Patent Publication Number: US-2023160696-A1

Title: 4-points phase and sensitivity estimation algorithm and related architecture

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Patent Application No. 63/282,050, filed Nov. 22, 2021, the disclosure of which is hereby incorporated by reference herein in its entirety. 
    
    
     BACKGROUND 
     Numerous items such as smartphones, smart watches, tablets, automobiles, aerial drones, appliances, aircraft, exercise aids, and game controllers utilize sensors during their operation (e.g., motion sensors, pressure sensors, temperature sensors, etc.). In commercial applications, microelectromechanical system (MEMS) sensors such as accelerometers and gyroscopes capture complex movements and determine orientation or direction. For example, smartphones are equipped with accelerometers and gyroscopes to understand the movement of the smartphone, to augment navigation systems that rely on Global Positioning System (GPS) information, and to perform numerous other functions. Wearable devices and internet-of-things (IoT) devices constantly measure movement and other characteristics of a person, animal, or electronic device. In another example, drones and aircraft determine orientation based on gyroscope measurements (e.g., roll, pitch, and yaw) and vehicles of all types implement assisted driving to improve safety (e.g., to recognize skid or roll-over conditions). 
     Accelerometers or gyroscopes of a MEMS system, when housed in a MEMS chip, may be subject to certain manufacturing or in-field external stresses. During manufacturing, typical tolerances may result in the MEMS chip experiencing certain imparted forces as stress. Component installation during production processes, such as soldering, can induce forces absorbed by the MEMS chip as stress. Other manufacturing processes with stress-inducing conditions may occur during packaging such as the MEMS system susceptibility to board-bending. In-field stress sources may also vary. For example, the MEMS device may experience displacement of moving mechanical parts from normal wear and bending conditions from transport. An external stress impact on a MEMS system can propagate to cause a corresponding impact on the internal sensors of the system. In another example, a stress with a bending effect on the MEMS chip may correspond to a related stress experienced by the accelerometer or the gyroscope of the MEMS chip. Externally induced stresses can introduce errors into the accelerometer and/or gyroscope measurements. Changes in environmental conditions such as temperature may result in stresses imparted on the MEMS chip as well. These stresses may result in measurement errors of a MEMS sensor such as a MEMS accelerometer or a MEMS gyroscope. Further, measurement error in a MEMS gyroscope and/or an accelerometer may occur because of drift or error in drive signal input. 
     Measurement errors and/or inconsistent performance may result from temporary or permanently induced stresses on the mechanical structure, e.g., caused by temperature changes and/or aging of a device. In general, sensitivity variations depend on variations of transfer function amplitude, while offset errors depend on variations of the transfer function phase. It is desired to minimize such errors and stabilize performance through variations in temperature and throughout a device lifecycle, including the process of manufacturing boards where the MEMS sensor is mounted or soldered. 
     SUMMARY 
     In an embodiment of the present disclosure, a method for estimating the variation of a microelectromechanical system (MEMS) sensor transfer function includes imparting a drive signal to one or more drive electrodes of a MEMS sensor, the drive signal having a drive frequency, applying a plurality of test signals to a proof mass sense signal to create a modified proof mass sense signal, where the plurality of test signals includes a plurality of frequencies, each of the plurality of frequencies different from the drive frequency, and driving a gyroscope of the MEMS sensor based on the modified proof mass sense signal, thereby injecting the plurality of test signals into a proof mass output sense signal. In some embodiments, any number of test signals and any type of test signal (e.g., sinusoidal tone, in-band tone, out-of-band tone, etc.) may be applied to modify the proof mass sense signal. The method further comprises receiving the proof mass output sense signal from the MEMS sensor, extracting an in-phase component and a quadrature component from the proof mass output sense signal, processing the in-phase component and the quadrature component based on the plurality of frequencies of the plurality of test signals, and determining a change in demodulation phase, or in a gain of the device, based on the processing of the in-phase component and the quadrature component. 
     In an embodiment of the present disclosure, a system for estimating the variation of a microelectromechanical system (MEMS) sensor transfer function comprises processing circuitry configured to impart a drive signal to one or more drive electrodes of a MEMS sensor, the drive signal having a drive frequency, apply a plurality of test signals to a proof mass sense signal to create a modified proof mass sense signal, where the plurality of test signals comprises a plurality of frequencies, each of the plurality of frequencies different from the drive frequency, and drive a gyroscope of the MEMS sensor based on the modified proof mass sense signal, thereby injecting the plurality of test signals into a proof mass output sense signal. The processing circuitry of the system is further configured to, after receiving the proof mass output sense signal from the MEMS sensor, extract an in-phase component and a quadrature component from the proof mass output sense signal, process the in-phase component and the quadrature component based on the plurality of frequencies of the plurality of test signals and the drive signal, and determine a change in demodulation phase, or in a gain of the device, based on the processing of the in-phase component and the quadrature component. 
     In an embodiment of the present disclosure, a gyroscope for estimating the variation of a microelectromechanical system (MEMS) sensor transfer function comprises test signal generation circuitry, a drive mass, and a proof mass. The gyroscope further comprises processing circuitry configured to impart a drive signal to one or more drive electrodes of a MEMS sensor, the drive signal having a drive frequency, apply a plurality of test signals to a proof mass sense signal to create a modified proof mass sense signal, where the plurality of test signals comprises a plurality of frequencies, each of the plurality of frequencies different from the drive frequency, and drive the proof mass of the MEMS sensor based on the modified proof mass sense signal, thereby injecting the plurality of test signals into a proof mass output sense signal. In further embodiments, processing circuitry receives the proof mass output sense signal from the MEMS sensor, extracts an in-phase component and a quadrature component from the proof mass output sense signal, processes the in-phase component and the quadrature component based on the plurality of frequencies of the plurality of test signals and the drive signal, and determines a change in demodulation phase, or in a gain of the device, based on the processing of the in-phase component and the quadrature component. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       The above and other features of the present disclosure, its nature, and various advantages will be more apparent upon consideration of the following detailed description, taken in conjunction with the accompanying drawings in which: 
         FIG.  1    shows an illustrative MEMS system in accordance with an embodiment of the present disclosure; 
         FIG.  2    shows an illustrative MEMS gyroscope in accordance with an embodiment of the present disclosure; 
         FIG.  3    shows an illustrative block diagram of a MEMS gyroscope and an in-phase and quadrature demodulation process in accordance with an embodiment of the present disclosure; 
         FIG.  4    shows a diagram depicting a sense transfer function of a MEMS before demodulation in accordance with an embodiment of the present disclosure, and its possible variations in two different instants of the sensor life; 
         FIG.  5    shows a diagram depicting a sense transfer function of a MEMS after demodulation in accordance with an embodiment of the present disclosure, and its possible variations in two different instants of the sensor life; 
         FIG.  6    shows an illustrative feed forward block diagram for estimating and compensating for sense transfer function variation in accordance with an embodiment of the present disclosure; 
         FIG.  7    shows an illustrative feedback block diagram for estimating and compensating for sense transfer function variation in accordance with an embodiment of the present disclosure; 
         FIG.  8    shows an illustrative block diagram of a multi-tone demodulator in accordance with an embodiment of the present disclosure; 
         FIG.  9    shows an illustrative block diagram of a single tone demodulator in accordance with an embodiment of the present disclosure; 
         FIG.  10    shows an illustrative block diagram of phase estimation in accordance with an embodiment of the present disclosure; 
         FIG.  11    shows an illustrative block diagram of amplitude estimation in accordance with an embodiment of the present disclosure; 
         FIG.  12    shows an illustrative feed forward block diagram including an equalizer for estimating and compensating for sense transfer function variation in accordance with an embodiment of the present disclosure; and 
         FIG.  13    shows an illustrative flowchart for reducing error in a MEMS device in accordance with an embodiment of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     Example approaches herein may generally minimize error and stabilize performance despite variations in temperature and aging of a MEMS device. In some examples, control mechanisms may be implemented (e.g., feedback and feed forward) to reduce error and stabilize performance. In at least some examples, changes of the sensor transfer function, both the amplitude and the phase, are measured and corrected to recover the relevant errors of the sensitivity and the offset, respectively. Among errors caused by stresses, phase errors may be more significant since relatively small errors may generate great variation of offset when the quadrature component coming from the MEMS device is very high. However, example methods herein are also suitable for sensitivity compensation, as they include a method for estimating the variation of the MEMS sensor transfer function. 
       FIG.  1    shows an illustrative MEMS system  100  in accordance with an embodiment of the present disclosure. Although particular components are depicted in  FIG.  1   , it will be understood that other suitable combinations of the MEMS, processing components, memory, and other circuitry may be utilized as necessary for different applications and systems. In certain embodiments of the present disclosure, the circuitry, devices, systems, and methods described herein may be described in the context of a system including control circuitry configured to apply a plurality of test signals to a proof mass sense signal to generate an output signal (e.g., a proof mass output sense signal), extract an in-phase component and a quadrature component from the output signal, and process the in-phase and quadrature components to determine a change in demodulation phase of the MEMS sense transfer function. It will be understood that the circuitry, devices, systems, and methods described herein may be applied to other types of MEMS devices or sensors. 
     Processing circuitry  104  may include one or more components providing necessary processing based on the requirements of the MEMS system  100 . In some embodiments, processing circuitry  104  may include hardware control logic that may be integrated within a chip of a sensor (e.g., on a base substrate of a MEMS gyroscope  102  or other sensor  108 , or on an adjacent portion of a chip to the MEMS gyroscope  102  or other sensor  108 ) to control the operation of the MEMS gyroscope  102  or other sensors  108  and perform aspects of processing for the MEMS gyroscope  102  or the other sensors  108 . In some embodiments, the MEMS gyroscope  102  and other sensors  108  may include one or more registers that allow aspects of the operation of hardware control logic to be modified (e.g., by modifying a value of a register). In some embodiments, processing circuitry  104  may also include a processor such as a microprocessor that executes software instructions, e.g., that are stored in memory  106 . The microprocessor may control the operation of the MEMS gyroscope  102  by interacting with the hardware control logic and processing signals received from MEMS gyroscope  102 . The microprocessor may interact with other sensors  108  in a similar manner. In some embodiments, some or all of the functions of the processing circuitry  104 , and in some embodiments, of memory  106 , may be implemented on an application specific integrated circuit (“ASIC”) and/or a field programmable gate array (“FPGA”). In some embodiments, MEMS gyroscope  102  may be referred to as a variety of MEMS sensors (e.g., an accelerometer, a barometer, an inertial measurement unit, a magnetometer, etc.). 
     Although in some embodiments (not depicted in  FIG.  1   ), the MEMS gyroscope  102  or other sensors  108  may communicate directly with external circuitry (e.g., via a serial bus or direct connection to sensor outputs and control inputs), in an embodiment the processing circuitry  104  may process data received from the MEMS sensor  102  and other sensors  108  and communicate with external components via a communication interface  110  (e.g., a serial peripheral interface (SPI) or I2C bus, in automotive applications a controller area network (CAN) or Local Interconnect Network (LIN) bus, or in other applications a suitably wired or wireless communications interface as is known in the art). The processing circuitry  104  may convert signals received from the MEMS gyroscope  102  and other sensors  108  into appropriate measurement units (e.g., based on settings provided by other computing units communicating over the communication bus  110 ) and perform more complex processing to determine measurements such as orientation or Euler angles, and in some embodiments, to determine from sensor data whether a particular activity (e.g., walking, running, braking, skidding, rolling, etc.) is taking place. In some embodiments, some or all of the conversions or calculations may take place on the hardware control logic or other on-chip processing of the MEMS gyroscope  102  or other sensors  108 . 
     In some embodiments, certain types of information may be determined based on data from multiple MEMS gyroscopes  102  and other sensors  108  in a process that may be referred to as sensor fusion. By combining information from a variety of sensors it may be possible to accurately determine information that is useful in a variety of applications, such as image stabilization, navigation systems, automotive controls and safety, dead reckoning, remote control and gaming devices, activity sensors, 3-dimensional cameras, industrial automation, and numerous other applications. 
     In accordance with the present disclosure, a phase-locked loop (PLL) actuates a drive signal having a drive frequency to displace a drive mass, via drive electrodes, and generates a proof mass sense signal. A plurality of test signals may be injected into MEMS gyroscope  102 , via test and calibration electrodes (e.g., self-test or quadrature electrodes), to create a modified proof mass sense signal, which drives a proof mass of the MEMS gyroscope  102  and generates an output signal (e.g., a proof mass output sense signal) detected by sense electrodes. In some embodiments, the one or more test signals may include a first test tone, f 1 , having a first frequency and a second test tone, f 2 , having a second frequency different from the first frequency. It will be understood that the first frequency of the first test tone, f 1 , and the second frequency of the second test tone, f 2 , may each represent offsets of the test tone frequencies from the drive frequency, f d , of the drive signal. In some embodiments, the first test tone, f 1 , and the second test tone, f 2 , may each have a lower frequency than the drive frequency, f d . After the modulation of the respective test tones in the MEMS gyroscope  102 , however, 4 frequencies (e.g., f d −f 2 , f d −f 1 , f d +f 1 , f d +f 2 ) are generated, where two frequencies are less than f d  (e.g., f d −f 2  and f d −f 1 ) and two frequencies are greater than f d  (e.g., f d +f 1  and f d +f 2 ). In some embodiments, the first frequency and the second frequency may be outside an intended signal bandwidth range for the MEMS gyroscope  102 . An in-phase and quadrature component are respectfully extracted from the output signal (e.g., proof mass output sense signal) and processed based on, e.g., the first frequency of the first test tone to determine a change in demodulation phase. In some embodiments, the in-phase and quadrature component may be processed based on the second frequency of the second test tone. In some embodiments, the one or more test signals may further include a third test tone having a third frequency, where the third frequency may be offset from the drive frequency of the drive signal. In some embodiments, the plurality of test signals may include a plurality of frequencies different from the drive frequency. A tone demodulator processes the in-phase component and the quadrature component of the output signal, which may include performing high frequency and low frequency signal processing. High frequency signal processing occurs when the single tone demodulator initially receives the output signal (e.g., proof mass output sense signal) generated by MEMS gyroscope  102  and integrates and downsamples the respective in-phase and quadrature components of the output signal based on, e.g. the first frequency of the first test tone with a velocity of variation of a sense transfer function. Low frequency signal processing is used to estimate both a phase and an amplitude of the sense transfer function after integration and downsample. In one embodiment, low frequency signal processing processes sense transfer function values (e.g., spectral points) via an inverse trigonometric (e.g., arctangent) function and a third order interpolation of the drive frequency to estimate phase. It will be understood that any interpolation order of the drive frequency may be used to estimate phase. In another embodiment, low frequency signal processing processes sense transfer function values (e.g., spectral points) via a square root of a sum of squares operation and the third order interpolation of the drive frequency to estimate amplitude (e.g., gain). In some embodiments, the processing of the in-phase and quadrature components of the output signal may further comprise an equalizer to flatten the gain and amplitude of the sense transfer function. 
