Unilateral Z-transformĪlternatively, in cases where x is defined only for n ≥ 0, the single-sided or unilateral Z-transform is defined as Where A is the magnitude of z, j is the imaginary unit, and ɸ is the complex argument (also referred to as angle or phase) in radians. Where n is an integer and z is, in general, a complex number: The bilateral or two-sided Z-transform of a discrete-time signal x is the formal power series X(z) defined as The Z-transform, like many integral transforms, can be defined as either a one-sided or two-sided transform. 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. 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. The modified or advanced Z-transform was later developed and popularized by E. It was later dubbed "the z-transform" by Ragazzini and Zadeh in the sampled-data control group at Columbia University in 1952. Hurewicz as a tractable way to solve linear, constant-coefficient difference equations. The basic idea now known as the Z-transform was known to Laplace, and re-introduced in 1947 by W. 9 Linear constant-coefficient difference equation.
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