Wavelets in DSP (excerpts)

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Excerpts from talk given 11/18/09 to IEEE San Diego Section by D. Lee Fugal, Chairman

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  • really useful to understand wavelets and its infinite aplications, i am trying to do a thesis on this subject, if any one know something about revolutionary methods to work noise in ECG signal using wavelets let me now, I will be really grateful
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Wavelets in DSP (excerpts)

  1. 1. ©2006-2010 ATICOURSES All Rights Reserved ©2010 D.L. Fugal WAVELETS: ANOTHER DIMENSION IN DIGITAL SIGNAL PROCESSING D. Lee Fugal, Chairman, IEEE Signal Processing Society IEEE San Diego Section Talk 11/18/09 1 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  2. 2. ©2006-2010 ATICOURSES All Rights Reserved WHAT IS A WAVELET? ©2010 D.L. Fugal Cosine Wave Db4 Wavelet • Sinusoids extend from minus to plus infinity. • Wavelet is waveform of limited duration (Starts & Stops) • Sinusoids are smooth and predictable. • Wavelets tend to be irregular and asymmetric. • Wavelets have an average value of zero • Wavelets are compared (correlated) with signals that have “events” in time like heartbeat, stock market, pulses. • Jargon Alert: This type of signal called “Non-Stationary” 2 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  3. 3. ©2006-2010 ATICOURSES All Rights Reserved EXAMPLES OF WAVELETS ©2010 D.L. Fugal • WAVE for Frequency, LET indicates Compact Support. • Jargon Alert*: Compact Support = having start & stop time • Some more localized in time, some more localized in freq. Haar Shannon or Sinc Daubechies 4 Daubechies 20 Gaussian or Spline Biorthogonal Mexican Hat Custom (arbitrary) 3 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  4. 4. ©2006-2010 ATICOURSES All Rights Reserved USES OF PARTICULAR WAVELETS ©2010 D.L. Fugal • Haar: Good for edge detection in images, for matching binary pulses, for very short phenomenon. • Shannon: Dual of Haar wavelet. Good frequency resolution and signal identification using frequency. Poor time resolution. • Daubechies: Robust, fast for identifying signals with both time and freq characteristics (use longer filters for better frequency resolution). Used in speech, fractals, non-symmetrical transients. Identifies polynomial signals or noise • Biorthogonal (2 wavelets). Symmetry and Linear Phase. Used extensively in Image Processing because human vision more tolerant of symmetrical errors and because images can be extended. Chosen by FBI and for JPEG. 4 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  5. 5. ©2006-2010 ATICOURSES All Rights Reserved APPLICATIONS OF WAVELETS ©2010 D.L. Fugal • Signal and Image Compression and Denoising. JPEG, FBI • Geology, Oceanography, Astronomy, Electrical Systems. • MRIs and similar non-invasive procedures. Mammogram enhancement to distinguish Tumors from calcifications. • EEG/EKG detection of transient “events”. • Finance for stock market patterns, quick variations of value. Internet Traffic. Biology. Metallurgy. Speech. • Radar and Sonar. Pulse detection by both time and frequency. Automatic signal and target recognition.* • Study of short-time phenomena as transient processes. • Non-Destructive Testing, SAR imagery • Motion Pictures (e.g. “A Bug’s Life”) • Rupture and Edge Detection (airport baggage screening). 5 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  6. 6. ©2006-2010 All Rights Reserved TRANSFORMS AND COMPARISONS ATICOURSES ©2010 D.L. Fugal COMPARISON OF WAVELET TRANSFORMS TO FOURIER TRANSFORMS (DFT/FFT) AND SHORT-TIME FOURIER TRANSFORMS (STFT) 6 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  7. 