     The aforementioned method and/or system allows for estimating the variation in the MEMS sensor transfer function in the presence of temporary (e.g., relating to temperature) and more permanent (e.g., relating to bend, soldering, attrition or aging over lifecycle, etc.) stresses impacting the MEMS gyroscope  102 . Estimating and tracking the variation in the phase and the amplitude of the MEMS sense transfer function enables compensation of such errors to correct them in a way to recover inaccuracies in sensitivity and offset respectively and improved stabilization of the MEMS gyroscope  102  over time. Otherwise, induced stresses on the mechanical structure of the MEMS gyroscope  102  may cause measurement errors and/or inconsistency. 
       FIG.  2    shows an illustrative MEMS gyroscope  200  in accordance with an embodiment of the present disclosure. The exemplary MEMS gyroscope  200  of  FIG.  2    is simplified for the purposes of illustration. It will be understood that a MEMS gyroscope as described in the present disclosure may include any suitable MEMS gyroscope design, including single-axis or multi-axis MEMS gyroscopes. Although portions of the present disclosure may be described in the context of a particular type of MEMS gyroscope configuration (e.g., a single-axis out-of-plane sensing gyroscope), it will be understood that the present disclosure may apply equally to other types and configurations of MEMS devices. 
     As illustrated in  FIG.  2   , the MEMS gyroscope  200  may include MEMS layer  202 , substrate layer  204  (e.g., a complementary metal-oxide-semiconductor (CMOS) substrate layer), and anchors  206   a ,  206   b  separating the layers and located within a gap between the two layers. Packaging and additional layers (e.g., a cap layer) are not shown in  FIG.  2    for ease of illustration but may be coupled to the MEMS layer  202  and/or substrate layer  204  to form a hermetically sealed cavity in which the movable MEMS components of a suspended spring-mass system (e.g., drive masses  212   a ,  212   b ), Coriolis masses (not depicted), proof masses  210   a ,  210   b , and additional springs and/or masses coupled thereto (not depicted) are able to move. The cavity may have a nominal pressure (e.g., at or near a vacuum pressure, or another suitable pressure for other particular designs). In the exemplary embodiment of  FIG.  2   , a bottom plane of the suspended spring-mass system of the MEMS layer  202  is located parallel to an upper plane of the substrate layer  204  and proof mass sense electrodes  208   a - 208   d  are located thereon. Drive mass sense electrodes  214   a ,  214   b  are located adjacent to the drive masses  212   a ,  212   b  for sensing movement thereof (e.g., imparted by drive electrodes, not depicted in  FIG.  2   ). 
     MEMS layer  202  includes a suspended spring-mass system including proof masses  210   a ,  210   b  and drive masses  212   a ,  212   b , which are suspended from anchors  206   a ,  206   b , respectively, by interconnected springs and/or masses (not visible in  FIG.  2   ). The components of the suspended spring-mass system are sized and configured in a manner to facilitate movement of the proof masses  210   a ,  210   b  in response to the movement of the drive masses  212   a ,  212   b  and an inertial force to be measured, e.g., angular velocity about an axis perpendicular to the drive axis. Although not depicted in  FIG.  2   , drive circuitry (e.g., a phase-locked loop (PLL)) may provide drive signals to the drive masses  212   a ,  212   b  of the suspended spring-mass system (e.g., via drive electrodes). For example, in a MEMS gyroscope, a drive signal may create a physical drive motion of one or more components (e.g., drive masses  212   a ,  212   b ) that in turn results in a Coriolis force experienced by the proof masses  210   a ,  210   b  when the gyroscope is rotated about an axis of interest. In an exemplary embodiment, the drive circuitry may provide the drive signal via one or more drive electrodes (e.g., a capacitive plate, comb electrode, etc.) located adjacent to components of the suspended spring-mass system (e.g., drive masses  212   a ,  212   b , etc.). Drive mass sense electrodes  214   a ,  214   b  are each located at a fixed location adjacent to a respective drive mass  212   a ,  212   b , and each outputs a signal (e.g., a capacitive signal) corresponding to the displacement of the respective drive mass  212   a ,  212   b  in response to the drive signal. 
     In the exemplary embodiment of  FIG.  2   , the proof masses are designed to move along the direction of the z-axis in response to the measured inertial force (e.g., rate of rotation or angular velocity for a gyroscope). For example, an illustrative MEMS gyroscope  200  includes a suspended spring-mass system including movable drive masses  212   a ,  212   b  and movable proof masses  210   a ,  210   b , springs, and additional components such as lever arms and Coriolis masses (not depicted in  FIG.  2   , but located within and patterned from MEMS layer  202 ) connecting the drive masses to the proof masses. The springs and other movable components of the spring-mass system are coupled to the drive masses  212   a ,  212   b  and proof masses  210   a ,  210   b  and are selectively patterned and positioned such that they are relatively rigid in response to forces in directions in which it is not desired to impart the drive motion, or measure the inertial force, and relatively flexible in a direction in which a force is to be imparted or measured. 
     Proof mass  210   a  is suspended over proof mass sense electrodes  208   a ,  208   b  and proof mass  210   b  is suspended over proof mass sense electrodes  208   c ,  208   d . In response to a z-axis movement of the proof masses due to an angular velocity experienced by a MEMS gyroscope (e.g., due to rotation of a device including a MEMS gyroscope about an axis perpendicular to the z-axis and the axis of the drive motion imparted by drive masses  212   a ,  212   b ), the proof masses  210   a ,  210   b  rotate out of the plane of the MEMS layer (e.g., about the y-axis) such that portions of the proof mass move closer to or farther away from respective proof mass sense electrodes, with the degree of rotation (e.g., how much the proof masses move with respect to the respective proof mass sense electrodes) based on the magnitude of the angular velocity and the motion imparted by the drive mass. The design of the suspended spring-mass system may be such that the proof masses  210   a ,  210   b  have minimal movement out of the MEMS plane in the absence of angular velocity about the sense axis. 
     In the exemplary embodiment of  FIG.  2   , the movement of the proof masses  210   a ,  210   b  out of the MEMS plane may be sensed using electrostatic sensing as depicted in  FIG.  2   . Fixed proof mass sense electrodes  208   a ,  208   b ,  208   c , and  208   d  are located parallel to the proof masses (e.g., on substrate layer  204  below proof masses  210   a ,  210   b ) to form capacitors with portions of the proof masses (e.g., electrode  208   a  forms a capacitor with a first portion of proof mass  210   a , electrode  208   b  forms a capacitor with a second portion of proof mass  210   a , electrode  208   c  forms a capacitor with a first portion of proof mass  210   b , and electrode  208   d  forms a capacitor with a second portion proof mass  210   b ). The capacitance of each of the proof masses may change based on the relative distance between each proof mass portion and its associated proof mass sense electrodes. In the embodiment of  FIG.  2   , these capacitances and the capacitances sensed by the drive mass sense electrodes are used by processing circuitry (e.g., in the substrate layer  204 ) to determine the inertial force (e.g., by demodulating and compensating the sensed movement of the proof masses to isolate the Coriolis force from other forces and signals, such as a quadrature signal of the MEMS gyroscope structure). Although electrostatic sensing is described in the embodiment of  FIG.  2   , it will be understood that other forms of sensing (e.g., piezoelectric, infrared, or magnetic) may be used in other embodiments. While some or all of the processing circuitry may be described as located within a substrate layer  204  (e.g., a CMOS substrate layer), in some embodiments a substrate may not include active processing components and may instead simply perform functions such as routing signals to other processing circuitry (e.g., on adjacent components to the MEMS sensor and/or stacked on layers above or below the substrate or cap of the MEMS sensor). 
       FIG.  3    shows an illustrative block diagram of a MEMS gyroscope and an in-phase and quadrature (I&amp;Q) demodulation process in accordance with an embodiment of the present disclosure. In the depicted embodiment, system  300  includes phase-locked loop (PLL) trim  302 , PLL and Phase trim  304 , drive loop  306 , a MEMS gyroscope  310 , drive electrodes  316 , drive mass  326 , drive sense electrodes  320 , quadrature electrodes  318 , self-test electrodes  322 , quadrature signal  330 , Coriolis signal  332 , proof mass  328 , and proof mass sense electrodes  324 . System  300  further includes demodulator  308 , digital gain  312   a ,  312   b , low pass filters  314   a ,  314   b , in-phase channel  336 , quadrature channel  338 , and digital gain trim  334 . In some embodiments, demodulator  308  may be referred to as an in-phase and quadrature (I&amp;Q) demodulator. In an example, the MEMS gyroscope  310  is MEMS gyroscope  120  or  200 . In the example illustrated in  FIG.  3   , an Amplification and an Analog to Digital conversion are not illustrated, although it will be understood that they may be implemented as part of the illustrated signal processing. 
     It will be understood that MEMS gyroscope  310  receives a drive signal, generated by a PLL loop (e.g., including PLL trim  302 , PLL and Phase trim  304 , and drive loop  306 ) of system  300 , and, in response to the drive signal and an appropriately applied external force, generates an output signal (e.g., a proof mass output sense signal) that is received by demodulator  308 , where an in-phase (e.g., Coriolis) component and a quadrature component are respectively extracted and processed. As described herein, one or more test signals are injected in the MEMS gyroscope  310  to modify a proof mass sense signal and a corresponding output signal (e.g., proof mass output sense signal), which in turn is used to estimate a variation in phase and gain (e.g., amplitude) of the MEMS sense transfer function with respect to an original or desired transfer function. Although particular components are depicted in certain configurations for system  300 , it will be understood that components may be removed, modified, or substituted and that additional components (e.g., electrodes, masses, filters, etc.) may be added in certain embodiments. 
     A plurality of errors, induced by external stresses, with different timescales may affect the operation of MEMS gyroscope  310 . In some embodiments, construction inaccuracies may cause errors that are normally measured at factory premises when the MEMS gyroscope  310  is offline (e.g., the MEMS gyroscope  310  is not working normally but it is in a special state dedicated to this testing and is then permanently compensated). In some embodiments, MEMS gyroscope  310  installation in its final application component (e.g., after soldering the MEMS gyroscope  310  on a printed circuit board (PCB) of a cellular phone, drone, gaming controller, etc.) may generate errors that may be measured after the installation and permanently compensated. This operation may be done when the MEMS gyroscope  310  is in a special state dedicated to a testing mode. In some embodiments, aging and usage of the MEMS gyroscope  310  may cause errors that need to be measured and compensated for several times during the lifespan of the MEMS gyroscope  310  (e.g., at each power on, once every month, etc.). Generally, the timescale for errors associated with MEMS gyroscope  310  attrition and aging may be very slow, e.g., in the range of many months and/or years. This operation may also be done when the MEMS gyroscope  310  is in a testing state dedicated to this testing feature and repeated on demand. In some embodiments, temperature may cause errors to occur during normal MEMS gyroscope  310  operation since temperature varies while the MEMS gyroscope  310  is normally in service. Accordingly, errors caused by temperature may need to be compensated with a mechanism working during normal MEMS gyroscope  310  operation with the ability to not affect MEMS gyroscope  310  functionality. A timescale for temperature errors may be every time the MEMS gyroscope  310  is requested to provide its normal functionality, or other suitable timescales based on number of uses, time of use, or passage of time without regard to usage. 
     MEMS gyroscope  310  includes a suspended spring-mass system, which further includes drive mass  326 , drive sense electrodes  320  for generating a drive sense signal corresponding to the displacement of drive mass  326 , and proof mass  328 . In addition, MEMS gyroscope  310  includes self-test electrodes  322 , Coriolis (e.g., in-phase) signal  332 , quadrature electrodes  318 , quadrature signal  330 , and proof mass sense electrodes  324  for generating an output signal (e.g., a proof mass output sense signal) based on the displacement of proof mass  328 . In some embodiments, MEMS gyroscope  310  may be an accelerometer or a variety of other sensors (e.g., a barometer, an inertial measurement unit, a magnetometer, etc.). 
     Phase-locked loop (PLL) &amp; Phase Trim  304  is a control system that generates an output signal (e.g., a proof mass output sense signal), via MEMS gyroscope  310 , with a phase related to a phase of a drive signal. PLL &amp; Phase Trim  304  generates and delivers a drive signal, via drive loop  306  (e.g., a 90° phase shifter), to drive electrodes  316 , which receive the drive signal at a particular drive frequency (e.g., 10 kHz) and displace drive mass  326  in accordance with the drive frequency. The displacement of drive mass  326  generates the drive sense signal, which is detected by drive sense electrodes  320 . In some embodiments, the drive loop  306  may shift the phase of the drive signal by any degree amount depending on the amount of offset MEMS gyroscope  310  experiences due to external stresses. In some embodiments, PLL &amp; Phase Trim  304  may be synchronized with the oscillation frequency of the drive system (e.g., the control system) and produce a higher frequency clock at a multiple (e.g., a harmonic) of the drive frequency. PLL and Phase trim  304  receives the drive sense signal, via drive sense electrodes  320 , and processes the drive sense signal (e.g., precisely adjusts the phase of the drive sense signal to closely match the drive frequency, f d ) to create reference signals (e.g., cos(2πf d t) and sin(2πf d t)) needed for demodulator  308 . PLL Trim  302  is a set of constants and parameters (e.g., a memory), that normally are adjusted at factory premises, that are needed by PLL &amp; Phase Trim  304  to correctly work. Demodulator  308  respectively receives the reference signals of the drive sense signal from PLL and Phase trim  304  at in-phase channel  336 , which feeds to digital gain  312   a , and quadrature channel  338 , which feeds to digital gain  312   b . It will be understood that the reference signals (e.g., cos(2πf d t) and sin(2πf d t)) serve as a baseline and contribute to identifying phase and/or gain variation when encoding with in-phase (e.g., Coriolis) and quadrature components of the MEMS gyroscope  310  output signal (e.g., proof mass output sense signal). Equation (1) describes the output signal (e.g., the proof mass output sense signal) of the MEMS gyroscope  310  below: 
         x   s ( t )= G   s ( f   d )[Ω( t )cos(2π f   d   t−φ   s ( f   d ))− Q   o  sin(2π f   d   t−φ   s ( f   d ))]  (1)
 
