Biomedical Engineering Reference
In-Depth Information
TABLE 14.1:
Convergence Table
f
(
t
)
JUMP-IN
IMPULSE-IN
DECREASING
a
n
WAVEFORM
f
(
t
)
Square wave
f
(
t
)
1/
n
Discontinuities
f
(
t
)
f
(
t
)
1/
n
2
Triangle wave
f
(
t
)
f
(
t
)
1/
n
3
Parabolic/
sinusoid
···
−−−
f
k
−1
(
t
)
f
k
(
t
)
n
k
1
/
Smoother
determining the rate of convergence, a basic expression was formulated called the Law
of Convergence.
The law covering the manner in which the Fourier coefficients diminish with
increasing “
n
” is expressed in the number of times a function must be differentiated to
produce a jump discontinuity. For the
k
th derivative, the convergence of the coefficients
will be on the order of 1
n
k
+1
. For example, if that derivative is the
k
th, then the
coefficients will converge at the rate shown in (13.4).
/
M
n
k
+1
M
n
k
+1
|
|
≤
|
|
≤
a
n
and
b
n
(13.4)
where
M
is a constant, which is dependent on
f
(
t
).
There are two ways to show convergence. Let us look at Table 14.1, the Con-
vergence Table. The left most column of the table denotes the type of waveform with
the right most column describing the waveform as going from with discontinuities to
smoother. The second and third columns labeled “Jump-in” and “Impulse-in,” respec-
tively.
Jump-in
means that discontinuity occurs in the function after successive differ-
entiation, where the discontinuities occur in the
square-like waveform
.
Impulse-in
means
what derivative (successive differentiation) of the function will result in an
impulse train
.
The fourth column gives the general expression for convergence of the coefficients. From
the table one may conclude that the smoother the function, the faster the convergence.
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