Digital Signal Processing Reference
In-Depth Information
The spectral analysis of deterministic signals consists of a decomposition based
on simpler signals (sine curves for example), similar to the way a point is marked in
space using its 3 coordinates. It is thus necessary to define an inner product, which is
used to measure the projection of a signal on a basic element.
Let
x
and
y
be two signals; their inner product 〈
x
,
y
〉
is defined continuously and
discretely respectively by:
+∞
+∞
=
∑
∫
() ()
() ()
x y
,
=
x t
y
*
t dt
x y
,
x k
y
*
k
[2.1]
−∞
k
=−∞
The energy of a signal
x
is defined by the inner product 〈
x
,
x
〉. The set of signals
(with continuous or discrete time) with finite energy, along with the inner product
defined above, constitutes a vector space. Very often, the basis elements selected
(such as the exponential basis of the Fourier transform) do not verify the finite
energy property. A rigorous mathematical processing requires the knowledge of the
theory of distributions; we will be content here with an intuitive introduction.
In the case of periodic signals with a known continuous period
T
or discrete
period
N
, the inner product has the following form:
N
−
1
1
T
1
=
∑
∫
() ()
() ()
x y
,
=
x t y
*
t dt
x y
,
x k
y
*
k
[2.2]
T
N
0
k
=
0
Thus the inner product 〈
x
,
x
〉
is the power of the periodic signal. The set of
periodic signals (with continuous or discrete time) with finite power, along with the
inner product defined above, constitute a vector space.
These concepts are dealt with in greater detail in section 2.2.2, particularly in the
case of the Fourier transform, in honor of the French mathematician J.B. Fourier
(1768-1830), which consists in taking cisoid functions as basis vectors. We will
present beforehand some functions and series required for the development of this
transform.
2.2. Transform properties
2.2.1.
Some useful functions and series
The unit constant
1
ℜ
is a function that is always equal to 1; for all
t
:
()
=
[2.3]
1
t
1
ℜ
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