# 2.1 Dsp00104-sampled time series  (Page 6/10)

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## The top two sinusoids

More important, however, is to note what has happened to the top two sinusoids. Because the frequencies of the top two sinusoids are above thefolding frequency, they no longer have a minimum of two samples per cycle. Thus, the apparent frequency of these two sinusoids has folded around the foldingfrequency and appears as a lower frequency.

## Top sinusoids match bottom sinusoids

In fact, the plot of the top sinusoid now looks exactly like the plot of the bottom sinusoid at a frequency of 0.25 cycles per second. This means that theenergy in the top sinusoid at 1.75 cycles per second has folded into a new frequency of 0.25 cycles per second.

The plot of the sinusoid immediately above the center looks exactly like the plot of the sinusoid immediately below the center at a frequency of 0.50 cyclesper second. This means that the energy in that sinusoid at 1.5 cycles per second has folded into a new frequency of 0.5 cycles per second.

## Bottom three plots are correct

The plots of the center sinusoid and the two sinusoids below the center are still correct (although not very well sampled). However, the frequency information embodied in the top two sinusoids has been lost. The top twosinusoids appear to be at a different frequency than the actual frequency of the corresponding analog signals. The fact that the frequencies of these twosinusoids were originally 1.75 and 1.50 cycles per second is now lost in the sampled data.

## A different approach

Now I'm going to illustrate the same folding phenomena from a different perspective using spectral analyses. First I will show you a case having nosampling problems. Then I will introduce a sampling problem and show you the impact that the problem has on the final results.

Figure 7 shows the result of performing spectral analyses on five different sinusoids (not the same five as in the previous discussion). Each plot in Figure 7 shows the spectrum of a different sinusoid. The spectrum is computed and displayed from zero frequency on the left to the folding frequency on theright.

Figure 7. Spectral analyses of five sinusoids with no sampling problems.

## Sampling frequency was four samples per second

The sampling frequency for the data in Figure 7 was four samples per second, giving a folding frequency of two cycles per second. Thus, the horizontal scaleon each plot represents the frequencies from zero on the left to two cycles per second on the right.

## The five sinusoids

Starting at the top, each of the five plots represents the frequency spectrum of a sinusoid having the amplitude and frequency shown in the following table.

Amplitudes and frequencies of sinusoids: ```Plot Amplitude Frequency 1 60 0.25 cycles per second2 70 0.50 cycles per second 3 80 0.75 cycles per second4 90 1.50 cycles per second 5 100 1.75 cycles per second```

## The heights of the spectral peaks

The height of each spectral peak in Figure 7 is consistent with the amplitude of the corresponding sinusoid given in the table.

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?
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
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
how did you get the value of 2000N.What calculations are needed to arrive at it
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