Numerical modulation in the space of marks them
Constellations of the numerical modulation
1) Origin of the constellations:
Every numerical process can be written in the 
shape where in the case of numerical modulations to
orthogonal signalling every addend smk(t) is expressed like linear combination of ortonormali
functions 
, using for every k always the same space
of it marks them, to every smk(t)
corresponds the 
carrier if then i coefficients Cjqk beyond to being discreet
belong also to with a discreet one {cq} then smk belong to
with discreet of 
carriers with n = 1,
2 …, MN that
forms the constellation upgrades them.
2) Modulation with memory and lacking in memory:
The modulations lacking in memory are those for which independence is had statistics between the Cjqk and Ciqh, the others are with memory. If independence statistics is worth always then all the carriers sn they are possible for which the constellation it upgrades them coincides with the constellation. In other cases in which also Cjqk they have null combined probability, the correspondents carriers have null probability and therefore cardinalità Mm of the constellation is inferior to the cardinalità MN of the constellation upgrades them.
3) Barycentre of the constellation:
4) medium Energy statistics of the constellation:
it is had that the medium energy statistics of the generic
addend smk(t) of the modulated process coincides with the medium
energy statistics and of the constellation therefore the power of the
process is 
.
5) Useful power of one constellation:
to the power of the process it comes embezzled the power associated to the barycentre which being famous it has not associated some information. In particular the constellations to null barycentre are said to minimal energy in how much the power are all profit.
6) particular Characteristics of the constellations:
to) the constellation equienergia it is such that all the carriers that compose it have the same energy
b) the antipodale constellation is constituted from two opposite carriers having had the same energy
c) the orthogonal constellation is constituted from carriers between orthogonal they, of it it derives that the barycentre is not null and therefore the constellation is not to minimal energy
d) the biorthogonal constellation is obtained from the orthogonal one admitting also the opposite carriers.
7) Characteristics of one bidimensional constellation:
The constellation upgrades them is composed from the
carriers 
, its barycentre is 
while its energy is 
where haq 2
andsaq are valor medium statistical
and the variance of the discreet process. Draft of the
associated typical constellation to the harmonic modulations, the
cardinalità is MN = M2 principle in the independence case
statistics between Tocqk andsqk while it becomes M in the
case of maximum correlation.
8) Characteristics of one monodimensional constellation:
The constellation upgrades them is composed from the
carriers 
in the uselessly complex case of uses
of carrying harmonic in quadrature, using instead single carrying a
constellation is 
whose barycentre is 
while the energy is .
It is the typical constellation which base is made reference in the case of modulation in band.
Shapes of base in the numerical modulation
9) Chosen of the base shape:
The choice of the base shape influences the
determination of the maker and of the quadripolo complementary, in any
case it must comply with the criterion of Nyquist on the
autocorrelationship 
per n ¹ 0 and with the
correspondent in the dominion of the frequency 
.
In kind a freedom Pò is demanded that it marks modulated them
occupies one small portion of band of trasmissivo means but in any
case pu² not to come down under 
the value
minimal tax from the theorem of the sampling, leaving has the
modulation to normal band in the case uses a shape of base with the
maximum frequency fyM that you respect the 
condition,
the phantoms that turn out from such condition are the phantoms to
raised cosine which the functions of autocorrelationship
correspond 
.
The shapes of having base to normal band are defined 
therefore the maximum frequency ftoM function of to 
in as far as belong to
the interval [ 0.1 ].
From the shapes of base to normal band the shapes of base
compressed in the time to extensive band can be obtained placing b= Lto have pertanto 
.
Numerical makers and demodulatores in band base
10) Maker to direct multiplication:
It comes used in order to obtain marks them elaborated
b(t) using shapes of base j(t)
to duration limited within T, is constituted from an ideal multiplier
having like incomes marks squared them to?q(t) and marks them
periodic to unitary power 
, is obtained marks them
elaborated 
.
11) Maker with quadripolo LTI:
Imposing that it marks them elaborated b(t) it obtains
give?q(t) for simple passage in quadripolo a LTI is obtained
that the function of transfer of this last one must be 
in which the sinc(fT) to denominator is cancelled
rendering the quadripolo not realizable for f = 2f0 , of it derives that the base shape
must have the maximum frequency fyM < 2f0 that is will have to be a shape of base to normal
band for which 
the evidences a gain function
that for to= 0 and = 1 demands one infinite selettività
of the quadripolo therefore usually it is opted for to = 0,5.
