Signals Basic Types

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Here are a few basic signals:

Unit Step Function

• It is used as best test signal.
• Area under unit step function is unity.

Unit Impulse Function

Impulse function is denoted by δ(t). and it is defined as δ(t) = $\left\{\begin{matrix}1 & t = 0\\ 0 & t\neq 0 \end{matrix}\right.$

$$ \int_{-\infty}^{\infty} δ(t)dt=u (t)$$

$$ \delta(t) = {du(t) \over dt } $$

Ramp Signal

Ramp signal is denoted by r(t), and it is defined as r(t) = $\left\{\begin {matrix}t & t\geqslant 0\\ 0 & t < 0 \end{matrix}\right. $

$$ \int u(t) = \int 1 = t = r(t) $$

$$ u(t) = {dr(t) \over dt} $$

Area under unit ramp is unity.

Parabolic Signal

Parabolic signal can be defined as x(t) = $\left\{\begin{matrix} t^2/2 & t \geqslant 0\\ 0 & t < 0 \end{matrix}\right.$

$$\iint u(t)dt = \int r(t)dt = \int t dt = {t^2 \over 2} = parabolic signal $$

$$ \Rightarrow u(t) = {d^2x(t) \over dt^2} $$

$$ \Rightarrow r(t) = {dx(t) \over dt} $$

Signum Function

Exponential Signal

Exponential signal is in the form of x(t) = $e^{\alpha t}$.

The shape of exponential can be defined by $\alpha$.

Case i: if $\alpha$ = 0 $\to$ x(t) = $e^0$ = 1

Case ii: if $\alpha$ < 0 i.e. -ve then x(t) = $e^{-\alpha t}$. The shape is called decaying exponential.

Case iii: if $\alpha$ > 0 i.e. +ve then x(t) = $e^{\alpha t}$. The shape is called raising exponential.

Rectangular Signal

Let it be denoted as x(t) and it is defined as

Triangular Signal

Let it be denoted as x(t)

Sinusoidal Signal

Sinusoidal signal is in the form of x(t) = A cos(${w}_{0}\,\pm \phi$) or A sin(${w}_{0}\,\pm \phi$)

Where T₀ = $ 2\pi \over {w}_{0} $

Sinc Function

It is denoted as sinc(t) and it is defined as sinc

$$ (t) = {sin \pi t \over \pi t} $$

$$ = 0\, \text{for t} = \pm 1, \pm 2, \pm 3 ... $$

Sampling Function

It is denoted as sa(t) and it is defined as

$$sa(t) = {sin t \over t}$$

$$= 0 \,\, \text{for t} = \pm \pi,\, \pm 2 \pi,\, \pm 3 \pi \,... $$

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