# 2.2 Dsp00108-averaging time series  (Page 8/14)

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## A short recap before continuing

Hopefully, by this point, you understand how multiplying two time series produces a new time series composed of the sum of all the products of theindividual sinusoids in the two original time series.

When each pair of sinusoids is multiplied together, they produce a new time series consisting of two other sinusoids whose frequencies are the sum anddifference of the original pair of frequencies.

## The error in the computed average

When an average is computed for a fixed number of points on the new time series, the error in the average tends to be greater for cases where theoriginal frequency values were close together. This is because the period of one of the new sinusoids becomes longer as the original frequencies become closer.In general, the longer the period of the sinusoid, the more points are requiredto get a good estimate of its average value.

## Does this matter?

There are many operations in DSP where this matters a lot. As mentioned earlier, the computational requirements for DSP frequently boil down to nothingmore than multiplying a pair of time series and computing the average of the product. You will see many examples of this as you continue studying the modulesin this series of tutorials on DSP.

## Spectral analysis

I am going to illustrate my point by showing you one such example in this module. This example will use a Fourier transform in an attempt to performspectral analysis and to separate two closely-spaced frequency components in a time series. As you will see, errors in the computed average can interfere withthis process in a significant way.

(This example will illustrate and explain the results using graphs. Future modules will provide more technical details on the DSP operationsinvolved.)

## Several steps are involved

I will provide this illustration in several steps.

## Spectral data for same frequency but different lengths

First, I will show you spectral data for several time series, each consisting of a single sinusoid. The time series will have different lengths but theindividual sinusoids will have the same frequency. This will serve as baseline data for the experiments that follow.

## Sum of two sinusoids

Then I will show you spectral data for several time series, each composed of the sum of two sinusoids. These time series will have different lengths. Thesinusoids in each time series will have the same frequencies. I will show you two cases that fall under this description. The frequency difference for the twosinusoids in each time series will be small in one case, and greater in another case.

## Sinusoids with different frequency differences

Finally, I will show you spectral data for several time series, each composed of the sum of two sinusoids. These time series will be different lengths, andthe sinusoids in each time series will have different frequencies. In particular, the frequency difference between the two sinusoids in each timeseries will be equal to the theoretical frequency resolution for a time series of that particular length.

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
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