12) Demodulatore to direct sampling:
In the case of shapes of base j(t) to duration limited within T it can be obtained
marks squared them to?(t) to leave from marks them elaborated b(t) simply
sampling this last one by means of a maker of marks squared them
followed from an amplifier with trasmettenza 
.
If the base shape has but greater practical duration of 2T this
method of demodulation from place to intersimbolica interference.
13) Demodulatore with filter adapted to the base shape:
The function of autocorrelationship andyy(t) of one whichever shape of
base y(t) satisfies the
criterion of Nyquist on the autocorrelationship
per n ¹ 0 therefore from marks them 
,
obtained from b(t) replacing the shape of base with its function of
autocorrelationship , pu² to gain itself marks squared them simply using
a demodulatore to direct sampling. Quadripolo that it carries
out the substitution of the shape of base with the its
autocorrelationship is called filter adapted to the base shape and has
function of 
transfer. The realizability
of the adapted filter is guaranteed for shapes of base j(t) to practically limited
duration within T if the impulsive answer is motive while in the case
of shapes of base to limited band a impulsive answer is had not motive
that comes rendered only practically such, the quadripolo that it
turns out some generates but linear distortion in band.
14) Diagram to eye in the numerical demodulation in band base:
It is obtained sending marks them of escape from the filter adapted to a having oscilloscope the axis of the times synchronized with the moments of reading, in the ideal case the opening of the eye is maximum and it is reduced in the case that the adapted filter, the channel or the maker discostino from the theoretical conditions giving place to intersimbolica interference.
15) Synchronization of the demodulatore in band base:
The demodulation in band base demands beyond the acquaintance of the shape of base j(t) also the temporal moments in which it it must be sampled, this sincrosegnale usually comes extracted from marks them elaborated b(t) which it is ciclostazionario and therefore contains this information.
Numerical modulation in band base
16) characterizing Elements the numerical modulation in band base:
It marks them elaborated it is 
, it
can therefore be represented by means of a monodimensional
constellation having 
barycentre and medium
energy statistics consequently the power of the
process is 
mind the useful power is 
.
To the aim to estimate the spectral occupation of
trasmissivo means it is a lot used the spectral efficiency 
where 
is the redundancy
of band of formation being fyM the present maximum frequency in
the phantom of the shape of employed base and 
its value minimal theoretical tax from the theorem of the sampling.
17) Phantom of power of the process modulated in band base:
in the case of constellation to not null barycentre haq ¹ 0 and shapes of base to normal
band the single discreet member to frequency zero in how much has
itself 
. The presence in the process of
modulation of one codifies preliminary matter with memory determines a
not null covarianza of the discreet process that influences on the
shape of the phantom.
18) spectral Efficiency in band base:
In the case shapes of base j (t)to limited duration within T are
used the mo-demodulatores are simple but bF >> 2 therefore the spectral efficiency is low 
therefore is demanded greater bands of
trasmissivi means therefore often is preferred to realize the makers
to normal band for which bF =
1 to and therefore the
spectral efficiency is 
that can be increased
choosing @to 0
but of the rest choosing to@1 the practical duration in how much diminishes
decreases like t-3 instead
that like t-1 .
19) binary Modulation in band base:
Symbols 0 and 1 come associate you to 2 various carriers of the monodimensional constellation, in the case of modulation lacking in memory have the two following constellations:
Modulation unipolare to1 = 0 > bit 0 to2 = d > bit 1
Modulation polare to1 = - d/2 > bit 0 to2 = d/2 > bit 1
They can use the same modulators and demodulatores and are distinguished for the useful power that in the case to unipolare it is the half of that is had in the polar case to parity of the maximum power. The two modulations introduce the same spectral density of power allotted to ridosso of the continuous one, pu² to move itself to frequency higher by means of codify it of Manche ster, in such a way the coaxial cable have better shielding property, diminish DRINKING but, being Rc = 0,5, the efficienxa is halved spectral hs .
20) ternary Modulations in band base:
The ternary modulations are obtained from the quaternarie for means of coders which make so that one of the symbols has one null probability of being emitted. The used ternary modulations more are:
ALTERNATED BIPOLAR MODULATION (LOVES)
To every entering elementx k of the binary sequence the coder associates one second word of 2 bit the rule
in short to xk = 0 come associated y = 00, than therefore it has ½ probability while to xk = 1 it comes alternatively associated y = 01 or y = 10, such symbols have therefore ¼ probability. Symbol 11 has probability 0.
