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This question is similar to several sets of previous questions dating back to when I joined the site 1, 2, 3. This time it regards a sine function of quadratic frequency. a.k.a. a "chirp signal". I know for the simpler case of constant frequency, we can use a generator which is infinitesimal rotation matrix:

$$R = \begin{bmatrix} \cos(\lambda\epsilon) & \sin(\lambda\epsilon)\\-\sin(\lambda\epsilon)&\cos(\lambda\epsilon) \end{bmatrix}$$

we can plot $\lambda \epsilon$ vs $\sin(\lambda \epsilon)$ and get any constant $\lambda$ frequency sine. $({R^k})_{1,2} = \sin(\lambda\epsilon k)$

But what about quadratic frequency? Like plotting $x$ vs $\sin(\lambda x^2)$. How can we create a generator matrix in the same sense for this situation?

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