WO2014040611A1

WO2014040611A1 - Method and device for determining the number of uncorrelated transmission channels in a mimo transmission system

Info

Publication number: WO2014040611A1
Application number: PCT/EP2012/067698
Authority: WO
Inventors: Adrian Schumacher
Original assignee: Rohde & Schwarz Gmbh & Co. Kg
Priority date: 2012-09-11
Filing date: 2012-09-11
Publication date: 2014-03-20

Abstract

A method for determining a number of uncorrelated transmission channels in a multiple-input-multiple-output transmission system determines the transmission coefficients h ₁₁,h ₁₂,..., h _ΝΜ of a transmission matrix H of the MIMO system and determines the number N of eigenvalues of the transmission matrix H. The number of uncorrelated transmission channels is determined by a rank k of the transmission matrix H, for which a quotient γ(k); γ(k)' between the sum of the k biggest eigenvalues and at least the next smaller eigenvalue of all N—k smaller eigenvalues is below a threshold value ρ_Th.

Description

Method and device for determining the number of uncorrelated transmission channels in a MIMO transmission system The invention relates to a method and a device for

determining the number of uncorrelated transmission channels in a MIMO transmission system.

Modern wireless communications systems are based on a multiple-^nput-multiple-output (MIMO) architecture

comprising multiple transmitter antennas and multiple receiver antennas. Fig. 1A shows a MIMO system with four transmitter antennas and four receiver antennas and the corresponding 16 transmissions paths leading to a

transmission matrix H with 16 transmission coefficients according to Fig. IB.

A MIMO receiver has to perform channel estimation in order to determine the transmission coefficients of the

transmission matrix. Furthermore, a MIMO receiver has to separate the signals transmitted from different

transmitter antennas by spatial equalization. Spatial equalization results in so called spatial data streams each corresponding to a data stream furnished to one transmitter antenna.

In the ideal case, if the transmitter antennas and the receiver antennas are sufficiently spaced apart, at least half the wavelength spaced apart, and if the signal is transmitted between the transmitter antenna and the receiver antenna in the shortest distance, i.e. in the _line-of-s_ight (LOS) path, all the transmission channels are uncorrelated to each other. The corresponding

transmission matrix has a full rank.

However, in case of a short distance between at least two transmitter antennas or between at least two receiver antennas or in case of obstacles in the LOS path resulting in multiple propagation paths, multiple transmission signals are correlated to each other, as figured out symbolically by means of a two correlated signals paths for the signals transmitted from the first and second transmitter antenna in Fig. 2 for a 4x4 MIMO system.

In such a case it will be worse to equalize, i.e. to separate, and to decode the signals each transmitted over a correlated signal path in a receiver. Correlation between different transmission signals results in a transmission matrix with deficient rank being smaller than the corresponding full rank.

The rank of a matrix describes the number of linear independent rows or linear independent columns of a matrix. The rank of a matrix is an integer number between 1 and the minimum Min {N,M} of N and M , whereby N represents the number of rows of the matrix and M

represents the number of columns of the matrix.

In US 2009/0103486 Al a method and a system is described, which examines, whether the transmission matrix of a MIMO system has a full rank corresponding to in total

uncorrelated signal paths or a deficient rank

corresponding to at least two correlated signal paths. The method and the systems determine the eigenvalues

MIMO system resulting in a deficient rank of the

transmission matrix, whereas a sum being higher than the threshold value p_th corresponds to fully uncorrelated signal paths in the MIMO system with a full ranked

transmission matrix. The method and the system in US 2009/0103486 Al does disadvantageously not determine the exact - deficient or full - rank of a transmission matrix in a MIMO system resulting in useful additional information about the amount of optimizing the MIMO transmission system, i.e. for example the number of transmitter antennas or receiver antennas to be repositioned for minimizing correlation or the volume of supplemental code rate for decoding

correlated signals.

Therefore the object of the invention is to develop a method and a system for determining the number of

correlated signals in a MIMO transmission system. The object can be solved by an inventive method for determining a number of uncorrelated transmission channels in a MIMO transmission system with the features according to claim 1 and by an inventive device for determining a number of uncorrelated transmission channels in a MIMO transmission system with the features according to claim 9. Additional technical advantages, a digital storage medium and a computer program are considered in the corresponding dependent claims. The number of uncorrelated transmission channels in a MIMO transmission system correlates with the rank k of the transmission matrix. Inventively, for different values k a corresponding quotient between the sum of the k biggest eigenvalues of the transmission matrix and at least the next smaller eigenvalue of all N - k smaller eigenvalues of the transmission matrix is compared with an

appropriately selected threshold value. The value k of the quotient, which is below, preferably nearest below, the threshold value, leads to the integer-valued rank k of the transmission matrix.

