Exponentials as eigenfunctions in LTI Systems

[Tanja]

I hope I catched the correct forum.

Under http://en.wikipedia.org/wiki/LTI_system_theory
e^{s t} is an eigenvalue.
I don't really understand that the following is an eigenvalue equation:
\quad = e^{s t} \int_{-\infty}^{\infty} h(\tau) \, e^{-s \tau} \, d \tau
\quad = e^{s t} H(s),

where

H(s) = \int_{-\infty}^\infty h(t) e^{-s t} d t .

The integrals H(s) and \int_{-\infty}^{\infty} h(\tau) \, e^{-s \tau} \, d \tau should be unique, isn't it? So there is only one solution for H(s) = \int_{-\infty}^{\infty} h(\tau) \, e^{-s \tau} \, d \tau with one e^{s t}, isn't it?

Thanks
Tanja

 

[EnumaElish]

$e^{s t}$ is an eigenvalue.
I don't really understand that the following is an eigenvalue equation:
$\quad = e^{s t} \int_{-\infty}^{\infty} h(\tau) \, e^{-s \tau} \, d \tau$
$\quad = e^{s t} H(s)$,

where

$H(s) = \int_{-\infty}^\infty h(t) e^{-s t} d t$.

The integrals H(s) and $\int_{-\infty}^{\infty} h(\tau) \, e^{-s \tau} \, d \tau$ should be unique, isn't it? So there is only one solution for $H(s) = \int_{-\infty}^{\infty} h(\tau) \, e^{-s \tau} \, d \tau$ with one $e^{s t}$, isn't it?

$F(x) = \int_a^xf(t)dt$ is unique by the fundamental theorem of calculus.

 

[tacman]

Hello Tanja,

Thank you for your question. I understand your confusion regarding the use of exponentials as eigenfunctions in LTI systems. Let me try to explain it in simpler terms.

In LTI (linear time-invariant) systems, the output of the system is related to the input by a linear and time-invariant function. This function can be represented by a differential equation, where the input and output are represented by functions of time, x(t) and y(t), respectively. In this case, the function h(t) represents the impulse response of the system, which is the output of the system when the input is a unit impulse, \delta(t).

Now, when we apply a complex exponential input, e^{st}, to the system, the output can be represented as y(t) = H(s)e^{st}, where H(s) is the transfer function of the system. This means that the output is a scaled version of the input, with the scaling factor given by the transfer function. This is similar to the concept of eigenvalues and eigenvectors in linear algebra, where the eigenvectors are scaled by the eigenvalues when they are transformed by a linear operator.

In this case, the exponential function e^{st} is considered an eigenfunction of the system, and the transfer function H(s) is the corresponding eigenvalue. This is because when we apply the exponential input to the system, the output is just a scaled version of the input, with no change in the shape or frequency content. This is a unique property of LTI systems, where complex exponentials are the only inputs that result in scaled outputs.

I hope this helps to clarify the concept of exponentials as eigenfunctions in LTI systems. If you have any further questions, please feel free to ask. Thank you.