Digital Filter Equivalence in a difference equation form

[tennisguy383]

Homework Statement

Find the digital filter equivalence of the
circuit in a difference equation form

Homework Equations

Is this the difference equation form?

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.

 

[rude man]

Homework Statement

Find the digital filter equivalence of the
circuit in a difference equation form

Homework Equations

Is this the difference equation form?
Yes. That is the generalized form, allowing any order, also recursive and non-recursive.

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.

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.

 

[aralbrec]

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.