5.3 Effect of Sampling in the Frequency Domain Two possible spectra of Gp(j)are shown below G(j) 1/T 一9m 0 291 39 27-8m G,(j) 1/T 0 29r 39T 2T-8m 9
§5.3 Effect of Sampling in the Frequency Domain • Two possible spectra of Gp(jΩ) are shown below
5.3 Effect of Sampling in the Frequency Domain It is evident from the top figure on the previous slide that if 2m,there is no overlap between the shifted replicas of Ga(j)generating Gp(j) On the other hand,as indicated by the figure on the bottom,if r<2 m,there is an overlap of the spectra of the shifted replicas of Ga(j)generating Gp(jS)
§5.3 Effect of Sampling in the Frequency Domain • It is evident from the top figure on the previous slide that if ΩT>2 Ωm , there is no overlap between the shifted replicas of Ga(jΩ) generating Gp(jΩ) • On the other hand, as indicated by the figure on the bottom, if ΩT<2 Ωm , there is an overlap of the spectra of the shifted replicas of Ga(jΩ) generating Gp(jΩ)
5.3 Effect of Sampling in the Frequency Domain If r>2 2m,ga(t)can be recovered exactly from gp(t)by passing it through an ideal lowpass filter H(j)with a gain T and a cutoff frequency c greater than m and less than r-m as shown below 8a() H,(02) 8a(0 p(t)
§5.3 Effect of Sampling in the Frequency Domain • If ΩT>2 Ωm , ga(t) can be recovered exactly from gp(t) by passing it through an ideal lowpass filter Hr(jΩ) with a gain T and a cutoff frequency Ωc greater than Ωm and less than ΩT - Ωm as shown below
5.3 Effect of Sampling in the Frequency Domain The spectra of the filter and pertinent signals are shown below H,() IT 2m <Oc<(OT-2m) 2 0 Ga(jQ) Gp(jQ) 1/T -m 0 Om -2m0 Om T 2m-2r a(j2) 1 -2m0 Om 2
§5.3 Effect of Sampling in the Frequency Domain • The spectra of the filter and pertinent signals are shown below
5.3 Effect of Sampling in the Frequency Domain Sampling theorem-Let g(t)be a band- limited signal with CTFT G.(j)=0 for 12>2m Then ga(t)is uniquely determined by its samples g(nT),-oo≤n≤ooif 2r≥22m where2n=2π/T
§5.3 Effect of Sampling in the Frequency Domain Sampling theorem - Let ga(t) be a bandlimited signal with CTFT Ga(jΩ)=0 for | Ω|> Ωm Then ga(t) is uniquely determined by its samples ga(nT) , -∞≤n≤∞ if ΩT ≥ 2 Ωm where ΩT=2π/T