Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


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)


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA