Method for analytical reconstruction of digital signals via stitched polynomial fitting

Patent metadata

Field	Value
Publication number	US-9715480-B1
Application number	US-201313847521-A
Country	US
Kind code	B1
Filing date	Mar 20, 2013
Priority date	Jun 24, 2011
Publication date	Jul 25, 2017
Grant date	Jul 25, 2017

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.

First claim

We 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.

2. The method of claim 1 , wherein the stitching step is performed by summing the polynomial functions of each interval.

3. The method of claim 2 , wherein the parameter is a time-dependent system variable x(t) of the time-domain proximity switch accelerometer.

4. The method of claim 2 , wherein the parameter is a measurable system variable y(z) which depends on a measurable system variable other than time.

5. The method of claim 1 , wherein the parameter is a closed switch state.

6. The method of claim 2 , wherein the intervals are not uniform but are of arbitrary length.

7. The method of claim 2 , wherein the intervals are of equal length.

8. The method of claim 2 , wherein the data points are collected at a non-constant sampling rate.

9. The method of claim 2 , wherein the data points are collected at a constant sampling rate.

10. The method of claim 2 , wherein the perturbation is a physical acceleration of the time-domain proximity switch accelerometer.

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:
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, [x i , t i ];
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 Δt, 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 k th order polynomial function P n k (t) to the measurements of each interval such that each interval has a corresponding polynomial function P n k (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.

12. The method of claim 11 , wherein the polynomial function P n k (t) is defined as follows:

P
n
k

⁡

(
t
)

=

(

A
n
0

)

+

(

A
n
1

)

⁢
t

+

(

A
n
2

)

⁢

t
2

+

…
⁢

⁢

(

A
n
k

)

⁢

t
k

=

∑

j
=
0

k

⁢

A
n
j

⁢

t
j

where

t
∈

[

-

Δ
⁢

⁢
t

Assignees

• Tally Charles H
• Swanson Paul David
• Waters Richard L
• Us Navy

Inventors

• Tally Charles H
• Swanson Paul David
• Waters Richard L

Classifications

• H03B5/30

  with frequency-determining element being electromechanical resonator · CPC title

• G01C7/04

  involving a vehicle which moves along the profile to be traced · CPC title

• G01P15/097 Primary

  by vibratory elements · CPC title

• G06F17/15 Primary

  Correlation function computation {including computation of convolution operations (arithmetic circuits for sum of products per se, e.g. multiply-accumulators G06F7/5443; digital filters, e.g. FIR, IIR, adaptive filters H03H17/00)} · CPC title

• G01P15/135

  by making use of contacts which are actuated by a movable inertial mass · CPC title