Time-frequency formulation and design of nonstationary Wiener filters

Kirchauer, H.; Hlawatsch, Franz; Kozek, W.

doi:10.1109/icassp.1995.479857

Cited by 26 publications

(9 citation statements)

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“…The result is a transfer function calculus ͑known as symbolic calculus in a quantummechanical context͒ that extends a previously introduced transfer function calculus [42][43][44] to a considerably wider and practically more relevant class of linear operators. This extended transfer function calculus provides a theoretical basis for several methods that have recently been proposed for nonstationary signal processing ͑specifically, to design and implement time-varying filters for signal enhancement, estimation, and detection [45][46][47][48][49][50][51][52][53] ͒. Furthermore, it has important implications in the theory of time-varying power spectra for nonstationary random processes.…”

Section: A Summary Of Results and Outline Of Papermentioning

confidence: 99%

“…[45][46][47][48][49][50][51][52][53] Furthermore, the calculus can immediately be applied to the theory of time-varying power spectra of nonstationary random processes. Here, the operators are either correlation operators or innovation systems ͑cf.…”

Section: Discussionmentioning

confidence: 99%

“…Hence, our subsequent results will show that a transfer function calculus ͑symbolic calculus͒ is valid for a significantly wider class of systems than that considered previously. [42][43][44] This calculus leads to simple and numerically efficient time-frequency methods for specific nonstationary signal processing applications [45][46][47][48][49][50][51][52][53] and it yields important results on time-varying power spectra of nonstationary random processes. 42,54,55 …”

Section: Transfer Function Calculus "Symbolic Calculus…mentioning

confidence: 99%

“…In a signal processing context, this is unfortunate since a multiplicative composition property of the generalized Weyl symbol ͑a multiplicative symbolic calculus͒ similar to the time-invariant/frequency-invariant case would allow very simple and intuitive formulations of ͑statistical͒ signal processing techniques in nonstationary environments. 42,[47][48][49][50][51][52][53][54]87 In the context of quantum mechanics, such a symbolic calculus-i.e., the approximation of the star product of symbols by the ordinary product-would establish an asymptotic correspondence ͑in the classical limit h→0, where h denotes Planck's constant͒ between pointwise products and Jordan products as well as between Poisson brackets and commutators 15 ͑i͒ For ␣ϭ1/2, if H 1 is a linear time-invariant operator with spectral transfer function G 1 ( f ) or H 2 is a linear frequency-invariant operator with temporal transfer function m 2 (t), it can be shown that…”

Section: Operator Productsmentioning

confidence: 99%

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Time-frequency transfer function calculus (symbolic calculus) of linear time-varying systems (linear operators) based on a generalized underspread theory

Matz

Hlawatsch

1998

Journal of Mathematical Physics

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Articles you may be interested inDynamic programming based time-delay estimation technique for analysis of time-varying time-delay Rev. Sci. Instrum. 81, 013501 (2010); 10.1063/1.3280161 Q-S (complete or anticipated) synchronization backstepping scheme in a class of discrete-time chaotic (hyperchaotic) systems: A symbolic-numeric computation approachWe introduce a generalized concept of underspread linear time-varying systems ͑linear operators͒ which contains a previous definition as a special case. We show that an existing approximate transfer function calculus ͑symbolic calculus͒ can be extended to this wider and practically more relevant class of underspread operators. As a mathematical underpinning of this calculus, we establish explicit bounds on various error quantities associated with it. The transfer function calculus provides a theoretical basis for various methods recently proposed for nonstationary signal processing, and it has important implications in the theory of time-varying power spectra.

