- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Advances in Mathematical Physics

Volume 2013 (2013), Article ID 315979, 3 pages

http://dx.doi.org/10.1155/2013/315979

## Power Spectrum of Generalized Fractional Gaussian Noise

School of Information Science & Technology, East China Normal University, No. 500 Dong-Chuan Road, Shanghai 200241, China

Received 29 August 2013; Accepted 10 September 2013

Academic Editor: Wen-Sheng Chen

Copyright © 2013 Ming Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Recently, we introduced a type of autocorrelation function (ACF) to describe a long-range dependent (LRD) process indexed with two parameters, which takes standard fractional Gaussian noise (fGn for short) as a special case. For simplicity, we call it the generalized fGn (GfGn). This short paper gives the power spectrum density function (PSD) of GfGn.

#### 1. Introduction

LRD time series increasingly gains applications to many fields of science and technologies; see, for example, Mandelbrot [1] and references therein. In this regard, standard fGn introduced by Mandelbrot and van Ness is a widely used tool for modeling LRD time series; see, for example, Beran [2], Abuzeid et al. [3, 4], and Liao et al. [5]. Following [1, H11], [2], its ACF is given by where is the Hurst parameter and . It implies three families of time series. In the case of , is nonintegrable, and a corresponding series is LRD. For , is integrable, and a corresponding series is short-range dependent (SRD). The case of corresponds to white noise. Note that statistics of LRD series substantially differ from SRD ones. From a practice view, SRD fGn may be less interesting in applications as can be seen from [1, 2]. This paper only considers LRD series unless otherwise stated.

Li [6] recently introduced an ACF form that is a generalization of ACF of fGn. Since ACF is an even function, we write ACF of GfGn by where and . We call a process whose ACF follows (2) GfGn for simplicity because it takes fGn as a special case of . Without loss of generality, the following considers the normalized ACF by letting . This paper aims at giving PSD of GfGn. The Fourier transform (FT) of is treated as a generalized function over Schwartz space of test functions since is nonintegrable.

#### 2. PSD of GfGn

Denote where , and , . Denote , where means FT and . Then, FT of is given by

Lemma 1 (see [7] or Gelfand and Vilenkin [8, Chapter 2]). * FT of is expressed by
**
where .*

Corollary 2. * equals .*

*Proof. *Note . Thus, doing with (5) yields Corollary 2.

Lemma 3 (binomial series). * and can be expanded as
**
where and are real number, and is binomial coefficient [9].*

Corollary 4. * and for can be expanded as
*

*Proof. * This corollary is straightforward from Lemma 3.

Corollary 5. * For , and can be expanded as
*

*Proof. * Since , according to , results. Similarly, follows due to and .

Corollary 6. * For and are given by (6), respectively,
*

*Proof. * Doing term by term for and with Lemma 1 yields (6), respectively.

Corollary 7. * For and are given by (7), respectively,
*

*Proof. * Doing FTs for and based on Lemma 1 results in (7).

The following proposition is a consequence of Corollaries 2, 6, and 7.

Proposition 8. * PSD of GfGn is given by
*

Considering the leading term of (8) results in the following proposition.

Proposition 9. *PSD of GfGn has the following approximate value:
*

From (9), we can easily get the two notes below.

*Note 1. * is divergent at the origin for , which is the LRD condition. This is the basic feature of LRD process.

*Note 2. * Recall . Then, the cases of and imply . This explains the range of for GfGn from a view in the frequency domain.

#### 3. Conclusions

We have derived PSD of GfGn. Its approximate expression has been given. The range of has been explained from a spectral view.

#### Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under the Project Grant nos. 61272402, 61070214, and 60873264, and by the 973 Plan under the Project Grant no. 2011CB302800.

#### References

- B. B. Mandelbrot,
*Gaussian Self-Affinity and Fractals*, Springer, 2001. - J. Beran,
*Statistics For Long-Memory Processes*, Chapman & Hall, New York, NY, USA, 1994. - O. M. Abuzeid, A. N. Al-Rabadi, and H. S. Alkhaldi, “Recent advancements in fractal geometric-based nonlinear time series solutions to the micro-quasistatic thermoviscoelastic creep for rough surfaces in contact,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 691270, 29 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus - O. M. Abuzeid, A. N. Al-Rabadi, and H. S. Alkhaldi, “Fractal geometry-based hypergeometric time series solution to the hereditary thermal creep model for the contact of rough surfaces using the Kelvin-Voigt medium,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 652306, 22 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus - Z. Liao, S. Hu, M. Li, and W. Chen, “Noise estimation for single-slice sinogram of low-dose X-ray computed tomography using homogenous patch,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 696212, 16 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus - M. Li, “Modeling autocorrelation functions of long-range dependent teletraffic series based on optimal approximation in Hilbert space-A further study,”
*Applied Mathematical Modelling*, vol. 31, no. 3, pp. 625–631, 2007. View at Publisher · View at Google Scholar · View at Scopus - S. Samko, A. A. Kilbas, and D. I. Maritchev,
*Integrals and Derivatives of the Fractional Order and Some of Their Applications*, Gordon and Breach, Armsterdam, The Netherlands, 1993. - I. M. Gelfand and K. Vilenkin,
*Generalized Functions*, vol. 1, Academic Press, New York, NY, USA, 1964. - G. Arfken,
*Mathematical Methods For Physicists*, Academic Press, Orlando, Fla, USA, 3rd edition, 1985.