1math::fourier(n)               Tcl Math Library               math::fourier(n)
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NAME

8       math::fourier - Discrete and fast fourier transforms
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SYNOPSIS

11       package require Tcl  8.4
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13       package require math::fourier  1.0.2
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15       ::math::fourier::dft in_data
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17       ::math::fourier::inverse_dft in_data
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19       ::math::fourier::lowpass cutoff in_data
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21       ::math::fourier::highpass cutoff in_data
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DESCRIPTION

26       The  math::fourier package uses the fast Fourier transform, if applica‐
27       ble, or the ordinary transform to implement the discrete Fourier trans‐
28       form.  It  also provides a few simple filter procedures as an illustra‐
29       tion of how such filters can be implemented.
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31       The purpose of this document is to describe the implemented  procedures
32       and  provide some examples of their usage. As there is ample literature
33       on the algorithms involved, we refer to relevant text  books  for  more
34       explanations.  We  also  refer to the original Wiki page on the subject
35       which describes some of the considerations behind the current implemen‐
36       tation.
37

GENERAL INFORMATION

39       The two top-level procedures defined are
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41       •      dft data-list
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43       •      inverse_dft data-list
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45       Both take a list of complex numbers and apply a Discrete Fourier Trans‐
46       form (DFT) or its inverse respectively to these lists  of  numbers.   A
47       "complex  number"  in this case is either (i) a pair (two element list)
48       of numbers, interpreted as the real and imaginary parts of the  complex
49       number, or (ii) a single number, interpreted as the real part of a com‐
50       plex number whose imaginary part is zero. The return value is always in
51       the  first  format. (The DFT generally produces complex results even if
52       the input is purely real.) Applying first one and  then  the  other  of
53       these procedures to a list of complex numbers will (modulo rounding er‐
54       rors due to floating point arithmetic) return the original list of num‐
55       bers.
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57       If the input length N is a power of two then these procedures will uti‐
58       lize the O(N log N) Fast Fourier Transform algorithm. If  input  length
59       is  not  a power of two then the DFT will instead be computed using the
60       naive quadratic algorithm.
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62       Some examples:
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65                  % dft {1 2 3 4}
66                  {10 0.0} {-2.0 2.0} {-2 0.0} {-2.0 -2.0}
67                  % inverse_dft {{10 0.0} {-2.0 2.0} {-2 0.0} {-2.0 -2.0}}
68                  {1.0 0.0} {2.0 0.0} {3.0 0.0} {4.0 0.0}
69                  % dft {1 2 3 4 5}
70                  {15.0 0.0} {-2.5 3.44095480118} {-2.5 0.812299240582} {-2.5 -0.812299240582} {-2.5 -3.44095480118}
71                  % inverse_dft {{15.0 0.0} {-2.5 3.44095480118} {-2.5 0.812299240582} {-2.5 -0.812299240582} {-2.5 -3.44095480118}}
72                  {1.0 0.0} {2.0 8.881784197e-17} {3.0 4.4408920985e-17} {4.0 4.4408920985e-17} {5.0 -8.881784197e-17}
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75       In the last case, the imaginary parts <1e-16 would have  been  zero  in
76       exact arithmetic, but aren't here due to rounding errors.
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78       Internally,  the procedures use a flat list format where every even in‐
79       dex element of a list is a real part and every odd index element is  an
80       imaginary  part. This is reflected in the variable names by Re_ and Im_
81       prefixes.
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83       The package includes two simple filters. They have an analogue  equiva‐
84       lent  in  a  simple electronic circuit, a resistor and a capacitance in
85       series. Using these filters requires the math::complexnumbers package.
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PROCEDURES

88       The public Fourier transform procedures are:
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90       ::math::fourier::dft in_data
91              Determine the Fourier transform of the  given  list  of  complex
92              numbers.  The  result  is a list of complex numbers representing
93              the (complex) amplitudes of the Fourier components.
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95              list in_data
96                     List of data
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99       ::math::fourier::inverse_dft in_data
100              Determine the inverse Fourier transform of  the  given  list  of
101              complex  numbers  (interpreted  as  amplitudes). The result is a
102              list of complex numbers representing the original (complex) data
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104              list in_data
105                     List of data (amplitudes)
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107
108       ::math::fourier::lowpass cutoff in_data
109              Filter the (complex) amplitudes so  that  high-frequency  compo‐
110              nents  are  suppressed.  The implemented filter is a first-order
111              low-pass filter, the discrete equivalent of a simple  electronic
112              circuit with a resistor and a capacitance.
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114              float cutoff
115                     Cut-off frequency
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117              list in_data
118                     List of data (amplitudes)
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121       ::math::fourier::highpass cutoff in_data
122              Filter the (complex) amplitudes so that low-frequency components
123              are suppressed. The implemented filter is a first-order low-pass
124              filter,  the  discrete equivalent of a simple electronic circuit
125              with a resistor and a capacitance.
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127              float cutoff
128                     Cut-off frequency
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130              list in_data
131                     List of data (amplitudes)
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BUGS, IDEAS, FEEDBACK

135       This document, and the package it describes, will  undoubtedly  contain
136       bugs  and  other  problems.  Please report such in the category math ::
137       fourier of the Tcllib Trackers  [http://core.tcl.tk/tcllib/reportlist].
138       Please  also  report any ideas for enhancements you may have for either
139       package and/or documentation.
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141       When proposing code changes, please provide unified diffs, i.e the out‐
142       put of diff -u.
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144       Note  further  that  attachments  are  strongly  preferred over inlined
145       patches. Attachments can be made by going  to  the  Edit  form  of  the
146       ticket  immediately  after  its  creation, and then using the left-most
147       button in the secondary navigation bar.
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KEYWORDS

150       FFT, Fourier transform, complex numbers, mathematics
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CATEGORY

153       Mathematics
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157tcllib                               1.0.2                    math::fourier(n)
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