LTI Discrete-Time Systems in the Transform domain An lti discrete-time system is completely characterized in the time-domain by its impulse response thin 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 lt Discrete-Time Systems We consider lti discrete -time systems characterized by linear constant-coefficient difference equations of the form aky{n-k]=∑pkxn-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 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 he加 (e0)=∑peK(e°) k=0 k=0 where Y(e/o)and X(e/)are the dfTs of vn] and xn], respectively 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 lt Discrete-Time Systems In developing the transform-domain representation of the difference equation. it has been tacitly assumed that X(e o) an Y(e/)exist The previous equation can be alternately written as N∑三 (0k e (e0)=∑pk eJ在 X( 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-DimensionallTi 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 N ∑dk=Y(-)=∑Pk=X(=) 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