The carriers associate sono to you to1 = - d > 01 to2 = 0 > 00 to3 = d > 10 to4 = 2d > 11 , of it derives that the barycentre is null and the power all useful one is distributed in way similar to how much is had for the polar coder with it codifies of Manche ster.
MODULATION DUOBINARIA
To every entering elementx k of the binary sequence the coder associates one second word of 2 bit the rule
in short to xk = 0 come associated y = 00, than therefore it has alternate ½ probability while tox k = 1 word (10 or 01)
regarding that emitting for previous the 1 to second that is a n° equal or uneven of 0 interposed. L or phantom that is obtained to ridosso of the continuous one but is nearly completely comprised under f = f0 .
Linear harmonic numerical modulation
21) Characteristics of the process of linear harmonic modulation:
It marks modulated them in traslata band is 
can therefore be represented by means of a bidimensional
constellation having 
barycentre and medium
energy statistics consequently the power of the
process is while the spectral efficiency . This generic modulation in traslata band is
called QASK.
22) Outline of principle of linear harmonic mo-demodulatores:
The modulator and the demodulatore can be realize the
classics second to you outlines in traslata band
 |
In alternative the demodulatore can be realized using an only filter adapted to the base shape and ollowing makers of mark squared them in band base
Such outline is but less used in how much the
having realization of the filter adapted in traslata band function of
transfer 
is problematic.
23) Spectral density of power of marks them with modulation QASK:
The spectral density of the complex envelope is
, it is relazionata alla spectral density del
modulated process Pm(f) second 
the , from they case h aq
¹ is evidencedthat nel 0 (…null
barycentre) and uses della shape of base to normal band, nell' complex
envelope has only the discreet member to frequency zero that nel
modulated process implies carrying presence della fc . The autocovarianza and the
covarianza, if present, determine the modification of the shape of the
phantom.
24) linear harmonic Modulations numerical lacking in memory:
to) Modulation QAM It is a QASK lacking in redundancy Rc = 1 in which the numerical elaborations in band base use the same one with of M symmetrical discreet elements and equispaziati, constellation to null barycentre therefore to minimal energy turns out some one.
b) Modulation PSK Is a QASK in which one codifies preliminary matter to
blocks ago that the single carriers have not null probability that are
found on a circle of 
beam, in such a way have in
fact the maximum power of mark modulated them.
To notice that the two modulations coincide for M = 4 in the which case are spoken about modulation QPSK, both the modifications then use codify it M-naria di Gray that consists in associating binary words that differ for a single bit to every brace of adjacent carriers situate to you at a distance minimal d.
25) numerical Modulations with carrying single:
The numerical information comes associated to a solo marks
them elaborated b(t) in band base, of it it derives that a single one
is necessary carrying and consequently the modulator and the
demodulatore are simplified. The constellation upgrades them of
this modulation dictates ASK is monodimensional with 
barycentre, medium energy 
statistics
and spectral efficiency .
To notice that in the case of carrying modulation ASK
sincrona with having exact multiple frequency of 
the demodulation happens simply with a filter adapted in followed
traslata band from a maker of it marks squared them. The
typology of ASK are following:
to) Modulation OOK it is come true multiplying harmonic carrying for marks them in band base obtained with a modulation to unipolare, the barycentre is not null therefore has a brace of discreet members to the frequency of carrying that fan that the useful power is alone 50% and moreover the spectral efficiency is much lowland, for these reasons comes used exclusively in the field of the radioaiuti to navigation.
b) Modulation BPSK is come true multiplying harmonic carrying for marks them
in band base obtained with a polar modulation, the barycentre is null
therefore not is discreet members in the phantom and the power is all
profit. Using shapes of base to normal band a spectral
efficiency good is obtained 
, moreover if the
symbols are equiprobabili the recovery of carrying happen less than an
addend np in the phase with n =
0 or 1. One becomes therefore necessary codifies
it differentiates them that, in case used, it suggests I use it for
the following demodulation of outline:
saying demodulatore differentiates them BPSK or DBPSK in how much marks them championship is positive in the case of permanence and negative in the variation case.
