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Hartley Transform and the Use of the Whitened Hartley Spectrum As A Hartley transform and the use of the Whitened Hartley spectrum as a tool for phase spectral processing Paraskevas, I, Barbarosou, M and Chilton, E http://dx.doi.org/10.1049/joe.2014.0350 Title Hartley transform and the use of the Whitened Hartley spectrum as a tool for phase spectral processing Authors Paraskevas, I, Barbarosou, M and Chilton, E Type Article URL This version is available at: http://usir.salford.ac.uk/id/eprint/50476/ Published Date 2015 USIR is a digital collection of the research output of the University of Salford. Where copyright permits, full text material held in the repository is made freely available online and can be read, downloaded and copied for non-commercial private study or research purposes. Please check the manuscript for any further copyright restrictions. For more information, including our policy and submission procedure, please contact the Repository Team at: [email protected]. Hartley transform and the use of the Whitened Hartley spectrum as a tool for phase spectral processing Ioannis Paraskevas1, Maria Barbarosou2, Edward Chilton3 1Academic Group of Engineering, Sports and Sciences, Centre for Advanced Performance Engineering (CAPE), University of Bolton, Bolton BL3 5AB, Lancashire, UK 2Department of Electronics, Electric Power, Telecommunications, Hellenic Air Force Academy, Dekelia Air Base, Tatoi 13671, Greece 3Faculty of Engineering and Physical Sciences, Centre for Vision, Speech and Signal Processing (CVSSP), University of Surrey, Guildford Surrey GU2 7XH, UK E-mail: [email protected] Published in The Journal of Engineering; Received on 28th December 2014; Accepted on 9th February 2015 Abstract: The Hartley transform is a mathematical transformation which is closely related to the better known Fourier transform. The prop- erties that differentiate the Hartley Transform from its Fourier counterpart are that the forward and the inverse transforms are identical and also that the Hartley transform of a real signal is a real function of frequency. The Whitened Hartley spectrum, which stems from the Hartley trans- form, is a bounded function that encapsulates the phase content of a signal. The Whitened Hartley spectrum, unlike the Fourier phase spectrum, is a function that does not suffer from discontinuities or wrapping ambiguities. An overview on how the Whitened Hartley spectrum encap- sulates the phase content of a signal more efficiently compared with its Fourier counterpart as well as the reason that phase unwrapping is not necessary for the Whitened Hartley spectrum, are provided in this study. Moreover, in this study, the product–convolution relationship, the time-shift property and the power spectral density function of the Hartley transform are presented. Finally, a short-time analysis of the Whitened Hartley spectrum as well as the considerations related to the estimation of the phase spectral content of a signal via the Hartley transform, are elaborated. 1 Introduction audio (gunshot) classification [19, 22], speech (phoneme) classifi- cation [23–25] and as a noise robust feature for signal analysis The Hartley transform, a close relative of the better known Fourier [26]. Please note that the terms ‘Whitened Hartley spectrum’ and transform, was first introduced in 1942 [1]. Its appealing properties ‘HPS’ are used in this paper interchangeably since both of them of symmetry (its forward and inverse transforms are identical) and convey the same meaning. Furthermore, the Hartley phase cep- that its transform of a real signal is also a real function of frequency, strum, which stems from the HPS, has been applied in: signal local- were seen, at that time, as having useful applications in the area of isation [27, 28], detection of transient events for power quality [29] communications theory [2]. Little was heard of this transform until and as a tool for improved phase spectral estimation [30, 31]. Bracewell published an account of the discrete Hartley transform This paper aims to provide a theoretical overview of the Hartley followed shortly by another on the fast Hartley transform. transform, to present its similarities with the Fourier transform as Bracewell observed that the real spectrum derived via the Hartley well as its attractive properties compared with its Fourier counter- transform from a real signal, contained phase information (as well part. Specifically, in Section 2 the Hartley transform as well as as magnitude information) and showed that analogue phase meas- the complementary Hartley transform are defined and its relation- urement was possible with suitable laboratory apparatus [3–7]. ship with the Fourier transform is stated. In the same section, the Published work related to the Hartley transform in the area of