# Mathematics of Signal Representations

## Math 4355 - Spring 2013 - Homework

Assignment 10, due Thursday, April 18

• Do Exercises on p. 186/187: 1; 2.

• (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 24-j consecutive coefficients in the initial signal by their group average. The resulting vector is the projection of the signal onto Vj. Write the corresponding audio files using wavwrite. Play them and describe what you hear as you change j.
Next, 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 Vj 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.