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Section: A Summary Of Results and Outline Of Papermentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Transfer Function Calculus "Symbolic Calculus…mentioning

confidence: 99%

Section: Operator Productsmentioning

confidence: 99%

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Time-frequency transfer function calculus (symbolic calculus) of linear time-varying systems (linear operators) based on a generalized underspread theory

Matz

Hlawatsch

1998

Journal of Mathematical Physics

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“…The underspread property is important in man other respects as well, such as nonstationary Wiener &ers [13], the Gabor expansion [7], and the short-time Fourier transform [SI. A class of time-varying spectrum estimators for underspread processes is studied in [14].…”

Section: Discussionmentioning

confidence: 99%

Correlative time-frequency analysis and classification of nonstationary random processes

Kozek

Hlawatsch²,

Kirchauer³

et al.

Proceedings of IEEE-SP International Symposium on Time- Frequency and Time-Scale Analysis

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The ezpected ambiguity function (EAF) is shown to provide a generalization of stationary correlation analysis to nonstationary random processes. Important properties of the EAF are discussed, and the EAFs of special processes are considered. Based on the EAF, a fundamental classification (underspread/overspread) of nonstationary processes is introduced and shown to be relevant to timevarying spectral analysis. I N T R O D U C T I O NThe correlative analysis of stationar processes using theis of fundamental importance [l]. In particular, the power spectrum S,(f) is the Fourier transform of the ACF ( Wiener-Khintchine relation)',For a nonstationary process x ( t ) , the ACF r z ( t i , t z ) = E { x ( t i ) x*(t2)} is a 2-D function [l], and the power spectrum S,(f) is replaced by a time-varying power spectrum T,(t, f ) such as the physical spectrum, the (generalized) Wagner-Ville spect m m , or the evolutionary spectrum [2, 3, 41. Nonstationary processes exhibit spectral correlation [5] as measured by the spectral ACF R Z ( f 1 , f 2 )= E{X(f')X*(fi)} (assuming existence of the Fourier transform X ( f ) of z(t)).We now ask if there exists a joint time-frequency (TF) correlation function which combines the temporal ACF r , ( t i , t 2 ) and the spectral ACF R , ( f l , f i in a meaningful way, and which is related to a meanin d ul time-varying spectrum by a Fourier transform (generalization of the Wiener-Khintchine relation (1)). In this paper, we show that a satisfactory answer to this question is provided by the expected ambiguity function (EAF) recently proposed in [S, 7 . We demonstrate that the effective support region nonstationarity inherent in a rocess. We then introduce a fundamental classification &nderspread/overspread) of nonstationary processes. For underspread processes, various timevarying spectra (such as the generdized WignerVille spectra and the evolutionary spectra) are shown to be effectively equivalent. Furthermore, the underspread property is relevant to timevarying spectrum estimation, and finally, the physical spectrum of an underspread process is a complete second-order description of the process. 'Funding by grant 4913 of the Jubilaumsfondf der Osterrei-'Integrals go from -cu to 00 unless specified otherwise. of the k A F provides useful indications about the type of chischen Nationalbank and FWF grant The (generalized) ambiguity function (AF) [SI of a signal x ( t ) is where a is a realvalued parameter. We define the ezpected (generalized) ambiguity function (EAF) of a nonstationary random process s(t) as the expectation of the AF, EA?)(^, e' E{A?)(~, VI} .It follows that the EAF is the Fourier transform of the a-parameterized ACF with respect to t , J tSince the ACF can be recovered from the EAF by inversion of the Fourier transform (2) followed by a simple substitution to obtain r , ( t l , t z ) from r p ) ( t , T ) , the EAF is a complete second-order description of the process.Interpretation as TF Correlation. An intuitively reasonable measure for the statistical correlation between ...

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Joint Approximate Diagonalization Method

Shi

2011

Blind Signal Processing

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Time-frequency formulation and design of nonstationary Wiener filters

Cited by 26 publications

References 11 publications

Time-frequency transfer function calculus (symbolic calculus) of linear time-varying systems (linear operators) based on a generalized underspread theory

Time-frequency transfer function calculus (symbolic calculus) of linear time-varying systems (linear operators) based on a generalized underspread theory

Correlative time-frequency analysis and classification of nonstationary random processes

Joint Approximate Diagonalization Method

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