7. ©2006-2010 ATICOURSES All Rights Reserved FFT CLASSIC EXAMPLE ©2010 D.L. Fugal • Noise in signal can be identified using FFT. • Can be removed using conventional filtering methods. • Here we remove 60-Hz noise “spike” or hum. • For this signal, the FFT is a better choice than Wavelets 2 HZ SIG WITH 60 HZ HIGH FREQ NOISE 2 HZ SIG WITH 60HZ HIGH FREQ NOISE 1.5 140 120 1 100 MAGNITUDE --> 0.5 MAGNITUDE --> 80 0 60 -0.5 40 -1 20 -1.5 0 0 50 100 150 200 250 0 50 100 150 FREQUENCY --> 7 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  8. 8. ©2006-2010 ATICOURSES All Rights Reserved FFT SIGNAL + NOISE ©2010 D.L. Fugal • De-noised signal shown below. Wavelets refers to this as the low-freq “Approximation” of the original signal. • Noise is also shown. Wavelets nomenclature refers to as high-frequency “Details”. • Note “Approximation” + “Details” = Original signal. DENOISED (LOW FREQ "APPROXIMATION") NOISE (HIGH FREQ "DETAILS") 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 0 50 100 150 200 250 0 50 100 150 200 250 8 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  9. 9. ©2006-2010 ATICOURSES All Rights Reserved FFT SHORTCOMINGS ©2010 D.L. Fugal LOW FREQ SIGNAL WITH HIGH FREQ NOISE 2 180 1.5 160 140 1 MAGNITUDE --> AMPLITUDE --> 120 0.5 100 0 80 -0.5 60 -1 40 -1.5 20 -2 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 FREQ --> LOW FREQ SIGNAL THEN HIGH FREQ SIGNAL LOW FREQ SIGNAL THEN HIGH FREQ SIGNAL 1 120 100 0.5 AMPLITUDE --> MAGNITUDE --> 80 0 60 40 -0.5 20 -1 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 TIME --> FREQ --> • Signal characteristics not seen in the FFT • Why wavelets are needed. Show both time & freq. 9 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  10. 10. ©2006-2010 ATICOURSES All Rights Reserved TIME/FREQ RESPONSE DEMO ©2010 D.L. Fugal • Compromise between the time- and frequency-based views of a signal. Provides some information about both. • Example of Discrete Fourier Transform (DFT) with piano strings and the word “Hello” heard in the 88 resonating piano-string frequencies (an “Audio-Based Discrete Fourier Transform”). • Example of Short Time Fourier Transform (STFT) by hearing the the piano-string DFT for the time-sequential words “Heh” and “Low” in succession. • Next look at Heisenberg Cells (boxes). • Jargon Alert: The Heisenberg Uncertainty Principle says you can’t know an exact frequency at an exact time* (it takes some time to oscillate--more for low notes, less for high). Thus a cell or box has the same area (next slide). 10 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  11. 11. ©2006-2010 ATICOURSES All Rights Reserved SHORT TIME FOURIER TRANSFORM ©2010 D.L. Fugal Amplitude 2 LONG WINDOWS PRECISION Frequency Time Time Heisenberg 4 SHORT WINDOWS PRECISION Cells (boxes). Note same Amplitude Frequency area in both shapes. Time Time • Looking at signal for long times (integration time) gives better frequency precision, but poorer time precision (when did it occur?). • Looking at signal for short times gives better time precision , but poorer frequency precision (what was it’s frequency at that time?). 11 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  12. 12. ©2006-2010 ATICOURSES All Rights Reserved WAVELET WINDOWING PATTERN ©2010 D.L. Fugal PRECISION “NATURAL” FREQUENCY PATTERN (don’t Amplitude need as much time to identify high freqs) Time TIME • Windowing technique with variable-sized regions. • Allows the use of long time intervals where we need more precise frequency information (low freqs), and shorter regions where we need precise time information. • An example of this “natural” pattern has been around for hundreds of years: Sheet Music 12 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  13. 