Proof )=x)28(-m X()=X(*F∑(-m7 F∑8-m)=r∑(f X()=∑X0 n n=-∞ The Fourier transform of the sampled signal is a replication of the Fourier transform of the original separated by 1/Ts intervals 2/TS Slide 10Proof ∞ x t () ∑δ(t − nTs ) δ () = x t n=−∞ ∞ X f ( ) * F[ ∑δ(t − nTs )] δ () = X f n=−∞ ∞ ∞ F[ ∑δ(t − nTs )] = 1 ∑ δ( f − n ) n=−∞ Ts n=−∞ Ts 1 ∞ n δ X f () = ∑ X f ( − ) Ts n=−∞ Ts • The Fourier transform of the sampled signal is a replication of the Fourier transform of the original separated by 1/Ts intervals -1/Ts -w w 1/Ts 2/Ts Eytan Modiano Slide 10