2.2.6. 4 Fourier transforms ofpower signals The transtorm of a sgn D a8g0分1ox(O) Where jo(O)=F26()]=2 Thus the transform of sgn(tis 2 X(O)= JO The transform pair 2 sgn(t)<> (2.113) JO
The transform of where Thus the transform of sgn(t) is The transform pair: 2.2.6.4 Fourier transforms of power signals sgn(t) dt d sgn(t) jX () dt d jX () = F2 (t)= 2 j X 2 ( ) = j t 2 sgn( ) (2.113)
2.2.6. 4 Fourier transforms ofpower signals sgn(t) o(jo) 丌/2 T/2 Fig. 2.43 sgn(t) and its spectra
2.2.6.4 Fourier transforms of power signals Fig. 2.43 sgn(t) and its spectra
2.2.6. 4 Fourier transforms ofpower signals 5. Periodic functions Periodic functions can be represented as a sum of complex exponentials; because we can transform complex exponentials by means of Eq (2.103), we should be able to represent a periodic function using the Fourier integral Assuming x(t is periodic of period T, then x()=∑ c eNos
5. Periodic functions Periodic functions can be represented as a sum of complex exponentials; because we can transform complex exponentials by means of Eq.(2.103), we should be able to represent a periodic function using the Fourier integral. Assuming x(t) is periodic of period T, then 2.2.6.4 Fourier transforms of power signals =− = n jn t n x t C e 0 ( )
2.2.6. 4 Fourier transforms ofpower signals The Fourier transform of x(t) X()=F[x() F∑Ce n=-0 ∑CnFe" (2117) 2z∑C,6(o-mo) n=-0 Where T x(te
The Fourier transform of x(t) : where 2.2.6.4 Fourier transforms of power signals =− =− =− = − = = = n n n j n t n n j n t n C n C F e F C e X F x t 2 ( ) [ ] [ ] ( ) [ ( )] 0 0 0 (2.117) − − = x t e dt T C jn t T n T 0 ( ) 1 2 2
2.2.6. 4 Fourier transforms ofpower signals Example 12: A special kind of periodic function is the unit impulse train shown in Figure 2.47. This function is useful in applications involving sampling of time waveforms 31-27-T 0T2T3 Fig. 2. 47 Periodic impulse sequence
Example 12: A special kind of periodic function is the unit impulse train shown in Figure 2.47. This function is useful in applications involving sampling of time waveforms. 2.2.6.4 Fourier transforms of power signals Fig. 2.47 Periodic impulse sequence