HOW TO TAKE ADVANTAGE OF ALIASING IN BANDLIMITED SIGNALS Patrick Vandewalle1 ∗, Luciano Sbaiz1, Joos Vandewalle2 † and Martin Vetterli1,3 1LCAV - School of Computer and Communication Sciences Ecole Polytechnique F´ed´erale de Lausanne (EPFL) - CH-1015 Lausanne, Switzerland 2SCD - Department of Electrical Engineering (ESAT) Katholieke Universiteit Leuven - Kasteelpark Arenberg 10, B-3001 Heverlee, Belgium 3Department of Electrical Engineering and Computer Science UC Berkeley - Berkeley CA94720 ABSTRACT pled signal is aliased, and perfect reconstruction is not pos- sible. In signal processing systems, aliasing is normally treated Vetterli et al. [1] showed that perfect reconstruction is as a disturbing signal. That motivates the need for effec- also possible for signals with finite rate of innovation. Cer- tive analog, optical and digital anti-aliasing filters. How- tain types of non-bandlimited signals (like streams of Diracs, ever, aliasing conveys also valuable information on the sig- piecewise polynomials, etc.) can be reconstructed from a fi- nal above the Nyquist frequency. Hence, an effective pro- nite set of samples. Vaidyanathan [2] considers some other cessing of the samples, based on a model of the input signal, sampling schemes for non-bandlimited signals, like the use would allow to virtually increase the sampling frequency of different sampling kernels. An overview of the current using slower and cheaper converters. In this paper, we pre- state in sampling is given by Unser [3]. sent such an algorithm for bandlimited signals that are sam- In this paper, we derive some results for bandlimited sig- pled below twice the maximum signal frequency. Using a nals that are sampled below twice the maximum frequency subspace method in the frequency domain, we show that of the signal. We will use multiple sets of regular samples these signals can be reconstructed from multiple sets of sam- to reconstruct the original signal exactly. A similar problem ples. The offset between the sets is unknown and can have for discrete-time signals is solved by Marziliano et al. [4] arbitrary values. This approach can be applied to the cre- using combinatorial optimization methods. ation of super-resolution images from sets of low resolution images. In this application, registration parameters have to 2. PROBLEM STATEMENT be computed from aliased images. We show that parameters and high resolution images can be computed precisely, even We define x(t) as a periodic, bandlimited and continuous- when high levels of aliasing are present on the low resolu- time signal with period 1 and maximum frequency L (since tion images. we consider a Fourier series L is integer). Its continuous time Fourier series is called X[l] (with −L ≤ l ≤ L). x(t) K K 1. INTRODUCTION is regularly sampled at frequency , with integer- valued. This results in a discrete-time signal y0[n]=x[n/k]. Y [k] In general, we say that a signal can be perfectly reconstructed Its discrete-time Fourier transform 0 can be represented X[k] from its samples if it is bandlimited and the sampling fre- as a function of the continuous-time Fourier coefficients quency satisfies the Nyquist criterion, i.e. it is larger than as twice the maximum signal frequency. If the signal is not ∞ K K − 1 bandlimited or the sampling frequency is too low, the sam- Y [k]= X[k + iK] − ≤ k ≤ . 0 with 2 2 i=−∞ ∗The work presented in this paper was supported by the National Com- petence Center in Research on Mobile Information and Communication (1) Systems (NCCR-MICS, http://www.mics.org), a center supported by the Y0[k] is a periodic signal with period K, so it is sufficient if Swiss National Science Foundation under grant number 5005-67322. † we only look at one period of this signal. Research supported by Research Council KUL: GOA-Mefi sto 666, If we want to reconstruct x(t) from its samples, we need Flemish Government- FWO: G.0407.02 (support vector machines), X[k] G.0197.02 (power islands), research communities (ICCoS, ANMMM); to know its Fourier coefficients . If the sampling fre- Belgian Federal Government: DWTC IUAP V-22. quency satisfies the Nyquist criterion (K>2L), X[k+iK] ,(((,,, ,&$663 is only different from zero for i =0and with x the largest integer value smaller than x.Thisal- lows us to compute S as K K − 1 Y0[k]=X[k] with − ≤ k ≤ . (2) L + K − 2 L − 1 2 2 S = + +1. K K (6) However, if the Nyquist criterion is not satisfied (K ≤ 2L), the copies of the continuous spectrum overlap in the sam- We can now make the important observation that the −1 pled spectrum. In equation (1) X[k + iK] is nonzero for modified sets of samples Dm Ym are all linear combina- i (i+1)K−1 multiple values of and therefore the Fourier coefficients tions of a set of S vectors XiK . Intuitively, it can X[k] of the continuous signal cannot be derived immedi- already be seen that by taking the number of sampling sets ately. M large enough, we will have enough equations to compute Now we take a second (regular) set of samples y1[n] at the Fourier coefficients X[k] as well as the unknown offsets the same sampling rate K, with an unknown offset t1 from tm. n the first set y0[n]: y1[n]=x K + t1 with t1 ∈ [0, 1). Y [k] Again, we can write its Fourier coefficients 1 as a func- 3. SOLUTION USING SUBSPACES tion of X[k]: ∞ 3.1. Offset estimation Y [k]= W k+iK X[k + iK] 1 t1 i=−∞ As mentioned before, the modified sets of Fourier coeffi- ∞ (3) cients of the samples are linear combinations of S vectors (i+1 ) K−1 = W k W iK X[k + iK], X . Therefore, they belong to an S-dimensional sub- t1 t1 iK i=−∞ space of the K-dimensional space and the subspace matrix j2πα −1 −1 with Wα = e and −K/2≤k ≤(K − 1)/2.This Y= Y0 D1 Y1 ··· DM−1YM−1 K X[k] ⎡ ⎤ gives us new equations in , but it adds also a new Y0[0] Y1[0] ··· YM−1[0] unknown t1. ⎢ ⎥ Y0[1] Wt1 Y1[1] ··· WtM−1 YM−1[1] ⎢ 2 2 ⎥ We can reformulate equation (1) and (3) using vector ⎢ Y0[2] W Y1[2] ··· W YM−1[2] ⎥ =⎢ t1 tM−1 ⎥ notation and using the period from 0 to K − 1 instead of ⎢ ⎥ ⎣ . ⎦ −K/2 to K/2: . ⎡ ⎤ ⎡ ⎤ Y [K −1] W K−1Y [K −1]··· W K−1 Y [K−1] 0 t1 1 tM−1 M−1 Ym[0] 10 0 0 ⎢ Y [1] ⎥ ⎢ 0 W 00⎥ (7) ⎢ m ⎥ ⎢ tm ⎥ ⎢ . ⎥ = ⎢ . ⎥ has rank S (if M ≥ S). We assume here that the rank is not ⎣ . ⎦ ⎣ 00 .. 0 ⎦ K−1 lower than S, meaning that there are no two sets of samples Ym[K − 1] 00 0W ⎡ tm ⎤ with offsets ti, tj for which (ti − tj)K ∈ . This would be X[iK] a degenerate case, because two such sets contain the same ∞ ⎢ ⎥ ⎢ X[iK +1] ⎥ samples. W iK ⎢ ⎥ tm ⎣ . ⎦ If the number of sampling sets M is at least S +1,it i=−∞ . X[(i +1)K − 1] is possible to find the relative offsets as the set of parameter values for which the (S +1)-th singular value of Y equals zero. ∞ iK (i+1)K−1 Ym = Dm W X tm iK 3.2. Computation of the Fourier coefficients i=−∞ Once the relative offsets tm are known, it is very easy to X[k] ∞ compute the Fourier coefficients of the original sig- −1 iK (i+1)K−1 S D Ym = W X , nal. Each of the equations in (3) is an equation in at most m tm iK M ≥ S +1 i=−∞ unknown Fourier coefficients. Because we have S +1 (4) sets of samples, there are also such equations in the same unknowns. Therefore, the unknown Fourier coeffi- with tm the relative offset from the set y0[n] (t0 =0)and cients are the solution of a linear set of equations. m the index of the sample set (0 ≤ m<M). Because the x(t) X(i+1)K−1 =0 original signal is bandlimited, iK only for 4. INTERPRETATION a finite number S of values i: −1 L + K − 2 L − 1 The unknown matrices D by which the vectors Ym are − ≤ i ≤ m K K (5) multiplied, are rotation matrices. They do not modify the ,,, energy in the different vectors (or equivalently, the Frobe- 1, 2, ..., S)forwhich Y nius norm of the matrix ), but they rather align the diffe- A= Y∗Y rent vectors such that they are contained in the S-dimensio- ⎡ ⎤ Y∗Y Y∗D−1Y ···Y∗D−1Y nal subspace. 0 0 0 1 1 0 M M ⎢ ∗ ∗−1 ∗ ∗ ∗−1 −1 ⎥ t ⎢Y1 D1 Y0 Y1 Y1 ···Y1 D1 DM YM⎥ As described in Section 3, we are looking for the 1, =⎢ ⎥ t t S +1 . 2,..., M−1 for which the ( )-th singular value is 0. ⎣. .. ⎦ Next to the global minimum, this function has also many ∗ ∗−1 ∗ ∗−1 −1 ∗ YM DM Y0 YM DM D1 Y1···YM YM local minima. It is not possible to apply a standard mini- (8) mization algorithm (like gradient descent) to the problem, because it will get stuck in one of the local minima. A pos- is rank deficient. Instead of computing singular values, we sibility would be to compute the (S+1)-th singular value on can then also simply compute the determinant of A.This a regular grid of KS values and apply a nonlinear minimiza- will require much less computations, but it will also be nu- tion algorithm to the minimum of those values.