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.

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