Biomedical Engineering Reference
In-Depth Information
CHAPTER
14
Truncation of the Infinite
Fourier Transform
Problems in the accuracy of the Fourier Transform occur when adherence to the “Dirichlet
Conditions” have not been met. Recall the Dirichlet condition requirements on the signal
to be transformed as
1.
a finite number of discontinuities,
2.
a finite number of maxima and minima, and
3.
that the signal be absolutely convergent (14.1)
f
(
t
)
dt
T
<
∞
(14.1)
o
It is impractical to solve an infinite number of transform coefficients; therefore,
the engineer must decide how many Fourier coefficients are needed to represent the
waveform accurately. How to make a decision on where to truncate the series is a very
important part of signal processing.
Truncation
means that all terms after the
n
th term are dropped, resulting in
n
finite
number of terms to represent the signal. However, not including all the coefficients results
in an error referred to as the “Truncation error.” The truncation error,
ε
n
, is defined as
the difference between the original function
f
(
t
) and the partial sum,
s
n
(
t
) of the inverse
transformed truncated Fourier terms.
ε
n
=
f
(
t
)
−
s
n
(
t
)
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