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
§4.2 The Frequency Response H(ei),in general,is a complex function of o with a period2π It can be expressed in terms of its real and imaginary parts H(e)=Hre(ej)+j Him(ei) or,in terms of its magnitude and phase, H(eio)=H(eio)川e(o) where θ(o)=argH(ejo)
§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)
§4.2 The Frequency Response The function H(ei)is called the magnitude response and the function 0(@)is called the phase response of the LTI discrete-time system Design specifications for the LTI discrete-time system,in many applications,are given in terms of the magnitude response or the phase response or both
§4.2 The Frequency Response • The function | H(ejω) | is called the magnitude response and the function θ(ω) is called the phase response of the LTI discrete-time system • Design specifications for the LTI discrete-time system, in many applications, are given in terms of the magnitude response or the phase response or both
§4.2 The Frequency Response n some cases,the magnitude function is specified in decibels as G(@)=20l0g10 H(ei)dB where G(@)is called the gain function The negative of the gain function A(O)=-G(0) is called the attenuation or loss function
§4.2 The Frequency Response • In some cases, the magnitude function is specified in decibels as G(ω) = 20log10| H(ejω) | dB where G(ω) is called the gain function • The negative of the gain function A(ω) = - G(ω) is called the attenuation or loss function
§4.2 The Frequency Response Note:Magnitude and phase functions are real functions of o,whereas the frequency response is a complex function of o If the impulse response h[n]is real then the magnitude function is an even function of o: H(eio)川=IH(e-jo)川 and the phase function is an odd function of o: 0(@)=-0(-0)
§4.2 The Frequency Response • Note: Magnitude and phase functions are real functions of ω, whereas the frequency response is a complex function of ω • If the impulse response h[n] is real then the magnitude function is an even function of ω: |H(ejω)| = |H(e - jω)| and the phase function is an odd function of ω: θ(ω) = - θ(-ω)
S 4.4 The Concept of Filtering One application of an LTI discrete-time system is to pass certain frequency components in an input sequence without any distortion (if possible)and to block other frequency components Such systems are called digital filters and one of the main subjects of discussion in this course
§4.4 The Concept of Filtering • One application of an LTI discrete-time system is to pass certain frequency components in an input sequence without any distortion (if possible) and to block other frequency components • Such systems are called digital filters and one of the main subjects of discussion in this course