Time-Frequency Localization and Sampling of Multiband Signals

Abstract

This paper begins with a review of some classical work of Landau, Slepian, Pollak and Widom concerning essentially time-and bandlimited signals and ends reviewing some recent work of Candès, Romberg and Tao that places specific but probabilistic limitations on essential time-and bandlimiting for finite signals and their discrete Fourier transforms. In between we outline some conceptual bridges from the continuous, singleband setting to the finite, multiband setting and pose a number of open problems whose solutions would solidify the connections outlined. These connections involve time-frequency localization of multiband signals and sampling theory for such signals.

Approximating Sampled Sinusoids and Multiband Signals Using Multiband Modulated DPSS Dictionaries

“…The first contribution of this paper is to investigate the spectrum of the matrix B N,W , which is equivalent 1 to a composed time-and multiband-limiting operator IN B W I * N defined in Section 2.2. In line with analogous results for time-frequency localization in the continuous-time domain [25,28], we extend some of the techniques from [25,28] for the discrete-time case and show that the number of dominant eigenvalues of IN B W I * N (and hence B N,W ) is essentially the time-frequency area N |W| = i 2N Wi, which also reveals the effective dimensionality of the union of curves M W . Furthermore, similar to the concentration behavior of the DPSS eigenvalues for a single frequency band, we show that the eigenvalues of the operator IN B W I * N have a distinctive behavior: the first ≈ N |W| eigenvalues tend to cluster near 1, while the remaining eigenvalues tend to cluster near 0 after a narrow transition, which has width proportional to the number of bands times log(N ).…”

“…Note that results similar to the above two theorems for time-frequency localization in the continuous domain have been established in [22,25,28]. Similar to the ideas used in [22], the key to proving Constructing an appropriate subspace with a carefully selected bandlimited sequence for the Weyl-Courant minimax characterization of eigenvalues is the key to proving Theorem 3.3.…”

“…Similar to the ideas used in [22], the key to proving Constructing an appropriate subspace with a carefully selected bandlimited sequence for the Weyl-Courant minimax characterization of eigenvalues is the key to proving Theorem 3.3. The proof techniques of [25,28] form the basis of our analysis in Appendix C, but some modifications are required to extend their results to the discrete domain.…”

“…and hence ι− = #L−. Let Sι − be the ι−-dimensional subspace of 2(Z) spanned by the functions E n N h, n ∈ L−, that is, Now using the same procedure as in (25), one has…”

On the Capacity Limit of Wireless Channels Under Colored Scattering

“…Also, the extension of Theorem 2 was initially pursued in [24] and the asymptotic distribution of the eigenvalues was derived. The composition of PSWFs for the multiple interval case can be found in [26], [27].…”