'IFFT' 태그의 글 목록

FFT를 이용한 inverse FFT

Image Recognition/Fundamental 2024. 7. 31. 12:11

Forward fft:

$f f t (x + i y) = X + i Y = \sum (x + i y) (W r + i W i) <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mi mathvariant="monospace">f</mi><mi mathvariant="monospace">f</mi><mi mathvariant="monospace">t</mi></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi><mo>+</mo><mi>i</mi><mi>Y</mi><mo>=</mo><mo data-mjx-texclass="OP">\sum</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>W</mi><mi>r</mi></msub><mo>+</mo><mi>i</mi><msub><mi>W</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math>$

$= \sum (x W r - y W i) + i (x W i + y W r) <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mo data-mjx-texclass="OP">\sum</mo><mo stretchy="false">(</mo><mi>x</mi><msub><mi>W</mi><mi>r</mi></msub><mo>-</mo><mi>y</mi><msub><mi>W</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mi>W</mi><mi>i</mi></msub><mo>+</mo><mi>y</mi><msub><mi>W</mi><mi>r</mi></msub><mo stretchy="false">)</mo></math>$

Inverse fft:

$ifft(X+iY)=1N∑(X+iY)(Wr−iWi)<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow data-mjx-texclass="ORD"><mi mathvariant="monospace">i</mi><mi mathvariant="monospace">f</mi><mi mathvariant="monospace">f</mi><mi mathvariant="monospace">t</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>i</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mo data-mjx-texclass="OP">∑</mo><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>i</mi><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>W</mi><mi>r</mi></msub><mo>−</mo><mi>i</mi><msub><mi>W</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math>$

$=1N∑(XWr+YWi)+i(−XWi+YWr)<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mo data-mjx-texclass="OP">∑</mo><mo stretchy="false">(</mo><mi>X</mi><msub><mi>W</mi><mi>r</mi></msub><mo>+</mo><mi>Y</mi><msub><mi>W</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mo stretchy="false">(</mo><mo>−</mo><mi>X</mi><msub><mi>W</mi><mi>i</mi></msub><mo>+</mo><mi>Y</mi><msub><mi>W</mi><mi>r</mi></msub><mo stretchy="false">)</mo></math>$

$=1N∑(XWr−(−Y)Wi)+(−i)(XWi+(−Y)Wr)<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>X</mi><msub><mi>W</mi><mi>r</mi></msub><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mi>W</mi><mi>i</mi></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mi>i</mi><mo stretchy="false">)</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>X</mi><msub><mi>W</mi><mi>i</mi></msub><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mi>Y</mi><mo stretchy="false">)</mo><msub><mi>W</mi><mi>r</mi></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow></math>$

$=1N[fft(X−iY)]∗=1N[fft((X+iY)∗)]∗<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="monospace">f</mi><mi mathvariant="monospace">f</mi><mi mathvariant="monospace">t</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>i</mi><mi>Y</mi><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>∗</mo></msup><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="monospace">f</mi><mi mathvariant="monospace">f</mi><mi mathvariant="monospace">t</mi></mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>i</mi><mi>Y</mi><msup><mo stretchy="false">)</mo><mo>∗</mo></msup><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>∗</mo></msup></math>$

또는

$1Nfft(Y+iX)=1N∑(Y+iX)(Wr+iWi)<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mn>1</mn><mi>N</mi></mfrac><mrow data-mjx-texclass="ORD"><mi mathvariant="monospace">f</mi><mi mathvariant="monospace">f</mi><mi mathvariant="monospace">t</mi></mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>+</mo><mi>i</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mo data-mjx-texclass="OP">∑</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>+</mo><mi>i</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>W</mi><mi>r</mi></msub><mo>+</mo><mi>i</mi><msub><mi>W</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math>$

$=1N∑(YWr−XWi)+i(YWi+XWr)=y+ix<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mo data-mjx-texclass="OP">∑</mo><mo stretchy="false">(</mo><mi>Y</mi><msub><mi>W</mi><mi>r</mi></msub><mo>−</mo><mi>X</mi><msub><mi>W</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mo stretchy="false">(</mo><mi>Y</mi><msub><mi>W</mi><mi>i</mi></msub><mo>+</mo><mi>X</mi><msub><mi>W</mi><mi>r</mi></msub><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><mo>=</mo><mi>y</mi><mo>+</mo><mi>i</mi><mi>x</mi></math>$

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FFT를 이용한 inverse FFT

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