A first look Ref: Walker (ch1) Jyun-Ming Chen, Spring 2001 Haar Wavelets A first look Ref: Walker (ch1) Jyun-Ming Chen, Spring 2001

Introduction Simplest; hand calculation suffice A prototype for studying more sophisticated wavelets Related to Haar transform, a mathematical operation

Haar Transform Assume discrete signal (analog function occurring at discrete instants) Assume equally spaced samples (number of samples 2n) Decompose the signal into two sub-signals of half its length Running average (trend) Running difference (fluctuation)

Haar transform, 1-level Running average Multiplication by is needed to ensure energy conservation (see later) Running difference Denoted by: Meaning of superscript explained later

Example

Inverse Transform

Small Fluctuation Feature Magnitudes of the fluctuation subsignal (d) are often significantly smaller than those of the original signal Logical: samples are from continuous analog signal with very short time increment Has application to signal compression Mallat order

Energy Concerns Energy of signals The 1-level Haar transform conserves energy

Proof of Energy Conservation

Haar Transform, multi-level

Compaction of Energy Compare with 1-level Can be seen more clearly by cumulative energy profile

Cumulative Energy Profile Definition

Algebraic Operations Addition & subtraction Constant multiple Scalar product

Haar Wavelets 1-level Haar wavelets “wavelet”: plus/minus wavy nature Translated copy of mother wavelet support of wavelet =2 The interval where function is nonzero Significance of “compact” support: Able to detect very sort-lived, transient, fluctuations in the signal (p. 35)…. What differs from FFT Property 1. If a signal f is (approximately) constant over the support of a Haar wavelet, then the fluctuation value is (approximately) zero.

Haar Scaling Functions 1-level scaling functions Graph: translated copy of father scaling function Support = 2

Haar Wavelets (cont) 2-level Haar scaling functions support = 4 2-level Haar wavelets support = 4 Regard these as “definitions”. (“defined as the following …”) Put these in matrix form …

Multiresolution Analysis (MRA) Natural basis: Therefore: The dimension of f (C3) is N.

MRA Note: the coefficient vectors 1. The dimension of C2 and D2 should be N/2 (seen from the number of basis functions, and number of coefficients to represent a function: c1, …, c(n/2)). 2. As to why the coefficients are determined by inner product is due to orthogonality of V’s and W’s: Write f=c2 +d2 then you’ll see.

MRA If do it all the way through, representing the average of all data Similarly, dimension of c1 and d1 is N/4 If do it all the way through, representing the average of all data

Example

Example (cont) Decomposition coefficients obtained by inner product with basis function In fact, all V and W are functions (V(x), W(x)). Here we use 8 numbers to represent sampling of such functions over the designated interval (as f(x)). Note: The coefficient of the basis represent some kind of time/frequency resolution … Q: What if the scaling fns and wavelets are not orthogonal? How would we obtain the decomposition coeff? How would this affect L2 norm of compression?

Haar MRA

More on Scaling Functions (Haar) They are in fact related Pj is called the synthesis filter (more later) Up to now, we haven’t mentioned how the scaling functions and wavelets are obtained.

Ex: Haar Scaling Functions Synthesis Filter P3

Ex: Haar Scaling Functions Synthesis Filter P2 Synthesis Filter P1

More on Wavelets (Haar) They are in fact related Qj is called the synthesis filter (more later)

Ex: Haar Wavelets Synthesis Filter Q3

Ex: Haar Wavelets Synthesis Filter Q1 Synthesis Filter Q2

Analysis Filters There is another set of matrices that are related to the computation of analysis/decomposition coefficient In the Haar case, they are the transpose of each other Later we’ll show that this is a property unique to orthogonal wavelets

Analysis/Decomposition (Haar) B2 Analysis Filter Aj Analysis Filter Bj B3 A1 B1

Synthesis Filters On the other hand, synthesis filters have to do with reconstructing the signal from MRA results

Synthesis/Reconstruction (Haar) Q2 P2 Q1 P1 Q3 P3 Synthesis Filter Pj Synthesis Filter Qj

Conclusion/Exercise Haar (N=8) j=3 j=2 j=1 j=0 In general N=2n support 4 8 2n-j translation