5.5 Exercises

1. Prove Equation 5.11, using the argument in Proposition 5.1.

2. Prove the estimate in Equation 5.51.

3. Prove that we can always conventionally normalize the high-pass filters in a biorthogonal set of CDFs, as claimed after Lemma 5.2.

4. Compute H*Hu( t), G*Gu( t), H*Gu( t), and G*Hu( t) for a pair H, G of orthogonal QFs acting on functions.

5. Prove Equation 5.23, the second part of Lemma 5.5.

6. Prove Equations 5.80 and 5.81.

7. Find ? _n( t) in the Shannon case. That is, compute the inverse Fourier transform of the function in Theorem 5.19. This is called the Shannon wavelet.

8. Write a pseudocode program that takes a low-pass QF sequence and produces a complete PQF data structure. Then write a second program that produces its conventionally indexed and normalized conjugate.

9. Implement adjoint periodic convolution-decimation using the "mod" operator. Compare its speed on your computer with the "non-mod" implementation.

10. Implement sequential output adjoint periodic convolution-decimation using Equation 5.5. Compare its results with the implementation of acdpo() given in the text.

11. Implement periodic convolution-decimation using FFT. For what length filters does this realize a speed gain?

12. Start with an aperiodic sequence supported in [ a, b] and apply the convolution-decimation F (supported in [0, R - 1]) a total of L times. What is the support of the resulting filtered sequence? Now apply F* L times to the filtered sequence. What is...