Methods for detection and characterization of signals in noisy data with the Hilbert-Huang Transform

Methods for detection and characterization of signals in noisy data with the Hilbert-Huang Transform Alexander Stroeer,∗John K. Cannizzo,† and Jordan B. Camp Laboratory for Gravitational Physics, Goddard Space Flight Center, Greenbelt, Maryland 20771 Nicolas Gagarin Starodub, Inc., 3504 Littledale Road, Kensington, MD, 20895 (Dated: October 29, 2018) The Hilbert-Huang Transform is a novel, adaptive approach to time series analysis that does not make assumptions about the data form. Its adaptive, local character allows the decomposi- tion of non-stationary signals with hightime-frequency resolution but also renders it susceptible to degradation from noise. We show that complementing the HHT with techniques such as zero-phase ltering, kernel density estimation and Fourier analysis allows it to be used eectively to detect and characterize signals with low signal to noise ratio. I. INTRODUCTION The Hilbert-Huang Transform (HHT) [1, 2] is a novel data analysis algorithm that adaptively decomposes time series data and derives the instantaneous amplitude (IA) and instantaneous frequency (IF) of oscillating signals. Because this transform operates locally on the data, and not as an integral in time over pre-selected basis functions, it can eectively decompose non-linear, non- stationary signals, and it is not limited by time-frequency uncertainty. Applications of the HHT include monitoring of heart rates[3], integrity of structures [4], and searching for gravitational waves [5]. The HHT proceeds in two steps [2]. The rst part of the algorithm, the empirical mode decomposition (EMD), decomposes the data into intrinsic mode func- tions (IMF), each representing a locally monochromatic frequency scale of the data, with the original data recov- ered by summing over all IMFs. EMD involves forming an envelope about the data maxima and minima with the use of a cubic spline, then taking the average of the two envelopes, and subtracting that from the time series to obtain the residual. An iteration of this procedure con- verges to an IMF, after which it is subtracted from the time series, and the procedure begins again. The second part applies the Hilbert transform to each individual IMF to construct an analytical complex time series represen- tation. The instantaneous frequency of the original IMF is obtained by taking the derivative of the argument of the complex time series, and the instantaneous amplitude by taking the magnitude. Many applications of the HHT to date have in- volved the decomposition of complicated mixings of non- stationary features, which may also be frequency modu- ∗Electronic address: Alexander.Stroeer@nasa.gov; Also at CRESST, Department of Astronomy, University of Maryland, Col- lege Park, Maryland 20742 †Also at CRESST, Physics Department, University of Maryland, Baltimore County, Baltimore, Maryland 21250 lated, but these generally have not been limited by low signal strength relative to the noise background. A dier- ent class of problems involves signal detection and charac- terization at low signal to noise ratio (SNR). The SNR of a signal h, as recorded discretely according to a sampling frequency with the individual time instances denoted by the subscript i, in white noise with standard deviation σn is dened as (matched lter denition): SNR = sX i h2 i /σn (1) An interesting question is the eectiveness of the HHT decomposition for low SNR. This is inuenced by what we describe as intrinsic and extrinsic eects. Intrinsic un- certainties are evident in the presence of noise within the bandwidth of the actual signal, so that the true wave- form of the signal is never visible to the data analysis method. Extrinsic uncertainties are induced by the data analysis algorithm in the form of errors in the processing of the data stream due to noise either inside or outside the signal bandwidth, leading to envelope undershoot or overshoot, with the error possibly magnied by the EMD iterations. Additional extrinsic uncertainties can be in- troduced in the application of the Hilbert transform if the IMF is not perfectly locally monochromatic, or due to limitations described in Bedrosian and Nuttal theorems [6]; or in the determination of the IF, as the numerical derivative of the instantaneous phase may be subject to uncertainties and error propagation. The length of the signal is also an important consid- eration in the accuracy of the HHT decomposition. The local character of the HHT implies a direct sensitivity of the decomposition to the local signal amplitude rela- tive to the noise (IA/σn). For a given SNR, the signal amplitude relative to the noise increases as the signal be- comes shorter in time (see Eq. 1). Thus shorter signals at a given SNR will be less subject to uncertainties, and more easily detected. We consider in this paper methods for enhancing the HHT performance in detecting and characterizing sig- arXiv:0903.4616v1 [physics.data-an] 26 Mar 2009 2 nals at low SNR (<20), and w

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