13. ©2006-2010 ATICOURSES All Rights Reserved TIME/FREQ EXAMPLE (MUSIC SCORE) ©2010 D.L. Fugal ff tempo 60 Frequency --> Frequency (inverse of Scale) 4 4 mf 4 4 pp Time --> Time • Frequency of musical notes are factors of 2 apart (octaves) • “Digital” in time (tempo, “4/4”), frequency, magnitude (ff) • “Low Notes” (lower frequency notes) need longer times to be correctly generated (tuba vs. piccolo). • Human ear requires longer time to determine frequency (pitch) and overtones of low notes. • (Display here is inverted from most wavelet displays). 13 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  14. 14. ©2006-2010 ATICOURSES All Rights Reserved MUSICAL EXAMPLE ©2010 D.L. Fugal • Top level of display has shorter times for higher frequencies (which don’t need as much time for good resolution). • Demonstration of Piccolo solo from John Philip Sousa’s “Stars and Stripes Forever” shows capability rapid changes at higher frequencies (lower scales). • Demonstration of Piccolo solo played on tuba shows not enough “integration time” for the lower frequency (higher scale) notes to be formed correctly (even if the musician does a perfect job of valve fingering). Piccolo solo from “Stars and Stripes Forever” 14 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  15. 15. ©2006-2010 ATICOURSES All Rights Reserved FFT-TYPE PULSE COMPARISON ©2010 D.L. Fugal Pulse Signal. 5 cycles in A 1/4 second = 20 Hz. Centered at 3/8 second. 40 Cycle per Second B (40 Hz) Sinusoid for comparison with pulse signal A. Poor correlation. Sinusoid stretched to 20 Hz for comparison. Good correlation. C Same frequency as pulse so peaks and valleys can align. Sinusoid stretched to 10 Cycles/Sec (10 Hz) D for comparison. Poor correlation again. Time (seconds) 0 1/4 1/2 3/4 1 15 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  16. 16. ©2006-2010 ATICOURSES All Rights Reserved ACTUAL FFT (DFT) OF PULSE ©2010 D.L. Fugal D C B EQUATION INDICATES THAT Magnitude THE SIGNAL IS MADE UP OF CONSTITUENT SINUSOIDS Frequency (Hz) 0 10 20 30 40 50 NOTE: Only frequency information is given by the FFT N −1 Χ ( k ) = ∑ x(n)e − j ( 2 π / N ) nk OR USING THE EULER IDENTITY n =0 N −1 N −1 = ∑ x (n) cos( 2 πnk / N ) − j ∑ x (n) sin( 2πnk / N ) n=0 n=0 16 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  17. 17. ©2006-2010 ATICOURSES All Rights Reserved CWT-TYPE PULSE COMPARISON (1) ©2010 D.L. Fugal Pulse Signal. 5 cycles in 1/4 second = 20 Hz. A Centered at 3/8 second. Roughly 40 Hz Daubechies 20 (Db20) Wavelet B for comparison with pulse signal A. Poor correlation. Roughly 40 Hz Db20 Wavelet shifted in time to line up with the pulse. Still a poor C comparison because the frequencies don’t match. Db20 Wavelet stretched D (“scaled”) by 2 to roughly 20 Hz and shifted for comparison. Good comparison (correlation). Time (seconds) 0 1/4 1/2 3/4 1 (If energy of wavelet and signal are both unity, values are correlation coefficients) 17 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  18. 18. ©2006-2010 ATICOURSES All Rights Reserved CWT-TYPE PULSE COMPARISON (2) ©2010 D.L. Fugal Pulse Signal. 5 cycles in 1/4 second = 20 Hz. Centered at 3/8 second. Db20 Wavelet stretched to roughly 20 Hz and shifted to where peaks begin to line up with peaks (or valleys). Weak correlation just past 1/4 second. Db20 Wavelet stretched to roughly 20 Hz and shifted to where more peaks line up. Stronger correlation just before 3/8 second. Db20 Wavelet stretched by 2 to roughly 20 Hz and shifted for comparison. Strongest correlation at 3/8 second. Time (seconds) 0 1/4 1/2 3/4 1 18 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  19. 