# 2.1 Elemental signals

 Page 1 / 1
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.

## Sinusoids

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

$s(t)=A\cos (2\pi {f}_{0}t+\phi )$
For this signal, $A$ is its amplitude, ${f}_{0}$ its frequency, and $\phi$ its phase.

## Complex exponentials

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

$s(t)=Ae^{i(2\pi {f}_{0}t+\phi )}=Ae^{i\phi }e^{i\times 2\pi {f}_{0}t}$
Here, $i$ denotes $\sqrt{-1}$ . $Ae^{i\phi }$ 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.
$\cos (2\pi ft)=\frac{e^{i\times 2\pi ft}+e^{-(i\times 2\pi ft)}}{2}$
$\sin (2\pi ft)=\frac{e^{i\times 2\pi ft}-e^{-(i\times 2\pi ft)}}{2i}$
$e^{i\times 2\pi ft}=\cos (2\pi ft)+i\sin (2\pi ft)$
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.
$A\cos (2\pi ft+\phi )=\Re (Ae^{i\phi }e^{i\times 2\pi ft})$
$A\sin (2\pi ft+\phi )=\Im (Ae^{i\phi }e^{i\times 2\pi ft})$

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 $\phi$ . 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 $\frac{1}{f}$ . 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.

$s(t)=e^{-\left(\frac{t}{\tau }\right)}$

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

$s(t)=Ae^{i\phi }e^{-\left(\frac{t}{\tau }\right)}e^{i\times 2\pi ft}=Ae^{i\phi }e^{(-\left(\frac{1}{\tau }\right)+i\times 2\pi f)t}$
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

$u(t)=\begin{cases}0 & \text{if t< 0}\\ 1 & \text{if t> 0}\end{cases}$

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: $s(t)=A\sin (2\pi ft)u(t)$ .

## Pulse

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

${p}_{\Delta }(t)=\begin{cases}0 & \text{if t< 0}\\ 1 & \text{if 0< t< \Delta }\\ 0 & \text{if t> \Delta }\end{cases}$
We will find that this is the second most important signal in communications.

## Square wave

The square wave $\mathrm{sq}(t)$ 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.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good