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 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 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