2.2.6. 4 Fourier transforms ofpower signals 1<>2丌o() (2.104) 0 Fig 2.39 The unity and its Fourier transform
2.2.6.4 Fourier transforms of power signals 1 2 () (2.104) Fig. 2.39 The unity and its Fourier transform
2.2.6. 4 Fourier transforms ofpower signals Furthermore, we have the following relationδ()=δ(t)*x(t)=x(1) (2.105) x(1)*C()=d(1)*x(t) d(Tx(t-rdr x(t) x()*(t-0)=d(t-t0)*x(1)=x(t-10)(2106) x(t) x(t)并b(t) 8(t) O x(t) X(t-t1)=X()*8(t-t1) O Fig. 2. 40 Convolution of an arbitrary function with a unit impulse
Furthermore, we have the following relation: 2.2.6.4 Fourier transforms of power signals x(t) (t) = (t) x(t) = x(t) (2.105) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x t x t d x t t t x t = = − = − ( ) ( ) ( ) ( ) ( ) 0 0 0 x t t − t = t − t x t = x t − t (2.106) Fig. 2.40 Convolution of an arbitrary function with a unit impulse
2.2.6. 4 Fourier transforms of power signals 2 Sinusoidal functions cOS印l÷ e J@oI (2.109) Using the transform pair e o <-> 28(0-Oo We see coSOot e>r[S(@-0o)+8(0+@o)1 (2.10) Similarly, sino2iz[6(o+o。)-6(a-0)(21
2. Sinusoidal functions Using the transform pair we see Similarly, 2.2.6.4 Fourier transforms of power signals 2 cos 0 0 0 j t j t e e t − + = (2.109) 2 ( ) 0 0 − j t e cos ( ) ( ) 0 −0 + +0 t (2.110) sin ( ) ( ) 0 +0 − −0 t j (2.111)
2.2.6. 4 Fourier transforms ofpower signals Xjw) cos wor xGl sin wot 们丌) A 丌 Fig. 2. 42 Sinusoidal functions and their spectra
2.2.6.4 Fourier transforms of power signals Fig. 2.42 Sinusoidal functions and their spectra
2.2.6. 4 Fourier transforms ofpower signals 3. The Signum Function The signum function, denoted by sgn(t) is defined as l,t<0 sg(1)={0,t=0 2.112 If x(t)>x(o) then dx(t) f>joX(a Suppose we differentiate the signum function. Its derivative is 20(t) Sgn(t)=20(1)
3. The Signum Function The signum function, denoted by sgn(t), is defined as If then Suppose we differentiate the signum function. Its derivative is 2δ(t): 2.2.6.4 Fourier transforms of power signals = − = 1, 0 0, 0 1, 0 sgn( ) t t t t (2.112) x(t) X () ( ) ( ) jX dt dx t sgn(t) 2 (t) dt d =