§5.2 Sampling of Continuous-time Signals Now, the frequency-domain representation of ga(t) is given by its continuos-time fourier transform (CTFT G(A2)=⊥g()dt The frequency-domain representation of gn is given by its discrete-time Fourier transform DTFT) °G(e10)=∑m=ml-0
§5.2 Sampling of Continuous-time Signals • Now, the frequency-domain representation of ga (t) is given by its continuos-time Fourier transform (CTFT): G j g t e dt j t a ∫ a ∞ −∞ − Ω ( Ω) = ( ) ∑ ∞ =−∞ ω − ω = n j j n G(e ) g[n]e • The frequency-domain representation of g[n] is given by its discrete-time Fourier transform (DTFT):
85.3 Effect of Sampling in the Frequency domain To establish the relation between gagjQ2 and g(ejo), we treat the sampling operation mathematically as a multiplication of ga(t by a periodic impulse train p(t: Em0=2-m)8(0+80 p()
§5.3 Effect of Sampling in the Frequency Domain • To establish the relation between Ga (jΩ) and G(ejω) , we treat the sampling operation mathematically as a multiplication of ga (t) by a periodic impulse train p(t): = ∑δ − ∞ n=−∞ p(t) (t nT) g (t) × a g (t) p p(t)
85.3 Effect of Sampling in the Frequency domain p(t consists of a train of ideal impulses with a period f as shown below +T卜 -T0 T2T The multiplication operation yields an impulse train 81()=8a()p()=∑g(m7)6(-mT) =-
§5.3 Effect of Sampling in the Frequency Domain • p(t) consists of a train of ideal impulses with a period T as shown below = = ∑ δ − ∞ n=−∞ p a a g (t) g (t) p(t) g (nT) (t nT) • The multiplication operation yields an impulse train:
85.3 Effect of Sampling in the Frequency domain g(t is a continuous-time signal consisting of a train of uniformly spaced impulses with the impulse at t=nt weighted by the sampled value ga nt)of ga(t at that instant t=nT ga) f 0 T-TO T2T. ga(47)
§5.3 Effect of Sampling in the Frequency Domain • gp (t) is a continuous-time signal consisting of a train of uniformly spaced impulses with the impulse at t = nT weighted by the sampled value ga (nT) of ga (t) at that instant t=nT
85.3 Effect of Sampling in the Frequency domain °· There are two different forms ofggjQ2): One form is given by the weighted sum of the CTFTS of 8(t-nT): Gp( Ont 2 1=-0a nl e To derive the second form, we note thatp(t can be expressed as a Fourier series: j(2T/T)kt Q的 kt where Q2T =2I/T
§5.3 Effect of Sampling in the Frequency Domain • There are two different forms of Gp (jΩ) : • One form is given by the weighted sum of the CTFTs of δ(t-nT): ∑ ∞ =−∞ − Ω Ω = n j nT p a G ( j ) g (nT)e ∑ ∑ ∞ =−∞ Ω ∞ =−∞ = = k j kt k j T kt T e T e T p t 1 1 ( ) 2( π / ) where ΩT = 2π /T • To derive the second form, we note that p(t) can be expressed as a Fourier series: