... sinusoid¹

The more accurate term here is ``quasi-sinusoidal signal'' though we use ``sinusoid'' for convenience.

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... definition²

When we cast these integrals in a slightly different form [6], we may obtain exact solutions by infinite sums of spherical Bessel functions of the first kind.

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... accuracy³

Efficient computation of numerical approximations of these sums has been a subject of mathematical research. In August of 2000, for example, Mielenz showed that a particular manipulation of the problem allowed an approximation accurate to within $5*10^{-10}$ as a sum of eleven precisely weighted terms.

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....⁴

Choosing to use $\alpha/2$ in equation 14 would allow a more direct interpretation of the chirp parameter as the frequency increase in radians per sample. Our current choice, however, allows cleaner notation in the exponential arguments.

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... integral.⁵

This reverses the paradigm often seen in introductory calculus, where the midpoint approximation is used to approximate integrals.

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...:⁶

We consider only the real part integral approximation though it should be apparent that the imaginary integral yields the same result.

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... tolerated.⁷

We note that in practice, this is not necessary, as we will find that specifying other parameters is a more useful approach.

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... sum.⁸

The imaginary part gives a similar visual cumulative sum result and is thus not shown.

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... domain.⁹

In actuality, we could consider the three-quarter height or some other fraction. As witnessed in the discussion above, however, choosing half height greatly facilitates comparison with the window transform to check model validity, even though it slightly restricts the $\alpha$ values for which our model is valid. We see that even in practical situations where $\alpha$ is smaller than that permitted by our assumptions, the current algorithm performs better than a phase inversion based version able to use $b<1$.

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... above.¹⁰

Even though our choice of $b$ may increase accuracy, we see that in practical situations, the magnitude inversion algorithm generally outperforms the phase inversion algorithm.

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... valid.¹¹

The small limits approximation applies there. This is covered in the next subsection.

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