# Homework Help: To determine whether it's LTI system

1. Apr 11, 2012

### glorimda

1. The problem statement, all variables and given/known data

this system satisfies the condition of initial rest.

Y[n] =(1/2)y[n-1] + x[n]

I'm trying to prove that this is linear and time invariant.

3. The attempt at a solution

I'm trying it this way.
To be LTI system,
It has to satisfy below.
aX1[n] -> aY1[n]
bX2[n] -> bY2[n]
aX1 + bX2 -> aY1 + bY2

1. firstly, I don't see
aX[n] -> aY[n]

Since if I put aX[n] then (1/2)Y[n-1] + aX[n] =! {aY[n]= a(1/2)Y[n-1]+aX[n]}
i.e., aX[n] -> aY[n] is not true in my view.

2. Secondly,
Just let me try this without multiplying constant

X1[n] -> Y1[n]
X2[n] -> Y2[n]
X1 + X2 -> Y1 + Y2

Y1[n] = (1/2)Y1[n-1] + X1[n]
Y2[n] = (1/2)Y2[n-1] + X2[n]

X3[n] = X1[n] + X2[n]

Y3[n] = (1/2)Y3[n-1] + X3[n] = (1/2)Y3[n-1] + {X1[n] + X2[n]}

After this I don't seem to find what to do with Y3[n-1] and since this, I can't prove
Y3[n] = Y1[n] + Y2[n]

Could somebody help me and tell me what I'm doing wrong?
Appreciate for it.