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
84.1 LTI Discrete-Time Systems in the Transform domain difference equation and making use or the e Applying the z-transform to both sides of the linearity and the time-invariance properties we arrive at ∑dkz-(z)=∑ PkE X() k=0 k=0 where y(z and x(z denote the z-transforms of yIn and xn 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
84.1 LTI Discrete-Time Systems in the Transform domain A more convenient form of the z-domain representation of the difiference equation is given by ∑4k=-k|y(=)=∑pk=-kX() 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 Most discrete-time signals encountered in practice can be represented as a linear combination of a very large, maybe infinite number of sinusoidal discrete time signals of dififerent angular frequencies Thus, knowing the response of the lti system to a single sinusoidal signal, we can determine its response to more complicated signals by making use of the superposition property
§4.2 The Frequency Response • Most discrete-time signals encountered in practice can be represented as a linear combination of a very large, maybe infinite, number of sinusoidal discretetime signals of different angular frequencies • Thus, knowing the response of the LTI system to a single sinusoidal signal, we can determine its response to more complicated signals by making use of the superposition property
§4.2 The Frequency Response The quantity H(ejo) is called the frequency response of the lti discrete time system H(ejo) provides a frequency-domain description of the system H(ejo) is precisely the dtft of the impulse response hn of the system
§4.2 The Frequency Response • The quantity H(ej) is called the frequency response of the LTI discretetime system • H(ej) provides a frequency-domain description of the system • H(ej) is precisely the DTFT of the impulse response {h[n]} of the system
§4.2 The Frequency Response H(eJo), in general, is a complex function of o with a period2兀 It can be expressed in terms of its real and imaginary parts H(ejo)=hre(ejo)+j Him(ejo) or, in terms of its magnitude and phase, H(ejo)=H(ejo )l ee(@) where B(o=argH(eJo)
§4.2 The Frequency Response • H(ej), in general, is a complex function of with a period 2p • It can be expressed in terms of its real and imaginary parts H(ej)= Hre(ej) +j Him(ej) or, in terms of its magnitude and phase, H(ej)=|H(ej)| e() where ()=argH(ej)
