Next |
Prev |
Up |
Top
|
Index |
JOS Index |
JOS Pubs |
JOS Home |
Search
By the modulation theorem for Fourier transforms,
![$\displaystyle \zbox {x(t)e^{-j\omega_0t}\;\longleftrightarrow\;X(\omega+\omega_0).}$](img1870.png) |
(11.30) |
This is proved in §B.6 as the dual of the shift-theorem. It
is also evident from inspection of the Fourier transform:
![$\displaystyle \int_{-\infty}^\infty \left[x(t)e^{-j\omega_0 t}\right] e^{-j\omega t} dt \eqsp \int_{-\infty}^\infty x(t)e^{-j(\omega+\omega_0) t} dt \isdefs X(\omega+\omega_0)$](img1871.png) |
(11.31) |
Applying the modulation theorem to the Gaussian transform pair above
yields
![$\displaystyle \zbox {e^{-pt^2} e^{-j\omega_0 t} \;\longleftrightarrow\;\sqrt{\frac{\pi}{p}} e^{-\frac{(\omega+\omega_0)^2}{4p}},\quad \forall p\in \mathbb{C}: \; \mbox{re}\left\{p\right\}>0.}$](img1872.png) |
(11.32) |
Thus, we frequency-shift a Gaussian chirp in the same way we
frequency-shift any signal--by complex modulation (multiplication by
a complex sinusoid at the shift-frequency).
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]