EI331 Signals and Systems Lecture 17 Bo Jiang John Hopcroft Center - PowerPoint PPT Presentation

EI331 Signals and Systems Lecture 17 Bo Jiang John Hopcroft Center for Computer Science Shanghai Jiao Tong University April 23, 2019

Contents 1. Sampling Theorem 2. Zero-order Hold and Linear Interpolation 3. Aliasing 1/30

Sampling CT Signals Sampling converts CT signals to DT signals CT signal DT signal x ( t ) x [ n ] = x ( nT ) n 0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 T 0 1 2 3 4 5 6 7 8 9 10 t T = sampling period, uniform sampling most common Allows use of digital electronics to process, record, transmit, store, and retrieve CT signals • MP3, digital camera, printer 2/30

Sampling CT Signals Sampling loses information, different signals may have same samples • x 1 ( t ) = cos( π 3 t ) , x 2 ( t ) = cos( 7 π 3 t ) , different • x 1 [ n ] = cos( π 3 n ) = x 2 [ n ] = cos( 7 π 3 n ) , identical x t n Under what conditions can we recover signal from samples? 3/30

Impulse-train Sampling Time domain x ( t ) CT signal x ( t ) t 0 impulse train p ( t ) 1 ∞ � p ( t ) = δ ( t − nT ) t − 5 T − 4 T − 3 T − 2 T − T 0 T 2 T 3 T 4 T 5 T n = −∞ x p ( t ) x p ( t ) = x ( t ) p ( t ) ∞ � = x ( nT ) δ ( t − nT ) t − 5 T − 4 T − 3 T − 2 T − T 0 T 2 T 3 T 4 T 5 T n = −∞ 4/30

Impulse-train Sampling X ( j ω ) Frequency domain A CT signal X ( j ω ) ω − ω M ω M P ( j ω ) impulse train ω s = 2 π ∞ T � P ( j ω ) = ω s δ ( ω − k ω s ) k = −∞ ω − ω s 0 ω s 2 ω s X p ( j ω ) = 1 2 π ( X ∗ P )( ω ) X p ( j ω ) ω s − ω M A ∞ = 1 T � X ( j ( ω − k ω s )) T ω − ω s 0 ω s 2 ω s k = −∞ − ω M ω M 5/30

Impulse-train Sampling X ( j ω ) Frequency domain A band-limited CT signal X ( j ω ) = 0 for | ω | > ω M ω − ω M ω M Sampling frequency ω s = 2 π T X p ( j ω ) ω s − ω M A Case 1: ω s > 2 ω M T no overlap between replicas ω − ω s ω s can recover X by lowpass filtering − ω M ω M X p ( j ω ) A Case 2: ω s < 2 ω M T replicas overlap ω − ω s − 2 ω s ω s 2 ω s cannot recover X − ω M ω M ω s − ω M 6/30

Sampling Theorem Band-limited CT signal x ( t ) whose spectrum X ( j ω ) = 0 for | ω | > ω M is uniquely determined by its samples x ( nT ) , n ∈ Z if ω s � 2 π T > 2 ω M , 2 ω M called Nyquist rate Given { x ( nT ) : n ∈ Z } , x ( t ) can be reconstructed as follows ∞ � 1. construct x p ( t ) = x ( nT ) δ ( t − nT ) n = −∞ 2. send x p through lowpass filter with gain T and cutoff frequency ω c ∈ ( ω M , ω s − ω M ) , i.e. H ( j ω ) = T [ u ( ω + ω c ) − u ( ω − ω c )] 3. filter output x r ( t ) with X r ( j ω ) = X p ( j ω ) H ( j ω ) is same as x ( t ) 7/30

Reconstruction in Frequency Domain X ( j ω ) x ( t ) × H ( j ω ) x r ( t ) A x p ( t ) ω − ω M ω M ∞ X p ( j ω ) � p ( t ) = δ ( t − nT ) ω s − ω M A n = −∞ T ω − ω s ω s − ω M ω M • Nyquist frequency ω M H ( j ω ) • Nyquist rate 2 ω M T Lowpass filter ω − ω c ω c • gain T X r ( j ω ) • cutoff frequency A ω M < ω c < ω s − ω M ω − ω M ω M 8/30

Reconstruction in Time Domain ∞ p ( t ) = � δ ( t − nT ) H ( j ω ) n = −∞ T x p ( t ) x ( t ) × x r ( t ) − ω c ω c Impulse response of lowpass filter h ( t ) = T sin( ω c t ) π t x recovered by band-limited interpolation using sinc function ∞ x ( nT ) T sin( ω c ( t − nT )) � x ( t ) = x r ( t ) = ( x p ∗ h )( t ) = π ( t − nT ) n = −∞ 9/30

Reconstruction in Time Domain x p ( t ) t − 2 T − T 0 T 2 T ω c = ω s 2 = π T x r ( t ) t − T T − 2 T 0 2 T 10/30

Reconstruction in Time Domain Setting ω c = ω s 2 yields Whittaker-Shannon interpolation formula ∞ � t − nT � where sinc( t ) = sin( π t ) � x r ( t ) = x ( nT ) sinc , T π t n = −∞ Since � t − nT � → Te − jnT ω [ u ( ω + π T ) − u ( ω − π F sinc ← − − T )] T Parseval’s identity (or multiplication property) implies π � t − nT � � t − mT � dt = T 2 � � T e j ( m − n ) T ω d ω = T δ [ n − m ] sinc sinc 2 π T T R − π T Whittaker-Shannon formula is orthogonal expansion 11/30

Reconstruction in Time Domain x p ( t ) t − 2 T − T 0 T 2 T ω c = ω s 4 = π 2 T > ω M x r ( t ) t − T T − 2 T 0 2 T 12/30

Reconstruction in Time Domain x p ( t ) t − 2 T − T 0 T 2 T ω c = 3 ω s < ω s − ω M 5 x r ( t ) t − T T − 2 T 0 2 T 13/30

Contents 1. Sampling Theorem 2. Zero-order Hold and Linear Interpolation 3. Aliasing 14/30

Other Interpolation Methods x [ n ] x ( t ) Zero-order hold n 0 1 2 3 4 5 6 7 8 9 10 0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 T t T = sampling period x ( t ) Linear interpolation 0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 T t 15/30

Zero-order Hold ∞ p ( t ) = � δ ( t − nT ) h 0 ( t ) n = −∞ 1 x p ( t ) x ( t ) × x 0 ( t ) t 0 T Reconstructed signal ∞ � x 0 ( t ) = ( x p ∗ h 0 )( t ) = x ( nT ) h 0 ( t − nT ) | H ( j ω ) | n = −∞ T ideal Zero-order hold filter zero-order hold H 0 ( j ω ) = e − j ω T / 2 2 sin( ω T / 2 ) − ω s − 2 ω s ω s 2 ω s − ω s ω s ω ω 2 2 16/30

Zero-order Hold x p ( t ) t − 2 T − T 0 T 2 T x 0 ( t ) t − T T − 2 T 0 2 T 17/30

Linear Interpolation (First-order Hold) ∞ p ( t ) = � δ ( t − nT ) h 1 ( t ) n = −∞ 1 x p ( t ) x ( t ) × x 0 ( t ) t − T T Reconstructed signal ∞ � x 1 ( t ) = ( x p ∗ h 1 )( t ) = x ( nT ) h 1 ( t − nT ) | H ( j ω ) | n = −∞ T ideal First-order hold filter first-order hold � 2 � sin( ω T / 2 ) H 1 ( j ω ) = 1 − ω s ω s − ω s ω s ω ω/ 2 T 2 2 18/30

Linear Interpolation x p ( t ) t − T T − 2 T 0 2 T x 1 ( t ) t − 2 T − T 0 T 2 T 19/30

Contents 1. Sampling Theorem 2. Zero-order Hold and Linear Interpolation 3. Aliasing 20/30

Aliasing output frequency Aliasing wraps ω s 2 frequencies input frequency ω s 2 x ( t ) = cos( ω 0 t + φ ) X ( j ω ) ω 0 < 1 π e − j φ π e j φ 2 ω s ω − ω 0 ω 0 P ( j ω ) − ω s − 2 ω s 0 ω s 2 ω s X p ( j ω ) x r ( t ) = cos( ω 0 t + φ ) − ω s ω s − ω s − 2 ω s ω 0 ω s 2 ω s 2 2 21/30

Aliasing output frequency Aliasing wraps ω s 2 frequencies input frequency ω s 2 x ( t ) = cos( ω 0 t + φ ) X ( j ω ) ω 0 < 1 π e − j φ π e j φ 2 ω s ω − ω 0 ω 0 P ( j ω ) − ω s − 2 ω s 0 ω s 2 ω s X p ( j ω ) x r ( t ) = cos( ω 0 t + φ ) − ω s ω s − ω s − 2 ω s ω 0 ω s 2 ω s 2 2 22/30

Aliasing output frequency Aliasing wraps ω s 2 frequencies input frequency ω s 2 x ( t ) = cos( ω 0 t + φ ) X ( j ω ) ω 0 > 1 π e − j φ π e j φ 2 ω s ω − ω 0 ω 0 P ( j ω ) − ω s − 2 ω s 0 ω s 2 ω s X p ( j ω ) x r ( t ) = cos(( ω s − ω 0 ) t − φ ) − ω s ω s − ω s − 2 ω s ω 0 ω s 2 ω s ω s − ω 0 2 2 23/30

Aliasing output frequency Aliasing wraps ω s 2 frequencies input frequency ω s 2 x ( t ) = cos( ω 0 t + φ ) X ( j ω ) ω 0 > 1 π e − j φ π e j φ 2 ω s ω − ω 0 ω 0 P ( j ω ) − ω s − 2 ω s 0 ω s 2 ω s X p ( j ω ) x r ( t ) = cos(( ω s − ω 0 ) t − φ ) − ω s ω s − ω s ω s − ω 0 − 2 ω s ω 0 ω s 2 ω s 2 2 24/30

Aliasing • x ( t ) = cos( 5 π 3 t ) , x [ n ] = cos( 5 π 3 n ) = cos( π 3 n ) • x r ( t ) = cos( π 3 t ) x ( t ) t n x r ( t ) t n Example. In movies, wheels often appear to rotate more slowly than they actually do and even in wrong direction 25/30

Aliasing output frequency Aliasing wraps ω s 2 frequencies input frequency ω s 2 x ( t ) = cos( ω 0 t + φ ) X ( j ω ) ω 0 = 1 π e − j φ π e j φ 2 ω s ω − ω 0 ω 0 P ( j ω ) − ω s − 2 ω s 0 ω s 2 ω s X p ( j ω ) x r ( t ) = (cos φ ) cos( ω 0 t ) − ω s ω s − ω s − 2 ω s ω s 2 ω s 2 2 26/30

Aliasing Aliasing for more complex signals also wraps frequencies X ( j ω ) ω − ω 0 ω 0 P ( j ω ) − ω s − 2 ω s 0 ω s 2 ω s X p ( j ω ) − ω s − ω s ω s − 2 ω s ω 0 ω s 2 ω s 2 2 ω s − ω 0 27/30

Aliasing Aliasing increases as sampling rate decreases X ( j ω ) ω − ω 0 ω 0 P ( j ω ) − ω s − 2 ω s 0 ω s 2 ω s X p ( j ω ) − ω s − ω s ω s − 2 ω s ω s 2 ω s 2 2 28/30

Anti-aliasing Filter Filter out frequencies above ω s 2 before sampling X ( j ω ) X a ( j ω ) − ω s ω s − ω 0 ω 0 − ω 2 ω 2 2 2 2 2 P ( j ω ) − ω s − 2 ω s ω s 0 2 ω s X p ( j ω ) − ω s − ω s ω s − ω 0 ω s − 2 ω s ω 0 ω s 2 ω s 2 2 29/30

Anti-aliasing Filter Filter out frequencies above ω s 2 before sampling X ( j ω ) X a ( j ω ) − ω s ω s − ω 0 ω 0 − ω 2 ω 2 2 2 2 2 P ( j ω ) − ω s − 2 ω s 0 ω s 2 ω s X p ( j ω ) − ω s − ω s ω s − ω 0 ω s − 2 ω s ω 0 ω s 2 ω s 2 2 30/30