Problem: Let (t) be a scaling function and (t) be its corresponding wavelet function in a multi-resolution analysis (MRA) for a wavelet basis {(t) = 2/2 (2³t-k)} where j = Z and k Є Z. 1. (a) Starting with the dilation equation for the scaling function (t): (t)=√2ho(2tk) kez derive the equivalent dilation equation for the wavelet function (t), given the relation: ψ(1) = 12 Σ 9 Φ(21 – 4) KEZ and express 9 in terms of the filter coefficients hk. 2. (b) Show that the orthogonality condition for the wavelet basis, (b) = jk, leads to the following relation for the filter coefficients: KEZ hkhk+2n = 8,0 and explain how this ensures that the scaling function (t) and wavelet function (t) form an orthonormal basis. 3. (c) For the Daubechies D4 wavelet, the filter coefficients ho, h1, h2, h3 are known. Compute the corresponding wavelet filter coefficients gk and verify that they satisfy the orthogonality condition derived in part (b). 4. (d) Given a signal f(t) = L² (IR), consider its approximation at scale j using the scaling function (t): fj(t) = c(k)x(t) KEZ where c; (k) are the scaling coefficients. Derive the recursive relation for obtaining cj-1(k) in terms of c; (k) using the scaling filter hk, and explain how this leads to the multi-resolution approximation. 5. (e) Discuss the vanishing moments property of wavelets. Prove that if a wavelet has N vanishing moments, then it can efficiently represent polynomials up to degree N - 1. Use the Daubechies D4 wavelet to illustrate this concept. Problem: Let (t) be a mother wavelet that satisfies the admissibility condition: Со = 10° /23 (4)/2 dw < ∞ || where (w) is the Fourier transform of (t). 1. (a) Prove that the admissibility condition ensures that the wavelet transform can perfectly reconstruct a signal f(t) from its wavelet coefficients. 2. (b) Consider a signal f(t) = L²(R). Show that the continuous wavelet transform W₁f (a, b), defined as: 1 Wof(a,b) = == $ (1-0). b dt a satisfies the following energy conservation property: |||||²(®) = ** | Woƒ (a,b) 2 dadb = 0 a² 3. (c) For the Mexican Hat wavelet (t) = (1 +²)e, compute its Fourier transform (w) and verify whether it satisfies the admissibility condition. 4. (d) Suppose you are given a discrete signal f[n] that is sampled from a continuous function. Derive the relation between the continuous wavelet transform and the discrete wavelet transform (DWT), and discuss the conditions under which the DWT can be used to reconstruct the continuous signal perfectly.