The Hilbert transform of a Fourier transformable signal m (t), denoted by m (t), is defined by below, in the frequency domain, we have M (?) = -j sgn (?) M (?), where m (t) ? M (?), m (t) ? M (?), and sgn (?) is the signum function. Using the definition of the Hilbert transform, show that a single-side-brand modulation signal resulting from the message signal m (t) and carrier cos (2??_ct) of the amplitude is given by s (t) = ? m (t) cos (2??_ct) + ? m (t) sin (2??_ct), where the minus sign corresponds to the transmission of the upper sideband and the plus sign corresponds to the transmission of the lower sideband.

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