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The function w(t)=δ(t+10)-2δ(t+1)+3δ(t-5) consisting of three weighted δ functions is represented graphically as three weighted arrows at t= -10, -1, 5, weighted by the appropriate constants.
The function w ( t ) = δ ( t + 10 ) - 2 δ ( t + 1 ) + 3 δ ( t - 5 ) consisting of three weighted δ functions is represented graphically as three weighted arrows at t = -10, -1, 5, weighted by the appropriate constants.

The discrete time counterpart of δ ( t ) is the (discrete) delta function

δ [ k ] = 1 k = 0 0 k 0 .

While there are a few subtleties (i.e., differences) between δ ( t ) and δ [ k ] , for the most part they act analogously. For example, the program specdelta.m calculates the spectrum of the (discrete) delta function.

time=2; % length of timeTs=1/100;                   % time interval between samples t=Ts:Ts:time;               % create time vectorx=zeros(size(t));           % create signal of all zeros x(1)=1;% delta function plotspec(x,Ts)              % draw waveform and spectrum
specdelta.m plots the spectrum of a delta function (download file)

The output of specdelta.m is shown in [link] . As expected from [link] , the magnitude spectrum of the delta function is equal to 1 at all frequencies.

Calculate the Fourier transform of δ ( t - t 0 ) from the definition. Now calculate it using the time shift property [link] . Are they the same?Hint: They had better be.

Use the definition of the IFT [link] to show that

δ ( f - f 0 ) e j 2 π f 0 t .

Mimic the code in specdelta.m to find the spectrum of the discrete delta function when:

  1. The delta does not occur at the start of x . Try x[10]=1 , x[100]=1 , and x[110]=1 . How do the spectra differ? Can you use the time shift property [link] to explain what you see?
  2. The delta changes magnitude x . Try x[1]=10 , x[10]=3 , and x[110]= 0.1 . How do the spectra differ? Can you use the linearity property [link] to explain what you see?
A (discrete) delta function at time 0 has a magnitude spectrum equal to 1 for all frequencies.
A (discrete) delta function at time 0 has a magnitude spectrum equal to 1 for all frequencies.

Mimic the code in specdelta.m to find the magnitude spectrum of the discrete delta function when:

  1. The delta does not occur at the start of x . Try x[10]=1 , x[100]=1 , and x[110]=1 . How do the spectra differ? Can you use the time shift property [link] to explain what you see?
  2. The delta changes magnitude x . Try x[1]=10 , x[10]=3 , and x[110]= 0.1 . How do the spectra differ? Can you use the linearity property [link] to explain what you see?

Modify the code in specdelta.m to find the phase spectrum of a signal that consists of a delta function when the nonzero termis located at the start ( x(1)=1 ), the middle ( x(100)=1 ) and at the end ( x(200)=1 ).

Mimic the code in specdelta.m to find the spectrum of a train of equally spaced pulses. For instance, x(1:20:end)=1 spaces the pulses 20 samples apart, and x(1:25:end)=1 places the pulses 25 samples apart.

  1. Can you predict how far apart the resulting pulses in the spectrum will be?
  2. Show that
    k = - δ ( t - k T s ) 1 T s n = - δ ( f - n f s ) ,
    where f s = 1 / T s . Hint: Let w ( t ) = 1 in [link] .
  3. Now can you predict how far apart the pulses in the spectrum are? Your answer should be in terms of how farapart the pulses are in the time signal.

In [link] , the spectrum of a sinusoid was shown to consist of two symmetrical spikes in the frequency domain,(recall [link] ). The next example shows why this is true by explicitly taking the Fouriertransform.

Let w ( t ) = A sin ( 2 π f 0 t ) , and use Euler's identity [link] to rewrite w ( t ) as

Questions & Answers

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
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
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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