Biomedical Engineering Reference
In-Depth Information
FIGURE 16.5
:
The Sinc function. Note the ripples
lobe spreading. More noise is introduced in the spectrum, which is a characteristic that
is most often forgotten. So the question an engineer must consider is “What is the best
solution or what is the best window?” According to the characteristics of the data, the
analyst must select a function that produces the least error and complications.
Let us begin with the most common window, which is the “Rectangular window.”
The simplest way to model a finite record length is through the usage of the rectangular
function (often called boxcar). The function equals one in the sampling interval and it is
zero everywhere else. The Window Weighting function is given by (16.1).
w
(
n
)
=
1;
For
n
=
0
to N
−
1
(16.1)
Consider a signal,
x
(
t
), with a Fourier Transform,
X
(
f
), and that the signal is
defined in some finite period of time zero (0) to
T
. The observed signal is multiplied
by the window Weighting Function
w
(
t
) in the time domain. The resulting function is
the product of
w
(
t
) and
x
(
t
). The major limitation of the new function is that it is zero
outside the time interval (observation window).
The Fourier transform of the window function
w
(
t
) is the well-known
SINC
function. The
SINC
(
x
) is defined as sin(
x
)
+
/
x
, which is given in Fig. 16.5.
16.3.2 The Rectangular Window and the Gibbs Phenomenon
The rectangular function truncates a signal in a discontinuous manner at its edges. The
Fourier transform approximates the function with a linear combination of sines and
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