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

Revision as of 17:54, 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 $du=dt\,\!$
	$=\int _{-\infty }^{\infty }x(u)\,e^{-j\,2\pi f\,(u+t_{0})}\,du$
	$=e^{-j\,2\pi f\,t_{0}}\int _{-\infty }^{\infty }x(u)\,e^{-j\,2\pi f\,u}\,du$
	$=e^{-j\,2\pi f\,t_{0}}\,F[x(t)]$

@@ Line 12: / Line 12: @@
 |<math>=a\,F[x(t)]+b\,F[x(t)]</math>
 |}
-===Time Invariance===
+===Time Invariance (Delay)===
 {| border="0" cellpadding="0" cellspacing="0"
 |-
-|<math>f(n+1)</math>
+|<math>F[x(t-t_0)]\,\!</math>
-|<math>=(n+1)^2</math>
+|<math>=\int_{-\infty}^{\infty} x(t-t_0)\,e^{-j\,2\pi f\,t}\,dt</math>
+|Let <math>u=t-t_0\,\!</math> and <math> du=dt\,\!</math>
 |-
 |
-|<math>=n^2 + 2n + 1</math>
+|<math>=\int_{-\infty}^{\infty} x(u)\,e^{-j\,2\pi f\,(u+t_0)}\,du</math>
+|-
+|
+|<math>=e^{-j\,2\pi f\,t_0}\int_{-\infty}^{\infty} x(u)\,e^{-j\,2\pi f\,u}\,du</math>
+|-
+|
+|<math>=e^{-j\,2\pi f\,t_0}\,F[x(t)]</math>
 |}