product–convolution and the time-shift properties of the Hartley signal processing can be found in [8–12]; the Hartley transform transform are also explained. In Section 3, the ‘Whitened Hartley has also found application in diverse areas such as geophysics spectrum’ is defined and its properties compared with the Fourier [13, 14], electrical power engineering [15] and pattern recognition phase spectrum are presented. In Section 4, the short-time analysis [16–18]. of the HPS is described, and finally in Section 5 the time-delay of a A real signal could be exactly represented, via the short-time dis- signal is evaluated based on the Whitened Hartley spectrum. crete Hartley transform, by two separate frequency domain func- tions, both real. One, the magnitude spectrum which is identical to that derived via the Fourier transform and represents the square- root of the power spectral density function of the signal, whereas the 2 Properties of the Hartley transform second function, rather clumsily called the ‘Whitened Hartley spec- 2.1 Some fundamental definitions trum’, is a function of phase only [19]; the term ‘whitened’ has been used since the derivation of the ‘Whitened Hartley spectrum’ is the The Hartley transform is an orthogonal transform with cosinusoidal result of the ‘whitening’ process [20]. This latter function, unlike its basis functions. It is a close relative of the widely used Fourier fi Fourier counterpart, is bounded and does not suffer from wrapping transform which is de ned as ambiguities thus avoiding the difficulties introduced by the discon- tinuities in the discrete phase spectrum when this is derived via the 1 = = −j2pft Fourier transform [21]. Thus, the ‘Whitened Hartley spectrum’ or Fourier Transform: F[s(t)] Fs(f ) s(t)e dt (1a) ‘Hartley Phase Spectrum (HPS)’ encapsulates the phase content −1 of the signal more efficiently, compared with its Fourier counter- part. Moreover, the HPS has already found useful application in: where s(t) is a continuous function. The inverse Fourier transform is J Eng, 2015, Vol. 2015, Iss. 3, pp. 95–101 This is an open access article published by the IET under the Creative Commons doi: 10.1049/joe.2014.0350 Attribution License (http://creativecommons.org/licenses/by/3.0/) given by the relation 1 = jvt v s(t) Fs(f )e d (1b) −1 Fig. 1 Self-inverse property of Hartley transform Alternatively, (1a) and (1b) can be expressed as presented in Fig. 3, whereas the Fourier transform of s(t)is 1 = v = √1 −jvt F[s(t)] Fs( ) s(t)e dt (2a) j1.5v 2p −1 e F(s(t)) = 1 + jv and which is a complex function of the angular frequency ω. 1 fi = √1 v jvt v Furthermore, it can be easily shown, using de nitions (2a) and s(t) Fs( )e d (2b) ’ 2p −1 (3a) and the Euler s formula, that the Hartley transform can be expressed in terms of the Fourier transform. Indeed, in case the ω The kernel function of the Fourier transform is the complex expo- signal s(t) is real, the Hartley transform is the real part SR( ) −jωt ω nential, e , whereas the kernel function of the Hartley transform minus the imaginary part SI( ) of the Fourier transform, that is is the cas(ωt) function. The cas function was introduced by Hartley fi ω ω ω = v − v in 1942 and is de ned as cas( t) = cos( t) + sin( t). Hence, the H[s(t)] SR( ) SI( ) (5a) Hartley transform of a function s(t)isdefined as while the real and the imaginary parts of the Fourier transform is the = v Hartley Transform: H[s(t)] Hs( ) even and the negative odd components of the Hartley transform, re- 1 spectively, that is 1 (3a) = √ s(t)( cos vt + sin vt)dt 2p −1 H(v) + H( − v) H(v) − H( − v) F[s(t)] = − j (5b) 2 2 where ω is the angular frequency in radians/second. Equivalently, fi the Hartley transform can be de ned using the linear frequency f Moreover, it is useful to observe the operation of the Hartley and the ω (units:√ 1/s) instead of the angular frequency . In this case, the Fourier transforms on a complex signal. Let s(t)=x(t)+jy(t), then / p fi 1 2 coef cient is omitted, that is directly from the linearity property we have = Hartley Transform: H[s(t)] Hs(f ) H[s(t)] = H[x(t)] + jH[y(t)] (6) 1 (4a) = p + p s(t)( cos 2 ft sin 2 ft)dt Thus, the Hartley transform of a complex signal is a complex func- −1 tion where the real and the imaginary components of the signal in Throughout this paper we use the definition in (3a), which means the time-domain map uniquely onto its real and imaginary compo- nents in the frequency domain. This may be compared with the case that H[s(t)] denotes Hs(ω), unless stated otherwise. The scaling and linearity properties of the Hartley transform are of the Fourier transform, where identical to those of the Fourier transform. However, the Hartley transform has two other useful properties that distinguish it from F[s(t)] = F[x(t)] + jF[y(t)] (7) its Fourier counterpart.
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