Question

Haar wavelets) Let 6 be a function defined by 0(x) = 1 for x € (0,1), and əla) = 0 otherwise. Let y(x) = º(2x) - (2x - 1). Then the Haar wavelets are the functions Umn(x) = 2m/2v(2" x - n), for m, n = 0, #1, #2, .... Sketch a graph of y(x), and then sketch a graph of Umn(x) for m, n = 0, +1, +2. Generally, what is the graph of Umn(x)? If f is square integrable and f(x) = Ï Ï cmnymn () m=-oon=- find coefficients Cmn.

Also, find Cmn.

Answer 1

Haar wavelets) Let 6 be a function defined by 0(x) = 1 for x € (0,1), and əla) = 0 otherwise. Let y(x) = º(2x) - (2x - 1). Th

Let & be a Function defined by d (G) 21 for xt(0,1), and f(x) = 0. Otherwise. Let WG) = $20-$ (2x-1) Then the Haas Wavelts are Functions Yan(n)= 21 462mman) Any: The The Haan Haan sequer sequence was proposed in 1990 by Alfred Example space unit Haan. of Haaa used these an orthonormal functions to give an system for the of Sanare - Integrable Functions on the Interval on The WCt) Haan Wavelets Mother Wavelet can be described as Function ostel WCt) =) - Ystal . dthoonwise Its scaling function Ych can be described os Wet) ostai otherwise Four Every Function pair nk Ynik is of Integers in 2, the Haar defined on the Real line R by the Formula Yuck (t) = 20 p (an t-k), ter

This Function is supported on the Right open Interval lnika [kan, (k+1) 2n] le, It Vanishes Integral o and L2(R) outside Nooma that interval. It has in the Hilbesit Space Je Inik (t) dt-o, Ilknoll this CH" dt = 1. 21 The Haas Functions are pairwise orthogonal, Je Yanko (t) Win, et dt = fm, na droite, Where dig Represents the Khonecked delta. Here is the Reason for oathogonality: When the two Supporting Intervaly Inick, and Imaika are Not Equal, then they are either disjoint, on else, the smallest of the two supports Say Inlik is contained in the lower or in the upper half of the other intervale on which the Function Phaika Remains . constant. It Follows in this case that the other interval, on kahich the function on

Product of these Two Haar Functions is a murple of the First Haar Function, hence the product has Integral o Maar system on the Real line is the of set functions. { Unik(t); NEZ, KEZY. It is the system on complete in L²CR): The Haan line is an orthonoomal basis Graph of Umn (x) in [2CR) Yemin (n) = 2 m/24 (2min) 02 Women Cn) = zala pommin)