LTI Discrete-Time Systems in the Transform domain An lti discrete-time system is completely characterized in the time-domain by its impulse response thin 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 transformdomain representations of an LTI system
Finite-Dimensional lt Discrete-Time Systems We consider lti discrete -time systems characterized by linear constant-coefficient difference equations of the form aky{n-k]=∑pkxn-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 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 he加 (e0)=∑peK(e°) k=0 k=0 where Y(e/o)and X(e/)are the dfTs of vn] and xn], respectively 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 lt Discrete-Time Systems In developing the transform-domain representation of the difference equation. it has been tacitly assumed that X(e o) an Y(e/)exist The previous equation can be alternately written as N∑三 (0k e (e0)=∑pk eJ在 X( 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-DimensionallTi 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 N ∑dk=Y(-)=∑Pk=X(=) 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 lt Discrete-Time Systems a more convenient form of the z-domain representation of the difference equation is given by ∑4k=k(二)=1∑Pkz-kx() 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 thin shown below hin Its input-output relationship in the time domain is given by the convolution sum y{n]=∑k]xn-k k=-0o 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 timedomain 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 on 0<1<00 then it follows that the output is given by y=∑Mke0)=∑小kk k k Let H(e10)=∑k]le k k -00 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