O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform", IEEE Trans. Signal Processing, vol. 46, pp. 1746--1757, May 1992. 19 -0.1 0 0.1  Home/Search   Document Not in Database   Summary   Related Articles   Check

....distribution (WD) of x, defined as [5, 6] W x (t; f) Z x(t =2)x (t Gamma =2)e Gammaj 2f d ; t; f) 2 IR 2 ; 2. 2) and Phi is a two dimensional kernel that completely characterizes the TFR P x (t; f ; Phi) Similarly, any bilinear TSR from the affine class can be expressed as [7] C x (t; a; Pi) Z Z W x (u; v) Pi( u Gamma t) a; av)dudv ; t; a) 2 IR Theta (0; 1) 2.3) 6 where again the kernel Pi completely characterizes the TSR C x (t; a; Pi) Cross TFRs P xy (t; f ; Phi) and cross TSRs C xy (t; a; Pi) will prove to be useful tools when multiple signals are ....

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform," IEEE Transactions on Signal Processing, vol. 46, pp. 1746--1757, May 1992.

....the mmse estimation problem can be posed. In fact, any class of bilinear signal representations which is characterized (linearly) by a kernel and includes the WD as a member can be used as a class of estimators. An example is the class of time scale representations proposed by Rioul and Flandrin [16], which may be useful in the case of processes exhibiting a 1=f type spectral structure. The corresponding globally optimal kernel will always be characterized by a linear equation; however, the characterization may not always be as simple and explicit as in the case of Cohen s class, and the ....

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform", IEEE Trans. Signal Processing, vol. 46, pp. 1746--1757, May 1992.

.... via (4) which were originally characterized in [3] Recently, Cohen extended his method to joint distributions of arbitrary variables [1, 10, 11] spurred by interest in variables like scale that was inspired by the wavelet transform [12] and the affine class of time scale representations [13]. 3 Such generalized joint signal representations in terms of arbitrary variables 1 Cohen originally used the (radian) frequency operator W = Gammaid=dt. 2 The procedure of associating functions of classical variables with operators, commonly referred to as quantization in physics, is ....

....corresponds to a change in the value of the variable. For time, frequency and scale, the notion of the shift operators is self evident because of our familiarity with the concepts, and many authors have explicitly or implicitly discussed it from a group theoretic perspective; see for example [26, 27, 28, 29, 30, 13, 31, 18]. However, it is not clear what shift means for an arbitrary variable. We will precisely define the concept of shift operators for arbitrary variables in Section 6, which is intimately related to the notion of duality (developed in Section 5) To discuss some of the other issues addressed in ....

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform", IEEE Trans. Signal Processing, vol. 46, pp. 1746--1757, May 1992.

....involving a nonlinear chirping behavior. To encompass a wider variety of signal characteristics, recently there has been significant interest in the development of generalized joint signal representations (JSRs) which analyze signals in terms of physical quantities other than time and frequency [3, 4, 5, 1, 6, 7, 8, 9, 10, 11]. The wavelet transform and generalizations are the best known, which analyze signals in terms of time and scale content [12, 3, 4, 5, 7] Owing to the recent interest in generalized JSRs, there has been substantial progress in the development of a general theory for generalized JSRs with respect ....

.... of generalized joint signal representations (JSRs) which analyze signals in terms of physical quantities other than time and frequency [3, 4, 5, 1, 6, 7, 8, 9, 10, 11] The wavelet transform and generalizations are the best known, which analyze signals in terms of time and scale content [12, 3, 4, 5, 7]. Owing to the recent interest in generalized JSRs, there has been substantial progress in the development of a general theory for generalized JSRs with respect to arbitrary variables. The most comprehensive theory to date is due to Cohen [1, 6] who has developed the generalization first proposed ....

[Article contains additional citation context not shown here]

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform", IEEE Trans. Signal Processing, vol. 40, pp. 1746--1757, May 1992.

....radar detection. However, quadratic TFRs, by virtue of their richer structure, hold promise for a wider variety of detection scenarios. Two important classes of such quadratic TFRs are Cohen s class [6] which generalizes the concept of the spectrogram (jSTFTj 2 ) and the affine class [7] which generalizes the concept of the scalogram (jCWTj 2 ) Recently, a comprehensive theory for optimum TFRbased detection has been developed that has put timefrequency detection on a firm footing [5] It also overcomes the limitations of previously proposed methods that were mostly ad hoc and ....

