The z-transform in the digital domain is the counterpart to the Laplace transform in the analog domain. The z-transform is an extremely useful tool for analyzing the stability of digital sequences, designing stable digital filters, and relating digital signal processing operations to the equivalent mathematics in the analog domain. The Laplace transform and the z-transform have many mathematical similarities, the most important of which are the properties of linearity and shift invariance. Bilinear transformations include a nonlinear mapping to the “w-plane” where the frequency part of the mapping follows a tangent, rather than linear, relationship. The literature on mapping between the s-plane and z-place often cites the use of a “bilinear transform” which maps the entire left-hand s-plane inside the unit circle on the z-plane. The z-transform is a useful tool for describing sampled-data systems mathematically. Its counterpart in the continuous signal analog domain is the Laplace transform.