Notes on Digital Signal Processing: Practical Recipes for Design, Analysis and Implementation

Note 51. Designing IIR Filters: Bilinear Transformation

The bilinear transformation is the most commonly used approach for designing IIR filters. As discussed in Note 50, the impulse invariance technique uses a many-to-one frequency mapping that maps multiple intervals from the jω axis in the s-plane repeatedly onto the same points on the unit circle in the z-plane. This mapping can introduce aliasing of the filter response, making the impulse invariance technique unsuitable for use in designing highpass or bandstop filters. The bilinear transformation uses a nonlinear frequency mapping such that the entire jω axis of the s-plane maps one-to-one onto points of the unit circle of the z-plane. The mapping avoids aliasing, but it does “warp” critical ...