Answer Exercise 9.7.3 using the Daubechies wavelets to compress the data. Compare your results.

Data From Exercise 9.7.3

In this exercise, we investigate the compression capabilities of the Haar wavelets. Let represent a signal defined on 0 ≤ x ≤ 1. Let s_r(x) denote the n^th partial sum, from j = 0 to r, of the Haar wavelet series (9.136).

(a) How many different Haar wavelet coefficients c_j,k appear in s_r(x)? If our criterion for compression is that ΙΙ f − s_r ΙΙ_∞ 

(b) Compare the Haar wavelet compression with the discrete Fourier method of Exercise 5.6.10.

Data From Exercise 5.6.10

Construct the discrete Fourier coefficients for   based on n = 128 sample points. Then graph the reconstructed function when using the data compression algorithm that retains only the 11 and 21 lowest-frequency modes. Discuss what you observe.

Transcribed Image Text:

f(x) = -x, X - -x+2, 0≤x≤, ≤x≤T, ≤ x ≤ 2π,