Question

A system is characterized by the input-output relation y(t) = x(2t) + x(3t) for all t, where y(t) is the output and x(t) is the input. It is
1. linear and causal
2. linear and non-causal
3. non-linear and causal
4. non-linear non-causal

Answer 1

Correct Answer - Option 2 : linear and non-causal

Concept:

Causal: A system is said to be causal if a system is dependent on present or past inputs only but not on the future input.

Linear: A system is said to be linear if it follows both superposition and homogeneity.

y{ax1[t] + bx2[t]} = a y{x1[t]} + b y{x2[t]}

Conditions to check whether the system is linear or not:

1) The output should be zero for zero input.

2) There should not be any non-linear operator present in the system.

Application:

y(t) = x(2t) + x(3t)

The given system is not causal because, for positive values of time, the output is depending upon the future values of the input.

Example: For t = 1, the output is:

y(1) = x(2) + x(3)

The given system is linear because:

1) For zero input, the output is also zero

2) It satisfies both additivity and homogeneity property, i.e.

y{ax1[t] + bx2[t]} = a y{x1[t]} + b y{x2[t]}