Extracting inertial information from nonlinear periodic signals

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

receiving, by an analog-to-digital converter, a first periodic analog signal from a first sensor that is responsive to motion of a proof mass;

converting, by the analog-to-digital converter, the first periodic analog signal to a first periodic digital signal;

generating, by signal-conditioning circuitry, a conditioned first periodic digital signal by at least:

scaling the first periodic digital signal to a predetermined amplitude; and

offsetting the scaled first periodic digital signal;

applying, by digital circuitry, an inverse trigonometric function to a quantity that is based on the conditioned first periodic digital signal to generate a result; and

determining the inertial parameter based on the result.

2. The method of claim 1 , wherein the first sensor is a first electrode interacting with the proof mass.

3. The method of claim 2 , wherein the first electrode electrostatically interacts with the proof mass.

4. The method of claim 1 , wherein:

the inertial parameter comprises a displacement of the proof mass; and

determining the inertial parameter comprises multiplying an unwrapped inverted signal by a geometric dimension to obtain the displacement of the proof mass.

5. The method of claim 1 , further comprising unwrapping the result, and wherein determining the inertial parameter based on the result comprises determining the inertial parameter based on the unwrapped result.

6. The method of claim 1 , wherein applying the inverse trigonometric function comprises applying an arcsine function to the quantity.

7. The method of claim 1 , wherein applying the inverse trigonometric function comprises applying an arccosine function to the quantity.

8. The method of claim 1 , wherein applying the inverse trigonometric function comprises using a lookup table to determine a table entry corresponding to the conditioned first periodic digital signal.

9. The method of claim 5 , wherein unwrapping comprises:

determining that a phase wrap has occurred by:

determining that a slope of the result has changed sign;

determining that the result has changed by a predetermined increment; and

determining that the sum of a prior slope and a prior value of the result would exceed a threshold; and

adjusting a present value of the result and one or more future values of the result.

10. The method of claim 4 , further comprising:

low-pass filtering the displacement to determine an inertial displacement; and

multiplying the inertial displacement by a square of a natural frequency of the proof mass to determine an acceleration of the proof mass.

11. The method of claim 1 , wherein the first periodic digital signal comprises a quotient of a first digital signal and a second digital signal.

12. The method of claim 1 , wherein offsetting comprises:

integrating the first periodic digital signal for a predetermined time interval to determine an integral; and

subtracting the integral from the first periodic digital signal.

13. The method of claim 1 , further comprising amplifying the first periodic analog signal to a first periodic analog voltage.

14. The method of claim 13 , further comprising:

receiving a second analog signal from a second sensor that is responsive to motion of the proof mass;

amplifying the second analog signal to a second analog voltage;

converting the second analog voltage to a second periodic digital signal; and

conditioning the second periodic digital signal by:

scaling the second periodic digital signal to the predetermined amplitude, and

offsetting the second periodic digital signal to the predetermined offset;

wherein the quantity is based on a quotient of the first and second conditioned digital signals.

15. The method of claim 13 , further comprising:

receiving a second analog signal from a second sensor that is responsive to motion of the proof mass; and

wherein amplifying the first analog signal to the first analog voltage comprises amplifying a difference between the first and second analog signals to the first analog voltage.

16. The method of claim 15 , further comprising:

receiving third and fourth analog signals from third and fourth sensors, respectively, each sensor responsive to motion of the proof mass;

amplifying a second difference between the third and fourth analog signals to a second analog voltage;

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

conditioning the second periodic digital signal by:

scaling the second periodic digital signal to the predetermined amplitude, and

offsetting the second periodic digital signal to the predetermined offset;

wherein the quantity comprises a quotient of the first and second conditioned digital signals.

17. The method of claim 16 , wherein the second, third, and fourth sensors are second, third, and fourth electrodes, respectively, each electrostatically interacting with the proof mass.

