R. G. Shenoy and T. W. Parks, "The Weyl correspondence and time-frequency analysis", IEEE Trans. Signal Processing, vol. 42, no. 2, pp. 318--332, Feb. 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....time frequency (TF) formulation of the p point uncertainty classes S, N and the robust time varying Wiener lter HR . Let LH (t; f) denote the Weyl symbol of a linear time varying system H [7] 9] and Wx(t; f) the WignerVille spectrum (WVS) of a nonstationary random process x(t) 10, 11] Using [7, 8] trfP i Rxg = LP i ; Wx = R t R f LP i (t; f)Wx(t; f) dtdf, a TF formulation of the p point uncertainty class S in (2) is obtained as Ws(t; f) LP i ; Ws = s i ; i = 1; 2; N , and similarly for N . For non sophisticated [12] subspaces X i , each X i corresponds to a TF ....

....(e) Figure 1. TF representations of signal and noise statistics and various Wiener lters: a) WVS of s(t) b) WVS of n(t) c) Weyl symbol of H 0 W , d) Weyl symbol of HR , and (e) Weyl symbol of e HR . Here, the impulse response p i (t; t 0 ) of P i is the inverse Weyl transform [7, 8] of the indicator function IR i (t; f ) which can be computed eciently using FFT methods. Note that P i approximates P i but is not exactly an orthogonal projection operator [12, 13] The prior knowledge necessary for designing this robust TF Wiener lter e HR is given by the mean regional ....

R. G. Shenoy and T. W. Parks, \The Weyl correspondence and time-frequency analysis," IEEE Trans. Signal Processing, vol. 42, pp. 318-331, Feb. 1994.

.... is known [4, 6, 7] that all quadratic test statistics can be rewritten as inner products in the TF domain, 1 IEEE 1998 (r) hHr; ri = h ; Wr i = Z t Z f (t; f) Wr (t; f) dt df: Here, t; f) LH (t; f) Z h t 2 ; t 2 e j2 f d is the Weyl symbol (WS) 13] [15] of H and Wr (t; f) Z r t 2 r t 2 e j2 f d is the Wigner distribution [16, 17] of r(t) Thus, any quadratic test statistic can be interpreted as a weighted integral of Wr (t; f ) However, instead of using this exact TF formulation, here we shall typically ....

R. G. Shenoy and T. W. Parks, \The Weyl correspondence and time-frequency analysis," IEEE Trans. Signal Processing, vol. 42, pp. 318-331, Feb. 1994.

....relation (9.3.13) We may say that the IWD is covariant to TF shifts of the system under analysis. Similar covariance properties of the IWD hold with respect to metaplectic unitary operators U a;b;c;d corresponding to area preserving linear TF coordinate transforms (t; f) at bf; ct df) [33, 37, 57], W (I) U a;b;c;d HU a;b;c;d (t; f) W (I) H (at bf; ct df) For example, W (I) CaHC a (t; f) W (I) H (at; f=a) where C a is the TF scaling operator de ned as (C a x) t) p jaj x(at) 6) Invariance property. The IWD is invariant to unitary system components on the ....

R. G. Shenoy and T. W. Parks, \The Weyl correspondence and time-frequency analysis," IEEE Trans. Signal Processing, vol. 42, pp. 318-331, Feb. 1994.

.... f) 10) calculated for least favorable pseudo WVS W L s ; W L n = arg max W s 2 e S W n2 e N emin(W s ; W n) with emin(W s ; W n) given by (9) This generalizes a similar result in the stationary case [4] From L e HR (t; f ) e HR can be obtained by an inverse Weyl transform [10, 11]. Next, we propose three di erent de nitions of TF uncertainty classes e S, e N and we provide closed form expressions for the respective robust TF Wiener lters e HR . p Point Model. Let fR i g i=1;2; N be a partition of the TF plane, i.e. S N i=1 R i = R 2 and R i R j = for i 6= j. ....

R. G. Shenoy and T. W. Parks, \The Weyl correspondence and time-frequency analysis," IEEE Trans. Signal Processing, vol. 42, pp. 318-331, Feb. 1994.

