Linear Discrete-Time Systems Accumulator-y1{n=∑x[(l,y2[m=∑x2[C For an input x[n]=ax[n]+Bx2n the output is y{n]=∑ax[(]+Bx2[ =a∑x(]+β∑x2[(]=ayfn]+By2n C=-0 Hence the above system is linear Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 11 Linear Discrete-Time Systems • Accumulator - • For an input the output is • Hence, the above system is linear = = =− =− n n y n x y n x [ ] [], [ ] [] 1 1 2 2 [ ] [ ] [ ] x n = x1 n + x2 n = ( + ) =− n y n x x [ ] [] [] 1 2 [ ] [ ] [ ] [ ] x1 x2 y1 n y2 n n n = + = + =− =−
Nonlinear discrete-Time System Consider n=x In-xIn-lxIn+ Outputs y[n]and y2ln for inputs xi[n and x2n]are given by n[n=x[n]-x1[n-1x[+1 y2m=x2[n]-x2[n-12m+11 Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 12 Nonlinear Discrete-Time System • Consider • Outputs and for inputs and are given by [ ] [ ] [ 1] [ 1] 2 y n = x n − x n − x n + y [n] 1 [ ] [ ] [ 1] [ 1] 1 1 2 y1 n = x1 n − x n − x n + y [n] 2 [ ] [ ] [ 1] [ 1] 2 2 2 y2 n = x2 n − x n − x n + x [n] 2 x [n] 1
Nonlinear discrete-Time System Output y[n] due to an input axn]+ Bx2n is given b yn]=a xn+B xn ax1[n-1+Bx2[n-1}{x1[m+1]+Bx2[n+1]} =a2{x[n]-x1[n-1]x[n+1 +B2{x2[]-x2[n-1]x2[n+1} +aB12x1[mx2[m]-x1[n-1x2[n+1]-x1[n+12n-1]} Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 13 Nonlinear Discrete-Time System • Output y[n] due to an input is given by x [n] x [n] 1 + 2 2 1 2 y n x n x n [ ] { [ ] [ ]} = + { [ 1] [ 1]}{ [ 1] [ 1]} − x1 n − + x2 n − x1 n + + x2 n + { [ ] [ 1] [ 1]} 1 1 2 1 2 = x n − x n − x n + { [ ] [ 1] [ 1]} 2 2 2 2 2 + x n − x n − x n + {2 [ ] [ ] [ 1] [ 1] [ 1] [ 1]} + x1 n x2 n − x1 n − x2 n + − x1 n + x2 n −
Nonlinear discrete-Time System On the other hand ayIn + By2ln =ax[m]-x1[n-1x[n+1]} +B(x2m-x2[n-1x2m+1]} ≠1n Hence the system is nonlinear Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 14 Nonlinear Discrete-Time System • On the other hand • Hence, the system is nonlinear y [n] y [n] 1 + 2 { [ ] [ 1] [ 1]} 1 1 2 = x1 n − x n − x n + { [ ] [ 1] [ 1]} 2 2 2 + x2 n − x n − x n + y[n]
Shift-Invariant System For a shift-invariant system, if y[n] is the response to an input xin], then the response to an input is simply yIn]=yln-no] Where no is any positive or negative integer The above relation must hold for any arbitrary input and its corresponding output Copyright C 2001, S K. Mitra
Copyright © 2001, S. K. Mitra 15 Shift-Invariant System • For a shift-invariant system, if is the response to an input , then the response to an input is simply where is any positive or negative integer • The above relation must hold for any arbitrary input and its corresponding output y [n] 1 x [n] 1 [ ] [ ] n no x n = x1 − [ ] [ ] n no y n = y1 − no