Continuous Wavelet Transforms Part I (Discrete to Follow) Rubin H - PowerPoint PPT Presentation

Continuous Wavelet Transforms Part I (Discrete to Follow) Rubin H Landau Sally Haerer, Producer-Director Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from the National Science Foundation Course: Computational Physics II 1 / 1

Time 9 0 1 2 3 4 5 6 7 8 Problem: Multiple Frequencies in Time Non Stationary Signals Numerical signal OK Amount ω i at each t ? Here analytic: ∆ number of ω ’s in t  sin 2 π t , for 0 ≤ t ≤ 2 ,     y ( t ) = 5 sin 2 π t + 10 sin 4 π t , for 2 ≤ t ≤ 8 ,     2 . 5 sin 2 π t + 6 sin 4 π t + 10 sin 6 π t , for 8 ≤ t ≤ 12 . 2 / 1

Why Not Fourier Analysis? Fourier Limitation: amount of sin ( n ω t ) pgflastimage No time resolution OK for stationary signals Fourier: correlated ω i ’s Not OK for Problem Poor data compression; Fourier: all ω i all time recompute c i 3 / 1

1.0 t 2 0 –2 –4 –6 t 0.5 0.0 –0.5 –1.0 ψ 4 6 0 –4 1.0 0.0 4 t 0 –4 1.0 0.0 –1.0 4 Wavelets in a Nutshell Three Wavelet Examples Extend Fourier Varied functional forms Wavelet basis expansion Nonstationary signals "let": small wave (pack) Fairly recent Each: finite & ∆ T Extensive applications Each: center different t E.g.: all oscillate 4 / 1

Y ω t –4 0 4 y Wave Packets = � Waves Wave Packet e.g. N Cycle Sine 0 10 Packet ⇒ y ( t ) = pulse ∆ t ⇒ Y ( ω ) = pulse ∆ ω  | t | < N T sin ω 0 t , for 2 ,  y ( t ) = | t | > N T 0 , for 2 ,  ⇒ ∆ t = NT = N 2 π ∆ ω ≃ ω 0 , N ω 0 5 / 1

t –4 0 4 y Y ω Uncertainty Principle (Theory) Fundamental Relation: ∆ t ↔ ∆ ω 0 10 N cycle example ⇒ general truth ∆ ω ≃ first 0’s of Y ( ω ) : ω − ω 0 = ± 1 ∆ ω ≃ ω − ω 0 = ω 0 ⇒ N N ω 0 N cycle ⇒ ∆ t ≃ NT = N 2 π ω 0 ⇒ ∆ t ∆ ω ≥ 2 π QM: "Heisenberg Uncertainty Principle" 6 / 1

Wave Packet Assessment (before break) Example Given three wave packets: y 1 ( t ) = e − t 2 / 2 , y 2 ( t ) = sin ( 8 t ) e − t 2 / 2 , y 3 ( t ) = ( 1 − t 2 ) e − t 2 / 2 For each wave packet: Estimate the width ∆ t . A good measure might be the full 1 width at half-maxima (FWHM) of | y ( t ) | . Evaluate and plot the Fourier transform Y ( ω ) . 2 Estimate the width ∆ ω of the transform. A good measure 3 might be the full width at half-maxima of | Y ( ω ) | . Determine the constant C for the uncertainty principle 4 ∆ t ∆ ω ≥ 2 π C . 7 / 1

Continuous Wavelet Transforms Part II (Discrete to Follow) Rubin H Landau Sally Haerer, Producer-Director Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from the National Science Foundation Course: Computational Physics II 8 / 1

Time 9 0 1 2 3 4 5 6 7 8 Aside: Wavelet Precursor Sets Stage Colored Boxes → Windows w ( t ) Seen: sin n ω t ∃ all t ’s ⇒ FT short time interval Overlap ⇒ correlated Boxes = windows = w ( t ) Dependent components ⇒ Y τ 1 ( ω ) , Y τ 2 ( ω ) , . . . Y τ N ( ω ) � + ∞ dt e i ω t w ( t − τ ) y ( t ) Y ( ST ) ( ω, τ ) = −∞ 9 / 1

