EEE 244 Laplace Transform Lecture 2 Flashcards

Flashcards covering definitions, properties, common transform pairs, and examples of the Laplace Transform from EEE 244 Lecture 2.

How is the bilateral Laplace transform of a signal $x(t)$ defined?

$$X(s) = \mathcal{L}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-st} dt$$

What is the formula for the complex variable $s$ used in the Laplace transform?

$$s = \sigma + j\Omega$$

In the complex variable $s = \sigma + j\Omega$, what is $\sigma$?

$\sigma$ is the real part.

In the complex variable $s = \sigma + j\Omega$, what is $\Omega$?

$\Omega$ is the imaginary part.

What is the mathematical definition of the unilateral Laplace transform?

$$X(s) = \mathcal{L}\{x(t)\} = \int_{0^{+}}^{\infty} x(t) e^{-st} dt$$

Why is the unilateral Laplace transform important for causal systems?

It is used in the analysis of causal systems described by constant coefficient linear differential equations with initial conditions.

What is the main difference between the limits of integration for bilateral and unilateral Laplace transforms?

The bilateral transform uses limits $\int_{-\infty}^{\infty}$ while the unilateral transform uses limits $\int_{0^{+}}^{\infty}$, depending only on the right-sided signal.

What is the Linearity property of the Laplace transform?

$$\mathcal{L}\{a_1 x_1(t) + a_2 x_2(t)\} = a_1 X_1(s) + a_2 X_2(s)$$

What is the formula for the inverse Laplace transform?

$$f(t) = \frac{1}{2\pi j} \int_{\sigma-j\infty}^{\sigma+j\infty} F(s) e^{st} ds$$

What is the Laplace transform of the impulse function $\delta(t)$?

$$1$$

What is the Laplace transform of the step function $u(t)$?

$$\frac{1}{s}$$

What is the Laplace transform of $e^{-at}$?

$$\frac{1}{s+a}$$

What is the Laplace transform of $\sin(\omega_0 t)$?

$$\frac{\omega_0}{s^2 + \omega_0^2}$$

What is the Laplace transform of $\cos(\omega_0 t)$?

$$\frac{s}{s^2 + \omega_0^2}$$

What is the Laplace transform of $t^n$?

$$\frac{n!}{s^{n+1}}$$

What is the Laplace transform of the $k$-th derivative $f^{(k)}(t)$?

$$s^k F(s) - s^{k-1} f(0^-) - s^{k-2} f'(0^-) - \dots - f^{(k-1)}(0^-)$$

What is the Laplace transform of the integral $\int_{-\infty}^{t} f(\tau) d\tau$?

$$\frac{F(s)}{s} + \frac{1}{s} \int_{-\infty}^{0^{-}} f(t) dt$$

What is the Laplace transform of $e^{-at} \sin(\omega t)$?

$$\frac{\omega}{(s+a)^2 + \omega^2}$$

What is the Laplace transform of $e^{-at} \cos(\omega t)$?

$$\frac{s+a}{(s+a)^2 + \omega^2}$$

According to the property table, what is the Laplace transform of $\frac{1}{\omega} e^{-at} \sin(\omega t)$?

$$\frac{1}{(s+a)^2 + \omega^2}$$

What is the Laplace Transform of a unit step function $f(t) = u(t)$?

$$F(s) = \frac{1}{s}$$

Find the Laplace Transform of $f(t) = u(t) + 2e^{-3t} + 2\sin(2t)$.

$$F(s) = \frac{1}{s} + \frac{2}{s+3} + \frac{4}{s^2 + 4}$$

What is the Time Scaling property for the Laplace transform?

$$\mathcal{L}\{x(at)\} = \frac{1}{|a|} X(\frac{s}{a})$$

If the Laplace Transform of $f(t)$ is $F(s) = \frac{1}{s+1}$, what is the LT of $f(3t)$?

$$\frac{1}{s+3}$$

What is the definition of a unit step function $u(t)$?

$u(t) = 1$ for $t > 0$ and $u(t) = 0$ for $t < 0$.

What is the value of the phase angle $\phi$ defined in the complex transform table for sine/cosine pairs?

$$\phi = \tan^{-1} (\frac{\omega}{a})$$

What is the Laplace transform of $\frac{\omega_n^2}{s^2 + 2\zeta \omega_n s + \omega_n^2}$?

$$\frac{\omega_n}{\sqrt{1-\zeta^2}} e^{-\zeta \omega_n t} \sin(\omega_n \sqrt{1-\zeta^2} t)$$

In the Assignment 2 problems, what is the first differential equation to transform?

$$\frac{dx(t)}{dt} + 7x(t) = 5\cos(2t)$$

In the Assignment 2 problems, what is the second differential equation to transform?

$$\frac{d^2 x(t)}{dt^2} + 6 \frac{dx(t)}{dt} + 8x(t) = 5\sin(3t)$$

What is the submission date for Assignment 2?

$$02/06/2025$$

What is the Course Code and Title for this lecture?

EEE 244: Signals and Systems

Who is the lecturer for this Course Code EEE 244?

Dr. Ernest Smith Mawuli

What university is associated with this lecture document?

Akenten Appiah-Menka University of Skills Training and Entrepreneurial Development

What is the Laplace transform of the function $1 - e^{-at}$?

$$\frac{a}{s(s+a)}$$

What is the Laplace transform of $[(a - \alpha)^2 + \omega^2]^{1/2} e^{-\alpha t} \sin(\omega t + \phi)$?

$$\frac{\omega}{(s+\alpha)^2 + \omega^2}$$

What does the abbreviation ROC stand for in the context of Laplace transforms?

Region of Convergence

At what lower limit is the unilateral Laplace transform $X(s)$ evaluated?

$$t = 0^+$$

In the linearity property proof, what are $R_1$ and $R_2$?

They are the Regions of Convergence (ROCs) of the signals $x_1(t)$ and $x_2(t)$ respectively.

Which transform depends on the entire signal $x(t)$ for all time?

The bilateral Laplace transform.

What is the term for the Laplace transform defined by Eq. (4.1) using the integral from $-\infty$ to $\infty$?

Bilateral Laplace transform.