Homework 3

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. The Taylor series expansion of the inverse sine function around zero is
   

   $$\arcsin{x} = x + \frac{1}{2} \frac{x^3}{3} + \frac{1 \... ... \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{x^7}{7} + \cdots$$ (1)

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

      Hint: Use the identity $1/Z-Z = 2 i \sin(\omega \Delta t)$.

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

      $$D(Z) = d_{-3}/Z^{3} + d_{-2}/Z^{2} + d_{-1}/Z + d_0 + d_1 Z + d_2 Z^2 + d_3 Z^3\;.$$ (2)

      
      

2. The parabolic B-spline $\beta_2(x)$ is a function defined as
   

   $$\beta_2(x) = \int\limits_{-\infty}^{\infty} \beta_1(t) \beta_0(x-t) d t\;,$$ (3)

   
   
   where
   

   $$\beta_0(x) = \left\{\begin{array}{lcl} 1 & \quad\mbox{for... ... &\quad \mbox{for}\quad& \vert x\vert > 1/2\end{array}\right.$$ (4)

   
   
   and
   

   $$\beta_1(x) = \int\limits_{-\infty}^{\infty} \beta_0(t) \... ... 0 & \quad \mbox{for}\quad& \vert x\vert > 1\end{array}\right.$$ (5)

   
   
   1. Find an explicit expression for $\beta_2(x)$.
   2. Show that decomposing a continuous data function $d(x)$ into the convolution basis with parabolic B-spines
      

      $$d(x) = \sum\limits_k c_k \beta_2(x-k)$$ (6)

      
      
      leads to an interpolation filter of the form
      

      $$Z^{\sigma} \approx B_2(Z) = \frac{a_0(\sigma) Z^{-1} + a_1(\sigma) + a_2(\sigma) Z}{b_0 Z^{-1} + b_1 + b_2 Z}\;.$$ (7)

      
      
      Define $a_0(\sigma)$, $a_1(\sigma)$, $a_2(\sigma)$, $b_0$, $b_1$, and $b_2$.

Homework 3