–4 4 0 –2 6 –6 1.0 0.5 0.0 –0.5 –1.0 ψ t 0 4 –4 1.0 0.0 4 t 0 –4 1.0 0.0 –1.0 t 2 The Wavelet Transform Y ( ω ) : exp ( i ω t ) → Y ( s , τ ) : ψ s ,τ ( t ) � + ∞ Y ( s , τ ) = dt ψ ∗ s ,τ ( t ) y ( t ) (wavelet transform) −∞ ∼ Short-time FT τ : time interval analyzed Wavelet localized in t s = scale = 2 π/ω ⇒ Own window t details ⇒ small s Oscillations ⇒ ∆ ω Small scale ⇒ high ω Y = amt ψ s ,τ ( t ) in y ( t ) 10 / 1

–6 –2 –2 0 2 4 6 t 0 –4 0 –6 2 4 6 t s฀=฀2,฀ τ ฀=฀0 –1.0 0.0 1.0 –4 s฀=฀1,฀ τ ฀=฀0 –2 –0.5 s฀=฀1,฀ τ฀ =฀6 t 10 8 6 4 2 –1.0 0.0 6 0.5 1.0 –6 –4 –2 0 2 4 –4 Generating Wavelet Basis Functions � t − τ 1 � Scale by s , Translate by τ : ψ s ,τ ( t ) = √ s Ψ s Ψ = mother of ψ Fixed # oscills; vary T , 0 Need fewer large s s <, > 1 → high, low ω Small s : details Large s : smooth envelope Need for hi resolution 11 / 1

Visualization: Transform of Chirp sin ( 60 t 2 ) � t − τ � + ∞ � 1 Y ( s , τ ) = √ s dt Ψ ∗ y ( t ) (Transform) s −∞ � + ∞ � + ∞ 1 ds s 3 / 2 ψ ∗ y ( t ) = d τ s ,τ ( t ) Y ( s , τ ) (Inverse) C 0 −∞ Convolute low scale Cover all ⇒ High res Expand ⇒ Shape 12 / 1

s 2 20 0 –20 12 8 4 0 Y 1 0 12 0 y(t) 1 Inverted฀Transform 0 ฀t Time 9 8 7 6 5 4 3 2 1 0 Input Solution to Problem Recall Nonstationary Signal 13 / 1

Required of Mother Wavelet Ψ For Math to Work Ψ( t ) is real 1 Ψ( t ) oscillates around 0 such that the average 2 � + ∞ Ψ( t ) dt = 0 −∞ Ψ( t ) is local (wave packet) & square integrable 3 � + ∞ | Ψ( t ) | 2 dt < ∞ −∞ The first p moments vanish (for details): 4 � + ∞ � + ∞ � + ∞ t 0 Ψ( t ) dt = t 1 Ψ( t ) dt = · · · = t p − 1 Ψ( t ) dt = 0 −∞ −∞ −∞ 14 / 1

Implementation: Visualizing Wavelet Transforms Example Convert your DFT program to a CWT one. 1 Examine different mother wavelets. Write methods for 2 a Morlet wavelet 1 a Mexican hat wavelet 2 a Haar wavelet 3 Test your transform on input: 3 y ( t ) = sin 2 π t , 1 y ( t ) = 2 . 5 sin 2 π t + 6 sin 4 π t + 10 sin 6 π t , 2 The nonstationary signal for our problem: 3  sin 2 π t , for 0 ≤ t ≤ 2 ,    y ( t ) = 5 sin 2 π t + 10 sin 4 π t , for 2 ≤ t ≤ 8 ,   2 . 5 sin 2 π t + 6 sin 4 π t + 10 sin 6 π t , for 8 ≤ t ≤ 12 .  Invert your CWT & compare to input. 4 15 / 1