Signals comprised of a stream of short pulses appear in many applications including bioimaging and radar. The recent finite rate of innovation framework, has paved the way to low rate sampling of such pulses by noticing that only a small number of parameters per unit time are needed to fully describe these signals. Unfortunately, for high rates of innovation, existing sampling schemes are numerically unstable. In this paper we propose a general sampling approach which leads to stable recovery even in the presence of many pulses. We begin by deriving a condition on the sampling kernel which allows perfect reconstruction of periodic streams from the minimal number of samples. We then design a compactly supported class of filters, satisfying this condition. The periodic solution is extended to finite and infinite streams and is shown to be numerically stable even for a large number of pulses. High noise robustness is also demonstrated when the delays are sufficiently separated. Finally, we process ultrasound imaging data using our techniques and show that substantial rate reduction with respect to traditional ultrasound sampling schemes can be achieved.
Another general family introduced in @cite_19 for the finite model is the class of exponential reproducing kernels. As a specific case, the authors propose E-spline sampling kernels. The CTFT of an E-spline of order @math is described by where @math are free parameters. In order to use E-splines as sampling kernels for pulse streams, the authors propose a specific structure on the @math 's, @math . Choosing exponents having a non-vanishing real part results in unstable weighting, as in the B-spline case. However, choosing the special case of pure imaginary exponents in the E-splines, already suggested by the authors, results in a reconstruction method based on Fourier coefficients, which demonstrates an interesting relation to our method. The Fourier coefficients are obtained by applying a matrix consisting of the exponent spanning coefficients @math , (see @cite_19 ), instead of our Vandermonde matrix relation . With this specific choice of parameters the E-spline function satisfies .
@cite_19
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We consider the problem of the reconstruction of a continuous-time function f(x) spl isin 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 sub 2 ) is considerably larger than the subspace of band-limited functions. Second, we use a more general specification of the reconstruction subspace V( spl psi ), so that the output of the system can take the form of a band-limited 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 approximation f spl isin V( spl psi ) 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( spl psi ). 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 filter bank. The modulation analysis, on the other hand, is useful in providing the connection with Papoulis' earlier results for the band-limited case. Finally, we illustrate the general applicability of our theory by presenting new examples of interlaced and derivative sampling using splines.