The X-transform and its use in sampled data system analysis

Abstract

Abstract This paper discusses the development of a new transform, called the x-transform, for a sampler-hold combination in open-loop and closed-loop sampled-data systems. The x-transform is a special form of the Laplace transform and is applicable to sampled-data systems which have all samplers followed by zero-order hold devices. The x-transform inherently eliminates a spectral requirement of the z-transform which is that all samplers must be followed by a hold device in order to be assured of accurate results. The x-transform is developed utilizing the change of variable, χn = e−nTs/s where T is the sampling period in seconds, s is the Laplace transform variable and n = 0,l,2, ⋯. The x-transform is defined as e(t) = E(s)/ (ϵ-nTs/s) = xn. It is noted that the definition of x differs from that of z by a sign change in the exponential and an s in the denominator.