CN110445600A - The method that secret communication receiving end synchronizes Rossler chaotic signal using linear system local generalized - Google Patents
The method that secret communication receiving end synchronizes Rossler chaotic signal using linear system local generalized Download PDFInfo
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- H—ELECTRICITY
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- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
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Abstract
A kind of method that secret communication receiving end synchronizes Rossler chaotic signal using linear system local generalized, only Rossler chaos is produced in local state space, but it is adjusted by parameter, range of this part can be adjusted arbitrarily to comprising entire Rossler attractor, therefore be equal to from effect and global to be generated Rossler chaos；Can also be adjusted by parameter makes it that can not generate Rossler chaos, improves its confidentiality in extreme circumstances；Finally there are a wide range of adjustable parameters, further increase secrecy ability.
Description
Technical field
The invention belongs to can be applied to private communication technology field, more particularly in receiving terminal of communication system from single input
The technology of linear system generalized synchronization Rossler chaotic signal.
Background technique
Chaotic motion is the branch of non-linear ambit, but its range being related to is well beyond traditional non-thread sexology
Section territorial limit, develop into comprehensive, intercrossing, cross-cutting subject branch it is very big widened people understanding it is non-
The ken of linear science, more deep to the understanding of nonlinear science
Chaos is also applied to secret communication.One typical application is chaotic modulation.The basic thought of chaotic modulation
It is to send original signal and a chaotic signal modulation together；And receiver is demodulated, according to chaotic signal point
Separate out original signal；To third party since it is unaware of the dynamic characteristic of the chaotic signal, can not decrypt.Just it is discussed herein
Technical solution for, original signal is modulated in Rossler chaotic signal by transmitting terminal, and receiving end carries out at signal appropriate
Reason, isolates original signal, this requires the linear system of receiving end that can generate corresponding Rossler chaos letter under certain control
Number.Receiving end is generally meant that using linear system, even if the receiver is obtained by adverse party, secrecy still with higher
Property, and be likely to that the related characteristic of transmitter will not be revealed.
The equation of Rossler system is as follows
Wherein x=(x1,x2,x3)TFor state vector, a, b and c are parameter.The system balancing point has
And
Totally two；When working as a=0.2, b=0.2 and c=5.7, two equalization point be (0.0070262, -0.035131,
0.035131)T(5.6930, -28.465,28.465)T, initial value is (5.0,5.0,5.0)TChaotic systems track as scheme
1。
Work before one of inventor and it is discussed the problem of in relation to (Lu J, Zhang D, Sun Y, et
al.Application of normal form in chaotic synchronization[C]//American Control
Conference.IEEE, 2005.), the generalized synchronization of controlled linear system to Rossler system, this process employs
Feature Rossler similar compared with linear system.
Through studying, above-mentioned technology is the most intuitive and simplest mode that linear system generates Rossler chaos, accordingly
Ground may be also most unclassified mode.
Summary of the invention
Confidentiality in order to overcome the shortcomings of existing linear system generalized synchronization Rossler chaos technology is poor, the present invention
A kind of preferable secret communication receiving end of confidentiality is provided using the synchronous Rossler chaotic signal of linear system local generalized
Method.
The technical solution adopted by the present invention to solve the technical problems is:
A kind of method that secret communication receiving end synchronizes Rossler chaotic signal using linear system local generalized, including
Following procedure:
If controlled Rossler system form is as follows:
Wherein k2And k3It is not all 0, system drifting vector field is
Input vector field is
Make Lie derivatives
Due to [k, adfK]=0, such as
For 0 can exact feedback linearization,It only may be implemented under special circumstances, work as k3≠
0 and k2When=0,Show that system can global feedback linearisation；If enabling k3=-k2,Solution be x1=a+c, i.e., in addition to plane x1Outside=a+c can feedback linearization, but the plane and system
Two equalization points it is too close, and equalization point is located at plane two sides, and the plane is necessarily frequently passed through in chaos system track, this
Sample can cannot carry out generalized synchronization using feedback linearization bring is convenient；
However, selection suitable parameters make equalization point be located at one sideCurved surface, and realize
It is possible that attractor is non-intersecting with the curved surface；
Adjustment parameter adjusts curved surfaceEnable its distance Rossler attractor farther out, this point passes through
It adjusts curved surface to realize to initial point distance, enable thusAnd
Assuming that having determined that parameter a, b, c, k2And k3, meet on curved surface to the coordinate at origin minimum distance
(x is solved from above-mentioned equation1,x2,x3), further calculate curved surface to origin the shortest distance；
As long as according to above-mentioned analysis selection parameter k2And k3Curved surface may be implementedIt is inhaled with Rossler
The separation of introduction, so that the regional area where Rossler chaotic signal can be with feedback linearization, and this regional area can
To pass through parameter k2And k3It is adjusted to arbitrarily large, then, feasibility has been verified, and further considers how to realize feedback linearization
Problem, for this purpose, enabling X=adfK makees Lie derivatives
It is noted thatWork as k2And k3When not being 0This also illustrates that system can be converted to p-normal form, utilizesRealize feedback
Linearisation, it is noted that
Inverse of a matrix is in above formula
Show to do state transformation
Then in this case
It therefore is lower triangular form, curved surface using the system equation that the coordinate writes outWith
Rossler attractor, so being communication system receiver selected parameter, under these parameters, is used without friendshipState write through system
Equation:
Further make state transformationI.e.
