CN107787002A - A kind of method of the rate of information throughput in rapid evaluation wireless power communication - Google Patents

A kind of method of the rate of information throughput in rapid evaluation wireless power communication Download PDF

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CN107787002A
CN107787002A CN201711008079.2A CN201711008079A CN107787002A CN 107787002 A CN107787002 A CN 107787002A CN 201711008079 A CN201711008079 A CN 201711008079A CN 107787002 A CN107787002 A CN 107787002A
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CN107787002B (en
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周思源
周全
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Hohai University HHU
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    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04WWIRELESS COMMUNICATION NETWORKS
    • H04W24/00Supervisory, monitoring or testing arrangements
    • H04W24/06Testing, supervising or monitoring using simulated traffic
    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04BTRANSMISSION
    • H04B7/00Radio transmission systems, i.e. using radiation field
    • H04B7/02Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/0413MIMO systems

Abstract

A kind of method of the rate of information throughput in being communicated the invention discloses rapid evaluation wireless power, it is characterized in that, comprise the following steps:1) MIMO transmission WET:PB is subjected to precoding processing, then obtains radio node energy signal by transmitting terminal mapping, MIMO transmission, receiving terminal mapping processing successively;2) MIMO transmission WIT:MIMO transmission information signal is used between radio node to access point, using with WET identical beam-forming methods, obtain the signal interference ratio for the information signal that radio node receives;3) rate of information throughput:By the desired value for the rate of information throughput that WIT is calculated.Advantage:The present invention can quickly and accurately evaluate the rate of information throughput of system and disclose influence of the systematic parameter of network to the rate of information throughput.

Description

A kind of method of the rate of information throughput in rapid evaluation wireless power communication
Technical field
The method of the rate of information throughput, belongs in radio communication in being communicated the present invention relates to a kind of rapid evaluation wireless power Energy acquisition technical field.
Background technology
The PB for considering to be distributed on a large scale carries out energy transmission for radio node, and radio node is carried out using the energy collected Information transfer, information is uploaded to access point.PB, radio node and access point have more antennas to be transmitted.Integrated communication Process is divided into two stages, and first stage is the WET processes from PB to radio node, second stage be from radio node to The WIT processes of access point.During the WIT of second stage, the co-channel interference that access point is subject to is contemplated in network mould In type, interference source is in other radio nodes of random distribution around access point.Based on network model described above, want to obtain The rate of information throughput for obtaining WIT is a complex target, it usually needs carries out Monte Carlo simulation using software to obtain Go out the desired value of the rate of information throughput.If network size is larger, this simulation process needs to consume substantial amounts of computing resource and fortune The row time
The wireless signal that radio frequency energy acquisition technique can receive wireless device is converted to electric energy, wireless device Power network, which need not be connected, or change battery can obtain continual and steady electric energy supply, and carry out signal using the electric energy changed Processing and signal transmitting.The process of energy acquisition is commonly referred to as wireless energy transfer (WET), because low work(can be significantly increased Consuming the life-span of wireless device simultaneously reduces the limitation bottleneck of the energy in radio communication, and WET has obtained the extensive of academia and industrial quarters Concern, and the technology for a great application prospect being considered as in next generation mobile communication system.
In order to realize merging for WET and wireless information transfer (WIT), energy beacon (PB) is as a kind of the special of energy signal With launch terminal, WET specially is carried out to radio node, radio node completes WIT using the energy of collection storage.Due to WET with WIT is split as the transmission of two steps, and the sending and receiving end of PB and information signal is independent mutually, and PB assists the deployment of formula WPC networks more Convenient, contacting for WET and WIT is also more close.Therefore, it is " double near can be suppressed in WPC networks it by large scale deployment by PB Far " problem, and realize WET high efficiency of transmission and WIT high rate data transmission.
Random geometry theory can effectively portray the random distribution of extensive node location information in network, when network section When random distribution is presented in point in certain area, the random distribution that poisson process (PPP) both accurate can must describe node is special Sign, and can is enough to provide condition to obtain the mathematical analysis solution of network performance index.
In order to improve the transfer rate of energy signal and information signal, multiple-input and multiple-output (MIMO) technology can be used in In wireless power communication network.By assembling more antennas, energy signal on the terminal devices such as PB, radio node, access point It will be greatly improved with the transmission Signal to Interference plus Noise Ratio of information signal.But due to the complexity of network scenarios, enter in computer platform The test of network information transfer speed under the configuration of row certain parameter can typically utilize Monte Carlo simulation, and this emulation is extensive Need to consume substantial amounts of operation time and calculation resources under network scenarios, and each network parameter can not be analyzed to information transfer The influence of rate is needed, therefore, finding a kind of method using the quick calculating network rate of information throughput of mathematical modeling will greatly carry High operator and equipment vendor carry out the speed of network configuration.
