Question

Consider the following systems where x is the input signal and y is the output signal. Determine and justify if the following systems are (i) Linear (ii) Causal (iii) Time invariant, specifying key conditions/assumptions as needed. (a) y(t)=x 2(t)+1

Answer 1

The system y(t) = x^2(t) + 1 is not linear because it doesn't satisfy superposition or homogeneity, it is causal as the output only depends on the input at the same time or earlier, and it is time-invariant because input time shifts result in identical output time shifts.

Step-by-step explanation:

To determine if the system represented by the equation y(t) = x^2(t) + 1 is linear, causal, and time-invariant, we evaluate it based on the definition and properties of each category:

• A system is linear if it satisfies the properties of superposition and homogeneity. However, the given system does not satisfy superposition because the output of the sum of two inputs is not the sum of the outputs of the individual inputs due to the squaring operation, and it fails the homogeneity test since scaling the input does not lead to a scaled output. Therefore, this system is not linear.
• A system is causal if the output at any time t depends only on the input at that time or earlier. Since the given equation for y(t) depends only on x(t) at the same time, the system is causal.
• A system is time-invariant if a time shift in the input signal x(t) results in an identical time shift in the output signal y(t). The given system is time-invariant since applying a time shift to x(t) will produce the same time shift in y(t), the squaring of the input and the addition of one are time-independent operations.

Therefore, the system is not linear due to the square term, causal as the output depends only on the current or past inputs, and time-invariant as shifting the input in time also shifts the output identically.