## General Theory and Applications of Y-transforms |

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1+Ty 2irj 2ttj a y transform F agreement with Eq approaches zero bilinear transformation CD H characteristic equation closed contour coefficient complex convolution theorem Complex differentiation theorem complex translation theorem corresponding denotes e u(t e"at e"atu(t essential singularities Example F y,m f(kT f(n+ f(nT f(oo Fa(y function f(t function f(t)u(t g(nT infinite one-sided integrand of Eq inverse y-transform Laplace domain Laplace transform Laurent expansion Laurent series lim F lim f(t lim lim member of Eq modified y modified y-transform oo oo pole q power series power series expansion real convolution theorem real translation theorem Residue of integrand residues gives right-hand member sampled-data control system sampled-data system sampling period separates the poles shown in Fig Substituting from Eqs t u(t Taking the limit thesis transform of f(t)u(t tu(t u(nT y-plane y-transform of f(t y+ l/T