Exponential Reproducing Kernels for Sparse Sampling Jose Antonio Urig¨ uen Imperial College of London jose.uriguen08@imperial.ac.uk Pier Luigi Dragotti Imperial College of London p.dragotti@imperial.ac.uk Thierry Blu The Chinese University of Hong Kong 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 this paper we review the ideal FRI sampling scheme and some techniques to combat noise. We then present alternative and more effective denoising methods for the case of exponential reproducing kernels. I. I NTRODUCTION In [1] and [2] it was shown how certain classes of non-bandlimited signals can be sampled and perfectly reconstructed. These signals can be completely characterised by their rate of innovation. In the presence of noise, the ideal approaches become unstable and alternative methods are required [3]. This paper focuses on the optimal use of exponential reproducing kernels introduced in [2] for the noisy scenario. II. SAMPLING SIGNALS WITH FRI Consider a stream of K Diracs at locations t k , with amplitudes a k and of duration τ seconds. If we sample the signal with an exponential reproducing kernel ϕ t T we obtain the measurements yn  xt,ϕ t T n  , for n 0, 1,...,N 1. Here N is the number of samples and we use a sampling period T τ N . An exponential reproducing kernel is any function ϕt that satis- fies n Z cm,0e αm n t ϕt n 1 with αm C for appropriate coefficients cm,n cm,0e αmn . Equivalently we can write cm,0 e αmt ϕt dt 1. (1) Furthermore, any composite function of the form ϕt γt β  α P t , where β  α P t is an E-Spline [4], is able to reproduce the set e αmt , m 0, 1,...,P . Reconstructing the input is a two step process [2]. First, the samples yn are linearly combined to get the new measurements sm N 1 n 0 cm,nyn. These are equivalent to a power series involving the locations t k and amplitudes a k for αm α0 mλ. Second, the unknown parameters can be retrieved using the classical Prony’s method. The key ingredient is the annihilating filter, for which the following holds [3]: Sh 0 (2) i.e. the Toeplitz matrix S is rank deficient. Note that we require P 2K 1. III. WORKING IN THE PRESENCE OF NOISE When the sampling process is not ideal we obtain a corrupted version of the measurements ˆ yn yn ǫn. The Toeplitz matrix of (2) then becomes ˆ S S B and is no longer rank deficient. When the noise term B is additive white Gaussian (AWGN) it is reasonable to look for a solution that minimises ˆ Sh 2 s.t. h 1 [3]. This is a classical total-least-square (TLS) problem that can be solved using singular value decomposition (SVD). The solution is further improved by denoising ˆ S using, for instance, Cadzow algorithm. Jose Antonio Urig¨ uen is sponsored by the non-profit organisation “Fun- daci´ on Caja Madrid” — Pier Luigi Dragotti is in part supported by a Global Research Award from the Royal Academy of Engineering. Modified TLS and E-Splines For exponential reproducing kernels B is due to coloured noise. In order for SVD to provide a reliable separation of the signal and noise subspaces it becomes necessary to “pre-whiten” the noise. If we know the covariance matrix of the noise R up to a constant factor λ, we can factor it: R λBB Q T Q and recover the appropriate subspaces by considering the SVD of ˆ S ˆ SQ 1 . It is also possible to control the term B by designing an appropriate sampling kernel. Consider the matrix C of size P 1 N with coefficients cm,n at locations m, n . If we want the noise to be white we need the matrix C to have orthonormal rows. This is achieved by making them orthogonal with αm jωm j 2πm N and then orthonormal by setting cm,0 1, which is achieved using (1): ˆ ϕωm ˆ γωm ˆ β  α P ωm 1, (3) where ˆ ϕ is the Fourier transform of ϕt . Among the kernels satis- fying (3), we are interested in the one with the shortest support. This kernel can be formed as a linear combination of various derivatives of the original E-Spline. It is a variation of the maximal-order minimal- support kernels of [5] and is still able to reproduce exponentials. Now, solving the problem in the Fourier domain we only need to determine a polynomial that interpolates (ωm, ˆ β  α P ωm 1 . IV. SIMULATION RESULTS Fig. 1 shows the modified E-Spline kernels (‘ME’) have the best performance, which improves with increasing order P . The modified Cadzow algorithm (‘MC’) marginally beats the original (‘C’). 0 5 10 15 20 30 10 -4 10 -3 10 -2 10 -1 SNR(dB) ∆ t 0 / τ P = 13, C P = 13, MC P = 13, ME P = 30, ME Figure 1. Retrieval of K 2 Diracs in the presence of noise. We use τ 1 seconds, N 31 samples and average over 1000 realisations. REFERENCES [1] M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Transactions on Signal Processing, vol. 50, pp. 1417–1428, 2002. [2] P. L. Dragotti, M. Vetterli, and T. Blu, “Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon Meets Strang-Fix,” IEEE Transactions on Signal Processing, vol. 55 (5), pp. 1741–1757, 2007. [3] T. Blu, P. L. Dragotti, M. Vetterli, P. Marziliano, L. Coulot, “Sparse Sampling of Signal Innovations,” IEEE Signal Processing Magazine, vol. 25 (2), pp. 31-40, 2008. [4] M. Unser and T. Blu, “Cardinal Exponential Splines: Part I — Theory and Filtering Algorithms,” IEEE Transactions on Signal Processing, vol. 53, pp. 1425-1438, 2005. [5] T. Blu, P. Thevenaz, and M. Unser, “MOMS: maximal-order interpolation of minimal support,” IEEE Transactions on Image Processing, vol. 10, pp. 1069-1080, 2001.