Home Signal And System Chapter
Post
Cancel

Signal And System Chapter

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

This post is licensed under CC BY 4.0 by the author.