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Geometric representation of signals provides a compact, alternative characterization of signals.

Geometric representation of signals can provide a compact characterization of signals and can simplify analysis of theirperformance as modulation signals.

Orthonormal bases are essential in geometry. Let s 1 t s 2 t s M t be a set of signals.

Define ψ 1 t s 1 t E 1 where E 1 t 0 T s 1 t 2 .

Define s 21 s 2 ψ 1 t 0 T s 2 t ψ 1 t and ψ 2 t 1 E 2 ^ s 2 t s 21 ψ 1 where E 2 ^ t 0 T s 2 t s 21 ψ 1 t 2

In general

ψ k t 1 E k ^ s k t j 1 k 1 s kj ψ j t
where E k ^ t 0 T s k t j 1 k 1 s kj ψ j t 2 .

The process continues until all of the M signals are exhausted. The results are N orthogonal signals with unit energy, ψ 1 t ψ 2 t ψ N t where N M . If the signals s 1 t s M t are linearly independent, then N M .

The M signals can be represented as

s m t n 1 N s mn ψ n t
with m 1 2 M where s mn s m ψ n and E m n 1 N s mn 2 . The signals can be represented by s m s m 1 s m 2 s mN

ψ 1 t s 1 t A 2 T
s 11 A T
s 21 A T
ψ 2 t s 2 t s 21 ψ 1 t 1 E 2 ^ A A T T 1 E 2 ^ 0

Dimension of the signal set is 1 with E 1 s 11 2 and E 2 s 21 2 .

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ψ m t s m t E s where E s t 0 T s m t 2 A 2 T 4

s 1 E s 0 0 0 , s 2 0 E s 0 0 , s 3 0 0 E s 0 , and s 4 0 0 0 E s

m n d mn s m s n j 1 N s mj s nj 2 2 E s
is the Euclidean distance between signals.

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Set of 4 equal energy biorthogonal signals. s 1 t s t , s 2 t s t , s 3 t s t , s 4 t s t .

The orthonormal basis ψ 1 t s t E s , ψ 2 t s t E s where E s t 0 T s m t 2

s 1 E s 0 , s 2 0 E s , s 3 E s 0 , s 4 0 E s . The four signals can be geometrically represented using the 4-vector of projection coefficients s 1 , s 2 , s 3 , and s 4 as a set of constellation points.

Signal constellation

d 21 s 2 s 1 2 E s
d 12 d 23 d 34 d 14
d 13 s 1 s 3 2 E s
d 13 d 24
Minimum distance d min 2 E s

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Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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
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Akash Reply
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
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I'm interested in nanotube
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Ramkumar Reply
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Source:  OpenStax, Digital communication systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10134/1.3
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