System equation is in this case
For the sake of convenient, x state is still used on the right side of above-mentioned 3rd equation equal sign, wherein
Simultaneously according to above formula, Rossler system itself is expressed as
Communication system transmitting terminal sends a signal alpha (x) or sends all 3 signals of y state, and receiving end is selected linear
The Brunovsky canonical form of system
Wherein v is scalar control input, defines error e=y-z, error system is
Design controller
V=α (x)+e1+3e2+3e3
It has been the whole negative real part poles of error system configuration, thus asymptotically stability error system according to lineary system theory
And realize Brunovsky canonical form to Rossler chaos generalized synchronization, if communication system transmitting terminal only sends a letter
Number α (x), receiving end carry out 3 integrals to the signal to obtain y state；If communication system transmitting terminal only sends the complete of y state
3, portion signal, to y3Differential obtains α (x).
A method of utilizing the synchronous Rossler chaotic signal of linear system local generalized, including following procedure:
If controlled Rossler system form is as follows:
Wherein k1,k2And k3It is not all 0, and meets k1(k1+ak2)=- k2(k2+k3), system drifting vector field is
Input vector field is
Assuming that k2≠ 0, and set k1=gk2, then k3=-(1+ag+g2)k2, at this point,
Make Lie derivatives calculating
Show that it meets the involution condition of feedback linearization,
The requirement of another regularity conditionsIt is computedI.e.
(1+ag+g2)x1-gx3=(a+c+g) (1+ag+g2)
It illustrates the plane in 3 dimension spaces, the plane can not be allowed to disappear, but can try to allow it far from Rossler attractor；
PlaneIt is in recently to initial point distance:
As g → ∞, there is x1' → ∞ and x3' → 1, thus
This means that can get sufficiently large local space by adjusting g and realize feedback linearization；
It enablesIt is canonical pairing vector field with span { k, X }, andIn space with span { k, adfK } only exist
It is different on one null set, it calculates
Due to Det (k, X, adf)=1, X thus span (k, X, adfIt X) is canonical vector field, InIn space withIt is only different on a null set, structural regime transformation
Under this state
It noticesAndSo using state transformationWithRealize feedback linearization, i.e.,
y1=-x1+gx2
y2=gx1+(1+ag)x2+x3
y3=(1+ag) x1+(a-g+a2g)x2-(c+g)x3+x1x3
System equation is
Rossler system equation is accordingly
Communication system transmitting terminal sends this signal of the equal sign right end of the 3rd equation of above formula, can also send y state
The Brunovsky canonical form of linear system is selected in all 3 signals, receiving end
Wherein v is scalar control input, defines error e=y-z, error system is
Design controller
V=-b (c+g-x1)+(a-g+a2g)x1+a(a-g+a2g)x2+(c-x1)(c+g-x1)x3-(x2+x3)(1+ag+x3)+
e1+3e2+3e3=α (x)+e1+3e2+3e3
According to lineary system theory, it is clear that be the whole negative real part poles of error system configuration, thus asymptotically stability error
System simultaneously realizes the generalized synchronization that can linearly control standard type to Rossler chaos, if communication system transmitting terminal only sends one
A signal alpha (x), receiving end carry out 3 integrals to the signal to obtain y state；If communication system transmitting terminal only sends y state
All 3 signals, to y3Differential obtains α (x).
