JOINT TIME FREQUENCY DOMAIN REFLECTOMETRY AND STATIONARITY INDEX FOR WIRE DIAGNOSTICS IN AIRCRAFT
This paper proposes a new technique for wire diagnostics based on joint time frequency analysis. The Continuous Wavelet Transform (CWT) is used to generate time frequency maps while a staionarity index is used to interpret such time frequency representation. Signals are generated using a high bandwidth Time Domain Reflectometry (TDR) device. The stationarity index is used as a direct indicator of defects in the wire under test. The index stays low if there is no defect but jumps suddenly if a defect is encountered. The detection method consists of comparing measures of the same global Time Frequency Representation (TFR) taken at different instants in the time frequency plan, where time is equated with distance. The global TFR is computed over the entire frequency span of the signature under test. The proposed approach has been applied to a database of wire signatures collected via the Goodrich Wire Integrity Tool (GWIT) a hand held TDR of 200 Pico second rise time and 2.5 GHZ bandwidth. Three different locations are used to collect data subject of this study. Data is collected at the Goodrich engineering lab in Vermont, at Sandia Lab in New Mexico, and over a Boeing 737 airplane at the FAA Tech Center in New Jersey. The detection algorithms are coded and implemented in the Goodrich hand held TDR. The unique input to the system is the TDR waveform; no other properties of the wire are required including wire type, gauge, or length. 1. Background Wire diagnostics has become a vital maintenance component to improve aircraft safety and readiness. Recent inspection of aging aircraft has uncovered a major role played by aircraft wiring systems in aircraft downtime. Some aircraft catastrophic incidents, such as the Swiss Air 111 in 1998, have been directly linked to wire problems. As such, an efficient solution for wire diagnostic is highly desirable. The FAA has been steadily encouraging research in this area. This study is funded by the FAA and is undertaken by Goodrich in response to a Broad Agency Announcement (BAA) to advance the Development of Electrical Wiring Interconnect System Test and Inspection Systems. Single wires are, by far, the most challenging type of wires for diagnosis mainly because of the uncontrolled distance between the wire and its return path. Distance between the wire under test and its return path affects the wire impedance, particularly when operating at high frequency. The distance between the wire and its return path in the case of single wires is expected to vary from one point over the wire to another given the anticipated bending and movement of wires. This is what makes single wires hard to diagnose. A system capable of adequately diagnosing single wires is expected to perform even Page 1 of 15 Sadok et al. 9 Joint FAA/DoD/NASA Aging Aircraft Conference better in the case of other types of wires. A specific study to quantify the effect of distance variation on the TDR based diagnosis of single wires has been submitted for publication elsewhere [1] Current techniques for wire inspection are based on the concept of Reflectometry [2 3] One of the big advantages of Reflectometry based techniques is that it requires connection to only one end of the wire. Most common Reflectometry techniques are either based on time domain (Time Domain Reflectometry, TDR) or on frequency (Frequency Domain Reflectometry, FDR) This paper attempts to take advantages of both domains and proposes detection techniques based on joint Time Frequency Domain Reflectometry (TFDR) 2. Time Frequency Domain Algorithms Wavelets are used to provide time frequency maps for TFDR analysis. Hilbert Huang Transform is another time frequency technique that was studied during this project. Finding of such a study is published in this conference as well [4] Wavelet transform is computed by projecting the signal of interest into a particular basis that is formed by dilating and translating a unique function commonly termed as the mother wavelet. Wavelet analysis depends heavily on the mother wavelet function. Different mother wavelets are appropriate for different applications. The mother wavelet is typically chosen such that the associated digital filter has a small order (to speed up the convolution) and such that the resulting analysis yields only few nonzero coefficients. This is criterion is mostly relevant in data compression. For the application at hand, the interest is in selecting a mother wavelet that resembles most embedded features in the TDR signature. Several types of wavelets such as Daubechies, Meyer, Morlet, Mexican hat, Symlets, and Biorthogonal wavelets have been tested. The Mexican hat wavelet is found to be most effective in extracting defectrelated features in TDR signatures; most likely because of the Gaussian nature of this wavelet function that resembles several high frequency natural artifacts. Implementing and testing the Mexican hat based wavelet transform constitute the first major part of the TFDR algorithms. The second part of the TFDR algorithms addresses the interpretation of the wavelet based time frequency representation. A Stationarity Index (SI) is used as a direct indicator of defects in the wire under test [5] The index stays very low if there is no defect but jumps suddenly if a defect is encountered. The detection method consists of comparing measures of the same global Time Frequency Representation (TFR) taken at different instants in the time frequency plan, where time is equated with distance. The stationarity index approach is based on interpreting the time frequency plan as a two dimensional probability distribution function already established in the literature. The formal analogy between TimeFrequency Representations (TFRs) and two dimensional probability density functions has been previously addressed [6 7] The analogy is drawn under the assumptions of total energy conservation and marginal properties in the TFR. Following is the mathematical foundation of the Mexican hat based continuous wavelet transform and the stationarity index method. 2.1. Wavelet Transform Let the "mother wavelet" function be noted (t ps The continuous wavelet transform of a function (i.e. space of finite energy functions) is defined as: f(t) L R 2 dt a b t t f t a 1 f(t) b) (a, W 2 1 ab f ps ps (1) Page 2 of 15 Sadok et al. 9 Joint FAA/DoD/NASA Aging Aircraft Conference where, ab a b t t =ps ps and stands for the conjugate of t ps (t ps The variable "a" represents the time information while the variable "b" represents the scale information that is inversely proportional to the frequency information. For (t ps to be a wavelet function, it has to meet the following condition [8] Ps p dw w w 2 h C (2) Where Ps is the Fourier transform of the mother wavelet (w (t ps The original signal f(t) can be then reconstructed from its wavelet transform: da db t a C t f ab h b) (a, W 1 1 f 2 ps (3) For the application at hand, the mother wavelet function,ps( t that satisfies both Equations (1) and (2) is given by: 2 2 4 1 2 1 3 2 t e t t Ps p (4) Equation 4 defines the Mexican hat mother wavelet while Figure 1 shows its general shape. 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Time Samples A m pl itu de General shape of the Mexican Hat mother wavelet Figure 1. General shape of the Mexican hat wavelet function. 2.2. Stationarity Index Formal analogy between Time Frequency Representations (TFRs) and two dimensional probability density functions is drawn under the assumptions of total energy conservation and marginal properties in the TFR, respectively, illustrated by Equations 5 6 [6 7] Energy conservation: dt t s dtdf f t Cs 2 (5) Page 3 of 15 Sadok et al. 9 Joint FAA/DoD/NASA Aging Aircraft Conference Marginal properties: 2 2 f S dt f t C t s df f t C