     In this equation:
         1) G s  (f d ) is the gain of the signal, i.e., the amplitude of the MEMS sense transfer function at the drive frequency f d      2) φ s  (f d ) is the phase delay of the signal, i.e., the phase of the MEMS sense transfer function at the drive frequency f d          

     In-phase (e.g., Coriolis) and quadrature components from the output signal (e.g., the proof mass output sense signal) are demodulated with reference signals (e.g., cos(2πf d t) and sin(2πf d t)) from the drive sense signal at in-phase (e.g., Coriolis) channel  336  and quadrature channel  338  to identify phase and/or gain offset, related to quadrature signal  330  and/or Coriolis (e.g., in-phase) signal  332 , before feeding into digital gain  312   a ,  312   b . Digital gain  312   a ,  312   b  respectively compensates for gain variation within the in-phase (e.g., Coriolis) signal received at in-phase channel  336  and the quadrature signal received at quadrature channel  338 . Digital gain trim  334  is incorporated into digital gain  312   a ,  312   b  to make precise changes to the amplitude (e.g., by either adding additional gain for a boost or attenuating the signal to reduce the gain) of either the in-phase (e.g., Coriolis) signal or the quadrature signal. In some embodiments, digital gain trim  334  may be actuated so that each of the in-phase (e.g., Coriolis) signal and the quadrature signal are more compatible with an external component (e.g., a matrix rotation, Tone demodulator, etc.) of system  300 . Low pass filters  314   a ,  314   b  respectively receive the outputs of digital gain  312   a ,  312   b , via in-phase channel  336  and quadrature channel  338 , and filter out components of either the in-phase signal or the quadrature signal with frequencies above a certain threshold frequency (e.g., 2*f d ). For example, the in-phase and quadrature components provided at the output of demodulator  308  may be described by equations (2) and (3) below: 
         x   sI ( t )=2 G   s ( f   d )[Ω( t )cos(2π f   d   t−φ   s ( f   d ))− Q   o  sin(2 πf   d   t−φ   s ( f   d ))]cos(2π f   d   t )  (2)
 
         x   sQ ( t )=−2 G   s ( f   d )[Ω( t )cos(2π f   d   t−φ   s ( f   d ))− Q   o  sin(2 πf   d   t−φ   s ( f   d ))]sin(2π f   d   t )  (2)
 
     Low pass filters  314   a ,  314   b  may remove certain frequency signal components at 2*f d  before the in-phase and quadrature signals are delivered to components external to system  300 . After filtering operation, equations (2) and (3) reduce to the following equations (4) and (5): 
         x   sI ( t )= G   s ( f   d )[Ω( t )cos(φ s ( f   d ))+ Q   o  sin(φ s ( f   d ))]  (4)
 
         x   sQ ( t )= G   s ( f   d )[−Ω( t )sin(φ s ( f   d ))− Q   o  cos(φ s ( f   d ))]  (5)
 
     Equations (4) and (5) depict explicit formulas for the in-phase (e.g., Coriolis) and quadrature components at baseband in absence of any compensation. The phase difference introduced by the MEMS sense transfer function, namely φ s (f d ), creates an unwanted crosstalk between the in-phase (e.g., Coriolis) component and the quadrature component. In some embodiments, if the quadrature signal Q o  is high, the in-phase (e.g., Coriolis) component measured on the in-phase channel  336  may be affected by a significant offset error (e.g., Q o  sin (φ s  (f d ))). 
     Processing circuitry within system  300  may compensate for phase shift φ s (f d ) within the output signal (e.g., the proof mass output sense signal), which creates crosstalk between the in-phase (e.g., Coriolis) and quadrature components with a phase adjustment capability that, e.g., may assign to the demodulation carriers a phase equal to φ s (f d ). In this way, equations (2) and (3) may be rewritten as equations (6) and (7) below, which represent the intermediate expressions of the in-phase (e.g., Coriolis) and quadrature components received by demodulator  308 : 
         x   sI ( t )=2 G   s ( f   d )[Ω( t )cos(2π f   d   t−φ   s ( f   d ))− Q   o  sin(2 πf   d   t−φ   s ( f   d ))]cos(2π f   d   t−φ   s ( f   d ))  (6)
 
         x   sQ ( t )=2 G   s ( f   d )[Ω( t )cos(2π f   d   t−φ   s ( f   d ))+− Q   o  sin(2 πf   d   t−φ   s ( f   d ))]sin(2π f   d   t−φ   s ( f   d ))  (7)
 
     According to this compensation for the phase shift φ s (f d ), the final expression of the in-phase (e.g., Coriolis) component and the quadrature component is achieved after the filtering operation performed by filters  314   a  and  314   b , so that the in-phase (e.g., Coriolis) component is received by the in-phase (e.g., Coriolis) channel  336  and the quadrature component is received by the quadrature channel  338 , and they are decoupled as shown by equations (8) and (9) below: 
         x   sI ( t )= G   s ( f   d )Ω( t )  (8)
 
         x   sQ ( t )= G   s ( f   d ) Q   o   (9)
 
