Z Transform

In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation. It is like a discrete equivalent of the Laplace transform. This similarity is explored in the theory of time scale calculus. The Z-transform and advanced Z-transform were introduced (under the Z-transform name) by E. I. Jury in 1958 in Sampled-Data Control Systems (John Wiley & Sons). The idea contained within the Z-transform was previously known as the “generating function method”. Definition The Z-transform, like many other integral transforms, can be defined as either a one-sided or two-sided transform.

Bilateral Z-transform The bilateral or two-sided Z-transform of a discrete-time signal x[n] is the function X(z) defined as

where n is an integer and z is, in general, a complex number: z = Aejφ (OR) z = A(cosφ + jsinφ) where A is the magnitude of z, and φ is the complex argument (also referred to as angle or phase) in radians.

Unilateral Z-transform Alternatively, in cases where x[n] is defined only for n ≥ 0, the single-sided or unilateral Z-transform is defined as

In signal processing, this definition is used when the signal is causal.

An important example of the unilateral Z-transform is the probability-generating function, where the component x[n] is the probability that a discrete random variable takes the value n, and the function X(z) is usually written as X(s), in terms of s = z − 1. The properties of Z-transforms (below) have useful interpretations in the context of probability theory.