19. ©2006-2010 ATICOURSES All Rights Reserved ACTUAL CWT OF PULSE ©2010 D.L. Fugal wavelet shifted to right by 3/8 second wavelet stretched Stretching or “scaling” to approx. (inverse of frequency) 20 Hz. unstretched basic wavelet at low scale. (poor results) 0 1/4 1/2 3/4 1 Time (seconds) NOTE: Both time AND frequency information of pulse given by the CWT! Also repeating this with various wavelets indicates the SHAPE of event! (Equation indicates that the signal is made up of constituent wavelets). 19 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  20. 20. ©2006-2010 ATICOURSES All Rights Reserved DEMO OF CWT CAPABILITY ©2010 D.L. Fugal TIME PLOT OF SIGNAL WITH SMALL DISCONTINUITY FFT PLOT OF SIGNAL WITH SMALL DISCONTINUITY 1 160 0.8 Hidden High frequency 140 0.6 discontinuity segments of 0.4 at time = 180 120 discontinuity too small 0.2 not visible on 100 to see on this 0 Amplitude vs. 80 Magnitude vs. -0.2 Time plot. 60 Frequency FFT plot, -0.4 -0.6 and would give no 40 -0.8 indication as to when 20 -1 they occurred anyway. 0 50 100 150 200 250 300 0 Time 0 50 100 150 200 250 300 Frequency WAVELET PLOT OF SIGNAL & DISCONTINUITY 20 Stretching (“Scaling” or “Level”) 18 16 Stretched “low frequency” Db4 14 wavelet compares better to 12 sinusoidal (wave) signal. It 10 “finds” peaks and valleys. 8 6 Small “high frequency” wavelet 4 compares well to discontinuity. 2 50 300 It “finds” it’s location at 180. Time 20 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  21. 21. ©2006-2010 All Rights Reserved FILTERS FROM WAVELETS ATICOURSES ©2010 D.L. Fugal OBTAINING REAL-WORLD DISCRETE FILTERS FROM WAVELETS WITH EXPLICIT MATHEMATICAL EXPRESSIONS (“CRUDE WAVELETS”) 21 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  22. 22. ©2006-2010 ATICOURSES All Rights Reserved WAVELET “LENGTH” FOR MEXH ©2010 D.L. Fugal • Wavelets in the real world of digital computers are also filters. First look at the the Mexican Hat “crude” wavelet:. • mexh(t) = 2/(sqrt(3)∗pi^0.25)∗exp(-t^2/2) ∗ (1-t^2) • Jargon Alert: 1 MEXICAN HAT WAVELET effective length “Crude” means 0.8 generated from explicit math 0.6 equation. AMPLITUDE --> 0.4 • Effective 0.2 Length from -8 to +8 0 (e.g. value at -0.2 time 5.1 = -0.4 -8 -6 -4 -2 0 2 4 6 8 3.6939e-06) (Relative) TIME --> 22 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  23. 23. ©2006-2010 ATICOURSES All Rights Reserved MEXH 17 POINTS ©2010 D.L. Fugal • Jargon Alert: Effective Length is often referred to as “Effective Support”. • Even with explicit mathematical expressions, we still must treat them as 17 POINTS ON MEXICAN HAT WAVELET 1 digital filters in convolving with 0.8 the signal in the 0.6 time domain. AMPLITUDE --> 0.4 • For the CWT, 0.2 start with short, 0 unstretched, HF filter. Values at -0.2 integers produce17 -0.4 -8 -6 -4 -2 0 2 4 6 8 points from -8 to +8. (Relative) TIME --> 23 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  24. 24. ©2006-2010 ATICOURSES All Rights Reserved MEXH 33 POINTS ©2010 D.L. Fugal • After comparing 17-point “filter” with signal (scale = a = 1), CWT software “stretches” it to 33 points corresponding to values of MEXH wavelet at the 1/2 integer points from - 8 to +8 (-8, -7.5, -7 . . . 0 . . . +8). This is scale = 2. • The next stretching (scale =3) is the 49 points corresponding to 1/3 integer values in the same interval. 33 POINTS ON MEXICAN HAT WAVELET 49 POINTS ON MEXICAN HAT WAVELET 1 1 0.8 0.8 0.6 0.6 AMPLITUDE --> AMPLITUDE --> 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -5 0 5 -5 0 5 TIME --> TIME --> 24 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  25. 25. ©2006-2010 All Rights Reserved WAVELETS FROM FILTERS ATICOURSES ©2010 D.L. Fugal WAVELET FILTERS OF SPECIFIC LENGTH THAT BUILD APPROXIMATIONS TO A “CONTINUOUS” WAVELET FUNCTION WHICH IN TURN CAN THEN PRODUCE FILTERS OF ANY DESIRED LENGTH. 25 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  26. 