Shift-Invariant System In the case of sequences and systems with indices n related to discrete instants of time the above property is called time-invariance property Time-invariance property ensures that for a specified input, the output is independent of the time the input is being applied 16 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 16 Shift-Invariant System • In the case of sequences and systems with indices n related to discrete instants of time, the above property is called time-invariance property • Time-invariance property ensures that for a specified input, the output is independent of the time the input is being applied
Shift-Invariant System Example-Consider the up-sampler with an input-output relation given by xn/L],n=0,+L2±2L, 0 otherwise For an input xin]=xin-no the output xluln is given by x1[n/,n=0,+L,+2L Alan u otherwise ∫x1(n-Ln0)/L],n=0,±L,±2L2 otherwise Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 17 Shift-Invariant System • Example - Consider the up-sampler with an input-output relation given by • For an input the output is given by = = , otherwise [ / ], , , ,..... [ ] 0 x n L n 0 L 2L xu n [ ] [ ] n no x1 n = x − [ ] x1,u n = = , otherwise [ / ], , , ,..... [ ] , 0 1 0 2 1 x n L n L L x u n − = = , otherwise [( )/ ], , , ,..... 0 x n Lno L n 0 L 2L
Shift-Invariant System However from the definition of the up-sampler 1-1 =x(m-m),n=mn土Lm士2L,… 0 otherwise ≠ Hence, the up-sampler is a time-varying system Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 18 Shift-Invariant System • However from the definition of the up-sampler • Hence, the up-sampler is a time-varying system [ ] u n no x − − = = , otherwise [( )/ ], , , ,..... 0 x n no L n no no L no 2L [ ] x1,u n
Linear Time-Invariant System Linear Time-Invariant (LTI System A system satisfying both the linearity and the time-invariance property LTI systems are mathematically easy to analyze and characterize Therefore, easy to design Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 19 Linear Time-Invariant System • Linear Time-Invariant (LTI) System - A system satisfying both the linearity and the time-invariance property • LTI systems are mathematically easy to analyze and characterize, • Therefore, easy to design