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D. Rocchesso: Sound Processing
independent variable
argument
dependent variable
codomain
inverse function
c = x + i y
0
x
y
-
c = x - i y
Figure 1: Geometric interpretation of a complex number
When every value of the variable x is associated with one and only one value
of another variable y we say that y is a function of x, and we write
y = f (x) .
(4)
x is said to be the independent variable (argument) while y is the dependent
variable, and the set of values that it takes for different assumed by x in its
domain is called the codomain. If, for each x
1
= x
2
, f (x
1
) = f (x
2
), then domain
and codomain have a biunivocal correspondence. In that case the roles of domain
and codomain can be inverted, and it is possible to define an inverse function
x = f
-1
(y). In general, functions can have more than one independent variable,
thus indicating a relation among many variables.
Often functions are defined by means of algebraic expressions, and associated
with domains and interpretations for the variables. For instance, the pitch h (in
Hz) of the note produced by an ideal string can be expressed by the function
h =
1
2l
t
d
,
(5)
where l is the length of the string in meters, t is the string tension in Newton,
and d is the density per unit length (Kg/m). This concise expression allows
to represent the pitch of a note whatever are the values of length, tension,
and density, as long as these values belong to the domain of non-negative real
numbers (indicated by R
+
).
Functions can be graphically represented in the cartesian plane. The abscissa
corresponds with an independent variable, and the ordinate corresponds to the
dependent variable. If we have more than one dependent variable, only one is
represented in abscissa, and the other ones are set to constant values.
For example, fig. 2 shows the function (5), with values of tension and den-
sity
set to 952N and 0.0367Kg/m, respectively. The domain of string lengths
ranges from 0.5m to 4.0m.
The chart of fig. 2 can be obtained by a simple script in Octave or Matlab:
r=0.0367; t=952;
% definitions of density and tension
l=[0.5:0.01:4.0]; % domain for the string length
3
These values are appropriate for the piano note C2.