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Maths

Laplace Transform

The following chapter shows fundamentals of the Laplace transform. The code used can be found here.

In [2]:
% run ./maths_functions.py

Definition

A Laplace transform of a function is denoted as $\mathcal{L}[f(t)]$ or $\bar{f}$. It is a transformation of a real function of real variable $t$ to a complex function of a complex variable $s$.

In the general case the transform has no limits, however in this chapter it will be assumed that it is causal and is applied for $t>=0$. This makes it useful for transformations from time-domain to frequency-domain. The transformation is computed with the following formula:

$\mathcal{L}[f(t)]=\int_{0}^{\infty}f(t).e^{-s.t}dt$

However that can get complicated, usually a list of common laplace transforms is used to do transformations. Such a table can be found here.

Properties

The transformation is linear:

$\mathcal{L}[f+g] = \mathcal{L}[f] + \mathcal{L}[g]$

$\mathcal{L}[k.f(t)] = k.\mathcal{L}[f(t)]$

and it is also invertible:

$\mathcal{L}^{-1}[\bar{f}(s)]=f(t)$

Time Derivatives and Integrals

Laplace transforms are particularly effective when transforming derivatives and ODE's.

Normal: $\mathcal{L}[f] = \bar{f}$

First Order: $\mathcal{L}[f'] = s.\bar{f} - f(0)$

Second Order: $\mathcal{L}[f''] = s^2.\bar{f} - f'(0) - s.f(0)$

An example first order DE takes the form $ay' + by = c$. The laplace tranform is then:

$a.(s\bar{y}+y(0)) +b\bar{y} = (b+as)\bar{y}+y(0) = \bar{c}$

After $\bar{y}$ is computed, inverse laplace transform is used to compute y.

Transforms of function integral also show an interesting connection:

$\mathcal{L}[\int_0^t f(\tau) d\tau] = \cfrac{\mathcal{L} [f(\tau)]}{s}$

Time-shifting Theorems

Since laplace transform is used for time- and frequency- operations one very useful transformation is one of the delayed signal $f(t-t_0)$ or frequency shifted transform $\bar{f}(s-s_0)$. The relationship can be shown with the following formulas where H is used to represent the Heavyside step-function.

$\mathcal{L}[f(t)] = \bar{f}(s)$

Time-shifting:

$\mathcal{L}[f(t-t_0).H(t-t_0)] = \bar{f}(s).e^{-st}$

$\mathcal{L^{-1}}[\bar{f}(s).e^{-st}] = f(t-t_0).H(t-t_0)$

Frequency-shifting:

$\mathcal{L}[f(t).e^{s_0.t}] = \bar{f}(s-s_0)$

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