Callbacks on mark them
IT MARKS DETERMINES THEM TO YOU IN THE DOMINION OF THE TIME
1) Energy of marks them:
The energy is the integral of the instantaneous power 
, if it assumes an ended value then marks is said
them to be of energy.
2) medium Power of marks them:
In the case it marks them s(t) is limitless in the time
and it does not have ended energy, considers the temporal medium power
of marks them 
essendo
marks cut them. If for a data it marks
them it assumes an ended value, then marks is said them to be of
power.
3) medium Valor of marks them:
, it is various from 0 solos for marks them of power.
4) Member alternated of marks them s(t):
5) Factor of peak:
It is reported to marks them symmetrical, which are
lacking in continuous member and have equal and oppositethe maximum value s M andthe minimal value s m |sm| = sM = sp ,
is had: 
6) Function of retort:
in practical it marks them retort is obtained adding marks them that they obtain traslando marks them x(t) of a multiple amount of the period T0 of the repetition, is observed that only if the period T0 is greater of the period TX of it marks them source obtains a course similar to it.
7) Power of marks them periodic obtained for repetition of marks them of ended duration:
being E(T0) the energy calculated to the inside of a period, it coincides with and only if TX < T0 .
8) Function of temporal intercorrelationship or temporal mutual correlation for marks them of energy:
characterizes the degree of likeness between 2 
funzioni, attention to the fact that does not
coincide with the convoluzione product.
9) mutual Energy:
Draft of the function of intercorrelationship calculated in t = 0 
.
10) Product to climb of 2 marks them of energy:
Draft of the value in the origin of the function of
intercorrelationship between two marks them 
,
measure the affinity of the two marks not traslati them.
11) It marks them parallels, antipodali, orthogonal:
Currency the index of intercorrelationship 
is had:
to) rxy = 1 it marks them parallels
b) rxy = 0 marks them orthogonal
c) rxy = -1 marks them antipodali
12) Function of intercorrelationship of marks them of power:
13) Function of temporal covarianza for marks them of power:
It is the function of temporal intercorrelationship of the members to valor medium null:
14) Relation between intercorrelationship and intercovarianza of two marks them of power:
15) Family of marks incorrelati them:
A family of marks them says incorrelata if all the functions of covarianza Kxy(t) is null.
16) Family of marks them incoherent:
She is with of marks them of power for which for every t the functions of intercorrelationship R are null allxy(t), say also that draft of a family of marks them orthogonal.
SHE MARKS DETERMINES THEM TO YOU IN THE DOMINION OF THE FREQUENCY
17) Transformed of Fourier:
where S(f) is the phantom of marks them, in particular in the case that s(t) is marks them real, the amplitude phantom is an equal function while the phase phantom is one uneven function.
18) Antitrasformata di Fourier:
that is it marks them s(t) is esprimibile by means of the sum of a number infinitely of complex harmonic functions andjwt of infinitesimal amplitude and frequency f distributed in continuous way on the real axis.
19) Property of the transformed one of Fourier:
 |
20) F [ 1 ]:
d(f)
21) F [ sgn(t) ]:
22) F [ u(t) ]:
23) F [ rect(t/T) ]:
24) F [ sinc(t/T) ]:
25) Series of Fourier of marks them periodic continuous time:
being Cn the having coefficients of Fourier 
expression, applying the transformed one of Fourier obtains 
that is has a discreet phantom and all the lines
are spaced of one amount 
.
26) Spectral density of energy:
from it the energy can be obtained of marks integrating
them on the axis of the 
frequencies.
27) Spectral density of power:
It is the transformed one of Fourier of the function of autocorrelationship for marks them of power, that is:
28) It marks limited them in band closely:
The phantom of E(f) energy is an equal real function for marks them real in the ideal case, extends between the maximum frequency fM and one minimal frequency fm , the band of marks them is therefore B = fM - fm , valid definition also in the real case where the phantom is limitless but can neglect the frequencies above fM and under fm .
29) It marks them in band base:
It is the typical one marks supplied them from a information source, its band is therefore allocata 0 £ fm £ fM, the single positive axle shaft is considered because for real and equal function marks them real the energy phantom is one.