The invention uses the fact that a transmission matrix having a deficient rank comprises one or more comparatively lower-valued eigenvalues and one or more comparatively bigger-valued eigenvalues. By ordering the determined eigenvalues according to their values in a decreasing order according to Fig. 8 and determining for each value of k a corresponding quotient between the sum of the k biggest eigenvalues and at least the next smaller eigenvalue of all N - k smaller eigenvalues, the determined quotients change with increasing value of k from a comparatively low value to a comparatively high value, if the k biggest eigenvalues comprise exactly all comparatively bigger-valued eigenvalues and not any further eigenvalue. This significant change of slope in the determined quotient values with increasing values of k can be detected by a comparison with a threshold value selected appropriately between the comparatively low values and the comparatively high values of the quotient.

In a first embodiment of the invention the quotient corresponding to the value k is created between the sum of the k biggest eigenvalues of the transmission matrix and the sum of the N - k smaller eigenvalues of the transmission matrix. The first embodiment is characterized by a steep slope of the curve showing the relation between the quotient and the value k enabling a precise detection of the rank of the transmission matrix.

In a second embodiment of the invention the quotient corresponding to the value k is built between the sum of the k biggest eigenvalues of the transmission matrix and the N - k ^th smallest eigenvalue of the transmission matrix. The second embodiment is characterized by determining only one sum resulting in a smaller calculation volume.

If any value k cannot be found for which the

corresponding quotient is below the threshold value, the corresponding deficient rank of the transmission matrix is 1 and thus the number of uncorrelated transmission

channels is 1. The comparison between the different quotient values and the threshold value results in an integer value k

corresponding to an integer value k for the rank of the transmission matrix. Normally, a distance between the threshold value and the detected quotient being nearest below to the threshold value exists, which indicates a probability for a correctness of the value k determined for the rank in comparison to a probability for a

correctness of the decremented value k - 1 for the rank.

Thus in the first embodiment of the invention a

probability value corresponding to the distance between the threshold value and the detected quotient being below, preferably nearest below, the threshold value and thus indicating the correctness of the value k for the rank of the transmission matrix is determined by the quotient between the sum of the N - k smallest eigenvalues

multiplied with the threshold value and the sum of the k biggest eigenvalues.

In the second embodiment of the invention the probability value indicating the correctness of the value k for the rank of the transmission matrix is determined by the quotient between the sum of the N - k smallest eigenvalues multiplied with the threshold and the N - k ^th smallest eigenvalue .

The probability value is zero, if the determined rank k equals N .

In a further embodiment of the invention the probability value indicating the correctness of the value k for the rank of the transmission matrix is determined by

calculating a fractional value k_fmh corresponding to the threshold value on a linearized curve of quotient values between the quotient corresponding to the value k and the quotient corresponding to the value k + 1 . Embodiments of the inventive method and the inventive system for determining the number of uncorrelated

transmission channels in a MIMO transmission system are described in more detail according to the drawings by examples only. The figures of the drawings show:

Fig. 1A a diagram showing a 4x4 MIMO transmission system and a corresponding transmission matrix,

Fig. IB the H-Matrix

Fig. 2 a diagram showing a 4x4 MIMO transmission system with correlated signal paths,

Fig. 3 a diagram showing an example of the

linearization of the curve of quotient values,

Fig. 4 a flowchart showing an embodiment of the

inventive method for determining the number of uncorrelated transmission channels in a MIMO transmission system,

Fig. 5 a block diagram showing an embodiment of the inventive system for determining the number of uncorrelated transmission channels in a MIMO transmission system,

Fig. 6A a first graphical presentations of an effective rank of a transmission matrix,

Fig. 6B a second graphical presentations of an effective rank of a transmission matrix, Fig. 6C a third graphical presentations of an effective rank of a transmission matrix, Fig. 6D a fourth graphical presentations of an effective rank of a transmission matrix,

Fig. 7A a diagram showing the quotient values in

dependency to the values k in the first embodiment of the invention,

Fig. 7B a diagram showing the quotient values in

dependency to the values k in the second

embodiment of the invention,

Fig. 8 a diagram showing descendent ordered eigenvalues and Fig. 9 a diagram showing the error vector magnitude in dependency of the fractional value k_Frac and several SNR parameter values.

Firstly, the mathematical basics being essential for the understanding of the inventive method and for the

inventive system for determining the number of

uncorrelated transmission channels in a MIMO transmission system are derived. According to equation (1) the transmission matrix H can be decomposed by means of a s_ingular-value-decomposition (SVD) in the diagonal matrix S containing the eigenvalues λ_ιι,...,λ_η,...,λ_{Ν N} of the transmission matrix H according to equation (2) and in the unitary matrices U

and V^H containing the corresponding eigenvectors.

