LTI Discrete-Time Systems in the Transform domain An lti discrete-time system is completely characterized in the time-domain by its impulse response hin i We consider now the use of the dtft and the z-transform in developing the transform domain representations of an LtI system Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 1 LTI Discrete-Time Systems in the Transform Domain • An LTI discrete-time system is completely characterized in the time-domain by its impulse response {h[n]} • We consider now the use of the DTFT and the z-transform in developing the transformdomain representations of an LTI system
Finite-Dimensional LTi Discrete-Time Systems We consider lti discrete-time systems characterized by linear constant-coefficient difference equations of the form k{n-k]=∑pkx[n-k] k=0 k=0 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 2 Finite-Dimensional LTI Discrete-Time Systems • We consider LTI discrete-time systems characterized by linear constant-coefficient difference equations of the form: = = − = − M k k N k k d y n k p x n k 0 0 [ ] [ ]
Finite-Dimensional LT Discrete-Time Systems Applying the dtFtto the difference equation and making use of the linearity and the time-invariance properties of Table 3.2 we arrive at the input-output relation in the transform-domain as ∑dkeo0y(e0)=∑peok(e) k=0 k=0 where y(e/u) and x(eu)are the dtfts of yIn] and x[n], respectivel Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 3 Finite-Dimensional LTI Discrete-Time Systems • Applying the DTFT to the difference equation and making use of the linearity and the time-invariance properties of Table 3.2 we arrive at the input-output relation in the transform-domain as where and are the DTFTs of y[n] and x[n], respectively ( ) ( ) 0 0 = − = − = j M k j k k j N k j k k d e Y e p e X e ( ) j Y e ( ) j X e
Finite-Dimensional LTi Discrete-Time Systems In developing the transform-domain representation of the difference equation, it has been tacitly assumed that X(eu)and Y(e/u)exist The previous equation can be alternately written as e/kY(e0)=、 pre oklo() ∑ k=0 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 4 Finite-Dimensional LTI Discrete-Time Systems • In developing the transform-domain representation of the difference equation, it has been tacitly assumed that and exist • The previous equation can be alternately written as ( ) j Y e ( ) j X e ( ) ( ) 0 0 = − = − = j M k j k k j N k j k k d e Y e p e X e
Finite-Dimensional LT Discrete-Time Systems Applying the z-transform to both sides of the difference equation and making use of the linearity and the time-invariance roperties of Table 3. 9 we arrive at N M ∑dkzY(=)=∑PkzX(=) k=0 k=0 where Y(z) and X(z)denote the z-transforms of yn] and xn] with associated RoCs, respectivel Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 5 Finite-Dimensional LTI Discrete-Time Systems • Applying the z-transform to both sides of the difference equation and making use of the linearity and the time-invariance properties of Table 3.9 we arrive at where Y(z) and X(z) denote the z-transforms of y[n] and x[n] with associated ROCs, respectively d z Y(z) p z X(z) M k k k N k k k = − = − = 0 0