.... In this paper, we provide an answer to this and several other questions of theoretical and practical importance: We show that for underspread [8 12] nonstationary processes the TF formulation in (2) is approximately valid if H(t; f) and S s (t; f ) Sn (t; f) are chosen as the Weyl symbol [13 16] and the Wigner Ville spectrum [12, 17 19] respectively. We present upper bounds on the associated approximation errors. We propose an ecient, intuitive, and nearly optimal TF design of signal estimators that is easily adapted to modi ed (weighted) error criteria. We discuss an ecient TF ....

....we review some fundamentals of timefrequency (TF) analysis that will be used in Sections IV and V for the TF formulation and TF design of optimal lters. A. Time Frequency Representations We rst review four TF representations on which our TF formulations will be based. The Weyl symbol (WS) [13 16] of a linear operator (linear, time varying system) H with kernel (impulse response) h(t; t 0 ) is de ned as LH (t; f) 4 = Z h t 2 ; t 2 e j2 f d ; 15) where t and f denote time and frequency, respectively. For underspread systems (see Subsection III B) the WS ....

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R. G. Shenoy and T. W. Parks, \The Weyl correspondence and time-frequency analysis," IEEE Trans. Signal Processing, vol. 42, pp. 318-331, Feb. 1994.

.... random process x(t) 16 19] W x (t; f) LRx (t; f) Z 1 1 r x t 2 ; t 2 e j2 f d : Note that the WVS is in one to one correspondence with the correlation R x and thus it constitutes a complete description of the second order statistics of x(t) Using the relation [13, 14] TrfP i R x g = LP i ; W x = Z 1 1 Z 1 1 LP i (t; f) W x (t; f) dt df ; 6 the uncertainty classes S and N in (4) can be reformulated as W s (t; f) LP i ; W s = s i ; i = 1; N and W n (t; f) LP i ; W n = n i ; i = 1; N , respectively. To ....

....expressing an equal TF weighting (cf. 15] of all process components lying in the same TF region R i . The robust TF Wiener lter e HR can be written as e HR = N X i=1 s i s i n i P i ; with the impulse response p i (t; t 0 ) of P i given by the inverse Weyl transform [13, 14] of I R i (t; f ) p i (t; t 0 ) Z 1 1 I R i t t 0 2 ; f e j2 f(t t 0 ) df : The inverse Weyl transform can be eciently computed using FFT methods. The systems P i are self adjoint and orthogonal (Tr P i P j = 0 for i 6= j) and their sum equals I. ....

R. G. Shenoy and T. W. Parks, \The Weyl correspondence and time-frequency analysis," IEEE Trans. Signal Processing, vol. 42, pp. 318-331, Feb. 1994.

....frequency response of time invariant systems in (2) is given by the generalized Weyl symbol [15 17] L ( H (t; f) Z h ( t; e j2 f d (5) with h ( t; h t 1 2 ; t 1 2 ; 6) where 2 R. Special cases are the Weyl symbol for = 0 [16 21], Zadeh s time varying frequency response for = 1=2 [16,17,20,22,23] and the Kohn Nirenberg symbol (equivalently, Bello s frequency dependent modulation function) for = 1=2 [18,23,24] the case = 0 has again certain advantages over other choices of [17,18,20] Note that the generalized ....

.... H ( is the 2 D Fourier transform of the generalized Weyl symbol in (5) The generalized spreading function S ( H ( is the coefficient function of an expansion of H into elementary time frequency shifts S ( where S ( x (t) x(t ) e j2 t e j2 ( 1=2) [15 18,20,21,23,30,31]. Hence, for a given ( jS H ( j indicates how much the time frequency shifted input signal S ( x (t) x(t ) e j2 t e j2 ( 1=2) contributes to the output signal. It follows that the timefrequency shifts caused by a linear time varying system are crudely characterized ....

R. G. Shenoy and T. W. Parks, "The Weyl correspondence and timefrequency analysis," IEEE Trans. Signal Processing, vol. 42, pp. 318--331, Feb. 1994.

