Signal and System
Signal
Signal is numbers with index. For example the signal \(x(t-t_0)\) actually means
1
2
def signal(t):
return x(t-t_0)
or
1
signal = lambda t: x(t-t_0)
Linear System
The response to \(x_1(t) + x_2(t)\) is \(y_1(t) + y_2(t)\)
Time Invariant System
The response to \(x(t-t_0)\) is \(y(t-t_0)\).
Is linear constant coefficient ODE LTI?
For example, consider an simple ODE
\[y'' + y' + y = x(t)\]if we let the system has all zero initial conditions before we apply any signal (initial rest), then it is LTI.
We can verify the response to \(x_1(t) + x_2(t)\) is \(y_1(t) + y_2(t)\)
\[y_1''(t) + y_1'(t) + y_1(t) = x_1(t)\] \[y_2''(t) + y_2'(t) + y_2(t) = x_2(t)\] \[y_1''(t) + y_2''(t) + y_1'(t)+y_2'(t) + y_1(t) + y_2(t) = x_1(t) + x_2(t)\]Since both \(y_1(t)\) and \(y_2(t)\) satisfy the initial rest, zero initial conditin at some time \(\tau\), their addition also satisfy zero inisital consition at \(\tau\).
We can also verify the response to \(x(t-t_0)\) is \(y(t-t_0)\)
\[y''(t) + y'(t) + y(t) = x(t)\] \[y''(t-t_0) + y'(t-t_0) + y(t-t_0) = x(t-t_0)\]before \(\tau\) either \(x(t)\) or \(x(t-t_0)\) is zero, thus \(y(t)\) and \(y(t-t_0)\) satisfy the initial condition.
$e^{j\omega t}$ is eigen signal for LTI system