ece 501 introduction to bme ece 501 dr. hang. part v biomedical signal processing introduction to...
Post on 22-Dec-2015
220 views
TRANSCRIPT
ECE 501 Dr. Hang
Fourier Analysis
Introduction
• A serious drawback: time information is lost
• Cannot handle transitory characteristics
0 200 400 600 800 1000 1200-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
ECE 501 Dr. Hang
Short-Time Fourier Analysis
Introduction
• 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
ECE 501 Dr. Hang
Wavelet Analysis
Introduction
• A windowing technique with variable-sized regions: long time interval for low-frequency information, shorter regions for high-frequency information
• Time-scale region
ECE 501 Dr. Hang
What is Wavelet Analysis
Introduction
• 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.
ECE 501 Dr. Hang
Fourier Analysis The sum over all time of the signal multiplied by a complex exponential
Continuous Wavelet Transform
ECE 501 Dr. Hang
CWT The sum over all time of the signal multiplied by scaled , shifted version of the wavelet function
Continuous Wavelet Transform
ECE 501 Dr. Hang
Scaling• Scaling a wavelet: stretching or compressing it• a: scaling factor
Continuous Wavelet Transform
ECE 501 Dr. Hang
Scaling• Low scale High frequency • High scale Low frequency
Continuous Wavelet Transform
ECE 501 Dr. Hang
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
ECE 501 Dr. Hang
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
ECE 501 Dr. Hang
Five Steps to a CWT
4. Scale the wavelet and repeat steps 1 through 3
Continuous Wavelet Transform
ECE 501 Dr. Hang
Five Steps to a CWT
5. Repeat steps 1 through 4 for all scales
Continuous Wavelet Transform
ECE 501 Dr. Hang
• Dyadic scales and positions:
• Mallat algorithm: fast algorithm via filtering
• Accurate analysis: compression, denoising
Discrete Wavelet Transform
ECE 501 Dr. Hang
One-Stage filtering: Approximations and Details
Discrete Wavelet Transform
Not Efficient!
ECE 501 Dr. Hang
One-Stage filtering: Approximations and Details
Discrete Wavelet Transform
Efficient!