     The decoupling of the in-phase (e.g., Coriolis) and quadrature components resolves the crosstalk issue introduced by equations (4) and (5) above. Evaluation of the phase shift φ s (f d ) that allows decoupling the in-phase (e.g., Coriolis) channel  336  and the quadrature component is normally done by some trimming procedure at factory premises, using some test equipment. The method disclosed herein proposes a way to measure the phase shift φ s (f d ) in the device itself, when the device is in normal operating conditions. Furthermore, the method is suitable also to measure the gain G s (f d ) and to compensate it. Furthermore, since the method is suitable to work during normal operations of the device, it allows to track the variation of both φ s (f d ) and G s (f d ) during the device life, when the initial trimming is no more accurate due to variations occurred after the factory trimming. 
     In accordance with some embodiments of the disclosed method, one or more test signals may be injected into MEMS gyroscope  310  via quadrature electrodes  318 , which contribute to the quadrature signal  330  by picking up MEMS gyroscope  310  inaccuracies and/or design impairments based on temporary stresses (e.g., temperature) and/or permanent stresses (e.g., bend in the MEMS gyroscope  310 , imprecise soldering, attrition over the lifespan of the MEMS gyroscope  310 , etc.), and/or via self-test electrodes  322 , which contribute to the Coriolis (e.g., in-phase) signal  332 . The test signals modify the proof mass sense signal (e.g., by adding some new information useful to evaluate the phase shift φ s (f d ) and the gain G s (f d )) and displace proof mass  328  with the modified proof mass sense signal to produce an output signal (e.g., a proof mass output sense signal), which is detected by proof mass sense electrodes  324  and delivered to demodulator  308  at in-phase (e.g., Coriolis) channel  336  and quadrature channel  338  (e.g., as separate, demodulated in-phase and quadrature components). In some embodiments, the proof mass output sense signal may be demodulated based on the drive frequency to create a baseband signal, from which a plurality of demodulated signals (e.g., demodulated in-phase and quadrature components) are created (e.g., by demodulating the baseband signal by each frequency of the plurality of test signals) and delivered to demodulator  308 . 
     In some embodiments, the injected test signals may either occur as in-phase (e.g., Coriolis) or quadrature signals that include variations to be monitored to identify occurring errors (e.g., gain offset, phase offset, etc.) within the output signal around the drive frequency f d . In some embodiments, the test signals injected in MEMS gyroscope  310  may be one or more sinusoidal tones, in-band or out-of-band, of the force signals that the MEMS gyroscope  310  is intended to detect. If a test signal is out-of-band, the disclosed method is suitable to work during normal sensor operations. In some embodiments, processing circuitry of MEMS gyroscope  310  may be configured to generate from the output signal (e.g., the proof mass output sense signal) a sense displacement quadrature signal and a sense displacement in-phase signal, where such displacement signals include the received test signals injected in the MEMS gyroscope  310  via self-test electrodes  322  and/or quadrature electrodes  318 . 
       FIG.  4    shows a diagram depicting a sense transfer function of a MEMS (e.g., MEMS gyroscope  310 ,  102 , and/or  200 ) before demodulation in accordance with an embodiment of the present disclosure, and possible variations of the sense transfer function as taken at two different instants of the sensor life (e.g., after factory setting and after being altered due to a stress applied to the MEMS device, such as from soldering the MEMS to a circuit board).  FIG.  4    includes intended signal bandwidth  402 , transfer function changed due to stress  404 , transfer function at factory setting  406 , transfer function variation  408 , first tone frequencies  412 ,  416 , second tone frequencies  410 ,  418 , and drive frequency  414 . In some embodiments, any number of sinusoidal tones, composing a test signal, may be injected into the MEMS to generate a sense transfer function provided that at least one tone is injected at a first frequency f 1 , causing the test signal to generate a first spectral sample of the sense transfer function at f d −f 1  and a second spectral sample to be generated at f d +f 1 , where f d  is the drive frequency  414 . In some embodiments, the first frequency f 1  may be any value that is convenient for the utility of the MEMS. It will be understood that drive frequency  414  may also be any value that is convenient for the utility of the MEMS. In an example, the first frequency f 1  is less than the drive frequency f d . Although particular features are depicted in certain configurations for  FIG.  4   , it will be understood that features may be removed, modified, or substituted and that additional features may be added in certain embodiments. 
     Intended signal bandwidth  402  represents a frequency range of forces the MEMS is intended to sense (e.g., the Coriolis force). It will be understood that forces received via test and calibration electrodes may or may not be included in intended signal bandwidth  402 . As depicted by  FIG.  4   , the test signal is a sum of two sinusoidal signals (e.g. Tone 1 at a first frequency f 1  and Tone 2 at a second frequency f 2 ). It will be understood that the injected tones may either be in-band or out-of-band with respect to the intended signal bandwidth  402 . After the MEMS intrinsic modulation process, where in-phase (e.g., Coriolis) and quadrature components are respectively generated, 4 frequency samples (e.g., first tone frequencies  412 ,  416  and second tone frequencies  410 ,  418 ) of the sense transfer function are provided around drive frequency (f d )  414 . It will be understood that drive frequency  414  marks the center of the intended signal bandwidth  402  at which the MEMS is supposed to operate. First tone frequencies  412  (e.g., f d −f 1 ),  416  (e.g., f d +f 1 ) and second tone frequencies  410  (e.g., f d −f 2 ),  418  (e.g., f d +f 2 ) may be chosen outside the intended signal bandwidth  402  (e.g., where the MEMS is supposed to operate) in such a way that the test signal does not affect the normal operations of the MEMS. In some embodiments, the test signal may be injected in the MEMS at any arbitrary moment when the MEMS is operating normally. Transfer function changed due to stress  404  conveys a change in transfer function characteristic (e.g., phase, amplitude, etc.) during MEMS usage or lifecycle (e.g., due to temperature, bend, attrition of the MEMS, imprecise soldering, etc.). It will be understood that certain transfer function changes (e.g., temperature) are temporary while other changes (e.g., soldering, bending, usury, etc.) are often permanent. Transfer function at factory setting  406  represents a transfer function characteristic immediately after factory trimming, where processing circuitry of the MEMS determines and compensates for relevant variations of the sense transfer function (e.g., related to amplitude and phase) with respect to one or more test signals to generate a more precise sense transfer function. Transfer function variation  408  conveys the difference in values, at drive frequency (f d )  414 , of the transfer function changed due to stress  404  and the transfer function at factory setting  406  to show the amount of inaccuracy temporary stresses (e.g., temperature) and/or more permanent stresses (e.g., bend, usury, soldering, etc.) may impose on the MEMS. It will be understood that  FIG.  4    is a general example and may be applied both to variations of the transfer function phase or to variations of the transfer function gain, namely the terms φ s  (f d ) and G s  (f d ) in equations (2), (3), (4), (5), (6), (7), (8), and (9) listed above. First tone frequencies  412 ,  416  represent a first sinusoidal tone injected into the MEMS (e.g., to create a modified proof mass sense signal), via test and calibration electrodes (e.g., self-test and/or quadrature electrodes), at a first frequency. Two spectral points are generated based on the first frequency (e.g., f d −f 1  at  412  and f d +f 1  at  416 ) and each of their corresponding values with the transfer function changed due to stress  404  and the transfer function at factory setting  406  are depicted in  FIG.  4   . Second tone frequencies  410 ,  418  represent a second sinusoidal tone injected into the MEMS (e.g., to create the modified proof mass sense signal), via test and calibration electrodes (e.g., self-test and/or quadrature electrodes), at a second frequency. Two spectral points are generated based on the second frequency (e.g., f d −f 2  at  410  and f d +f 2  at  418 ) and each of their corresponding values with the transfer function changed due to stress  404  and the transfer function at factory setting  406  are depicted in  FIG.  4   . 
       FIG.  5    shows a diagram depicting a sense transfer function of a MEMS (e.g., MEMS gyroscope  310 ,  102 , and/or  200 ) after demodulation in accordance with an embodiment of the present disclosure, and possible variations of the sense transfer function as taken at two different instants of the sensor life (e.g., after factory setting and after being altered due to a stress applied to the MEMS device, such as from soldering the MEMS to a circuit board).  FIG.  5    includes transfer function changed due to stress  504  and transfer function at factory setting  506  after the demodulation process (i.e., shifted to the left by a quantity f d ) in addition to intended signal bandwidth  502  (corresponding to intended signal bandwidth  402  of  FIG.  4   ), transfer function variation  508 , first tone frequencies  512 ,  516 , second tone frequencies  510 ,  518 , and null frequency  514 . In some embodiments, a test signal comprising, e.g., any number of sinusoidal tones, may be injected into the MEMS to generate a sense transfer function provided that at least one tone is injected at a first frequency f 1 , causing, after demodulation, the test signal to generate a first spectral sample of the sense transfer function at −f 1  and a second spectral sample to be generated at +f 1 . In some embodiments, the first frequency f 1  may be any value that is convenient for the utility of the MEMS. It will be understood that null frequency  514  is equal to 0 Hz. Although particular features are depicted in certain configurations for  FIG.  5   , it will be understood that features may be removed, modified, or substituted and that additional features may be added in certain embodiments. 
     Intended signal bandwidth  502  represents a frequency range of forces where the MEMS is intended to sense (e.g., the Coriolis force). It will be understood that forces received via test and calibration electrodes may or may not be included in intended signal bandwidth  502 . The test signal is, in this example, a sum of two sinusoidal signals (e.g., Tone 1 at a first frequency f 1  and Tone 2 at a second frequency f 2 ), and  FIG.  5    depicts the two sinusoidal signals in the frequency domain after demodulation has been performed by an in-phase and quadrature (I&amp;Q) demodulator (e.g., demodulator  308 ). As a result, the frequency axis of  FIG.  5    has been shifted to the left by a factor of the drive frequency, f d , such that the transfer function variation  508  is evaluated at null frequency  514  (e.g., 0 Hz). It will be understood that the injected tones may either be in-band or out-of-band with respect to the intended signal bandwidth  502 . After the demodulation process (e.g., an in-phase and quadrature (I&amp;Q) demodulation process), 4 frequency samples (e.g., first tone frequencies  512 ,  516  and second tone frequencies  510 ,  518 ) of the sense transfer function are provided around null frequency  514 , which marks the center of the intended signal bandwidth  502 . First tone frequencies  512  (e.g., −f 1 ),  516  (e.g., +f 1 ) and second tone frequencies  510  (e.g., −f 2 ),  518  (e.g., +f 2 ) may be chosen outside the intended signal bandwidth  502  (e.g., where the MEMS is supposed to operate) in such a way that the test signal does not affect the normal operations of the MEMS. In some embodiments, the test signal may be injected in the MEMS at any arbitrary moment when the MEMS is operating normally. As described above, transfer function changed due to stress  404  conveys a change in transfer function characteristic (e.g., phase, amplitude, etc.) during MEMS usage or lifecycle (e.g., due to temperature, bend, attrition of the MEMS, imprecise soldering, etc.). It will be understood that certain transfer function changes (e.g., temperature) are temporary while other changes (e.g., soldering, bending, usury, etc.) are often permanent. As described above, transfer function at factory setting  406  represents a transfer function characteristic immediately after factory trimming, where processing circuitry of the MEMS determines and compensates for relevant variations of the sense transfer function (e.g., related to amplitude and phase) with respect to one or more test signals to generate a more precise sense transfer function. Transfer function variation  508  conveys the difference in values, at null frequency  514  (e.g., 0 Hz), of the transfer function changed due to stress  404  and the transfer function at factory setting  406  to show the amount of inaccuracy temporary stresses (e.g., temperature) and/or more permanent stresses (e.g., bend, usury, soldering, etc.) may impose on the MEMS. It will be understood that  FIG.  5    is a general example and may be applied to both variations of the transfer function phase or to variations of the transfer function gain, namely the terms φ s (f d ) and G s (f d ) in equations (2), (3), (4), (5), (6), (7), (8), and (9) listed above. First tone frequencies  512 ,  516  represent a first sinusoidal tone injected into the MEMS, via self-test and/or quadrature electrodes, at a first frequency. Two spectral points are generated after demodulation based on the first frequency (e.g., at  512  and +f 1  at  516 ) and each of their corresponding values with the transfer function changed due to stress  404  and the transfer function at factory setting  406  are depicted in  FIG.  5   . Second tone frequencies  510 ,  518  represent a second sinusoidal tone injected into the MEMS, via self-test and/or quadrature electrodes, at a second frequency. Two spectral points are generated after demodulation based on the second frequency (e.g., −f 2  at  510  and +f 2  at  518 ) and each of their corresponding values with the transfer function changed due to stress  404  and the transfer function at factory setting  406  are depicted in  FIG.  5   . 
     In some embodiments, at I&amp;Q demodulator (e.g., demodulator  308 ) output, it may be necessary to distinguish between positive and negative frequencies, in which case the I&amp;Q demodulator provides complex signals to satisfy this embodiment. Otherwise, by using a real demodulator providing the in-phase channel only and not the quadrature channel, the negative and the positive frequencies would not be distinguishable from each other, such that after demodulation the information required for interpolation at 0 Hz would be lost. Positive and negative frequencies may be distinguished from each other because a positive frequency has a phasor (i.e., the complex number built by the real part, or “I” component, and the imaginary part, or the “Q” component) that rotates counter-clockwise, while a negative frequency has a phasor that rotates clockwise. If the Q component is not recovered, there is no way to distinguish among the positive and the negative frequencies. In some embodiments, as opposed to estimating the entire parameters of the sense transfer function, it may only be necessary to measure the sense transfer function variations in a few spectral points (e.g., the offset in values between the transfer function changed to stress  404  and the transfer function at factory setting  406  at spectral values  510 ,  512 ,  516 , and  518 ) from a known status (e.g., the related values at factory trim vs their actual values), which may improve the precision of the estimations and/or compensation of the sense transfer function and reduce the complexity of the computation/hardware required in relevant digital signal processing. In some embodiments, untrimmed, non-fully compensated in-phase and quadrature components of an output signal (e.g., a proof mass output sense signal), as provided by equations (4) and (5) above, may be recovered to a trimmed form, as presented by equation 10 below by a matrix rotation: 
     
