Extracting inertial information from nonlinear periodic signals

1. A method for determining an inertial parameter, comprising:

receiving first and second analog signals from first and second sensors, each sensor responsive to motion of a proof mass;

determining a difference between the first and second analog signals;

determining a plurality of timestamps corresponding to times at which the difference crosses a threshold;

determining a plurality of time intervals based on the timestamp;

determining a result of applying a trigonometric function to a quantity, the quantity based on the plurality of time intervals; and

determining the inertial parameter based on the result.

2. The method of claim 1 , wherein the first and second sensors are first and second electrodes, respectively, each interacting with the proof mass.

3. The method of claim 2 , wherein the first and second electrodes electrostatically interact with the proof mass.

4. The method of claim 1 , wherein determining the difference comprises amplifying the difference between the first and second analog signals to an analog voltage.

5. The method of claim 4 , wherein determining the timestamps comprises:

converting the analog voltage to a digital representation to generate a first periodic digital signal;

interpolating to generate an upsampled digital signal; and

determining timestamps corresponding to times at which the upsampled digital signal crosses the threshold.

6. The method of claim 4 , wherein determining the timestamps comprises:

comparing the analog voltage to a second threshold;

based on determining that the analog voltage has crossed the second threshold, toggling a rectangular-wave signal between a first value and a second value; and

determining the timestamps based on times at which the rectangular-wave signal toggles between the first value and the second value.

7. The method of claim 6 , wherein the quantity comprises a phase shift.

8. The method of claim 1 , wherein the quantity comprises a first quotient of two of the time intervals.

9. The method of claim 8 , further comprising:

determining a second result of applying a second trigonometric function to a second quantity comprising a second quotient of a second two of the time intervals; and

determining a second difference of the result and the second result.

10. The method of claim 9 , further comprising:

determining the multiplicative inverse of the second difference;

determining a product of a pitch of the proof mass and the multiplicative inverse; and

determining an estimate of displacement amplitude of the proof mass based on the product;

wherein the inertial parameter is the estimate of displacement amplitude.

11. The method of claim 9 , further comprising:

determining a sum of the result and the second result;

determining a third quotient of the sum and the difference;

determining a fourth quotient of a first scale factor and a sum of a third two of the time intervals;

determining a square of the fourth quotient;

determining a product of the third quotient, the square, and a second scale factor; and

determining an acceleration of an inertial device comprising the proof mass;

wherein the inertial parameter comprises the acceleration.

12. The method of claim 11 , wherein the second scale factor comprises a pitch of the proof mass.

13. The method of claim 1 , wherein the quantity comprises a phase shift.

14. The method of claim 1 , wherein each of the plurality of time intervals is a difference between a respective two of the plurality of timestamps.

15. A system for determining an inertial parameter, comprising circuitry configured for:

receiving first and second analog signals from first and second sensors, each sensor responsive to motion of a proof mass;

determining a difference between the first and second analog signals;

determining a plurality of timestamps corresponding to times at which the difference crosses a threshold;

determining a plurality of time intervals based on the timestamp;

determining a result of applying a trigonometric function to a quantity, the quantity based on the plurality of time intervals; and

determining the inertial parameter based on the result.

16. The system of claim 15 , wherein the first and second sensors are first and second electrodes, respectively, each interacting with the proof mass.

17. The system of claim 16 , wherein the first and second electrodes electrostatically interact with the proof mass.

18. The system of claim 15 , wherein determining the difference comprises amplifying the difference between the first and second analog signals to an analog voltage.

19. The system of claim 18 , wherein determining the timestamps comprises:

converting the analog voltage to a digital representation to generate a first periodic digital signal; and

interpolating to generate an upsampled digital signal;

determining timestamps corresponding to times at which the upsampled digital signal crosses the threshold.

20. The system of claim 18 , wherein determining the timestamps comprises:

comparing the analog voltage to a second threshold;

based on determining that the analog voltage has crossed the second threshold, toggling a rectangular-wave signal between a first value and a second value; and

determining the timestamps based on times at which the rectangular-wave signal toggles between the first value and the second value.

21. The system of claim 20 , wherein the quantity comprises a phase shift.

22. The system of claim 15 , wherein the quantity comprises a first quotient of two of the time intervals.

23. The system of claim 22 , further comprising circuitry configured for:

determining a second result of applying a second trigonometric function to a second quantity comprising a second quotient of a second two of the time intervals; and

determining a second difference of the result and the second result.

24. The system of claim 23 , further comprising circuitry configured for:

determining the multiplicative inverse of the second difference;

determining a product of a pitch of the proof mass and the multiplicative inverse; and

determining an estimate of displacement amplitude of the proof mass based on the product;

wherein the inertial parameter is the estimate of displacement amplitude.

25. The system of claim 23 , further comprising circuitry configured for:

determining a sum of the result and the second result;

determining a third quotient of the sum and the difference;

determining a fourth quotient of a first scale factor and a sum of a third two of the time intervals;

determining a square of the fourth quotient;

determining a product of the third quotient, the square, and a second scale factor; and

determining an acceleration of an inertial device comprising the proof mass;

wherein the inertial parameter comprises the acceleration.

26. The system of claim 25 , wherein the second scale factor comprises a pitch of the proof mass.

27. The system of claim 15 , wherein the quantity comprises a phase shift.

28. The system of claim 15 , wherein each of the plurality of time intervals is a difference between a respective two of the plurality of timestamps.

29. The system of claim 15 , further comprising the proof mass and the first and second sensors.