....d d d d ; which implies that Phi( Phi ( ffi Gamma Gamma Delta ffi Gamma Gamma Delta which can not be true. A.3.c Symplectic Transformations. The Wigner distribution is the only quadratic distribution covariant to symplectic transformations [42]. For the quartic class, the following are, respectively, sufficient kernel constraints for TFD s in the quartic class to be covariant to scalings, Fourier transforms, and chirp multiplications Phi( Phi(a ; a; a; a) Phi( Phi( Gamma ; Gamma ; ....

R.G. Shenoy and T.W. Parks, "The Weyl correspondence and time-frequency analysis", IEEE Trans. on Signal Processing, vol. 42, no. 2, pp. 318--331, Feb. 1994.

....Weyl symbol [74] see Chapter 4) a family of time varying transfer functions defined as: L (ff) H (t; f) Z h(t Gamma 1 2 Gamma ff Delta ; t Gamma Gamma 1 2 ff Delta )e Gammaj 2f d . 3. 12) Zadeh s transfer function corresponds to ff = 1=2, and the Weyl symbol [111, 118] to ff = 0. The generalised evolutionary spectrum, ES (ff) x (t; f ) is defined as a direct extension of (3.11) ES (ff) x (t; f) fi fi fi L (ff) p Rx (t; f) fi fi fi 2 . 3.13) It includes Priestley s evolutionary spectrum (ff = 1=2) and the transitory evolutionary spectrum ....

....dependent modulation function, is defined by Bello [13] as: BH (t; f) Z h(t ; t)e Gammaj 2f d : By introducing a symmetry to the form of the kernel in (4. 4) the Weyl symbol (WS) originally introduced in quantum mechanics [117] is recognised as a TF parameterised transfer function [47, 64, 73, 97, 103, 111, 118]: LH (t; f) Z h(t =2; t Gamma =2)e Gammaj 2f d : Kozek [74] incorporates all three representations in a single framework via the introduction of the generalised Weyl symbol (GWS) L (ff) H (t; f) Z h(t Gamma 1 2 Gamma ff Delta ; t Gamma Gamma 1 2 ff Delta )e ....

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R.G. Shenoy and T.W. Parks. The Weyl correspondence and time-frequency analysis. IEEE Trans. Signal Processing, 42:318--331, 1994.

....of the test statistic that is a function of the observations is of the form L ( ff) x) hQ ( ff) x; xi; 2.16) where Q is the positive definite linear operator given by R Gamma1 n R s R Gamma1 n . The connection to TFRs and TSRs is made through the use of the Weyl correspondence [10, 11] which relates inner products, positive definite linear operators, and the Wigner distribution through the following relation: hQx; yi = Z Z W xy (u; v)WSQ (u; v)dudv; 2.17) where WSQ is the Weyl symbol [10, 11] of the operator Q (or its kernel Q) defined as WSQ (u; v) Z Q(u =2; u ....

....The connection to TFRs and TSRs is made through the use of the Weyl correspondence [10, 11] which relates inner products, positive definite linear operators, and the Wigner distribution through the following relation: hQx; yi = Z Z W xy (u; v)WSQ (u; v)dudv; 2. 17) where WSQ is the Weyl symbol [10, 11] of the operator Q (or its kernel Q) defined as WSQ (u; v) Z Q(u =2; u Gamma =2)e Gammaj 2v d : 2.18) Using this relation, the test statistic in (2.16) can be expressed as L ( ff) x) Z Z W x (u; v)WS Q ( ff) u; v)dudv: 2.19) The Weyl symbol s covariance to time shifts, ....

R. G. Shenoy and T. W. Parks, "The Weyl correspondence and time-frequency analysis," IEEE Transactions on Signal Processing, vol. 42, no. 2, pp. 318--332, February 1994.

....bank of quadratic detectors. We wish to show that TFRs and TSRs provide a natural structure with which to implement the optimal array processor. The connection between quadratic detectors and bilinear TFRs and TSRs, originally derived in [1] is made through the use of the Weyl correspondence [8, 9] which relates inner products, positive definite linear operators, and the Wigner distribution through the following relation: hQx; yi = Z Z W xy (u; v)WSQ (u; v)dudv; 24) where WSQ is the Weyl symbol [8, 9] of the operator Q (or its kernel Q) defined as WSQ (u; v) Z Q(u =2; u Gamma ....

