R. J. Barton and H. V. Poor. Signal detection in fractional Gaussian noise. IEEE Transactions on Information Theory, 34(5):943-955, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....H 2 (0; 1) and H 2 (0; 1) respectively, and suppose that H 6= H fBH 0 (t)g t2R 0 (t; u)dBH (u) 1.3) t; u) C 1 (H) H 1=2) C 1 (H ) H 1=2) 1 (H H 1) 1.4) with C 1 (1=2) 1) A representation similar to (1. 3) was obtained by Barton and Poor ([1], p. 958) Their result, however, does not cover Theorem 1.1 because the integral R f(u)dBH (u) in Barton and Poor [1] is de ned for a smaller class of integrands f (f 2 L (R) L (R) bounded and continuous) and, moreover, for H 2 (1=2; 1) only. By setting H = 1=2 in the representation (1.3) ....

....(t; u) C 1 (H) H 1=2) C 1 (H ) H 1=2) 1 (H H 1) 1.4) with C 1 (1=2) 1) A representation similar to (1.3) was obtained by Barton and Poor ( 1] p. 958) Their result, however, does not cover Theorem 1. 1 because the integral R f(u)dBH (u) in Barton and Poor [1] is de ned for a smaller class of integrands f (f 2 L (R) L (R) bounded and continuous) and, moreover, for H 2 (1=2; 1) only. By setting H = 1=2 in the representation (1.3) one recovers the moving average representation (1.1) When H 6= 1=2, however, the representation (1.3) is signi cantly ....

R. J. Barton and H. V. Poor. Signal detection in fractional Gaussian noise. IEEE Transactions on Information Theory, 34(5):943-955, 1988.

....and beta functions, respectively. Remark 2. The integrals (2.1) and (2.2) are de ned in the mean square sense as well as improper by Riemann Stieltjes integrals. Let us also note that the representation (2. 1) rst appeared in Molchan and Golosov [14] already in the late sixties (see also [2, 7, 17]) Remark 3. Actually, the kernel z does not appear in Norros et al. 15] However, it is easy to derive it by the same methods that yield the kernel z: Using the notions of fractional calculus we can write z and z neatly as z(t; s) c 3 s (s) 2.3) t; s) c 3 s ....

Barton, R. J. and Poor, H. V. Signal detection in fractional Gaussian noise. IEEE Trans. Inf. Theory 34, 943959, 1988.

....H 2 (0; 1) RH (s; t) can be written as KH (s; r)KH (t; r)dr; 8) in operator notations, RH = KHK H , where KH is the HilbertSchmidt operator introduced in Theorem [2.1] We hereafter identify an operator and its kernel. Proof. For H 1=2, it is easy to see that du dr Moreover(see [2]) ru) Z ru (r Gamma v) u Gamma v) dv: Hence for H 1=2, 8) holds with (t; r) Gamma(H Gamma 1=2) u Gamma r) du 1 [0;t] r) A change of variable in this equation transforms the integral term in 1 Gamma (1 Gamma t=r)u j H Gamma1=2 du: By the ....

R. J. Barton and H. Vincent Poor. Signal detection in fractional gaussian noise. IEEE trans. on information theory, 34(5):943959, September 1988.

....model Rewriting (10) we consider the model y jk = jk ffl ff w jk (12) w jk = Z jk dBH : 13) In this section, we draw conclusions about the reduced dependence structure of the wavelet coefficients w jk which will form the basis for the proof of Theorem 1. Following for example Barton and Poor (1988), the stochastic integrals defining w jk have mean zero and covariances given by E Z fdBH Z gdBH = 1=2) Z f( g( jj Gamma(1 Gammaff) d (14) In this section, we deduce some properties of the error process j fw jk g that will be used in the proof of Theorem 2. First, we use scaling ....

Barton, R. J. and Poor, H. V. (1988), `Signal detection in fractional gaussian noise', IEEE Transactions on Information Theory 34, 943--959.

....of H: Remark 3.1. Note that when H = 1 = 2 ; 1 is nothing but I 1 0 ; the quadrature operator. Moreover, according to (1) for any H; the predecessor of R(t; by 1 is K(t; Another representation, well de ned only for H 1 = 2 ; is given by the following theorem : Theorem 3. 2 ([2, 8, 16]) For any H 1 = 2 ; consider L 2 equipped with the twisted scalar product : f; g = ZZ [0;1] 2 f(s)g(t)jt sj 2H 2 ds dt: First de ne the linear map 2 on step functions by : 2 : L 2 ; H 1 [0;t] 7 R(t; Skohorod Stratonovitch integral for the fBm 5 Denote by ....

R. J. Barton and H. Vincent Poor. Signal Detection in Fractional Gaussian Noise. IEEE trans. on information theory, 34(5):943959, September 1988.

