Chapter 4 Frequency-domain Representation of LTI Discrete-Time Systems
Chapter 4 Frequency-domain Representation of LTI Discrete-Time Systems
S 4.1 LTI Discrete-Time Systems in the Transform Domain This course is concerned with LTI discrete-time systems characterized by linear constant coefficient difference equations of the form: N M ∑dkyIn-k]=∑pkx[n-k] k=0 k=0
§4.1 LTI Discrete-Time Systems in the Transform Domain • This course is 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 [ ] [ ]
4.1 LTI Discrete-Time Systems in the Transform Domain Applying the z-transform to both sides of the difference equation and making use of the linearity and the time-invariance properties we arrive at M dzkY(e)=PzkX(e） k=0 k=0 where Y(z)and X(z)denote the z-transforms of y[n]and x[n]with associated ROCs,respectively
§4.1 LTI Discrete-Time Systems in the Transform Domain • Applying the z-transform to both sides of the difference equation and making use of the linearity and the time-invariance properties we arrive at d z Y(z) p z X (z) M k k k N k k ∑ k ∑ = − = − = 0 0 where Y(z) and X(z) denote the z-transforms of y[n] and x[n] with associated ROCs, respectively
S 4.1 LTI Discrete-Time Systems in the Transform Domain A more convenient form of the z-domain representation of the difference equation is given by 三ae总me k=0 k=0
§4.1 LTI Discrete-Time Systems in the Transform Domain • A more convenient form of the z-domain representation of the difference equation is given by d z Y(z) p z X (z) M k k k N k k k = ∑ ∑ = − = − 0 0
§4.2 The Frequency Response The quantity H(ei)is called the frequency response of the LTI discrete- time system H(ei)provides a frequency-domain description of the system H(ei)is precisely the DTFT of the impulse response (h[n]}of the system
§4.2 The Frequency Response • The quantity H(ejω) is called the frequency response of the LTI discretetime system • H(ejω) provides a frequency-domain description of the system • H(ejω) is precisely the DTFT of the impulse response {h[n]} of the system