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One easy way to see the validity of the theorem is to consider the corresponding analytic signals. Let a signal $s(t)$ be defined by $$s(t)=m(t)c(t)\tag{1}$$ where $m(t)$ is a low pass (message) signal, and $c(t)$ is a bandpass (carrier) signal. Note that we don't need to assume that $c(t)$ is sinusoidal. Let $\hat{x}(t)$ denote the Hilbert transform of \$x(t)...