....because, by the above theorem, e is not complete. The integral I so defined satisfies the relations (3.3) and (3.4) for f; g 2 e . We conclude this section with a proposition which characterizes a large subset of e in the case 0 1=2 (this result is mentioned in Barton and Poor [2], p. 945) Proposition 3.1 For 0 1=2, f 2 L 1 (R) L 2 (R) implies f 2 e . 7 Proof: The proposition follows from the estimate Z R j b f(x)j 2 jxj Gamma2 dx = Z jxj1 j b f(x)j 2 jxj Gamma2 dx Z jxj 1 j b f(x)j 2 jxj Gamma2 dx kfk 2 L 1 (R) Z jxj1 ....

....a standard fBm. Then the RKHS H ( Gamma ) of B is H ( Gamma ) g : g(t) Gamma( 1) 2 c 1 ( 2 Z R g (u) I Gamma 1 [0;t) u)du; for some g 2 L 2 (R) 6.10) with the inner product (6.5) The result (6. 10) coincides with the characterization of Barton and Poor [2], who studied the case 0 1=2 and expressed H ( Gamma ) as H ( Gamma ) ae g : g(t) 1 Gamma( Z t 0 Z s Gamma1 (s Gamma u) Gamma1 e g(u)duds; for some e g 2 L 2 (R) oe : 6.11) Indeed, by using (3.21) and the fractional integration by parts formula (3.20) we can write ....

R. J. Barton and H. V. Poor. Signal detection in fractional Gaussian noise. IEEE Transactions on Information Theory, 34(5):943--955, 1988.

....to occur in a wide variety of physical processes such as [7] voltages and currents in semiconductors, resistances in electronic components, frequencies of quartz crystal oscillators and geophysical records. This 1=f noise is therefore of great interest in communications systems; see for example [1] for a discussion of the signal detection problem in 1=f noise. These noises also occur in situations where accurate modelling of them is important for controlling a process. Such examples include [7] rate of insulin uptake by diabetics, economic data, rate of Traffic flow and more recently it has ....

R. J. Barton and H. V. Poor, Signal detection in fractional Gaussian noise, IEEE Transactions on Information Theory, IT-34 (1988), pp. 943--959.

.... Brownian motion (fBm) and fractional Gaussian noise (fGn) 15, 14, 33, 38, 41, 5, 11] A large part of the impetus for such work has been the problem of dealing with so called 1=f stochastic processes which have become of growing importance to physicists and the signal processing community [24, 2, 28, 37, 42], and more recently to the control theory community [27] To be more specific, 1=f noise is the colloquial term given to a stochastic process fx k g whose sample spectral density, or periodogram 1 , jbx N ( j 2 is of the form jbx N ( j 2 This work was supported by the Australian ....

R. J. Barton and H. V. Poor, Signal detection in fractional Gaussian noise, IEEE Transactions on Information Theory, IT-34 (1988), pp. 943--959.

....H , where KH is the HilbertSchmidt operator introduced in Theorem [2.1] We hereafter identify an operator and its kernel. Stochastic Analysis of the fBm 7 Proof. For H 1=2, it is easy to see that RH (s; t) VH 4H(2H Gamma 1) Z t 0 Z s 0 jr Gamma uj 2H Gamma2 du dr Moreover(see [2]) VH 4H(2H Gamma 1) jr Gamma uj 2H Gamma2 = ru) H Gamma1=2 Z ru 0 v 1=2 GammaH (r Gamma v) H Gamma3=2 (u Gamma v) H Gamma3=2 dv: Hence for H 1=2, 8) holds with KH (t; r) r 1=2 GammaH Gamma(H Gamma 1=2) Z t r u H Gamma1=2 (u Gamma r) H Gamma3=2 du 1 [0;t] ....

R. J. Barton and H. Vincent Poor. Signal detection in fractional gaussian noise. IEEE trans. on information theory, 34(5):943959, September 1988.

....differentiable, and so on. Another useful, and easily proved, property is that the functions span ; that is, 126) Several other useful properties can be found in the literature; the reprint volume of Weinert [135] is a useful source for the papers up to 1982; for later work, see for example, [2] [4] The basic idea is that if we express as then (127) and in carrying this idea to the limit. In other words, as emphasized by Parzen, if a function can be represented in terms of linear operations on the family , including the operations of differentiation and integration, then belongs to ....

....generalizations of the RKHS to handle nonlinear operations of linear processes. Several new results can be obtained in this way, but the machinery to obtain these is quite heavy, and we may refer only to [24] Explicit RKHS results for self similar (or fractal) signals and noise are found in [2]. The RKHS has several other important applications, e.g. to the study of intersymbol interference (see Messerschmidt [70] robustness in signal estimation [3] and spline approximation (Kimeldorf and Wahba [61] Sidhu and Weinert [109] density estimation (see Wahba [131] performance ....

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R. J. Barton and H. V. Poor, "Signal detection in fractional Gaussian noise," IEEE Trans. Inform. Theory, vol. 34, pp. 943--959, Sept. 1988.

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Barton, R. J. and H. V. Poor: 1988, `Signal Detection in Fractional Gaussian Noise'. IEEE Transactions on Information Theory 34(5), 943-959.

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BARTON, R. J. and POOR, H. V. (1988). Signal detection in fractional Gaussian noise. IEEE Trans. Inform. Theory 34 943-959.

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