... by^1.1

For clarity I have assumed that x[t] & X[f] have indices between ${\frac{-N}{2}}$ and ${\frac{N}{2}}$ - 1 although in practice the indices will be between 1 and N.

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... however)^1.2

This restriction could be relaxed. Let us know if it would be useful to do so.

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... provided^1.3

The natural width is $\sigma$ in the expression exp(${\frac{-(x - x_0)^2}{\sigma^2}}$)

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... constant^1.4

${\frac{1}{\sqrt{\ln(16)}}}$

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... provided^1.5

The natural width is $\sigma$ in the expression exp(${\frac{-(x - x_0)^2}{\sigma^2}}$)

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... constant^1.6

${\frac{1}{\sqrt{\ln(16)}}}$

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