From Fourier to Wavelets

Assignment 8, due Wednesday, April 29

• Do Exercises §4.5: 1; 2. Read the supplementary notes to see a worked example.
  
• (Matlab project) Haar wavelets vs. Fourier. Read the file tqbfjotld.wav using wavread (it is sampled at 11025 Hz, make sure you store the sample rate). Assuming the coefficients in the vector belong to a signal in the subspace of piecewise constant functions V₄, let Matlab compute the coefficients a^(j)_k of its Haar decomposition for j=3,2,1 and 0. How many coefficients do you need to specify the projection of the signal onto the subspaces V₃, V₂, V₁ and V₀? Investigate what is lost when discarding detail information: For a given j=3,2,1,0, replace each group of 2^4-j coefficients in the initial signal by their group average. The resulting vector is the projection of the signal onto V_j. Write the corresponding audio files using wavwrite. Play them and describe what you hear.
  Now use a Discrete Fourier Transform to compute approximate Fourier coefficients. (Matlab ONLY computes these for positive indices, but since you know the signal is real, you can deduce the values of the Fourier coefficients with negative indices.) Reconstruct the signal from a partial sum of the Fourier series, keeping the same number of (low-frequency) Fourier coefficients as for the spaces V_j and setting the others to zero. Write the audio files corresponding to j=3,2,1,0. Play them and describe what you hear in comparison with the Haar wavelet decomposition. Attach your Matlab code to your descriptions.