The content of the invention
The technical problems to be solved by the invention are the defects of overcoming prior art, there is provided a kind of rapid evaluation wireless power The method of the rate of information throughput in communication, can not only quickly obtain the performance indications of system, and and can enough discloses network parameter Influence for the rate of information throughput.
In order to solve the above technical problems, the present invention provides the rate of information throughput in a kind of rapid evaluation wireless power communication Method, it is characterized in that, comprise the following steps:
1) MIMO transmission WET (MIMO transmission wireless energy transfer), WET is realized by MINO technologies:By PB Precoding processing is carried out, then obtains radio node energy by transmitting terminal mapping, MIMO transmission, receiving terminal mapping processing successively Signal;
2) MIMO transmission WIT (MIMO transmission wireless information transfer), WIT is realized by MINO technologies:From nothing Line node to using MIMO transmission information signal between access point, using with WET identical beam-forming methods, wirelessly saved The signal interference ratio for the information signal that point receives;
3) rate of information throughput:By the desired value for the rate of information throughput that WIT is calculated.
Further, the step 1), it is assumed that i-th of PB can obtain instant channel information with radio node, i.e., Channel matrix Hi, i=1,2,3 ..., PB carry out precoding processing, the energy signal x for preparing transmission are mapped into ViShape on vector Into energy transmission signal ViX, wherein, ViFor transmission matrix HiThe right singular vector of eigenvalue of maximum is corresponded to after singular value decomposition;Energy Measure transmission signal ViX is sent to radio node after mimo channel, forms energy acceptance signal HiViX, radio node utilize Transmission matrix HiLeft singular vector Ui, the energy acceptance signal H that will receiveiViX is mapped to UiOn, connecing after being processed The collection of letters number is UiHiVix;The electromagnetic energy that the radio node receives isThe energy of radio node is adopted Integrate efficiency as μ, τ is that WET time quantum, λ are distributed in a time frameiRepresent transmission matrix HiEigenvalue of maximum, its In, Φ represents the set of all energy beacons, P represents the transmission power of energy beacon, riRepresent from i-th of energy beacon to The geometric distance of wireless terminal, α are path loss coefficient.
Further, the eigenvalue of maximum
Further, the step 2), MIMO transmission information signal is used between radio node to access point, transmission Mimo channel can be expressed as matrix W, using with WET identical beam-forming methods, the information signal that radio node receives Signal interference ratio beWherein λ0For the eigenvalue of maximum corresponding to channel matrix W, r0Believe for information Number transmission range, I be co-channel interference power.
Further, the step 3), the desired value R of the WIT rate of information throughput are obtained by following formula:
Wherein,
In above equation, MnAnd M (s)d(s) represent to be directed to molecule and denominator in reception signal signal interference ratio expression formula respectively Power produces equation (MGF), and ρ and ρ ' represents the distribution density of PB and radio node, 2F respectively1Hypergeometric equation is represented, b, c are by nothing The transmitting antenna number of line node is NsTransmitting antenna number with access point is NdDefinition, db,cCorresponding each b is represented, c pairs Coefficient, T are N by the transmitting antenna number at PB endspTransmitting antenna number with radio node is NsDefinition, rcFor interfering nodes away from From the minimum distance of access point, α ' is the path loss coefficient of interference signal, and P ' is the transmission power of interference signal.
Further, in the Mn(s) in, b increases to min (N from 1 successivelyd,Ns), c is successively from max (Nd,Ns)-min (Nd,Ns) increase to (max (Nd,Ns)+min(Nd,Ns))b-2b2,db,cThe corresponding each b of expression, c pairs of coefficient,
The T:
U increases to min (N from 1 successivelyp,Ns), v is successively from max (Np,Ns)-min(Np,Ns) increase to (max (Np,Ns)+ min(Np,Ns))u-2u2, k increases to v+1, d from 1 successivelyu,vThe corresponding each u of expression, v pairs of coefficient,Represent from v+1 The number of the combination of different ks is found out in number, Beta represents Beta equations.
The beneficial effect that the present invention is reached:
The present invention carries out information transfer speed compared with the emulation based on large scale network model, using formula set forth above The test of rate, the numerical solution very close with Monte Carlo simulation result can be obtained, and the calculating time is significantly reduced, and is reduced Degree increase as the change of network size is big.
