Example Problem Set 6

The bulk of Example Problem Set 6 (all except part 1) is material that has been moved to the Information Theory and Coding course.

1.

Define Gabor wavelets in one dimension in three parameter form, and explain the meaning of each of the three parameters.

2.

Explain the Gabor-Heisenberg-Weyl ``Uncertainty Principle". Express it as an inequality and define the quantities involved. Explain what is special about Gabor wavelets in terms of this principle.

3.

If you construct an Information Diagram whose axes are time and frequency, how small an area in this plane can be occupied? Illustrate several differently parameterized Gabor wavelets in this plane, and explain what property they all share. How will they compare with all other possible functions in this plane?

4.

Explain in what sense Gabor wavelets unify both the time domain and the frequency domain, constructing a continuous deformation between the two. Explain how the Fourier Transform is just a special case of the Gabor Transform, and what price is paid for this unification in terms of the computational difficulty of obtaining the Gabor Transform coefficients.