CN1298120C - Detection method for multiple input multiple output system channel capacity - Google Patents

Abstract

The present invention relates to a detection method for the channel capacity of a multi-input multi-output system. The method mainly aims to analyze the influence of fading correlation and channel physical parameters on the channel capacity of the multi-input multi-output system. A channel matrix is resolved into the product of a channel fading correlation matrix and a matrix U; the elements of the matrix U are N(0, 1) variables obeying independence and the same distribution. The influence of the size of scattering angles and the distance between antennas on the channel capacity under the condition that fading correlation exists is analyzed by reestablishing a fading correlation mathematical model. The closed type expression of the channel capacity of the M*N multi-input multi-output system is analyzed and derived by the cyclic matrix theory. The influence of the fading correlation on the channel capacity of the multi-input multi-output system can be obtained.

Description

A kind of detection method of channel capacity of multi-input multi-output system

Technical field

The present invention relates to a kind of detection method of channel capacity of multi-input multi-output system, this method mainly is in order to detect the relevant influence that reaches the channel physical parameter to channel capacity of multi-input multi-output system of decline.

Background technology

MIMO technique will be one of main path of future mobile communication system raising spectrum efficiency.The traditional wireless communication theory is considered as causing the interference of signal fadeout with multipath transmisstion, and multi-input multi-output system adopts many antenna arrays to transmit and receive information at transmitting terminal and receiving terminal, makes full use of space resources and improves systematic function, increases power system capacity.Adopt MIMO technique as the subchannel of on original frequency range, having set up a plurality of independent parallels, utilize the spatial character of signal and multiuser detection to separate exactly, transmitted at receiving terminal.Adopt MIMO technique, transmitting terminal and receiving terminal all use many antennas, if the channel between the every pair of dual-mode antenna is separate rayleigh fading channel, then when reception antenna number during more than or equal to the transmitting antenna number, power system capacity will be with the number of transmit antennas linear growth.The channel capacity of multi-input multi-output system analytical method is mainly contained following several: 1) suppose transmitting antenna and reception antenna to decline be independent identically distributed Rayleigh decline.Analyze transmitting terminal known channel information in this case, receiving terminal known channel information and transmitting terminal the unknown, channel capacity of multi-input multi-output system under two kinds of conditions of receiving terminal known channel information.If channel is a slow fading channel, then receiving terminal can have time enough to give transmitter with feedback of channel information.Transmitting terminal can distribute transmitting power on the every antenna effectively according to channel information, so that obtain higher bit rate.Though adopt the best power allocative decision can improve the channel capacity of the performance of system, raising system effectively, transmitting terminal acquisition channel information needs certain time delay but also need take certain resource.During transmitting terminal Unknown Channel information, the every antenna that the common method that adopts is a transmitting terminal all adopts the constant power emission.1996, hierarchy when Bell Laboratory has proposed a kind of diagonal angle sky was called D-BLAST (diagonally Bell Laboratories Layered Space-time).Adopt this structure can make full use of space resources, improve the channel capacity of system significantly.For the D-BLAST system, all use many antennas as transmitting terminal and receiving terminal, and suppose that the channel between every pair of dual-mode antenna is separate rayleigh fading channel, then when reception antenna number during more than or equal to the transmitting antenna number, power system capacity will be with the number of transmit antennas linear growth.Channel capacity can be expressed from the next when adopting the constant power emission

$C_{\mathrm{eq}}={\log }_2\det [I_{n_R}+\frac{P}{n_T{\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{\mathrm{HH}}^H]$

Wherein, n _TAnd n _RRepresent transmitting antenna and reception antenna number respectively; H is n _R* n _TThe dimension channel matrix, element h _IjRepresent the multiple path gain between i root reception antenna and j transmit antennas.H is the conjugate transpose of matrix; σ _n ²Variance for additive white Gaussian noise n; Det is a determinant of a matrix.Adopt the advantage of constant power emission to be: receiver needn't can reduce the complexity that whole system realizes to the transmitter feedback channel information; 2) in the actual environment, the independence between limited scattering thing number is difficult to guarantee to decline around antenna distance and the antenna array.Exist correlation can cause the reduction of channel capacity of multi-input multi-output system between decline.One piece of by name " decline relevant and to the influence of multiaerial system channel capacity " (IEEE Transaction onCommunications, vol48, No.3, pp.502-512, in March, 2000) adopt " one-ring " model that the spatial fading correlation is carried out mathematical modeling in the literary composition, from channel capacity of multi-input multi-output system bound angle analysis the influence of decline correlation to channel capacity.But do not provide the closed solutions of channel capacity, do not analyze of the influence of channel physical parameter simultaneously yet channel capacity of multi-input multi-output system.One piece of " having MIMO wireless system channel capacity under the relevant decline situation " by name (IEEE Transactionson Information Theory, vol48, No.3, pp.637-650, in March, 2002) analyzed the channel capacity of single user's multi-input multi-output system under decline relevant environment situation in the literary composition.The method that adopts is with the relevant relevant consideration respectively with emission of reception.That is, $E\left[H_{\mathrm{pj}}{H}_{\mathrm{qk}}^{*}\right]={\mathrm{\&Psi;}}_{\mathrm{jk}}^T{\mathrm{\&Psi;}}_{\mathrm{pq}}^R.$ H wherein _IjBe the multiple path gain between transmitting antenna j and reception antenna i.ψ ^TAnd ψ ^RIt is n * n correlation matrix.At this decline model, done following hypothesis: (1) transmitting antenna p and q and the relevant ψ of being of decline between same reception antenna _Pq ^T, irrelevant with reception antenna; Equally, ψ ^TWhat portray is that emission is relevant.(2) same transmit antennas is correlated with the decline between different reception antenna j and k and is ψ _Jk ^R, irrelevant with transmitting antenna.ψ ^RWhat portray is to receive to be correlated with.(3) two different antennae to decline relevantly be the relevant product relevant of corresponding emission with reception.(4) ψ ^TAnd ψ ^RThe experience of the characteristic value into F that distributes _TAnd F _RProvided the progressive expression formula of channel capacity of multi-input multi-output system under two kinds of situations of transmitting terminal known channel parameter and Unknown Channel parameter in the literary composition, analyzed the influence of antenna distance progressive channel capacity.But the influence of channel capacity is not provided corresponding analysis for the scattering environments parameter.

Summary of the invention

The object of the present invention is to provide in a kind of detection fading environment, channel physical parameter and antenna distance are to the detection method of the channel capacity of multi-input multi-output system of the influence of channel capacity.

