<< Chapter < Page Chapter >> Page >
Complex signals can be built from elemental signals, including the complex exponential, unit step, pulse, etc. This module presents the elemental signalsin brief.

Elemental signals are the building blocks with which we build complicated signals. By definition, elemental signals have a simple structure. Exactly what wemean by the "structure of a signal" will unfold in this section of the course. Signals are nothing more thanfunctions defined with respect to some independent variable, which we take to be time for the most part. Very interestingsignals are not functions solely of time; one great example of which is an image. For it, the independent variables are x and y (two-dimensional space). Video signals are functions of three variables: two spatialdimensions and time. Fortunately, most of the ideas underlying modern signal theory can be exemplified with one-dimensional signals.


Perhaps the most common real-valued signal is the sinusoid.

s t A 2 f 0 t φ
For this signal, A is its amplitude, f 0 its frequency, and φ its phase.

Complex exponentials

The most important signal is complex-valued, the complex exponential.

s t A 2 f 0 t φ A φ 2 f 0 t
Here, denotes -1 . A φ is known as the signal's complex amplitude . Considering the complex amplitude as a complex numberin polar form, its magnitude is the amplitude A and its angle the signal phase. The complex amplitude is also known as a phasor . The complex exponential cannot be further decomposed into more elemental signals, and is the most important signal in electrical engineering ! Mathematical manipulations at first appear to be more difficult because complex-valued numbers areintroduced. In fact, early in the twentieth century, mathematicians thought engineers would not be sufficientlysophisticated to handle complex exponentials even though they greatly simplified solving circuit problems. Steinmetz introduced complex exponentials to electrical engineering, and demonstrated that "mere" engineers could use them to goodeffect and even obtain right answers! See Complex Numbers for a review of complex numbers and complex arithmetic.

The complex exponential defines the notion of frequency: it is the only signal that contains only one frequency component. The sinusoid consists of two frequencycomponents: one at the frequency f 0 and the other at f 0 .

This decomposition of the sinusoid can be traced to Euler's relation.
2 f t 2 f t 2 f t 2
2 f t 2 f t 2 f t 2
2 f t 2 f t 2 f t
The complex exponential signal can thus be written in terms of its real and imaginary parts using Euler's relation. Thus,sinusoidal signals can be expressed as either the real or the imaginary part of a complex exponential signal, the choicedepending on whether cosine or sine phase is needed, or as the sum of two complex exponentials. These two decompositions aremathematically equivalent to each other.
A 2 f t φ A φ 2 f t
A 2 f t φ A φ 2 f t

Graphically, the complex exponential scribes a circle in the complex plane as time evolves. Its real and imaginary partsare sinusoids. The rate at which the signal goes around the circle is the frequency f and the time taken to go around is the period T . A fundamental relationship is T 1 f .

Using the complex plane, we can envision the complex exponential's temporal variations as seen in the above figure( [link] ). The magnitude of the complex exponential is A , and the initial value of the complex exponential at t 0 has an angle of φ . As time increases, the locus of points traced by the complexexponential is a circle (it has constant magnitude of A ). The number of times per second we go around the circle equals the frequency f . The time taken for the complex exponential to go around the circle once is known asits period T , and equals 1 f . The projections onto the real and imaginary axes of the rotating vector representing the complex exponentialsignal are the cosine and sine signal of Euler's relation ( [link] ).

Real exponentials

As opposed to complex exponentials which oscillate, real exponentials decay.

s t t τ

The real exponential.

The quantity τ is known as the exponential's time constant , and corresponds to the time required for the exponential to decrease by afactor of 1 , which approximately equals 0.368 . A decaying complex exponential is the product of a real and a complex exponential.

s t A φ t τ 2 f t A φ 1 τ 2 f t
In the complex plane, this signal corresponds to an exponential spiral. For such signals, we can define complex frequency as the quantity multiplying t .

Unit step

The unit step function is denoted by u t , and is defined to be

u t 0 t 0 1 t 0

The unit step.

This signal is discontinuous at the origin. Its value at the origin need not be defined, and doesn't matter in signaltheory.
This kind of signal is used to describe signals that "turn on" suddenly. For example, tomathematically represent turning on an oscillator, we can write it as the product of a sinusoid and a step: s t A 2 f t u t .


The unit pulse describes turning a unit-amplitude signal on for a duration of Δ seconds, then turning it off.

p Δ t 0 t 0 1 0 t Δ 0 t Δ
The pulse.
We will find that this is the second most important signal in communications.

Square wave

The square wave sq t is a periodic signal like the sinusoid. It too has an amplitude and a period, which must be specified tocharacterize the signal. We find subsequently that the sine wave is a simpler signal than the square wave.

The square wave.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Answers please
Nikki Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of electrical engineering i' conversation and receive update notifications?