University:
California State University, Northridge
Course:
ECE 501 | Introduction to Biomedical Engineering
Academic year:

2024
Views:

422

Pages:

32
Author:

Cerkvenikkop

ECE 501 Introduction to BME Part V Biomedical Signal Processing Introduction to Wavelet Transform Introduction Fourier Analysis Introduction Fourier Analysis • A serious drawback: time information is lost • Cannot handle transitory characteristics 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 200 400 600 800 1000 1200 Introduction Short-Time Fourier Analysis • A compromise between the time- and frequency-based views of a signal: analyze a small section of the signal at a time • A drawback: The window is the same for all frequencies Introduction Wavelet Analysis • A windowing technique with variable-sized regions: long time interval for low-frequency information, shorter regions for high-frequency information • Time-scale region Introduction What is Wavelet Analysis • A wavelet is a waveform of effectively limited duration that has an average value of zero • Wavelet analysis is the breaking up of a signal into shifted and scaled versions of the original (mother) wavelet. Continuous Wavelet Transform Fourier Analysis The sum over all time of the signal multiplied by a complex exponential Continuous Wavelet Transform CWT The sum over all time of the signal multiplied by scaled , shifted version of the wavelet function Continuous Wavelet Transform Scaling • Scaling a wavelet: stretching or compressing it • a: scaling factor Continuous Wavelet Transform • Low scale • High scale Scaling High frequency Low frequency Continuous Wavelet Transform Shifting Continuous Wavelet Transform Five Steps to a CWT 1. Take a wavelet and compare it to a section at the start of the original signal 2. Calculate the wavelet coefficient C Continuous Wavelet Transform Five Steps to a CWT 3. Shift the wavelet to the right and repeat steps 1 and 2 until the whole signal is covered. Continuous Wavelet Transform Five Steps to a CWT 4. Scale the wavelet and repeat steps 1 through 3 Continuous Wavelet Transform Five Steps to a CWT 5. Repeat steps 1 through 4 for all scales Continuous Wavelet Transform Plot CWT coefficients Continuous Wavelet Transform Plot CWT coefficients Discrete Wavelet Transform • Dyadic scales and positions: • Mallat algorithm: fast algorithm via filtering • Accurate analysis: compression, denoising Discrete Wavelet Transform One-Stage filtering: Approximations and Details Not Efficient! Discrete Wavelet Transform One-Stage filtering: Approximations and Details Efficient! Discrete Wavelet Transform One-Stage filtering: Approximations and Details Discrete Wavelet Transform One-Stage filtering: Approximations and Details Discrete Wavelet Transform Multiple-Level Decomposition Discrete Wavelet Transform Multiple-Level Decomposition Discrete Wavelet Transform Wavelet Reconstruction Up Sampling Discrete Wavelet Transform Wavelet Reconstruction Discrete Wavelet Transform Wavelet Reconstruction Wavelet Families Daubechies family Wavelet Families Symlets Denoising 1. Decompose 2. Threshold detail coefficients 3. Reconstruct Denoising Two thresholding method: (1) Soft (2) Hard

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