Bilinear Transform Visualizer

[edoering]

Overview: Interactively visualize the bilinear transform that maps the continuous-time s-plane onto the discrete-time z-plane with the equation z = (1+s)/(1-s). The bilinear transform is used to transform continuous-time filter prototypes such as Butterworth and Chebyshev into discrete-time filters, and to do so free of aliasing because the entire left half of the s-plane is contained inside the unit circle of the z-plane.

Details: The s-plane on the left side of the display is the continuous-time interpretation of the complex plane. The horizontal real axis is sigma (s) and the vertical imaginary axis is omega (W); the latter is normally called the "jay-omega" axis. The symbol s defined as s = s+jW represents a specific location in the s-plane. The z-plane on the right side of the display is the discrete-time interpretation of the complex plane. No special names are used for the real and imaginary axes in the z-plane.

The VI front panel controls establish a grid in the s-plane which is then mapped to the z-plane according to the bilinear transform z = (1+s)/(1-s). Each s-plane line (or contour) maps to a corresponding contour in the z-plane. The "limit" controls establish the bounds on the s-plane grid while the "step" controls adjust the density of contours. Note that the display is limited to the left-half s-plane, that is, the region where transfer function poles must be located to ensure stable filter operation; this translates to the interior of the unit circle in the z-plane.

Demo: This tutorial video explains the bilinear transform in more detail and includes two demonstrations of this VI in action at 7:00 and 9:04:

Block Diagram: The VI uses an event structure to handle front-panel control changes in real time. The block diagram follows: