Chapter 4 requency-domain Representation of LtI 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 ystems in the transform-domain for certain applications 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: ∑ koln-k]=∑Dm-k k=0 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 applying the dtft to the difference equation and making use of the linearity and the time invariance properties we arrive at the input- output relation in the transform-domain as iok ∑de~0Y(e0)=∑p he yok X(e0) k=0 k=0 where Y(ej)and X(ej@)are the tfTs of yln andx四l respectiv rely
§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 ω = ω − ω = − ω ∑ = ∑ j M k j k k j N k j k k d e Y e p e X e where Y(ejω) and X(ejω) 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(ej) and Y(ejo) exist The le previous equation can be alternatel written as e/0 ∑ (l)=∑ h?e圆oh LX(e/o 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(ejω) and Y(ejω) exist • The previous equation can be alternately written as ( ) ( ) 0 0 ω = ω − ω = − ω = ∑ ∑ j M k j k k j N k j k k d e Y e p e X e