SIGNAL PROCESSING - Biomechanics and Motor Control of Human Movement - page 28

Figure 2.10 Relationship between a signal as seen in the time domain and its equiv-
alent in the frequency domain.

of the fundamental frequency f 0 . These higher frequencies are called harmon-
ics . The third harmonic is 3 f 0 , and the tenth harmonic is 10 f 0 . Any perfectly
periodic signal can be broken down into its harmonic components. The sum
of the proper amplitudes of these harmonics is called a Fourier series .
Thus, a given signal V (t ) can be expressed as:

V (t ) = V dc + V 1 sin ( ω 0 t + θ 1 ) + V 2 sin (2 ω 0 t + θ 2 ) +···

+ V n sin ( nω 0 t + θ n ) (2.10)

where ω 0 = 2 π f 0 , and θ n is the phase angle of the n th harmonic.
For example, a square wave of amplitude V can be described by the Fourier
series of odd harmonics:

sin ω 0 t

4 V
π

1
3 sin 3 ω 0 t

1
5 sin 5 ω 0 t

V (t ) = + + +··· (2.11)

A triangular wave of duration 2 t and repeating itself every T seconds is:

1
2 +

2
π

2 2

2 Vt
T

2
3 π

V (t ) = cos ω 0 t + cos 3 ω 0 t +··· (2.12)

Several names are given to the graph showing these frequency components:
spectral plots, harmonic plots , and spectral density functions . Each shows the
amplitude or power of each frequency component plotted against frequency;
the mathematical process to accomplish this is called a Fourier transformation