Chapter 4 F Frequency-domain Representation of LTI e Discrete-Time Systems
Chapter 4 Frequency-domain Representation of LTI Discrete-Time Systems
84.1 LTI Discrete-Time Systems in the Transform domain Such transform-domain representations provide additional insight into the behavior of such systems It is easier to design and implement these systems in the transform-domain for certain applications 一· We consider now the use of the dtft and the z-transform in developing the transform domain representations of an Lti system
§4.1 LTI Discrete-Time Systems in the Transform Domain • Such transform-domain representations provide additional insight into the behavior of such systems • It is easier to design and implement these systems in the transform-domain for certain applications • We consider now the use of the DTFT and the z-transform in developing the transformdomain representations of an LTI system
84.1 LTI Discrete-Time Systems in the Transform domain In this course we shall be concerned with lti discrete-time systems characterized by linear constant coefficient difference equations of the form: ∑dky{n-k]=∑pkxn-k] k=0
§4.1 LTI Discrete-Time Systems in the Transform Domain • In this course we shall be concerned with 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 [ ] [ ]
84.1 LTI Discrete-Time Systems in the Transform domain a. Applying the dtft to the diffe erence equation and making use of the linearity and the time invariance properties we arrive at the input- output relation in the transform-domain as k ≌ evoke(eo k Pk Y(e1) k=0 e where Y(eo)and x(eo) are the dfts of yin and x n, respectively
§4.1 LTI Discrete-Time Systems in the Transform Domain • Applying the DTFT to the difference equation and making use of the linearity and the timeinvariance properties we arrive at the inputoutput relation in the transform-domain as ( ) ( ) 0 0 = − = − = j M k j k k j N k j k k d e Y e p e X e where Y(ej) and X(ej) are the DTFTs of y[n] and x[n], respectively
84.1 LTI Discrete-Time Systems in the Transform domain In developing the transform-domain representation of the difference equation, it has been tacitly assumed that X( Jo) and Y(ejo ) exist The previous equation can be alternately written as k e k y(e0)=∑pk e Jok X(e/) k=0 k=0
§4.1 LTI Discrete-Time Systems in the Transform Domain • In developing the transform-domain representation of the difference equation, it has been tacitly assumed that X(ej) and Y(ej) exist • The previous equation can be alternately written as ( ) ( ) 0 0 = − = − = j M k j k k j N k j k k d e Y e p e X e