       
         
           
             
               
                 
                   
                     
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     Any further compensation of the demodulation phase coming from stresses (e.g., temperature, bend, soldering, usury, etc.) the MEMS may be subjected to during its lifecycle may be performed after the I&amp;Q demodulation process, fully digitally, by using digital rotation algorithms (e.g., a CORDIC—coordinate rotation digital computer). Digital rotation algorithms simplify processing circuitry architecture by avoiding complex mixed signal interaction between the digital signal processor and the phase-locked loop (PLL), enabling a fully digital phase compensation in a feed forward fashion. 
       FIG.  6    shows an illustrative feed forward block diagram for estimating and compensating for sense transfer function variation in accordance with an embodiment of the present disclosure. In the depicted embodiment, system  600  includes test and calibration electrodes  602  (corresponding to self-test electrodes  322  and/or quadrature electrodes  318  of  FIG.  3   ), MEMS gyroscope  310  and demodulator  308  of  FIG.  3   , capacitance-to-voltage (C2V) converter  606 , matrix rotation  610 , digital gain  612   a ,  612   b , transfer function errors estimation  614 , Tone 1 and Tone 2 demodulator  616 , signal generator  618 , digital-to-analog converter (DAC)  620 , driver  622 , and reference clock  624 . It will be understood that any number of tones may be processed by system  600 . Although particular components are depicted in certain configurations for system  600 , it will be understood that components may be removed, modified, or substituted and that additional components (e.g., electrodes, converters, filters, etc.) may be added in certain embodiments. 
     Test and calibration electrodes  602  may receive an analog test signal (e.g., a voltage or a current, etc.) from driver  622 , and may convert the test signal to either the in-phase (e.g., Coriolis) channel or the quadrature channel of MEMS gyroscope  310  to displace a proof mass internal to the MEMS gyroscope  310  and generate an output signal (e.g., a proof mass output sense signal). In some embodiments, injecting the test signal in the in-phase (e.g., Coriolis) channel results in in-phase (e.g., Coriolis) signals with a phase near 0° or −180° based on whether the drive resonant frequency is less than or greater than the sense resonant frequency. In some embodiments, injecting the test signal in the quadrature channel results in quadrature signals with a phase near −90° (e.g., mostly visible on the Quadrature channel). In some embodiments, injection of the test signal, via quadrature electrodes, may result in more accuracy (e.g., less phase and amplitude variation) as opposed to injection of the test signal via other types of calibration electrodes (e.g., self-test electrodes). It will be understood that test signals injected via the quadrature channel are less prone to jeopardize signals travelling along the in-phase (e.g., Coriolis) channel (e.g., the intended measurement channel). MEMS gyroscope  310  may include a suspended-spring mass system, a proof mass, the Coriolis (e.g., in-phase) channel (e.g., corresponding to an in-phase signal), the quadrature channel (e.g., corresponding to a quadrature signal), and proof mass sense electrodes for generating an output signal (e.g., a proof mass output sense signal) based on the displacement of the proof mass. In some embodiments, MEMS gyroscope  310  may be a variety of other sensors (e.g., an accelerometer, a barometer, an inertial measurement unit, a magnetometer, etc.). One or more test signals are injected into MEMS gyroscope  310 , via test and calibration electrodes  602 , to encode, via modulation, the in-phase signal and/or the quadrature signal and drive the proof mass such that the proof mass sense electrodes detect the displacement of the proof mass and generate an output signal (e.g., a proof mass output sense signal), which includes the in-phase signal and/or the quadrature signal. In some embodiments, MEMS gyroscope  310  may include processing circuitry that, e.g., is configured to receive the output signal (e.g., the proof mass output sense signal), via proof mass sense electrodes (e.g., moving capacitors), and generate additional signals (e.g., in-phase and quadrature displacement signals) from the output signal. Capacitance-to-voltage (C2V) converter  606  receives the output signal (e.g., the proof mass output sense signal) from MEMS gyroscope  310 , specifically from proof mass sense electrodes acting as moving capacitors, converts the received capacitance signal to a voltage signal, and feeds the voltage signal to demodulator  308 . The demodulator  308  extracts respective in-phase and quadrature components from the output signal. Additionally, demodulator  308  converts the in-phase and quadrature components from the analog to the digital domain. It will be understood that the order in which the demodulation and the analog-to-digital conversion are executed by demodulator  308  is not pertinent to the disclosure as described herein. It will be understood that demodulator  308  concurrently delivers the in-phase and quadrature components of the MEMS gyroscope  310  output signal to Tone 1 and Tone 2 demodulator  616  and matrix rotation  610 . Demodulator  308  also receives reference signals (e.g., cos(ω d ) and sin(ω d )) from signal generator  618  to serve as a baseline and contribute to identifying phase and/or gain offset when encoded with the in-phase (e.g., Coriolis) and quadrature components of the MEMS gyroscope  310  output signal. 
     Matrix rotation  610  (e.g., CORDIC—coordinate rotation digital computer) applies matrix rotation, as described by equation (10) above, to compensate for calculated phase error φ s (f d ) in accordance with phase estimation calculated by transfer function errors estimation  614 . The application of the compensation value by matrix rotation  610  updates with the same timescale of the transfer function errors estimation  614 . As a result, the processing circuitry of system  600  may be managed with a low-complexity hardware algorithm/unit working in a sequential fashion (e.g., using a CORDIC). Digital gain  612   a ,  612   b  respectively compensates for gain variation within the in-phase (e.g., Coriolis) component received at digital gain  612   a  and the quadrature component received at digital gain  612   b . Specifically, digital gain  612   a ,  612   b  uses data from transfer function errors estimation  614  to compensate for the variation gain of the term G s (f d ) described in equation (10) above. Transfer function errors estimation  614  receives, e.g., 8 real numbers generated by Tone 1 and Tone 2 demodulator  616  and generates an estimation of phase and amplitude or of their variations. In some embodiments, this operation may include computational complexity since it involves complex calculations (e.g., inverse trigonometric function—arctangent, squaring and square root, interpolation, etc.). Transfer function errors estimation  614  is intended to track variations of the sense transfer function due to stresses (e.g., bend, usury, soldering, etc.) or temperature in the order of the temperature gradient. The update rate of these computations may typically be slow (e.g., on the order of few times per seconds, or slower). In some embodiments, these operations may be managed in a sequential way by using a middle-low complexity computation unit (e.g., a single arithmetic logic unit (ALU)). Once the demodulation of the entire test signal from the drive frequency, f d , to the null frequency, 0 Hz, is executed, the test signal spectrum is similar to the diagram depicted in  FIG.  4   . To recover phase and amplitude information from the output signal (e.g., proof mass output sense signal) regarding the two test tones f 1  and f 2  after the demodulator  308 , a further step of demodulation is implemented using the in-phase (e.g., Coriolis) and quadrature components related to f 1  and f 2  generated by signal generator  618 . Tone 1 and Tone 2 demodulator  616  provides 4 complex signals, or equivalently 8 real signals, representing spectral samples of the sense transfer function H s , e.g., H s (ω d −ω 2 ), H s (ω d −ω 1 ), H s (ω d +ω 1 ), H s (ω d +ω 2 ), to transfer function errors estimation  614 . It will be also understood that, after demodulation, the sensor transfer function H s  is shifted to the left by a quantity f d , becoming the baseband sensor transfer function, thereinafter denoted by H bb , such that the following equation (100) holds: 
         H   bb (2π f )= H   bb (ω)= H   s (ω−ω d )= H   s (2π[ f−f   d ])  (100)
 
     It will be also understood that, for Tone 1 and Tone 2 spectral samples, the following equations (101) hold: 
         H   s (ω d −ω 2 )= H   bb (−ω 2 )
 
         H   s (ω d −ω 1 )= H   bb (−ω 1 )
 
         H   s (ω d +ω 1 )= H   bb (+ω 1 )
 
         H   s (ω d +ω 2 )= H   bb (+ω 2 )
 
         H   s (ω d )= H   bb (0)  (101)
 
