The AFD methods to compute Hilbert transform

In the literature adaptive Fourier decomposition is abbreviated as AFD that addresses adaptive rational approximation, or alternatively adaptive Takenaka-Malmquist system approximation. The AFD type approximations may be characterized as adaptive approximations by linear combinations of parameterized Szegö and higher order Szegö kernels. This note proposes two kinds of such analytic approximations of which one is called maximal-energy AFDs, including core AFD, Unwending AFD and Cyclic AFD; and the other is again linear combinations of Szegö kernels but generated through SVM methods. The proposed methods are based on the fact that the imaginary part of an analytic signal is the Hilbert transform of its real part. As consequence, when a sequence of rational analytic functions approximates an analytic signal, then the real parts and the imaginary parts of the functions in the sequence approximate, respectively, the original real-valued signals and its Hilbert transform. The two approximations have the same errors in the energy sense due to the fact that Hilbert transformation is an unitary operator in the L space. This paper for the first time promotes the complex analytic method for computing Hilbert transforms. Experiments show that such computational methods are as effective as the commonly used one based on FFT. Index Terms Hilbert Transform, Hardy Space, Takenaka-Mulmquist System, Orthogonal Rational System, Adaptive Fourier Decomposition I. PREPARATION Computation of Hilbert transform is a difficult task due to the singularity of the complex Cauchy kernel at the origin (see for instance [17] and [8], and the references thereafter). In this paper we introduce a set of AFD type analytic methods to compute Hilbert transform. As abbreviation of adaptive Fourier decomposition, AFD, in general, addresses all types of adaptive approximations in the form of finite linear combinations of Szegö and higher order Szegö kernels. In this paper we deal with two kinds of AFDs. One can be phrased as Maximal-Energy-AFD that, in its core algorithm part, is based on a maximal selection principle to adaptively and optimally select the relevant parameters. Such AFD method initiated in [12] and further developed in [13], [9], [10]. The main variations of this kind include Core AFD, Unwending AFD and Cyclic AFD. They have been found to have effective applications in system identification, as well as in signal analysis ([5], [6], [7]). The algorithm of Core AFD is studied in [14]. The other kind of AFD concerned in this paper is a complex type SVM, phrased as SVM-AFD. In this paper we also call it “complex SVM” although it is not just by using complex numbers, but complex analytic functions, and especially Szegö kernels. The complex SVM is also found to have effective applications in system identification ([3], [4]). The relevant algorithm codes are found in the web address http://www.fst.umac.mo/en/staff/fsttq.html. The importance of Hilbert transformation lies in the facts that with Cs(z) = 1 2πi ∫ ∞ −∞ s(t) t− z dt, the Cauchy integral of s, one has lim y→0+ Cs(x+ iy) = 1 2πi ∫ ∞ −∞ s(t) t− (x+ iy) dt = s(t), where s(t) = 1 2 s(t) + 1 2 iHs(t) is the analytic signal associated with the given real-valued signal s of finite energy, and Hs is its Hilbert transform. This result is known as the Plemelj formula ([1]). If s(t) is a given real-valued signal, then we have the easy relations s = 2Res, Hs = 2Ims. Yan Mo, Tao Qian and Wei-Xiong Mai are with the Department of Mathematics, University of Macau, Macao. e-mails: [email protected]., [email protected]., [email protected]. Qiuhui Chen is with the Cisco School of Informatics, Guangdong University of Foreign Studies. e-mail: [email protected]. The study was supported by University of Macau MYRG116(Y1-L3)-FST13-QT and Macao Government FDCT 098/2012/A3