In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex valued frequency-domain (the z-domain or z-plane) representation. It can be considered a discrete-time counterpart of the Laplace transform (the s-domain or s-plane). This similarity is explored in the theory of time-scale calculus. While the continuous-time Fourier transform is evaluated on the s-domain's vertical axis (the imaginary axis), the discrete-time Fourier transform is evaluated along the z-domain's unit circle. The s-domain's left half-plane maps to the area inside the z-domain's unit circle, while the s-domain's right half-plane maps to the area outside of the z-domain's unit circle. In signal processing, one of the means of designing digital filters is to take analog designs, subject them to a bilinear transform which maps them from the s-domain to the z-domain, and then produce the digital filter by inspection, manipulation, or numerical approximation. Such methods tend not to be accurate except in the vicinity of the complex unity, i.e. at low frequencies.
History The basic idea now known as the Z-transform was known to Laplace, and it was re-introduced in 1947 by W. Hurewicz and others as a way to treat sampled-data control systems used with radar. It gives a tractable way to solve linear, constant-coefficient difference equations. It was later dubbed "the z-transform" by Ragazzini and Zadeh in the sampled-data control group at Columbia University in 1952. The modified or advanced Z-transform was later developed and popularized by E. I. Jury. The idea contained within the Z-transform is also known in mathematical literature as the method of generating functions which can be traced back as early as 1730 when it was introduced by de Moivre in conjunction with probability theory. From a mathematical view the Z-transform can also be viewed as a Laurent series where one views the sequence of numbers under consideration as the (Laurent) expansion of an analytic function.
Definition The Z-transform can be defined as either a one-sided or two-sided transform (similarly to the one-sided Laplace transform and the two-sided Laplace transform).
Bilateral Z-transform The bilateral or two-sided Z-transform of a discrete-time signal x [ n ] {\displaystyle x[n]} is the formal power series X ( z ) {\displaystyle X(z)} defined as:
where n {\displaystyle n} is an integer and z {\displaystyle z} is, in general, a complex number. In polar form, z {\displaystyle z} may be written as:
z = A e i ϕ = A ⋅ ( cos ϕ + i sin ϕ ) {\displaystyle z=Ae^{i\phi }=A\cdot (\cos {\phi }+i\sin {\phi })}
where A {\displaystyle A} is the magnitude of z {\displaystyle z} , i {\displaystyle i} is the imaginary unit, and ϕ {\displaystyle \phi } is the complex argument (also referred to as angle or phase) in radians.
Unilateral Z-transform Alternatively, in cases where x [ n ] {\displaystyle x[n]} is defined only for n ≥ 0 {\displaystyle n\geq 0} , the single-sided or unilateral Z-transform is defined as:
This is essentially the same as Dirichlet series of ( x [ n ] ) n ∈ N {\displaystyle (x[n])_{n\in \mathbb {N} }} . In signal processing, the above definition can be used to evaluate the Z-transform of the unit impulse response of a discrete-time causal system. An important example of the unilateral Z-transform is the probability-generating function, where the component x [ n ] {\displaystyle x[n]} is the probability that a discrete random variable takes the value n {\displaystyle n} . The properties of Z-transforms (listed in § Properties) have useful interpretations in the context of probability theory.
Inverse Z-transform The inverse Z-transform is:
where C {\displaystyle C} is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC). In the case where the ROC is causal (see Example 2), this means the path C {\displaystyle C} must encircle all of the poles of X ( z ) {\displaystyle X(z)} . A special case of this contour integral occurs when C {\displaystyle C} is the unit circle. This contour can be used when the ROC includes the unit circle, which is always guaranteed when X ( z ) {\displaystyle X(z)} is stable, that is, when all the poles are inside the unit circle. With this contour, the inverse Z-transform simplifies to the inverse discrete-time Fourier transform, or Fourier series, of the periodic values of the Z-transform around the unit circle:
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