Approximating Sampled Sinusoids and Multiband Signals Using Multiband Modulated DPSS Dictionaries

Many signal processing problems--such as analysis, compression, denoising, and reconstruction--can be facilitated by expressing the signal as a linear combination of atoms from a well-chosen dictionary. In this paper, we study possible dictionaries for representing the discrete vector one obtains when collecting a finite set of uniform samples from a multiband analog signal. By analyzing the spectrum of combined time- and multiband-limiting operations in the discrete-time domain, we conclude that the information level of the sampled multiband vectors is essentially equal to the time-frequency area. For representing these vectors, we consider a dictionary formed by concatenating a collection of modulated Discrete Prolate Spheroidal Sequences (DPSS's). We study the angle between the subspaces spanned by this dictionary and an optimal dictionary, and we conclude that the multiband modulated DPSS dictionary--which is simple to construct and more flexible than the optimal dictionary in practical applications--is nearly optimal for representing multiband sample vectors. We also show that the multiband modulated DPSS dictionary not only provides a very high degree of approximation accuracy in an MSE sense for multiband sample vectors (using a number of atoms comparable to the information level), but also that it can provide high-quality approximations of all sampled sinusoids within the bands of interest.