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So frequency/time scaling property of Fourier transform says that: fourier transform of $|c|f(ct)$ is $F(\omega/c)$. (I am using angular frequency $\omega = 2\pi f$ here)

However, this doesn't seem to make sense for periodic signals: for example, $e^{i3t}$ (Fourier transform: $\delta(\omega-3)$). If we are to frequency-scale to $e^{i6t}$ (Fourier transform $\delta(\omega-6)$, by the above, it seems that we need to multiply by 2 to obtain $2e^{i6t}$, which is inconsistent.

Or am I getting something wrong here?

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You're right. That's because in the periodic case the doubling in a period "stays within a period" due to wraparound, there is no length dilation that needs to be compensated.

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