1. Details, details… Intro to Discrete Wavelet Transform The Story of Wavelets Theory and Engineering Applications

2. Discrete Wavelet Transform  CWT computed by computers is really not CWT, it is a discretized version of the CWT.  The resolution of the time-frequency grid can be controlled (within Heisenberg’s inequality), can be controlled by time and scale step sizes.  Often this results in a very redundant representation  How to discretize the continuous time-frequency plane, so that the representation is non-redundant?  Sample the time-frequency plane on a dyadic (octave) grid

3. Discrete Wavelet Transform  Dyadic sampling of the time –frequency plane results in a very efficient algorithm for computing DWT:  Subband coding using multiresolution analysis  Dyadic sampling and multiresolution is achieved through a series of filtering and up/down sampling operations H x[n] y[n]

4. Recall: Orthogonal Projection X 3 E R 3 X 2 E R 2 X 1 E R 1 e 2 E W 2 e 1 E W 1 V3V3 V2V2 Coarse approximation of X 3 at level 2

5. Vector Spaces  V N-1 is the next coarser vector space, it is a subspace of V N.  W N-1 is orthogonal to V N-1, and  In general  and

6. Details vs. Approximations

7. Discrete Wavelet Transform Implementation G H 2 2 G H 2 2 2 2 G H + 2 2 G H + x[n] Decomposition Reconstruction ~ ~ ~ ~ 2-level DWT decomposition. The decomposition can be continues as long as there are enough samples for down-sampling. G H Half band high pass filter Half band low pass filter 2 2 Down-sampling Up-sampling

8. DWT - Demystified Length: 512 B: 0 ~  g[n]h[n] g[n]h[n] g[n]h[n] 2 d 1 : Level 1 DWT Coeff. Length: 256 B: 0 ~  /2 Hz Length: 256 B:  /2 ~  Hz Length: 128 B: 0 ~  /4 Hz Length: 128 B:  /4 ~  /2 Hz d 2 : Level 2 DWT Coeff. d 3 : Level 3 DWT Coeff. …a 3 …. Length: 64 B: 0 ~  /8 Hz Length: 64 B:  /8 ~  /4 Hz 2 22 2 2 |H(jw)| w  /2 -  /2 |G(jw)| w  --  /2 -  /2 a2a2 a1a1 Level 3 approximation Coefficients

9. Quadrature Mirror Filters  It can be shown that that is, h[] and g[] filters are related to each other: in fact, that is, h[] and g[] are mirrors of each other, with every other coefficient negated. Such filters are called quadrature mirror filters. For example, Daubechies wavelets with 4 vanishing moments…..

10. DB-4 Wavelets h = -0.0106 0.0329 0.0308 -0.1870 -0.0280 0.6309 0.7148 0.2304 g = -0.0106 -0.0329 0.0308 0.1870 -0.0280 -0.6309 0.7148 -0.2304 h = 0.2304 0.7148 0.6309 -0.0280 -0.1870 0.0308 0.0329 -0.0106 g = -0.0106 -0.0329 0.0308 0.1870 -0.0280 -0.6309 0.7148 -0.2304 ~ ~ L: filter length (8, in this case)

11. Implementation of DWT on MATLAB Load signal Choose wavelet and number of levels Hit Analyze button Level 1 coeff. Highest freq. Approx. coef. at level 5 s=a 5 +d 5 +…+d 1 (Wavedemo_signal1)