Brief description of the drawings
Fig. 1 is the graph of a relation of WPC system information transmissions Mean Speed and PB distribution densities.
Embodiment
The invention will be further described below in conjunction with the accompanying drawings.Following examples are only used for clearly illustrating the present invention Technical scheme, and can not be limited the scope of the invention with this.
A kind of method of the rate of information throughput in rapid evaluation wireless power communication, it is characterized in that, comprise the following steps:
1) MIMO transmission WET (MIMO transmission wireless energy transfer), WET is realized by MINO technologies:By PB Precoding processing is carried out, then obtains radio node energy by transmitting terminal mapping, MIMO transmission, receiving terminal mapping processing successively Signal;
2) MIMO transmission WIT (MIMO transmission wireless information transfer), WIT is realized by MINO technologies:From nothing Line node to using MIMO transmission information signal between access point, using with WET identical beam-forming methods, wirelessly saved The signal interference ratio for the information signal that point receives;
3) rate of information throughput:By the desired value for the rate of information throughput that WIT is calculated.
The step 1), it is assumed that i-th of PB can obtain instant channel information, i.e. channel matrix with radio node Hi, i=1,2,3 ..., PB carry out precoding processing, the energy signal x for preparing transmission are mapped into ViEnergy hair is formed on vector Penetrate signal ViX, wherein, ViFor transmission matrix HiThe right singular vector of eigenvalue of maximum is corresponded to after singular value decomposition;Energy transmitting letter Number ViX is sent to radio node after mimo channel, forms energy acceptance signal HiViX, radio node utilize transmission matrix HiLeft singular vector Ui, the energy acceptance signal H that will receiveiViX is mapped to UiOn, the reception signal after being processed is UiHiVix;The electromagnetic energy that the radio node receives isThe energy acquisition efficiency of radio node is μ, τ are that WET time quantum, λ are distributed in a time frameiRepresent transmission matrix HiEigenvalue of maximum, wherein, Φ represent The set of all energy beacons, P represents the transmission power of energy beacon, riRepresent from i-th of energy beacon to wireless terminal Geometric distance, α are path loss coefficient.
The eigenvalue of maximum
The step 2), MIMO transmission information signal, the mimo channel of transmission are used between radio node to access point Matrix W can be expressed as, using with WET identical beam-forming methods, the signal interference ratio for the information signal that radio node receives ForWherein λ0For the eigenvalue of maximum corresponding to channel matrix W, r0For the transmission of information signal Distance, I are the power of co-channel interference.
The step 3), the desired value R of the WIT rate of information throughput are obtained by following formula:
Wherein,
In above equation, MnAnd M (s)d(s) represent to be directed to molecule and denominator in reception signal signal interference ratio expression formula respectively Power produces equation (MGF), and ρ and ρ ' represents the distribution density of PB and radio node, 2F respectively1Represent hypergeometric equation, it is assumed that PB The transmitting antenna number at end is Np, the transmitting antenna number of radio node is Ns, the transmitting antenna number of access point is Nd, then exist Mn(s) in, b increases to min (N from 1 successivelyd,Ns), c is successively from max (Nd,Ns)-min(Nd,Ns) increase to (max (Nd,Ns)+ min(Nd,Ns))b-2b2,db,cRepresent corresponding each b, c pairs of coefficient (this coefficient document [G.Amarasuriya, Etc, IEEE Trans.Commun., Vol.60, No.7, pp.1823-1837, July 2012] in provide), and constant T's In expression formula, u increases to min (N from 1 successivelyp,Ns), v is successively from max (Np,Ns)-min(Np,Ns) increase to (max (Np,Ns)+ min(Np,Ns))u-2u2, k increases to v+1, d from 1 successivelyu,vThe corresponding each u of expression, v pairs of coefficient,Represent from v+1 The number of the combination of different ks is found out in number, Beta represents Beta equations, rcIt is interfering nodes apart from the nearest of access point Distance, α ' are the path loss coefficient of interference signal, and P ' is the transmission power of interference signal, and s, x represent the variable being integrated.