     In some embodiments, Tone 1 and Tone 2 demodulator  616  may provide any number of complex signals, in accordance with signals received via signal generator  618  (e.g., sin(ω1t), cos(ω1t), sin(ω2t), cos(ω2t), where ω1=2πf1 and (ω2=2πf2), to transfer function errors estimation  614 . By interpolation of these four complex values at transfer function errors estimation  614 , or any other information derived thereof, the value of the sense transfer function at the drive frequency, fd, may be estimated, e.g., H s (ω d ), and from the sense transfer function the variation of the phase and/or gain of H s (ω d ) may be estimated, where ω d =2πf d . 
     Signal generator  618  is fed by a phase-locked loop (PLL) and serves to generate the test signal (e.g., cos(ω 1 t)+cos(ω 2 t), where ω 1 =2πf 1  and ω 2 =2πf 2 ), which is received by digital-to-analog converter (DAC)  620 . It&#39;ll be understood that signal generator  618  generates the low frequency signals needed to recover the relevant phase and amplitude information (e.g., cos(2πf 1 t), cos(2πf 2 t), sin(2πf 1 t), and sin(2πf 2 t)) after the demodulation process. Signal generator  618  may also generate reference signals (e.g., cos(ω d t) and sin(ω d t), where ω d =2πf d ) necessary for demodulator  308  to encode with the in-phase (e.g., Coriolis) and quadrature components of the MEMS gyroscope  310  output signal (e.g., proof mass output sense signal). DAC  620  converts the received test signal from the digital domain to the analog domain and feeds the analog converted test signal to driver  622  (e.g., a digital driver), which amplifies the test signal for driving the test and calibration electrodes  602 . In some embodiments, the noise added to the test signal by DAC  620  and driver  622 , the accuracy of the digital-to-analog conversion of the test signal by DAC  620 , and the delay DAC  620  and driver  622  impose on the path of injection into the MEMS gyroscope  310  may be significant due to any error, delay, or inaccuracy introduced on top of the test signal being reflected in a related inaccuracy in the estimation of sense transfer function coefficients {b k } and {c k }, as described by equations (11) through (18) below. Accordingly, DAC  620  and driver  622  provide high linearity and low delay, or at least a delay stable over temperature and with the device lifecycle stresses (e.g., bend, soldering, attrition, etc.). In some embodiments, a solution based on the digital drivers may be beneficial for the objectives of the disclosure described herein, which may include a sigma delta (SD) or a pulse-width modulated (PWM) DAC that enables the use of low delay/low noise digital drivers for which stability over a variety of temperatures and linearity is generally better than an analog drive or conversion architecture. Reference clock  624  drives the PLL, which then drives the clock distribution of system  600 . The clock distribution is balanced such that the clock arrives at every endpoint simultaneously, including the PLL&#39;s feedback input. 
       FIG.  7    shows an illustrative feedback block diagram for estimating and compensating for sense transfer function variation in accordance with an embodiment of the present disclosure. In the depicted embodiment, system  700  includes test and calibration electrodes  602  (corresponding to self-test electrodes  322  and/or quadrature electrodes  318  of  FIG.  3   ), MEMS gyroscope  310  and demodulator  308  of  FIG.  3   , capacitance-to-voltage (C2V) converter  606 , matrix rotation  610 , digital gain  612   a ,  612   b , Tone 1 and Tone 2 demodulator  616 , signal generator  618 , digital-to-analog converter (DAC)  620 , driver  622 , and reference clock  624  from system  600 . In addition, system  700  includes transfer function errors estimation  708  (e.g., a loop filter). It will be understood that any number of injected tones may be processed by system  700 . Although particular components are depicted in certain configurations for system  700 , it will be understood that components may be removed, modified, or substituted and that additional components (e.g., electrodes, converters, filters, etc.) may be added in certain embodiments. 
     The feedback block diagram of  FIG.  7    depicts ADC/I&amp;Q demodulator  308  receiving a voltage signal (e.g., an output signal of MEMS gyroscope  310 ) from C2V converter  606 , where ADC/I&amp;Q demodulator  308  converts the received voltage signal from the analog domain to the digital domain and proceeds to extract respective in-phase (e.g., Coriolis) and quadrature components from the voltage signal. It will be understood that the demodulation and the analog-to-digital conversion may be executed in any order that is convenient. It will be understood that ADC (analog-to-digital converter)/I&amp;Q demodulator  308  receives reference signals (e.g., cos(ω d ) and sin(ω d )) from signal generator  618  to serve as a baseline and contribute to identifying phase and/or gain variation when encoded with the in-phase (e.g., Coriolis) and quadrature components of the MEMS gyroscope  310  output signal. ADC/I&amp;Q demodulator  308  feeds the in-phase component and the quadrature component of the MEMS gyroscope  310  output signal (e.g., proof mass output sense signal) to matrix rotation  610  (e.g., CORDIC—coordinate rotation digital computer), which applies matrix rotation, as described by equation (10) above, to compensate for calculated phase error φ s (f d ) in accordance with phase estimation calculated by transfer function errors estimation  708 . The application of the compensation value by matrix rotation  610  updates with the same timescale of the transfer function errors estimation  708 . As a result, the processing circuitry of system  700  may be managed with a low-complexity hardware algorithm/unit working in a sequential fashion (e.g., using a CORDIC). Digital gain  612   a ,  612   b  respectively compensates for gain variation within the in-phase (e.g., Coriolis) component received at digital gain  612   a  and the quadrature component received at digital gain  612   b . Specifically, digital gain  612   a ,  612   b  uses data from transfer function errors estimation  708  to compensate for the variation gain of the term G s (f d ) described in equation (10) above. As opposed to system  600 , where Tone 1 and Tone 2 demodulator  616  receives the in-phase and quadrature components of the MEMS gyroscope  310  output signal from demodulator  308  (e.g., including phase and gain variation), in system  700 , Tone 1 and Tone 2 demodulator  616  receives the in-phase and quadrature components from digital gain  612   a ,  612   b  (e.g., after being compensated for both phase and gain variation). Once the demodulation of the entire test signal from the drive frequency, f d , to the null frequency, 0 Hz, is executed, the test signal spectrum is similar to the diagram depicted in  FIG.  5   . To recover phase and amplitude information from the output signal (e.g., proof mass output sense signal) regarding the two test tones f 1  and f 2  after the ADC/I&amp;Q demodulator  308 , a further step of demodulation is implemented using the in-phase (e.g., Coriolis) and quadrature components related to f 1  and f 2  generated by signal generator  618 . Tone 1 and Tone 2 demodulator  616  provides 4 complex signals, or equivalently 8 real signals, e.g., H s (ω d −ω 2 ), H s (ω d −ω 1 ), H s (ω d +ω 1 ), H s (ω d +ω 2 ), to transfer function errors estimation  708 . In some embodiments, Tone 1 and Tone 2 demodulator  616  may provide any number of complex signals, in accordance with signals received via signal generator  618  (e.g., sin(ω 1 t), cos(ω 1 t), sin(ω 2 t), cos(ω 2 t), where ω 1 =2πf 1  and ω 2 =2πf 2 ), to transfer function errors estimation  708 . By interpolation of these four complex values at transfer function errors estimation  708  (e.g., loop filter), or any other information derived thereof, the value of the sense transfer function at the drive frequency, f d , may be estimated, e.g., H s (f d ), and from the sense transfer function the variation of the phase and/or gain of H s (f d ) may be estimated. In order to implement the Loop Filter  708 , the gain and phase may be calculated, then the error of gain and phase with respect to some reference values (e.g., the related values at factory trim) may be processed by means of a PID (Proportional-Integral-Derivative) controller, such to get the correct values of the gain and phase that null the errors. 
       FIG.  8    shows an illustrative block diagram of a multi-tones demodulator in accordance with an embodiment of the present disclosure. Tones demodulator  800  (e.g., a multiple test signal demodulator) includes in-phase channel  804 , quadrature channel  806 , Tone 1 demodulator  808   a , Tone 2 demodulator  808   b , Tone 1 transfer function coefficients  812   a ,  812   b , and Tone 2 transfer function coefficients  814   a ,  814   b . In some embodiments, tones demodulator  800  may receive more than two tones to process and generate sense transfer function coefficients (e.g., spectral points). Although particular components are depicted in certain configurations for tones demodulator  800 , it will be understood that components may be removed, modified, or substituted and that additional components may be added in certain embodiments. 
     Tones demodulator  800  (e.g., a test signal demodulator) calculates the transfer function coefficients (e.g., the spectral points) for a MEMS sense transfer function, based on an injected test signal (e.g., a first tone with a first frequency, f 1 , and a second tone with a second frequency, f 2 ), at Tone 1 demodulator  808   a  and Tone 2 demodulator  808   b  respectively. Tone demodulator  800  corresponds with Tone 1 &amp; Tone 2 Demodulator  616  in  FIG.  6    and with Tone 1 &amp; Tone 2 Demodulator  616  in  FIG.  7   . In-phase channel  804  feeds an in-phase (e.g., Coriolis) component from an external I&amp;Q demodulator to Tone 1 demodulator  808   a  and Tone 2 demodulator  808   b , while quadrature channel  806  feeds a quadrature component from the external I&amp;Q demodulator to Tone 2 demodulator  808   b  and Tone 1 demodulator  808   a . In some embodiments, as depicted by  FIG.  7   , tones demodulator  800  may receive in-phase and quadrature components from digital gain outputs. Tone 1 demodulator  808   a , including a first internal I&amp;Q demodulator, uses the first tone with the first frequency, f 1 , as a demodulation frequency and executes integration and downsampling during a suitable time window in accordance with the fastest velocity of variation of the sense transfer function that the tones demodulator  800  is intended to track. Tone 2 demodulator  808   b , including a second internal I&amp;Q demodulator, uses the second tone with the second frequency, f 2 , as a demodulation frequency and executes integration and downsampling during a suitable time window in accordance with the fastest velocity of variation of the sense transfer function that the tones demodulator  800  is intended to track. Tones demodulator  800  outputs, via Tone 1 demodulator  808   a,  2 complex values, or equivalently 4 real values, in the form of Tone 1 transfer function coefficients  812   a ,  812   b . In addition, Tones demodulator  800  outputs, via Tone 2 demodulator  808   b,  2 complex values, or equivalently 4 real values, in the form of Tone 2 transfer function coefficients  814   a ,  814   b . Tones demodulator  800  feeds Tone 1 transfer function coefficients  812   a ,  812   b  and Tone 2 transfer function coefficients  814   a ,  814   b  to a transfer function errors estimation component, which uses the received spectral values to estimate a sense transfer function, e.g., H s (ω d ) and from the sense transfer function the variation of the phase and/or gain of H s (ω d ) may be estimated. 
       FIG.  9    shows an illustrative block diagram of a single tone demodulator in accordance with an embodiment of the present disclosure. Single tone demodulator  808  (e.g., a single test signal demodulator) includes high frequency signal processing  902 , low frequency signal processing  904 , in-phase channel  906 , quadrature channel  908 , sine component  910 , cosine component  912 , integration and downsampling  914   a - 914   d , and transfer function coefficients  916   a ,  916   b . Although particular components are depicted in certain configurations for single tone demodulator  808 , it will be understood that components may be removed, modified, or substituted and that additional components may be added in certain embodiments. 
     In-phase channel  906  travels from an external I&amp;Q demodulator (e.g., demodulator  308 ) to single tone demodulator  808  carrying an in-phase (e.g., Coriolis) signal, which includes sine component  910  (e.g., sin(ω n t)) and cosine component  912  (e.g., cos(ω n t)), and quadrature channel  908  travels from the I&amp;Q demodulator to single tone demodulator  808  carrying a quadrature signal, which also includes sine component  910  (e.g., sin(ω n t)) and cosine component  912  (e.g., cos(ω n t)). In some embodiments, as depicted by  FIG.  7   , single tone demodulator  808  may receive in-phase and quadrature components from digital gain outputs. In some embodiments, single tone demodulator  808  may either process a first tone with a first frequency, f 1  as demodulation frequency or a second tone with a second frequency, f 2 , as demodulation frequency such that single tone demodulator  808  actuates integration and downsampling  914   a - 914   d  on a suitable time window in accordance with the fastest velocity of variation of a sense transfer function that the single tone demodulator  808  is intended to track. It will be understood that single tone demodulator  808  calculates transfer function coefficients  916   a ,  916   b  (e.g., spectral points) based on either the first tone with the first frequency, f 1 , or the second tone with the second frequency, f 2 . In some embodiments, the velocity of variation is dictated by a temperature gradient specified for the single tone demodulator  808  and may be in a range of some degrees (e.g., Kelvin) per minute. Accordingly, the integration and downsampling  914   a - 914   d  may be in the order of seconds or very few Hz. It will be understood that high frequency signal processing  902  includes the mixing of the sine component  910  (e.g., sin(ω n t)) and the cosine component  912  (e.g., cos(ω n t)) with the in-phase (e.g., Coriolis) signal, fed by the in-phase channel  906  to integration and downsampling  914   a ,  914   b , and the quadrature signal, fed by the quadrature channel  908  to integration and downsampling  914   c ,  914   d , providing demodulation of both the in-phase (e.g., Coriolis) signal and the quadrature signal. High frequency signal processing  902  additionally includes integration and downsampling  914   a - 914   d , which, e.g., may integrate and downsample received in-phase (e.g., Coriolis) and quadrature signals based on a first frequency, f 1 , or a second frequency, f 2 , with the velocity of variation of the sense transfer function. Low frequency signal processing  904  receives the outputs of integration and downsampling  914   a - 914   d  and processes the respective in-phase (e.g., Coriolis) and quadrature components, via an inverse trigonometric function (e.g., arctangent) and third order interpolation, to estimate phase. In addition, low frequency signal processing  904  estimates gain by computing a square root of a sum of squares and a third order interpolation of the received in-phase (e.g., Coriolis) and quadrature signals. Single tone demodulator  808  outputs, via low frequency signal processing  904 , transfer function coefficients  916   a ,  916   b , which include 2 complex values, or equivalently 4 real values, to a transfer function errors estimation component that uses the received spectral values to estimate the sense transfer function, e.g., H s (2πf d ) and any phase and/or gain variation associated with H s (2πf d ). 
       FIG.  10    shows an illustrative block diagram of phase estimation in accordance with an embodiment of the present disclosure. Phase estimation  1000  includes low frequency signal processing  904 , transfer function coefficients  1002   a - 1002   d , phase calculation  1004   a - 1004   d , phases  1008   a - 1008   d , and third order interpolation  1006 . It will be understood that phase estimation  1000  may receive transfer function coefficients (e.g., spectral points) driven by any number of tones having a commensurate number of frequencies. Although particular components are depicted in certain configurations for phase estimation  1000 , it will be understood that components may be removed, modified, or substituted and that additional components may be added in certain embodiments. 
     Phase estimation  1000  partially composes low frequency signal processing  904 , which receives transfer function coefficients  1002   a - 1002   d , e.g., 4 spectral points H s (ω d −ω 2 ), H s (ω d −ω 1 ), H s (ω d +ω 1 ), H s (ω d +ω 2 ), or equivalently, according to equation (100) and (101), H bb (−ω 2 ), H bb (−ω 1 ), H bb (+ω 1 ), H bb (+ω 2 ), based on either a first tone with a first frequency, f1, or a second tone with a second frequency, f 2 , from high frequency signal processing. Transfer function coefficients  1002   a - 1002   d  include 4 complex values (e.g., real {H bb (−ω 2 )}, imaginary {H bb (−ω 2 )}  1002   a , real {H bb (−ω 1 )}, imaginary {H bb (−ω 1 )}  1002   b , real {H bb (+ω 1 )}, imaginary {H bb (+ω 1 )}  1002   c , and real {H bb (+ω 2 )}, imaginary {H bb (+ω 2 )}  1002   d ), or equivalently 8 real values. Low frequency signal processing  904  estimates phase  1008   a - 1008   d  by receiving each of transfer function coefficients  1002   a - 1002   d  and performing a phase calculation  1004   a - 1004   d  (e.g., with an inverse trigonometric function—arctangent). Third order interpolation  1006  of drive frequency, f d , receives each of phases  1008   a - 1008   d  and calculates a missing phase complex value, e.g., in the middle of the received spectral points, e.g., φ s (f d )—phase error. It will be understood that third order interpolation  1006  determines 4 coefficients, as depicted by equations (11) below: 
       Arg{ H   s (2π[ f   d   −f   2 ])}= c   0   +c   1 ( f   d   −f   2 )+ c   2 ( f   d   −f   2 ) 2   +c   3 ( f   d   −f   2 ) 3  
 
