# Special Functions

#### Unit Step Function:

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

The constant $a$ shifts the unit function to the right. Somtimes $u(t)$ is also refered to as the function $\epsilon(t)$.

Special property: $\int_{-\infty}^{\infty}u(t-a)x(t)dt=\int_{a}^{\infty}x(t)dt$

#### Square Pulse Function:

$u(t)-u(t-a)$

The constant $a$ determines the duration of the pulse.

#### Cardinal Sine Function:

$sinc(t)=\frac{sin(\pi t)}{\pi t}$

#### Rectangular Function:

$rect(t/a)=\begin{cases}
1,&\lvert t/a\rvert<1/2\\
0,&\lvert t/a\rvert>1/2
\end{cases}$

#### Triangular Function:

$\lambda(t)=\begin{cases}
0,&\lvert t\rvert>1\\
1-t,&\lvert t\rvert<1
\end{cases}$

#### Ramp Function:

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

#### Dirac Delta Function:

The Dirac delta function can be represented heuristically:

$\delta(x)=\begin{cases}
0,&x\neq0\\
\infty,&x=0
\end{cases}$ such that the integral is always: $\int_{-\infty}^{\infty}\delta(x)dx=1$

# LODE n-th Order

**L**inear **O**rdinary **D**ifferential **E**quation n-th Order:

$$\sum_{i=0}^nf_i(x)y^{(i)}=g(x)$$

Where $\max i$ is the order of the differential equation. The *LODE* is called homogeneous if and only if $g(x)=0$. For the case $g(x)\neq0$, the *LODE* is called inhomogeneous. The function $g(x)$ is called disturbance function or forcing function.

The general solution of a *LODE* is $y=y_h+y_p$.

# Filters

#### 1. Order Lowpass: $\frac{dU_{Out}}{dt}+\frac{1}{\tau}U_{Out}=\frac{1}{\tau}U_{In}$
#### 1. Order Highpass: $\frac{dU_{Out}}{dt}+\frac{1}{\tau}U_{Out}=\frac{dU_{In}}{dt}$

Time constant $\tau$ depends on the circuitry values:
- RC & CR Circuit: $\tau=RC$
- RL & LR Circuit: $\tau=\frac{L}{R}$

#### 2. Order Lowpass: $\frac{d^2U_{Out}}{dt^2}+\omega^2U_{Out}=\omega^2U_{In}$
#### 2. Order Highpass: $\frac{d^2U_{Out}}{dt^2}+\omega^2U_{Out}=\frac{d^2U_{In}}{dt^2}$

Resonance Frequency $\omega$ depends on the circuitry values:
- LC & CL Circuit: $\omega^2=\frac{1}{LC}$

# Terms & Notations

Only true for *LTI* Systems:

| Function | Notation |
| ------------------ | ------------ |
| Input signal | $x(t)$ |
| Output signal | $y(t)$ |
| Impulse response | $h(t)$ |
| Frequency response | $H(j\omega)$ |
| Transfer function | $H(s)$ |

#### Impulse response:

The impulse response of a system is the system response (output signal) evaluated using the dirac delta function as input: $h(t)=y(t)\big|_{x(t)=\delta(t)}$

Knowing that $h(t)=\mathcal T\{\delta(t)\}$ leads to the general system response: $y(t)=x(t)\circledast h(t)$