What are Elementary Signals?

By Aina Parasher|Updated : August 26th, 2022

What are Elementary Signals? The signal is a detectable quantity that contains specified information in it. In our perspective i.e., in engineering, it is a description of how one parameter is varying with the change in another parameter.

Further, we have provided the basic information regarding elementary signals and the types of elementary signals in the upcoming sections. In this article, we will discuss the elementary continuous-time signals and their mathematical expressions in brief.

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What are Elementary Signals?

Before discussing the elementary signals, we must know their significance. In any engineering application, if we want to analyze the characteristics of any system that can be a process or a physical device, we must test or analyze it with a certain set of inputs. But every time it is not possible to apply these directly to that physical device. Hence, we model the physical system into a suitable mathematical model and analyze it with a set of signals that replicate the character of the input we want to apply to this system and record its response by using various mathematical tools such as Fourier transforms, Laplace transforms, eigenfunctions, etc.

But how do we model these signals? It is not possible to model every random input, but certain sets can replicate most inputs we want to apply to the system. For example, if the nature of your input is a sudden blow in a short duration, we can model this as an impulse signal; if it is a sudden input with a finite magnitude that longs for an infinite(long) duration we can model as a step signal; if it is oscillatory, then we can model it as a sinusoidal signal. Like this, some signals those found extensive use in our applications such as

  • Sinusoidal signal
  • Exponential signal
  • Impulse signal
  • Unit step signal
  • Ramp signal
  • Parabolic signal

Sinusoidal Signal

The sinusoidal signal has found a versatile application in the fields like mathematics, physics, signal processing, engineering, etc, the reason behind its extensive application is that we can relate this to so many wave patterns that occur in nature like sound, wind, and the light wave. In addition to these, we can relate every physical oscillation with a sinusoidal signal. Most importantly it has a unique property that is it retains its shape when we add it with another sinusoid of the same frequency and arbitrary phase and magnitude, this property leads to the invention of the Fourier series.

its mathematical expression is


Where A is amplitude andΦ is phase difference.

Sinusoidal Signal

Exponential Signal

The mathematical expression for the exponential signal is given by x(t)=eαt

The shape of this signal depends on the value of α.

If α=0 then x(t) = e0 = 1

Exponential Signal

If α is less than zero, then x(t)= e-αt, and the shape is decaying exponential.

Exponential Signal 2

Ifα is greater than zero then x(t)= eαt, the shape is called raising exponentially.


Impulse Signal

If we want to model the input of an ample magnitude that disappears in no time (very short duration) we must use the impulse signal. The ideal impulse signal is zero everywhere but infinitely high at the instant zero. But the area of the ideal impulse signal is a finite value only.

-∞δ(t)dt= 1

The unit impulse signal finds extensive use in the analysis of signals and systems.

The mathematical representation of continuous-time unit impulse signal is given by:

δ(t) = 1; for t = 0

andδ(t) = 0 ; for t ≠ 0


Unit Step Signal

If we want to model any signal that changes its magnitude suddenly and remains with the same magnitude for an infinite period. The mathematical expression of the unit step signal is given below.

u(t) = 1; for t≥ 0

and u(t) = 0 ; for t < 0


Ramp Signal

Starting at the instant t=0, the amplitude of the continuous-time ramp signal increases linearly with time. The mathematical expression of a continuous-time ramp signal is

r(t) = 1; for t≥ 0

and r(t) = 0 ; for t < 0

Ramp signal can also be expressed as r(t)= tu(t).


Parabolic Signal

The mathematical representation of the continuous-time parabolic signal is

x(t) = t2/2; for t≥ 0

and x(t) = 0 ; for t < 0


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FAQs on Elementary Signals

  • There are some elementary signals those feature regularly in the analysis of signals and systems. They are namely, impulse signal, unit step signal, sinusoidal signal, exponential signal, ramp signal, and parabolic signal. Along with these, we use some signals derived from these basic signals like sinc.

  • If the signal’s value is defined for every instant of time in a considered interval, however small the interval is, it is called a continuous-time signal. A discrete-time signal is derived from a continuous-time signal by a procedure called uniform sampling.

  • The basic expression of the sinusoidal signal is  x(t) = Asin(ωt+Φ)

    Where A is amplitude and Φ is phase difference. This is one of the signals that are used in the analysis of various fields like physics, mathematics, and engineering due to its unique property that retains its shape when we add another sinusoidal signal of the same frequency, this property leads to an important development in signal processing which is Fourier series analysis. Apart from this, we can also relate various phenomena that occur in nature like sound, wind, and light with the sinusoidal wave.

  • The basic expression of an impulse signal is 

    δ(t) = ; for t = 0 and

    δ(t) = 0 ; for t ≠ 0

    However, the area under the impulse signal is finite, which is equal to unity or 1. 

    ∫δ(t)dt= 1 

    The unit impulse signal is expressed as 

    δ(t) = 1; for t = 0 and

    δ(t) = 0 ; for t ≠ 0

    This is one of the most widely used standard signals in the analysis of signals and systems. We can model sudden and short-sustained inputs on any system with the impulse signal. 

  • The expression for the unit-step signal is given by 

    u(t)= 1; for t 0 and

    u(t) = 0 ; for t < 0

    If we need to obtain only the right-side portion of any continuous-time signal, then we can multiply that signal with a unit step signal. We can model a sudden input that lasts for a very long time with a unit-step signal. 

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