In the present invention, Rossler chaos only is produced in local state space, but adjust by parameter, the model of this part
Enclosing can arbitrarily adjust to comprising entire Rossler attractor, therefore be equal to from effect that global to generate Rossler mixed
It is ignorant；Can also be adjusted by parameter makes it that can not generate Rossler chaos, improves its confidentiality in extreme circumstances；Finally, depositing
In a wide range of adjustable parameter, secrecy ability is further increased.
Beneficial effects of the present invention are mainly manifested in: improving confidentiality.
Detailed description of the invention
Fig. 1 is the schematic diagram of Rossler chaos.
Fig. 2 is k2=6.0 and k3Curved surface when=1.0The signal intersected with Rossler attractor
Figure.
Fig. 3 is k2=0.06 and k3Curved surface when=1.0Attract the schematic diagram that do not hand over Rossler
Fig. 4 is the schematic diagram of the Rossler chaos under the state transformation of embodiment 1.
Fig. 5 is the linear system trajectory diagram of embodiment 1.
Fig. 6 is the error dynamics figure of embodiment 1.
Plane when Fig. 7 is g=0.1The schematic diagram intersected with Rossler attractor.
Plane when Fig. 8 is g=6Attract disjoint schematic diagram with Rossler.
Fig. 9 is the schematic diagram of the Rossler chaos under the state transformation of embodiment 2.
Figure 10 is the linear system trajectory diagram of embodiment 2.
Figure 11 is the error dynamics figure of embodiment 2.
Specific embodiment
The invention will be further described below in conjunction with the accompanying drawings.
Embodiment 1
Referring to Fig.1~Fig. 6, a kind of secret communication receiving end are believed using the synchronous Rossler chaos of linear system local generalized
Number method, including following procedure:
If controlled Rossler system form is as follows:
Wherein k2And k3It is not all 0, system drifting vector field is
Input vector field is
Make Lie derivatives
Due to [k, adfK]=0, such as
It can not exact feedback linearization for 0.It only may be implemented under special circumstances, work as k3≠
0 and k2When=0,Show system can global feedback linearisation, this is also that related document has discussed
The case where (Lu J, Zhang D, Sun Y, et al.Application of normal form in chaotic
synchronization[C]//American Control Conference.IEEE,2005.).If enabling k3=-k2,Solution be x1=a+c, i.e., in addition to plane x1Outside=a+c can feedback linearization, but the plane and system
Two equalization points it is too close, and equalization point is located at plane two sides, and the plane is necessarily frequently passed through in chaos system track, this
Sample can cannot carry out generalized synchronization using feedback linearization bring is convenient.If taking Rossler system parameter a=0.2, b=
0.2 and c=5.7, and dominant vector parameter k2=6.0 and k3=1.0, curved surfaceAnd initial value (5.0,
5.0,5.0) intersection is presented in track such as Fig. 2, and illustrating also can not be same using the convenient progress broad sense of feedback linearization bring
Step；
However, selection suitable parameters make equalization point be located at one sideCurved surface, and realize
It is possible that attractor is non-intersecting with the curved surface, for example takes k2=0.06 and k3=1.0, curved surface and system trajectory are non-intersecting, such as
Fig. 3.
Indeed, it is possible to which adjustment parameter adjusts curved surfaceEnable its distance Rossler attractor compared with
Far, this point is realized by adjusting curved surface to initial point distance.It enables thusAnd
Assuming that having determined that parameter a, b, c, k2And k3, meet on curved surface to the coordinate at origin minimum distance
(x is solved from above-mentioned equation1,x2,x3), curved surface can be further calculated to the shortest distance of origin for example, to a=
0.2, b=0.2 and c=5.7 and k2=0.01 and k3=1, on curved surface to origin most nearby for (- 2.7907, -99.99966,
0.999997), be greater than 100 to the distance of far point, illustrate by the centre of sphere 100 of origin for can feedback linearization in the sphere of radius
Change, the track in comparison diagram 1 can not interfere.Change k2=0.001, shortest distance point be (- 62.610355 ,-
), 991.9859,0.9919859 can the region of feedback linearization be at least expanded in the ball using origin as the centre of sphere 1000 for radius
Body.Change k2=0.0001, shortest distance point be (660.824, -9912.1,0.99121), can feedback linearization region at least
It is expanded in the sphere using origin as the centre of sphere 10000 for radius.Change k2=-0.01, shortest distance point be (10.5078,
), it 98.2831,0.982831 can be at least by the centre of sphere 100 of origin be the sphere of radius the region of feedback linearization.Change k2=-
0.001, shortest distance point be (70.3266,990.269,0.990269), can feedback linearization region be at least expanded to
Origin is the sphere that the centre of sphere 1000 is radius.Change k2=-0.0001, shortest distance point be (668.5407,9910.39,
0.991039), can the region of feedback linearization be at least expanded in the sphere using origin as the centre of sphere 10000 for radius.
