华南理工大学：《数字信号处理》（双语版） Chapter 9 LTI Discrete-Time Systems in the Transform domain

LTI Discrete-Time Systems in the Transform domain An lti discrete-time system is completely characterized in the time-domain by its impulse response hin i We consider now the use of the dtft and the z-transform in developing the transform domain representations of an LtI system Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 1 LTI Discrete-Time Systems in the Transform Domain • An LTI discrete-time system is completely characterized in the time-domain by its impulse response {h[n]} • We consider now the use of the DTFT and the z-transform in developing the transform￾domain representations of an LTI system

Finite-Dimensional LTi Discrete-Time Systems We consider lti discrete-time systems characterized by linear constant-coefficient difference equations of the form k{n-k]=∑pkx[n-k] k=0 k=0 Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 2 Finite-Dimensional LTI Discrete-Time Systems • We consider 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 [ ] [ ]

Finite-Dimensional LT Discrete-Time Systems Applying the dtFtto the difference equation and making use of the linearity and the time-invariance properties of Table 3.2 we arrive at the input-output relation in the transform-domain as ∑dkeo0y(e0)=∑peok(e) k=0 k=0 where y(e/u) and x(eu)are the dtfts of yIn] and x[n], respectivel Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 3 Finite-Dimensional LTI Discrete-Time Systems • Applying the DTFT to the difference equation and making use of the linearity and the time-invariance properties of Table 3.2 we arrive at the input-output relation in the transform-domain as where and are the DTFTs of y[n] and x[n], respectively ( ) ( ) 0 0  =  −  = −   =  j M k j k k j N k j k k d e Y e p e X e ( ) j Y e ( ) j X e

Finite-Dimensional LTi Discrete-Time Systems In developing the transform-domain representation of the difference equation, it has been tacitly assumed that X(eu)and Y(e/u)exist The previous equation can be alternately written as e/kY(e0)=、 pre oklo() ∑ k=0 Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 4 Finite-Dimensional LTI Discrete-Time Systems • In developing the transform-domain representation of the difference equation, it has been tacitly assumed that and exist • The previous equation can be alternately written as ( ) j Y e ( ) j X e ( ) ( ) 0 0  =  −  = −         =        j M k j k k j N k j k k d e Y e p e X e

Finite-Dimensional LT Discrete-Time Systems Applying the z-transform to both sides of the difference equation and making use of the linearity and the time-invariance roperties of Table 3. 9 we arrive at N M ∑dkzY(=)=∑PkzX(=) k=0 k=0 where Y(z) and X(z)denote the z-transforms of yn] and xn] with associated RoCs, respectivel Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 5 Finite-Dimensional LTI Discrete-Time Systems • Applying the z-transform to both sides of the difference equation and making use of the linearity and the time-invariance properties of Table 3.9 we arrive at where Y(z) and X(z) denote the z-transforms of y[n] and x[n] with associated ROCs, respectively d z Y(z) p z X(z) M k k k N k k  k  = − = − = 0 0

Finite-Dimensional LTi Discrete-Time Systems a more convenient form of the z-domain representation of the difference equation is given ∑ k ()=∑Pk=6X(=) k=0 k=0 Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 6 Finite-Dimensional LTI Discrete-Time Systems • 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

The Frequency Response Most discrete-time signals encountered in practice can be represented as a linear combination of a very large, possibly 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 Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 7 The Frequency Response • Most discrete-time signals encountered in practice can be represented as a linear combination of a very large, possibly 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

The Frequency Response An important property of an LTI system is that for certain types of input signals, called eigen functions, the output signal is the input signal multiplied by a complex constant We consider here one such eigen function as the input Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 8 The Frequency Response • An important property of an LTI system is that for certain types of input signals, called eigen functions, the output signal is the input signal multiplied by a complex constant • We consider here one such eigen function as the input

The Frequency Response Consider the lti discrete-time system with an impulse response hn shown below xIn Its input-output relationship in the time domain is given by the convolution sum yrm]=∑hkxm-k] Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 9 • Consider the LTI discrete-time system with an impulse response {h[n]} shown below • Its input-output relationship in the time￾domain is given by the convolution sum The Frequency Response x[n] h[n] y[n]   =− = − k y[n] h[k]x[n k]

The Frequency Response If the input is of the form x[n]=ej <n<0 then it follows that the output is given by yn=∑Mho(n)=∑ kle-jok eJon k=-0 el H(e/)=∑列klek k=-0 10 Copyright C 2001, S K Mitra

Copyright © 2001, S. K. Mitra 10 The Frequency Response • If the input is of the form then it follows that the output is given by • Let = −      x n e n j n [ ] , j n k j k k j n k y n h k e h k e e   =− −   =−  −       [ ] =  [ ] =  [ ] ( ) =   =−  −  k j j k H(e ) h[k]e