ON THE EXPONENTIAL REPRODUCING KERNELS FOR SAMPLING SIGNALS WITH FINITE RATE OF INNOVATION Jose Antonio Urig¨ uen , Pier Luigi Dragotti and Thierry Blu Imperial College of London The Chinese University of Hong Kong jose.uriguen08@imperial.ac.uk, p.dragotti@imperial.ac.uk, thierry.blu@m4x.org ABSTRACT The theory of Finite Rate of Innovation (FRI) broadened the traditional sampling paradigm to certain classes of parametric signals. In the pres- ence of noise, the original procedures are not as stable, and a different treatment is needed. In this paper we review the ideal FRI sampling scheme and some of the existing techniques to combat noise. We then present alternative denoising methods for the case of exponential reproducing kernels. We first vary existing subspace-based approaches. We also discuss how to design exponential reproducing kernels that are most robust to noise. Keywords— FRI, Sampling, Noise, Subspace, SVD 1. INTRODUCTION Recently, Vetterli et al. demonstrated how certain classes of non- bandlimited signals can be sampled and perfectly reconstructed using the sinc and the Gaussian kernels [1]. These signals are completely determined by a finite number of degrees of freedom and are called signals with Finite Rate of Innovation (FRI). In [2], these results were extended to the case of sampling kernels with compact support and, in particular, to exponential reproducing kernels such as E-Splines [3]. In the presence of noise, however, these approaches become unstable. In [4] and [5] improved alternatives to the original methods were pre- sented, focused on a subspace perspective for signal retrieval. This paper focuses on the use of exponential reproducing kernels in the noisy scenario. Our contribution is twofold. First, we discuss varia- tions of the algorithms considered in [5] when exponential reproducing kernels are involved. Second, we present a methodology to design ex- ponential reproducing kernels that are most robust against noise. The outline of the paper is as follows. In Section 2 we review the noiseless scenario of [2]. Then, in Section 3 we give an overview of the denoising techniques of [5]. We also introduce our modified pro- cedures. In Section 4 we connect the Sum of Sincs kernel of [6] with the exponential reproducing kernels. Finally, in Section 5 we show simulation results, to then conclude in Section 6. 2. SAMPLING SIGNALS WITH FRI For the sake of clarity we consider that xt is a stream of K Diracs with amplitudes a k K 1 k 0 located at instants of time t k K 1 k 0 0,τ xt K 1 k 0 a k δt t k . (1) Jose Antonio Urig¨ uen is sponsored by the non-profit organisation “Fun- daci´ on Caja Madrid” We assume the sampling period is T τ N . The measurements obtained sampling with ϕ t T , for n 0, 1,...,N 1, are yn  xt,ϕ t T n  K 1 k 0 a k ϕ t k T n . (2) In [1, 2] it was shown that, with a proper choice of the acquisition kernel, a perfect reconstruction of xt from the samples yn is possible. In this paper we concentrate on a specific class of kernels, used in [2], that are able to reproduce exponentials. An exponential reproducing kernel is any function ϕt that satisfies n Z cm,nϕt n e αmt with αm C, (3) for a proper choice of coefficients cm,n. The coefficients cm,n in the above equation are given by cm,n e αmt ˜ ϕt n dt, (4) where ˜ ϕt is chosen to form with ϕt a quasi-biorthonormal set [2]. Note that the coefficients cm,n are discrete-time exponentials. This can be shown by making a change of variable in equation (4), which yields: cm,n e αmx e αmn ˜ ϕx dx e αmn cm,0. (5) Exponential splines (E-Splines) [3] are central to the exponential re- production property. The Fourier transform of the P -th order E-Spline is: ˆ β  α P ω P m 0 1 e αm jω jω αm . (6) The above function is able to reproduce the exponentials e αmt , m 0, 1,...,P . Moreover, since the exponential reproduction formula is preserved through convolution [3], any composite function of the form γt β  α P t is also able to reproduce exponentials. In the reconstruction scheme of [2] the samples yn are first com- bined linearly with the coefficients cm,n to obtain the new measure- ments sm N 1 n 0 cm,nyn ˆ a k u m k (7) for m 0,...,P , and where ˆ a k a k e α 0 t k T and u k e λ t k T . Here we have used the fact that the original signal is a stream of Diracs, we have combined equations (2) and (4), and used αm α0 mλ. The values sm represent the exact exponential moments of the continuous- time signal xt .