On Accumulated Cohen’s Class Distributions and Mixed-State Localization Operators

Abstract

Recently we introduced mixed-state localization operators associated to a density operator and a (compact) domain in phase space. We continue the investigations of their eigenvalues and eigenvectors. Our main focus is the definition of a time-frequency distribution which is based on the Cohen class distribution associated to the density operator and the eigenvectors of the mixed-state localization operator. This time-frequency distribution is called the accumulated Cohen class distribution. If the trace class operator is a rank-one operator, then the mixed-state localization operators and the accumulated Cohen class distribution reduce to Daubechies' localization operators and the accumulated spectrogram. We extend all the results about the accumulated spectrogram to the accumulated Cohen class distribution. The techniques used in the case of spectrograms cannot be adapted to other distributions in Cohen's class since they rely on the reproducing kernel property of the shorttime Fourier transform. Our approach is based on quantum harmonic analysis on phase space which also provides the tools and notions to introduce the analogues of the accumulated spectrogram for mixed-state localization operators; the accumulated Cohen's class distributions.1991 Mathematics Subject Classification. 81S30, 94A12, 42C25 .

Local Structure and effective Dimensionality of Time Series Data Sets

“…[6], and is defined by applying smoothing kernels to the so-called Wigner distribution. For details on the connection between the two distinct definitions, see [25]. We give two examples of Cohen class analysis in Figure 19.…”

“…In light of the above, this means that we want to know what prevents the essential dimension of χΩ |Ω| S from being small, or by (23) what prevents the Ω-augmentation to be completely described by a few functions. To clarify this, we use the following result which is an easy consequence of [25,Lem. The reader will of course have noticed that ALC( S, Ω) = 1 |Ω| P (χ Ω S) by Proposition 3.6. Nevertheless, we single out ALC( S, Ω) in order to discuss its interpretation.…”

“…Nevertheless, we single out ALC( S, Ω) in order to discuss its interpretation. We start by considering the integrand in (25) for fixed z: As we know that R 2d S(z ) dz = 1, (26) measures how far S is from being completely concentrated in Ω − z -i.e. the lack of concentration.…”

“…In this work, we make a connection between the ubiquitous local averaging of TF-representations of signals in a given data set, data augmentation of original time-domain data and their effective dimensionality [28] via tools from quantum harmonic analysis developed in [23][24][25].…”

“…Another novelty of our approach is the use of tools such as, mixed state localization operators and the accumulated Cohen's class of an operator, which have recently been developed in [24,25] in work aimed to connect time-frequency analysis and quantum harmonic analysis [29]. In addition, it leads us to introduce notions, like the total correlation function that captures all the time-frequency localization between the different data points.…”