Fourier transform property: Modulation

Statement: if $g(t)\leftrightarrow G(f)$, then multiplying $g(t)$ by a sinusoidal carrier shifts its spectrum to $\pm f_c$:

$$g(t)\cos(2\pi f_ct)\leftrightarrow\frac{1}{2}\left[G(f-f_c)+G(f+f_c)\right]$$

Explanation: write $\cos2\pi f_ct=\frac{1}{2}(e^{j2\pi f_ct}+e^{-j2\pi f_ct})$. By the frequency-shift property, $g(t)e^{j2\pi f_ct}\leftrightarrow G(f-f_c)$, so each exponential term moves the whole spectrum by $\pm f_c$, and the factor $\frac{1}{2}$ halves its height.

Significance: this is the basis of amplitude modulation (DSB-SC). A baseband message of bandwidth $W$ is moved to a band $f_c-W$ to $f_c+W$; multiplying again by the carrier at the receiver moves it back (coherent demodulation).