The Design of Infinite Impulse Response Filter

An Infinite Impulse Response system is described by the following difference equation:

And its system function is given by,

Design of IIR Filter: Bilinear Transformation

Let us consider a system function as shown below:

which is characterized by the differential equation

The solution to this differential equation can be approximated using Trapezoidal rule as

At and , where is the sampling time period, the above equation becomes,

At , the equation becomes,

With and , becomes,

And becomes,

Substituting in ,

Taking -transform on both sides of ,

which can be written as,

Comparing and , the mapping from the -plane to the -plane is obtained as

Design Procedure

To design an IIR filter given the filter specificatoins (, , etc.), the following steps need to be followed:

1. Choose (usually 1) and determine the analog frequencies and
2. Design an analog filter using the specifications (Butterworth, Chebyshev, or Elliptic) to meet the specifications.
3. Set

Example

Example 1: Design a DT filter with the following spectifications: , , and .

The required DT filter is designed below:

1. Choose and determine the analog frequencies

2. Design an analog filter using the specifications to meet the specifications.

   1. For Butterworth design, the squared magnitude response is given by, where, is the cutoff frequency and is the number of poles in the filter.
   2. Finding and :
      1. Substitute the specs into equation ,
      2. Taking the inequalities and writing them as equalities as shown below,
      3. Taking on both sides and simplyfying,
      4. Solving for , we get . However, must be an integer, so we take .
      5. Now, we can decide whether to pick to match the passband spec or match the stopband spec. Since bilinear transforms don’t suffer from aliasing, we often choose the latter. Substituting into equation and solving for , we get .
   3. The normalized Butterworth filter for poles is given by,
   4. Then the unnormalized Butterworth filter is obtained by substituting by in ,

3. Set