Homework 3 |

You can either write your answers to theoretical questions on paper or
edit them in the file `hw3/paper.tex`. Please show all the
mathematical derivations that you perform.

- Show that, using the helix transform and imposing helical boundary conditions, it is possible to compute a 2-D digital Fourier transform using 1-D FFT program. Assuming that the input data is of size , would this approach have any computational advantages?
- The Taylor series expansion of the inverse sine function around zero is

- Show how one can use expansion (1) to design a
digital filter that approximates the derivative
operator.
**Hint:**Use the identity . - In particular, find a seven-point derivative filter of the form

- Show how one can use expansion (1) to design a
digital filter that approximates the derivative
operator.
- The parabolic B-spline is a function defined as

where

and

- Find an explicit expression for .
- Show that decomposing a continuous data function into the convolution basis
with parabolic B-spines

leads to an interpolation filter of the form

Define , , , , , and .

Homework 3 |

2014-10-02