Trigonometry: Trigonometric functions
Arbitrary periodic signals
An arbitrary signal is periodic with period (also known as vibration time) when is the smallest positive number with the property that
For such a signal, the frequency is again defined as
As a rule, is the number of vibrations per second; one measures the frequency in hertz (Hz).
The equilibrium value of such a signal is understood ad the mean value over one period.
The amplitude of is the maximum deviation from the equilibrium value.
The graph of the signal
The amplitude can be calculated exactly: the function takes on the displayed time interval the maximum value for and this value is equal to . The amplitude of the signal is thus equal to .
The above example may seem artifical, but you come across many periodic signals in practice. We give two examples, one from biomechanics and the other from tidal analysis.
This example comes from biomechanics and is about the knee angle (in degrees) during hopping.
Measurements on video images of the movement are plotted in the below figure along with the graph of an approximation with two sinusoids and a constant. The mathematical formula of this approach is
This is an example of tidal analysis. We have chosen here for the analysis of the tide in Sewells Point (Virginia, USA) on March 23-25 March 2006. Measured data are plotted in the below figurealong with the graph of an approximation with three sinusoids and a constant. The mathematical formula for this approach is