As long as reasonably selecting parameter k according to above-mentioned analysis2And k3Curved surface may be implementedWith
The separation of Rossler attractor so that regional area where Rossler chaotic signal can with feedback linearization, and this
Regional area can pass through parameter k2And k3It is adjusted to arbitrarily large.Then, feasibility has been verified, it is further contemplated that such as
What realizes feedback linearization problem.For this purpose, enabling X=adfK makees Lie derivatives
It is noted thatWork as k2And k3When not being 0This also illustrates that system can be converted to p-normal form, but the object here is not to make this
The conversion of sample, but utilizeRealize feedback linearization.It notices
Inverse of a matrix is in above formula
Show to do state transformation
Then in this case
Therefore the system equation write out using the coordinate should be lower triangular form.To avoid equation form excessively complicated,
Specific value is substituted into all parameters in following discussion, but its process and method are general.Determine Rossler system
Parameter a=0.2, b=0.2 and c=5.7, and dominant vector parameter k2=0.06 and k3=1.0, according to discussed above known at this
Under a little parameters, curved surfaceWith Rossler attractor without friendship, so being that communication system receiver can select
With this group of parameter.Under these parameters, it usesState write through system equation
Further make state transformationI.e.
System equation is in this case
For the sake of convenient, x state is still used on the right side of above-mentioned 3rd equation equal sign, wherein
Simultaneously according to above formula, Rossler system itself can be expressed as
Communication system transmitting terminal can send a signal alpha (x) or send all 3 signals of y state.Receiving end
Select the Brunovsky canonical form of linear system
Wherein v is scalar control input.Error e=y-z is defined, error system is
Design controller
V=α (x)+e1+3e2+3e3
It has been the whole negative real part poles of error system configuration, thus asymptotically stability error system according to lineary system theory
And realize Brunovsky canonical form to Rossler chaos generalized synchronization.If communication system transmitting terminal only sends a letter
Number α (x), receiving end can carry out 3 integrals to the signal to obtain y state；If communication system transmitting terminal only sends y state
All 3 signals, can be to y3Differential obtains α (x).
To verify this technology, generalized synchronization of the Matlab software emulation linear system to Rossler system, parameter are utilized
A=0.2, b=0.2, c=5.7, k2=0.06 and k3=1, initial value is (5.0,5.0,5.0)TRossler chaotic systems track
Fig. 1 is seen；Fig. 4 is same track under y state, at this time corresponding initial value be (- 494.6073, -497.953,
1025.3435)T；Fig. 5 be system trajectory of the linear system at control amount v, Initial State of Linear Systems be (- 694.6073 ,-
697.953,825.3435)T；The track difference of Fig. 4 and Fig. 5 is shown in Fig. 6, shows that generalized synchronization has been realized.
Embodiment 2
Referring to Fig.1, Fig. 7~Figure 11, a kind of secret communication receiving end are mixed using the synchronous Rossler of linear system local generalized
The method of ignorant signal, including following procedure:
If controlled Rossler system form is as follows:
Wherein k1,k2And k3It is not all 0, and meets k1(k1+ak2)=- k2(k2+k3) system drifting vector field is
Input vector field is
Assuming that k2≠ 0, and set k1=gk2, then k3=-(1+ag+g2)k2, at this point,
Make Lie derivatives calculating
Show that it meets the involution condition of feedback linearization.
The requirement of another regularity conditionsIt is computedI.e.