.... and TSRs, originally derived in [1] is made through the use of the Weyl correspondence [8, 9] which relates inner products, positive definite linear operators, and the Wigner distribution through the following relation: hQx; yi = Z Z W xy (u; v)WSQ (u; v)dudv; 24) where WSQ is the Weyl symbol [8, 9] of the operator Q (or its kernel Q) defined as WSQ (u; v) Z Q(u =2; u Gamma =2)e Gammaj 2v d : 25) The Weyl symbol s covariance to time shifts, frequency shifts, and scale offsets [10] are captured by the relations WS Q ( u; v) WSQ (u Gamma ; v Gamma ) 26) WS Q ( c) u; ....

R. G. Shenoy and T. W. Parks, "The Weyl correspondence and time-frequency analysis," IEEE Transactions on Signal Processing, vol. 42, no. 2, pp. 318--332, February 1994.

.... by an affine smoothing of the WD [9] C s (t; c; Pi) j Z Z W s (u; v) Pi( u Gamma t c ; cv)dudv (5) where the kernel Pi completely characterizes the TSR C s ( Pi) Cohen s class and the affine class can be expressed as a weighted sum of spectrograms and scalograms, respectively, as [11] P s (t; f ; Phi) X k k j(STFT s (t; f ; u k )j 2 (6) C s (t; f ; Pi) X k k j(CWT s (t; f ; v k )j 2 (7) where the k s and u k s represent the eigenvalues and orthonormal eigenvectors, respectively, of the linear operator defined by the kernel Phi. This same relationship is ....

R. G. Shenoy and T. W. Parks, "The Weyl correspondence and time-frequency analysis," IEEE Transactions on Signal Processing, vol. 42, pp. 318--332, February 1994.

....are dual operators, and similarly Gamma and Upsilon G l are dual operators. 10 Note that A Pi ff = e j2 ffA and B Pi fi = e j2 fiB are unitary representations of (IR; on L 2 (IR; dx) where OE is the 2d kernel. Another formulation is given by the Weyl correspondence [26, 1, 27, 28] M W (ff;fi) j e j2 (ffA fiB) 23) P (OE) can be then be recovered via (20) and the marginals (17) and (18) are satisfied if the kernel satisfies OE(ff; 0) 1, for all ff, and OE(0; fi) 1, for all fi, respectively. However, for arbitrary pairs of variables (operators) the kernel ....

R. G. Shenoy and T. W. Parks, "The Weyl correspondence and time-frequency analysis", IEEE Trans. Signal Processing, vol. 42, no. 2, pp. 318--332, Feb. 1994.

....types of signals. The original class will be renamed the type I Cohen class and the other three classes will be denoted the type II, III, and IV Cohen classes. There are three common methods for deriving TFDs for type I signals. The first uses operator theory [1] 2] the second uses group theory [6], and the third uses covariance properties [3] 4] 5] In this paper we choose to use the covariance based approach to investigate TFDs for signals of types II, III, and IV, because of the simplicity and directness of the mathematics. Narayanan, McLaughlin, Atlas, and Droppo [7] 8] have ....

R.G. Shenoy and T.W. Parks, "The Weyl correspondence and time-frequency analysis", IEEE Trans. on Signal Processing, vol. 42, no. 2, pp. 318--331, Feb. 1994.

.... that if we represent the channel by an operator H, it is composed of a linear combination of time and frequency shift operators H = Z Z H( F T d d : 10) It is well known that an arbitrary time varying linear system admits such a representation in terms of time and frequency shifts [5, 6]. 1 3.1 Statistical Channel Parameters The time variant channel impulse response h(t; is best modeled as a stochastic process and a realistic model in many situations is the wide sense stationary uncorrelated scatterer 1 The special class of linear time invariant systems can be represented ....

R. G. Shenoy and T. W. Parks, "The Weyl correspondence and time-frequency analysis, " IEEE Trans. Signal Processing, vol. 42, pp. 318--332, Feb. 1994.