....0) The window g is called the mother wavelet which is usually bandpass. The squared magnitude of the STFT is known as the spectrogram, and that of the CWT is known as the scalogram. Bilinear TFRs provide a richer structure than linear ones, and two important classes are Cohen s and affine classes [6, 7]. Both classes can be defined as smoothed versions of the Wigner distribution (WD) which is defined as Ws(t; f) j Z s i t 2 j s i t Gamma 2 j e Gammaj2 f d : 3) Cohen s class, which is a generalization of the spectrogram, can be expressed as a convolutional smoothing ....

[Article contains additional citation context not shown here]

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform", IEEE Trans. Signal Processing, vol. 40, pp. 1746--1757, May 1992.

....linear representations. I. Introduction Recognizing the limitations of time frequency representations (TFRs) generalized joint signal representations which analyze signals in terms of physical quantities other than time and frequency have recently been investigated by a number of authors [1], 2] 3] 4] 5] 6] 7] For example, joint time scale representations analyze signals in terms of time and scale content [1] 2] In existing literature, the construction of joint signal representations has been based on two main approaches. Cohen s pioneering method of constructing ....

.... signal representations which analyze signals in terms of physical quantities other than time and frequency have recently been investigated by a number of authors [1] 2] 3] 4] 5] 6] 7] For example, joint time scale representations analyze signals in terms of time and scale content [1], 2] In existing literature, the construction of joint signal representations has been based on two main approaches. Cohen s pioneering method of constructing bilinear TFRs interprets the TFRs as quasi energy distributions which satisfy certain marginal constraints analogous to probability ....

[Article contains additional citation context not shown here]

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform," IEEE Trans. Signal Processing, vol. 46, pp. 1746--1757, May 1992.

.... distribution (WD) of x, defined as [26, 12] W x (t; f) Z x(t =2)x (t Gamma =2)e Gammaj2 f d ; t; f) 2 IR 2 ; 17) and the two dimensional kernel Phi completely characterizes the TFR P x (t; f ; Phi) Similarly, any bilinear TSR from the affine class can be expressed as [27] C x (t; a; Pi) Z Z W x (u; v) Pi( u Gamma t) a; av)dudv ; t; a) 2 IR Theta (0; 1) 18) where again the kernel Pi completely characterizes the TSR C x (t; a; Pi) We note that both P x ( Phi) and C x ( Pi) are characterized as averaged versions of the WD, the difference being in the ....

....weighted sum of a bank of spectrograms or scalograms, respectively. The short time Fourier transform (STFT) of a signal x is defined as [12, 31] STFT x (t; f ; h) Z x(u)h (u Gamma t)e Gammaj2 f u du ; 69) where h is the analysis window, and the wavelet transform (WT) of x is defined as [31, 27] WT x (t; a; g) Z x(u)jaj Gamma1=2 g u Gamma t a du ; 70) where g is called the analysis or mother wavelet. By comparing (69) with (40) 37) and (38) it can be easily verified that L ( O (x) 1 N 0 X k k k N 0 jSTFT x ( u k )j 2 X k log N 0 k N ....

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform", IEEE Trans. Signal Processing, vol. 40, pp. 1746--1757, May 1992.

....highlighted by its simple structure and interpretation, and naturally extends the concept of the corresponding linear representations. 1. INTRODUCTION Joint signal representations in terms of physical quantities other than time and frequency have recently been investigated by a number of authors [1, 2, 3, 4, 5, 6]. For example, joint time scale representations (TSRs) 1, 2] analyze signal characteristics in terms of time and scale content. The motivation for studying such generalized joint representations is to develop tools that can provide a useful description for a broad class of nonstationary signal ....

....the concept of the corresponding linear representations. 1. INTRODUCTION Joint signal representations in terms of physical quantities other than time and frequency have recently been investigated by a number of authors [1, 2, 3, 4, 5, 6] For example, joint time scale representations (TSRs) [1, 2] analyze signal characteristics in terms of time and scale content. The motivation for studying such generalized joint representations is to develop tools that can provide a useful description for a broad class of nonstationary signal characteristics. In existing literature, the construction of ....