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

receiving, by an analog-to-digital converter, a first periodic analog signal from a first sensor that is responsive to motion of a proof mass;

converting, by the analog-to-digital converter, the first periodic analog signal to a first periodic digital signal;

generating, by signal-conditioning circuitry, a conditioned first periodic digital signal by at least:

scaling the first periodic digital signal to a predetermined amplitude; and

offsetting the first periodic digital signal;

determining a result of applying an inverse trigonometric function to a quantity, that is based on the conditioned first periodic digital signal; and

determining the inertial parameter based on the result.

19. The system of claim 18 , wherein the first sensor is a first electrode interacting with the proof mass.

20. The system of claim 19 , wherein the first electrode electrostatically interacts with the proof mass.

21. The system of claim 18 , wherein:

the inertial parameter comprises a displacement of the proof mass; and

determining the inertial parameter comprises multiplying an unwrapped inverted signal by a geometric dimension to obtain the displacement of the proof mass.

22. The system of claim 18 , further comprising circuitry configured for unwrapping the result, and wherein determining the inertial parameter based on the result comprises determining the inertial parameter based on the unwrapped result.

23. The system of claim 18 , wherein determining the result of applying the inverse trigonometric function to the quantity comprises determining the result of applying an arcsine function to the quantity.

24. The system of claim 18 , wherein determining the result of applying the inverse trigonometric function to the quantity comprises determining the result of applying an arccosine function to the quantity.

25. The system of claim 18 , wherein determining the result of applying the inverse trigonometric function to the quantity comprises determining the result using a lookup table, the result corresponding to the conditioned digital signal.

26. The system of claim 22 , wherein unwrapping comprises:

determining that a phase wrap has occurred by:

determining that a slope of the result has changed sign;

determining that the result has changed by a predetermined, increment; and

determining that the sum of a prior slope and a prior value of the result would exceed a threshold; and

adjusting a present value of the result and one or more future values of the result.

27. The system of claim 21 , further comprising circuitry configured for:

low-pass filtering the displacement to determine an inertial displacement; and

multiplying the inertial displacement by a square of a natural frequency of the proof mass to determine an acceleration of the proof mass.

28. The system of claim 18 , wherein the first periodic digital signal comprises a quotient of a first digital signal and a second digital signal.

29. The system of claim 18 , wherein offsetting comprises:

integrating the first periodic digital signal for a predetermined time interval to determine an integral; and

subtracting the integral from the first periodic digital signal.

30. The system of claim 18 , further comprising circuitry configured for amplifying the first periodic analog signal to a first periodic analog voltage.

31. The system of claim 30 , further comprising circuitry configured for:

receiving a second analog signal from a second sensor that is responsive to motion of the proof mass;

amplifying the second analog signal to a second analog voltage;

converting the second analog voltage to a second periodic digital signal; and

conditioning the second periodic digital signal by:

scaling the second periodic digital signal to the predetermined amplitude, and

offsetting the second periodic digital signal to the predetermined offset;

wherein the quantity is based on a quotient of the first and second conditioned digital signals.

32. The system of claim 30 , further comprising circuitry configured for:

receiving a second analog signal from a second sensor that is responsive to motion of the proof mass; and

wherein amplifying the first analog signal to the first analog voltage comprises amplifying a difference between the first and second analog signals to the first analog voltage.

33. The system of claim 32 , further comprising circuitry configured for:

receiving third and fourth analog signals from third and fourth sensors, respectively, each sensor responsive to motion of the proof mass;

amplifying a second difference between the third and fourth analog signals to a second analog voltage;

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

conditioning the second periodic digital signal by:

scaling the second periodic digital signal to the predetermined amplitude, and

offsetting the second periodic digital signal to the predetermined offset;

wherein the quantity comprises a quotient of the first and second conditioned digital signals.

34. The system of claim 33 , wherein the second, third, and fourth sensors are second, third, and fourth electrodes, respectively, each electrostatically interacting with the proof mass.

35. The system of claim 18 , further comprising the proof mass and the first sensor.

36. The system of claim 32 , further comprising the second sensor.

37. The system of claim 33 , further comprising the third and fourth sensor.