NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
2017-09-28 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
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We repeat the derivation of the preceding section, but this time
we treat the error term more carefully.
Again we want to approximate 
with
an 
th-order
polynomial:
is the ''remainder term'' which we will no longer assume is zero.
Our problem is to find 
so as to minimize over some interval 
containing 
.
There are many ''optimality criteria'' we could choose. The one
that falls out naturally here is called ''Padé'' approximation.
Padé approximation sets the error value and its first
derivatives to zero at a single chosen point, which we take to be
. Since
all 
''degrees of freedom'' in the polynomial coefficients 
are
used to set derivatives to zero at one point, the approximation is
termed ''maximally flat'' at that point. Padé approximation comes
up often in signal processing. For example, it is the sense in
which Butterworth lowpass filters
are optimal. (Their frequencyreponses
are maximally flat at dc.) Also,
Lagrange interpolation filters can be shown to maximally flat
at dc in the frequency domain.
Setting in
the above
polynomial approximation produces
where we have used the fact that the error is to be exactly zero at
.
Differentiating the polynomial approximation and setting
gives
where we have used the fact that we also want the slopeof
the error to be exactly zero at .
In the same way, we find
for 
, and the first derivatives of the remainder term are all zero. Solving these
relations for the desired constants yields the th-order
Taylor series expansion of about
the point
as before, but now we better understand the remainder term.
From this derivation, it is clear that the approximation error
(remainder term) is smallest in the vicinity of .
All degrees of freedomin the polynomial coefficients were
devoted to minimizing the approximation error and its derivatives
at . As
you might expect, the approximation error generally worsens as
gets
farther away from 
.
To obtain a more uniform approximation over some interval
in, other
kinds of error criteria may be employed. This is classically called
''economization of series,'' but nowadays we may simply call
itpolynomial approximation
under different error criteria. In
Matlab, the function polyfit(x,y,n) will find the
coefficients of a polynomial 
of
degree n that fits the data y over the points
x in a
least-squares sense. That is, it minimizes
where 
.
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