For achieving the above object, the technical solution used in the present invention is:

1) at first makes up the up link of 3 * 3 multi-input multi-output systems

A) transmitting of the mobile you of setting to separate, it is UCA (the uniform Circular array) antenna array of r that reception antenna adopts radius, the received signal vector of arrowband multi-input multi-output system is

y＝Hx+n (1)

In the formula, x=[x ₁, x ₂, x ₃] ^HBe 3 dimension emission signal vector; Y=[y ₁, y ₂, y ₃] ^HBe 3 dimension received signal vectors; H is 3 * 3 channel matrixes; N is that average is 0, and variance is 1 additive white Gaussian noise;

B) only consider up link, channel matrix H is decomposed,

H＝R ^1/2U (2)

R is 3 * 3 reception correlation matrixes in the formula; The element of U is that average is 0, and variance is 1 the multiple Gaussian random variable of independence;

C) coefficient correlation between any two antennas of measurement

If it is A that multipath signal arrives the angle of scattering of UCA antenna array, and the multipath signal arrival direction evenly distributes in [-A, +A], is that coefficient correlation is between any two antennas of d for spacing

In the formula $k=\frac{2\mathrm{\&pi;}}{\mathrm{\&lambda;}}$ Be wave number,  is the average arrival direction angle of ripple, and when A=π, (3) Shi Kejian is

r _i.l＝J ₀(kd) (4)

For up link, antenna for base station set up than around high many of scattering thing, so angle of scattering is smaller, (3) formula can be reduced to like this

Be without loss of generality, make =0, then following formula can be reduced to

$r_{i,l}=\frac{\sin \left(\mathrm{kdA}\right)}{\mathrm{kdA}}=\sin c\left(\frac{2\mathrm{\&pi;dA}}{\mathrm{\&lambda;}}\right)---\left(5\right)$

So it is a circulation symmetrical matrix that multiple-input and multiple-output 3 * 3 channels receive correlation matrix R

$R=\left[\begin{array}{ccc}1& \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)\\ \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& 1& \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)\\ \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& 1\end{array}\right]=\left[\begin{array}{ccc}{c}_0& c_1& c_2\\ c_2& c_0& c_1\\ c_1& c_2& c_0\end{array}\right]---\left(6\right)$

D) measure the characteristic value that multiple-input and multiple-output 3 * 3 channels receive correlation matrix R as follows;

If circular matrix C has following form

$C=\left[\begin{array}{cccccccc}{c}_0& c_1& c_2& c_3& \mathrm{\&Lambda;}& \mathrm{\&Lambda;}& \mathrm{\&Lambda;}& c_{n-1}\\ c_{n-1}& c_0& c_1& c_2& & & & \\ c_{n-2}& & & & O& & & \\ & & & O& O& & & M\\ M& & \mathrm{\&Lambda;}& & O& O& & \\ & & & & & & & c_2\\ c_2& & & & & & c_0& c_1\\ c_1& & & \mathrm{\&Lambda;}& \mathrm{\&Lambda;}& & & c_0\end{array}\right]$

Be that C itself is a special Toeplitz matrix, the characteristic value ψ of C _k, characteristic vector y _kBe separating of following formula,

Cy＝ψy

Following formula is equivalent to a following n equation

$\underset{k=0}{\overset{m-1}{\mathrm{\&Sigma;}}}{c}_{n-m+k}{y}_k+\underset{k=m}{\overset{n-1}{\mathrm{\&Sigma;}}}{c}_{k-m}{y}_k={\mathrm{\&psi;y}}_m,m=\mathrm{0,1},\mathrm{\&Lambda;},n-1$

Get y _k=exp (j2 π mk/n) separates for following formula, so the characteristic value of C is

${\mathrm{\&psi;}}_m=\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{c}_k\exp (-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;mk}/n)$

The characteristic value that promptly gets circular matrix R is

${\mathrm{\&psi;}}_m=\underset{k=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_k\exp (-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;mk}/3),m=\mathrm{0,1,2}---\left(7\right)$

So have

${\mathrm{\&psi;}}_0=\underset{k=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_k=1+c_1+{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">c</figure-callout>}}_2=1+2\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)---\left(8\right)$

${\mathrm{\&psi;}}_1=\underset{k=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_k\exp (-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;k}/3)=1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)---\left(9\right)$

${\mathrm{\&psi;}}_2=\underset{k=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_k\exp (-j4\mathrm{\&pi;k}/3)=1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)---\left(10\right)$

2) ergodic capacity of measurement 3 * 3 multi-input multi-output systems

Suppose the receiver known channel state, and transmitter the unknown, transmitting terminal adopts the constant power emission, and total transmission power setting is P, and then the transmitting power of every transmit antennas is P/3, and channel capacity can be expressed as

$C={\log }_2\left(\det (I+\frac{P}{3}{\mathrm{HH}}^H)\right)---\left(11\right)$

With (2) formula substitution (11) Shi Kede

$C={\log }_2\left(\det (I+\frac{P}{3}{R}^{1/2}{\mathrm{UU}}^H{R}^{1/2})\right)$

$={\log }_2\left(\det (I+\frac{P}{3}{\mathrm{RUU}}^H)\right)---\left(12\right)$

Make W=UU ^H, R and W are carried out characteristic value decomposition respectively get

$C={\log }_2\left(\det (I+\frac{P}{3}{\mathrm{\&Lambda;}}_R{\mathrm{\&Lambda;}}_W)\right)=\underset{i=1}{\overset{3}{\mathrm{\&Sigma;}}}{\log }_2(1+\frac{P}{3}{\mathrm{\&psi;}}_i{\mathrm{\&lambda;}}_i)---\left(13\right)$

In the formula,

${\mathrm{\&Lambda;}}_R=\left[\begin{array}{ccc}{\mathrm{\&psi;}}_0& 0& 0\\ 0& {\mathrm{\&psi;}}_1& 0\\ 0& 0& {\mathrm{\&psi;}}_2\end{array}\right]=\left[\begin{array}{ccc}1+2\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& 0& 0\\ 0& 1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& 0\\ 0& 0& 1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)\end{array}\right]$

${\mathrm{\&Lambda;}}_W=\left[\begin{array}{ccc}{\mathrm{\&lambda;}}_1& 0& 0\\ 0& {\mathrm{\&lambda;}}_2& 0\\ 0& 0& {\mathrm{\&lambda;}}_3\end{array}\right]$

In the formula (13), λ is the characteristic value of W, λ ₁Probability density function be

In the formula,

$L_k^{n-m}\left(x\right)=\frac{1}{k!}{e}^x{x}^{m-n}\frac{d^k}{{\mathrm{dx}}^k}\left(e^{-x}{x}^{n-m+k}\right)$

Can push away by formula (14)

$P_{{\mathrm{\&lambda;}}_1}\left({\mathrm{\&lambda;}}_1\right)=\frac{1}{3}[1+{(1-{\mathrm{\&lambda;}}_1)}^2+\frac{1}{4}{({\mathrm{\&lambda;}}_1^2-{4\mathrm{\&lambda;}}_1+2)}^2]\exp (-{\mathrm{\&lambda;}}_1)---\left(15\right)$

Promptly getting the multi-input multi-output system ergodic capacity is

$E\left(C\right)=E[{\log }_2(1+\frac{P}{3}(1+\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)){\mathrm{\&lambda;}}_1)+2E[{\log }_2(1+\frac{P}{3}(1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)){\mathrm{\&lambda;}}_1)]$

