4.1 Haar Wavelets. Haar Wavelet. The Haar Scaling Function

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1 4 Haar Wavelets Wavelets were first aplied in geophysics to analyze data from seismic surveys, which are used in oil and mineral exploration to get pictures of layering in subsurface roc In fact, geophysicssts rediscovered them; mathematicians had developed them to solve abstract problems some twenty years earlier but had not anticipated their applications in signal processing The ey to an accurate seismic survey is a proper analysis of each trace The Four transform is not a good tool here It can only provide frequency information (the oscillations that comprise the signal) It does not provide direct informaton about when an oscillation occurred Wavelets can eep trac of time and frequency information Haar Wavelet The Haar Scaling Function There are two functions that play a primary role in wavelet analysis, the scaling function and the wavelet These two functions generate a family of functions that can be used to brea up or reconstruct a signal To emphasize the marriage involved in building this family, is sometimes called the father wavelet and, the mother wavelet The simplest wavelet analysis is based on the Haar scaling function, whose graph is given in Figure 2

2 Figure 2 The graph of the Haar scaling function Definition The Haar scaling function is defined as, if x ( x), elsewhere The building blocs are translations and dilations (both in height and width) of this basic graph We wish to illustrate the basic ideas involved in such an analysis Consider the signal shown below Fig 3 oltage from a faulty meter

3 We may thin of this as a measurement of some physical quantity perhaps line voltage over a single cycle---as a function of time The two sharp spies in the graph might represent noise coming from a loose connection in the volt meter, and we want to filter out this undesirable noise The graph in Figure 4 shows one possible approximation to the signal using Haar building blocs The high frequency noise sows up as tall, then blocs The high frequency noise shows up as tall, thin bloc An algorithm that deletes the thin blocs will eliminate the noise and not disturb the rest of the signal Figure 4 Approximation of voltage signal in Fig 2 by Haar functions The building blocs generated by the Haar scaling function are particularly simple and illustrate the general ideas underling a multiresolution analysis, which we will discuss in detail The disadvantage of the Haar wavelets is that they are discontinuous and therefore do not approximate continuous signals very well The function ( x- ) has the same graph as but translated to the right by units (assuming is positive) Let be the space of all functions of the form Z a ( x ) a R where can range over any finite set of positive or negative integers Since ( x ) is discontinuous at x and x, an alternative description of is that it consists of all piecewise constant functions whose discontinuities are contained in the set of integers Since ranges over a finite set, each element of is zero outside a bounded set Such a function is said to have finite or compact support The graph of a typical element of is given in the following graph

4 Graph of typical element in Note that a function in may not have discontinuities at all the integers ( for example, if a a2, then the preceding sum is continuous at x 2 ) Example : Graph f ( x) 2 ( x) 3 ( x ) 3 ( x 2) ( x 3) Graph of (2 x)

5 Example 2: Graph of function f ( x) 4 (2 x) 2 (2x ) 2 (2x 2) (2x 3) Definition 2 Suppose is any nonnegative integer The space of step functions at level, denoted by, is defined to be the space spanned by the set over the real numbers {, (2 x ), (2 x), (2 x ), (2 x 2), } discontinuities are contained in the set is the space of piecewise constant functions of finite support whose {,,,,,, } A function in is a piecewise constant function with discontinuities contained in the set of integers Any function in is also contained in, which consists of piecewise constant functions whose discontinuities are contained in the set of half-integers,/ 2,,/ 2,,3/ 2, The same applies for 2 and so forth: This containment is strict For example, the function (2 x) does not belong to Question: Prove Basic Properties of the Haar Scaling Function Theorem (a) A function f( x ) belongs to f(2 x ) belongs to (b) A function f( x ) belongs to f(2 x) belongs to

6 Proof (a) if f( x ) belongs to, then f ( x) a ( x ) a R, so f (2 x) a(2 x ) a R, this means that f(2 x ) belongs to Z For (b), we can similarly prove Z /2 /2 Theorem 2 The set of functions 2 (2 x ), Z is an orthonormal basis of Question: Prove Theorem 2 The Haar Wavelet Having an orthonormal basis of is only half of the picture In order to solve our noise-filtering problem, we need to have a way of isolating the spies that belong to This is where the wavelet enters the picture but that are not members of The idea is to decompose as an orthogonal sum of and its complement Again, let start with and identify the orthogonal complement of in Since is generated by and its translates, it is reasonable to expect that the orthogonal complement of is generated by the translates of some function We need, and is orthogonal to Consider the following function Definition 3 The Haar wavelet is the function ( x) (2 x) (2x ) The graph of Haar wavelet is below Question: Prove, and is orthogonal to

7 We find out that any function f ( x) a (2 x ) Z is orthogonal to, that is, orthogonal to each ( x l), l Z if and only if a a, a a, 3 2 (prove the above result as an exercise) In this case, we have f ( x) a ( (2 x ) (2x )) a ( x ) () 2 2 Z Z In other words, a function in is orthogonal to if and only if it is of the form a ( x ) (relabeling a 2 by a ) Definition 4 Let U and be subspaces of W Then W is said to be the direct sum of U and, and we write W U if W U U and Let W be the space of all functions of the form a ( x ) a Z R where, again, we assume that only a finite number of the that W is the orthogonal complement of in Z a are nonzero What we have ust shown is W or, in other words, In a similar manner, the following more general result can be established Theorem 3 Let W be the space of functions of the form a (2 x ) a Z assume that only a finite number of the and a are nonzero (2) W Proof To show (), we have to prove two facts: R Each function in W is orthogonal to every function in 2 Any function in that is orthogonal to W is the orthogonal complement of must belong to W in

8 For the first requirement, suppose that g( x) a (2 x ) W and f, we need to show g, f g( y) f ( y) dy 2 L Z Since f, the function f (2 x) So g, f g( y) f ( y) dy 2 L 2 a (2 y ) f ( y) dx (let y 2 x) Z 2 a ( x ) f (2 x) dx (because is orthogonal to ) Z Thus, g is orthogonal to any f and thus we have proved the first one requirement The discussion leading to equation () established the second requirement when since we proved that any function in that is orthogonal to must be a linear combination of ( x ), Z The argument of general is very similar to the case when By successively decomposing, and so on, we have W W W 2 2 W W W W 2 2 So each f can be decomposed uniquely as a sum f w w w v 2 where ww,, and v Intuitively, w represents the spies of f of width / 2 l that can t be represented as linear combinations of spies of other widths When, we have the following: Theorem 4 The space L 2 ( R ) can be decomposed as an infinite orthogonal direct sum 2 L ( R) W W W2 In other words, each f L 2 ( R) can be written uniquely as

9 f v w where v and w W The infinite sum should be thought of as a limit of finite sums In other words, N lim N f v w where the limit is taen in the sense of 2 L The proof is beyond the scope of this course The approximation by step functions