How to interpret evaluating L(f(t))(σ+jω) for a specific σ for a *system*

Question

The question is not about mathematics; that part is clear. I understand the relation between Laplace and differential equations and the multiplication/convolution.

I have a system's impulse response f(t).

With a Fourier transform, it is clear: if I want to know the frequency response of the system, I calculate $F(f(t))(ω) or L(f(t))(jω)$.

• Evaluating $\mathcal{f}(f(t))(jω)$ is the same as applying the input signal $sin(ωt)$

• Evaluating $\mathcal{L}(f(t))(jω)$ is the same as applying the input signal $sin(ωt)$

I want to complete the sentence below:

• Evaluating $\mathcal{L}(f(t))(σ+jω)$ for a specific $\sigma$ is the same as ...
  (What is the status of the system I create by multiplying its impulse response with $e^−\sigma t$)

This question is a new try for the closed question: interpretation of the relation between Laplace transform and Fourier transform

Answer 1

Evaluating $\mathcal{L}(f(t))(σ+jω)$ for a specific $\sigma$ is the same as applying $e^{st}h_0(t)$ as input.

based on jnez71's answer to How can I expand function into a series of exponentially decaying (co)sines?:

the unilateral Laplace-transform of the impulse-response is the ratio of the output to the input when the input is $e^{st}h_0(t)$ where $h_0(t)$ is the unit-step function and $s=\sigma + j \omega$ or a signal of the format $e^{\sigma t}\sin(\omega t) h_0(t)$. $h_0(t)$ is there to let the signal start at $t=0$.