Lecture 3: The Sampling Theorem Eytan Modiano AA Dept. Eytan Modiano
Lecture 3: The Sampling Theorem Eytan Modiano AA Dept. Eytan Modiano Slide 1
Sampling Given a continuous time waveform, can we represent it using discrete samples? How often should we sample? Can we reproduce the original waveform?
Sampling • Given a continuous time waveform, can we represent it using discrete samples? – How often should we sample? – Can we reproduce the original waveform? � � � � � � � � � � Eytan Modiano Slide 2
The fourier transform Frequency representation of signals · Definition: X(= x()e /2midt x(1)=X(0l2d Notation X()=F[X( X(t)=F-1[X(f)] x()分X(f) Eytan Modiano
The Fourier Transform • Frequency representation of signals ∞ () = ∫−∞ x(t)e− j ft • Definition: X f dt 2π ∞ 2 π x t() = ∫−∞ X( f )e j ft df • Notation: X(f) = F[x(t)] X(t) = F-1 [X(f)] x(t) � X(f) Eytan Modiano Slide 3
Unit impulse 8(t) 6(t)=0,Vt≠0 6(t)=1 6()x(t)=x(0) 6(t-c)x()=x(t) F[6()=|60-/2=2=1 6(t) F6(t) 0()台1 Eytan Modiano
Unit impulse δ(t) δ() t = ∀ 0, t ≠ 0 ∞ δ() =1 ∫−∞ t ∞ δ() () = x(0) ∫−∞ t x t ∫ ∞ δ(t − τ )x(τ ) = x(t) −∞ δ ∞ 2π t F[ ( δ t)] F[ (t)] = ∫−∞δ(t)e− j ftdt = e0 = 1 δ() δ() t ⇔ 1 0 Eytan Modiano Slide 4 1
Rectangle pulse tk1/2 I()={1/2|t=1/2 0 otherwise FIn(]= II(oe/2gd-_[/2 e 4g dt ing Sin(f) Sinc(f jtF 1/2 Eytan Modiano
j t j f Rectangle pulse 1 | | t < 1/ 2 Π() t = 12 / | t |= 12 / 0 otherwise 12 Π ∞ 2π / F[ (t)] = ∫−∞Π(t)e− j ftdt = ∫−12 e− j2πftdt / e − π − e π πf = = Sin() = Sinc f − j f 2π πf () Π( )t 1 1/2 1/2 Eytan Modiano Slide 5