Math 414-502 — Spring 2012

Assignment 8

• Read sections 5.1 and 5.2
• Problems.
  1. Chapter 5 exercises: 2, 5, 8(c,d,e,f)
  2. Let the p_k's be the scaling coefficients in Example 5.8, p. 195. For these, the scaling and wavelet relations are
     φ(x) = p₀φ(2x) + p₁φ(2x−1) + p₂φ(2x−2) + p₃φ(2x−3),
     ψ(x) = p₃φ(2x+2) − p₂φ(2x+1) + p₁φ(2x) − p₀φ(2x−1).
     1. Show these p_k's satisfy the four properties in Theorem 5.9, p. 196.
     2. Find all four filters corresponding to these coefficients: low pass and high pass decomposition and reconstruction filters.
     3. Use Theorem 5.9 and the wavelet ψ(x) above to show that ∫ ψ(x)dx = 0.
     4. BONUS: Show that ∫ xψ(x)dx = 0.