Part 9 - Laplace transform
Introduction
In this part we’ll cover the Laplace transform and how we’ll use it.
Before that, let’s understand why we’ll use it.
What the Laplace transform will allow us to solve are linear differential equations with constant coefficients.
From what we’ve learned so far, we have no way to synthesize exponentially growing signals.
Therefore: $$ e^{j\omega t} \rarr e^{st} $$
Where $s = \sigma + j\omega$.
Definition
The Laplace transformation is a unilateral transform, which just means it is one-sided.
Given a function that is defined for $t \geq 0$ and it is locally integrable, meaning that its integral exists in every finite interval of $[0, \infty)$
$$ F(s) = \int_0^{\infty} f(t)e^{-st}\ dt \\ F(s) = \mathcal{L}[f(t)] $$
Since this is a complex integral, whether we get a complex answer or not depends on $s$.
We call the set of values that $s$ can take, such that the Laplace transform yields a finite number, for the Region of Convergence (Roc).
We can describe the RoC in terms of a vertical line on the complex plane. This is given by the real part of, $s$. We call this for $\sigma_0$. This line is called the abscissa of convergence. So we usually want to have $\sigma > \sigma_0$.
Of course there exists an inverse transform: $$ f(t) = \dfrac{1}{2\pi j} \int_{c - j\infty}^{c + j\infty} F(s)e^{st}\ ds $$
Since we won’t cover integration for complex functions, we’ll always use a table for this.
Properties
Just like any operations, we have some properties. Let’s see what the Laplace transform has in for us.
Linearity
$$ f_1(t) \iff F_1(s) \\ f_2(t) \iff F_2(s) \\ $$
$$ a_1 f_1(t) + a_2 f_2(t) \iff a_1 F_1(s) + a_2 F_2(s) $$
For example, if we want to find the inverse Laplace transform for: $$ F(s) = \dfrac{7s - 6}{s^2 - s - 6} $$
We will have to split up this fraction.
Time shifting
$$ f(t) \iff F(s) $$
$$ f(t - t_0) \iff F(s)e^{-st_0} \ | \ t_0 \geq 0 $$
$$ f(t)u(t) \iff F(s) $$
$$ f(t - t_0)u(t - t_0) \iff F(s)e^{-st_0} \ | \ t_0 \geq 0 $$
Frequency shifting
$$ f(t) \iff F(s) $$
$$ f(t)e^{s_0 t} \iff F(s - s_0) $$
Time-differentiation
$$ f(t) \iff F(s) $$
$$ \dfrac{df}{dt} \iff sF(s) - f(0^-) $$
This is also repeating: $$ \dfrac{d^n f}{dt^n} \iff s^n F(s) - s^{n - 1} f(0^-) - s^{n - 2} f^\prime(0^-) - \ldots - f^{n - 1}(0^-) $$
Where $f^{i}(0^-)$ is $\dfrac{d^{i} f}{d t^{i}}$ at $t = 0^-$ $$ tf(t) \iff -\dfrac{d}{ds} F(s) $$
Time integration
$$ f(t) \iff F(s) $$
$$ \int_{0^-}^t f(\tau)\ d\tau \iff \dfrac{F(s)}{s} $$
$$ \int_{-\infty}^t f(\tau)\ d\tau \iff \dfrac{F(s)}{s} + \dfrac{\int_{-\infty}^{0^-} f(\tau)\ d\tau}{s} $$
$$ \dfrac{f(t}{t} \iff \int_s^{\infty} F(z)\ dz $$
Scaling
$$ f(t) \iff F(s) $$
$$ f(at) \iff \dfrac{1}{a} F\left(\dfrac{s}{a}\right) \ | \ a > 0 $$
Convolution
$$ f_1(t) \iff F_1(s) \\ f_2(t) \iff F_2(s) \\ $$
Time convolution: $$ f_1(t) * f_2(t) = F_1(s) F_2(s) $$
Frequency convolution $$ f_1(t) f_2(t) = \dfrac{1}{2\pi j} \left[F_1(s) * F_2(s)\right] $$