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The output of a continuous-time, linear time-invariant system is denoted by T{x(t)} where x(t) is the input signal. A signal z(t) is called eigen-signal of the system T, if T{z(t)} = y z(t), where y is a complex number, in general, and is called an eigenvalue of T. Suppose the impulse response of the system T is real and even. Which of the following statements is TRUE?

1. cos(t) is an eigen-signal but sin(t) is not
2. cos(t) and sin(t) are both eigen-signals but with different eigenvalues
3. sin(t) is an eigen-signal but cos(t) is not
4. cos(t) and sin(t) are both eigen-signals with identical eigenvalues

Correct answer: cos(t) and sin(t) are both eigen-signals with identical eigenvalues

Solution

Both cos(t) and sin(t) are eigen-signals of the system because the system's impulse response is real and even, which allows both signals to be transformed into scaled versions of themselves. Since the system is linear and time-invariant, the eigenvalues for these signals will be identical due to their orthogonal relationship.

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