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? ? ? ? ? ? ? ? ? ? ? Eytan Modiano Slide 2 The Fourier Transform ? Frequency representation of signals ∞ () = ∫ ?∞ x ( t ) e ? jf t d t ? Definition: Xf 2 π ∞ 2 π xt () = ∫ ?∞ X ( f ) e jf t d f ? Notation: X(f) = F[x(t)] X(t) = F-1 [X(f)] x(t) ? X(f) Eytan Modiano Slide 3 δδδ Unit impulse δ (t) t 0 , δ () =? t ≠ 0 ∞ δ () = 1 ∫ ?∞ t ∞ δ () () = x ( 0 ) ∫ ?∞ tx t ∫ ∞ δ ( t ? τ ) x ( τ ) = x ( t ) ?∞ δ ∞ 2 π t F [( t ) ] δ F [( t ) ] = ∫ ?∞ δ ( t ) e ? jf t dt = e 0 = 1 δ () t δ () ? 1 0 Eytan Modiano Slide 4 1 jt jf Rectangle pulse t ? 1 || < 1 / 2 ? t / / Π () = ? 12 | t | = 12 ?? 0 otherwise 12 Π ∞ 2 π / F [( t ) ] = ∫ ?∞ Π ( t ) e ? jf t d t = ∫ ? 12 e ? j 2 π f t d t / e ? π ? e π π f = = Sin () = Sinc f ? jf π f 2 π () Π () t 1 1/2 1/2 Eytan Modiano Slide 5 ααα βββ ααα βββ τττ πππ τττ Properties of the Fourier transform ? Linearity – x1(t) <=> X1(f), x2(t) <=>X2(f) => α x1(t) + β x2(t) <=> α X1(f) + β X2(f) ? Duality – X(f) <=> x(t) => x(f) <=> X(-t) and x(-f)<=> X(t) ? Time-shifting: x(t- τ ) <=> X(f)e -j2 π f τ ? Scaling: F[(x(at)] = 1/|a| X(f/a) ? Convolution: x(t) <=> X(f), y(t) <=> Y(f) then, – F[x(t)*y(t)] = X(f)Y(f) – Convolution in time corresponds to multiplication in frequency and visa versa ∞ xt xt ? τ ) y ( τ ) d τ () * y ( t ) = ∫ ?∞ ( Eytan Modiano Slide 6 ΠΠΠ πππ ΠΠΠ ΠΠΠ Fourier transform properties (Modulation) 2 π xt e jf o t ? X ( f ? f o ) () e jx + e ? jx Now , cos( x ) = 2 xt e jf o t + x t e ? j 2 π f o t xt () cos( 2 π f o t ) = () 2 π () 2 ( x t Hence ,( ) cos( 2 π f o t ) ? Xf ? f o ) + X ( f + f o ) 2 ? Example: x(t)= sinc (t), F[sinc (t)] = Π (f) ? Y(t) = sinc (t)cos(2 π f o t) <=> ( Π (f-f o )+ Π (f+f o ))/2 1/2 Eytan Modiano -f o +f o Slide 7 |( More properties ? Power content of signal ? Autocorrelation ? Sampling Eytan Modiano Slide 8 ∫ ∞ ∞ xt ?∞ |( ) | 2 d t = ∫ ?∞ Xf ) | 2 d f ∞ * R x () = ∫ ?∞ x ( t ) x ( t ? τ ) d t τ R x () ? | X ( f ) | 2 τ xt o ( ) δ ( t ? t o ) () = xt ∞ xt () ∑ δ ( t ? n t o ) = sampled version of x(t) n =? ∞ ∞ ∞ F [ ∑ δ ( t ? n t o ) ] = 1 ∑ δ ( f ? n ) ] t t n =? ∞ o n =? ∞ o The Sampling Theorem Xf () () = 0 , for all f , | f | ≥ W ? Band-limited signal Xf – Bandwidth < W -w w Sampling Theorem: If we sample the signal at intervals Ts where Ts <= 1/ 2W then signal can be completely reconstructed from its samples using the formula ∞ xt () = ∑ 2 W ? T s x ( n T s ) sin c [ 2 W ? ( t ? n T s )] n =? ∞ Where , W ≤ W ? ≤ 1 ? W T s ∞ 1 t With T = = > xt s () = ∑ x ( n T s ) sin c [( ? n )] 2 W T n =? ∞ s ∞ n n () = ∑ x ( ) sin [ 2 xt cW ( t ? )] 2 W 2 W n =? ∞ Eytan Modiano Slide 9 Proof ∞ xt () ∑ δ ( t ? n T s ) δ () = x t n =? ∞ ∞ Xf () * F [ ∑ δ ( t ? n T s ) ] δ () = X f n =? ∞ ∞ ∞ F [ ∑ δ ( t ? n T s ) ] = 1 ∑ δ ( f ? n ) n =? ∞ T s n =? ∞ T s 1 ∞ n δ () = ∑ Xf ? ) Xf ( Ts n =? ∞ T s ? 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 1 0 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 Hf T () =Π ( ) s 2 W ? Now the recovered signal after low pass filtering f Xf f T () = X δ () s Π ( ) 2 W f xt () = F ? 1 [ X δ ( f ) T s Π ( ) ] 2 W ∞ t () = ∑ xn T s ) Sinc ( ? n ) xt ( n =? ∞ T s Eytan Modiano Slide 1 1 Notes about Sampling Theorem ? When sampling at rate 2W the reconstruction filter must be a rectangular pulse – Such a filter is not realizable – For perfect reconstruction must look at samples in the infinite future and past ? In practice we can sample at a rate somewhat greater than 2W which makes reconstruction filters that are easier to realize ? Given any set of arbitrary sample points that are 1/2W apart, can construct a continuous time signal band-limited to W Eytan Modiano Slide 1 2