$=\frac{1}{3}{\&Integral;}_0^{\&infin;}{\log }_2[1+\frac{P}{3}(1+2\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)){\mathrm{\&lambda;}}_1]\left[e^{{-\mathrm{\&lambda;}}_1}(1+{(1-{\mathrm{\&lambda;}}_1)}^2+{({\mathrm{\&lambda;}}_1^2-{4\mathrm{\&lambda;}}_1+2)}^2)\right]{\mathrm{d\&lambda;}}_1$

$+\frac{2}{3}\underset{0}{\overset{\&infin;}{\&Integral;}}{\log }_2[1+\frac{P}{3}(1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)){\mathrm{\&lambda;}}_1]\left[e^{{-\mathrm{\&lambda;}}_1}(1+{(1-{\mathrm{\&lambda;}}_1)}^2+{({\mathrm{\&lambda;}}_1^2-{4\mathrm{\&lambda;}}_1+2)}^2)\right]{\mathrm{d\&lambda;}}_1---\left(16\right)$

$=\frac{{-22a}^3+{14a}^2-5a+1}{{12a}^3\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">ln</figure-callout>}2}+e^{\frac{1}{a}}(\frac{1}{3\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">ln</figure-callout>}2}+\frac{{8a}^2-4a+1}{12\ln 2a^4})\mathrm{\&zeta;}+\frac{a-1}{3a\ln 2}-\frac{1+{2a}^2}{{3a}^2\ln 2}{e}^{\frac{1}{a}}\mathrm{\&zeta;}$

$+\frac{{-22b}^3+{14b}^2-5b+1}{{6\ln 2b}^3}+e^{\frac{1}{b}}(\frac{2}{3\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">ln</figure-callout>}2}+\frac{{8b}^2-4b+1}{{6b}^4\ln 2})\mathrm{\&zeta;}+\frac{b-1}{3b\ln 2}-\frac{1+{2\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}{{3\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">b</figure-callout>}}^2\ln 2}{e}^{\frac{1}{b}}\mathrm{\&zeta;}---\left(17\right)$

In the formula, $a=\frac{P}{3}(1+2\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)),b=1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)$

$\mathrm{\&zeta;}=\mathrm{\&gamma;}-\ln a+\underset{i=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{1}{i*i!}{(-\frac{1}{a})}^i,$ Wherein γ is an Euler's constant.

The present invention is that channel fading correlation matrix and matrix U are long-pending with channel matrix decomposition, and the element of matrix U is to obey independent identically distributed N (0,1) variable, by having rebulid the Mathematical Modeling of decline correlation.Analyzed and existed under the decline correlation circumstance angle of scattering size and antenna distance the influence of channel capacity; And utilize the derived enclosed of M * N channel capacity of multi-input multi-output system of circular matrix theory analysis to express.The influence of correlation to channel capacity of multi-input multi-output system can obtain declining.

Description of drawings

Fig. 1 is that the present invention receives the relation between coefficient correlation and antenna distance and angle of scattering, and wherein abscissa is the spacing/wavelength that bursts at the seams, and ordinate is for receiving coefficient correlation;

Fig. 2 is channel capacity of the present invention and antenna distance and angle of scattering relation, and wherein abscissa is an antenna distance, and ordinate is a channel capacity;

Fig. 3 is the relation of channel capacity of the present invention and angle of scattering and transmission rate, and wherein abscissa is a transmission rate, and ordinate is a channel capacity;

Fig. 4 is antenna number of the present invention and channel capacity relation, and wherein abscissa is an antenna distance, and ordinate is a channel capacity;

Fig. 5 is received signal to noise ratio of the present invention and channel capacity relation, and wherein abscissa is a signal to noise ratio, and ordinate is a channel capacity.

Embodiment

Below The whole analytical process is provided and be described in detail.

1) the multi-input multi-output system channel model is set up

At first make up the up link of 3 * 3 multi-input multi-output systems.Suppose transmitting of mobile you, through behind the spatial transmission, because influence of fading makes signal have certain correlation at reception antenna to separate.Reception antenna adopts the UCA antenna array, and establishing radius is r.

Received signal vector is

y(t)＝H(t)*x(t)+n(t)

In the formula, H (t) is the channel impulse response matrix; X (t) is an emission signal vector; Y (t) is a received signal vector; N (t) is additive white Gaussian noise (AWGN); * represent convolution algorithm.Only consider the arrowband multi-input multi-output system, then following formula is reduced to

y＝Hx+n (1)

In the formula, x=[x ₁, x ₂, x ₃] ^HBe 3 dimension emission signal vector; Y=[y ₁, y ₂, y ₃] ^HBe 3 dimension received signal vectors; H is 3 * 3 channel matrixes; N is that average is 0, and variance is 1 additive white Gaussian noise.Only consider up link.

Channel matrix H is decomposed, can obtain

H＝R ^1/2U (2)

R is 3 * 3 reception correlation matrixes in the formula; The element of U is that average is 0, and variance is 1 the multiple Gaussian random variable of independence.If it is A that multipath signal arrives the angle of scattering of UCA antenna array, and the evenly distribution in [-A, +A] of multipath signal arrival direction, promptly probability density is

Be that coefficient correlation is between any two antennas of d for spacing

In the formula $k=\frac{2\mathrm{\&pi;}}{\mathrm{\&lambda;}}$ Be wave number,  is the average arrival direction angle of ripple.When A=π, (3) Shi Kejian is

r _i.l＝J ₀(kd) (4)

For up link, antenna for base station set up than around high many of scattering thing.Therefore angle of scattering is smaller.(3) formula can be reduced to like this

Be without loss of generality, make =0.Then following formula can be reduced to

$r_{i,l}=\frac{\sin \left(\mathrm{kdA}\right)}{\mathrm{kdA}}=\sin c\left(\frac{2\mathrm{\&pi;dA}}{\mathrm{\&lambda;}}\right)---\left(5\right)$

So multiple-input and multiple-output 3 * 3 channels receive correlation matrix and can be written as

$R=\left[\begin{array}{ccc}1& \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)\\ \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& 1& \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)\\ \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& 1\end{array}\right]=\left[\begin{array}{ccc}{c}_0& c_1& c_2\\ c_2& c_0& c_1\\ c_1& c_2& c_0\end{array}\right]---\left(6\right)$

Analyze following formula as can be known, R is a circulation symmetrical matrix.Can adopt circular matrix to ask eigenvalue method to obtain its characteristic value.