       Arg{ H   s (2π[ f   d   −f   1 ])}= c   0   +c   1 ( f   d   −f   1 )+ c   2 ( f   d   −f   1 ) 2   +c   3 ( f   d   −f   1 ) 3  
 
       Arg{ H   s (2π[ f   d   +f   1 ])}= c   0   +c   1 ( f   d   +f   1 )+ c   2 ( f   d   +f   1 ) 2   +c   3 ( f   d   +f   1 ) 3  
 
       Arg{ H   s (2π[ f   d   +f   2 ])}= c   0   +c   1 ( f   d   +f   2 )+ c   2 ( f   d   +f   2 ) 2   +c   3 ( f   d   +f   2 ) 3  
 
     Once third order interpolation  1006  solves the system depicted by equations (11), the phase error at f d  is determined as: 
       Arg{ H   s (2π f   d )}=φ s ( f   d )= c   0   +c   1 ( f   d )+ c   2 ( f   d ) 2   +c   3 ( f   d ) 3   (12)
 
     It will be understood that this formula applies not only to f d , but to any other frequency comprised within [f d −f 2 ] and [f d +f 2 ]. 
     It will be understood that low frequency signal processing  904  may receive and process any number of transfer function coefficients. In some embodiments, phase estimation  1000  is not required to calculate the entire spectrum of transfer function coefficients, but only their variations with respect to the injected test signal, e.g., including the first tone of the first frequency, f 1 , and the second tone of the second frequency, f 2 . To track the relevant phase variations (e.g., due to bend, soldering, attrition, temperature, etc.) occurring during the lifecycle of the MEMS, equations (11) above may be modified as follows, where H s0  (f d ) denotes the sense transfer function after factory trimming: 
       Arg{ H   s0 (2π[ f   d   −f   2 ])}−Arg{ H   s (2π[ f   d   −f   2 ])}= c   0   +c   1 ( f   d   −f   2 )+ c   2 ( f   d   −f   2 ) 2   +c   3 ( f   d   −f   2 ) 3  
 
       Arg{ H   s0 (2π[ f   d   −f   1 ])}−Arg{ H   s (2π[ f   d   −f   1 ])}= c   0   +c   1 ( f   d   −f   1 )+ c   2 ( f   d   −f   1 ) 2   +c   3 ( f   d   −f   1 ) 3  
 
       Arg{ H   s0 (2π[ f   d   +f   1 ])}−Arg{ H   s (2π[ f   d   +f   1 ])}= c   0   +c   1 ( f   d   +f   1 )+ c   2 ( f   d   +f   1 ) 2   +c   3 ( f   d   +f   1 ) 3  
 
       Arg{ H   s0 (2π[ f   d   +f   2 ])}−Arg{ H   s (2π[ f   d   +f   2 ])}= c   0   +c   1 ( f   d   +f   2 )+ c   2 ( f   d   +f   2 ) 2   +c   3 ( f   d   +f   2 ) 3   (13)
 
     Once third order interpolation  1006  solves the system depicted by equations (13), the phase error at f d  is determined as: 
       Arg{ H   s0 (2π f   d )}−Arg{ H   s (2π f   d )}=Δφ s ( f   d )= c   0   +c   1 ( f   d )+ c   2 ( f   d ) 2   +c   3 ( f   d ) 3   (14)
 
     Δφ s (f d ) represents the additional phase correction to be added to the phase factory trimming to compensate for additional phase impairments that may occur after factory trimming. The transfer function coefficients  1002   a - 1002   d  (e.g., spectral points) of the sense transfer function at factoring trimming, e.g., H s (ω d −ω 2 ), H s (ω d −ω 1 ), H s (ω d +ω 1 ), H s (ω d +ω 2 ), or any related information required by the method, may be stored in memory (e.g., one-time programmable (OTP) memory, flash memory, etc.) to be read and used when needed to calculate φ s (f d ) and Δφ s (f d ). It will be understood that equation (14) applies not only to f d , but to any other frequency comprised within [f d −f 2 ] and [f d +f 2 ]. 
       FIG.  11    shows an illustrative block diagram of amplitude estimation in accordance with an embodiment of the present disclosure. Amplitude estimation  1100  includes low frequency signal processing  904 , transfer function coefficients  1002   a - 1002   d , amplitude calculations  1104   a - 1104   d , amplitudes  1108   a - 1108   d , and third order interpolation  1106 . It will be understood that amplitude estimation  1100  may receive transfer function coefficients (e.g., spectral points) driven by any number of tones having a commensurate number of frequencies. Although particular components are depicted in certain configurations for amplitude estimation  1100 , it will be understood that components may be removed, modified, or substituted and that additional components may be added in certain embodiments. 
     Amplitude estimation  1100  partially composes low frequency signal processing  904 , which receives transfer function coefficients  1002   a - 1002   d , e.g., 4 spectral points H s (ω d −ω 2 ), H s (ω d −ω 1 ), H s (ω d +ω 1 ), H s (ω d +ω 2 ), based on either a first tone with a first frequency, f 1 , or a second tone with a second frequency, f 2 , from high frequency signal processing. Transfer function coefficients  1002   a - 1002   d  include 4 complex values (e.g., real {H bb (−ω 2 )}, imaginary {H bb (−ω 2 )}  1002   a , real {H bb (−ω 1 )}, imaginary {H bb (−ω 1 )}  1002   b , real {H bb (+ω 1 )}, imaginary {H bb (+ω 1 )}  1002   c , and real {H bb (+ω 2 )}, imaginary {H bb (+ω 2 )}  1002   d ), or equivalently 8 real values. Low frequency signal processing  904  estimates amplitude  1108   a - 1108   d  by receiving each of transfer function coefficients  1002   a - 1002   d  and performing an amplitude calculation  1104   a - 1104   d , respectively (e.g., with a square root of a sum of squares). Third order interpolation  1106  of drive frequency, f d , receives each of amplitudes  1108   a - 1108   d  and calculates a missing complex value in the middle of the received spectral points, e.g., G s  (f d )—amplitude variation. It will be understood that third order interpolation  1106  determines 4 coefficients, as depicted by equations (15) below: 
       | H   s (2π[ f   d   −f   2 ])|= b   0   +b   1 ( f   d   −f   2 )+ b   2 ( f   d   −f   2 ) 2   +b   3 ( f   d   −f   2 ) 3  
 
       | H   s (2π[ f   d   −f   1 ])|= b   0   +b   1 ( f   d   −f   1 )+ b   2 ( f   d   −f   1 ) 2   +b   3 ( f   d   −f   1 ) 3  
 
       | H   s (2π[ f   d   +f   1 ])|= b   0   +b   1 ( f   d   +f   1 )+ b   2 ( f   d   +f   1 ) 2   +b   3 ( f   d   +f   1 ) 3  
 
       | H   s (2π[ f   d   +f   2 ])|= b   0   +b   1 ( f   d   +f   2 )+ b   2 ( f   d   +f   2 ) 2   +b   3 ( f   d   +f   2 ) 3   (15)
 
     Once third order interpolation  1106  solves the system depicted by equations (15), the amplitude variation at f d  is determined as: 
       | H   s (2 πf   d   |=G   s ( F   d )= b   0   +b   1 ( f   d )+ b   2 ( f   d ) 2   +b   3 ( f   d ) 3   (16)
 
     It will be understood that this formula applies not only to f d , but to any other frequency comprised within [f d −f 2 ] and [f d +f 2 ]. 
     In some embodiments, amplitude estimation  1100  is not required to calculate the entire spectrum of transfer function coefficients, but only their variations with respect to the injected test signal, e.g., including the first tone of the first frequency, f 1 , and the second tone of the second frequency, f 2 . To track the relevant amplitude variations (e.g., due to bend, soldering, attrition, temperature, etc.) occurring during the lifecycle of the MEMS, equations (15) above may be modified as follows, where H s0 (f d ) denotes the sense transfer function after factory trimming: 
       | H   s0 (2π[ f   d   −f   2 ])|−| H   s (2π[ f   d   −f   2 ])|= b   0   +b   1 ( f   d   −f   2 )+ b   2 ( f   d   −f   2 ) 2   +b   3 ( f   d   −f   2 ) 3  
 
       | H   s0 (2π[ f   d   −f   1 ])|−| H   s (2π[ f   d   −f   1 ])|= b   0   +b   1 ( f   d   −f   1 )+ b   2 ( f   d   −f   1 ) 2   +b   3 ( f   d   −f   1 ) 3  
 
       | H   s0 (2π[ f   d   +f   1 ])|−| H   s (2π[ f   d   +f   1 ])|= b   0   +b   1 ( f   d   +f   1 )+ b   2 ( f   d   +f   1 ) 2   +b   3 ( f   d   +f   1 ) 3  
 
       | H   s0 (2π[ f   d   +f   2 ])|−| H   s (2π[ f   d   +f   2 ])|= b   0   +b   1 ( f   d   +f   2 )+ b   2 ( f   d   +f   2 ) 2   +b   3 ( f   d   +f   2 ) 3   (17)
 
