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hwsol-0403 - ECE 178 Hwk#3 Solutions 1(a y[n = T[x[n = x[n...

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ECE 178 Hwk #3 Solutions 1) (a) y[n] = T[x[n]] = x[n]+2x[n+1]+3 Test Linearity: check if T[ax 1 [n]+ bx 2 [n]] = ay 1 [n]+ by 2 [n] T[ax 1 [n]+ bx 2 [n]] = ax 1 [n]+ bx 2 [n]+ 2ax 1 [n+1]+ 2bx 2 [n+1]+3 Eq. 1 ay 1 [n]+ by 2 [n] = ax 1 [n]+ 2ax 1 [n+1]+3a+ bx 2 [n]+ 2bx 2 [n]+3b Eq. 2 Since Eq. 1 does not equal Eq. 2 the system is not linear. Test Shift Invariance: check if T[x[n-n 0 ]] = y[n-n 0 ] T[x[n-n 0 ]] = x[n-n 0 ] +2 x[n-n 0 +1]+3 Eq. 3 y[n-n 0 ] = x[n-n 0 ] +2 x[n-n 0 +1]+3 Eq. 4 Since Eq. 3 equals Eq. 4 then the system is shift invariant. (b) y[m,n] = exp(-x[m,n]) Test Linearity: check if T[ax 1 [m,n]+ bx 2 [m,n]] = ay 1 [m,n]+ by 2 [m,n] T[ax 1 [m,n]+ bx 2 [m,n]] = exp(-ax 1 [m,n]- bx 2 [m,n]) Eq. 5 ay 1 [m,n]+ by 2 [m,n] = a*exp(-x 1 [m,n])+b*exp(-x 2 [m,n]) Eq. 6 Since Eq. 5 does not equal Eq. 6 the system is not linear. Test Shift Invariance: check if T[x[m-m 0 ,n-n 0 ]] = y[m-m 0 ,n-n 0 ] T[x[m-m 0 ,n-n 0 ]] = exp(-x[m-m 0 ,n-n 0 ]) Eq. 7 y[m-m 0 ,n-n 0 ] = exp(-x[m-m 0 ,n-n 0 ]) Eq. 8 Since Eq. 7 equals Eq. 8 the system is shift invariant. (c) y[m,n] = m*n*x[m,n] Test Linearity: check if T[ax 1 [m,n]+ bx 2 [m,n]] = ay 1 [m,n]+ by 2 [m,n] T[ax 1 [m,n]+ bx 2 [m,n]] = a*m*n*x 1 [m,n]+ b*m*n*x 2 [m,n] Eq. 9 ay 1 [m,n]+ by 2 [m,n] = a*m*n*x 1
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