Circular matrix asks the characteristic value eigenvector method as follows

If circular matrix C has following form

$C=\left[\begin{array}{cccccccc}{c}_0& c_1& c_2& c_3& \mathrm{\&Lambda;}& \mathrm{\&Lambda;}& \mathrm{\&Lambda;}& c_{n-1}\\ c_{n-1}& c_0& c_1& c_2& & & & \\ c_{n-2}& & & & O& & & \\ & & & O& O& & & M\\ M& & \mathrm{\&Lambda;}& & O& O& & \\ & & & & & & & c_2\\ c_2& & & & & & c_0& c_1\\ c_1& & & \mathrm{\&Lambda;}& \mathrm{\&Lambda;}& & & c_0\end{array}\right]$

As can be seen from the above equation, C itself is a special Toeplitz matrix.The characteristic value ψ of C _k, characteristic vector y _kBe separating of following formula,

Cy＝ψy

Following formula is equivalent to a following n equation

$\underset{k=0}{\overset{m-1}{\mathrm{\&Sigma;}}}{c}_{n-m+k}{y}_k+\underset{k=m}{\overset{n-1}{\mathrm{\&Sigma;}}}{c}_{k-m}{y}_k={\mathrm{\&psi;y}}_m,m=\mathrm{0,1},\mathrm{\&Lambda;},n-1$

Be easy to checking, y _k=exp (j2 π mk/n) separates for following formula.So the characteristic value of C is

${\mathrm{\&psi;}}_m=\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{c}_k\exp (-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;mk}/n).$

By above analysis as can be known, the characteristic value of circular matrix R is

${\mathrm{\&psi;}}_m=\underset{k=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_k\exp (-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;mk}/3),m=\mathrm{0,1,2}---\left(7\right)$

So have

${\mathrm{\&psi;}}_0=\underset{k=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_k=1+c_1+c_2=1+2\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)---\left(8\right)$

${\mathrm{\&psi;}}_1=\underset{k=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_k\exp (-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;k}/3)=1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)---\left(9\right)$

${\mathrm{\&psi;}}_2=\underset{k=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_k\exp (-j4\mathrm{\&pi;k}/3)=1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)---\left(10\right)$

2) 3 * 3 channel capacity of multi-input multi-output system analyses

Suppose the receiver known channel state, and transmitter the unknown.In this case, transmitting terminal adopts the constant power emission.Total transmission power setting is P, and then the transmitting power of every transmit antennas is P/3.Channel capacity can be expressed as

$C={\log }_2\left(\det (I+\frac{P}{3}{\mathrm{HH}}^H)\right)---\left(11\right)$

With (2) formula substitution (11) Shi Kede

$C={\log }_2\left(\det (I+\frac{P}{3}{R}^{1/2}{\mathrm{UU}}^H{R}^{1/2})\right)$

$={\log }_2\left(\det (I+\frac{P}{3}{\mathrm{RUU}}^H)\right)---\left(12\right)$

Make W=UU ^HR and W are carried out characteristic value decomposition respectively to be got

$C={\log }_2\left(\det (I+\frac{P}{3}{\mathrm{\&Lambda;}}_R{\mathrm{\&Lambda;}}_W)\right)=\underset{i=1}{\overset{3}{\mathrm{\&Sigma;}}}{\log }_2(1+\frac{P}{3}{\mathrm{\&psi;}}_i{\mathrm{\&lambda;}}_i)---\left(13\right)$

In the formula,

${\mathrm{\&Lambda;}}_R=\left[\begin{array}{ccc}{\mathrm{\&psi;}}_0& 0& 0\\ 0& {\mathrm{\&psi;}}_1& 0\\ 0& 0& {\mathrm{\&psi;}}_2\end{array}\right]=\left[\begin{array}{ccc}1+2\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& 0& 0\\ 0& 1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& 0\\ 0& 0& 1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)\end{array}\right]$

${\mathrm{\&Lambda;}}_W=\left[\begin{array}{ccc}{\mathrm{\&lambda;}}_1& 0& 0\\ 0& {\mathrm{\&lambda;}}_2& 0\\ 0& 0& {\mathrm{\&lambda;}}_3\end{array}\right]$

In the formula (13), λ is the characteristic value of W.Joint probability density function without ordering λ is

$P_{\mathrm{\&lambda;}}({\mathrm{\&lambda;}}_1,{\mathrm{\&lambda;}}_2,{\mathrm{\&lambda;}}_3)={\left(m{!K}_{m,n}\right)}^{-1}\exp \left({-\mathrm{\&Sigma;}}_i{\mathrm{\&lambda;}}_i\right)\underset{i}{\mathrm{\&Pi;}}{\mathrm{\&lambda;}}_i^{n-m}\underset{i<j}{\mathrm{\&Pi;}}{({\mathrm{\&lambda;}}_i-{\mathrm{\&lambda;}}_j)}^2$

In the formula, n and m are respectively reception antenna and number of transmit antennas purpose maximum and minimum value.K _{M, n}Be normalization coefficient.N=m=3 herein.

For analyzing the multiple-input and multiple-output ergodic capacity, need the marginal probability density of computation of characteristic values λ.

$P_{{\mathrm{\&lambda;}}_1}\left({\mathrm{\&lambda;}}_1\right)=\&Integral;\mathrm{\&Lambda;}\&Integral;P_{\mathrm{\&lambda;}}({\mathrm{\&lambda;}}_1,\mathrm{\&Lambda;\&Lambda;},{\mathrm{\&lambda;}}_m){\mathrm{d\&lambda;}}_2\mathrm{\&Lambda;}{\mathrm{\&lambda;}}_m$

Can obtain λ through deriving ₁Probability density function be

In the formula,

$L_k^{n-m}\left(x\right)=\frac{1}{k!}{e}^x{x}^{m-n}\frac{d^k}{{\mathrm{dx}}^k}\left(e^{-x}{x}^{n-m+k}\right)$

Can push away by formula (14)

$P_{{\mathrm{\&lambda;}}_1}\left({\mathrm{\&lambda;}}_1\right)=\frac{1}{3}[1+{(1-{\mathrm{\&lambda;}}_1)}^2+\frac{1}{4}{({\mathrm{\&lambda;}}_1^2-{4\mathrm{\&lambda;}}_1+2)}^2]\exp (-{\mathrm{\&lambda;}}_1)---\left(15\right)$

The multi-input multi-output system ergodic capacity is

$E\left(C\right)=E[{\log }_2(1+\frac{P}{3}(1+\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)){\mathrm{\&lambda;}}_1)+2E[{\log }_2(1+\frac{P}{3}(1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)){\mathrm{\&lambda;}}_1)]$

$=\frac{1}{3}{\&Integral;}_0^{\&infin;}{\log }_2[1+\frac{P}{3}(1+2\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)){\mathrm{\&lambda;}}_1]\left[e^{{-\mathrm{\&lambda;}}_1}(1+{(1-{\mathrm{\&lambda;}}_1)}^2+{({\mathrm{\&lambda;}}_1^2-{4\mathrm{\&lambda;}}_1+2)}^2)\right]{\mathrm{d\&lambda;}}_1$

$+\frac{2}{3}\underset{0}{\overset{\&infin;}{\&Integral;}}{\log }_2[1+\frac{P}{3}(1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)){\mathrm{\&lambda;}}_1]\left[e^{{-\mathrm{\&lambda;}}_1}(1+{(1-{\mathrm{\&lambda;}}_1)}^2+{({\mathrm{\&lambda;}}_1^2-{4\mathrm{\&lambda;}}_1+2)}^2)\right]{\mathrm{d\&lambda;}}_1---\left(16\right)$

