Power Spectrum of Generalized Fractional Gaussian Noise

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 𝜌(𝜏)()=𝜎=𝑟𝜏;𝐻₂2(|𝜏|+1)^2𝐻−2|𝜏|^2𝐻+||𝜏|−1|^2𝐻, (1) where 𝐻 is the Hurst parameter and 𝜎²=Γ(1−2𝐻)cos(𝐻𝜋)/𝐻𝜋. It implies three families of time series. In the case of 𝐻∈(0.5,1), 𝜌 is nonintegrable, and a corresponding series is LRD. For 𝐻∈(0,0.5), 𝑟 is integrable, and a corresponding series is short-range dependent (SRD). The case of 𝐻=0.5 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 𝐶(𝜏)=𝐶(𝜏;𝐻,𝛼)=0.5𝜎²|𝜏|^𝛼+1^2𝐻−2|𝜏|^𝛼2𝐻+|𝜏|^𝛼−1^2𝐻, (2) where 𝐻∈(0.5,1) and 𝛼∈(0,1]. We call a process whose ACF follows (2) GfGn for simplicity because it takes fGn as a special case of 𝐶(𝜏;𝐻,1)=𝜌(𝜏;𝐻). Without loss of generality, the following considers the normalized ACF by letting 𝑟(𝜏)=𝐶(𝜏)/𝜎². This paper aims at giving PSD of GfGn. The Fourier transform (FT) of 𝑟(𝜏) is treated as a generalized function over Schwartz space of test functions since 𝑟(𝜏) is nonintegrable.

2. PSD of GfGn

Denote 𝑟(𝜏)=0.5𝑟₁(𝜏)−2𝑟₂(𝜏)+𝑟₃(𝜏), (3) where 𝑟₁=(|𝜏|^𝑎+1)^2𝐻, and 𝑟₂=(|𝜏|^𝛼)^2𝐻, 𝑟₃=||𝜏|^𝑎−1|^2𝐻. Denote 𝑆_𝑚(𝜔)=𝐹(𝑟_𝑚), where 𝐹 means FT and 𝑚=1,2,3. Then, FT of 𝑟(𝜏) is given by 𝑆(𝜔)=0.5𝑆₁(𝜔)−2𝑆₂(𝜔)+𝑆₃(𝜔). (4)

Lemma 1 (see [7] or Gelfand and Vilenkin [8, Chapter 2]). FT of |𝑡|^𝜆 is expressed by 𝐹|t|^𝜆=−sin𝜆𝜋2Γ(𝜆+1)|𝜔|^−𝜆−1, (5) where 𝜆≠−1,−3,….

Corollary 2. 𝑆₂(𝜔) equals −sin(𝐻𝛼𝜋)Γ(2𝐻𝛼+1)|𝜔|^{−2𝐻𝛼−1}.

Proof. Note 2𝐻𝛼≠−1,−3,…. Thus, doing 𝐹(|𝜏|^2𝐻𝛼) with (5) yields Corollary 2.

Lemma 3 (binomial series). (1+𝑥)^𝜈 and (1−𝑥)^𝜈 can be expanded as (1+𝑥)^𝜈=^∞_𝑘=0𝜈𝑘𝑥^𝑘=^∞_𝑘=0Γ(𝜈+𝑘)Γ(𝜈)Γ()𝑥1+𝑘^𝑘𝑓𝑜𝑟|𝑥|<1,(1−𝑥)^𝜈=^∞_𝑘=0𝜈𝑘(−𝑥)^𝑘=^∞_𝑘=0(−1)^𝑘Γ(𝜈+𝑘)Γ(𝜈)Γ()𝑥1+𝑘^𝑘𝑓𝑜𝑟|𝑥|<1, (6a)(6b) where 𝑥 and 𝜈 are real number, and (^𝜈_𝑘) is binomial coefficient [9].

Corollary 4. 𝑟₁(𝜏) and 𝑟₃(𝜏) for |𝜏|<1 can be expanded as 1+|𝜏|^𝛼2𝐻=^∞_𝑘=0𝑘|𝜏|2𝐻_𝛼𝑘=^∞_𝑘=0Γ(2𝐻+𝑘)Γ()Γ()|𝜏|2𝐻1+𝑘^𝛼𝑘,1−|𝜏|^𝛼2𝐻=^∞_𝑘=0(−1)^𝑘𝑘|𝜏|2𝐻_𝛼𝑘=^∞_𝑘=0(−1)^𝑘Γ(2𝐻+𝑘)Γ()Γ()|𝜏|2𝐻1+𝑘^𝛼𝑘. (7a)(7b)

Proof. This corollary is straightforward from Lemma 3.

