EE 451

Homework Assignment 10
Due Nov. 1, 2000

  1. We wish to design an FIR differentiator which has the ideal frequency response:


    \begin{displaymath} H(\omega) = j\omega e^{-j\omega\frac{M-1}{2}}, -\pi < \omega \le \pi \end{displaymath}

    1. Take the inverse Fourier transform of $H(\omega)$ to show that:


      \begin{displaymath} h(n) = \left\{ \begin{array}{cl} \frac{cos\left[\pi\left(n-\... ...\frac{M-1}{2} \\ 0 & n = \frac{M-1}{2} \\ \end{array}\right. \end{displaymath}

    2. Use the above equation with a Hamming window to find $h(n)$ for a differentiator where $M = 20$. Plot $h(n)$ and plot the pole-zero diagram for $h(n)$.
    3. Plot the gain $\vert H(\omega)\vert$ and phase $\angle H(\omega)$ of the filter.
    4. In MATLAB, generate a signal $x(n) = \cos(\omega_0 n)$, where $\omega_0 = \pi/16$. Find 101 terms for $x(n)$: 
              wo = pi/16;
        n = 0:100;
        x = cos(wo*n);

      Use the above $h(n)$ to filter the signal $x(n)$: y = filter(h,1,x);. Use MATLAB to plot $x(n)$. Over the top of this, plot $y(n-9.5)$. (Remember that the output was delayed by $(M-1)/2$ samples.) Does $y(n)$ look like the derivative of $x(n)$?

    5. Repeat Part (e) for $w_0 = \pi/4$.

  2. We wish to design a Hilbert transformer which has the ideal frequency response:


    \begin{displaymath} H(\omega) = \left\{ \begin{array}{rl} -j e^{-j\omega\frac{M-... ...mega\frac{M-1}{2}}, & 0 < \omega \le \pi\\ \end{array}\right. \end{displaymath}

    1. Take the inverse Fourier transform of $H(\omega)$ to show that:


      \begin{displaymath} h(n) = \left\{ \begin{array}{cl} \frac{1-cos\left[\pi\left(n... ...\frac{M-1}{2} \\ 0 & n = \frac{M-1}{2} \\ \end{array}\right. \end{displaymath}

    2. Use the above equation with a Hamming window to find $h(n)$ for a Hilbert transformer where $M = 20$. Plot $h(n)$ and plot the pole-zero diagram for $h(n)$.
    3. Plot the gain $\vert H(\omega)\vert$ and phase $\angle H(\omega)$ of the filter.
    4. In MATLAB, generate a signal $x(n) = \cos(\omega_0 n)$, where $\omega_0 = \pi/16$. Find 101 terms for $x(n)$: 
              wo = pi/16;
        n = 0:100;
        x = cos(wo*n);

      Use the above $h(n)$ to filter the signal $x(n)$: y = filter(h,1,x);. Use MATLAB to plot $x(n)$. Over the top of this, plot $y(n-9.5)$. (Remember that the output was delayed by $(M-1)/2$ samples.) Does $y(n)$ look like the ouput you would expect for a Hilbert transformer?

    5. Repeat Part (e) for $w_0 = \pi/4$.


Bill Rison, <rison@ee.nmt.edu >