....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 nature of the averaging. Using Weyl correspondence [28, 29, 30], the test statistic in (15) can be expressed as 8 L (ff;fi) x) hQ (ff;fi) x; xi = Z Z W x (u; v)WS Q (ff;fi) u; v)dudv (19) where WS Q (ff;fi) is the Weyl symbol [28, 29, 30] of the operator Q (ff;fi) or its kernel Q (ff;fi) defined as WS Q (ff;fi) u; v) Z Q (ff;fi) u ....

....averaged versions of the WD, the difference being in the nature of the averaging. Using Weyl correspondence [28, 29, 30] the test statistic in (15) can be expressed as 8 L (ff;fi) x) hQ (ff;fi) x; xi = Z Z W x (u; v)WS Q (ff;fi) u; v)dudv (19) where WS Q (ff;fi) is the Weyl symbol [28, 29, 30] of the operator Q (ff;fi) or its kernel Q (ff;fi) defined as WS Q (ff;fi) u; v) Z Q (ff;fi) u =2; u Gamma =2)e Gammaj2 v d : 20) Comparing (19) and (16) we note that by using a time frequency varying kernel, any test statistic L (ff;fi) corresponding to any ....

R. G. Shenoy and T. W. Parks, "The Weyl correspondence and time-frequency analysis", IEEE Trans. Signal Processing, vol. 42, no. 2, pp. 318--332, Feb. 1994.

....a direct consequence of the group theoretic approach used to derive this distribution. This approach is analogous to that used to find the continuous Wigner distribution. To summarize, the continuous distribution can be obtained by studying the Heisenberg group generated by the real numbers (see [13] for a detailed derivation of this, as 0 50 100 0.5 0 0.5 30 20 10 0 10 20 30 40 Time Freq (a) 0 50 100 0.5 0 0.5 30 20 10 0 10 20 30 40 Time Freq (b) Figure 1. The discrete Wigner distribution (a) and samples of the continuous distribution (b) of a length 65 cosine signal with ....

....group of continuous time shifts and continuousfrequency shifts. It follows that one would obtain the discrete distribution by studying the Heisenberg group generated by the integers modulo N , which corresponds to the group of discrete time shifts and discrete frequency shifts [8, 9] Summarizing [13], define the frequency modulation operator, M , and the time advance operator, S , for a signal x(t) to be (M x) t) e j2 t x(t) 1) S x) t) x(t ) 2) These operators allow us to define the symmetric form of the continuous cross ambiguity function, A x;y , A x;y ( aeM S ....

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R. G. Shenoy and T. W. Parks, "The Weyl Correspondence and Time-Frequency Analysis, " IEEE Trans. Signal Process., vol. 42, pp. 318--331, 1994.

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R. G. Shenoy and T. W. Parks, "The Weyl correspondence and time-frequency analysis", IEEE Trans. Signal Processing, vol. 42, no. 2, pp. 318--332, Feb. 1994.

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R. G. Shenoy and T. W. Parks, "The Weyl correspondence and time-frequency analysis", IEEE Trans. Signal Processing, vol. 42, no. 2, pp. 318--332, Feb. 1994.

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R. G. Shenoy and T. W. Parks, "The Weyl correspondence and time-frequency analysis," IEEE Transactions on Signal Processing, vol. 42, pp. 318--332, February 1994.

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R. G. Shenoy and T. W. Parks, "The Weyl correspondence and timefrequency analysis," IEEE Trans. Signal Processing, vol. 42, pp. 318-- 331, Feb. 1994.

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R. G. Shenoy and T. W. Parks, \The Weyl correspondence and time-frequency analysis," IEEE Trans. Signal Processing, vol. 42, pp. 318-331, Feb. 1994.

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R. G. Shenoy and T. W. Parks, "The Weyl correspondence and time-frequency analysis," IEEE Trans. Signal Processing, vol. 42, pp. 318--331, Feb. 1994.

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R. G. Shenoy and T. W. Parks, "The Weyl correspondence and time-frequency analysis," IEEE Trans. Signal Processing, vol. 42, pp. 318--331, Feb. 1994.

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R. G. Shenoy und T. W. Parks, "The Weyl correspondence and time-frequency analysis," IEEE Trans. Signal Processing, vol. 42, no. 2, pp. 318-331, Feb. 1994.

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