Abstract
A time–frequency transform is a sesquilinear mapping from a suitable family of test functions to functions on the time–frequency plane. The goal is to quantify the energy present in the signal at any given time and frequency. The transform is further specified by imposing conditions such as those stemming from basic transformations of signals and those which an energy density should satisfy. We present a systematic study on how properties of a time–frequency transform are reflected in the associated evaluation at time–frequency origin, integral kernel and quantization and discuss some examples of time–frequency transforms.
Highlights
Representing a signal u : Rn → C in terms of simpler components is the basic problem in signal analysis
We study characterizations of quadratic time–frequency transforms in the space Rn
X ∈Rn for all multi-indices α and β. This space equipped with the topology given by the family of seminorms defined by the left-hand side of (3) is called the Schwartz space S (Rn) of rapidly decreasing smooth functions, known as the space of Schwartz test functions
Summary
Representing a signal u : Rn → C in terms of simpler components is the basic problem in signal analysis. We consider quadratic time–frequency transforms (u, v) → Q(u, v), where Q(u, u) acts as an energy density of the signal u in the time–frequency plane Rn × Rn. For an ideal energy density (x, η) → Q(u, u)(x, η) the value at the point (x, η) could be interpreted as the energy content of the component at frequency η at time x. Time–frequency transforms (x, η) → Q(u, v)(x, η) may be characterized by properties related to basic transformations of signals If they are considered as functions on the Heisenberg group, the correspondence between automorphisms of the Heisenberg group and certain unitary operators on signals u ∈ L2(Rn) yields these so-called covariance properties. (x, η) → Q[u](x, η) should yield the total energy of the signal if integrated over the whole time–frequency plane
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