# 13.1 An introduction to the analysis of brain waves  (Page 2/8)

 Page 2 / 8

## 1.

Frequency measures how often a wave passes. We can make a wave with frequency $\omega$ by writing:

$sin\left(2,\pi ,\omega ,t\right)$

Aside : The wave described by $sin\left(t\right)$ has frequency $1/2\pi$ . If we instead write $sin\left(2\pi t\right)$ , we will have a wave with frequency 1, which is easier to work with.

We can express the same information in terms of wavelength. Wavelength is how close neighboring waves are to each other. It is inversely proportional to frequency, which means that the higher the frequency, the smaller the wavelength. If $\ell$ is wavelength, we write this in the following way:

$\ell =\frac{1}{\omega }$

A wave with wavelength $\ell$ is written as $sin\left(\frac{2\pi t}{\ell }\right)$ .

## 2.

Amplitude measures how high the wave is. We can make a wave of amplitude $a$ by writing:

$a·sin\left(2\pi t\right)$

Sometimes $a$ will be negative. In this case, we still say that the wave has amplitude $a$ , but note that the function will be flipped across the x-axis.

## 3.

Phase describes how far the wave has been shifted from center. To create a wave with phase $p$ , we write:

$sin\left(2\pi \omega t+p\right)$

Since a sin wave repeats every $2\pi$ , the following is always true (that is, for any $\varphi$ ):

$sin\left(\varphi \right)=sin\left(\varphi +2\pi \right)$

Aside : A cosine wave is a sine wave shifted back by a $\pi /2$ , a quarter of the standard wave:

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

Every sine (cosine) wave can be described completely by these characteristics. These are shown in [link] (phase $=0$ for simplicity):

To code [link] in MATLAB, use:

>>t = 0:.01:1;>>amp = 3;>>freq = 1;>>phase = 0>>y = amp*sin(freq*2*pi*t + phase);

## Hearing sine waves

The sine wave represents a pure tone. To hear one, we use the MATLAB function sound() , which converts a vector into sound. Find the frequency that is specified, and compare to human range of hearing. Should we be able to hear this sound?

>>freq = 1000;>>samp_rate = 1e4;>>duration = 1;>>samples = 0 : (1/samp_rate) : duration;>>sound_wave = sin(2 * pi * samples * freq);>>sound(sound_wave, samp_rate);

Enter the above code into Matlab to hear the sound.

## Adding sine waves

We can add together multiple sin waves to accomplish different shapes. For example, if we add the wave $\left[4·sin\left(2\pi t\right)\right]$ and the wave $\left[sin\left(6·\left(2\pi t\right)\right)\right]$ we get:

Look at the figure and try to identify the effect of each wave. The first wave has frequency 1 and amplitude 4. This accounts for the large up-and-down motion that only goes through one cycle in the figure. The second wave has frequency 6 (wavelength $\frac{1}{6}$ ) and amplitude 1. This accounts for the small wiggles that happen many times in the figure.

Figure 4 is implemented in MATLAB with the following code:

>t = 0:.01:1;>>y = 4*sin(2*pi*t)+sin(6*2*pi*t);>>plot(t,y)

A wave of any shape can be expressed as a sum of sin and cosine waves, although it may take infinitely many. In the end of this module we will find interesting uses for the this fact.

## 1.1

[link] two shows a sum of two sine waves (with phase 0). Try to replicate it, and report what the amplitudes and frequencies of each wave is. The compound wave for exercise 1.1

## 1.2

Use the identities $cos\left(\varphi \right)=sin\left(\varphi +\pi /2\right)$ and $sin\left(\varphi \right)=sin\left(\varphi +2\pi \right)$ to solve for $p$ in the following equations by making the above substitutions. Check your answer visually against the figures of sine and cosine waves:

#### Questions & Answers

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
characteristics of micro business
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
Got questions? Join the online conversation and get instant answers! By Danielrosenberger By    By Eric Crawford By Mistry Bhavesh By  By Anindyo Mukhopadhyay