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)