Corollary 5. For |𝜏|^𝛼>1, 𝑟₁(𝜏) and 𝑟₃(𝜏) can be expanded as 1+|𝜏|^𝛼2𝐻=|𝜏|^∞2𝐻𝛼_𝑘=0𝑘|𝜏|2𝐻_−𝛼𝑘=^∞_𝑘=0Γ(2𝐻+𝑘)Γ()Γ()|𝜏|2𝐻1+𝑘^{𝛼(2𝐻−𝑘)},|𝜏|^𝑎−1^2𝐻=|𝜏|^∞2𝐻𝛼_𝑘=0(−1)^𝑘𝑘|𝜏|2𝐻_−𝛼𝑘=^∞_𝑘=0(−1)^𝑘Γ(2𝐻+𝑘)Γ()Γ()|𝜏|2𝐻1+𝑘^{𝛼(2𝐻−𝑘)}. (8a)(8b)

Proof. Since (1+|𝜏|^𝛼)^2𝐻=|𝜏|^2𝐻𝛼(1+|𝜏|^−𝛼)^2𝐻, according to (7a), (8a) results. Similarly, (8b) follows due to ||𝜏|^𝑎−1|^2𝐻=|𝜏|^2𝐻𝛼|1−|𝜏|^−𝛼|^2𝐻 and (7b).

Corollary 6. For |𝜏|<1,𝑆₁ and 𝑆₃ are given by (6), respectively, 𝑆₁(𝜔)=^∞_𝑘=0−Γ(2𝐻+𝑘)Γ(𝛼𝑘+1)Γ(2𝐻)Γ(1+𝑘)×sin𝛼𝑘𝜋2|𝜔|^{−𝛼𝑘−1},𝑆₃(𝜔)=^∞_𝑘=0(−1)^𝑘+1Γ(2𝐻+𝑘)Γ(𝛼𝑘+1)Γ(2𝐻)Γ(1+𝑘)×sin𝛼𝑘𝜋2|𝜔|^{−𝛼𝑘−1}. (6)

Proof. Doing 𝐹[|𝜏|^𝛼𝑘] term by term for (7a) and (7b) with Lemma 1 yields (6), respectively.

Corollary 7. For |𝜏|^𝛼>1,𝑆₁ and 𝑆₃ are given by (7), respectively, 𝑆₁(𝜔)=^∞_𝑘=0−Γ(2𝐻+𝑘)Γ[(𝛼(2𝐻−𝑘)+1]Γ()Γ()𝛼()𝜋2𝐻1+𝑘×sin2𝐻−𝑘2|𝜔|^{−𝛼(2𝐻−𝑘)−1},𝑆₃(𝜔)=^∞_𝑘=0(−1)^𝑘+1Γ(2𝐻+𝑘)Γ[𝛼(2𝐻−𝑘)+1]Γ()Γ()𝛼()𝜋2𝐻1+𝑘×sin2𝐻−𝑘2|𝜔|^{−𝛼(2𝐻−𝑘)−1}. (7)

Proof. Doing FTs for (8a) and (8b) 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 𝑆(𝜔)=sin(𝐻𝛼𝜋)Γ(2𝐻𝛼+1)|𝜔|^{−2𝐻𝛼−1}+0.5∑^∞_𝑘=0(−1)^𝑘+1−1Γ(2𝐻+𝑘)Γ(𝛼𝑘+1)Γ(2𝐻)Γ(1+𝑘)×sin𝛼𝑘𝜋2|𝜔|^{−𝛼𝑘−1},|𝜏|()Γ(𝜔|<1,sin𝐻𝛼𝜋2𝐻𝛼+1)|^{−2𝐻𝛼−1}+0.5∑^∞_𝑘=0(−1)^𝑘−1Γ(2𝐻+𝑘)Γ[𝛼(2𝐻−𝑘)+1]Γ()Γ()𝛼()𝜋2𝐻1+𝑘×sin2𝐻−𝑘2|𝜔|^{−𝛼(2𝐻−𝑘)−1},|𝜏|^𝛼>1. (8)

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

Proposition 9. PSD of GfGn has the following approximate value: 𝑆(𝜔)≈sin(𝐻𝛼𝜋)Γ(2𝐻𝛼+1)|𝜔|^{−2𝐻𝛼−1}. (9)

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

Note 1. 𝑆(𝜔) is divergent at the origin for 0<2𝐻𝛼+1<1, which is the LRD condition. This is the basic feature of LRD process.

Note 2. Recall 2𝐻𝛼+1>0. Then, the cases of 2𝐻𝛼+1<1 and 𝐻∈(0.5,1) imply 𝛼∈(0,1]. This explains the range of 𝛼 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 𝛼 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.