Innovative Solution Of The Time Domain Deconvolution Problem
Signal measurement is an extremely important aspect of scientific research. However, many quantities of interest are not directly measurable using instruments with which we are familiar. Instead, we often must measure a reaction to the desired quantity and infer the original signal using such a measurement. Examples include the reconstruction of force using strain measurement or heat transfer using temperature measurement.
In the simplest case of static equilibrium, the desired measurement (e.g. force or heat transfer) is simply a scale multiple of the measurement. However, in situations where the system of interest cannot achieve static equilibrium, the problem becomes more difficult and more complex relationships exist between the two quantities. For the purposes of this discussion, these scenarios will be referred to as “dynamic.” This is often the case in hypersonic wind tunnel testing where the short duration and high forcing prevent any static equilibrium. As the worldwide interest in hypersonic flight continues to grow, so too will the interest in performing dynamic measurement in hypersonic wind tunnels.
The motivation for this work is dynamic force measurement. For example, the quantification of control forcing on the millisecond scale will be a vital contribution to hypersonic flight development. One common approach to measuring dynamic force is the use of the Time Domain Deconvolution Method (TDDM). In the TDDM, one deconvolves a measured, time domain signal such as acceleration or strain with the Impulse Response Function (IRF) of the system to obtain a dynamic applied force. Due to the accuracy and broad-reaching applications of the TDDM, much research has been dedicated to performing accurate and efficient deconvolution.
Although there are many approaches to solve this problem, many begin with a zero order discretization of the convolution integral. This assumes that a sampled quantity holds constant until a new sample is taken. Although this simplifies many continuous equations, it imparts an unnatural, step-like representation of the measured signal. In this research, we sought a more natural representation of the signals and instead assume the distribution between measured points is linear.
Furthermore, the deconvolution problem is ill-posed. To solve a discrete, time domain deconvolution problem, the inversion of a system matrix is required. However, the near-singularity of such matrices can cause trace amounts of noise in the measurement to become large errors. This is referred to as an ill-posed problem. To remedy this shortcoming, typical solution invokes regularization; the process of introducing additional information into the problem to improve the conditioning of the inverted matrix. Many methods exist to regularize this inversion and again even more research is focused on the selection of a regularization parameter for a given method. This is an effective approach but often must impart arbitrary smoothness into the reconstructed force.
In this work, we sought a simpler solution that did not invoke additional assumptions into the problem. We achieve this by reducing the requested number of applied force points in the problem. This allows for a system matrix with many more rows than columns and is therefore suitable for an accurate solution via the least squares pseudo-inverse; a simple and commonly used technique. Using this approach, we may circumvent the entire complexity of regularization and parameter selection by simply selecting a resolution parameter to better condition the inversion.
During bench top testing on a dynamically similar test article to that seen in a large-scale hypersonic facility, this method proved highly effective at reconstructing applied hammer pulses in all three directions. Using a more natural representation of the unknown applied load (i.e. linear instead of zero order sampling) and simpler reduction of the resolution to improve the conditioning, we accurately reconstructed dynamic applied loads.
In future work, we will apply these methods to a wind tunnel data set. A hypersonic wind tunnel test is planned to measure the dynamic response of a hypersonic test article subjected to a deployed flap in a hypersonic flow. These methods will be applied to that dataset. The results may provide an integral example test for quantifying the controllability of such vehicles. We hope that our simplified TDDM will serve as a more approachable solution methodology for these scenarios so dynamic force measurement becomes more common in the wind tunnel community.
These findings are described in the article entitled Numerical construction of impulse response functions and input signal reconstruction, recently published in the Journal of Sound and Vibration. This work was conducted by J.W. Draper III from AEDC White Oak Wind Tunnel 9 and the University of Maryland, S.W. Lee from the University of Maryland, and E.C. Marineau from AEDC White Oak Wind Tunnel 9.