Proof, continued If 1/Ts> 2W then the replicas of X(f) will not overlap and can be recovered How can we reconstruct the original signal? Low pass filter the sampled signal Ideal low pass filter is a rectangular pulse H(=/n2w) Now the recovered signal after low pass filtering X()=X(I( 2W x()=F[x()m( ()=∑x(nSin(-n)Proof, continued • If 1/Ts > 2W then the replicas of X(f) will not overlap and can be recovered • How can we reconstruct the original signal? – Low pass filter the sampled signal f • Ideal low pass filter is a rectangular pulse H f () = ΠT ( ) s 2W • Now the recovered signal after low pass filtering f X f () = X f T δ () sΠ( ) 2W f x t() = F−1[ Xδ ( f )TsΠ( )] 2W ∞ t () = ∑ x nTs x t ( )Sinc( − n) n=−∞ Ts Eytan Modiano Slide 11