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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) = $\left\{\begin{matrix}1 & t \geqslant 0\\ 0 & t<0 \end{matrix}\right.$
![Unit Step Function](/signals_and_systems/images/unit_step_function.png)
- 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) = $\left\{\begin{matrix}1 & t = 0\\ 0 & t\neq 0 \end{matrix}\right.$
![Unit Impulse Function](/signals_and_systems/images/unit_impulse_function.png)
$$ \int_{-\infty}^{\infty} δ(t)dt=u (t)$$
$$ \delta(t) = {du(t) \over dt } $$
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. $
![Ramp Signal](/signals_and_systems/images/ramp_signal.png)
$$ \int u(t) = \int 1 = t = r(t) $$
$$ u(t) = {dr(t) \over dt} $$
Area under unit ramp is unity.
Parabolic Signal
Parabolic signal can be defined as x(t) = $\left\{\begin{matrix} t^2/2 & t \geqslant 0\\ 0 & t < 0 \end{matrix}\right.$
![Parabolic Signal](/signals_and_systems/images/parabolic_signal.png)
$$\iint u(t)dt = \int r(t)dt = \int t dt = {t^2 \over 2} = parabolic signal $$
$$ \Rightarrow u(t) = {d^2x(t) \over dt^2} $$
$$ \Rightarrow r(t) = {dx(t) \over dt} $$
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. $
![Signum Function](/signals_and_systems/images/signum_function.png)
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
![Rectangular signal](/signals_and_systems/images/rectangular_signal.png)
Triangular Signal
Let it be denoted as x(t)
![Triangular signal](/signals_and_systems/images/triangular_signal.png)
Sinusoidal Signal
Sinusoidal signal is in the form of x(t) = A cos(${w}_{0}\,\pm \phi$) or A sin(${w}_{0}\,\pm \phi$)
![Sinusoidal signal](/signals_and_systems/images/sinusoidal_signal.png)
Where T0 = $ 2\pi \over {w}_{0} $
Sinc Function
It is denoted as sinc(t) and it is defined as sinc
$$ (t) = {sin \pi t \over \pi t} $$
$$ = 0\, \text{for t} = \pm 1, \pm 2, \pm 3 ... $$
![Sinc Function](/signals_and_systems/images/sinc_function.png)
Sampling Function
It is denoted as sa(t) and it is defined as
$$sa(t) = {sin t \over t}$$
$$= 0 \,\, \text{for t} = \pm \pi,\, \pm 2 \pi,\, \pm 3 \pi \,... $$
![Sampling Function](/signals_and_systems/images/sampling_function.png)