《测试与检测技术基础》课程电子教案（PPT教学课件）Fundamentals of Measurement Technology（17/28）

Fundamentals of Measurement Technology (3) Prof Wang Boxiong

Fundamentals of Measurement Technology (3) Prof. Wang Boxiong

2.2.6. 4 Fourier transforms of power signals 1. Unit impulse function Assuming a rectangular pulse pa(t)of a width A and an amplitude 1/4, its area is equal to1.As△→0, the limit of p4()is called the unit impulse function or delta function P, ( 8(t) △△ Fig. 2. 36 Rectangular pulse function and delta function S(t)

1. Unit impulse function Assuming a rectangular pulse pΔ (t) of a width Δ and an amplitude 1/Δ, its area is equal to 1. As Δ→0, the limit of pΔ (t) is called the unit impulse function or delta function denoted by δ(t). 2.2.6.4 Fourier transforms of power signals Fig. 2.36 Rectangular pulse function and delta function δ(t)

2.2.6. 4 Fourier transforms of power signals o(t is a pulse with unbounded amplitude and zero time duration This impulse function must be treated as a so-called generalized function ☆ Properties: 0V)≈J∞,t=0 (2.96) 0.t≠0 (t)kl(t)=1 (297) The two properties for the impulse function can be conveniently summarized into one defining equation for o(t) x(t)6(t-t0)d=x(t0) provided x(t is continuous at t=to

δ(t) is a pulse with unbounded amplitude and zero time duration. This impulse function must be treated as a so-called generalized function. ❖ Properties: 1) 2) The two properties for the impulse function can be conveniently summarized into one defining equation for δ(t). provided x(t) is continuous at t=t0 . 2.2.6.4 Fourier transforms of power signals      = = 0, 0 , 0 ( ) t t  t (2.96) ( ) ( ) = 1   −  t d t (2.97)   − ( ) ( − ) = ( ) 0 0 x t  t t dt x t (2.99)

2.2.6. 4 Fourier transforms ofpower signals The Fourier transform of the impulse function a(t) X(O)=F[6(1)=6()em (2.100) Fourier transform pair 6(t)<>1 2.101) XLa Fig. 2.37 d(t) and its Fourier transform

The Fourier transform of the impulse function δ(t): Fourier transform pair: 2.2.6.4 Fourier transforms of power signals ( ) =  ( ) = ( ) =1   − − X F t t e dt jt    (2.100)  (t) 1 (2.101) Fig. 2.37 δ(t) and its Fourier transform

2.2.6. 4 Fourier transforms ofpower signals 6(t-t0)4>e (2.102) ↑△G) slope =-to Fig. 2.38 8(t-to)and its Fourier transform Using the symmetry property, we can derive the transform pairs Foot >27(-O0) (2.103)

2.2.6.4 Fourier transforms of power signals 0 ( ) 0 j t t t e    − − (2.102) Fig. 2.38 δ(t-t0 ) and its Fourier transform 2 ( ) 0 0       − j t e Using the symmetry property, we can derive the transform pairs: (2.103)

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,tx(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 () ( ) ( ) jX  dt dx t  sgn(t) 2 (t) dt d = 