Integration

Integration is the operation that integrates a signal according to time.

Note: As the primary application of this method is to convert acceleration data into velocity, the method which is implemented contains several steps that are specifically designed to avoid or limit unwanted effects generated by the integration step itself.

The processing steps applied to the signal when using Integration are as follows:

  1. Remove DC: the average value (overall mean value) of the time signal is removed from the signal by subtraction.

    A non-zero mean can lead to incorrect high magnitude values during the integration step.

  2. Up-sample 4x: the time signal is resampled to four times its original sampling frequency.

    Integration algorithms are known to create artifacts at high frequencies. By up-sampling the signal before integration, you ensure that the artifacts will be created outside the frequency band of interest.

  3. Integrate: the time signal is then integrated using the cumulative trapezoidal method.

    This method appears to be the most appropriate when dealing with acceleration and velocity time data.

  4. Down-sample x1/4: the current time signal is resampled to the sampling frequency of the original signal.

    To avoid aliasing, the resampling function applies a low-pass filter before the resampling itself. This ensures that the potential high-frequency artifacts are removed from the signal.

  5. High-pass filter 20 Hz: a 6th order high-pass Butterworth filter with a 20 Hz cutoff frequency is applied.

    This filtering step aims to remove potential constant components introduced by the integration step (the integration method can introduce a low frequency trend evolving all along the signal, potentially hiding the real phenomenon in which you are interested).

Note: Because of the different processing steps applied before and after the integration algorithm itself, using integration then differentiation on a signal will not revert exactly to the original signal.