Question

Which of the following system is linear?

• A. $y(t) = x^2(t)$
• B. $y(t) = 3x(t) + 2$
• C. $y(t) = x(t) + x(t - 1)$
• D. $y(t) = \sin(x(t))$

Hint:

A quick way to check linearity is to look for non-linear operations on the input signal, such as squaring ($x^2$), trigonometric functions ($\sin(x)$), logarithms ($\ln(x)$), or independent constant terms ($+c$).
If none of these are present and the operations consist only of scaling, shifting, addition, integration, or differentiation, the system is linear.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks us to identify which of the given continuous-time systems satisfies the mathematical property of linearity.

Step 2: Key Formula or Approach:
A system $T$ is linear if and only if it satisfies the principle of superposition, which combines both additivity and homogeneity:
\[ T\{a_1 x_1(t) + a_2 x_2(t)\} = a_1 T\{x_1(t)\} + a_2 T\{x_2(t)\} \]
for any arbitrary inputs $x_1(t)$, $x_2(t)$ and constants $a_1$, $a_2$.

Step 3: Detailed Explanation:

• Let us analyze each option individually to test for linearity:

• Option A: $y(t) = x^2(t)$.
If we scale the input by a constant $a$, the new output is $T\{a x(t)\} = (a x(t))^2 = a^2 x^2(t)$.
For the system to be linear, the output must be $a y(t) = a x^2(t)$. Since $a^2 x^2(t) \neq a x^2(t)$ for $a \neq 1$, the homogeneity property is violated, making the system non-linear.

• Option B: $y(t) = 3x(t) + 2$.
Let $y_1(t) = 3x_1(t) + 2$ and $y_2(t) = 3x_2(t) + 2$.
For a linear combination of inputs $x_3(t) = a_1 x_1(t) + a_2 x_2(t)$, the output is:
\[ y_3(t) = 3(a_1 x_1(t) + a_2 x_2(t)) + 2 = 3a_1 x_1(t) + 3a_2 x_2(t) + 2 \]
However, a linear combination of the individual outputs yields:
\[ a_1 y_1(t) + a_2 y_2(t) = a_1(3x_1(t) + 2) + a_2(3x_2(t) + 2) = 3a_1 x_1(t) + 3a_2 x_2(t) + 2a_1 + 2a_2 \]
Since $y_3(t) \neq a_1 y_1(t) + a_2 y_2(t)$, the system is non-linear. Any system with a non-zero constant bias is non-linear (it is an affine system).

• Option C: $y(t) = x(t) + x(t - 1)$.
Let $y_1(t) = x_1(t) + x_1(t-1)$ and $y_2(t) = x_2(t) + x_2(t-1)$.
For the input $x_3(t) = a_1 x_1(t) + a_2 x_2(t)$, the corresponding output is:
\[ y_3(t) = x_3(t) + x_3(t-1) = (a_1 x_1(t) + a_2 x_2(t)) + (a_1 x_1(t-1) + a_2 x_2(t-1)) \]
Rearranging the terms:
\[ y_3(t) = a_1 [x_1(t) + x_1(t-1)] + a_2 [x_2(t) + x_2(t-1)] = a_1 y_1(t) + a_2 y_2(t) \]
Since the superposition principle is fully satisfied, this system is linear.

• Option D: $y(t) = \sin(x(t))$.
Because the sine function is a non-linear operator, $T\{a_1 x_1(t) + a_2 x_2(t)\} = \sin(a_1 x_1(t) + a_2 x_2(t)) \neq a_1 \sin(x_1(t)) + a_2 \sin(x_2(t))$. Thus, it is non-linear.

Step 4: Final Answer
Thus, the only linear system among the options is $y(t) = x(t) + x(t - 1)$, which corresponds to option (C).