§52 Sampling of Continuous-time Signals Now, the frequency-domain representation of ga (t is given by its continuos-time fourier transform (CTFT): Ga(92) -oo oa (t)e dt The frequency-domain representation of e gln is given by its discrete-time Fourier transform DTFT): G(e10)=∑8nlel0n
§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 − − ( ) = ( ) =− − = 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: p(t)=∑8(t-nT) g0(0)--8,() p()
§5.3 Effect of Sampling in the Frequency Domain • To establish the relation between Ga (j) and G(ej) , we treat the sampling operation mathematically as a multiplication of ga (t) by a periodic impulse train p(t): = − 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 T as shown below 刘T -2T-T0 T 2T The multiplication operation yields an e impulse train: 8n(t)=ga(tp(t)=28a(nr)b(t-nT n-
§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 = = − n=− g p (t) ga (t) p(t) ga (nT) (t nT) • The multiplication operation yields an impulse train:
85.3 Effect of Sampling in the Frequency Domain gn(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 tnT 8a0 8a(4T
§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 of g gQ2) One form is given by the weighted sum of the CTFTS of 8(t-nT): Gn(D)=∑n=8(m)e-bm To derive the second form, we note that p(t)can be expressed as a Fourier series: p()=n∑ j(2T/T)K k=-0 T k: where Q2T=2T/T
§5.3 Effect of Sampling in the Frequency Domain • There are two different forms of Gp (j) : • One form is given by the weighted sum of the CTFTs of (t-nT): =− − = n j nT p a G ( j ) g (nT)e =− =− = = k j kt k j T kt T e T e T p t 1 1 ( ) (2 / ) where T = 2 /T • To derive the second form, we note that p(t) can be expressed as a Fourier series: