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Overview

WAVELET NOISE REMOVAL FROM BASEBAND DIGITAL SIGNALS IN BANDLIMITED CHANNELS Dr. Robert Barsanti SSST March 2010, University of Texas At Tyler. Overview. Introduction Baseband Digital Signals Wavelet Domain Filtering Matched Filtering Wavelet Coefficients Simulations and Results Summary.

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Overview

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  1. WAVELET NOISE REMOVAL FROM BASEBAND DIGITAL SIGNALS IN BANDLIMITED CHANNELSDr. Robert BarsantiSSST March 2010, University of Texas At Tyler

  2. Overview • Introduction • Baseband Digital Signals • Wavelet Domain Filtering • Matched Filtering Wavelet Coefficients • Simulations and Results • Summary

  3. s(t) Digital Input Data ak Pulse Amplitude Modulator Transmit Filter g(t) Baseband Digital Signals The simplest digital baseband systems transmit binary data made up of pulses. These Pulse Amplitude Modulated (PAM) signals can be represented as The symbol ‘a’ represents the signal amplitude, and T is the bit duration. Block diagram of PAM signal generation [6].

  4. Baseband Digital Signals The power spectrum of the PAM signal is Where the power spectrum of the uncorrelated symbols is given by , and G(f) is the spectrum of the pulse g(t)

  5. y(nT) r(t) an Decision Device s(t) Channel c(t) Receive Filter h(t) + Sample t = T n(t) Baseband Digital Signals Block diagram of PAM signal reception [6]. x(t) Transmitted symbol Filtered Noise ISI

  6. Raised Cosine Pulse Shape This figure shows an example of a raised cosine pulse used to combat ISI.

  7. Antipodal Signal Probability of Bit Error • Assuming the received signal at the input to the correlator is corrupted with Gaussian noise of zero mean and variance No/2. The probability of bit error can be computed to be [5] • It can be seen that the probability does not depend on the detailed signal and noise characteristics, but only upon the signal to noise ratio per bit (SNR) [6].

  8. The Classical Cross Correlation Receiver for two transmitted signals Detector (choose largest) X Output Symbol ri So X Sample at t = T S1

  9. Time Domain Matched Filter Receiver for two transmitted signals Detector (choose largest) Receive Filter h1(t) Output Symbol ri Receive Filter h2(t)

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

  11. Wavelet Filtering

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

  13. Wavelet Filtering of a PAM Signal DWT of a noise free PAM signal. DWT of noisy PAM signal.

  14. Wavelet Domain Filtering • transform prototype signal into the wavelet domain and pre-stored DWT coefficients • transform received signal into the wavelet domain via the DWT, • apply a non-linear threshold to the DWT coefficients (to remove noise), • correlate the noise free DWT coefficients of the signal, and the pre-stored • DWT coefficients of the prototype signal.

  15. Wavelet Domain Correlation Receiver Bank of Cross-Correlation Receivers Detector (choose largest) Wavelet De-noise r i DWT Output Symbol So DWT S1 DWT

  16. y(Nk) Output Symbol r(t) w(t) w’(k) Wavelet Receive Filter h(k) Decision Device DWT Wavelet De-noise Wavelet Domain Matched Filter Receiver

  17. Simulation • Pulse signals were generated using 256 samples per symbol • Monte Carlo Runs at each SNR with different instance of AWGN • 10 SNR’s between -6 and +10 dB • Symmlet 8 wavelet & hard threshold • Threshold set to σ/2. • Only 64 coefficients retained

  18. Bit Error Rate Curve for PAM

  19. Results • Both the WD and classical TD receivers provided similar results. • The WDC provides improvement in processing speed since only • 64 vice 256 coefficients were used in the correlations.

  20. Summary • Receiver for PAM signals in the presence of AWGN. • Uses the matched filtering of DWT coefficients • Procedure is enhanced by using standard wavelet noise removal techniques • Simulations of the performance of the proposed algorithm were presented.

  21. Wavelets Some S8 Symmlets at Various Scales and Locations 9 8 7 6 5 Scale j 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 time index k 1. Can be defined by a wavelet function (Morlet & Mexican hat) 2. Can be defined by an FIR Filter Function (Haar, D4, S8)

  22. EFFECTIVENESS OF WAVELET ANALYSIS • Wavelets are adjustable and adaptable by virtue of large number of possible wavelet bases. • DWT well suited to digital implementation. ~O (N) • Ideally suited for analysis non-stationary signals [ Strang, 1996] • Has been shown to be a viable denoising technique for transients [Donoho, 1995] • Has been shown to be a viable detection technique for transients [Carter, 1994] • Has been shown to be a viable TDOA technique for transients [Wu, 1997]

  23. Wavelet Implementation Response LPF HPF HP Filter Details X(n) LP Filter Frequency Averages F/2 Pair of Half Band Quadrature Mirror Filters (QMF) [Vetterli, 1995]

  24. Signal Reconstruction Two Channel Perfect Reconstruction QMF Bank Analysis + Synthesis = LTI system

  25. Wavelet Implementation [Mallat, 1989] 2 LPLPLP J = 4 2 2 LP LP LP 2 HP LPLPHP J = 3 2 LPHP J = 2 HP 2 HP HP J = 1 2J samples LP HP LPHP LPLPHP LPLPLP

  26. Symmlet Wavelet vs. Time and Frequency

  27. Calculating a Threshold Let the DWT coefficient be a series of noisy observations y(n) then the following parameter estimation problem exists: y(n) = f(n) +s z(n), n = 1,2,…. z ~N(0,1) and s = noise std. s is estimated from the data by analysis of the coefficients at the lowest scale. s = E/0.6475 where E is the absolute median deviation [Kenny]

  28. Thresholding Techniques * Hard Thresholding (keep or kill) * Soft Thresholding (reduce all by Threshold) The Threshold Value is determined as a multiple of the noise standard deviation, eg., T = ms where typically 2< m <5

  29. Hard vs. Soft Thresholds

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