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
84.1 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 ∑dkzY()=∑pk=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 diffe Terence equation is given by -k )=∑pk=-X() 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
°§42 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 different angular frequencies Thus. knowing the response of the ltl 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 discrete-time 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
°§42 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(elo) is precisely the dtft of the impulse response (hn 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
°§42 The Frequency Response H(ej@), in general, is a complex function of@ with a period2兀 It can be expressed in terms of its real and imaginary parts H(ej@=hre(ejo +j him(ejoy or, in terms of its magnitude and phase e H(ejo)l ee(a) where 6(0)=rgH(
§4.2 The Frequency Response • H(ejω), in general, is a complex function of ω with a period 2π • It can be expressed in terms of its real and imaginary parts H(ejω)= Hre(ejω) +j Him(ejω) or, in terms of its magnitude and phase, H(ejω)=|H(ejω)| eθ(ω) where θ(ω)=argH(ejω)
