System Analysis: Linearity and Time-Invariance

The question asks us to classify a continuous-time system given by the equation \(y(t) = x(t)\cos(t)\) based on its linearity and time-invariance properties. To do this, we need to apply the definitions of linear and time-invariant systems.

Linearity of the System

A system is considered linear if it satisfies the superposition principle, which combines two properties: homogeneity (scaling) and additivity.

• Homogeneity: If the input is scaled by a constant factor, the output should also be scaled by the same factor.
  Let input be \(x_1(t)\) producing output \(y_1(t) = x_1(t)\cos(t)\).
  If the input is \(ax_1(t)\), the output \(y_{new}(t) = (ax_1(t))\cos(t) = a(x_1(t)\cos(t)) = ay_1(t)\). This property holds.
• Additivity: If the input is a sum of two signals, the output should be the sum of the outputs produced by each signal individually.
  Let input \(x_1(t)\) produce \(y_1(t) = x_1(t)\cos(t)\).
  Let input \(x_2(t)\) produce \(y_2(t) = x_2(t)\cos(t)\).
  If the input is \(x(t) = x_1(t) + x_2(t)\), the output \(y(t)\) would be:
  \[y(t) = (x_1(t) + x_2(t))\cos(t)\] \[y(t) = x_1(t)\cos(t) + x_2(t)\cos(t)\] Since \(y_1(t) = x_1(t)\cos(t)\) and \(y_2(t) = x_2(t)\cos(t)\), we have:
  \[y(t) = y_1(t) + y_2(t)\] This property holds.

Since both homogeneity and additivity (or the combined superposition principle) are satisfied, the system is linear.

Time-Invariance of the System

A system is considered time-invariant if a time delay or advance in the input signal results in an identical time delay or advance in the output signal. In other words, the system's characteristics do not change with time.

• Let the original input be \(x(t)\) producing the output \(y(t) = x(t)\cos(t)\).
• Now, let's consider a time-shifted input, \(x(t - t_0)\). The output due to this shifted input, let's call it \(y_{shifted\_input}(t)\), would be:
  \[y_{shifted\_input}(t) = x(t - t_0)\cos(t)\]
• Next, let's consider the original output \(y(t)\) and shift it by the same amount, \(t_0\). This would be \(y(t - t_0)\):
  \[y(t - t_0) = x(t - t_0)\cos(t - t_0)\]
• For the system to be time-invariant, \(y_{shifted\_input}(t)\) must be equal to \(y(t - t_0)\).
  Comparing the two:
  \(x(t - t_0)\cos(t)\) vs \(x(t - t_0)\cos(t - t_0)\)
  Since \(\cos(t)\) is generally not equal to \(\cos(t - t_0)\) (unless \(t_0 = 0\)), the two expressions are not equal.
  Thus, \(y_{shifted\_input}(t) \neq y(t - t_0)\).

The presence of the \(\cos(t)\) term, which is a function of time multiplying the input, causes the system's behavior to change with time. Therefore, the system is time-varying.

System Characteristics Conclusion

Based on our analysis:

• The system satisfies the superposition principle, making it linear.
• The system's output changes differently when the input is shifted versus when the entire output is shifted, due to the time-dependent multiplier \(\cos(t)\). This makes the system time-varying.

Therefore, the given continuous-time system is linear and time-varying.