26. ©2006-2010 ATICOURSES All Rights Reserved BASIC WAVELET & SCALING FUNCT. ©2010 D.L. Fugal • Here is the basic Db4 wavelet filter -0.1294 -0.2241 0.8365 -0.4830 and the lowpass filter or scaling function filter 0.4830 0.8365 0.2241 -0.1294 • Note similarities in the filter values. PRQMFs. 4 PT BASIC HP DB4 WAVELET FILTER 4 PT BASIC LP SCALING FUNCTION FILTER 1 1 0.8 AMPLITUDE --> AMPLITUDE --> 0.5 0.6 0.4 0 0.2 0 -0.5 -0.2 1 2 3 4 1 2 3 4 NUMBER OF POINTS (n) --> NUMBER OF POINTS (n) --> 26 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  27. 27. ©2006-2010 ATICOURSES All Rights Reserved DB4 UPSAMPLED AND LPF (STRETCH) ©2010 D.L. Fugal • Here is the basic wavelet filter upsampled with zeros between the existing points. • After lowpass filtering we have a “stretched” 10-point filter (length = 7 pts of upsampled filter + 4 pts LPF -1 = 10) UPSAMPLED STRETCHED 1 1.5 1 0.5 AMPLITUDE --> AMPLITUDE --> 0.5 0 0 -0.5 -0.5 -1 0 2 4 6 8 0 2 4 6 8 10 NUMBER OF POINTS (n) --> NUMBER OF POINTS (n) --> 27 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  28. 28. ©2006-2010 ATICOURSES All Rights Reserved DB4 STRETCHED TO MORE POINTS ©2010 D.L. Fugal • We continue the process of upsampling and lowpass filtering to produces increasingly stretched wavelet filters with 22, 46 (shown below), 94, 190 (shown below), 382 and finally 766 points. • We now have an approximation of a Db4 “continuous” wavelet function built from the original 4 points. 1.5 1.5 1 1 AMPLITUDE --> AMPLITUDE --> 0.5 0.5 0 0 -0.5 -0.5 -1 -1 0 10 20 30 40 50 0 50 100 150 200 NUMBER OF POINTS (n) --> NUMBER OF POINTS (n) --> 28 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  29. 29. ©2006-2010 ATICOURSES All Rights Reserved 4 + 2 FILTER PTS SUPERIMPOSED ©2010 D.L. Fugal • We superimpose the original 4 Db4 filter points used to build this wavelet function. As we convert our 766 point “continuous” function to a “length” of 0 to 3, the points -0.1294 -0.2241 0.8365 -0.4830 are found at 2/6, 5/6, 8/6 and 11/6 or 1/2 integer apart starting at 1/3. They are overplotted on the wavelet function along with the zero values at 14/6 and 17/6. WAVELET FUNCTION PSI • Like the “crude” 1.5 wavelet filters, can be 1 used with a CWT. AMPLITUDE --> 0.5 • Unlike the crude filters 0 they can be used with -0.5 a Discrete Wavelet Transform (DWT) -1 0 0.5 1 1.5 2 2.5 3 "TIME" (t) --> 29 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  30. 30. ©2006-2010 All Rights Reserved DISCRETE WAVELET TRANSFORMS ATICOURSES ©2010 D.L. Fugal THE UNDECIMATED DISCRETE WAVELET TRANSFORM (UDWT). Also called Stationary, Shift Invariant, “A’ Trous”, or “redundant” (but not near as redundant as the CWT) 30 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  31. 31. ©2006-2010 ATICOURSES All Rights Reserved 1 LEVEL UDWT SYSTEM ©2010 D.L. Fugal • The Scaling Funtion filters L and L’ produce a Halfband Lowpass Filter while the Wavelet Filters H and H’ produce a Halfband Highpass Filter. • Jargon Alert: Halfband filters cut the frequency band in half as shown below--with some symmetrical overlap. • Summing the results of the highpass and lowpass halfband filters produces a constant in the frequency domain • Final result, S’, is the H H’ same as the original signal, S, except for a D1 delay and/or a scaling constant S S’ D1 L L’ A1 A1 Frequency Spectrum 31 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  32. 32. ©2006-2010 ATICOURSES All Rights Reserved 2-LEVEL UDWT ©2010 D.L. Fugal • The filter in the RED oval is essentially the original Lowpass “Scaling Function Filter” stretched (“scaled”) in time by a factor of 2. • Recall the Haar Scaling Function (lowpass) Filter L = [1 1] convolved with the upsampled Wavelet (highpass) Filter Hup = [1 0 -1] produced [1 1 -1 -1] H cD1 H D1 S S’ Hup cD2 Hup cA1 cA1 A1 L L cA2 Lup Lup 32 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  33. 