DSP fundamentals — The Tustin Transform

This article walks through the Tustin transform, the fundamental equation required to change a filter represented using Laplace transform (s-domain, continuous time) to Z-transform (z-domain, discrete time).

Relationship between Laplace and Z-transform

The equation for the Laplace transforms X(s) of a continuous time signal x(t) is given by the equation

Laplace transform equation

For the nth sample of the continuous-time sample x(t) sampled at a time interval T, the above equation becomes

Conversion of continuous time Laplace transform to discrete time Laplace transform

The equation for Z-transform of a discrete time signal is given by

Expression for Z-transform

Comparing the above equations gives

Relation between S-domain and Z-domain

Using the first-order Taylor series expansion of e^x gives

Re-arranging the above equation gives

The above equation is the final expression for Tustin transform. Using the above equation, any analog filter can be converted to its digital form provided that Nyquist theorem holds (Nyquist–Shannon sampling theorem — Wikipedia). The above linearity holds only if the sampling frequency is significantly higher than the input signal frequency since (sT/2) << 1 for the Taylor series to hold.

Frequency wrapping

Reference: Frequency Warping | Introduction to Digital Filters

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The s term and z term can be represented in the frequency domain too. Let Ω represent the frequency in the digital domain and ω the frequency in the analog domain. The Tustin transform equation becomes

The digital domain frequency can be calculated as

If T → 0, then, the digital frequency becomes equal to the analog frequency. So, in-order to design a digital filter with the same cut-off frequency as that of the analog filter, multiply the target digital cut-off frequency with