Time-frequency Representations Application in
Psychological Testing
REIZ Romulus
1, GORDAN Cornelia
11 University of Oradea, Romania,
Department of Electronics and Telecommunications, Faculty of Electrical Engineering and Information Technology, Universitatii Str. No. 1, 410087 Oradea, Romania, E-Mail1: rreiz@uoradea.ro
Abstract – A psychological test is a test that is designed to measure one aspect of human behavior. These tests are usually designed to evaluate a person’s ability to complete tasks that were individual's performance on certain tasks that have usually been requested in advance. Usually a test score is used to compare with other results to measure the individual’s performance regarding cognitive ability, aptitude, personality, etc. One such test is the so called “finger tapping” test, designed to measure the integrity of the neuromuscular system and examine motor control. There are several ways to perform such a test. The purpose of this paper isn’t to study the finger tapping test which is well documented in the literature, but to develop if possible a simple way of performing such a test. Using the method presented in the paper a nonstationary signal was obtained and it was analyzed using the Short-time Fourier time frequency representation to obtain the signals frequency and its variation in time. The results presented in the paper show that this method can be used to perform the test and the frequency and spatial amplitude of the obtained tapping signal can be determined easily.
Keywords: finger tapping test; time frequency representations; instantaneous frequency;
I. INTRODUCTION
Time frequency representations are useful signal processing tools that have applications in many various research fields like: speech recognition, seismic wave analysis, medical signal processing, and so on. In the field of medical research, many times the tests produce a signal that can be analyzed with signal processing tools. One such test is the so called “Finger tapping” test[1],[2]. The tapping test is a neurological test that tries to evaluate the muscular control and motor ability of the tested individual. The test is used also in evaluating the motor control change in cases of brain damage. The test was used from the 19th century. The test is executed with the subject putting the hand on a flat surface having the palm fixed and not moving. The fingers should be all extended. The index finger is the one that is used in the test. The subject is asked to tap
the index finger on the surface as fast as possible within a time interval. The amplitude and frequency of the fingers movement are both important in the test.
There are several methods developed to measure the finger tapping speed [3],[4], and the fingers position over time. Some use a simple switch that is activated by the finger hitting the table. The switch is connected to a counter that is used to find the frequency of the tapings. Other methods use magnetic tracking systems to record the finger’s position over time. There are also methods based on optical recording of the finger’s motion using video cameras. In this case infrared markers on the fingers are used and tracked by the software [4].
Some of these test methods produce an electronic signal that can be analyzed using signal processing tools. This signal has three parameters that are important in the finger tapping test, namely time, amplitude and frequency. Usual signal processing tools such as the Fourier transform can provide only a limited amount of information about the signal. In the case of a stationary signal that information is usually enough. A signal is stationary if its characteristics are not changing with time. However, in the case of the tapping test the signal that is obtained can vary in time, namely its amplitude and frequency are both time dependant. These types of signals are called non-stationary and usually have a more complex time-frequency spectrum.
Time frequency representations are best suited to analyze these types of signals, because they provide information about the signals frequency and amplitude at a certain moment in time, so the temporal information is not lost as is the case with the classic Fourier transform. Nonstationary signals can be analyzed with the Short-time Fourier transform by dividing the signal into small blocks, and considering the signal as being a stationary one in one of these blocks. Then the spectrum is calculated for each of the blocks. The blocks can be obtained using a window function that is localized in time. The calculated spectrums of the signal blocks give us the Short-time Fourier transform (STFT) that is a two-dimensional representation of the signal in the time-frequency plane [5]:
(
)
( ) (
)
STFT jȦIJ
x
TF t,Ȧ ∞ x IJ w IJ t e− dIJ
−∞
=
³
− (1)The window function that is used to obtain the representation can be a Hamming, Hanning …… It is often a unit energy founction:
( )
2L2w t =1 (2)
The peak values of the obtained Short-time Fourier representation can be used to estimate the instantaneous frequency of the signal [6].
II. ANALYSIS METHOD
As it was mentioned before, the “tapping test” can be made in many different ways. The method that was used in this paper has the advantage of simplicity. There are not required specialized hardware or software tolls to perform the test; all that was used was a piezoelectric microphone and a computer soundcard to acquire the signal. The tested person has his hand laid on a table with the palm down and fingers extended. The subject is required to hit the table’s surface with his index finger over a period of time as fast as possible. Accordingly, a simple microphone was used to record the sound of the finger hitting the table, and that signal was then processed to obtain a signal that is an approximation of the finger’s position over time, similar to the finger tapping signal obtained using passive markers placed on the finger and image based motion detectors. The microphone was a piezoelectric one placed on the table near the subject’s hand. A sampling frequency of 8 kHz was used, which is enough because the main interest is in the lower frequency components of the recorded signal. Usual finger tapping speeds have frequencies of only a few Hz so a lower sampling frequency can be used to record the signal. We used a higher sampling frequency and then performed a downsampling, obtaining a signal that has a sampling frequency of about 400 Hz. In figure 1 is presented an example of recorded signal, the spikes in the plot representing the recording of the sound produced when the finger hits the table.
