Ulterior rappresentazioni of mark them
1) 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 
.
2) It marks them analytical left:
It is marks them that it is obtained antitransforming the single part of the phantom that extends in the axle shaft negative.
3) It marks them analytical:
s reaches this
definition in how much since the energy phantom is an equal function,
has 
and therefore antitransforming 
from which 
. The relation is had
moreover 
.
4) Transformed of Hilbert of x(t):
it is a transformation between marks them in how much is
remained in the dominion of the time. She marks them and
transformed its of Hilbert has the same phantom of energy. She
reaches herself this definition from the expression of marks them
analytical obtained antitransforming the positive phantom 
.
5) Antitrasformata di Hilbert:
6) Relation between the X(f) phantom of marks them x(t)
and the 
phantom of the transformed one of Hilbert 
of marks them x(t):
it is obtained applying the property of the convoluzione to the transformed definition of of Hilbert.
7) 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
that can
also be written in the 
shape and remembering that 
has 
.
8) analogic Member of vlf in phase:
9) analogic Member of vlf in quadrature:
10) Distance between x(t) and y(t):
11) Relation of ortonormalità:
12) Representation of marks them x(t) through one base:
where {yk(t)} is with discreet of functions and tok they are
chosen so as to to diminish the square of the distance that is the
medium quadratic error 
.
13) Equality of Parseval:
when it is verified, the entirety {yk(t)} is
thought satisfactory in order to represent x(t), in short in fact is
had that to growing of the number of the functions used, improves the
accuracy.
14) Theorem of Nyquist-Shannon:
It marks them limitless in the time but limited in band, it can be reconstructed to leave from the acquaintance of its champions finds T to you cbetween they at a distance, in particular the sampling frequency must be at least the double quantity of the maximum present frequency to the inside of marks them, is had:
15) Sampling of marks them in traslata band:
The representation is considered 
where xc and xs are expressed in the sampled shape 
.
These champions must are capture you to a double frequency regarding the case of mark them in band base.
The minimal number of champions is N=2BT.
16) Relation between the phantom of marks them analogic and the phantom of the sequence obtained from its sampling:
The phantom of the sequence is an infinite repetition of the phantom of marks them analogic. If the sampling frequency is sufficient, the two phantoms coincide in [ -p , p] otherwise superimposition between the members to higher frequency and the members to lower frequency is had aliasing that is, and it is not more possible to reconstruct exactly marks them analogic that championship beginning from.
17) Formula of interpolation in order to reconstruct marks them to continuous time to leave from its champions:
18) Ortogonalizzazione di Gram-Schmidt:
When an entirety ended is had of marks them pu², through the process of ortonormalizzazione of Gram-Schmidt to create a ortonormale base in which they can be represents to you. In short the first one is taken marks them and it is standardized, dopodichè takes the second it marks them, it embezzles its orthogonal projection along the direction of the first one marks obtaining them in such a way marks them to orthogonal it, dopodichè marks them therefore obtained comes standardized. It is proceeded in such a way also for the others marks them, characterizing therefore one ortonormale base.
19) 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.
