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 jt (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)