H U-S-V (1)

(2) o

The eigenvalues λ_ΎΙ,...,λ_η,...,λ_{Ν N} are ordered in a descending order resulting in a vector λ, of descending ordered eigenvalues λι,.., λΝ- as shown

exemplary in Fig. 8.

In the first embodiment of the invention for each value k a corresponding quotient y(k) between the sum of the k biggest eigenvalues and the sum of the N—k smallest eigenvalues is calculated according to equation (3) .

k

∑X

Y(k) = -^! with k = \, 2,..., N -\ (3) ∑ X

i=k+l

Each quotient y(k) calculated for a corresponding value k is compared to a threshold value p_Th . An appropriately selected value has to be used for the threshold value p_Th . A recommended value for the threshold value p_Th is p_Th = N⁴ , whereby N is the number of diagonal elements in the diagonal matrix S .

The value k_Int corresponding to a quotient y(k_Int) , which is nearest below to the threshold value p_Th of all calculated quotient values y(k) (with k = 1, 2,..., N - 1 ) according to equation (4), represents the integer-valued rank_Int of the transmission matrix (see equation (5) ) : k_lnt = Min {p_n - r(k) > 0 I Vk = 1, 2, N - 1} (4) rank_Int = k_Int (5)

A fractional value k_Fmc corresponding to the distance between the threshold value p_Th and the detected quotient y{k_Int) being nearest below to the threshold value p_Th represents a probability value, which indicates the correctness of the determined value k_Int for the rank of the transmission matrix and is determined in the first embodiment of the invention by the quotient between the sum of the N—k smallest eigenvalues multiplied with the threshold p_Th and the sum of the k biggest eigenvalues according to the lower side of equation (6) :

If a full rank for the transmission matrix ( rank_M = N ) is determined, fractional value k_Fmc representing a

probability value, which indicates the correctness of the determined value k_Int for the rank of the transmission matrix, is zero according to the upper side of equation (6) .

In the first embodiment of the invention an effective rank_Eff sums the integer-valued rank rank_Int and the

fractional value k_Fmc according to equation (7). ranker = rank_M + k Frac ( :

In the second embodiment of the invention for each value k a corresponding quotient y(k)^' between the sum of the k biggest eigenvalues and the N -k ^th smallest eigenvalue is calculated according to equation (8).

(8)

Each quotient γ(Κ)' calculated for a corresponding value k is compared to a threshold value p_Th . The value k_Int ' corresponding to a quotient y(k_M 'y , which is nearest below to the threshold value p_Th of all calculated quotient values y(k)' (with k = 1, 2, ..., N - l ) according to equation (9), represents the integer-valued rank_Int ' of the transmission matrix (see equation (10)): k_Int ^'= Min {p_Th - r(ky> 0 I Vk = 1, 2, N -1} ( 9 ) rank_M = k_M (10) A fractional value k_Frac' corresponding to the distance between the threshold value p_Th and the detected quotient y(k_M 'y being nearest below the threshold value p_Th

represents a probability value, which indicates the correctness of the determined value k_Int ' for the rank of the transmission matrix and is determined in the second embodiment of the invention by the quotient between the sum of the N—k smallest eigenvalues multiplied with the threshold value p_Th and the sum of the k biggest

eigenvalues according to the lower side of equation (11) :

If a full rank for the transmission matrix { rank_Int '= N ) is determined, fractional value k_Frac' representing a

probability value, which indicates the correctness of the determined value k_Int ' for the rank of the transmission matrix, is zero according to the upper side of equation (11) .

In the second embodiment of the invention an effective rank rank_Eff ' sums the integer-valued rank rank_Int ' and the fractional value k_Frac' according to equation (12). rank_Eff '= rank_Int '+k_Fmc ' (12)

In a third embodiment of the invention a probability value as the fractional value k_Frac" is determined by

linearization of the curve of quotient values

y(k)" , y(k + Ϊ)" , y(k + 2)" and so on in the range between the quotient values y{k)" being nearest below the threshold value p_Th and the quotient values y{k + \)" being nearest above the threshold value p_Th and shifting the curve of quotient values y(k)" , y(k + Ϊ)" , y(k + 2)" and so on downwards by the factor y(k)" , until the quotient value y(k)" is located in the origin of the coordination system according to Fig. 3. The fractional value k_Frac" corresponds to the value p_Th - y(k))" of the shifted curve of quotient values

y{k)^" , y(k + iy , ( + 2)"and so on. The shifted curve of quotient values y(k)" , y(k + Y)" , y(k + 2)" and so on in Fig. 3 corresponds to the general linear equation (13) with the gradient a according to equation (14), the ordinate value y according to equation (15) and the abscissa value x according to equation (16) and the center point b being zero. y = a - x + b (13) a = /(k + l)^"-/(k)^" (14) y = p_n - r(k)^" (15)

Frac (16)

By inserting equations (14), (15) and (16) in equation (13) the fractional value k_Frac" in the third embodiment be determined according to equation (17) .

