Special Functions

Unit Step Function:

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

The constant

$a$ shifts the unit function to the right. Somtimes

$u(t)$ is also refered to as the function

$ϵ(t)$.

Special property:

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}u(t-a)x(t)dt={\int }_a^{\mathrm{\infty }}x(t)dt$

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:

$rect(t/a)=\{\begin{array}{ll}1,& |t/a|<1/2\\ 0,& |t/a|>1/2\end{array}$

Triangular Function:

$\lambda (t)=\{\begin{array}{ll}0,& |t|>1\\ 1-t,& |t|<1\end{array}$

Ramp Function:

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

Dirac Delta Function:

The Dirac delta function can be represented heuristically:

$\delta (x)=\{\begin{array}{ll}0,& x\ne 0\\ \mathrm{\infty },& x=0\end{array}$ such that the integral is always:

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)dx=1$

LODE n-th Order

Linear Ordinary Differential Equation n-th Order:

$$\sum _{i=0}^n{f}_i(x)y^{(i)}=g(x)$$

Where

$maxi$ is the order of the differential equation. The LODE is called homogeneous if and only if

$g(x)=0$. For the case

$g(x)\ne 0$, 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:

$\frac{d{U}_{Out}}{dt}+\frac{1}{\tau }{U}_{Out}=\frac{1}{\tau }{U}_{In}$

1. Order Highpass:

$\frac{d{U}_{Out}}{dt}+\frac{1}{\tau }{U}_{Out}=\frac{d{U}_{In}}{dt}$

Time constant

$\tau$ depends on the circuitry values:

• RC & CR Circuit:
  $\tau =RC$
• RL & LR Circuit:
  $\tau =\frac{L}{R}$

2. Order Lowpass:

$\frac{d^2{U}_{Out}}{d{t}^2}+{\omega }^2{U}_{Out}={\omega }^2{U}_{In}$

2. Order Highpass:

$\frac{d^2{U}_{Out}}{d{t}^2}+{\omega }^2{U}_{Out}=\frac{d^2{U}_{In}}{d{t}^2}$

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:

$h(t)=y(t){|}_{x(t)=\delta (t)}$

Knowing that

$h(t)=\mathcal{T}\{\delta (t)\}$ leads to the general system response:

$y(t)=x(t)⊛h(t)$