30) It marks them in traslata band:
It is obtained elaborating marks them in band base to the aim to adapt it to trasmissivo means, its band is therefore allocata:
0 < fm £ fc £ fM
a lot often the extension of the band is approximately equal to the maximum frequency fM , although minimal frequency fm is various from zero.
31) It marks them in tightened traslata band:
It is marks them that it respects the condition 
or B < fm .
32) It marks them in traslata band much grip:
It is marks them that it respects the condition 
where fc is a
contained frequency to the inside of the B band, this condition
includes 
the .
ULTERIOR RAPPRESENTAZIONI OF MARK THEM
33) Phantom of marks them of real energy:
Draft of an equal function, is therefore sufficient to
study the phantom that extends in the axle shaft 
negative or the phantom that extends in the positive axle
shaft 
.
34) It marks them analytical:
For the linearity of the antitransformed one of Fourier it
is had that to 
the phantom 
corresponds of the rest of marks them real is hermitiano
therefore 
which corresponds 
replacing has 
being 
marks them analytical associated to s(t). In truth
the complete shape of marks them analytical is 
where 
is the transformed one of Hilbert of s(t).
35) Transformed of Hilbert of s(t):
is the transformed one of Hilbert of s(t), from it
riottiene s(t) by means of the antitransformed one
. In the dominion of the frequency the
relation between the phantoms is
.
36) complex Envelope of marks them:
Draft of the antitransformed one of the phantom of marks
them analytical 
traslato so as to to carry fthe c in the origin, ha
from which ha 
and therefore 
. The complex envelope has
therefore the scope to concur of dealing marks them in traslata band
s(t) by means of two members in band base, they evincono writing
itself in terms of real part and imaginary part 
has in fact the member in 
phase and the member
in 
quadrature, by means of these members marks
them in traslata band can be written in the shape 
.
37) Relation of ortonormalità:
38) Representation of marks them s(t) through one base:
where {yk(t)} is with discreet of ortonormali functions
and i coefficients ck are
chosen so as to to diminish the medium quadratic error 
.
39) Theorem of Nyquist - Shannon:
Admitting for it marks them s(t) limitless in the time and
limited in band base a representation in the dominion of the
frequency by means of the base 
obtains
that it can be reconstructed to leave from the acquaintance of i its
champions Ck = s(kTN) provided that obtained with an at
least double frequency of sampling regarding the maximum present
frequency to the inside of it marks them, has infatti 
, the interval of Nyquist is in this 
case marks them in traslata band
can be instead reconstructed leaving from the acquaintance of the
champions of its members in band base sc(t) and ss(t)
having frequency maximum 
has in fact 
with 
ed x = c, s and the
interval of Nyquist is in this halved case 
.
40) N° of functions necessary in order to represent whichever marks them in its definition interval:
Of it it is necessary an infinite number, however for some types of it marks them can be caught up a good accuracy also with a number ended of marks them, it is the case of marks them physicists practically limits to you in band and in duration for which a sampling with a number of champions N=2BT being B is sufficient the band of marks them physicist and T its duration.
SEQUENCES And ELEMENTS Of MARK THEM NUMERICAL
41) Sequence:
With it is ordered of values that can be obtained from
mark them continuous s(t) considering instead of the variable one
continue t variable the discreet NT where n it is the succession of
the entire numbers, the generic sequence has expression 
.
42) Energy for sequences:
43) Power for sequences:
44) Sequences of intercorrelationship between sequences to ended energy:
45) temporal Sequences of intercorrelationship for power sequences:
46) Sequences of intercovarianza for power sequences:
47) M-nario Alphabet:
The elements of a numerical sequence can only assume pertaining values to with discreet {sq}, to everyone of these values can be associated one of the elements of a M_nario alphabet {zq} of cardinalità M that in kind is a power of 2 ossia M = 2b so as to to be able to use alphabets with long binary words b bit.
48) numerical Flow:
Draft of the succession of symbols zq pertaining to the M-nario alphabet, every associate to an element of the sequence {sq} and that they are repeated with the same temporizzazione kT.