33. ©2006-2010 ATICOURSES All Rights Reserved 2-LEVEL EQUIVALENT UDWT ©2010 D.L. Fugal • The filters in the GREEN oval are also stretched versions of the highpass and lowpass filters. The output is D2 and A2. Added together they produce D2+A2=A1. • Thus the reconstructed signal, S’, is given by S’ = D1+A1 = D1+D2+A2 H cD1 H D1 S S’ Hup cD2 Hup D2 L cA1 A1 L cA2 A2 Lup Lup L 33 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  34. 34. ©2006-2010 ATICOURSES All Rights Reserved FREQUENCY ALLOCATION ©2010 D.L. Fugal • Freq. allocation for a 2 and a 4-level UDWT shown here • Beginning to see utility and flexibility. Remember that, unlike the FFT we can adjust Details and Approximations for any desired part of the total time. 2-level S MAGNITUDE UDWT A1 freq. NORMALIZED FREQUENCY bands A2 D2 D1 (NYQUIST = 1) FREQUENCY S A1 4-level A2 UDWT A3 MAGNITUDE freq. Nyquist Frequency bands A4 D4 D3 D2 D1 FREQUENCY 34 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  35. 35. ©2006-2010 All Rights Reserved DISCRETE WAVELET TRANSFORMS ATICOURSES ©2010 D.L. Fugal • THE DISCRETE (Conventional, Decimated) WAVELET TRANSFORM (Usually called “The DWT”) 35 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  36. 36. ©2006-2010 ATICOURSES All Rights Reserved 1: STRETCH WAVELET BY 2 - UDWT ©2010 D.L. Fugal 36 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  37. 37. ©2006-2010 ATICOURSES All Rights Reserved 2: SHRINK SIGNAL BY 2 - DWT* ©2010 D.L. Fugal Scale 1 or level 0 Scale 2 or level 1 Scale 4 or level 2 37 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  38. 38. ©2006-2010 ATICOURSES All Rights Reserved 1 LEVEL DWT SYSTEM ©2010 D.L. Fugal H H’ D1 cD1 cD1 S S’ L cA1 cA1 L’ A1 “ANALYSIS” “SYNTHESIS” A1 D1 Frequency Spectrum A = Approximation or lower frequency components D = Details or higher frequency components • Problem: Downsampling can produce aliasing! • Solution: Proper design of filters can eliminate aliasing under certain conditions (see downsampling/aliasing 101) 38 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  39. 39. ©2006-2010 ATICOURSES All Rights Reserved DWT APPROXIMATIONS, DETAILS ©2010 D.L. Fugal 500 pts Cd1 d1 250 pts Cd2 1000 pts S’ Cd3 d2 25 Ca2 d3 a1 0 Ca3 125 pts a2 Ca1 500 pts a3 S’ = d1+a1 a1= d2+a2 a2=d3+a3 • Signal, S, can be decomposed into various Approximations and Details using the Analysis portion • Signal can then be reconstructed from these Approximations and Details at the end of the Syntheses portion. S’ = d1 + a1 = d1 + d2 + a2 = d1 + d2 + d3 + a3 • Same relationship can be seen in frequency domain below A1 A2 A3 D3 D2 D1 Frequency Spectrum of Signal, S 39 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  40. 40. ©2006-2010 ATICOURSES All Rights Reserved 2-LEVEL WAVELET PACKET SYS (1) ©2010 D.L. Fugal • 2-Level Wavelet Packet Analysis System • Jargon Alert: Analysis portion is the left half of the DWT that decomposes the signal into coefficients. Synthesis portion is the right half of DWT that rebuilds the signal. • In Wavelet Packets, Details and Approximations are split. Underline shows 1 of 4 ways signal can be S decomposed A1 D1 AA2 AD2 DA2 DD2 40 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  41. 41. ©2006-2010 ATICOURSES All Rights Reserved 2-LEVEL WAVELET PACKET SYS (2) ©2010 D.L. Fugal Cdd2 Cd1 Cad2 Cd1 Cda2 Ca1 Ca1 Caa2 41 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  42. 42. ©2006-2010 ATICOURSES All Rights Reserved 2 LEVEL TRANSMULTIPLEXER SYS ©2010 D.L. Fugal H H SIG 1 SIG 1 H H SIG 2 TRANSMU. SIG 2 L SIGNAL L + H SIG 3 H SIG 3 L L SIG 4 L SIG 4 L 42 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  43. 43. ©2006-2010 All Rights Reserved APPLICATIONS & CASE STUDIES ATICOURSES ©2010 D.L. Fugal • CASE STUDIES OF APPLICATIONS OF WAVELETS TO REAL-LIFE PROBLEMS (Optional Song Demo) 43 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  44. 44. ©2006-2010 ATICOURSES All Rights Reserved TIME DEPENDENT THRESHOLD - 1 ©2010 D.L. Fugal • Pure Binary Signal (BPSK PNRZ) with no noise and its FFT PURE SIGNAL FFT OF PURE SIGNAL 10 1200 8 1000 6 4 800 MAGNITUDE--> AMPLITUDE--> 2 0 600 -2 400 -4 -6 200 -8 -10 0 2000 4000 6000 8000 0 2000 4000 6000 8000 10000 TIME--> FREQ--> 44 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  45. 45. ©2006-2010 ATICOURSES All Rights Reserved TIME DEPENDENT THRESHOLD - 2 ©2010 D.L. Fugal • Binary Signal buried in 10000x chirp noise at left. • 100 times closeup shows bit pattern overplotted, but bits are actually buried in the 1x104 noise. x 10 4 CHIRP JAMMER WITH BURRIED SIGNAL 200xCLOSEUP 1 100 0.8 80 0.6 60 0.4 40 AMPLITUDE--> 0.2 20 0 0 -0.2 -20 -0.4 -40 -0.6 -60 -0.8 -80 -1 -100 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 2000 2500 3000 3500 4000 4500 5000 TIME--> 45 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  46. 46. ©2006-2010 ATICOURSES All Rights Reserved TIME DEPENDENT THRESHOLD - 3 ©2010 D.L. Fugal • 4 Level DWT (frequency allocation shown here) used to decompose noisy signal • Signal is 2 13 points long. 7 levels of decomposition plenty. S A1 A2 AMPLTITUDE A3 Nyquist Frequency A4 D4 D3 D2 D1 FREQUENCY 46 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  47. 47. ©2006-2010 ATICOURSES All Rights Reserved TIME DEPENDENT THRESHOLD - 5 ©2010 D.L. Fugal Highest frequency portion, D1, of noisy signal as a function of time. Noise is 10000 x as large as signal but is confined to end of time sequence. Note scale is +/- 15,000 x 10 4 x 10 4 d1 d1 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 0 -1.5 1000 2000 3000 0 20004000 5000 6000 4000 7000 8000 6000 9000 8000 10000 47 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  48. 48. ©2006-2010 ATICOURSES All Rights Reserved TIME DEPENDENT THRESHOLD - 6 ©2010 D.L. Fugal D1 “scrap” with noisy portion zeroed out. Note change of scale from +/- 15000 to +/- 0.6. Signal “remnants” can now be clearly seen. We use Time-Dependant Thresholding on the other levels as well and then reconstruct the signal from the remnants d1 0.8 Note that the 0.6 time/scale property 0.4 of wavelet analysis 0.2 allows us to do 0 this. FFT would -0.2 not work. -0.4 -0.6 -0.8 0 1000 2000 3000 4000 5000 6000 7000 8000 48 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  49. 49. ©2006-2010 ATICOURSES All Rights Reserved TIME DEPENDENT THRESHOLD - 8 ©2010 D.L. Fugal • Main portion of original noiseless binary signal (top) • Wavelet Time-Dependant Thresholding de-noised signal (from 10000x or 80 dB noise) shown superimposed on noiseless signal (bottom). • We have exploited DWT knowledge of both time AND frequency 49 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  50. 50. ©2006-2010 ATICOURSES All Rights Reserved BPSK PNRZ CWT EXAMPLE - 2 ©2010 D.L. Fugal • First look at DWT of noiseless signal • Note there is no information to be had by adding Details from levels 1 through 4 on noiseless test case. • Can thus threshold out levels 1 through 4 on noisy Signal. 50 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459
  51. 51. ©2006-2010 ATICOURSES All Rights Reserved BPSK PNRZ CWT EXAMPLE - 3 ©2010 D.L. Fugal As • Levels 1 - 4 thresholded out and now don’t contribute to the reconstructed signal at all. • Note thresholded coeffs. exist only for levels >= 5. • Reconstructed signal seen in upper left graph. • Knowing signal is +/- 1, we can reconstruct signal exactly 51 ©2006 Spac e & Signals Te chnologies, LLC. All Rights Re serve d. www.Conc eptualWave le ts.com 877-845-6459

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