FourierTransforms

# FourierTransforms - f(t = ∑ n=0 ∞ A n cos(2 π n ν o t...

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Colby College Fourier Transformation Any periodic wave can be contructed as a sum of sine and cosine waves. The free induction decay, FID, in NMR is a combination of all the line frequencies in the corresponding NMR spectrum. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.0005 0.001 0.0015 0.002 0.0025 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.0005 0.001 0.0015 0.002 0.0025 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.0005 0.001 0.0015 0.002 0.0025 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.0005 0.001 0.0015 0.002 0.0025 Fourier series:
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Unformatted text preview: f(t) = ∑ n=0 ∞ A n cos(2 π n ν o t) + ∑ n=0 ∞ B n sin(2 π n ν o t) with ν o = lowest frequency = 1/L. L is the period. To find the Fourier coefficients use: A n = 2 ⌡ ⌠ L f(t) cos(2 π n ν o t) dt B n = 2 ⌡ ⌠ L f(t) sin(2 π n ν o t) dt The FID has Fourier coefficients at 1 and 2 kHz, but not 3 kHz. The spectrum from the FID: 1 kHz 2 kHz 3 kHz FID frequency (sec-1 ) B n 1 2 3 L...
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