Homework 3

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# Theoretical part

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.

1. 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?

2. The Taylor series expansion of the inverse sine function around zero is
 (1)

1. Show how one can use expansion (1) to design a digital filter that approximates the derivative operator.

Hint: Use the identity .

2. In particular, find a seven-point derivative filter of the form
 (2)

3. The parabolic B-spline is a function defined as
 (3)

where
 (4)

and
 (5)

1. Find an explicit expression for .
2. Show that decomposing a continuous data function into the convolution basis with parabolic B-spines
 (6)

leads to an interpolation filter of the form
 (7)

Define , , , , , and .

 Homework 3

Next: Fourier compression Up: Homework 3 Previous: Prerequisites

2014-10-02