ece431FTtable

ece431FTtable - Fourier Transform X( f ) = Inversion...

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Unformatted text preview: Fourier Transform X( f ) = Inversion Formula x(t) = x(t) I[-T,T ] (t) 2 fc sinc(2 fct) - - x(t)e- j2 f t dt X( f )e j2 f t d f X( f ) 2T sinc(2T f ) I[- fc , fc ] ( f ) T sinc2 (T f ) (1 - | f |/ fc )I[- fc , fc ] ( f ) 1 + j2 f 2 (1 - |t|/T )I[-T,T ] (t) fc sinc2 ( fct) e- t u(t) e- |t| 2 + (2 f )2 2 + t2 e-(t/ ) 2 /2 e-2 | f | 2 2 2 e- (2 f ) /2 Note. The indicator function I[a,b] (t) := 1 for a t b and I[a,b] (t) := 0 otherwise. In particular, u(t) := I[0,) (t) is the unit step function. Also, sinc(t) := [sin( t)]/( t) for t = 0 and sinc(0) := 1. Series Formulas N-1 zk = k=0 1 - zN , 1-z 1 , 1-z z=1 ez := zk k=0 k! z n n (a + b)n = k=0 n n k n-k ab k zk = k=0 |z| < 1 lim 1 + n = ez k k=1 n = n(n + 1) 2 ...
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This note was uploaded on 04/15/2008 for the course ECE 431 taught by Professor Gubner during the Spring '08 term at Wisconsin.

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