0 2 4 6 8 10 12 14
x 104 -0.5
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
Samples
Fig. 1: Recorded signal during the test
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
Time(s)
Fig. 2: Signal after downsampling
In figure 2 it is presented a portion of 5 seconds extracted from the initial signal presented in figure 1, after the operation of downsampling. As it can be seen in this figure, in one second the signal has about 3 spikes, which means that the finger’s frequency was at about 3 Hz. This frequency can be obtained from the signal by calculating its spectrum, however the waveform presented in figure 2 has a complex spectrum having both low frequency and high frequency components. Since after the operation of downsampling the new sampling frequency was 400 Hz, the signal had frequency components spanning from 0 Hz to 200 Hz. As it was mentioned before, we are only interested in the low frequency components of this spectrum, so the high frequency components can be eliminated. These high frequency components represent useless information, mainly from the noise that the microphone and the soundcard produce.
The simplest way to eliminate these components is to use a low-pass filter. We used a direct-form transposed filter with coefficients from an 8–th order lowpass Butterworth design. The low pass cutoff frequency was chosen as 15 Hz. The low-pass filtering and the downsampling before were performed using a Matlab script. The low-pass filtered signal corresponding to the waveform from figure 2 is presented in figure 3.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
Time(s)
A
m
pl
it
ude
Fig. 3: Low-pass filtered signal
The signal presented in figure 3 is similar to a signal that is obtained using other finger movement tracking methods. Its amplitude corresponds to the spatial amplitude of the finger movement, and its frequency shows the tapping speed. The obtained signal from the low pass filter was then used to generate a time-frequency representation. A Hamming window function used to calculate the time-frequency representation. Other windowing functions provided similar results. Since low frequency components were the ones that were important, a long window function was used, this way improving low frequency resolution. The Short time Fourier representation of the low-pass filtered signal is presented in figure 4, where it is obvious that the signal has a frequency of about 3 Hz, clearly visible in the time-frequency plane. The use of a time-frequency representation has the advantage of having both the time and frequency information available and also the intensity of the signal’s energy is related to its amplitude. The representation in figure 4 is also an image that can be used to compare it with previously recorded finger tapping patterns. Maybe some pattern recognition algorithms [7] could be used to develop an automatic diagnostic system, where the time-frequency representation can be compared to previously recorded ones corresponding to the subject’s age and gender. Any change of the patterns in the time-frequency plane could provide useful information about motor function problems or brain damage. Also, a time-frequency representation is very useful when there are changes of the signals frequency. These changes in frequency can occur during the test for instance when an external stimulus (optical acoustical etc.) is applied to the subject. To test if the frequency changes can be observed in the time-frequency plane, a signal was generated, where a deliberate change in the finger tapping frequency occurred. A time plot of this test signal is presented in figure 5, the change in tapping frequency being observable in the middle of the waveform.
Time [s]
F
requenc
y
[
H
z
]
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0
2 4 6 8 10 12 14 16 18 20
Fig. 4: Short time Fourier representation of the tapping signal
0 1 2 3 4 5 6 7 8 9 10
x 104 -0.6
-0.4 -0.2 0 0.2 0.4 0.6 0.8
Samples
A
m
p
lit
u
d
e
Fig. 5: Recorded signal with a tapping frequency change
In figure 6 is presented a part of this signal where the actual tapping frequency takes place. It can be observer that in the first part of the signal the frequency is higher then it becomes lower. This signal was then processed using the presented algorithm. The low past filtered signal is presented in figure 7 where the change in tapping speed can also be observed.
The Short-time Fourier representation of this signal is presented in figure 8 and the change of frequency can now be even measured in the time-frequency plane. There are several ways to obtain the instantaneous frequency of a signal from it’s time-frequency representation [8]. One method is to use the peaks of the time-frequency representation. This method was used to obtain the instantaneous frequency of the signal and its change in time. The instantaneous frequency is presented in figure 9 where the change of the signal frequency is clearly visible. The signal had initially a frequency of 5 Hz then it decreases to about 3 Hz which is in accordance to the test signals parameters. So the time-frequency representation can be used to observe the changes in tapping frequency. As finger tapping frequency is an indicator of the severity of brain injuries and is used to evaluate the recovery from such injuries, time-frequency representations could provide an easy way to obtain information about this parameter.
0 0.5 1 1.5 2 2.5 3 3.5
x 104 -0.6
-0.4 -0.2 0 0.2 0.4 0.6 0.8
Samples
A
m
pl
it
ude
Fig. 6: The frequency change is noticeable in the downsampled signal
0 0.5 1 1.5 2 2.5 3 3.5 4 -0.08
-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
Time(s)
Fig. 7: Low-pass filtered signal with frequency change
Time [s]
F
requ
enc
y
[
H
z
]
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0 1 2 3 4 5 6 7 8
Fig. 8: Short time Fourier representation of the signal with a change in frequency
0 200 400 600 800 1000 1200 1400 1600 1800
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2
F
req
uenc
y
(H
z
)
Fig. 9: Instantaneous frequency of the test signal, detected from the time-frequency representation
III. CONCLUSIONS
In this paper is presented a method for estimating the instantaneous frequency of the signals obtained during the so called “finger tapping” neurological tests. A simple method was used to obtain these signals and time frequency representations were used to obtain the results. Due to the nonstationary nature of these signals, using time-frequency representations, namely the Short-time Fourier transform has many advantages. An analysis is performed regarding the ability of the Short-time Short-time-frequency representation to detect frequency changes in the finger tapping speed. The relations that exist between IF and time-frequency distributions is used to obtain the instantaneous frequency of the recorded tapping signal. Further work can be done using more advanced time-frequency representations, such as smoothed and reassigned versions of time frequency distributions or even the wavelet transform. Also more signals are required to create a database with signals corresponding to subjects having different ages, genders and presenting symptoms related to neurological disabilities.
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