What exactly is Laplace transform?

I've been working on Laplace transform for a while. I can carry it out on calculation and it's amazingly helpful. But I don't understand what exactly is it and how it works. I google and found out that it gives "less familiar" frequency view.

My question is how does Laplace Transform give frequency view?

I don't understand the connection between $f(t)$ and $\mathscr{L} (f(t))$. For example:- let $f(t) = t$, $\mathscr{L}(t)={1 \over s^2}$

$f(t) $ gives time view but how does $1 \over s^2$ give the frequency view? Somebody help me to understand what exactly is it. Thank you!!

Can anyone explain it in some physical phenomenon? Like harmonic oscillator? $$ \ddot {x} + \omega_n x = f(t)$$

Solution 1:

The Laplace transform is a useful tool for dealing with linear systems described by ODEs. As mentioned in another answer, the Laplace transform is defined for a larger class of functions than the related Fourier transform.

The 'big deal' is that the differential operator ('$\frac{d}{dt}$' or '$\frac{d}{dx}$') is converted into multiplication by '$s$', so differential equations become algebraic equations. In other words, convolution in the time or space-domain becomes multiplication in the s-domain. Another, often unspoken, 'big deal' is that the transform is unique in some sense (eg, if the transforms of two continuous functions agree, then the functions agree in the original domain). So if you can solve the problem in the s-domain, then you have solved it, in some sense, in the original domain. There is a formula for inversion, although tables are typically used for inversion. However, the inversion formula shows how the poles of the transformed functions manifest themselves in the time or space domain.

The Laplace transform comes in a few varieties; for engineering applications the most usual is the unilateral transform (behavior for $t<0$ is not relevant). Fourier transforms are often used to solve boundary value problems, Laplace transforms are often used to solve initial condition problems. Also, the Laplace transform succinctly captures input/output behavior or systems described by linear ODEs.

Regarding the 'frequency view'; instead of thinking of frequency as the $\omega$ in $\sin \omega t$, think of it as a collection of points in $\mathbb{C}$ that characterizes the behavior of $\hat{f} = \mathscr{L} f$. For example, look at the Laplace transform of $f(t) = e^{\alpha t}$, which is $\hat{f}(s) = \frac{1}{s - \alpha}$. The single point $\alpha$ (which may be complex) completely characterizes the time domain behavior. More generally, the poles and zeros of $\hat{f}$ characterize the time domain behavior of $f$. Very loosely speaking, if $\hat{f}$ has poles $p_1,...,p_n$, then we expect $f$ to have time-domain 'behaviors' of the forms $e^{p_1 t},...,e^{p_n t}$ (the zeros, and pole multiplicities of $\hat{f}$ complicate this simplistic viewpoint somewhat). So, think of the frequencies (ie, poles & zeros) as characterizing the structure of $\hat{f}$.

In your question, I think you meant the system $\ddot {x} + \omega_n^2 x = f(t)$. The unilateral transform gives $$s^2 \hat{x}(s) -s x(0) - x'(0)+ \omega_n^2 \hat{x}(s) = \hat{f}(s),$$ where $\hat{x}, \hat{f}$ are the Laplace transforms of $x,f$ respectively. This equation is typically written in the following form, which shows the relationship between the input $\hat{f}$, the (time) initial conditions $x(0), x'(0)$, and the output $\hat{x}$: $$\hat{x}(s) = \frac{s x(0) + x'(0)}{s^2 + \omega_n^2} + \frac{\hat{f}(s)}{s^2 + \omega_n^2}.$$ We can see that the $\frac{1}{s^2 + \omega_n^2}$ term 'contributes' two poles (at $s = \pm i \omega_n$) lying on the imaginary axis to $\hat{x}$. So, we expect (at least) behaviors involving $t \mapsto \sin \omega_n t$ and $t \mapsto \cos \omega_n t$.

If we take $f = 0$, you can see (meaning look up a table of transforms) that the initial conditions translate into a time function of the form $x(t) = x(0) \cos \omega_n t + \frac{x'(0)}{\omega_n} \sin \omega_n t$. So, in this particular problem, the initial conditions 'remain' forever.

If we take the system to be at rest initially (ie, take the initial conditions to be zero), then we need to know $\hat{f}$ in order to compute $\hat{x}$. If we take $f(t) = e^{i \omega t}$ (admittedly not real, but easier to compute), we have $\hat{f}(s) = \frac{1}{s-i\omega}$, which gives $\hat{x}(s) = \frac{1}{(s-i\omega)(s^2 + \omega_n^2)}$. If we take $w \neq w_n$, then using a partial fraction expansion we can write $\hat{x}(s) = \frac{1}{\omega_n^2 - \omega^2}(\frac{1}{s-i\omega} - \frac{s+i \omega}{s^2 + \omega_n^2})$, which gives $x(t) = \frac{1}{\omega_n^2 - \omega^2}(e^{i \omega t} - \cos \omega_n t - i \frac{\omega}{\omega_n} \sin \omega_n t)$. If $w = w_n$, then we obtain $\hat{x}(s) = \frac{i}{2 \omega_n}(\frac{1}{s^2+\omega_n^2} - \frac{1}{(s-i\omega_n)^2} ) $, which corresponds to $x(t) = \frac{1}{2 w_n^2} \sin \omega_n t - \frac{i}{2 \omega_n} t e^{i \omega_n t}$. Notice the response in this case is unbounded, even though the input is bounded. The '$t$' term arises because of the pole of multiplicity 2 at $s = i \omega_n$.

Solution 2:

What is your background? Are you a Mathematics major, or a Physics / Engineering major?

The purpose of the Laplace Transform is to transform ordinary differential equations (ODEs) into algebraic equations, which makes it easier to solve ODEs. However, the Laplace Transform gives one more than that: it also does provide qualitative information on the solution of the ODEs (the prime example is the famous final value theorem).

Do note that not all functions have a Fourier Transform. The Laplace Transform is a generalized Fourier Transform, since it allows one to obtain transforms of functions that have no Fourier Transforms. Does your function $f (t)$ grow exponentially with time? Then it has no FT. No problem, just multiply it by a decaying exponential that decays faster than $f$ grows, and you now have a function that has a Fourier Transform! The FT of that new function is the LT (evaluated on a line on the complex plane parallel to the imaginary axis).

If you are an engineering student who first encountered Laplace Transforms in your Signals & Systems class, then think about the name "signals & systems". Linear time-invariant (LTI) systems can be fully described by an impulse response, say $h (t)$. Laplace-transform the impulse response, and you obtain the transfer function, $H(s)$. What is the point? The point is that exponential functions (including complex exponential functions and, thus, sines and cosines) have simple Laplace Transforms. Thus, you can take a signal $x (t)$, and obtain its Laplace transform $X (s)$. What is $X (s)$? It is the transfer function of the LTI system whose impulse response is $x (t)$ itself! You have an LTI system that serves as a "signal generator", so to speak. Do you want to know how an LTI system responds to a sinusoid? Laplace-transform the sinusoid, Laplace-transform the system's impulse response, multiply the two (which corresponds to cascading the "signal generator" with the given system), and compute the inverse Laplace Transform to obtain the response. To summarize: the Laplace Transform allows one to view signals as the LTI systems that can generate them.

What is the Laplace Transform? Quoting Tim Gowers: "a mathematical object is what it does" ;-)