SIGNAL PROCESSING - Biomechanics and Motor Control of Human Movement - page 29

or harmonic analysis . Figure 2.10 shows plots of time-domain signals and
their equivalents in the frequency domain.
Care must be used when analyzing or interpreting the results of any har-
monic analysis. Such analyses assume that each harmonic component is
present with a constant amplitude and phase over the total analysis period.
Such consistency is evident in Equation (2.10), where amplitude V n and phase
θ n are assumed constant. However, in real life each harmonic is not constant
in either amplitude or phase. A look at the calculation of the Fourier coef-
ficients is needed for any signal x (t ) . Over the period of time T ,usingthe
discrete Fourier transform , we calculate n harmonic coefficients.

T

2
T

a n = x (t ) cos nω 0 tdt (2.13)

0

T

2
T

b n = x (t ) sin nω 0 tdt (2.14)

0

a n +

c n = b n

tan − 1 a n
b n

θ n = (2.15)

It should be noted that a n and b n are calculated average values over the
period of time T . Thus, the amplitude c n and the phase θ n of the n th harmonic
are average values as well. A certain harmonic may be present only for part of
the time T , but the computer analysis will return an average value, assuming
that it is present over the entire time. The fact that a n and b n are average
values is important when we attempt to reconstitute the original signal as is
demonstrated in Section 2.2.4.5.
The digital equivalent of the Fourier transform is important to review
because it gives us some insight into the number of calculations that are
necessary. In digital form, Equations (2.13) and (2.14) for N samples during
the period T :

N

2
N

a n = x i cos (nω 0 i /N ) (2.16)

=

i 0

N

2
N

b n = x i sin (nω 0 i /N ) (2.17)

i = 0

For each of the n harmonics, N calculations are necessary. The number
of harmonics that can be analyzed is from the fundamental ( n 1) up to the

Nyquist frequency, which is when there are two samples per sine or cosine
wave or when n

=

N / 2. Therefore, for N / 2 harmonics, there are N 2 / 2 calcu-
lations necessary for each of the sine or cosine coefficients. The total number

=