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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 trans-form transform
       form and the fast Fourier transform. It also provides a few simple filter procedures as an  illustra-tions illustrations
       tions 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 implementation.

GENERAL INFORMATION
       The two top-level procedures defined are

       •      dft data-list

       •      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 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.

       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  com-puted computed
       puted 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 cir-
       cuit,  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 components are suppressed.  The  imple-mented implemented
              mented 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  imple-mented implemented
              mented 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)