1 An Investigation of Impropriety and Noncircularity in High Frequency Radar Data Khalid El-Darymli ∗ , Member, IEEE, Wei Wang † , Member, IEEE, Eric W. Gill ‡ , Senior Member, IEEE, Weimin Huang § , Senior Member, IEEE, and Barry Dawe ¶ Abstract—Discarding the phase content of signals from single- channel high-frequency (HF) radar is commonplace among practitioners in the field. From the perspective of complex- valued statistics, this practice implicitly implies that the complex- valued HF data is second-order proper or circular. This paper presents a preliminary investigation into the validity of this assumption using HF surface wave radar (HF-SWR) field data. It is found that the HF-SWR data is indeed second-order improper or noncircular. This negates the common belief regarding the non-informativness of the phase, and it warrants more in- depth analysis into characterizing impropriety/noncircularity for relevant HF radar applications. Index Terms—HF radar, phase, impropriety, noncircularity, nonstationarity I. I NTRODUCTION D ISCARDING the phase content of signals from single- channel coherent radar, such as high-frequency (HF) radar, is commonplace in the radar community. This practice is justified by conventional radar resolution theory, strictly relevant to point targets, and based on various simplifying as- sumptions such as linearity and stationarity [1, 2]. In previous investigations, the insufficiency for one of these assumptions - i.e., linearity -, which leads to the use of the second-order perturbation theory, has been demonstrated [3–5]. This paper presents a preliminary investigation into the validity of another often implicit assumption that leads to the discarding of the phase. From the perspective of complex- valued statistics, the practice of discarding the phase may be justified by assuming that the underlying random variables are second-order proper or circular in nature [6–8]. Propriety implies that the complex-valued HF data is uncorrelated with its complex-conjugate. Circularity means that the complex- valued HF data has a probability density function (PDF) that ∗ Dr. Khalid El-Darymli is a senior research engineer with Northern Radar Inc., 25 Anderson Av, St. John’s, NL, Canada, A1B 3E4, e-mail: k.el- darymli@northernradar.com. † Dr. Wei Wang is a post-doctoral researcher with the Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL, Canada, A1B 3X5, e-mail: weiw@mun.ca ‡ Dr. Eric W. Gill is a professor in the Dept. of Electrical and Computer En- gineering, Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL, Canada, A1B 3X5, e-mail: ewgill@mun.ca § Dr. Weimin Huang is an associate professor in the Dept. of Electrical and Computer Engineering, Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL, Canada, A1B 3X5, e-mail: weimin@mun.ca ¶ Mr. Barry Dawe is the vice president, major projects, with Northern Radar Inc., 25 Anderson Av, St. John’s, NL, Canada, A1B 3E4, e-mail: bdawe@northernradar.com is invariant under rotation in the complex plane. Accord- ingly, discarding the phase content implicitly implies that the aforementioned assumptions are satisfied. Otherwise, valuable information about the targets in the complex-valued data is lost. This information can be of significant importance for various HF radar applications. In terms of physical properties, a necessary but insufficient condition for impropriety or non- circularity is that the random process be at least nonstationary [9]. This means that if the impropriety/noncircularity of the HF data can be estimated, one can characterize the nonstationrity of the underlying oceanic/ionospheric processes from which the HF signals are backscattered. The remainder of this paper is organized as follows. In Section II, the formal definitions of propriety/impropriety and circularity/noncircularity are presented. In Section III, the HF datasets utilized in this paper are described. In Section IV, the analysis procedures proposed in this paper are outlined. Results and discussion are given in Section V, and concluding remarks appear in Section VI. II. DEFINITIONS OF PROPRIETY/I MPROPRIETY AND CIRCULARITY/NONCIRCULARITY Formally, a zero-mean complex-valued random variable (i.e., X = X R + jX I ) is said to be second-order proper when its pseudo-covariance is zero [6]; i.e., when Ψ= E  X 2  =0. (1) For a random vector X, propriety implies [6, 7] σ X R = σ X I , and E {X R X I } =0, (2) E  X R X T R  = E  X I X T I  , (3) and, E  X R X T I  = -E  X I X T R  , (4) where σ X R and σ X I are the standard deviations of the real and imaginary parts of X, respectively; X R and X I are the real and imaginary parts of X, respectively; E {.} is the expectation; and T denotes the transpose. A stronger condition for propriety is based on the PDF of the random variable. A complex-random variable X is termed circular if X and X exp (jθ) have the same PDF (i.e., the PDF is rotation invariant) [8]. This means that the phase of the 978-1-4673-9724-7/16/$31.00 ©2016 IEEE