$=\frac{{-22a}^3+{14a}^2-5a+1}{{12a}^3\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">ln</figure-callout>}2}+e^{\frac{1}{a}}(\frac{1}{3\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">ln</figure-callout>}2}+\frac{{8a}^2-4a+1}{12\ln 2a^4})\mathrm{\&zeta;}+\frac{a-1}{3a\ln 2}-\frac{1+{2a}^2}{{3a}^2\ln 2}{e}^{\frac{1}{a}}\mathrm{\&zeta;}$

$+\frac{{-22b}^3+{14b}^2-5b+1}{{6\ln 2b}^3}+e^{\frac{1}{b}}(\frac{2}{3\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">ln</figure-callout>}2}+\frac{{8b}^2-4b+1}{{6b}^4\ln 2})\mathrm{\&zeta;}+\frac{b-1}{3b\ln 2}-\frac{1+{2\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}{{3\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">b</figure-callout>}}^2\ln 2}{e}^{\frac{1}{b}}\mathrm{\&zeta;}---\left(17\right)$

In the formula, $a=\frac{P}{3}(1+2\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)),b=1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)$

$\mathrm{\&zeta;}=\mathrm{\&gamma;}-\ln a+\underset{i=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{1}{i*i!}{(-\frac{1}{a})}^i,$ Wherein γ is an Euler's constant.

3) test results and analysis

We only consider the situation of up link flat fading.If the scattering thing all is positioned at the far field of receiving antenna array, because multi-path influence, the angle of scattering that signal arrives each antenna is A.Receiving antenna array is UCA, and radius of circle is r.Reception antenna and number of transmit antennas are 3.Produce 1000 accidental channels, ask it on average to obtain channel matrix H then.

Referring to Fig. 1, can see that by Fig. 1 angle of scattering is big more, it is more little to receive the required antenna distance of first zero crossing of coefficient correlation.Along with the reception antenna spacing increases, receive coefficient correlation and converge to zero gradually.Through after first zero point, coefficient correlation amplitude smaller (＜0.3) is very little to power system capacity and performance impact, can ignore.This point will obtain proof in the l-G simulation test of back.

Referring to Fig. 2, can obtain by formula (5), first zero crossing that receives coefficient correlation appears at $d_{\min }=\frac{1}{2A}$ The place reaches maximum in this point channel capacity.Can see that from Fig. 2 antenna distance reaches d _MinThe time, channel capacity is got maximum.It is very little to the channel capacity influence to increase antenna distance again.Angle of scattering is big more in addition, and it is just fast more that channel capacity converges to maximum.Comparison diagram 1 and Fig. 2 can find out that also after channel capacity reached maximum, the decline correlation was very little to the influence of capacity.

Referring to Fig. 3, antenna distance is λ/2.As can be seen from Figure, angle of scattering is to the obvious effect of channel capacity.Influence is that it has changed the channel gain of each subchannel to the decline correlation to channel capacity.When angle of scattering was enough big, it is uncorrelated that channel fading is approximately, and receives correlation matrix and converge to unit matrix, and channel capacity obtains maximum.

As an embodiment of this invention, under the limited condition of receiving terminal antenna placement space that we have adopted methods analyst of the present invention, the channel capacity of multi-input multi-output system when existing decline relevant.Here the analytical method with 3 * 3 channel capacity of multi-input multi-output system of front has been generalized to more ordinary circumstance of M * N, has analyzed the influence to channel capacity of angle of scattering size and antenna distance.

(multi-input multi-output system of M * N), promptly number of transmit antennas and reception antenna number are respectively M and N to consider one.Signal model is identical with the front.N is that average is 0, and variance is $E\left[{\mathrm{nn}}^H\right]={\mathrm{\&sigma;}}_n^2{I}_N$ Additive white Gaussian noise.Gross power is P if transmit, and is irrelevant with number of transmit antennas, and transmitting terminal adopts the constant power emission, so have $E\left[{\mathrm{xx}}^H\right]={\mathrm{\&sigma;}}_x^2{I}_M=\frac{P}{M}{I}_M.$ Other supposes that signal is uncorrelated with noise, i.e. E[xn ^H]=0.If the transmitting terminal antenna distance is enough big, then transmits on each antenna and can be approximately separate.Receiving terminal adopts the UCA array at restriceted envelope.Signal is through spatial transmission, because the influence of decline and the finiteness of receiving terminal antenna distance make to have certain correlation between signal.Equally, channel matrix decomposition is obtained formula (2).According to top derivation and analysis, can obtain for spacing is that decline coefficient correlation between two antennas of d is

A hour, following formula can be reduced to

For the UCA antenna array, establishing its radius is r, and antenna number is N.Then the spacing of any antenna and other antennas is on the antenna array

$d_i=2r\sin \left(\frac{\mathrm{\&pi;i}}{N}\right)i=\mathrm{0,1},\mathrm{\&Lambda;\&Lambda;},N-1---\left(19\right)$

Therefore can obtain receiving correlation matrix according to formula (18) and (19) is

$R=\left[\begin{array}{ccccccccc}R\left(0\right)& R\left(1\right)& R\left(2\right)& \mathrm{\&Lambda;\&Lambda;}& R\left(\frac{N-1}{2}\right)& R\left(\frac{N-1}{2}\right)& \mathrm{\&Lambda;\&Lambda;}& R\left(2\right)& R\left(1\right)\\ R\left(1\right)& R\left(0\right)& R\left(1\right)& R\left(2\right)& \mathrm{\&Lambda;\&Lambda;}& R\left(\frac{N-1}{2}\right)& R\left(\frac{N-1}{2}\right)& \mathrm{\&Lambda;\&Lambda;}& R\left(2\right)\\ M& & & & \mathrm{\&Lambda;}& \mathrm{\&Lambda;}& & & M\\ R\left(2\right)& \mathrm{\&Lambda;\&Lambda;}& R\left(\frac{N-1}{2}\right)& R\left(\frac{N-1}{2}\right)& \mathrm{\&Lambda;\&Lambda;}& R\left(2\right)& R\left(1\right)& R\left(0\right)& R\left(1\right)\\ R\left(1\right)& R\left(2\right)& \mathrm{\&Lambda;\&Lambda;}& R\left(\frac{N-1}{2}\right)& R\left(\frac{N-1}{2}\right)& \mathrm{\&Lambda;\&Lambda;}& R\left(2\right)& R\left(1\right)& R\left(0\right)\end{array}\right]---\left(20\right)$

In the formula, R (i)=R (d _i) calculate by formula (18) and (19).From formula (20) as can be seen, receiving correlation matrix R is a circulation symmetrical matrix.Can adopt above-mentioned circular matrix to ask eigenvalue method to obtain its characteristic value.

The characteristic value of circular matrix R is

${\mathrm{\&lambda;}}_m=\underset{k=0}{\overset{N-1}{\mathrm{\&Sigma;}}}R\left(k\right)\exp (-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;mk}/N)$

Because correlation matrix R is a real symmetric matrix, so formula (21) can be reduced to

${\mathrm{\&lambda;}}_m=\underset{k=0}{\overset{N-1}{\mathrm{\&Sigma;}}}R\left(k\right)\cos (2\mathrm{\&pi;km}/N)$

Correlation matrix R is the positive semidefinite circular matrix, so its characteristic value is all more than or equal to zero, and all is real.

