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  litera-
       ture on the algorithms involved, we refer to relevant text books for more explanations. We also refer to the original Wiki page on the sub-
       ject which describes some of the considerations behind the current implementation.

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 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 com-
       plex 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.

       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.

       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 equivalent 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 (com-
	      plex) 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 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)

       ::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://sourceforge.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

math								       1.0.2							  math::fourier(n)