Method for analytical reconstruction of digital signals via stitched polynomial fitting
First Claim
1. A method for approximating the behavior of a periodic parameter of a time-domain proximity switch accelerometer in response to a perturbation, the method comprising the following steps:
- generating a triggering event when a first element of a switch within the time-domain proximity switch accelerometer passes a second element of the switch, wherein the first element is in oscillation and the second element is located at a fixed trigger position of one or more fixed trigger positions within the amplitude range of the oscillation, wherein the fixed trigger positions are predefined physical locations with respect to a reference position of the first element;
determining at least one time period between successive triggering events;
using the at least one time period and the fixed trigger position of the second element to determine the periodic parameter;
collecting data points at discrete times, wherein each data point represents a measurement of the parameter;
forming a preliminary approximation of the behavior of the system parameter by calculating an offset bias of the time-domain proximity switch accelerometer due to the perturbation;
dividing the time domain into intervals, each given interval containing at least one data point sampled during the given interval;
fitting a polynomial function to the data points of each interval such that each interval has a corresponding polynomial function that is time-centered on the interval'"'"'s center point and that accurately describes the parameter behavior over that interval;
stitching together the polynomial functions piece-wise to create an analytic approximation of the behavior of the periodic parameter over the entire time domain; and
altering a navigational course of a moving vehicle based on the preliminary approximation and correcting the altered navigational course based on the analytic approximation.
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Abstract
A method for approximating the behavior of a periodic parameter of a time-domain device in response to a perturbation, the method comprising the following steps: collecting data points at discreet times, wherein each data point represents a measurement of the parameter; dividing the time domain into intervals, each given interval containing at least one data point sampled during the given interval; fitting a polynomial function to the data points of each interval such that each interval has a corresponding polynomial function that is time-centered on the interval'"'"'s center point and that accurately describes the parameter behavior over that interval; and stitching together the polynomial functions piece-wise to create an analytic approximation of the behavior of the system parameter over the entire time domain.
10 Citations
18 Claims
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1. A method for approximating the behavior of a periodic parameter of a time-domain proximity switch accelerometer in response to a perturbation, the method comprising the following steps:
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generating a triggering event when a first element of a switch within the time-domain proximity switch accelerometer passes a second element of the switch, wherein the first element is in oscillation and the second element is located at a fixed trigger position of one or more fixed trigger positions within the amplitude range of the oscillation, wherein the fixed trigger positions are predefined physical locations with respect to a reference position of the first element; determining at least one time period between successive triggering events; using the at least one time period and the fixed trigger position of the second element to determine the periodic parameter; collecting data points at discrete times, wherein each data point represents a measurement of the parameter; forming a preliminary approximation of the behavior of the system parameter by calculating an offset bias of the time-domain proximity switch accelerometer due to the perturbation; dividing the time domain into intervals, each given interval containing at least one data point sampled during the given interval; fitting a polynomial function to the data points of each interval such that each interval has a corresponding polynomial function that is time-centered on the interval'"'"'s center point and that accurately describes the parameter behavior over that interval; stitching together the polynomial functions piece-wise to create an analytic approximation of the behavior of the periodic parameter over the entire time domain; and altering a navigational course of a moving vehicle based on the preliminary approximation and correcting the altered navigational course based on the analytic approximation. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 17, 18)
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11. A method for approximating the behavior of a system parameter x(t) of a time-domain proximity switch accelerometer in response to a perturbation, the method comprising the following steps:
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generating a triggering event when a first element of a switch within the time-domain proximity switch accelerometer passes a second element of the switch, wherein the first element is in oscillation and the second element is located at a fixed trigger position of one or more fixed trigger positions within the amplitude range of the oscillation, wherein the fixed trigger positions are predefined physical locations with respect to a reference position of the first element; determining at least one time period between successive triggering events; measuring the parameter at discrete times, [xi, ti]; forming a preliminary approximation of the behavior of the system parameter by calculating an offset bias of the time-domain proximity switch accelerometer due to the perturbation; dividing the time domain into n time intervals of length Δ
t, each given interval containing at least k+1 discrete measurements of parameter x(t), wherein k and n are both positive integers greater than 1;fitting a kth order polynomial function Pnk(t) to the measurements of each interval such that each interval has a corresponding polynomial function Pnk (t), thereby approximating parameter x(t) during each interval; stitching together the polynomial functions piece-wise to create a smooth, analytic approximation of the behavior of the system parameter x(t) over the entire time domain; and altering a navigational course of a moving vehicle based on the preliminary approximation and correcting the altered navigational course based on the analytic approximation. - View Dependent Claims (12, 13)
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14. A method for approximating the behavior of a system parameter of a time-domain proximity switch accelerometer in response to a perturbation, the method comprising the following steps:
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generating a triggering event when a first element of a switch within the time-domain proximity switch accelerometer passes a second element of the switch, wherein the first element is in oscillation and the second element is located at a fixed trigger position of one or more fixed trigger positions within the amplitude range of the oscillation, wherein the fixed trigger positions are predefined physical locations with respect to a reference position of the first element; determining at least one time period between successive triggering events; collecting data points at discrete times while the first element is at a known reference position and while the device is subjected to the perturbation, wherein each data point represents a measurement of the parameter; forming a preliminary approximation of the behavior of the system parameter by calculating an offset bias of the time-domain proximity switch accelerometer due to the perturbation; dividing the time domain into intervals, each given interval containing at least three data points sampled during the given interval; fitting a polynomial function to the data points of each interval such that each interval has a corresponding polynomial function; stitching together the polynomial functions piece-wise to create an analytic approximation of the behavior of the system parameter over the entire time domain; correcting the preliminary behavior approximation by comparing it to the analytic behavior approximation; and altering a navigational course of a moving vehicle based on the preliminary approximation and correcting the altered navigational course based on the analytic approximation. - View Dependent Claims (15)
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Specification