1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Digital Filter Equivalence in a difference equation form

  1. Nov 6, 2012 #1
    1. The problem statement, all variables and given/known data
    33502d6969074b993192b567831be33a.png
    Find the digital filter equivalence of the
    circuit in a difference equation form


    2. Relevant equations
    1f0adc005bb6a09f953ccb49d52c8600.png
    Is this the difference equation form?


    3. The attempt at a solution
    If that is the proper form.. I think this is how to solve it? I got this example from my textbook I'm just not sure if its solving for the question asked.
    ef702936f8c7e98bf158457775b5d353.png
     
  2. jcsd
  3. Nov 7, 2012 #2

    rude man

    User Avatar
    Homework Helper
    Gold Member

    Your attempt at a solution looks exactly right and appropriate.

    Realize that there is no single difference equation that exactly represents the analog network. All are approximations. Entire courses are devoted to formulating optimal difference equations. A widely used type is the Runge-Kutta method which itself has several orders etc.
     
  4. Nov 7, 2012 #3
    As mentioned, there is more than one way to find a difference equation that is approx equivalent to a continuous time system.

    What you did is called a backward difference integrator, where you've approximated a continuous time differentiation with a single backward difference. This effectively replaces s by (1 - z-1)/T. Higher order approximations for differentiation are also possible.

    But there are two main methods you may or may not have studied yet :- impulse invariance is one where you try to keep the impulse response the same in the discrete time domain. You do this by sampling the continuous time impulse response. This method can suffer from aliasing. Another is the bilinear transform which does not suffer from aliasing but does suffer from frequency warping. The bilinear transform maps the entire jw axis onto the unit circle which means you are squashing an infinite length axis onto a semicircle which will inevitably lead to frequency warping. What is normally done with the bilinear transform is the response is pre-warped to make sure important frequencies in the s-domain (eg bandwidth) appear at the same place (frequency wise) in the z-domain.
     
    Last edited: Nov 7, 2012
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Digital Filter Equivalence in a difference equation form
Loading...