(1+ag+g2)x1-gx3=(a+c+g) (1+ag+g2)
Illustrate the plane in 3 dimension spaces, we can not allow the plane to disappear, but can try that it is allowed to inhale far from Rossler
Introduction.For example, plane intersects with Rossler attractor as g=0.1；Work as g=6, it is non-intersecting with Rossler attractor, such as
Shown in Fig. 7 and Fig. 8.
PlaneIt is in recently to initial point distance:
As g → ∞, there is x '1→ ∞ and x '3→ 1, thus
This means that can get sufficiently large local space by adjusting g and realize feedback linearization.Such as to a=0.2 and
The case where c=5.7, to makePlane separation initial point distance at least 100, from
Solve g=94.1112 or g=-105.909.
It is now discussed with the feedback linearization how realized on above-mentioned local space.It enables thusObviously with span k,
X } it is canonical pairing vector field, andIn space with span { k, adfK } it is only different on a null set.It calculates
Due to Det (k, X, adf)=1, X thus span (k, X, adfIt X) is canonical vector field, InIn space withIt is only different on a null set.Structural regime transformation
Under this state
It noticesAndSo state transformation can be used WithRealize feedback linearization, i.e.,
y1=-x1+gx2
y2=gx1+(1+ag)x2+x3
y3=(1+ag) x1+(a-g+a2g)x2-(c+g)x3+x1x3
System equation is
Rossler system equation is accordingly
Communication system transmitting terminal can send this signal of equal sign right end of the 3rd equation of above formula or send y shape
All 3 signals of state.The Brunovsky canonical form of receiving end selection linear system
Wherein v is scalar control input, defines error e=y-z, error system is
Design controller
V=-b (c+g-x1)+(a-g+a2g)x1+a(a-g+a2g)x2+(c-x1)(c+g-x1)x3-(x2+x3)(1+ag+x3)+
e1+3e2+3e3=α (x)+e1+3e2+3e3
According to lineary system theory, it is clear that be the whole negative real part poles of error system configuration, thus asymptotically stability error
System simultaneously realizes the generalized synchronization that can linearly control standard type to Rossler chaos.If communication system transmitting terminal only sends one
A signal alpha (x), receiving end can carry out 3 integrals to the signal to obtain y state；If communication system transmitting terminal only sends y
All 3 signals of state, can be to y3Differential obtains α (x).
To verify this technology, generalized synchronization of the Matlab software emulation linear system to Rossler system, parameter are utilized
A=0.2, b=0.2, c=5.7 and g=6.0, initial value are (5.0,5.0,5.0)TRossler chaotic systems track seen
Fig. 1；Fig. 9 is the same track under y state, and corresponding initial value is (25.0,46.0,50.1) at this timeT；Figure 10 is linear system
System trajectory at control amount v, initial value are (5.0,21.0,30.1)T；The track difference of Fig. 9 and Figure 10 is shown in Figure 11, shows broad sense
Synchronization has been realized.
Claims (2)
1. a kind of secret communication receiving end utilizes the method for the synchronous Rossler chaotic signal of linear system local generalized, feature
It is, the method includes following procedure:
If controlled Rossler system form is as follows:
Wherein k2And k3It is not all 0, system drifting vector field is
Input vector field is
Make Lie derivatives
Due to [k, adfK]=0, such as
For 0 can exact feedback linearization,It only may be implemented under special circumstances, work as k3≠ 0 and k2
When=0,Show that system can global feedback linearisation；If enabling k3=-k2,Solution be x1=a+c, i.e., in addition to plane x1Outside=a+c can feedback linearization, but the plane and system
Two equalization points it is too close, and equalization point is located at plane two sides, and the plane is necessarily frequently passed through in chaos system track, this
Sample can cannot carry out generalized synchronization using feedback linearization bring is convenient；
However, selection suitable parameters make equalization point be located at one sideCurved surface, and realize attraction
Son non-intersecting with the curved surface is possible；
Adjustment parameter adjusts curved surfaceEnable its distance Rossler attractor farther out, this point passes through adjusting
Curved surface is realized to initial point distance, is enabled thusAnd
Assuming that having determined that parameter a, b, c, k2And k3, meet on curved surface to the coordinate at origin minimum distance
(x is solved from above-mentioned equation1,x2,x3), further calculate curved surface to origin the shortest distance；
As long as according to above-mentioned analysis selection parameter k2And k3Curved surface may be implementedWith Rossler attractor
Separation so that the regional area where Rossler chaotic signal can be with feedback linearization, and this regional area can lead to
Cross parameter k2And k3It is adjusted to arbitrarily large, then, feasibility has been verified, it further considers how to realize feedback linearization problem,
For this purpose, enabling X=adfK makees Lie derivatives
It is noted thatWork as k2And k3When not being 0This also illustrates that system can be converted to p-normal form, utilizesadfK, k realize feedback
Linearisation, it is noted that
Inverse of a matrix is in above formula
Show to do state transformation
Then in this case
It therefore is lower triangular form, curved surface using the system equation that the coordinate writes outIt is inhaled with Rossler
Introduction, so being communication system receiver selected parameter, under these parameters, is used without friendshipState write through system equation:
Further make state transformationI.e.
