# 1.5 Sinusoidal signals

 Page 1 / 1

## Sinusoidal signals

Sinusoidal signals are perhaps the most important type of signal that we will encounter in signal processing. There are two basic types of signals, the cosine :

$x\left(t\right)=Acos\left(\Omega t\right)$

and the sine :

$x\left(t\right)=Asin\left(\Omega t\right)$

where $A$ is a real constant. Plots of the sine and cosine signals are shown in [link] . Sinusoidal signals are periodic signals. The period of the cosine and sine signals shown above is given by $T=2\pi /\Omega$ . The frequency of the signals is $\Omega =2\pi /T$ which has units of rad/sec . Equivalently, the frequency can be expressed as $1/T$ , which has units of $se{c}^{-1}$ , cycles/sec , or Hz . The quantity $\Omega t$ has units of radians and is often called the phase of the sinusoid. Recalling the effect of a time shift on the appearance of a signal, we can observe from [link] that the sine signal is obtained by shifting the cosine signal by $T/4$ seconds, i.e.

$sin\left(\Omega t\right)=cos\left(\Omega \left(t-T/4\right)\right)$

and since $T=2\pi /\Omega$ , we have

$sin\left(\Omega t\right)=cos\left(\Omega t-\pi /2\right)\right)$

Similarly, we have

$cos\left(\Omega t\right)=sin\left(\Omega t+\pi /2\right)\right)$

Using Euler's Identity, we can also write:

$Acos\left(\Omega t\right)=\frac{A}{2}\left({e}^{j\Omega t},+,{e}^{-j\Omega t}\right)$

and

$Asin\left(\Omega t\right)=\frac{A}{2j}\left({e}^{j\Omega t},-,{e}^{-j\Omega t}\right)$

The quantity ${e}^{j\Omega t}$ is called a complex sinusoid and can be expressed as

${e}^{±j\Omega t}=cos\left(\Omega ,t\right)±jsin\left(\Omega ,t\right)$

There are a number of trigonometric identities which are sometimes useful. These are shown in [link] . [link] shows some basic calculus operations on sine and cosine signals.

 $sin\left(\theta \right)=cos\left(\theta -\pi /2\right)$ $cos\left(\theta \right)=sin\left(\theta +\pi /2\right)$ $sin\left({\theta }_{1}\right)sin\left({\theta }_{2}\right)=\frac{1}{2}\left[cos,\left({\theta }_{1}-{\theta }_{2}\right),-,cos,\left({\theta }_{1}+{\theta }_{2}\right)\right]$ $sin\left({\theta }_{1}\right)cos\left({\theta }_{2}\right)=\frac{1}{2}\left[sin,\left({\theta }_{1}-{\theta }_{2}\right),-,sin,\left({\theta }_{1}+{\theta }_{2}\right)\right]$ $cos\left({\theta }_{1}\right)cos\left({\theta }_{2}\right)=\frac{1}{2}\left[cos,\left({\theta }_{1}-{\theta }_{2}\right),+,cos,\left({\theta }_{1}+{\theta }_{2}\right)\right]$ $acos\left(\theta \right)+bsin\left(\theta \right)=\sqrt{{a}^{2}+{b}^{2}}cos\left(\theta ,-,{tan}^{-1},\left(\frac{b}{a}\right)\right)$ $cos\left({\theta }_{1}±{\theta }_{2}\right)=cos\left({\theta }_{1}\right)cos\left({\theta }_{2}\right)\mp sin\left({\theta }_{1}\right)sin\left({\theta }_{2}\right)$ $sin\left({\theta }_{1}±{\theta }_{2}\right)=sin\left({\theta }_{1}\right)cos\left({\theta }_{2}\right)±sin\left({\theta }_{1}\right)cos\left({\theta }_{2}\right)$
 $\frac{d}{dt}cos\left(\Omega t\right)=-\Omega sin\left(\Omega t\right)$ $\frac{d}{dt}sin\left(\Omega t\right)=\Omega cos\left(\Omega t\right)$ $\int cos\left(\Omega t\right)dt=\frac{1}{\Omega }sin\left(\Omega t\right)$ $\int sin\left(\Omega t\right)dt=-\frac{1}{\Omega }cos\left(\Omega t\right)$ ${\int }_{0}^{T}sin\left(k{\Omega }_{o}t\right)cos\left(n{\Omega }_{o}t\right)dt=0$ ${\int }_{0}^{T}sin\left(k{\Omega }_{o}t\right)sin\left(n{\Omega }_{o}t\right)dt=0,k\ne n$ ${\int }_{0}^{T}cos\left(k{\Omega }_{o}t\right)cos\left(n{\Omega }_{o}t\right)dt=0,k\ne n$ ${\int }_{0}^{T}{sin}^{2}\left(n{\Omega }_{o}t\right)dt=T/2$ ${\int }_{0}^{T}{cos}^{2}\left(n{\Omega }_{o}t\right)dt=T/2$

Now suppose that we have a sum of two sinusoids, say

$x\left(t\right)=cos\left({\Omega }_{1}t\right)+cos\left({\Omega }_{2}t\right)$

It is of interest to know what the period $T$ of the sum of 2 sinusoids is. We must have

$\begin{array}{ccc}\hfill x\left(t-T\right)& =& cos\left({\Omega }_{1}\left(t-T\right)\right)+cos\left({\Omega }_{2}\left(t-T\right)\right)\hfill \\ \hfill & =& cos\left({\Omega }_{1}t-{\Omega }_{1}T\right)+cos\left({\Omega }_{2}t-{\Omega }_{2}T\right)\hfill \\ \hfill \end{array}$

It follows that ${\Omega }_{1}T=2\pi k$ and ${\Omega }_{2}T=2\pi l$ , where $k$ and $l$ are integers. Solving these two equations for $T$ gives $T=2\pi k/{\Omega }_{1}=2\pi l/{\Omega }_{2}$ . We wish to select the shortest possible period, since any integer multiple of the period is also a period. To do this we note that since $2\pi k/{\Omega }_{1}=2\pi l/{\Omega }_{2}$ , we can write

$\frac{{\Omega }_{1}}{{\Omega }_{2}}=\frac{k}{l}$

so we seek the smallest integers $k$ and $l$ that satisfy [link] . This can be done by finding the greatest common divisor between $k$ and $l$ . For example if ${\Omega }_{1}=10\pi$ and ${\Omega }_{2}=15\pi$ , we have $k=2$ and $l=3$ , after dividing out 5, the greatest common divisor between 10 and 15. So the period is $T=2\pi k/{\Omega }_{1}=0.4$ sec. On the other hand, if ${\Omega }_{1}=10\pi$ and ${\Omega }_{2}=10.1\pi$ , we find that $k=100$ and $l=101$ and the period increases to $T=2\pi k/{\Omega }_{1}=20$ sec. Notice also that if the ratio of ${\Omega }_{1}$ and ${\Omega }_{2}$ is not a rational number, then $x\left(t\right)$ is not periodic!

If there are more than two sinusoids, it is probably easiest to find the period of one pair of sinusoids at a time, using the two lowest frequencies (which will have a longer period). Once the frequency of the first two sinusoids has been found, replace them with a single sinusoid at the composite frequency corresponding to the first two sinusoids and compare it with the third sinusoid, and so on.

what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!