when 0 < s < 1. The main result: let E be a family of exponentials exp(iλ n t) and E forms an unconditional basis in L 2 (−π, π). Then there exist two number s 0 , s 1 such that E forms an unconditional basis in H s for s < s 0 , E forms an unconditional basis in its span with codimension 1 in H s for s 1 < s. For s 0 ≤ s ≤ s 1 the exponential family is not an unconditional basis in its span.
2010
Sampling and reconstruction of functions is a central tool in science. A key result is given by the sampling theorem for bandlimited functions attributed to Whittaker, Shannon, Nyquist, and Kotelnikov. We develop an analogous sampling theory for operators which we call bandlimited if their Kohn-Nirenberg symbols are bandlimited. We prove sampling theorems for such operators and show that they are extensions of the classical sampling theorem.
Interpolation of subspaces and applications to exponential bases in Sobolev spaces
2001
We give precise conditions under which the real interpolation space [Y_0,X_1]_s,p coincides with a closed subspace of the corresponding interpolation space [X_0,X_1]_s,p when Y_0 is a closed subspace of X_0 of codimension one. This result is applied to study the basis properties of nonharmonic Fourier series in Sobolev spaces H^s on an interval when 0<s<1. The main result: let E be a family of exponentials exp(i λ_n t) and E forms an unconditional basis in L^2 on an interval. Then there exist two number s_0, s_1 such that E forms an unconditional basis in H^s for s<s_0, E forms an unconditional basis in its span with codimension 1 in H^s for s_1<s. For s in [s_0,s_1] the exponential family is not an unconditional basis in its span.