# 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 V
_{4}, 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
V_{3}, V_{2}, V_{1} and V_{0}?

Investigate
what is lost when discarding detail information: For a given j=3,2,1,0, replace each group of 2^{4-j}
consecutive 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
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 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.