Connection with system analysis and laplace&Z transform

Question

Laplace&Z transform is just frequency analysis of a system with a multiplication of decaying exponential. We analyse frequency with varying Laplace Exponential, which can be seen in the formula below for laplace transfrom.

From my simple aspect I can't make the theoretical connection, how inspecting frequency with decaying exponential helps with understanding how system will behave? And how it helps with coping with differential functions for example "Analysis of Electric Circuits". And even with IIR filters.

What further surprises me the things laplace&Z transforms used. They seem to me completely different .But somehow multiplying the system with decaying exponential works for them all. The only similarity I see between; for example RLC curcuits and IIR filters is; these systems are not memoryless(output depends on prev. outputs).

When I search the net I see only laplace-table's, questions etc.. I don't want to solve a problem, I want to understand the problem. If somebody can explain or direct me to a link I will be thankful.

Answer 1

You are right that the (bilateral) Laplace transform can be interpreted as the Fourier transform of $e^{-\sigma t}f(t)$. However, I think that the significance of the Laplace transform only becomes clear when $s=\sigma+j\omega$ is viewed as a complex variable because then we can study the analytic properties of the system function. E.g., electrical networks are characterized by the poles and zeros (in the complex $s$-plane) of their system function.

The practical importance of the Laplace transform for the analysis of continuous systems lies in the fact that it transforms linear differential equations with constant coefficients into algebraic equations, which are obviously much easier to solve. This is achieved by the fact that in the Laplace transform domain, integration is equivalent to multiplication by $1/s$, and differentiation becomes multiplication with $s$.

Any linear time-invariant (LTI) system, such as the RLC-circuit you mentioned, will result in linear differential equations with constant coefficients. This is why any LTI system can be analyzed by the Laplace transform. Note that in practice usually the unilateral transform is used (where the integral starts at $t=0$ instead of $t=-\infty$), because practical systems are causal, and because in this way initial conditions (such as charged capacitors at $t=0$) can be easily taken into account. The unilateral Laplace transform is thus an appropriate tool to solve initial value problems.

Everything said so far carries over to the $\mathcal{Z}$-transform if you replace 'continuous system' by 'discrete system', and 'differential equation' by 'difference equation'. Linear time-invariant discrete systems consisting of adders, multipliers and delay elements are treated by the $\mathcal{Z}$-transform in the same way as LTI electrical networks are treated by the Laplace transform. And indeed, IIR filters are analogous to RLC circuits in the sense that both have an infinitely long impulse response and that both can be characterized by the poles and zeros of their system functions.

Answer 2

the exponential input:

$$ x(t) = e^{s t} $$

is an eigenfunction to any Linear and Time-Invarient (LTI) system. that means if the input is $x(t)$ above, the output $y(t)$ is (for some constant $s$):

$$ y(t) = h(t) \circledast x(t) = h(t) \circledast e^{s t} = H(s) e^{s t} $$

("$\circledast$" means convolution.)

$H(s)$ is a constant because it's a function of $s$ which is constant. so the output $y(t)$ is a scaled version of the input $x(t)$, so that's what makes $x(t)$ an eigenfunction of the LTI system that is fully described by either the impulse response $h(t)$ or transfer function $H(s)$. the transfer function $H(s)$ is the Laplace Transform of the impulse response $h(t)$.

$s$ can be anything, real (like $s=\sigma$), imaginary ($s=j \omega$), or complex ($s=\sigma+j\omega$).

it turns out that, due to some theorem in complex analysis, that it suffices to know how the system (if it's "analytic") responds to $x(t)=e^{j\omega t}$ for all real $\omega$ to fully describe the input/output relationship of the LTI system.