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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.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38
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