Consider a discrete-time system which satisfies the additivity property, i.e., if the output for u1[n] is y1[n] and that for u2[n] is y2[n], then output for u1[n] + u2[n] is y1[n] + y2[n]. Such a system is : 1. Linear 2. Non-linear 3. Sometimes Linear and Sometimes non-linear?

Option 4 : 3 only

Concept:

Linearity property: it is a combination of Additivity + Homogeneity

Assume signals x₁(t) and x₂(t) gives the responses to a system y₁(t) and y₂(t) respectively.

Now if the input x₁(t) + x₂(t) gives the response as y₁(t) + y₂(t) then it obeys additivity property.

Assume signal α x(t) is the input and α y(t) is the output then it obeys Homogeneity.

If a system obeys both Additivity and Homogeneity then it is said to be Linear.

Calculation:

Given that response for u₁ is y₁ that of u₂ is y₂.

And for the input u₁ + u₂ the response is y₁ + y₂

Additivity property is satisfied. We don’t know about scaling property so the system may be linear or non-linear.

But if we assume α = 1 then the system will be linear.