10/10,13,16,17 - Fourier Transform Properties: Difference between revisions

Revision as of 18:08, 21 November 2008

$F\left[a\,x(t)+b\,x(t)\right]$	$=\int _{-\infty }^{\infty }\left[a\,x(t)+b\,x(t)\right]e^{-j\,2\pi f\,t}\,dt$
	$=a\int _{-\infty }^{\infty }\,x(t)\,e^{-j\,2\pi f\,t}\,dt+b\int _{-\infty }^{\infty }\,x(t)\,e^{-j\,2\pi f\,t}\,dt$
	$=a\,F[x(t)]+b\,F[x(t)]$

$F[x(t-t_{0})]\,\!$	$=\int _{-\infty }^{\infty }x(t-t_{0})\,e^{-j\,2\pi f\,t}\,dt$	Let $u=t-t_{0}\,\!$ and $du=dt\,\!$
	$=\int _{-\infty }^{\infty }x(u)\,e^{-j\,2\pi f\,(u+t_{0})}\,du$
	$=e^{-j\,2\pi f\,t_{0}}\int _{-\infty }^{\infty }x(u)\,e^{-j\,2\pi f\,u}\,du$
	$=e^{-j\,2\pi f\,t_{0}}\,F[x(t)]$

$F\left[e^{j\,2\pi f\,t}x(t)\right]$	$=\int _{-\infty }^{\infty }\left[e^{j\,2\pi f_{0}\,t}x(t)\right]e^{-j\,2\pi f\,t}\,dt$
	$=\int _{-\infty }^{\infty }x(t)\,e^{-j\,2\pi (f-f_{0})\,t}\,dt$
	$=X(f-f_{0})\,\!$

@@ Line 27: / Line 27: @@
 |
 |<math>=e^{-j\,2\pi f\,t_0}\,F[x(t)]</math>
+|}
+===??===
+{| border="0" cellpadding="0" cellspacing="0"
+|-
+|<math>F\left[e^{j\,2\pi f\,t}x(t)\right]</math>
+|<math>=\int_{-\infty}^{\infty}\left[e^{j\,2\pi f_0\,t}x(t)\right]e^{-j\,2\pi f\,t}\,dt</math>
+|-
+|
+|<math>=\int_{-\infty}^{\infty}x(t)\,e^{-j\,2\pi (f-f_0)\,t}\,dt</math>
+|-
+|
+|<math>=X(f-f_0)\,\!</math>
 |}