Modulation Theorem (Shift Theorem Dual)

The Fourier dual of the shift theorem is often called the modulation theorem:

$\displaystyle \zbox {x(t)e^{j\nu t}\;\longleftrightarrow\;X(\omega-\nu)}$ (B.13)

This is proved in the same way as the shift theorem above by starting with the inverse Fourier transform of the right-hand side:

\begin{eqnarray*}
\hbox{\sc IFT}_\omega(\hbox{\sc Shift}_\nu(X)) &\isdef &
\frac{1}{2\pi}\int_{-\infty}^\infty X(\omega-\nu) e^{j\omega t}dt\qquad\mbox{(define $\sigma=\omega-\nu$)}\\
&=& \frac{1}{2\pi}\int_{-\infty}^\infty X(\sigma) e^{j(\sigma+\nu)t}d\sigma\\
&=& e^{j\nu t}\frac{1}{2\pi}\int_{-\infty}^\infty X(\sigma) e^{j\sigma t}d\sigma\\
&\isdef & e^{j\nu t} x(t)
\end{eqnarray*}

or,

$\displaystyle \zbox {e^{j\nu t} x(t) \;\longleftrightarrow\; X(\omega-\nu).}$ (B.14)


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Shift Theorem