If receiving terminal known channel parameter, and transmitting terminal Unknown Channel parameter.In this case, can adopt the constant power emission.Total transmission power setting is P, so the transmitting power on every antenna is So can obtain following formula

$I(y;x)=H\left(y\right)-H\left(y\right|x)=\frac{1}{2}\log \frac{\left|\mathrm{var}\left(y\right)\right|}{\left|\mathrm{var}\left(y\right|x)\right|}---\left(22\right)$

In the formula, var (y|x)=var (n).Suppose that each element of emission signal vector is independent identically distributed multiple gaussian variable, then

var(y)＝var(Hx+n)

＝E[Hxx ^HH ^H]+E[nn ^H] (23)

With H=R ^1/2In the U substitution formula (23), have

$\mathrm{var}\left(y\right)=E\left[R^{1/2}{\mathrm{Uxx}}^H{U}^H{R}^{1/2}\right]+{\mathrm{\&sigma;}}_n^2{I}_N$

$=R^{1/2}E\left[\mathrm{Ux}{\left(\mathrm{Ux}\right)}^H\right]R^{1/2}+{\mathrm{\&sigma;}}_n^2{I}_N---\left(24\right)$

Again with in formula (24) the substitution formula (22), can obtain channel capacity and be

$C=\frac{1}{2}\log \det \frac{R^{1/2}E\left[\mathrm{Ux}{\left(\mathrm{Ux}\right)}^H\right]R^{1/2}+{\mathrm{\&sigma;}}_n^2{I}_N}{{\mathrm{\&sigma;}}_n^2{I}_N}---\left(25\right)$

In order to try to achieve the channel capacity closed solutions, key issue is to obtain E[Ux (Ux) ^H].Concrete derivation is as follows

Need use following theorem in the derivation: establish two-dimentional continuous random variable (X, probability density Y) be f (x, y), function

$\left\{\begin{array}{c}u=g_1(x,y)\\ v=g_2(x,y)\end{array}\right.$

Unique inverse function is arranged

$\left\{\begin{array}{c}x=h_1(u,v)\\ y=h_2(u,v)\end{array}\right.$

And inverse function has continuous single order partial derivative, then stochastic variable Z ₁=g ₁(X, Y), Z ₂=g ₂(X, joint probability density Y) is

Wherein $J=\frac{\&PartialD;(x,y)}{\&PartialD;(u,v)}=\left|\begin{array}{cc}\frac{\&PartialD;x}{\&PartialD;u}& \frac{\&PartialD;x}{\&PartialD;v}\\ \frac{\&PartialD;y}{\&PartialD;u}& \frac{\&PartialD;y}{\&PartialD;v}\end{array}\right|$ Be function x=h ₁(u, v), y=h ₂(u, Jacobian v),

D＝{(u，v)|u＝g ₁(x，y)v＝g ₂(x，y)}。#

Order $\mathrm{\&Omega;}=\mathrm{Ux}=\left[\begin{array}{ccccc}{u}_{11}& u_{12}& \mathrm{\&Lambda;\&Lambda;}& u_{1M-1}& u_{1M}\\ u_{21}& u_{22}& \mathrm{\&Lambda;\&Lambda;}& u_{2M-1}& u_{2M}\\ & M& \mathrm{\&Lambda;}& & M\\ u_{N-11}& u_{N-12}& \mathrm{\&Lambda;\&Lambda;}& u_{N-1M-1}& u_{N-1M}\\ u_{N1}& {\mathrm{<figure-callout\; id="2"\; label="u\; N"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">u</figure-callout>}}_N& \mathrm{\&Lambda;\&Lambda;}& u_{\mathrm{NM}-1}& u_{\mathrm{NM}}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ M\\ x_{M-1}\\ x_M\end{array}\right]$

$=\left[\begin{array}{c}{u}_{11}{x}_1+u_{12}{x}_2+\mathrm{\&Lambda;\&Lambda;}+u_{1M-1}{x}_{M-1}+u_{1M}{x}_M\\ u_{21}{x}_1+u_{22}{x}_2+\mathrm{\&Lambda;\&Lambda;}+u_{2M-1}{x}_{M-1}+u_{2M}{x}_M\\ M\\ u_{N-11}{x}_1+u_{N-12}{\mathrm{<figure-callout\; id="2"\; label="x"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">x</figure-callout>}}_2+\mathrm{\&Lambda;\&Lambda;}+u_{N-1M-1}{x}_{M-1}+u_{\mathrm{NM}}{x}_M\\ u_{N1}{x}_1+u_{N2}{x}_2+\mathrm{\&Lambda;\&Lambda;}+u_{\mathrm{NM}-1}+u_{\mathrm{NM}}{x}_M\end{array}\right]=\left[\begin{array}{c}{a}_1\\ a_2\\ M\\ a_{N-1}\\ a_N\end{array}\right]---\left(26\right)$

Separate between each element of hypothesis matrix U and x, have

$E\left[{\mathrm{\&Omega;\&Omega;}}^H\right]=E\left[\begin{array}{ccccc}{a}_1{a}_1^{*}& & & & \\ & a_2{a}_2^{*}& & 0& \\ & & O& & \\ & 0& & a_{N-1}{a}_{N-1}^{*}& \\ & & & & a_N{a}_N^{*}\end{array}\right]---\left(27\right)$

Therefore in order to obtain E[Ω Ω ^H], must obtain z=xy x～N (0,1) y～N (0, σ ²) probability density function.

Can get by theorem

$\left\{\begin{array}{c}u=\mathrm{xy}\\ v=x\end{array}\right.---\left(28\right)$ Its inverse function is $\left\{\begin{array}{c}y=\frac{u}{v}\\ x=v\end{array}\right.---\left(29\right)$

The Jacobian that can get function x and y according to (28) and (29) is

$J=\left|\begin{array}{cc}\frac{1}{v}& -\frac{\mathrm{<figure-callout\; id="2"\; label="u\; v"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">u</figure-callout>}}{v^{}}\\ 0& 1\end{array}\right|=\frac{1}{v}$

So can obtain the probability density of z=xy be

$f_z\left(u\right)=\frac{1}{\mathrm{\&pi;\&sigma;}}{\&Integral;}_0^{\&infin;}\exp (-\frac{{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">v</figure-callout>}}^2}{2})\exp (-\frac{{\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">u</figure-callout>}}^2}{2{\mathrm{\&sigma;}}^2{v}^2})\frac{1}{v}\mathrm{dv}---\left(30\right)$

We can ask for the variance of z by top analysis

${\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">z</figure-callout>}}^2=\frac{1}{\mathrm{\&pi;\&sigma;}}{\&Integral;}_{-\&infin;}^{\&infin;}{\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">u</figure-callout>}}^2{\&Integral;}_0^{\&infin;}\exp (-\frac{{\mathrm{<figure-callout\; id="2"\; label="v"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">v</figure-callout>}}^2}{2})\exp (-\frac{{\mathrm{<figure-callout\; id="2"\; label="u"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">u</figure-callout>}}^2}{{2\mathrm{\&sigma;}}^2{v}^2})\frac{1}{v}\mathrm{dvdu}$