49) Time of symbol:
It is the interval of T time that elapses between a symbol
and the successive one of the numerical flow, its inverse one 
is the symbol rhythm that in the case to every symbol comes
associated a binary word constituted from 
bit
comes defined binary rhythm 
whose inverse it is
the time of bit Tb .
50) It marks them numerical multilevel:
It marks samples them to you numerical introduce of dthe (t) that for being
transmitted they demand average to infinite band, in order to obviate
to ci² replaces dthe (t)
with a impulsive function of having energyf (t) ended duration and for which a limitation
practical in band can be obtained in such a way obtaining marks them
numerical multilevel 
, as an example it is
obtained marks them numerical squared 
in the
case uses the rectangular impulse to unitary energy 
not to confuse with marks them analogic squared 
.
51) Speed of modulation:
It marks them numerical asynchronous 
are such for which the intervals tk between a symbol and the successive one the minimum of the
intervals between a successive symbol and the pertaining ones to the
sequence are not constant therefore 
cannot be more spoken than to rhythm than symbol but
about speed of modulation being T V. The
modulation speed is expressed in Baud and represents the maximum
number of veicolabili symbols from marks them in the time unit.
GENERALITY ON PROCESSES STOCASTICI
52) Amount of information:
In the case of a numerical source the information amount
is the largeness associated to the choice of a symbol, zq , between M the possible elements that
they form a M-nario alphabet, it is worth 
being P(zq) the probability
that comes emitted the symbol zq . The definition is coherent if the 0.1)
equiprobabili [ p(0) = p(1) = 0,5 ] amounts of information are
believed that for a binary alphabet with symbols (are unitary.
53) continuous stocastico Process:
Its realizations are continuous functions x(t) that they can be emitted from the analogic source. For a data T moment the process is reduced to one v.a. continues Xt = X(t) with distribution density p(Xt , t).
54) discreet stocastico Process to discreet values:
It is the case of an emitted numerical flow from a numerical source, is the time that the value of the realizations is not continuous but discreet.
55) Density of combined probability of 2° the order:
expresses the probability that to time t1 is X1 comprised between x1 ed x1 dx1 and to time t2 X are2 comprised between x2 ed x2 dx2 , where X1 and X2 is
two variable aleatory ones that are obtained from the stocastico
process fixing the time in moments t1 and t2 .
56) Density of conditioned probability of order n:
in short variable the Xn to the time tn it is conditioned from the acquaintance of the v.a. emitted in the previous moments.
57) Process of Markoff of order n:
It is a process of which the full acquaintance has itself if the density of combined probability of order is known n 1.
58) statistical medium Value:
59) medium Power statistics:
60) Variance:
.
61) Relation between variance and power:
62) Function of autocorrelationship statistics:
63) Function of autocovarianza statistics:
for t1 = t2 the autocovarianza coincides with the variance.
64) Relation between the autocovarianza and the autocorrelationship :
65) stationary Process in tight sense:
It is a process for which the density of invariant probabilities is respect to one arbitrary temporal translation.
66) stationary Process broadly speaking:
It is a process for which the density of invariant probabilities is respect to an arbitrary temporal translation, while the density of combined probability of 2° order only depends on t , is had that is:
to) p(x;t) = p(x) b) p2(x1,x2;t1,t2) = p2(x1,x2;t) essendo t = t2 - t1
for a stationary process broadly speaking it is had that the valor medium statistical, the power and the variance are constant and correspond to the expected values of the correspondents temporal largenesses for a single realization of the process while the functions of autocorrelationship R(t) and autocovarianza K(t) depend solo from t.
67) Relation between the autocovarianza and the autocorrelationship for stationary processes:
68) First relation of Wiener - Khinchine:
that discreet spectral member in the origin evidences therefore one.
69) ergodico stationary Process:
It is a process for which the single realization, observed on the entire axis of the times, adds all property statistics of the aleatory process, therefore that the temporal largenesses converge to the largenesses statistics.
70) Spectral density of intercrossed power:
71) Second relation of Wiener - Khinchine:
72) Characteristic of the process sum of two processes stazionari a(t) = x(t) y(t) :
Considering process a(t) = x(t) y(t) the valor is
had medium statistical hto = hx hy while the autocorrelationship is 
therefore the process stationary sum is also it.
