We consider three different versions of the Zak transform (ZT) for discrete-time signals, namely, the discretetime ZT, the polyphase transform, and a cyclic discrete ZT. In particular, we show that the extension of the discrete-time ZT to the complex z-plane results in the polyphase transform, an important and well-known concept in multirate signal processing and filter bank theory. We discuss ...

Weyl-Heisenberg frames are complete signal representations corresponding to rectangular tiling of the time-frequency plane. Extensions of these frames are obtained in the rotated time-frequency planes by using the fractional Fourier transformation. It is shown that rotation does not affect the frame bounds. For some specific angles, lattices in rotated coordinates will map to the lattices in th...

We present a notion of frame multiresolution analysis on the Heisenberg group, abbreviated by FMRA, and study its properties. Using the irreducible representations of this group, we shall define a sinc-type function which is our starting point for obtaining the scaling function. Further, we shall give a concrete example of a wavelet FMRA on the Heisenberg group which is analogous to the Shannon...

A discrete version of the Zak transform is defined and used to analyze discrete Weyl–Heisenberg frames, which are nonorthogonal systems in the space of square-summable sequences that, although not necessarily bases, provide representations of square-summable sequences as sums of the frame elements. While the general theory is essentially similar to the continuous case, major differences occur w...

We consider the long standing problem of constructing d equiangular lines in C, i.e., finding a set of d unit vectors (φj) in C d with |〈φj , φk〉| = 1 √ d + 1 , j 6= k. Such ‘equally spaced configurations’ have appeared in various guises, e.g., as complex spherical 2–designs, equiangular tight frames, isometric embeddings `2(d) → `4(d), and most recently as SICPOVMs in quantum measurement theor...

We give an introduction to irregular Weyl-Heisenberg frames showing the latest developments and open problems. We provide several new results for semiirregular WH-frames as well as giving new and more accessable proofs for several results from the literature.

We apply the theory of Weyl-Heisenberg frames (WHFs) to oversampled FIR and IIR DFT filter banks (FBs). We show that the polyphase matrices provide a matrix representation of the frame operator, and we find conditions on a DFT FB to provide a WHF expansion. We also show that paraunitary and biorthogonal DFT FBs correspond to tight and exact WHFs, respectively, and that the frame bounds can be o...

A Weyl-Heisenberg frame for L2(R) is a frame consisting of modulates Embg(t) = e 2πimbtg(t) and translates Tnag(t) = g(t − na), m,n ∈ Z, of a fixed function g ∈ L2(R), for a, b ∈ R. A fundamental question is to explicitly represent the families (g, a, b) so that (EmbTnag)m,n∈Z is a frame for L2(R). We will show an interesting connection between this question and a classical problem of Littlewoo...