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The real part of the spectrum is symmetrical

The real part of the complex spectrum is shown in the second plot from the top in Figure 1 . It will become important later to note that the real part of the spectrum is symmetrical about the folding frequency near the center of theplot (at the eighth tick mark) .

Without attempting to explain why, I will simply tell you that the real part of the Fourier transform of a complex series whose imaginary part is all zeros (like a typical sampled time series for real-world data) is always symmetrical about the folding frequency.

The imaginary part of the spectrum is asymmetrical

The imaginary part of the complex spectrum is shown in the third plot from the top. Again, it will become important later to note that the imaginary partof the spectrum is asymmetrical about the folding frequency.

Once again, without attempting to explain why, the imaginary part of the Fourier transform of a complex series whose imaginary part is all zeros (like a typical sampled time series for real-world data) is always asymmetrical about the folding frequency.

The converse is also true

It is also true that the values of the imaginary part of the Fourier transform of a complex spectrum whose real part is symmetrical about the foldingfrequency and whose imaginary part is asymmetrical about the folding frequency will all be zero. I will take advantage of these facts later to simplify thecomputing and plotting process.

The amplitude spectrum

The amplitude spectrum is shown in the fourth plot down from the top. Recall from previous modules that the amplitude values are always positive, consistingof the square root of the sum of the squares of the real and imaginary parts.

The output time series

The output time series, produced by performing an inverse Fourier transform on the complex spectrum is shown in the bottom plot in Figure 1 . Compare the bottom plot to the top plot. As you can see, they are the same, demonstratingthe reversible nature of the Fourier transform.

The program named Dsp035

I will discuss this program in fragments. A complete listing of the program is provided in Listing 14 near the end of the module.

The beginning of the class named Dsp035

The beginning of the class for Dsp035 , including the declaration of some variables and the creation of some array objects is shown in Listing 1 . This code is straightforward.

Listing 1. The beginning of the class named Dsp035.
class Dsp035 implements GraphIntfc01{ final double pi = Math.PI;int len = 256; double[]timeDataIn = new double[len];double[] realSpect = new double[len]; double[]imagSpect = new double[len];double[] angle = new double[len];//unused double[]magnitude = new double[len];double[] timeDataOut = new double[len];

Beginning of the constructor

The constructor begins in Listing 2 . The code in Listing 2 creates the input time series data shown in the top plot of Figure 1 .

Listing 2. Beginning of the constructor.
public Dsp035(){//constructor //Create the raw data pulsestimeDataIn[0] = 0;timeDataIn[1] = 50;//... //code deleted for brevity//... timeDataIn[254]= -80; timeDataIn[255]= -80; //Create raw data sinusoidfor(int x = len/3;x<3*len/4;x++){ timeDataIn[x]= 80.0 * Math.sin( 2*pi*(x)*1.0/20.0);}//end for loop

Questions & Answers

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
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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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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