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1. Consider a Gaussian function:
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It's Fourier transform can be found as follows:
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Note that the integral in the last equation = 1 (from integral tables), so that the gaussian transforms to a gaussian, a remarkable result.
2. Next, consider a so-called PI function which has the value of 1 from x=-1/2 to x=1/2 and is zero elsewhere:
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The PI function describes truncation of the data. For example, the process of turning a pulse on and off instantaneously, or of starting and stopping the encoding of the MR signal, can be though of as multiplying by a PI function. After Fourier transformation, this results in sinc modulation of the data.
3. Finally, consider a decaying exponential:
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The final function is a so-called Lorentzian, which is the canonical line shape in magnetic resonance of a decaying exponential after Fourier transformation. For example, it is the frequency domain representation of a free induction decay (FID). It is also the blur of a pixel before the before the truncation (see 1) that gives the characteristic sinc-shaped point spread function.
And here is what these functions look like:
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| Gaussian |
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| sinc |
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| Lorentzian |