     Once third order interpolation  1106  solves the system depicted by equations (17), the amplitude variation at f d  is determined as: 
       | H   s0 (2π f   d )|−| H   s (2π f   d )|=Δ G   s ( f   d )= b   0   +b   1 ( f   d )+ b   2 ( f   d ) 2   +b   3 ( f   d ) 3   (18)
 
     ΔG s (f d ) represents the additional gain correction to be added to the gain factory trimming to compensate for gain impairments that may occur after the factory trimming. The transfer function coefficients  1002   a - 1002   d  (e.g., spectral points) of the sense transfer function at factory trimming, e.g., H s0 (ω d −ω 2 ), H s0 (ω d −ω 1 ), H s0 (ω d +ω 1 ), H s0 (ω d +ω 2 ) or any related information required by the method, may be stored in memory (e.g., one-time programmable (OTP) memory, flash memory, etc.) to be read and used when needed to calculate G s (f d ) and ΔG s (f d ). It will be understood that this equation (18) applies not only to f d , but to any other frequency comprised within [f d −f 2 ] and [f d +f 2 ]. 
       FIG.  12    shows an illustrative feed forward block diagram including an equalizer for estimating and compensating for sense transfer function variation in accordance with an embodiment of the present disclosure. In the depicted embodiment, system  1200  includes test and calibration electrodes  602  (corresponding to self-test electrodes  322  and/or quadrature electrodes  318  of  FIG.  3   ), MEMS gyroscope  310  and demodulator  308  of  FIG.  3   , capacitance-to-voltage (C2V) converter  606 , matrix rotation  610 , digital gain  612   a ,  612   b , transfer function errors estimation  614 , Tone 1 and Tone 2 demodulator  616 , signal generator  618 , and reference clock  624  from system  600 . In addition, system  1200  includes equalizer  1202 , pulse-width modulated (PWM)/sigma delta (SD) digital-to-analog converter (DAC)  1204 , and low delay driver  1206 . It will be understood that any number of injected tones may be processed by system  1200 . Although particular components are depicted in certain configurations for system  1200 , it will be understood that components may be removed, modified, or substituted and that additional components (e.g., electrodes, converters, filters, etc.) may be added in certain embodiments. 
     Demodulator  308  extracts an in-phase (e.g., Coriolis) component and a quadrature component from a received voltage signal and converts the respective in-phase and quadrature (I&amp;Q) components from the analog domain to the digital domain. It will be understood that the order in which the demodulation and the analog-to-digital conversion are executed by demodulator  308  is not limited in the disclosure described herein. Demodulator  308  also receives reference signals (e.g., cos(ω a ) and sin(ω d )) from signal generator  618  (e.g., a numerically-controlled oscillator) to serve as a baseline and contribute to provide reference signals for phase and/or gain offset when demodulating the in-phase and quadrature components of the MEMS gyroscope  310  output signal (e.g., proof mass output sense signal). Demodulator  308  feeds the in-phase and quadrature components to equalizer  1202 , which flattens the gain and linearizes the phase of the in-phase and quadrature components and, as a result, makes the interpolation process (e.g., the third order interpolation of drive frequency, f d , used for phase and amplitude estimation) simpler and more accurate. In some embodiments, equalizer  1202  may be implemented in a feedback architecture. In some embodiments, the frequency separation between the drive resonant frequency and the sense resonant frequency is small (e.g., near the test tone frequencies f 1  and f 2 ) such to cause a large phase variation of the sense transfer function near the frequency separation that may jeopardize the interpolation process. System  1200  further includes PWM/SD DAC  1204 , which enables the use of low delay driver  1206  (e.g., a digital driver) for which stability over a variety of temperatures and linearity is generally better than an analog driver or conversion architecture. 
       FIG.  13    shows an illustrative flowchart for reducing error in a MEMS device (e.g., a MEMS gyroscope such as MEMS gyroscope  310 , MEMS gyroscope  200 , and/or MEMS gyroscope  102 ) in accordance with an embodiment of the present disclosure. Although particular steps are depicted in certain configurations for  FIG.  13   , it will be understood that steps may be removed, modified, or substituted and that additional steps may added in certain embodiments. At step  1302 , processing circuitry (e.g., a control system including phase-locked loop (PLL) trim  302 , PLL and Phase trim  304 , and drive loop  306 ) drives drive electrodes (e.g., drive electrodes  316 ) of a microelectromechanical system (MEMS) sensor (e.g., MEMS gyroscope  310 ) with a drive signal having a drive frequency, fa. It will be understood that the drive signal displaces a drive mass (e.g., drive mass  326 ), which produces a drive sense signal that is detected by drive sense electrodes (e.g., drive sense electrodes  320 ). In some embodiments, a drive loop (e.g., drive loop  306 ) of the processing circuitry may shift the phase of the drive signal by any degree amount depending on an amount of offset the MEMS (e.g., MEMS gyroscope  310 ) experiences due to external stresses. In some embodiments, a PLL &amp; Phase Trim (e.g., PLL &amp; Phase Trim  304 ) of the processing circuitry may be synchronized with the oscillation frequency of the control system and produce a higher frequency clock at a multiple (e.g., a harmonic) of the drive frequency, f d . At step  1304 , processing circuitry applies (e.g., injects) a plurality of test signals to a proof mass sense signal, via test and calibration electrodes (e.g., quadrature electrodes and/or self-test electrodes  322 ), to create a modified proof mass sense signal. The plurality of test signals includes a quadrature component (e.g., quadrature signal  330 ), injected by quadrature electrodes (e.g., quadrature electrodes  318 ) or by self-test electrodes (e.g., self-test electrodes  322 ), which picks up design impairments based on temporary stresses (e.g., temperature) and/or permanent stresses (e.g., bend in the MEMS, stress after soldering, attrition and/or aging over the lifespan of the MEMS, etc.), and an in-phase (e.g., Coriolis) component (e.g., Coriolis signal  332 ), detected by self-test electrodes (e.g., self-test electrodes  322 ), which is the intended measurement component. In some embodiments, the plurality of test signals may include a first tone with a first frequency, f 1 , and a second tone with a second frequency, f 2 , among a plurality of frequencies. It will be understood that the plurality of test signals creates a modified proof mass sense signal by mixing, via modulation, with the proof mass sense signal. At step  1306 , processing circuitry drives a gyroscope (e.g., proof mass  328 ) of the MEMS sensor (e.g., MEMS gyroscope  310 ) based on the modified proof mass sense signal, which displaces the gyroscope and generates, as described by step  1308 , an output signal (e.g., a proof mass output sense signal), including in-phase (e.g., Coriolis) and quadrature components from the plurality of test signals, from the MEMS sensor. Sense electrodes (e.g., moving capacitors such as sense electrodes  324 ) detect the displacement of the proof mass (e.g., proof mass  328 ) with respect to the sense electrodes, generating a capacitive signal (e.g., the output signal), and feed the output signal (e.g., the proof mass output sense signal) to an in-phase and quadrature (I&amp;Q) demodulator (e.g., demodulator  308 ). 
     At step  1310 , processing circuitry, via the I&amp;Q demodulator (e.g., demodulator  308 ), extracts an in-phase (e.g., Coriolis) component and a quadrature component from the output signal (e.g., proof mass output sense signal). In addition, the I&amp;Q demodulator (e.g., demodulator  308 ) converts the in-phase component and the quadrature component from the analog domain to the digital domain. It will be understood that the order in which the demodulation and the analog-to-digital conversion are executed by the I&amp;Q demodulator (e.g., demodulator  308 ) is not limited by the disclosure described herein. In some embodiments, the I&amp;Q demodulator (e.g., demodulator  308 ) may receive reference signals (e.g., cos(ω d ) and sin(ω d )) from a signal generator (e.g., signal generator  618 ) to serve as a baseline and contribute to identifying phase and/or gain variation when encoded with the in-phase (e.g., Coriolis) and quadrature components of the MEMS (e.g., MEMS gyroscope  310 ) output signal (e.g., proof mass output sense signal). In at least some example approaches, an in-phase component and/or quadrature component may be compensated or corrected. For example, as illustrated in  FIG.  13   , at step  1312  processing circuitry compensates the in-phase component and the quadrature component, via matrix rotation (e.g., matrix rotation  610 ), for calculated phase error in accordance with phase estimation and, via digital gain (e.g., digital gain  612   a ,  612   b ), for gain variation. Matrix rotation (e.g., matrix rotation  610 ) applies matrix rotation to compensate for calculated phase error φ s (f d ) in accordance with phase estimation calculated by transfer function errors estimation (e.g., transfer function errors estimation  614 ). The application of the compensation value by matrix rotation (e.g., matrix rotation  610 ) updates with the same timescale of transfer function errors estimation (e.g., transfer function errors estimation  614 ). Digital gain (e.g., digital gain  612   a ,  612   b ) uses data from transfer function errors estimation (e.g., transfer function errors estimation  614 ) to compensate for variation gain of the term G s (f d ). At step  1314 , processing circuitry processes the in-phase component and the quadrature component based on a plurality of frequencies of the plurality of test signals. In some embodiments, the processing of the in-phase component and the quadrature component may include Tone 1 and Tone 2 demodulator (e.g., Tone 1 and Tone 2 demodulator  616 ) to receive, in some embodiments, in-phase and quadrature components from the I&amp;Q demodulator (e.g., demodulator  308 ) and deliver, in accordance with signals (e.g., sin(ω 1 t), cos(ω 1 t), sin(ω 2 t), cos(ω 2 t), where ω 1 =2πf 1  and ω 2 =2πf 2 ) received via the signal generator (e.g., signal generator  618 ), a number of complex and/or real numbers to the transfer function errors estimation. At step  1316 , via the transfer function errors estimation (e.g., transfer function errors estimation  614 ), processing circuitry determines a change in demodulation phase and/or sensitivity based on the processing of the in-phase component and the quadrature component. It will be understood that the processing circuitry may also determine a change in demodulation gain based on the processing of the in-phase component and the quadrature component. Transfer function errors estimation (e.g., transfer function errors estimation  614 ) interpolates received complex and/or real numbers to estimate the value of the sense transfer function at the drive frequency (e.g., H s (f d )) and the variation of the phase and/or gain of H s (f d ). In some embodiments, the processing may include a matrix rotation (e.g., matrix rotation  610 —CORDIC) to compensate for calculated phase error φ s (f d ) in accordance with phase estimation calculated by a transfer function errors estimation (e.g., transfer function errors estimation  614 ), and a digital gain (e.g. digital gain  612   a ,  612   b ) to compensate for gain variation of the term G s (f d ) within the in-phase (e.g., Coriolis) component and the quadrature component. 
     The foregoing description includes exemplary embodiments in accordance with the present disclosure. These examples are provided for purposes of illustration only, and not for purposes of limitation. It will be understood that the present disclosure may be implemented in forms different from those explicitly described and depicted herein and that various modifications, optimizations, and variations may be implemented by a person of ordinary skill in the present art, consistent with the following claims.