Extrapolation Of Bandlimited, Time-concentrated Sequences

ABSTILKT This presentation addresses the problem of extrap lating discrete-time, bandlimited signals, which have been observed over a short time-interval, and whose energy is known to be concentrated in an other finite time-interval. The approximate length and location of this concentration interval, and the band limits are both assumed to be known. The criteria used to pick the extrapolation is that it be consistent with the given data, and have the least energy outside the concentration interval. 1. INTRQDUCTION The problem of reconstructing a bandlimited signal from a finite number of samples is a central problem in many signal processing applications, including synthetic aperture radar, limited-angle tomography, image restoration, x-ray crystallography, limited-view image reconstruction, etc. We will assume that we are interested in reconstructing the bandlimited signal at integral multiples of a given sampling interval, which we normalize to 1, and that the signal's normalized bandwidth is much less than T radians. We will also assume that the signal has been observed at a finite number of possibly nonconsecutive integer multiples of the sample-spacing. So the problem is one of reconstructing a bandlimited sequence, when the normalized band limits (less than n radians) are known, and we are given a finite number of possibly non-consecutive samples. It requires not only extrapolation outside the observation window, but also interpolation between measurements. If the signal bandwidth is much less than n radians, then theoretically the sequence can be reconstructed from fewer than 1 sample per time-unit. For instance, if the sequence is lowpass, and its bandwidth is 5 radians (where m is an integer greater than I), the desired sequence can be reconstructed uniquely from only 1 observation (on the average) for every set of m samples. However, the sequence is necessarily of infinite length (since it is limited in frequency), and as a result, in general, an infinite number of observations is needed for an exact reconstruction. No finite collection of samples is sufficient to reconstruct the sequence uniquely, even if the sampling rate in the finite observation window is much higher than the Nyquist rate. m What this means is that there are intlnitely many sequences, all of which exactly match the given finite set of measurements, and are bandlimited to the given frequency band. It is obvious that we need some additional apriori information to pick the correct extrapolation, or at least to narrow the choice down. It was shown by Jain and Ranganathl that amongst the infinite set of bandlimited sequences that are consistent with the given data, most extrapolation algorithms2* 3. 4* pick the sequence that has the smallest energy. The minimum-energy extrapolation is unique, but there is no reason to believe that the underlying signal has the smallest energy amongst all the consistent extrapolations. The minimum-energy criterion is difficult to justify in a practical situation, and it seems to be an artificial constraint imposed on the extrapolation for the sole purpose of obtaining a unique solution in closed form. Instead of imposing the minimum-energy constraint on the extrapolation, this presentation deals with incorporating some additional apriori knowledge about the "effective duration" of the signal. As we have said before, the bandlimited signal cannot be time-limited as well; however, most of its energy might well be concentrated in a finite interval, which we will refer to either as the effective-duration or the concentration-interval. In many practical signal processing problems, it is known apriori that the underlying bandlimited signal is time-concentrated, and it is not unreasonable to assume that the length of the concentration-interval is known approximately. As an illustration, consider the so-called missing cone problem in limited-angle tomography. The problem here is of reconstructing a spatially-limited image from samples of its two-dimensional Fourier transform within a conical region in the frequency plane. There is no information about the signal content in the complement of this frequency cone. Since the complement is also conical, hence the term "missing cone".6 This problem is essentially that of extrapolating Fourier-domain data corresponding to a spatiallylimited image, with the additional information that the Fourier transform of an image has 95% of its energy concentrated in a small low-frequency region around the origin. Thus the missing-cone reconstruction problem is the Fourier-dual of the problem of extrapolation of bandlimited, time-concentrated signals, where the measurements of the signal are not necessarily in the high energy concentration interval.