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Consider the following statements with respect to the bilinear transformation method of digital filter design:

1. It preserves the number of poles and thereby the order of the filter.

2. It maintains the phase response of the analog filter.

3. The impulse response of the analog filter is not preserved.

Which of the above statements are correct?


1. 1, 2 and 3
2. 1 and 2 only 
3. 1 and 3 only 
4. 2 and 3 only 

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Correct Answer - Option 3 : 1 and 3 only 

Bilinear Transformation:

  • It is used for transforming an analog filter into a digital filter.
  • It uses a trapezoidal rule for integrating a continuous-time function.
  • It can be regarded as a correction of the backward difference method.
  • It is a mapping from the s plane to the z plane.

 

In the bilinear transformation of an analog filter to a digital filter, using the trapezoidal rule, the substitution for ‘s’ is given as:

\(s = \frac{2}{T}\frac{{\left( {1 - {z^{ - 1}}} \right)}}{{1 + {z^{ - 1}}}}\)

Properties:

  • It preserves the stability and maps every point of the frequency response of the continuous-time filter to a corresponding g point in the frequency response of the discrete-time filter. 
  • Another valuable property of the bilinear transform is that order is preserved, i.e. an Nth-orde s-plane transfer function carries over to an Nth-order z-plane transfer function.
  • The bilinear transformation is a rational function that maps the left half s plane inside the unit circle and maps the jω axis in a one to one manner onto the unit circle.
  • Bilinear Transformation only preserves the magnitude response of the analog filter.

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