Electrical – How to check a discrete time system if the following is linear and time invariant

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.

I wasn't really able to follow your reasoning, but I can tell you how you can simplify your second equation:

$$9 H(e^{j\pi})$$

Note that $$e^{j\pi} = -1$$ (Euler's formula).

Inserting that into the equation above gives:

$$ \frac{9}{(1-\frac{1}{2}(-1))^2} = \frac{9}{1.5^2}= 4$$ I hope this helps ;)