math::fourier(n) Tcl Math Library math::fourier(n) ______________________________________________________________________________
NAME
math::fourier - Discrete and fast fourier transforms
SYNOPSIS
package require Tcl 8.4
package require math::fourier 1.0.2
::math::fourier::dft in_data
::math::fourier::inverse_dft in_data
::math::fourier::lowpass cutoff in_data
::math::fourier::highpass cutoff in_data
_________________________________________________________________
DESCRIPTION
The math::fourier package implements two versions of discrete Fourier
transforms, the ordinary transform and the fast Fourier transform. It
also provides a few simple filter procedures as an illustrations of how
such filters can be implemented.
The purpose of this document is to describe the implemented procedures
and provide some examples of their usage. As there is ample literature
on the algorithms involved, we refer to relevant text books for more
explanations. We also refer to the original Wiki page on the subject
which describes some of the considerations behind the current implemen-
tation.
GENERAL INFORMATION
The two top-level procedures defined are
o dft data-list
o inverse_dft data-list
Both take a list of complex numbers and apply a Discrete Fourier Trans-
form (DFT) or its inverse respectively to these lists of numbers. A
"complex number" in this case is either (i) a pair (two element list)
of numbers, interpreted as the real and imaginary parts of the complex
number, or (ii) a single number, interpreted as the real part of a com-
plex number whose imaginary part is zero. The return value is always in
the first format. (The DFT generally produces complex results even if
the input is purely real.) Applying first one and then the other of
these procedures to a list of complex numbers will (modulo rounding
errors due to floating point arithmetic) return the original list of
numbers.
If the input length N is a power of two then these procedures will uti-
lize the O(N log N) Fast Fourier Transform algorithm. If input length
is not a power of two then the DFT will instead be computed using a the
naive quadratic algorithm.
Some examples:
% dft {1 2 3 4}
{10 0.0} {-2.0 2.0} {-2 0.0} {-2.0 -2.0}
% inverse_dft {{10 0.0} {-2.0 2.0} {-2 0.0} {-2.0 -2.0}}
{1.0 0.0} {2.0 0.0} {3.0 0.0} {4.0 0.0}
% dft {1 2 3 4 5}
{15.0 0.0} {-2.5 3.44095480118} {-2.5 0.812299240582} {-2.5 -0.812299240582} {-2.5 -3.44095480118}
% inverse_dft {{15.0 0.0} {-2.5 3.44095480118} {-2.5 0.812299240582} {-2.5 -0.812299240582} {-2.5 -3.44095480118}}
{1.0 0.0} {2.0 8.881784197e-17} {3.0 4.4408920985e-17} {4.0 4.4408920985e-17} {5.0 -8.881784197e-17}
In the last case, the imaginary parts <1e-16 would have been zero in
exact arithmetic, but aren't here due to rounding errors.
Internally, the procedures use a flat list format where every even
index element of a list is a real part and every odd index element is
an imaginary part. This is reflected in the variable names by Re_ and
Im_ prefixes.
The package includes two simple filters. They have an analogue equiva-
lent in a simple electronic circuit, a resistor and a capacitance in
series. Using these filters requires the math::complexnumbers package.
PROCEDURES
The public Fourier transform procedures are:
::math::fourier::dft in_data
Determine the Fourier transform of the given list of complex
numbers. The result is a list of complex numbers representing
the (complex) amplitudes of the Fourier components.
list in_data
List of data
::math::fourier::inverse_dft in_data
Determine the inverse Fourier transform of the given list of
complex numbers (interpreted as amplitudes). The result is a
list of complex numbers representing the original (complex) data
list in_data
List of data (amplitudes)
::math::fourier::lowpass cutoff in_data
Filter the (complex) amplitudes so that high-frequency compo-
nents are suppressed. The implemented filter is a first-order
low-pass filter, the discrete equivalent of a simple electronic
circuit with a resistor and a capacitance.
float cutoff
Cut-off frequency
list in_data
List of data (amplitudes)
::math::fourier::highpass cutoff in_data
Filter the (complex) amplitudes so that low-frequency components
are suppressed. The implemented filter is a first-order low-pass
filter, the discrete equivalent of a simple electronic circuit
with a resistor and a capacitance.
float cutoff
Cut-off frequency
list in_data
List of data (amplitudes)
BUGS, IDEAS, FEEDBACK
This document, and the package it describes, will undoubtedly contain
bugs and other problems. Please report such in the category math ::
fourier of the Tcllib SF Trackers [http://source-
forge.net/tracker/?group_id=12883]. Please also report any ideas for
enhancements you may have for either package and/or documentation.
KEYWORDS
FFT, Fourier transform, complex numbers, mathematics
CATEGORY
Mathematics
math 1.0.2 math::fourier(n)
Mac OS X 10.8 - Generated Fri Sep 7 05:44:52 CDT 2012