A
Anonymous
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On 9 and 10 January 2002, Eric Foster published some log-linear plots of the object waveforms of non-ferrous and ferrous targets, as seen by a pulse induction metal detector. The curves in these plots appear to be the sum of one or more exponential functions, and to have a wide dynamic range, although the amplitude scale was not marked. Other postings have quoted articles on the theory of pulse induction which conclude that the response will be the sum of exponentials, just as we here observe.
The purpose of this posting is to disclose a neat trick, a simple but unobvious way to process such log-linear data to smooth the noise in the low-signal parts of the plot. I developed this method while analyzing time glitches in a compuetr system being used to run a large phased-array radar, but the method applies to all sum-of-exponentials functions. (Ref: "Some Measurements of Timeline Gaps in VAX/VMS", Joe Gwinn, ACM Operating System Review, V.28, N.2, April 1994, Pp.92-96.)
The trick is to go backwards. One generates the reverse cumulative function, by integrating backwards, from +Infinity to x, where x varies. It is a property of the sum of exponential function that reverse integration preserves the shape of the original sum of exponential function, but displaced upward a constant amount in a log-linear plot.
In practice, with binned data, one computes the cumulative sum from +Infinty back towards zero. The parameters of the individual exponentials are estimated one by one, starting with the rightmost segment, and working inward to the first segment. After each exponential is estimated, it is subtracted from the experimental data, and the next exponential in turn is estimated based on this reduced data. In other words, one peels the exponentials off one by one, starting with the one having the largest time constant.
This approach has a number of advantages: First, cumulative functions are smoother and therefore yield better fits. Second, summing smaller before larger numbers is better numerically, as one can avoid adding small numbers to big numbers. Third, for exponential functions, one may quite simply compute the parameters of the function from the fitted parameters of the reverse cumulative function.
Joe Gwinn
The purpose of this posting is to disclose a neat trick, a simple but unobvious way to process such log-linear data to smooth the noise in the low-signal parts of the plot. I developed this method while analyzing time glitches in a compuetr system being used to run a large phased-array radar, but the method applies to all sum-of-exponentials functions. (Ref: "Some Measurements of Timeline Gaps in VAX/VMS", Joe Gwinn, ACM Operating System Review, V.28, N.2, April 1994, Pp.92-96.)
The trick is to go backwards. One generates the reverse cumulative function, by integrating backwards, from +Infinity to x, where x varies. It is a property of the sum of exponential function that reverse integration preserves the shape of the original sum of exponential function, but displaced upward a constant amount in a log-linear plot.
In practice, with binned data, one computes the cumulative sum from +Infinty back towards zero. The parameters of the individual exponentials are estimated one by one, starting with the rightmost segment, and working inward to the first segment. After each exponential is estimated, it is subtracted from the experimental data, and the next exponential in turn is estimated based on this reduced data. In other words, one peels the exponentials off one by one, starting with the one having the largest time constant.
This approach has a number of advantages: First, cumulative functions are smoother and therefore yield better fits. Second, summing smaller before larger numbers is better numerically, as one can avoid adding small numbers to big numbers. Third, for exponential functions, one may quite simply compute the parameters of the function from the fitted parameters of the reverse cumulative function.
Joe Gwinn