TABLE 14.1 CT Fourier Transform Pairs Signal Fourier Transform urier Series Coefficients(if periodic) ak=0, otherwise cos oor 0. otherwise sin oo ap=0, otherwise for any choice of T. >0) t<T 2 sin ko sn如o五)=sino and xt+To)=x0) all k 2 sin o-1 >T, w, Wt sin Wr w 1 10 8(o) 9t}>0 e-t){ The above properties are particularly useful in CT system analysis and design, especially when the system haracteristics are easily specified in the frequency domain, as in linear filtering. Note that Properties 1,6, and 7 are useful for solving differential or integral equations. Property 4(time-domain convolution) provides the c 2000 by CRC Press LLC© 2000 by CRC Press LLC The above properties are particularly useful in CT system analysis and design, especially when the system characteristics are easily specified in the frequency domain, as in linear filtering. Note that Properties 1, 6, and 7 are useful for solving differential or integral equations. Property 4 (time-domain convolution) provides the TABLE 14.1 CT Fourier Transform Pairs Signal Fourier Transform Fourier Series Coefficients (if periodic) — — 1 — — — — — — a ek k jk t = • +• – w0 2 0 p a k kd w w k ( ) - =-• +•  ak ejw t0 2 0 pd( ) w - w a ak 1 1 0 = = , otherwise cos w0 t p[d( ) w - w0 0 + + d( ) w w ] a a ak 1 1 1 2 0 = = = - , otherwise sin w0 t p d w w d w w j [ ( ) - 0 0 - + ( )] a a j ak 1 1 1 2 0 = - = = - , otherwise x t( ) = 1 2pd( ) w a a k 0 k 1 0 0 0 = = ¹ > , , ( ) has this Forier series representation for any choice of T0 Periodic square wave and x t t T T t T x t T x t ( ) = < < £ Ï Ì Ô Ó Ô ( ) + = ( ) 1 0 2 1 1 0 0 , , 2 0 1 0 sin k T k k k w d w( ) - w =-• +•  w p w p w p 0 1 T 0 1 0 1 k T k T k sincÊ Ë Á ˆ ¯ ˜ = sin d t nT n ( ) - =-• +•  2p 2 d w p T k T k - Ê Ë Á ˆ ¯ ˜ =-• +•  a T k k = 1 for all x t t T t T ( ) = < > Ï Ì Ô Ó Ô 1 0 1 1 , , 2 2 1 1 T T sinc wT1 p w w Ê Ë Á ˆ ¯ ˜ = sin W Wt Wt p p pt sincÊ Ë Á ˆ ¯ ˜ = sin X W W w w w ( ) = < > Ï Ì Ô Ó Ô 1 0 , , d( )t u t( ) 1 jw + pd( ) w d( ) t - t 0 e - wj t0 e u t a -at ( ), 5e { } > 0 1 a + wj te u t a -at ( ), 5e { } > 0 1 2 ( ) a + jw t n e u t a n at - - ( ) - ( ) { } > 1 1 0 ! , 5e 1 a j n ( ) + w