[Article contains additional citation context not shown here]

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform", IEEE Trans. Signal Processing, pp. 1746--1757, May 1992.

....of nonstationary signals [1] 11] 19] Generalized JSRs represent nonstationary signal characteristics in terms of quantities other than time and frequency (time and scale, for example) and have the ability to be matched to signals with radically different characteristics. For example, TSRs [20] such as the wavelet transform and the affine class perform a constant Q 1 analysis, and the hyperbolic class [21] is useful for Doppler invariant analysis. Despite the proliferation of generalized JSRs, their development has again been primarily directed towards exploratory data analysis. ....

....trivially characterized in Cohen s recipe, the covariance properties are neither guaranteed nor easily characterized. As we will see, covariance properties are crucial in a JSR based detection framework and thus for such applications a covariance based generalization is needed (the affine class [20] was constructed using covariance arguments) In such generalizations, which have also been proposed recently [17, 18, 19] 2 The discussion also applies to cases in which the signal space of interest is a closed (and hence complete) subspace of L 2 (IR) for example, the space of analytic ....

[Article contains additional citation context not shown here]

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform," IEEE Trans. Signal Processing, vol. 40, pp. 1746--1757, May 1992.

.... Recently, in an attempt to tailor joint signal representations to a broader class of signals, there has been substantial progress in the development of joint distributions of variables other than time and frequency [2] 7] Joint time scale representations constituted first such generalizations [2, 3], spurred by the interest in the wavelet transform [8] In view of this recent trend, general theories for joint distributions of arbitrary variables have been proposed by many authors [1, 5, 9, 10, 11] The first such generalization was proposed by Scully and Cohen [12] and developed by Cohen ....

.... Moreover, we are interested in a characterization of bilinear distributions in which the kernel is not a function of the variables, as is true for all covariance based generalizations [10, 11] and for Cohen s class of bilinear TFRs [1] and the affine class of bilinear time scale representations [3, 2], in particular. 4 We use the fact that if A is a linear operator on a complex inner product space H, then hAs; si = 0 for all s 2 H , A j 0; see, for example, 15, p. 374] 5 An operator U is unitary if hUs; Usi = hs; si for all s. 6 Which follows from the fact that e jA is a unitary ....

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform," IEEE Trans. Signal Processing, vol. 46, pp. 1746--1757, May 1992.

.... ) 4) L TS (x) max ( c) Cy) 1=c; Pi) F TS ( c) 5) for some functions F TF and F TS independent of x, where P ( Phi) is a bilinear TFR from Cohen s class characterized by the kernel Phi [5] and C( Pi) is a bilinear TSR from the affine class characterized by the kernel Pi [6]. In (4) and (5) y = R Gamma1 n x, where Rn is the operator defined by the noise correlation function as (Rnx) t) j Z Rn (t; u)x(u)du ; 6) and we assume that R Gamma1 n exists. 5 Moreover, the characterizing kernels Phi and Pi are completely characterized by fRn ; R TF g and fRn ; ....

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform", IEEE Trans. Signal Processing, pp. 1746--1757, May 1992.

No context found.

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform", IEEE Trans. Signal Processing, vol. 46, pp. 1746--1757, May 1992. 19 -0.1 0 0.1

No context found.

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform," IEEE Trans. Signal Processing, vol. 46, pp. 1746--1757, May 1992.

No context found.

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform", IEEE Trans. Signal Processing, vol. 40, pp. 1746--1757, May 1992.

No context found.

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform," IEEE Trans. Signal Processing, vol. 46, pp. 1746--1757, May 1992.

No context found.

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform", IEEE Trans. Signal Processing, vol. 40, pp. 1746--1757, May 1992.

No context found.

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform", IEEE Trans. Signal Processing, pp. 1746--1757, May 1992.

No context found.

O. Rioul and P. Flandrin, "Time-scale distributions: A general class extending the wavelet transform," IEEE Trans. Signal Processing 40, pp. 1746--1757, May 1992.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC