Wavelet dönüşümü ve işaret işlemedeki uygulamaları

Abstract

Bu tezde wavelet fonksiyonları ve wavelet dönüşümünün tanıtımı yapılmış, işaret isleme alanındaki en yaygın uygulaması olan kodlama konusu incelenmiştir. Wavelet -fonksiyonları ve dönüşümü kullanılarak herhangi bir işaretin seriye açılımı gösterilmiş, bu açılım wavelet serisi olarak tanımlanmıştır. Çok önerimli veya çok çözümlü ( Multi resolution ) işaret ayrıştırma metodu olarak bilinen wavelet serisi yöntemindeki multi- resolution kavramı açılım derecesi olarak adlandırılmış ve ilgili bölümde bu kavramın özellikleri incelenmiştir Farklı dereceli açılımlar arasındaki ilişki gösterilerek, birbirini izleyen iki açılım arasında geçiş yapma olanağı sağlayan bir algoritma üzerinde durulmuştur. Wavelet dönüşümü ve iki kanallı bir QMF (Quadrature Mirror Filter) 'de işaret işlenmesi arasındaki ilişki incelenerek iki kanallı QMF'de kullanılan -filtrelerin sağlaması gereken koşullar gösterilmiştir. Bu koşulları yerine getiren bir örnek olan Binomsal filtreler incelenmiştir.

As in the case of vector space, also in the signal space, an arbitrary signal f(t) can be represented as a superposition of an orthogonal -function set. The most usual decomposition technique is the met hod o-f Fourier Serial. But, in the analysis of tran sient signals such as those encountered in speech, or in certain kinds of image processing, standard Fourier analysis is often not satisfactory. This is because the basis functions of Fourier analysis (sines, cosines ) extend over infinite time whereas the signals to be analysed are short- time transients. A new method for dealing with transient signals is Wavelet Transform. The basis functions are referred to as wavelets, and they employ time compression (or dilation) rather than a variation of frequency of the modulated sinusoid. Hence all the wavelets have the same number of cycles. The analysing wavelets must satisfy a few simple conditions, but are not otherwise specified. There is therefore a wide latitude in the choice of these functions and they can be taylored to specific applications. A wavelet family consists of functions generated from a single function q(t) by dilations and transla tions as q»to(t> = 1/lal *^sqC(t-b)/aD W; This generating -function satisfies a» J -a> q(t)dt ? 0 Similar to the purpose of the bitrary signal of the -family, have di-f-ferent other known decomposition techniques the wavelet transform is to represent any ar f (t) as a superposition of the wavelets It is clear that the wavelets of a set scale levels. In practise discrete sets of a and b preferred. This means that the grid structures of a and b should be defined. If a = am«, b = nbeaffla » m,n.= Z, Z is set of integers and ao>l j let decomposition becomes bo>0 and fixed, the wave- N/2' f (t) d (m.n) q =? = f - q(2~mt-n )f - pm-l,2n+l(t)] where c(m.,n) are the projection coefficients of f(t) onto pml-»(t) functions. If the input signal f (t) is a discrete-time signal f (n) these samples can be taken as the highest resolu tion coefficients, f (n) =» a(0,n). In this case the input signal is decomposed into two band. This two band (VJ.11 ) decomposition technique satisfies the perfect recons truction condition. This case can be expressed by using the two subband signal components : N/2m c so that 1 T(z) - - (""""."? """"-') Therefore, the perfect reconstruction requirement redu ces to finding an H(z) = Hi + R(-z> This condition, can be readily recast in an alternate time domain -form ; R(z) = a(N)zN + alN-DzN"1 +... + a(0)z° +... a(N)z~N R(-z> = -a(n)sM + a(N~l)zN-* -... +a(0)z° +...-a(N)z- Therefore Q(z) consist only of even powers of z. To force Q(z) = constanat, it suffices to make all even indexed coefficient in R(z) equal to zero. However, the a(n) coefficients in R(s) ar& sim ply the samples of auto correlation r(n) given by ; r( N n) ? ÎL h(k)h(k+n) where r(n) is the convolution of h(n) with h(-n), or equivalently, the time autocorrelation. We need to set r(n) = 0 for n even, and n=0. Therefore, N r(2n) « L* h(k)h(k+2n) - 0 n=fO k=0 If the normalization is imposed 2- |h(k) k=0 (}<> Therefore ; N 2- h(k)h(k+2n) ? Sin) k=0 Binomial-QMF 's, which are given as example, having some processes applied on them, shows the same charac teristics as the h(n) and g(n) -filters which are des cribed by using wavelet -functions.