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 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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text: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.
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