y2=87.3515x1+11.7018x2-4.94442x1x2-0.702108x3+0.296665x1x3
System equation is in this case
For the sake of convenient, x state is still used on the right side of above-mentioned 3rd equation equal sign, wherein
Simultaneously according to above formula, Rossler system itself is expressed as
Communication system transmitting terminal sends a signal alpha (x) or sends all 3 signals of y state, and linear system is selected in receiving end
Brunovsky canonical form
Wherein v is scalar control input, defines error e=y-z, error system is
Design controller
V=α (x)+e1+3e2+3e3
It has been error system configuration all negative real part poles according to lineary system theory, thus asymptotically stability error system and real
Showed Brunovsky canonical form to Rossler chaos generalized synchronization, if communication system transmitting terminal only sends a signal alpha
(x), receiving end carries out 3 integrals to the signal to obtain y state；If communication system transmitting terminal only sends whole the 3 of y state
A signal, to y3Differential obtains α (x).
2. a kind of secret communication receiving end utilizes the method for the synchronous Rossler chaotic signal of linear system local generalized, feature
It is, the method includes following procedure:
If controlled Rossler system form is as follows:
Wherein k1,k2And k3It is not all 0, and meets k1(k1+ak2)=- k2(k2+k3), system drifting vector field is
Input vector field is
Assuming that k2≠ 0, and set k1=gk2, then k3=-(1+ag+g2)k2, at this point,
Make Lie derivatives calculating
Show that it meets the involution condition of feedback linearization, it is another
A regularity conditions requirementIt is computedI.e.
(1+ag+g2)x1-gx3=(a+c+g) (1+ag+g2)
It illustrates the plane in 3 dimension spaces, the plane can not be allowed to disappear, but can try to allow it far from Rossler attractor；
PlaneIt is in recently to initial point distance:
As g → ∞, there is x '1→ ∞ and x '3→ 1, thus
This means that can get sufficiently large local space by adjusting g and realize feedback linearization；
It enablesIt is canonical pairing vector field with span { k, X }, andIn space with span { k, adfK } only at one
It is different on null set, it calculates
Due to Det (k, X, adf)=1, X thus span (k, X, adfIt X) is canonical vector field, InIn space withIt is only different on a null set, structural regime transformation
Under this state
It noticesAndSo using state transformationWithRealize feedback linearization, i.e.,
y1=-x1+gx2
y2=gx1+(1+ag)x2+x3
y3=(1+ag) x1+(a-g+a2g)x2-(c+g)x3+x1x3
System equation is
Rossler system equation is accordingly
Communication system transmitting terminal sends this signal of the equal sign right end of the 3rd equation of above formula, can also send the whole of y state
The Brunovsky canonical form of linear system is selected in 3 signals, receiving end
Wherein v is scalar control input, defines error e=y-z, error system is
Design controller
V=-b (c+g-x1)+(a-g+a2g)x1+a(a-g+a2g)x2+(c-x1)(c+g-x1)x3-(x2+x3)(1+ag+x3)+e1+
3e2+3e3=α (x)+e1+3e2+3e3
According to lineary system theory, it is clear that be the whole negative real part poles of error system configuration, thus asymptotically stability error system
And the generalized synchronization that can linearly control standard type to Rossler chaos is realized, if communication system transmitting terminal only sends a letter
Number α (x), receiving end carry out 3 integrals to the signal to obtain y state；If communication system transmitting terminal only sends the complete of y state
3, portion signal, to y3Differential obtains α (x).
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