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This paper (Page 657, Section 2.3) says:

It is well known that for a given function there might be more than one representation. For example, a purely harmonic function can also be represented as a function having time varying amplitude and time varying phase: $\cos(2\pi t) = \{1 + a(t)\} \cos[2\pi\{t + b(t)\}]$, where $b'(t)$ and $a(t)$ might be 'large' compared with $1$.

I was wondering what could be an example of such a pair of $a(t)$ and $b(t)$. Any help would be greatly appreciated.

Shanks
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Here's an example: pick a very large numbers $N$. Set \begin{align*}b(t)&=\Big(1/4-1/N\Big)\cos(2\pi N t)-t\\ a(t)&=-1+\frac{\cos(2 \pi t)}{\cos(2\pi(t+b(t)))} \end{align*}

Notice that $t+b(t)$ always stays strictly between $-1/4$ and $1/4$. This means that $\cos(2\pi(t+b(t)))$ will never be zero. Hence $a(t)$ is always defined. Moreover, the way we built $a(t)$ ensures the desired identity.

Note that $b(t)$ will have very fast oscillations for large $N$ and will hence have a large derivative. Also, while $\cos(2\pi(t+b(t)))$ never crosses zero, it gets very close for large $N$, since $t+b(t)$ will get very close to $1/4$ (and do so very often). Thus, $a(t)$ will have a large magnitude.

AAA
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