Filtering Data

Filtering Data

Definitions

Cut-off frequency: Frequency where the gain is 0.5 or -6 dB. In filter design, the cut-off frequency is normally where the gain is (sqrt(0.5), so that the power gain (which is proportionally to the square of the Voltage amplitude) is 0.5 or 10log(0.5)=-3 dB.

Pass band: Frequency range where the gain is ideally 1

Stop band: Frequency range where the gain is ideally 0

Transition band: Frequency range where the gain makes a transition between pass band and stop band.

FFT Low Pass filter

Filtering is performed in the frequency domain and the filter stops the high frequency components in the data, that are higher than the cutoff frequency. The frequency components of the signal are extracted using Fast Fourier Transform (FFT). This method is computationally faster and allows sharpening and smoothing.

A filter using The filtering is performed as follows:

The trailing pen-up part is removed.
A resting signal is prepended and appended (consisting of the first and the last sample, respectively, for: SampleFrequency/(FilterFrequency (1 - 1/Sharpness).
A half cosine wave of at least 0.25 * number of samples of the trial will be appended. Appending will be such that the new number of samples will be the smallest power of 2. Appending ensures also a cyclic movement. Subsequently, the Fast Fourier Transform (FFT) is performed.
The resulting complex spectrum is then multiplied with the desired number of zero-lag differentiators (2*pi*f for velocity, (2*pi*f)**2 for acceleration, (2*pi*f)**3 for jerk)
Then the complex spectrum is multiplied with the specified zero-lag, zero-ripple low-pass filter frequency characteristic
Finally, the reverse FFT is performed and the prepended and appended data are removed.

Transition band = Filter Frequency * (1-1/Sharpness) to Filter Frequency * (1+1/Sharpness)

Example: If the FilterFrequency is 7.0 Hz, the sinusoidal transition band is between 3.0 Hz and 11.0 Hz.

Filter Frequency selection

To select an appropriate low-pass filter frequency, consider:

~ The typical handwriting (velocity) frequency spectrum has a peak around 5 Hz and tapers to zero at 10 Hz. This frequency can be estimated by 2 / DeltaT, where DeltaT = Average Stroke Duration.

~ Compare the (velocity) frequency spectra before and after filtering and verify that the frequency peak associated with the movement signal is little affected and the generally flat motor noise and equipment noise spectra are suppressed. An optimal Wiener filter suppresses 50% where signal and noise amplitude are equal.

~ Start with a filter frequency of 10 Hz, which is appropriate in most normal cases.

~ When velocity, acceleration, or jerk signals curves look "too wild", DECREASE the filter frequency (See > Charting_Processed_Trials)

~ Change filter frequency and verify that the end results vary only marginally, otherwise INCREASE the filter frequency.

~ Do not increase the filter frequency beyond the Sampling Frequency / 4.

~ Fluent, complex handwriting movements may require lower filter frequencies than single, discrete strokes.

~ Slower movements may require lower filter frequencies.

~ Less accurate digitizers may require lower filter frequencies.

~ Higher sampling rates may require lower filter frequencies.

~ Use the same filter frequency when comparing different participants or groups.

Filter Frequency (Hz)	Half Width of the Attenuation Band (Hz)	Pass Band from 0 to .. (Hz)	Stop Band from .. to Sampling Frequency/2 (Hz)	Application
Sampling Frequency/2	0	Sampling Frequency/2	None	No filtering
23.5	13.5	10	37	The least low-pass filtering
18	10	8	28
15	8.5	6.5	23.5
12	6.8	5.3	20.8
10	5.7	4.3	15.7	Default filter frequency
8	4.5	3.5	12.5
7	4	3	11	Fluent, slower movements in patients and children
3	1.7	1.3	4.7
0.5	0.3	0.2	0.8	Leaves only the left-to-right progression in handwritten words
0	0	0	0	Complete suppression, only the average location remains