Power Spectrum of Generalized Fractional Gaussian Noise

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