The power density is 
therefore the two
processes simply sommabili addends cone in power if they are
orthogonal or if 
.
73) Characteristic of complex process b(t) = x(t) j y(t):
Considering process b(t) = x(t) jy(t) the x
are had valor medium statisticalh b =h jhy while the autocorrelationship
is 
therefore the complex process is also
stationary it. The power density is 
.
PROCESSES STOCASTICI CICLOSTAZIONARI
74) ciclostazionario stocastico Process:
Draft of a process whose function of autocorrelationship is periodic in t, in particular is spoken about ciclostazionari processes of 1° the order if the regularity is present also in valor medium statistical and the ciclostazionari processes of 2° the order if it is present single in the autocorrelationship . The analysis can be led back to that one of the stationary processes carrying out a translation z on the axis of the times and carrying out one medium independent but uniform such statistics also on v.a. in the period, in such a way hs and Rss(t) is estimated like medium values of the respective periodic functions in t .
75) Processes represent to you through complex envelope:
Having 
the process 
realization is stationary if the two component processes
are jointly stationary while it is ciclostazionario if it it is also
one only of they, the autocorrelationship is
therefore 
while the 
process is in a generalized manner ciclostazionario
in how much its autocorrelationship mediated on a period is 
and has intensity of 
power.
76) stationary Process in traslata band:
Being stationary the S(t) process in traslata band it is
had that also the processes in band base Sc(t) and Ss(t) is stationary broadly speaking with identical functions
of autocorrelationship 
and valor medium null.
It is gained that the power is not divided on the two processes
but is found again identical on everyone of they.
77) real Processes with aleatory factors represent to you by means of temporal series:
A S(t) process is considered having 
realization being zk continuous aleatory variable determination of the Zk that they form with their Z(n)
sequence a stationary real discreet process while p(t) real function
is one having practical duration Tp . In a generalized manner S(t) is a ciclostazionario
process having valor medium 
, autocorrelationship 
and spectral density of 
power. The power of the process is simply 
in the case is orthogonal the v.a. zk or it marks them energy addends that
constitute the periodic shape.
78) Processes sample in band base to you:
It is a particular case of the previous situation,
considers in fact a process in band having S(t) base 
realization, obtains that the valor medium statistical h he is equal to valor medium hthe c of the discreet process constituted from the sequence of
the aleatory champions while the spectral density of power is 
.
79) complex Processes with aleatory factors:
We consider a continuous process real ciclostazionario in
traslata band which realization is associated the process complex envelope 
having ,
the spectral density of associated power to it is:
80) Process sum of real processes with aleatory factors:
A S(t) process is had with representation in the 
shape, the turning out correlation is 
.
81) Gaussian continuous Process:
It is a process of which the full acquaintance has itself
single statistics based on the acquaintance of the probability density
function of 2° the order, in the stazionarietà case the expression of
the density of probability of 1° the order is 
.
One important property of this process is that the sum of
independent Gaussian processes is still a Gaussian process with valor
medium sum of the medium values and variance sum of the variances
moreover the sum of n arbitrary but independent processes is a
Gaussian process for n ® ¥ second asserted how much from
the theorem of the limit centers them.
82) Gaussian Noise:
It is the Gaussian process turning out from the sum of
numerous marks them aleatory points out to you, in particular we have
a Gaussian noise white man if the spectral density of power is N(f) =
constantN 0 = which the
autocorrelationship corresponds 
.
83) stationary Gaussian Noise in traslata band:
We consider a generic representation of the noise in
band traslata 
, the processes in band base Nc(t) and Ns(t) turn out Gaussian stationary and the power of
the noise in traslata band is found again identical on everyone of
they while the spectral density introduces a various bandwidth to
second of as it comes chosen fc , in particular is maximum and equal to 2B if fc is to one of the ends of the traslata
band while he is minimal and equal to B if fc is to the center of the traslata band.
84) Gaussian Noise white man in the space of marks them:
In the space with N dimension tending to the infinite, a
realization of the noise white man is rappresentabile with a having carrier n members
ciascuna with density of probability of 1° ordine 
the .
