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1 ============================================================================== |
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2 |
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3 FFTReal |
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4 Version 2.00, 2005/10/18 |
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5 |
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6 Fourier transformation (FFT, IFFT) library specialised for real data |
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7 Portable ISO C++ |
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8 |
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9 (c) Laurent de Soras <laurent.de.soras@club-internet.fr> |
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10 Object Pascal port (c) Frederic Vanmol <frederic@fruityloops.com> |
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11 |
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12 ============================================================================== |
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13 |
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14 |
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15 |
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16 1. Legal |
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17 -------- |
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18 |
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19 This library is free software; you can redistribute it and/or |
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20 modify it under the terms of the GNU Lesser General Public |
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21 License as published by the Free Software Foundation; either |
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22 version 2.1 of the License, or (at your option) any later version. |
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23 |
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24 This library is distributed in the hope that it will be useful, |
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25 but WITHOUT ANY WARRANTY; without even the implied warranty of |
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26 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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27 Lesser General Public License for more details. |
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28 |
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29 You should have received a copy of the GNU Lesser General Public |
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30 License along with this library; if not, write to the Free Software |
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31 Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA |
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32 |
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33 Check the file license.txt to get full information about the license. |
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34 |
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35 |
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36 |
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37 2. Content |
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38 ---------- |
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39 |
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40 FFTReal is a library to compute Discrete Fourier Transforms (DFT) with the |
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41 FFT algorithm (Fast Fourier Transform) on arrays of real numbers. It can |
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42 also compute the inverse transform. |
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43 |
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44 You should find in this package a lot of files ; some of them are of interest: |
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45 - readme.txt: you are reading it |
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46 - FFTReal.h: FFT, length fixed at run-time |
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47 - FFTRealFixLen.h: FFT, length fixed at compile-time |
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48 - FFTReal.pas: Pascal implementation (working but not up-to-date) |
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49 - stopwatch directory |
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50 |
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51 |
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52 |
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53 3. Using FFTReal |
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54 ---------------- |
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55 |
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56 Important - if you were using older versions of FFTReal (up to 1.03), some |
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57 things have changed. FFTReal is now a template. Therefore use FFTReal<float> |
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58 or FFTReal<double> in your code depending on the application datatype. The |
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59 flt_t typedef has been removed. |
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60 |
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61 You have two ways to use FFTReal. In the first way, the FFT has its length |
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62 fixed at run-time, when the object is instanciated. It means that you have |
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63 not to know the length when you write the code. This is the usual way of |
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64 proceeding. |
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65 |
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66 |
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67 3.1 FFTReal - Length fixed at run-time |
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68 -------------------------------------- |
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69 |
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70 Just instanciate one time a FFTReal object. Specify the data type you want |
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71 as template parameter (only floating point: float, double, long double or |
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72 custom type). The constructor precompute a lot of things, so it may be a bit |
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73 long. The parameter is the number of points used for the next FFTs. It must |
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74 be a power of 2: |
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75 |
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76 #include "FFTReal.h" |
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77 ... |
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78 long len = 1024; |
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79 ... |
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80 FFTReal <float> fft_object (len); // 1024-point FFT object constructed. |
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81 |
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82 Then you can use this object to compute as many FFTs and IFFTs as you want. |
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83 They will be computed very quickly because a lot of work has been done in the |
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84 object construction. |
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85 |
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86 float x [1024]; |
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87 float f [1024]; |
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88 |
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89 ... |
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90 fft_object.do_fft (f, x); // x (real) --FFT---> f (complex) |
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91 ... |
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92 fft_object.do_ifft (f, x); // f (complex) --IFFT--> x (real) |
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93 fft_object.rescale (x); // Post-scaling should be done after FFT+IFFT |
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94 ... |
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95 |
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96 x [] and f [] are floating point number arrays. x [] is the real number |
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97 sequence which we want to compute the FFT. f [] is the result, in the |
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98 "frequency" domain. f has the same number of elements as x [], but f [] |
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99 elements are complex numbers. The routine uses some FFT properties to |
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100 optimize memory and to reduce calculations: the transformaton of a real |
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101 number sequence is a conjugate complex number sequence: F [k] = F [-k]*. |
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102 |
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103 |
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104 3.2 FFTRealFixLen - Length fixed at compile-time |
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105 ------------------------------------------------ |
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106 |
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107 This class is significantly faster than the previous one, giving a speed |
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108 gain between 50 and 100 %. The template parameter is the base-2 logarithm of |
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109 the FFT length. The datatype is float; it can be changed by modifying the |
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110 DataType typedef in FFTRealFixLenParam.h. As FFTReal class, it supports |
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111 only floating-point types or equivalent. |
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112 |
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113 To instanciate the object, just proceed as below: |
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114 |
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115 #include "FFTRealFixLen.h" |
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116 ... |
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117 FFTRealFixLen <10> fft_object; // 1024-point (2^10) FFT object constructed. |
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118 |
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119 Use is similar as the one of FFTReal. |
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120 |
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121 |
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122 3.3 Data organisation |
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123 --------------------- |
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124 |
