134
D. Rocchesso: Sound Processing
kernel of the Fourier
transform
spectrum
magnitude spectrum
phase spectrum
Z transform
A.8.2
The Fourier Transform
The Fourier transform of y(t), t R, can be obtained as a specialization of the
Laplace transform in the case that the latter is defined in a region comprising
the imaginary axis. In such case we define
Y () = Y
L
(j) ,
(54)
or, in detail,
Y () =
+

y(t)e
jt
dt ,
(55)
where j indicates a generic point on the imaginary axis. Since the kernel of the
Fourier transform is the complex sinusoid (i.e., the complex eponential) having
radial frequency , we can interpret each point of the transformed function as
a component of the frequency spectrum of the function y(t). In fact, given a
value =
0
and considered a signal that is the complex sinusoid y(t) = e
j
1
t
,
the integral (55) is maximized when choosing
0
=
1
, i.e., when y(t) is the
complex conjugate of the kernel
. The codomain of the transformed function
Y () belongs to the complex field. Therefore, the spectrum can be decomposed
in a magnitude spectrum and in a phase spectrum.
A.8.3
The Z Transform
The domains of functions can be classes of numbers of whatever kind and nature.
If we stick with functions defined over rings, particularly important are the
functions whose domain is the ring of integer numbers. These are called discrete
variable functions, to distinguish them from functions of variables defined over
R or C, which are called continuousvariable functions.
For discretevariable functions the operators derivative and integral are re
placed by the simplest operators difference and sum. This replacement brings a
new definition of transform for a function y(n), n Z:
Y
Z
(z) =
+
n=
y(n)z
n
, z C .
(56)
The transform (56) is called Z transform and the region of convergence is a
ring
of the complex plane. Within this ring the transform can be inverted.
Example 3. The Z transform of the discretevariable causal exponential
Y
Z
(z) =
+
n=
y(n)z
n
=
+
n=0
e
z
0
n
z
n
=
+
0
(e
z
0
z
1
)
n
=
1
1  e
z
0
z
1
,
(57)
14
Often the Fourier transform is defined as a function of f , where 2f =
15
Exercise: find the Fourier transform of the causal complex exponential (48), with s
0
=
+ j
0
, and show that it has maximum magnitude for =
0
.
16
A ring here is the area between two circles and not an algebraic structure.
17
The latter equality in (57) is due to the identity
+
n=0
a
n
=
1
1  a
, a < 1, which can be
verified by the reader with a = 1/2.