Can get through computing and abbreviation

${\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="z"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">z</figure-callout>}}^2={\mathrm{\&sigma;}}^2$

So

$E\left[{\mathrm{\&Omega;\&Omega;}}^H\right]=\left[\begin{array}{ccccc}{\mathrm{M\&sigma;}}^2& & & & \\ & {\mathrm{M\&sigma;}}^2& & 0& \\ & & O& & \\ & 0& & {\mathrm{M\&sigma;}}^2& \\ & & & & {\mathrm{M\&sigma;}}^2\end{array}\right].---\left(32\right)$

To get in its substitution formula (25)

$C=\frac{1}{2}\log \det [I_N+\frac{\mathrm{RE}\left({\mathrm{\&Omega;\&Omega;}}^H\right)}{{\mathrm{\&sigma;}}_n^2{I}_N}]---\left(33\right)$

R is carried out substitution formula (31) after the characteristic value decomposition,

$C=\frac{1}{2}\log \det [I_N+{\mathrm{V\&Lambda;}}_R{V}^H\frac{E\left({\mathrm{\&Omega;\&Omega;}}^H\right)}{{\mathrm{\&sigma;}}_n^2{I}_N}]$

$=\frac{1}{2}\log \det [I_N+{\mathrm{\&Lambda;}}_R\frac{E\left({\mathrm{\&Omega;\&Omega;}}^H\right)}{{\mathrm{\&sigma;}}_n^2{I}_N}]$

$=\frac{1}{2}\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\log (1+{\mathrm{\&lambda;}}_{i}M\frac{{\mathrm{\&sigma;}}^2}{{\mathrm{\&sigma;}}_n^2})---\left(34\right)$

In the formula

${\mathrm{\&Lambda;}}_R=\left[\begin{array}{ccccc}{\mathrm{\&lambda;}}_1& & & & \\ & {\mathrm{\&lambda;}}_2& & 0& \\ & & O& & \\ & 0& & {\mathrm{\&lambda;}}_{N-1}& \\ & & & & {\mathrm{\&lambda;}}_N\end{array}\right],$

σ ²Be the variance that transmits, be known as ${\mathrm{\&sigma;}}^2=\frac{P}{M}.$ So formula (17) can be reduced to

$C=\frac{1}{2}\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\log (1+{\mathrm{\&lambda;}}_i\frac{P}{{\mathrm{\&sigma;}}_n^2})$

$=\frac{1}{2}\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\log (1+\mathrm{\&lambda;},\mathrm{\&kappa;})---\left(36\right)$

In the formula,

$\mathrm{\&kappa;}=\frac{P}{{\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="C20031012220900265.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}.$

Result of the test is as follows

Fig. 4 has provided the relation between antenna number, angle of scattering size and antenna distance.UCA antenna array radius is r=0.5 λ, signal to noise ratio snr=20dB.We only consider to receive relevant, receive coefficient R and are calculated by formula (5).As can be seen from Figure 4, under the certain situation in receiving terminal space, because the relevant influence of decline, channel capacity and antenna number are not entirely linear relationship.As shown in Figure 4, regardless of the angle of scattering size, each bar curve all is made up of two parts: one section straight line and one section curve.This explanation receives space (being r) regularly, and channel capacity reached capacity after antenna number increased to a certain degree.It is very little to the channel capacity influence to increase antenna number again.Along with the increase of angle of scattering, channel capacity also increases gradually.This explanation, angle of scattering has increased, and the coefficient correlation between decline has decline to a certain degree.

Fig. 5 has provided the relation between signal to noise ratio, antenna number, angle of scattering size and channel capacity.UCA antenna array radius is r=0.5 λ.As can be seen from the figure, when antenna number was identical, angle of scattering was big more, and it is obvious more that channel capacity increases amplitude with signal to noise ratio.The angle of scattering size is certain, and the big more channel capacity of antenna number changes obvious more with signal to noise ratio.This explanation, when scattering environments one timing, particularly for indoor wireless local area network, suitable increase antenna number can increase substantially channel capacity.And for outdoor environment, the size of angle of scattering to the influence of channel capacity also clearly.Therefore, seem particularly important according to the suitable antenna number of different environmental selections.

Claims (1)

1, a kind of detection method of channel capacity of multi-input multi-output system is characterized in that:

1) at first makes up the up link of 3 * 3 multi-input multi-output systems

A) transmitting of the mobile you of setting to separate, it is the UCA antenna array of r that reception antenna adopts radius, the received signal vector of arrowband multi-input multi-output system is

y＝Hx+n (1)

In the formula, x=[x ₁, x ₂, x ₃] ^HBe 3 dimension emission signal vector; Y=[y ₁, y ₂, y ₃] ^HBe 3 dimension received signal vectors; H is 3 * 3 channel matrixes; N is that average is 0, and variance is 1 additive white Gaussian noise;

B) only consider up link, channel matrix H is decomposed,

H＝R ^1/2U (2)

R is 3 * 3 reception correlation matrixes in the formula; The element of U is that average is 0, and variance is 1 the multiple Gaussian random variable of independence;

C) coefficient correlation between any two antennas of measurement

If it is A that multipath signal arrives the angle of scattering of UCA antenna array, and the multipath signal arrival direction evenly distributes in [-A, +A], is that coefficient correlation is between any two antennas of d for spacing

In the formula $k=\frac{2\mathrm{\&pi;}}{\mathrm{\&lambda;}}$ Be wave number,  is the average arrival direction angle of ripple, and when A=π, (3) Shi Kejian is

r _i，J＝J ₀(kd) (4)

For up link, antenna for base station set up than around high many of scattering thing, so angle of scattering is smaller, (3) formula can be reduced to like this

Be without loss of generality, make =0, then following formula can be reduced to

$r_{i,l}=\frac{\sin \left(\mathrm{kdA}\right)}{\mathrm{kdA}}=\sin c\left(\frac{2\mathrm{\&pi;dA}}{\mathrm{\&lambda;}}\right)$

So it is a circulation symmetrical matrix that multiple-input and multiple-output 3 * 3 channels receive correlation matrix R

$R=\left[\begin{array}{ccc}1& \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)\\ \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& 1& \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)\\ \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& \sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& 1\end{array}\right]=\left[\begin{array}{ccc}{c}_0& c_1& c_2\\ c_2& c_0& c_1\\ c_1& c_2& c_0\end{array}\right]---\left(6\right)$

D) measure the characteristic value that multiple-input and multiple-output 3 * 3 channels receive correlation matrix R as follows;

If circular matrix C has following form

$C=\left[\begin{array}{cccccccc}{c}_0& c_1& c_2& c_3& \mathrm{\&Lambda;}& \mathrm{\&Lambda;}& \mathrm{\&Lambda;}& c_{n-1}\\ c_{n-1}& c_0& c_1& c_2& & & & \\ c_{n-2}& & & & O& & & \\ & & & O& O& & & O\\ M& & \mathrm{\&Lambda;}& & O& O& & \\ & & & & & & & c_2\\ c_2& & & & & & c_0& c_1\\ c_1& & & \mathrm{\&Lambda;}& \mathrm{\&Lambda;}& & & c_0\end{array}\right]$