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125 Mathematically speaking, the formulas below show what does FFTReal: |
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126 |
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127 do_fft() : f(k) = sum (p = 0, N-1, x(p) * exp (+j*2*pi*k*p/N)) |
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128 do_ifft(): x(k) = sum (p = 0, N-1, f(p) * exp (-j*2*pi*k*p/N)) |
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129 |
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130 Where j is the square root of -1. The formulas differ only by the sign of |
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131 the exponential. When the sign is positive, the transform is called positive. |
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132 Common formulas for Fourier transform are negative for the direct tranform and |
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133 positive for the inverse one. |
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134 |
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135 However in these formulas, f is an array of complex numbers and doesn't |
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136 correspound exactly to the f[] array taken as function parameter. The |
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137 following table shows how the f[] sequence is mapped onto the usable FFT |
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138 coefficients (called bins): |
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139 |
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140 FFTReal output | Positive FFT equiv. | Negative FFT equiv. |
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141 ---------------+-----------------------+----------------------- |
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142 f [0] | Real (bin 0) | Real (bin 0) |
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143 f [...] | Real (bin ...) | Real (bin ...) |
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144 f [length/2] | Real (bin length/2) | Real (bin length/2) |
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145 f [length/2+1] | Imag (bin 1) | -Imag (bin 1) |
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146 f [...] | Imag (bin ...) | -Imag (bin ...) |
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147 f [length-1] | Imag (bin length/2-1) | -Imag (bin length/2-1) |
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148 |
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149 And FFT bins are distributed in f [] as above: |
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150 |
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151 | | Positive FFT | Negative FFT |
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152 Bin | Real part | imaginary part | imaginary part |
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153 ------------+----------------+-----------------+--------------- |
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154 0 | f [0] | 0 | 0 |
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155 1 | f [1] | f [length/2+1] | -f [length/2+1] |
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156 ... | f [...], | f [...] | -f [...] |
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157 length/2-1 | f [length/2-1] | f [length-1] | -f [length-1] |
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158 length/2 | f [length/2] | 0 | 0 |
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159 length/2+1 | f [length/2-1] | -f [length-1] | f [length-1] |
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160 ... | f [...] | -f [...] | f [...] |
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161 length-1 | f [1] | -f [length/2+1] | f [length/2+1] |
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162 |
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163 f [] coefficients have the same layout for FFT and IFFT functions. You may |
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164 notice that scaling must be done if you want to retrieve x after FFT and IFFT. |
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165 Actually, IFFT (FFT (x)) = x * length(x). This is a not a problem because |
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166 most of the applications don't care about absolute values. Thus, the operation |
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167 requires less calculation. If you want to use the FFT and IFFT to transform a |
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168 signal, you have to apply post- (or pre-) processing yourself. Multiplying |
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169 or dividing floating point numbers by a power of 2 doesn't generate extra |
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170 computation noise. |
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171 |
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172 |
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173 |
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174 4. Compilation and testing |
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175 -------------------------- |
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176 |
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177 Drop the following files into your project or makefile: |
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178 |
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179 Array.* |
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180 def.h |
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181 DynArray.* |
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182 FFTReal*.cpp |
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183 FFTReal*.h* |
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184 OscSinCos.* |
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185 |
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186 Other files are for testing purpose only, do not include them if you just need |
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187 to use the library ; they are not needed to use FFTReal in your own programs. |
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188 |
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189 FFTReal may be compiled in two versions: release and debug. Debug version |
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190 has checks that could slow down the code. Define NDEBUG to set the Release |
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191 mode. For example, the command line to compile the test bench on GCC would |
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192 look like: |
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193 |
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194 Debug mode: |
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195 g++ -Wall -o fftreal_debug.exe *.cpp stopwatch/*.cpp |
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196 |
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197 Release mode: |
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198 g++ -Wall -o fftreal_release.exe -DNDEBUG -O3 *.cpp stopwatch/*.cpp |
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199 |
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200 It may be tricky to compile the test bench because the speed tests use the |
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201 stopwatch sub-library, which is not that cross-platform. If you encounter |
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202 any problem that you cannot easily fix while compiling it, edit the file |
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203 test_settings.h and un-define the speed test macro. Remove the stopwatch |
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204 directory from your source file list, too. |
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205 |
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206 If it's not done by default, you should activate the exception handling |
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207 of your compiler to get the class memory-leak-safe. Thus, when a memory |
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208 allocation fails (in the constructor), an exception is thrown and the entire |
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209 object is safely destructed. It reduces the permanent error checking overhead |
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210 in the client code. Also, the test bench requires Run-Time Type Information |
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211 (RTTI) to be enabled in order to display the names of the tested classes - |
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212 sometimes mangled, depending on the compiler. |
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213 |
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214 The test bench may take a long time to compile, especially in Release mode, |
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215 because a lot of recursive templates are instanciated. |
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216 |
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217 |
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218 |
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219 5. History |
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220 ---------- |
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221 |
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222 v2.00 (2005.10.18) |
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223 - Turned FFTReal class into template (data type as parameter) |
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224 - Added FFTRealFixLen |
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225 - Trigonometric tables are size-limited in order to preserve cache memory; |
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226 over a given size, sin/cos functions are computed on the fly. |
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227 - Better test bench for accuracy and speed |
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228 |
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229 v1.03 (2001.06.15) |
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230 - Thanks to Frederic Vanmol for the Pascal port (works with Delphi). |
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231 - Documentation improvement |
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232 |
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233 v1.02 (2001.03.25) |
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234 - sqrt() is now precomputed when the object FFTReal is constructed, resulting |
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235 in speed impovement for small size FFT. |
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236 |
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237 v1.01 (2000) |
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238 - Small modifications, I don't remember what. |
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239 |
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240 v1.00 (1999.08.14) |
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241 - First version released |
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242 |