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Overview

COMPARING NOISE REMOVAL IN THE WAVELET AND FOURIER DOMAINS Dr. Robert Barsanti SSST March 2011, Auburn University. Overview. Introduction Transform Domain filtering Basis Selection Simulations and Results Summary. Introduction. It is widely known that the DFT has it shortcomings.

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Overview

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  1. COMPARING NOISE REMOVAL IN THE WAVELET AND FOURIER DOMAINSDr. Robert BarsantiSSST March 2011, Auburn University

  2. Overview • Introduction • Transform Domain filtering • Basis Selection • Simulations and Results • Summary

  3. Introduction • It is widely known that the DFT has it shortcomings. • We look at using the DWT on these signals. • We also use entropy to explain why one basis may be best. • Simulations of the performance of the proposed algorithm arepresented.

  4. Signal TRANSFORMATION Noisy Signal Noise Noise Removal • Separate the signal from the noise

  5. Fourier Analysis The DFT Wavelet Analysis The DWT FOURIER vs. WAVELETS

  6. Some Typical Wavelets

  7. Signal in the Time, Fourier, & Wavelet Domain

  8. Signal + Noise in the Time, Fourier, & Wavelet Domain

  9. Threshold De-noising Use Thres = Threshold Method -hard -soft

  10. Wavelet Based Filtering THREE STEP DENOISING 1. PERFORM DWT 2. THRESHOLD COEFFICIENTS 3. PERFORM INVERSE DWT

  11. Basis Selection Best Basis will concentrate signal energy into the fewest coefficients. Use Signal Entropy H(x) defined in [9] Where pi is normalized energy of ith component

  12. Entropy The entropy H(x) is bounded such that; H(x) = 0 only if all the signal energy is concentrated in one coefficient. H(x) = log(N), only if pi = 1/N for all i. The decomposition with the smaller entropy corresponds to the better basis for threshold filtering.

  13. Simulation

  14. Simulation • 3 simulated signal waveforms using 2^10 points. • Many trials using different instances of AWGN were conducted at signal to noise ratios ranging from -5 dB to 10 dB. • A sufficient number of trials were conducted to produce a representative MSE curve. Simulations for the all the filters used the same noise scale.

  15. Entropy Table

  16. Wavelets vs. Fourier Filtering signal 1 at 10 dB using the DFT MSE vs. SNR for signal 1.

  17. Wavelets vs. Fourier Filtering signal 3 at 10 dB using the DFT MSE vs. SNR for signal 3.

  18. Wavelets vs. Fourier Filtering signal 3 at 10 dB using the DFT MSE vs. SNR for signal 3.

  19. Summary • Discussed noise removal on signals using DFT and DWT. • Use of signal entropy as a measure of the best basis. • Simulations compared performance on simple signals.

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