Be that C itself is a special Toeplitz matrix, the characteristic value ψ of C _k, characteristic vector y _kBe separating of following formula,

Cy＝ψy

Following formula is equivalent to a following n equation

$\underset{k=0}{\overset{m-1}{\mathrm{\&Sigma;}}}{c}_{n-m+k}{y}_k+\underset{k=m}{\overset{n-1}{\mathrm{\&Sigma;}}}{c}_{k-m}{y}_k={\mathrm{\&psi;y}}_m---m=\mathrm{0,1},\mathrm{\&Lambda;},n-1$

Get y _k=exp (j2 π mk/n) separates for following formula, so the characteristic value of C is

${\mathrm{\&psi;}}_m=\underset{k=0}{\overset{n-1}{\mathrm{\&Sigma;}}}{c}_k\exp (-j2\mathrm{\&pi;mk}/n)$

The characteristic value that promptly gets circular matrix R is

${\mathrm{\&psi;}}_m=\underset{k=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_k\exp (-j2\mathrm{\&pi;mk}/3)---m=\mathrm{0,1,2}---\left(7\right)$

So have

${\mathrm{\&psi;}}_0=\underset{k=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_k=1+c_1+c_2=1+2\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)---\left(8\right)$

${\mathrm{\&psi;}}_1=\underset{k=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_k\exp (-j2\mathrm{\&pi;k}/3)=1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)---\left(9\right)$

${\mathrm{\&psi;}}_2=\underset{k=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_k\exp (-j4\mathrm{\&pi;k}/3)=1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)---\left(10\right)$

2) ergodic capacity of measurement 3 * 3 multi-input multi-output systems

Suppose the receiver known channel state, and transmitter the unknown, transmitting terminal adopts the constant power emission, and total transmission power setting is P, and then the transmitting power of every transmit antennas is P/3, and channel capacity can be expressed as

${\log }_2\left(\det (I+\frac{P}{3}{\mathrm{HH}}^H)\right)---\left(11\right)$

With (2) formula substitution (11) Shi Kede

$C={\log }_2\left(\det (I+\frac{P}{3}{R}^{1/2}{\mathrm{UU}}^H{R}^{1/2})\right)$

$={\log }_2\left(\det (I+\frac{P}{3}{\mathrm{RUU}}^H)\right)---\left(12\right)$

Make W=UU ^H, R and W are carried out characteristic value decomposition respectively get

$C={\log }_2\left(\det (I+\frac{P}{3}{\mathrm{\&Lambda;}}_R{\mathrm{\&Lambda;}}_W)\right)=\underset{i=1}{\overset{3}{\mathrm{\&Sigma;}}}{\log }_2(1+\frac{P}{3}{\mathrm{\&psi;}}_i{\mathrm{\&lambda;}}_i)---\left(13\right)$

In the formula,

${\mathrm{\&Lambda;}}_r=\left[\begin{array}{ccc}{\mathrm{\&psi;}}_0& 0& 0\\ 0& {\mathrm{\&psi;}}_1& 0\\ 0& 0& {\mathrm{\&psi;}}_2\end{array}\right]=\left[\begin{array}{ccc}1+2\sin \left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& 0& 0\\ 0& 1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)& 0\\ 0& 0& 1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)\end{array}\right]$

${\mathrm{\&Lambda;}}_W=\left[\begin{array}{ccc}{\mathrm{\&lambda;}}_1& 0& 0\\ 0& {\mathrm{\&lambda;}}_2& 0\\ 0& 0& {\mathrm{\&lambda;}}_3\end{array}\right]$

In the formula (13), λ is the characteristic value of W, λ ₁Probability density function be

In the formula,

$L_k^{n-m}\left(x\right)=\frac{1}{k!}{e}^x{x}^{m-n}\frac{d^k}{{\mathrm{dx}}^k}\left(e^{-x}{x}^{n-m+k}\right)$

Can push away by formula (14)

$P_{{\mathrm{\&lambda;}}_1}\left({\mathrm{\&lambda;}}_1\right)=\frac{1}{3}[1+{(1-{\mathrm{\&lambda;}}_1)}^2+\frac{1}{4}{({\mathrm{\&lambda;}}_1^2-{4\mathrm{\&lambda;}}_1+2)}^2]\exp \left({-\mathrm{\&lambda;}}_1\right)---\left(15\right)$

Promptly getting the multi-input multi-output system ergodic capacity is

$E\left(C\right)=E\left[{\log }_2(1+\frac{P}{3}(1+\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)){\mathrm{\&lambda;}}_1)\right]+2E\left[{\log }_2(1+\frac{P}{3}(1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)){\mathrm{\&lambda;}}_1)\right]$

$=\frac{1}{3}{\&Integral;}_0^{\&infin;}{\log }_2[1+\frac{P}{3}(1+2\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)){\mathrm{\&lambda;}}_1)\left]\right[e^{-{\mathrm{\&lambda;}}_1}(1+{\left({1-\mathrm{\&lambda;}}_1\right)}^2+{({\mathrm{\&lambda;}}_1^2-4{\mathrm{\&lambda;}}_1+2)}^2)]d{\mathrm{\&lambda;}}_1$

$=\frac{2}{3}{\&Integral;}_0^{\&infin;}{\log }_2[1+\frac{P}{3}(1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)){\mathrm{\&lambda;}}_1)\left]\right[e^{-{\mathrm{\&lambda;}}_1}(1+{\left({1-\mathrm{\&lambda;}}_1\right)}^2+{({\mathrm{\&lambda;}}_1^2-4{\mathrm{\&lambda;}}_1+2)}^2)]d{\mathrm{\&lambda;}}_1$

$=\frac{-22a^3+14a^2-5a+1}{12a^3\ln 2}+e^{\frac{1}{a}}(\frac{1}{3\ln 2}+\frac{{8a}^2-4a+1}{12\ln {2a}^4})\mathrm{\&zeta;}+\frac{a-1}{3a\ln 2}-\frac{1+{2a}^2}{{3a}^2\ln 2}{e}^{\frac{1}{a}}\mathrm{\&zeta;}---\left(16\right)$

$+\frac{-22b^3+14b^2-5b+1}{6\ln 2b^3}+e^{\frac{1}{b}}(\frac{2}{3\ln 2}+\frac{{8b}^2-4b+1}{{6b}^4\ln 2})\mathrm{\&zeta;}+\frac{b-1}{3b\ln 2}-\frac{1+{2b}^2}{{3b}^2\ln 2}{e}^{\frac{1}{b}}\mathrm{\&zeta;}---\left(17\right)$

In the formula,

$a=\frac{P}{3}(1+2\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right))---b=1-\sin c\left(\frac{2\mathrm{dA}}{\mathrm{\&lambda;}}\right)$

$\mathrm{\&zeta;}=\mathrm{\&gamma;}-\ln a+\underset{i=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{1}{i*i!}{(-\frac{1}{a})}^i,$ Wherein γ is an Euler's constant.

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