'\"
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.TH "math::fourier" n 1\&.0\&.2 tcllib "Tcl Math Library"
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.SH NAME
math::fourier \- Discrete and fast fourier transforms
.SH SYNOPSIS
package require \fBTcl  8\&.4\fR
.sp
package require \fBmath::fourier  1\&.0\&.2\fR
.sp
\fB::math::fourier::dft\fR \fIin_data\fR
.sp
\fB::math::fourier::inverse_dft\fR \fIin_data\fR
.sp
\fB::math::fourier::lowpass\fR \fIcutoff\fR \fIin_data\fR
.sp
\fB::math::fourier::highpass\fR \fIcutoff\fR \fIin_data\fR
.sp
.BE
.SH DESCRIPTION
.PP
The \fBmath::fourier\fR 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\&.
.PP
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
implementation\&.
.SH "GENERAL INFORMATION"
The two top-level procedures defined are
.IP \(bu
dft data-list
.IP \(bu
inverse_dft data-list
.PP
Both take a list of \fIcomplex numbers\fR and apply a Discrete Fourier
Transform (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 complex 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\&.
.PP
If the input length N is a power of two then these procedures will
utilize 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\&.
.PP
Some examples:
.CS


    % 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}

.CE
.PP
In the last case, the imaginary parts <1e-16 would have been zero in exact
arithmetic, but aren't here due to rounding errors\&.
.PP
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\&.
.PP
The package includes two simple filters\&. They have an analogue
equivalent in a simple electronic circuit, a resistor and a capacitance
in series\&. Using these filters requires the
\fBmath::complexnumbers\fR package\&.
.SH PROCEDURES
The public Fourier transform procedures are:
.TP
\fB::math::fourier::dft\fR \fIin_data\fR
Determine the \fIFourier transform\fR of the given list of complex
numbers\&. The result is a list of complex numbers representing the
(complex) amplitudes of the Fourier components\&.
.RS
.TP
list \fIin_data\fR
List of data
.RE
.sp
.TP
\fB::math::fourier::inverse_dft\fR \fIin_data\fR
Determine the \fIinverse Fourier transform\fR of the given list of
complex numbers (interpreted as amplitudes)\&. The result is a list of
complex numbers representing the original (complex) data
.RS
.TP
list \fIin_data\fR
List of data (amplitudes)
.RE
.sp
.TP
\fB::math::fourier::lowpass\fR \fIcutoff\fR \fIin_data\fR
Filter the (complex) amplitudes so that high-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\&.
.RS
.TP
float \fIcutoff\fR
Cut-off frequency
.TP
list \fIin_data\fR
List of data (amplitudes)
.RE
.sp
.TP
\fB::math::fourier::highpass\fR \fIcutoff\fR \fIin_data\fR
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\&.
.RS
.TP
float \fIcutoff\fR
Cut-off frequency
.TP
list \fIin_data\fR
List of data (amplitudes)
.RE
.sp
.PP
.SH "BUGS, IDEAS, FEEDBACK"
This document, and the package it describes, will undoubtedly contain
bugs and other problems\&.
Please report such in the category \fImath :: fourier\fR of the
\fITcllib Trackers\fR [http://core\&.tcl\&.tk/tcllib/reportlist]\&.
Please also report any ideas for enhancements you may have for either
package and/or documentation\&.
.PP
When proposing code changes, please provide \fIunified diffs\fR,
i\&.e the output of \fBdiff -u\fR\&.
.PP
Note further that \fIattachments\fR are strongly preferred over
inlined patches\&. Attachments can be made by going to the \fBEdit\fR
form of the ticket immediately after its creation, and then using the
left-most button in the secondary navigation bar\&.
.SH KEYWORDS
FFT, Fourier transform, complex numbers, mathematics
.SH CATEGORY
Mathematics