Discrete-Time Systems: Examples sEnf=2(n 0.9 dnf-fandom signal xn] 10 50 Time index n yIn Time index n Copyright C 2001, S.K. Mitra
Copyright © 2001, S. K. Mitra 6 [ ] 2[ (0.9) ], d[n] - random signal n s n = n Discrete-Time Systems:Examples 0 10 20 30 40 50 0 1 2 3 4 5 6 7 Time index n Amplitude s[n] y[n]
Discrete-Time Systems: Examples Linear interpolation -employed to estimate sample values between pairs of adjacent sample values of a discrete-time sequence Factor-of-4 interpolation xnI x团n j 6789101112 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 7 Discrete-Time Systems:Examples • Linear interpolation - Employed to estimate sample values between pairs of adjacent sample values of a discrete-time sequence • Factor-of-4 interpolation
Discrete-Time Systems Examples Factor-of-2 interpolator yn=x小]+5(xn-1]+xn+) Factor-of-3 interpolator =x]+( 3xln-1]+x[n+2]) +4(xnn-2]+xn+1) 8 Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 8 Discrete-Time Systems: Examples • Factor-of-2 interpolator - • Factor-of-3 interpolator - ( [ 1] [ 1]) 2 1 y[n] = xu [n]+ xu n − + xu n + ( [ 1] [ 2]) 3 1 y[n] = xu [n]+ xu n − + xu n + ( [ 2] [ 1]) 3 2 + xu n − + xu n +
Discrete-Time Systems Classification Linear Systems Shift-Invariant Systems Causal systems Stable systems Passive and Lossless systems Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 9 Discrete-Time Systems: Classification • Linear Systems • Shift-Invariant Systems • Causal Systems • Stable Systems • Passive and Lossless Systems
Linear Discrete-Time Systems Definition-If yin]is the output due to an input xin and y2n is the output due to an input x,n then for an input x[n]=axn+Bx2n the output is given b yIn]=ayIn]+By2ln Above property must hold for any arbitrary constants a and B, and for all possible inputs x[n]and x2[n Copyright C 2001, S K Mitra
Copyright © 2001, S. K. Mitra 10 Linear Discrete-Time Systems • Definition - If is the output due to an input and is the output due to an input then for an input the output is given by • Above property must hold for any arbitrary constants and and for all possible inputs and [ ] y1 n [ ] x1 n [ ] x2 n [ ] 2 y n [ ] [ ] [ ] x n = x1 n + x2 n [ ] [ ] [ ] y n = y1 n + y2 n , [ ] x1 n [ ] x2 n