Embodiment:In Simulation Test Environment, a wireless sensor network is simulated, if being disposed to random distribution in network The dry PB for meeting PPP random geometry models, and using multiple antennas by beam forming to wireless senser transmission energy signal, One wireless senser first gathers the energy signal launched from the PB of surrounding, recycles the electric energy of storage on WAP Pass information.Assuming that the energy conversion efficiency of rectification circuit is μ=60%, path loss factor alpha=3, rc=800m, P '= 0.5W。
It is as follows by using heretofore described method, its simulated effect:
Assuming that PB ends are configured with 2 antennas, and 2 antennas are respectively configured in radio node, gradually increase PB distribution density, By 10-5m-210 are increased to successively-3m-2, and as a control group in change system PB transmission power, it can be seen that base in Fig. 1 In the analytic solutions and the result for the numerical solution for emulating to obtain by a large amount of Monte Carlo of the rate of information throughput that the method provides It is highly identical, in figure, based on the result that the method provides with point be, based on Monte Carlo emulation result solid line table Show.Also, along with the increase of PB distribution densities, when carrying out energy transmission using the inventive method, the information transfer speed of system Rate also significantly improves.
Described above is only the preferred embodiment of the present invention, it is noted that for the ordinary skill people of the art For member, without departing from the technical principles of the invention, some improvement and deformation can also be made, these are improved and deformation Also it should be regarded as protection scope of the present invention.

Claims (6)

1. a kind of method of the rate of information throughput in rapid evaluation wireless power communication, it is characterized in that, comprise the following steps:
1) MIMO transmission WET, WET is realized by MINO technologies:PB is subjected to precoding processing, then reflected successively by transmitting terminal Penetrate, MIMO transmission, receiving terminal mapping processing, obtain radio node energy signal;
2) MIMO transmission WIT, WIT is realized by MINO technologies:MIMO transmission information is used between radio node to access point Signal, using with WET identical beam-forming methods, obtain the signal interference ratio for the information signal that radio node receives;
3) rate of information throughput:By the desired value for the rate of information throughput that WIT is calculated.
2. the method for the rate of information throughput, its feature in a kind of rapid evaluation wireless power communication according to claim 1 It is the step 1), it is assumed that i-th of PB can obtain instant channel information, i.e. channel matrix H with radio nodei, i= 1st, 2,3 ..., PB carries out precoding processing, and the energy signal x for preparing transmission is mapped into ViEnergy transmitting letter is formed on vector Number ViX, wherein, ViFor transmission matrix HiThe right singular vector of eigenvalue of maximum is corresponded to after singular value decomposition;Energy transmission signal ViX is sent to radio node after mimo channel, forms energy acceptance signal HiViX, radio node utilize transmission matrix Hi Left singular vector Ui, the energy acceptance signal H that will receiveiViX is mapped to UiOn, the reception signal after being processed is UiHiVix;The electromagnetic energy that the radio node receives isThe energy acquisition efficiency of radio node is μ, τ are that WET time quantum, λ are distributed in a time frameiRepresent transmission matrix HiEigenvalue of maximum, wherein, Φ represent The set of all energy beacons, P represents the transmission power of energy beacon, riRepresent from i-th of energy beacon to wireless terminal Geometric distance, α are path loss coefficient.
3. the method for the rate of information throughput, its feature in a kind of rapid evaluation wireless power communication according to claim 2 It is the eigenvalue of maximum
4. the method for the rate of information throughput, its feature in a kind of rapid evaluation wireless power communication according to claim 2 It is the step 2), MIMO transmission information signal is used between radio node to access point, the mimo channel of transmission can be with table Matrix W is shown as, is using with WET identical beam-forming methods, the signal interference ratio for the information signal that radio node receivesWherein λ0For the eigenvalue of maximum corresponding to channel matrix W, r0For the transmission distance of information signal From I is the power of co-channel interference.
