Electronic – determine if the system is linear, time-invariant, and/or causal

mathsignal

Here is my problem:

$$\frac{d^2y}{dt^2} + 5\frac{dy}{dt} + 8y(t) = \frac{dx}{dt} + 3x(t)$$

I have to prove if this system is homogeneous and additive in order for the function to be linear, but can I prove that? It says the output should be as the input, and again how can I solve the above problem? For example, \$x(t)\$ must equal \$y(t)\$ how can I implement that rule into the function?

Best Answer

The input to this system is denoted by \$x(t)\$ and its output by \$y(t)\$.

  • To prove that the system is homogeneous, what you need to show is that

    if input \$x(t)\$ produces output \$y(t)\$, then the output produced by input \$\alpha\cdot x(t)\$ is \$\alpha\cdot y(t)\$.

    Here, \$\alpha\$ is an arbitrarily chosen real number, that is, the highlighted statement must hold regardless of the value of \$\alpha\$. Similarly, the requirement must be satisfied regardless of the choice of input \$x(t)\$. In other words, finding one pair of input and output signals \$x(t)\$ and \$y(t)\$ and one real number \$\alpha\$ for which the highlighted statement is true is not sufficient; it has to hold for all such choices.

  • To prove that the system is additive, what you need to show is that

    if inputs \$x_1(t)\$ and \$x_2(t)\$ produce outputs \$y_1(t)\$ and \$y_2(t)\$ respectively, then the output produced by input \$x_1(t)+x_2(t)\$ is \$y_1(t)+y_2(t)\$.

    Once again, cherry-picking is not allowed; the statement has to hold for all choices of input signals and corresponding output signals.

It is not too hard to verify that your system is both homogeneous and additive.

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