1. COMPARING NOISE REMOVAL IN THE WAVELET AND FOURIER DOMAINS Dr. Robert Barsanti SSST March 2011, Auburn University

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

3. Introduction (1)It is widely known that the DFT has it shortcomings. (2)We look at using the DWT on these signals. (3)We also use entropy to explain why one basis may be best. (4)Simulations of the performance of the proposed algorithm are presented.

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

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

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 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 p i is normalized energy of i th 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 p i = 1/N for all i. The decomposition with the smaller entropy corresponds to the better basis for threshold filtering.

13. Simulation

14. -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 DomainSignal 1Signal 2Signal 3 Time6.634.866.46 Fourier0.6934.883.50 Wavelet2.143.742.52

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

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

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

19. Summary (1)Discussed noise removal on signals using DFT and DWT. (2) Use of signal entropy as a measure of the best basis. (3)Simulations compared performance on simple signals.