Wavelet Transform n Fourier Transform n n Gives

Wavelet Transform n Fourier Transform n n Gives only frequency component No time information n We don’t know where the frequency change during the time Good for stationary signal (no sudden change in frequency) Wavelet Transform n Gives both frequency component + time n We know where the frequency change in time

Fourier Transform Weakness

Introduction to Wavelet Transform (1) n Objectives n Gives information about n n Allow us to detect features that might go underdetected at one resolution but may be easy to spot at another Basis function n Short-period wave (wavelet) n n n what frequencies contain in the signal (frequency component) Where the frequency change in time (time information) Mother wavelet Scale and Translation wavelet Example of Wavelet Application n Music example n Extract the frequency of notes and when to play that note

Time-Frequency Division

Introduction to Wavelet Transform (2) n Wavelet Transform n is similar to Fourier Transform n Wavelet Transform

Introduction to Wavelet Transform (3) n Example of Wavelet Basis Function n Mother wavelet n Scale & Translated wavelet

Introduction to Wavelet Transform (4) n Ex. Haar Wavelet n Mother wavelet n Scaled and Translated Wavelet

Forward Wavelet Transform Scaling function (Approximation function) Wavelet function (Detail function)

DWT Example f (x) = [1 2 3 4] Scaling function j 0 = 0

DWT Example j = j 0 = 0; Wavelet function k=0

DWT Example Wavelet function j = 1; k=0 j = 1; k=1

Wavelet Coefficients Scaling function Wavelet function

Inverse DWT Scaling function

Inverse DWT Wavelet function

Generation of New Mother wavelet from simple rectangular pulse n Using two-scale relation n Mother wavelet n New Mother Wavelet

Wavelet (Proof for fast wavelet) Mother wavelet Wavelet

Wavelet Coefficients n=2 k, k>=0

Wavelet Blockdiagram 2

Approximation and Detail Wavelet 2 2

Multi-resolution Wavelet Implementation

2 D Wavelet (Separable) (LL) (LH) (HL) (HH) Row Wavelet Column Wavelet

Example of Wavelet Coefficients LL LH HL HH

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