Biomedical Engineering Reference
In-Depth Information
=
+
0
time (s)
0.003
Fig. 1.5.
Components of a complex oscillation. The sound wave at the top is not a
simple or
pure
oscillation. Instead, it is the sum of two simple oscillations called its
components
, shown below. The components of a complex sound are usually enumer-
ated in order of decreasing period (or increasing frequency): the first component
is the one with the largest period of all the components, the second component
is the one with the second largest period, and so on. Note that the period of the
complex sound is equal to the period of its first component. The frequency of the
first component is also called the
fundamental frequency
F
n
=
n/T
are called the
harmonics
. The frequencies of the harmonic com-
ponents are multiples of the fundamental frequency, i.e.,
F
n
=
nF
1
, and are
known as
harmonic frequencies
.
The timbre of a sound is determined by the quantities and relative weights
of the harmonic components present in the signal. This constitutes what is
usually referred to as the
spectral content
of a signal.
1.3.2 Adding up Waves
We can create strange signals by adding simple waves. How strange? In
Fig. 1.6, we show a fragment of a periodic signal of a very particular shape,
known as a triangular function or
sawtooth
. In the figure, we show how we
can approximate the triangular function by superimposing and weighting six
harmonic functions. The simulated triangular function becomes more similar
to the original function as we keep on adding the right harmonic components
to the sum.
A mathematical result widely used in the natural sciences indicates that
a large variety of functions of time (for example, that representing the vari-
ations of pressure detected by a microphone when we record a note) can be
expressed as the sum of simple harmonic functions such as the ones illustrated
in Fig. 1.6, with several harmonic frequencies. This means that if the period
characterizing our complex note
f
(
t
)is
T
, we can represent it as a sum of
harmonic functions of frequencies
F
1
=1
/T
,
F
2
=2
/T
, ...,
F
n
=
nF
1
,that
is, the fundamental frequency and its harmonics:
