EI331 Signals and Systems Lecture 20 Bo Jiang John Hopcroft Center - PowerPoint PPT Presentation

EI331 Signals and Systems Lecture 20 Bo Jiang John Hopcroft Center for Computer Science Shanghai Jiao Tong University May 7, 2019

Contents 1. Magnitude-phase Representation of Fourier Transform 2. Uncertainty Principle 3. Relations Among Fourier Representations 1/22

Magnitude-phase Representation of Fourier Transform X ( e j ω ) = | X ( e j ω ) | e j arg X ( e j ω ) X ( j ω ) = | X ( j ω ) | e j arg X ( j ω ) , Recall Fourier transform is decomposition of signal into superposition of complex exponentials (“waves”) • | X | gives magnitudes of components • arg X gives phases of components Phase arg X contains substantial information about signal • determines whether components add constructively or destructively • small change can lead to very differential-looking signals for same magnitude spectrum 2/22

Importance of Phase Information x ( t ) = 1 + 1 2 cos( 2 π t + φ 1 ) + cos( 4 π t + φ 2 ) + 2 3 cos( 6 π t + φ 3 ) x 1 ( t ) φ 1 = 0 , φ 2 = 0 , φ 3 = 0 t x 2 ( t ) φ 1 = 4 , φ 2 = 8 , φ 3 = 12 t x 3 ( t ) φ 1 = 6 , φ 2 = − 2 . 7 , φ 3 = 0 . 93 t x 4 ( t ) φ 1 = 1 . 2 , φ 2 = 4 . 1 , φ 3 = − 7 . 02 t 3/22

6000000 5000000 4000000 3000000 2000000 1000000 0 0 200 400 600 800 4 3 2 1 0 1 2 3 4 0 200 400 600 800 frequency (Hz) Magnitude vs. Phase Waveform x for Chinese word “ ” Magnitude and phase spectra | X | , arg X (DFT) 6000000 5000000 4000000 3000000 2000000 1000000 0 0 5000 10000 15000 20000 4 3 2 1 0 1 2 3 4 0 5000 10000 15000 20000 frequency (Hz) 4/22

Magnitude vs. Phase Waveform x for Chinese word “ ” Magnitude and phase spectra | X | , arg X (DFT) 6000000 6000000 5000000 5000000 4000000 4000000 3000000 3000000 2000000 2000000 1000000 1000000 0 0 0 2000 4000 6000 8000 10000 0 200 400 600 800 4 4 3 3 2 2 1 1 0 0 1 1 2 2 3 3 4 4 0 2000 4000 6000 8000 10000 0 200 400 600 800 frequency (Hz) frequency (Hz) 4/22

Magnitude vs. Phase Waveform x for Chinese word “ ” Waveform reconstructed by magnitude spectra only F − 1 {| X |} Waveform reconstructed by phase spectra only F − 1 { e j arg X } 5/22

Magnitude vs. Phase Top row X , | X | , arg X Bottom row F − 1 {| X |} F − 1 { e j arg X } 6/22

Magnitude-phase Representation of Frequency Response For LTI systems Y ( e j ω ) = H ( e j ω ) X ( e j ω ) Y ( j ω ) = H ( j ω ) X ( j ω ) , Thus | Y | = | H | · | X | , | H | called gain of system and arg Y = arg H + arg X , arg H called phase shift of system Effects of LTI system may or may not be desirable • want specific effects for filtering • if undesirable, effects called distortion Example. Distortionless transmission • ideally, H ( j ω ) = 1 , but noncausal • H ( j ω ) = Ke − j ω t 0 , preserves shape, only scaling + delay 7/22

Linear Phase For CT LTI system with unit gain and linear phase H ( j ω ) = e − j ω t 0 = ⇒ y ( t ) = x ( t − t 0 ) output is delayed version of input For DT LTI system with unit gain and linear phase H ( e j ω ) = e − j ω n 0 , output � π ∞ y [ n ] = 1 � X ( e j ω ) e − j ω n 0 e j ω n d ω = x [ m ] sinc( n − n 0 − m ) 2 π − π m = −∞ • for integer n 0 , y [ n ] = x [ n − n 0 ] is delayed version of input • for non-integer n 0 , y [ n ] = y c ( n − n 0 ) is sample of delayed ∞ version of envelope y c ( t ) = � x [ m ] sinc( t − m ) of x m = −∞ 8/22

Linear Phase H ( j ω ) = e − j ω/ 2 For input 3 � x ( t ) = cos( 2 k π t ) = 1 + cos( 2 π t ) + cos( 4 π t ) + cos( 6 π t ) k = 0 output is 3 cos( 2 k π t − k π ) = x ( t − 1 � y ( t ) = 2 ) k = 0 x ( t ) t y ( t ) t 9/22

Linear Phase H ( e j ω ) = e − j ω/ 2 Half-sample delay For input x [ n ] = cos( π 3 n ) output is ∞ cos( π 3 m ) sinc( n − n 0 − m ) = cos( π 3 [ n − 1 � y [ n ] = 2 )] m = −∞ x [ n ] n y [ n ] n 10/22

Nonlinear Phase H ( j ω ) = e − j arctan ω For input 3 � x ( t ) = cos( 2 k π t ) = 1 + cos( 2 π t ) + cos( 4 π t ) + cos( 6 π t ) k = 0 output is 3 � y ( t ) = cos[ 2 k π t − arctan( 2 k π )] k = 0 x ( t ) t y ( t ) t 11/22

Group Delay For narrowband input x centered at ω 0 , i.e. X ( j ω ) = 0 , for | ω − ω 0 | > ∆ ω, where ∆ ω ≪ 1 Use linear approximation for phase arg H ( j ω ) ≈ arg H ( j ω 0 ) − τ ( ω 0 )( ω − ω 0 ) = φ 0 − τ ( ω 0 ) ω where group delay at ω is τ ( ω ) = − d d ω arg H ( j ω ) If | H ( j ω ) | ≈ | H ( j ω 0 ) | for | ω − ω 0 | ≤ ∆ ω , Y ( j ω ) ≈ | H ( j ω 0 ) | X ( j ω ) e j φ 0 − j τ ( ω 0 ) ω = ⇒ y ( t ) ≈ | H ( j ω 0 ) | e j φ 0 x ( t − τ ( ω 0 )) 12/22

Group Delay ⇒ τ ( ω ) = d 1 H ( j ω ) = e − j arctan ω = d ω arctan( ω ) = 1 + ω 2 For input (sum of two narrowband signals z , ¯ z centered at ± π ) 11 11 cos( k π 10 t ) = Re z ( t ) = 1 2 z ( t )+ 1 e j k π � � 10 t x ( t ) = 2 ¯ z ( t ) , where z ( t ) = k = 9 k = 9 output y ( t ) ≈ 1 2 e j φ 0 z ( t − τ 0 ) + 1 2 e − j φ 0 ¯ e j φ 0 z ( t − τ 0 ) � � z ( t − τ 0 ) = Re π 1 where φ 0 = − arctan π + 1 + π 2 , τ 0 = 1 + π 2 . x ( t ) t y ( t ) t 13/22

Contents 1. Magnitude-phase Representation of Fourier Transform 2. Uncertainty Principle 3. Relations Among Fourier Representations 14/22

Uncertainty Principle Assume CT signal x ∈ L 2 ( R ) , so X = F { x } ∈ L 2 ( R ) Define normalized power density in time and frequency | x ( t ) | 2 | X ( j ω ) | 2 p ( t ) = P ( ω ) = R | x ( τ ) | 2 d τ, � � R | X ( j θ ) | 2 d θ NB. p and P can be interpreted as probability densities, as done in quantum mechanics x and X are centered at t 0 and ω 0 resp. in the sense � � t 0 = tp ( t ) dt , ω 0 = ω P ( ω ) d ω R R “Standard deviation” measures energy spread around center � 1 � 1 �� �� 2 2 ( t − t 0 ) 2 p ( t ) dt ( ω − ω 0 ) 2 P ( ω ) d ω ∆ t = ∆ ω = , R R 15/22

Uncertainty Principle Theorem. If x ( t ) ∈ L 2 ( R ) with Fourier transform X ( j ω ) , then ∆ t ∆ ω ≥ 1 2 with equality iff x is Gaussian In fact, the following slightly more general relation holds D a ( x ) D b ( X ) ≥ 1 2 � x � 2 · � X � 2 where for g ∈ L 2 ( R ) and a ∈ R , � 1 �� 2 ( ξ − a ) 2 | g ( ξ ) | 2 d ξ D a ( g ) = R NB. Roughly speaking, signals cannot be localized in both time and frequency; short pulse has large bandwidth, narrowband signal has long duration 16/22

Proof of Uncertainty Principle First assume a = b = 0 . � 1 � 1 �� �� 2 2 ω 2 | X ( j ω ) | 2 dt | j ω X ( j ω ) | 2 d ω D 0 ( X ) = = R R F Since x ′ ( t ) ← − − → j ω X ( j ω ) , Parseval’s identity yields � 1 � � 2 | x ′ ( t ) | 2 dt D 0 ( X ) = 2 π R By Cauchy-Schwarz inequality � 1 � 1 √ �� 2 �� 2 | tx ( t ) | 2 dt | x ′ ( t ) | 2 dt D 0 ( x ) D 0 ( X ) = 2 π R R √ √ � � � � � � � � � � tx ∗ ( t ) x ′ ( t ) dt tx ∗ ( t ) x ′ ( t ) dt ≥ 2 π � ≥ 2 π � Re � � � � � � R R 17/22

Proof of Uncertainty Principle Note � � � � td | x ( t ) | 2 tx ∗ ( t ) x ′ ( t ) dt = tx ∗ ( t ) x ′ ( t ) dt + tx ( t ) x ′ ( t ) dt = 2 Re R R R R Integration by parts yields � � tx ∗ ( t ) x ′ ( t ) dt = t | x ( t ) | 2 � ∞ | x ( t ) | 2 dt 2 Re −∞ − � � R R Since inequality is trivial if D 0 ( x ) = ∞ , can assume D 0 ( x ) < ∞ , so t | x ( t ) | 2 → 0 as t → ∞ . Thus √ 2 π 2 = 1 � x � 2 D 0 ( x ) D 0 ( X ) ≥ 2 � x � 2 · � X � 2 2 For a , b � = 0 , note D a ( x ) = D 0 ( y ) and D b ( X ) = D 0 ( Y ) for F → Y ( j ω ) = X ( j ( ω + b )) e j ( ω + b ) a y ( t ) = x ( t + a ) e − jbt ← − − 18/22

Contents 1. Magnitude-phase Representation of Fourier Transform 2. Uncertainty Principle 3. Relations Among Fourier Representations 19/22

Four Fourier Representations CT Fourier series CT Fourier transform x [ k ] = 1 � � x ( t ) e − j 2 π T kt dt x ( t ) e − j ω t dt ˆ X ( j ω ) = T T R x ( t ) = 1 � � x [ k ] e j 2 π T kt x ( t ) = x ( t + T ) = ˆ X ( j ω ) e j ω t d ω 2 π k ∈ Z R DT Fourier series DT Fourier transform x [ k ] = 1 � X ( e j ω ) = x [ n ] e − j ω n � x [ n ] e − j 2 π N kn ˆ N n ∈ Z n ∈ [ N ] � x [ n ] = 1 x [ k ] e j 2 π X ( e j ω ) e j ω n d ω � N kn x [ n ] = x [ n + N ] = ˆ 2 π 2 π k ∈ [ N ] DFT is one period of DTFS 20/22

Relations among Four Fourier Representations time frequency CTFS continuous periodic discrete aperiodic CTFT continuous aperiodic continuous aperiodic DTFS discrete periodic discrete periodic DTFT discrete aperiodic continuous periodic Observations • periodic in one domain ⇐ ⇒ discrete in other domain • discretization by sampling in one domain ⇐ ⇒ periodic extension in other domain • continualization by interpolation in one domain ⇐ ⇒ extraction of one period in other domain 21/22

Relations among Four Fourier Representations T → ∞ (extract one period) CTFS CTFT periodic extension (sampling in frequency) sample interpolate sample interpolate N → ∞ (extract one period) DTFS DTFT periodic extension (sampling in frequency) NB. Conditions apply in some cases. 22/22