Laplace Transform Table

1. Unit Step Function $u(t)$

The Unit Step Function models a sudden "turn ON" switch at time $t = 0$.

$$u(t) = \begin{cases} 0, & t < 0 \\ 1, & t > 0 \end{cases}$$

(At $t=0$, $u(0)$ is often defined as $1/2$ or left undefined).

Key Properties:

• Time Shift $u(t - a)$: Turns ON at time $t = a$ instead of $t = 0$.
• Windowing: Multiplying a function $f(t)$ by $u(t - a)$ zeroes out the signal before $t = a$.
• Laplace Transform:

$$\mathcal{L}\{u(t)\} = \frac{1}{s}$$ $$\mathcal{L}\{u(t - c)\} = \frac{e^{-cs}}{s}$$

2. Dirac Delta Function $\delta(t)$

The Dirac Delta Function is a generalized function (or distribution) representing an infinitely tall, infinitely narrow impulse at $t = 0$ with total area equal to 1.

\infty, & t = 0 \end{cases} \quad \text{such that} \quad \int_{-\infty}^{\infty} \delta(t) \, dt = 1$$ ![[dirac_delta_func.png|579]] ### Key Properties: - **Sifting Property (Sampling):** Integrating $\delta(t - a)$ against a continuous signal $f(t)$ extracts the single value $f(a)$:

\int_{-\infty}^{\infty} f(t) \delta(t - a) , dt = f(a)

- **Scaling Property:** $\delta(at) = \frac{1}{\vert{}a\vert{}} \delta(t)$ - **Laplace Transform:**

\mathcal{L}{\delta(t)} = 1

$$$$

\mathcal{L}{\delta(t - a)} = e^{-as}

## The Fundamental Relationship Between $\delta(t)$ and $u(t)$ The Dirac Delta Function is the **generalized derivative** of the Unit Step Function:

\frac{d}{dt} u(t) = \delta(t)

$$Conversely, the Unit Step Function is the **running integral** of the Dirac Delta Function:$$

u(t) = \int_{-\infty}^{t} \delta(\tau) , d\tau

### Physical Intuition: - **Unit Step $u(t)$:** Represents a continuous force or voltage that suddenly steps up (e.g., throwing a switch to 1V). - **Dirac Delta $\delta(t)$:** Represents an instantaneous force spike or impulse (e.g., striking a baseball with a bat or a instantaneous voltage surge).