A fast time-frequency multi-window analysis using a tuning directional kernel

Czarnecki, Krzysztof; Fourer, Dominique; Auger, François; Rojewski, M.

doi:10.1016/j.sigpro.2018.01.019

Cited by 26 publications

(17 citation statements)

References 29 publications

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“…The self‐tuning CT was proposed by Czarnecki et al in [28]. This method is closely related to that which was presented in the previous section, because both rely on similar methodology and equations.…”

Section: Self‐tuning Ctmentioning

confidence: 99%

“…In accordance with this, a window is fitted to the instantaneous signal parameters by changing a rotation of the TF cell. In general, the CT is a convolution product of a signal

x false(t false)

with a function

g_{α}

given in this way [28]

\begin{matrix} right leftthickmathspace.5em \\ C_{x}^{g} (t, ω) & = \int_{double-struckR} x (τ) g_{α} (t - τ, ω) d τ \\ = \int_{double-struckR} x (τ) h (t - τ) {normale}^{j α (t^{2} / 2)} {normale}^{j ω (t - τ)} d τ, \end{matrix}

where

h false(t false)

is an analysis window (in general any differentiable window) modulated by the chirp component providing the transformation kernel

g_{α} false(t, ω false) = h false(t false) e^{normalj α false(t^{2} / 2 false)} e^{normalj ω t} = h_{α} false(t false) e^{normalj ω t},

where

α

stands for the CR of the window. In the optimal case, the window CR is the same as the signal CR, which provides an energy concentration around a component with corresponding frequency rate.…”

Section: Self‐tuning Ctmentioning

confidence: 99%

“…Finally, the ACT is given in the following manner:

{\overset{´}{C}}_{x}^{g} false(t, ω false) = \int_{R} x false(τ false) h false(t - τ false) e^{normalj \overset{false~}{α} false(t false) false({false(t - τ false)}^{2} / 2 false)} e^{normalj ω false(t - τ false)} normald τ .

For expressions describing the self‐tuning CT upper symbol

\overset{´}{□}

is dedicated in this paper. In [28], further considerations related to this topic were carried out. Namely, a multi‐window technique was proposed to maximise adaptability, but this method significantly increases computational complexity.…”

Section: Self‐tuning Ctmentioning

confidence: 99%

“…In this paper, this method is significantly improved and extended and, as it turned out, the TF resolution can be increased if an additional analysis window parameter is added to the consideration – its CR. For this purpose, the author was inspired by the adaptive (self‐tuning) chirplet transform (CT) proposed in [28]. According to this method, the directional kernel of transform is defined by CR and is characterised by the properties of adjusting to local signal parameters by changing its rotation on the TF plane.…”

Section: Introductionmentioning

confidence: 99%

“…Section 2 presents the algorithm of the adaptive STFT (ASTFT) proposed in [27]. Section 3 is devoted to the self‐tuning CT known from [28], while Section 4 presents the idea behind the novel combination of these methods. Section 5 collects the results for real‐life data, and finally, a summary closes the paper.…”

Section: Introductionmentioning

confidence: 99%

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Section: Self‐tuning Ctmentioning

confidence: 99%

“…In accordance with this, a window is fitted to the instantaneous signal parameters by changing a rotation of the TF cell. In general, the CT is a convolution product of a signal

x false(t false)

with a function

g_{α}

given in this way [28]

\begin{matrix} right leftthickmathspace.5em \\ C_{x}^{g} (t, ω) & = \int_{double-struckR} x (τ) g_{α} (t - τ, ω) d τ \\ = \int_{double-struckR} x (τ) h (t - τ) {normale}^{j α (t^{2} / 2)} {normale}^{j ω (t - τ)} d τ, \end{matrix}

where

h false(t false)

is an analysis window (in general any differentiable window) modulated by the chirp component providing the transformation kernel

g_{α} false(t, ω false) = h false(t false) e^{normalj α false(t^{2} / 2 false)} e^{normalj ω t} = h_{α} false(t false) e^{normalj ω t},

where

α

stands for the CR of the window. In the optimal case, the window CR is the same as the signal CR, which provides an energy concentration around a component with corresponding frequency rate.…”

Section: Self‐tuning Ctmentioning

confidence: 99%

“…Finally, the ACT is given in the following manner: