132
D. Rocchesso: Sound Processing
primitive function
transforms
Laplace Transform
exponential function
which is valid for regular functions. The reader can justify the (44) intuitively
by thinking of the derivative of F (x) as a ratio of increments. The increment at
the numerator is given by the difference of two areas obtained by shifting the
right edge by dx. The increment at the denominator is dx itself. Called m the
average value taken by f () in the interval having length dx, such value converges
to f (x) as dx approaches zero.
F (x) is also called a primitive function of f (x), where the article a subtends
the property that indefinite integrals can differ by a constant. This is due to the
fact that the derivative of a constant is zero, and it justifies the fact that the po-
sition of the first integration edge doesn't come into play in the relationship (44)
between a function and its primitive.
At this point, it is easy to be convinced that the availability of a primitive
F (x) for a function f (x) allows to compute the definite integral between any
two edges a and b by the formula
b
a
f (u)du = F (b) - F (a) .
(45)
We encourage the reader to find the primitive functions of polynomials,
sinusoids, and exponentials. To acquire better familiarity with the techniques
of derivation and integration, the reader without a background in calculus is
referred to chapter VIII of the book [25].
A.8
Transforms
The analysis and manipulation of functions can be very troublesome opera-
tions. Mathematicians have always tried to find alternative ways of expressing
functions and operations on them. This research has expressed some transforms
which, in many cases, allow to study and manipulate some classes of functions
more easily.
A.8.1
The Laplace Transform
The Laplace Transform was introduced in order to simplify differential calculus.
The Laplace transform of a function y(t), t R is defined as a function of the
complex variable s:
Y
L
(s) =
+
-
y(t)e
-st
dt, s C ,
(46)
where is the region where the integral is not divergent. The region is always
a vertical strip in the complex plane, and within this strip the transform can be
inverted with
y(t) =
1
2j
+j
-j
Y
L
(s)e
st
ds, t R .
(47)
The edges of the integration (47) indicate that the integration is performed
along a vertical line with abscissa .
Example 1. The most important transform for the scope of this book is
that of the causal complex exponential function, which is defined as
y(t) =
e
s
0
t
t 0 , s
0
C
0
t < 0
.
(48)
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