- Signals and Systems Tutorial
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- Signals & Systems Overview
- Signals Basic Types
- Signals Classification
- Signals Basic Operations
- Systems Classification
- Signals Analysis
- Fourier Series
- Fourier Series Properties
- Fourier Series Types
- Fourier Transforms
- Fourier Transforms Properties
- Distortion Less Transmission
- Hilbert Transform
- Convolution and Correlation
- Signals Sampling Theorem
- Signals Sampling Techniques
- Laplace Transforms
- Laplace Transforms Properties
- Region of Convergence
- Z-Transforms (ZT)
- Z-Transforms Properties
- Signals and Systems Resources
- Signals and Systems - Resources
- Signals and Systems - Discussion
Signals Basic Types
Here are a few basic signals:
Unit Step Function
Unit step function is denoted by u(t). It is defined as u(t) = $\left\{\begin{matrix}1 & t \geqslant 0\\ 0 & t<0 \end{matrix}\right.$
- 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
Signum function is denoted as sgn(t). It is defined as sgn(t) = $ \left\{\begin{matrix}1 & t>0\\ 0 & t=0\\ -1 & t<0 \end{matrix}\right. $
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
$$sa(t) = {sin t \over t}$$
$$= 0 \,\, \text{for t} = \pm \pi,\, \pm 2 \pi,\, \pm 3 \pi \,... $$
