Pavel Holoborodko
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Central Differences
The most common way of computing numerical derivative of a function
at any point
is to approximate
by some
polynomial
in the neighborhood of . It is expected that if selected neighborhood of
is sufficiently small then
approximates
near
well and we can assume that
.
Let’s consider this approach in details (or go directly to the table of formulas).
At first, we sample
at the
(
is odd) equidistant points around
:
,
where
is some step.
Then we interpolate points
by polynomial of
degree:
. Its coefficients
are found as a solution of system of linear equations:
By our assumption
can be approximated by the derivative of the constructed interpolating polynomial:
To illustrate the process let’s consider case when
for simplicity):
. Here interpolating parabola are defined by the system (we assume
After subtracting first equation from the last we get final result:
In the same way we can obtain expressions for any
. Formulas for
are listed below:
These expressions are very widely used in numerical analysis and commonly refered as central(finite) differences. As it can be
clearly seen they have simple anti-symmetric structure and in general difference of -th order can be written as:
,
where
are coefficients derived by procedure described above.
It is easy to see that if
is a polynomial of a degree
, then central differences of order
give precise values for
derivative at any point. This follows from the fact that central differences are result of approximating
by polynomial.
If
is a polynomial itself then approximation is exact and differences give absolutely precise answer.
To differentiate a digital signal
we need to use h=1/SamplingRate and replace
case derivative of a signal
is found by
Frequency response of central differences is:
Magnitude responses for
are drawn below:
by
in the expressions above. In this
Red dashed line is the magnitude response of an ideal differentiator
differentiator response (by increasing tangency order with
at
. As
grows
moves closer to ideal
).
In practice there is no need for ideal differentiators because usually signals contain noise at high frequencies which should be
suppresed. This means that response of practical differentiator should be close to
on some passband interval
and should tend to zero in the upper part of Nyquist interval
.
From the plot we can see that central differences don’t resemble such behavior, all they care about is to get as closer as possible
to the response of ideal differentiator, without supression of noisy high frequencies. As a consequence they perform well only
on exact values, which contain no noise. Different technique is needed for robust derivative estimation of noisy signals. Such
approaches are considered in the next sections:
Savitzky-Golay/Low-noise Lanczos differentiators
Smooth noise-robust differentiators
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80 Comments
Kem
Posted February 19, 2009 at 8:32 pm | #
Hi Pavel, great tutorial!
How can this be done for the second derivatives f”(x) and also for the 2D case , especially how can we approximate d^2 f(x,y) / dxdy with different orders of
approximation
Thank you!
Reply
Pavel Holoborodko
Posted February 20, 2009 at 5:42 pm | #
Thanks, Kem.