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Types of Signals – Rectangular, Triangular, Signum, Sinc and Gaussian Functions
A single valued function of one or more independent variables which contains some information is known as a signal.
Basic Types of Signals
There are several basic signals which play an important role in the study of signals and systems. These basic signals are the basic building blocks for the construction of more complex signals. These elementary signals are also called standard signals.
Rectangular Signal
A signal that produces a rectangular shaped pulse with a width of τ (where 𝜏 = 1 for the unit rectangular function) centred at 𝑡 = 0 is known as rectangular signal. The rectangular signal pulse also has a height of 1. Mathematically, the unit rectangular signal is defined as,
$$\mathrm{\prod \left ( \frac{t}{\tau} \right )=\left\{\begin{matrix} 1\: for\: \left | t \right |\leq \left ( \frac{\tau }{2} \right )\ 0\; otherwise\ \end{matrix}\right.}$$
The rectangular signal is also known as the unit pulse, gate function or normalised boxcar function. Also, the rectangular function is an even function of time. The graphical representation of a rectangular pulse signal is shown in Figure-1.
Triangular Signal
A function whose graph takes the shape of a triangle is known as triangular signal. The triangular signal is also known as hat function or tent function. Mathematically, the unit triangular pulse signal Δ(t/τ) is defined as,
$$\mathrm{\Delta\left ( \frac{t}{\tau} \right )=\left\{\begin{matrix} 1-\left ( \frac{2\left | t \right |}{\tau } \right )\; for\left | t \right |< \left ( \frac{\tau }{2} \right )\ 0\; for\left | t \right |> \left ( \frac{\tau}{2} \right )\ \end{matrix}\right.}$$
The triangular signal is also an even function of time. The graphical representation of a triangular signal is shown in Figure-2.
Signum Function
The Signum function or sign function is an odd mathematical function which extracts the sign of a real number. The signum function is represented as sgn. Mathematically, the unit signum function is defined as,
$$\mathrm{sgn(t)=\left\{\begin{matrix} 1\; for\: t> 0\ -1\; for\: t< 0\ \end{matrix}\right.}$$
The graphical representation of the unit signum function is shown in Figure-3.
The signum function can also be expressed in terms of the unit step function as follows −
𝑠𝑔𝑛(𝑡) = −1 + 2𝑢(𝑡)
Sinc Function
The sinc function is defined as,
$$\mathrm{sinc(t)=\frac{sint}{t}\: for -\infty< t<\infty}$$
From the definition, it is clear that the sinc function oscillates with a time period of 2π and decays with increasing time (t). The value of a sinc function is zero at 𝑡 = 𝑛𝜋, where 𝑛 = ±1, ±2, ±3 … The sinc function is also an even function of time.
The graphical representation of the unit sinc function is shown in Figure-4.
Gaussian Function
The Gaussian function is very useful in probability theory. The Gaussian function is defined as,
𝑔𝑎(𝑡) = 𝑒−𝑎𝑡2 for − ∞ < 𝑡 < ∞
The graphical representation of the unit Gaussian function is shown in Figure-5.
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