


math::fourier(n)         Tcl Math Library        math::fourier(n)


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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

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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



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math::fourier(n)         Tcl Math Library        math::fourier(n)


       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.




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math::fourier(n)         Tcl Math Library        math::fourier(n)


              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
























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