Z-transform

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.

The foundational concept now recognized as the Z-transform, which is a cornerstone in the analysis and design of digital control systems, was not entirely novel when it emerged in the mid-20th century.

However, the explicit formulation and application of what we now understand as the Z-transform were significantly advanced in 1947 by Witold Hurewicz and colleagues.

Their work was motivated by the challenges presented by sampled-data control systems, which were becoming increasingly relevant in the context of radar technology during that period.

The Z-transform provided a systematic and effective method for solving linear difference equations with constant coefficients, which are ubiquitous in the analysis of discrete-time signals and systems.

[4][5] The method was further refined and gained its official nomenclature, "the Z-transform," in 1952, thanks to the efforts of John R. Ragazzini and Lotfi A. Zadeh, who were part of the sampled-data control group at Columbia University.

Their work not only solidified the mathematical framework of the Z-transform but also expanded its application scope, particularly in the field of electrical engineering and control systems.

Jury's work extended the applicability and robustness of the Z-transform, especially in handling initial conditions and providing a more comprehensive framework for the analysis of digital control systems.

This advanced formulation has played a pivotal role in the design and stability analysis of discrete-time control systems, contributing significantly to the field of digital signal processing.

[8][9] Interestingly, the conceptual underpinnings of the Z-transform intersect with a broader mathematical concept known as the method of generating functions, a powerful tool in combinatorics and probability theory.

This connection was hinted at as early as 1730 by Abraham de Moivre, a pioneering figure in the development of probability theory.

De Moivre utilized generating functions to solve problems in probability, laying the groundwork for what would eventually evolve into the Z-transform.

In signal processing, this definition can be used to evaluate the Z-transform of the unit impulse response of a discrete-time causal system.

is a counterclockwise closed path encircling the origin and entirely in the region of convergence (ROC).

By integrating around a closed contour in the complex plane, the residues at the poles of the Z-transform function inside the ROC are summed.

This method is widely used for its efficiency and simplicity, especially when the original function can be easily broken down into recognizable components.

Note: B) Determine the inverse Z-transform of the following by series expansion method, Eliminating negative powers if

In this case the ROC is the complex plane with a disc of radius 0.5 at the origin "punched out".Let

If we need both stability and causality, all the poles of the system function must be inside the unit circle.

, known as the unit circle, we can express the transform as a function of a single real variable

-periodic function is the periodic summation of a Fourier transform, which makes it a widely used analysis tool.

Due to periodicity, there are only a finite number of unique amplitudes, which are readily computed by the much simpler discrete Fourier transform (DFT).

(the sampling parameter): The inverse Laplace transform is a mathematical abstraction known as an impulse-sampled function.

(Notation conventions typically use capitalized letters to refer to the z-transform of a signal denoted by a corresponding lower case letter, similar to the convention used for notating Laplace transforms.)

Rearranging results in the system's transfer function: From the fundamental theorem of algebra the numerator has

By factoring the denominator, partial fraction decomposition can be used, which can then be transformed back to the time domain.

Doing so would result in the impulse response and the linear constant coefficient difference equation of the system.