1math::fourier(n) Tcl Math Library math::fourier(n)
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8 math::fourier - Discrete and fast fourier transforms
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11 package require Tcl 8.4
12 package require math::fourier 1.0.2
14 ::math::fourier::dft in_data
16 ::math::fourier::inverse_dft in_data
18 ::math::fourier::lowpass cutoff in_data
20 ::math::fourier::highpass cutoff in_data
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24
26 The math::fourier package implements two versions of discrete Fourier
27 transforms, the ordinary transform and the fast Fourier transform. It
28 also provides a few simple filter procedures as an illustrations of how
29 such filters can be implemented.
30 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
39 The two top-level procedures defined are
40 · dft data-list
42 · inverse_dft data-list
44 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
54 errors due to floating point arithmetic) return the original list of
55 numbers.
56 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 a the
60 naive quadratic algorithm.
61 Some examples:
63 % 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}
73 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.
77 Internally, the procedures use a flat list format where every even
79 index element of a list is a real part and every odd index element is
80 an imaginary part. This is reflected in the variable names by Re_ and
81 Im_ prefixes.
82 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.
86
88 The public Fourier transform procedures are:
89 ::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.
94 list in_data
96 List of data
97 ::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
103 list in_data
105 List of data (amplitudes)
106 ::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.
113 float cutoff
115 Cut-off frequency
116 list in_data
118 List of data (amplitudes)
119 ::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.
126 float cutoff
128 Cut-off frequency
129 list in_data
131 List of data (amplitudes)
132
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.
140 When proposing code changes, please provide unified diffs, i.e the out‐
142 put of diff -u.
143 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.
148
150 FFT, Fourier transform, complex numbers, mathematics
151
153 Mathematics
154tcllib 1.0.2 math::fourier(n)