5. the method for the rate of information throughput, its feature in a kind of rapid evaluation wireless power communication according to claim 3 It is the step 3), the desired value R of the WIT rate of information throughput is obtained by following formula:
<mrow> <mi>R</mi> <mo>=</mo> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <mi>&amp;infin;</mi> </msubsup> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>M</mi> <mi>n</mi> </msub> <mo>(</mo> <mi>s</mi> <mo>)</mo> <mo>)</mo> </mrow> <msub> <mi>M</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>s</mi> </mrow>
Wherein,
<mrow> <msub> <mi>M</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mo>&amp;Sigma;</mo> <mrow> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mrow> </msub> <mfrac> <mrow> <msup> <mi>b</mi> <mrow> <mi>c</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>d</mi> <mrow> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mrow> </msub> </mrow> <mrow> <mi>c</mi> <mo>!</mo> </mrow> </mfrac> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <mi>&amp;infin;</mi> </msubsup> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> <mi>&amp;rho;</mi> </mrow> <mi>&amp;alpha;</mi> </mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi>s</mi> <mi>&amp;mu;</mi> <mi>&amp;tau;</mi> <mi>P</mi> </mrow> <mo>)</mo> </mrow> <mfrac> <mn>2</mn> <mi>&amp;alpha;</mi> </mfrac> </msup> <msup> <mi>Tx</mi> <mfrac> <mn>2</mn> <mi>&amp;alpha;</mi> </mfrac> </msup> <mo>-</mo> <mi>b</mi> <mi>x</mi> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mi>c</mi> </msup> <mi>d</mi> <mi>x</mi> </mrow>
<mrow> <mi>T</mi> <mo>=</mo> <msub> <mo>&amp;Sigma;</mo> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> </msub> <msubsup> <mi>C</mi> <mrow> <mi>v</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <msup> <mi>u</mi> <mrow> <mo>-</mo> <mfrac> <mn>2</mn> <mi>&amp;alpha;</mi> </mfrac> </mrow> </msup> <mi>B</mi> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mfrac> <mn>2</mn> <mi>&amp;alpha;</mi> </mfrac> <mo>,</mo> <mi>v</mi> <mo>+</mo> <mn>1</mn> <mo>-</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo>/</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>M</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msup> <mi>&amp;pi;&amp;rho;</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>&amp;tau;</mi> <mo>)</mo> </mrow> <msubsup> <mi>r</mi> <mn>0</mn> <mi>&amp;alpha;</mi> </msubsup> <msubsup> <mi>r</mi> <mi>c</mi> <mrow> <mn>2</mn> <mo>-</mo> <msup> <mi>&amp;alpha;</mi> <mo>&amp;prime;</mo> </msup> </mrow> </msubsup> <msup> <mi>sP</mi> <mo>&amp;prime;</mo> </msup> </mrow> <mrow> <msup> <mi>&amp;alpha;</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mn>2</mn> </mrow> </mfrac> <mn>2</mn> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mfrac> <mn>2</mn> <msup> <mi>&amp;alpha;</mi> <mo>&amp;prime;</mo> </msup> </mfrac> <mo>;</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mn>2</mn> <msup> <mi>&amp;alpha;</mi> <mo>&amp;prime;</mo> </msup> </mfrac> <mo>;</mo> <mo>-</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <mi>&amp;tau;</mi> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>r</mi> <mn>0</mn> <mi>&amp;alpha;</mi> </msubsup> <msup> <mi>sP</mi> <mo>&amp;prime;</mo> </msup> <msubsup> <mi>r</mi> <mi>c</mi> <mrow> <mo>-</mo> <msup> <mi>&amp;alpha;</mi> <mo>&amp;prime;</mo> </msup> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow>
In above equation, MnAnd M (s)d(s) represent that being directed to the power of molecule and denominator in reception signal signal interference ratio expression formula produces respectively Raw equation, ρ and ρ ' represent the distribution density of PB and radio node, 2F respectively1Hypergeometric equation is represented, b, c are by radio node Transmitting antenna number is NsTransmitting antenna number with access point is NdDefinition, dB, cRepresent corresponding each b, c pairs of coefficient, T by The transmitting antenna number at PB ends is NpTransmitting antenna number with radio node is NsDefinition, rcIt is interfering nodes apart from access point Minimum distance, α ' be interference signal path loss coefficient, P ' be interference signal transmission power.
6. the method for the rate of information throughput, its feature in a kind of rapid evaluation wireless power communication according to claim 5 It is, in the Mn(s) in, b increases to min (N from 1 successivelyd, Ns), c is successively from max (Nd, Ns)-min(Nd, Ns) increase to (max(Nd, Ns)+min(Nd, Ns))b-2b2, dB, cThe corresponding each b of expression, c pairs of coefficient,
The T:
<mrow> <mi>T</mi> <mo>=</mo> <msub> <mo>&amp;Sigma;</mo> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> </msub> <msubsup> <mi>C</mi> <mrow> <mi>v</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <msup> <mi>u</mi> <mrow> <mo>-</mo> <mfrac> <mn>2</mn> <mi>&amp;alpha;</mi> </mfrac> </mrow> </msup> <mi>B</mi> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mfrac> <mn>2</mn> <mi>&amp;alpha;</mi> </mfrac> <mo>,</mo> <mi>v</mi> <mo>+</mo> <mn>1</mn> <mo>-</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo>/</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> </mrow>
U increases to min (N from 1 successivelyp, Ns), v is successively from max (Np, Ns)-min(Np, Ns) increase to (max (Np, Ns)+min (Np, Ns))u-2u2, k increases to v+1, d from 1 successivelyU, vThe corresponding each u of expression, v pairs of coefficient,Represent from v+1 number In find out different ks combination number, Beta represents Beta equations.
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