The quotients y{k)" and y{k + \)" each can be calculated as a quotient /(k) in the first embodiment of the invention according to equation (3) or as a quotient y(k)' in the second embodiment of the invention according to equation (8) .

In a fourth embodiment of the invention the fractional value k_Frac'" can be calculated according to equation (18).

On the basis of this mathematical background some

embodiments of the inventive method and the inventive system for determining the number of uncorrelated

transmission channels in a MIMO transmission system ar described in the following.

In the first method step S10 of the embodiment of the inventive method for determining the number of

uncorrelated transmission channels in a MIMO transmission system according to the flowchart shown in Fig. 4 all the transmission coefficients \_γ , \₂ , ..., 1α_ΝΜ of the NxM

transmission matrix H in the MIMO system are measured in a measurement unit 1 according to the block diagram in Fig. 5. The transmission coefficients \_γ , \₂ , ..., 1α_ΝΜ of the NxM transmission matrix H are measured by use of well known channel estimation algorithms.

In the next method step S20 the eigenvalues λ _Χ,...,λ_ιι,...,λ_{Ν N} of the transmission matrix H are determined in a

processing unit 2 using a s_ingular-value-decomposition (SVD) according to equation (1) . The number N of

eigenvalues of the transmission matrix H corresponds to the minimum Min {N,M} of the N rows and M columns of the transmission matrix H . Another algorithm can also be used to obtain the singular values.

In the following method step S30 the eigenvalues

λ _Χ,...,λ_ιι,...,λ_{Ν N} of the transmission matrix H are ordered in a descending order resulting in a vector

Λ of descending ordered eigenvalues

. On the basis of these descending ordered eigenvalues I , .., I , .., N the quotients y(k) resp. y(k)^' between the sum of the k biggest eigenvalues and at least the next smaller eigenvalue of all N—k smaller

eigenvalues for each value of k according to the first resp. second embodiment of the invention are determined. In the first embodiment of the invention the quotients /(k) are calculated between the sum of the k biggest eigenvalues and the sum of the N—k smallest eigenvalues for each value of k according to equation (3) . Fig. 7A shows the quotients /(k) for each value of k with the characterizing slope in the neighborhood of the value k_Int for the integer-valued rank rank_lnt of the transmission matrix H . In the second embodiment of the invention the quotients γ(Κ)' are calculated between the sum of the k biggest eigenvalues and the N - k ^th smallest eigenvalue for each value of k according to equation (8) . Fig. 7B shows the quotients γ(Κ)' for each value of k with the

characterizing slope in the neighborhood of the value k_Int' for the integer-valued rank rank_Int' of the transmission matrix H being flatter than in the first embodiment.

The following method step S40 contains the comparison of the quotients /(k) resp. γ(Κ)' with an appropriately selected threshold value p_Th as shown in Fig. 7A resp. 7B.

In the following method step S50 the integer value k_Int resp. k_Int' of the quotient y(k) resp. y(k)' being nearest below to the threshold value p_Th and representing the integer-valued rank rank_Int resp. rank_Int ^r are determined according to equation (4) for the first embodiment of the invention resp. according to equation (9) for the second embodiment of the invention.

In the next method step S60 a fractional value k_Fmc , k_Frac' , ^Frac " ^and k_Pmc ^"' representing a probability value for indicating a probability for a correctness of the

determined integer-value k_Int resp. k_Int' for the rank rank_M resp. rank_Int' of the transmission matrix H is determined.

The probability value is a value between zero and 1 and is proportional to the distance between the threshold value p_Th and the quotient y(k_I ) resp. y{k_Int')' at the determined integer-value k_Int resp. k_Int' .

In case of a probability being bigger than 0.5 the

incremented integer-value k_Int + \ resp. k_Int'+\ probably represents a correct value for the rank rank_Int resp. rank_Int ^r of the transmission matrix H . Whereas in case of a probability being smaller than 0.5 the determined integer- value k_Int resp. k_Int' probably represents a correct value for the rank rank_lnt resp. rank_Int' of the transmission matrix H .

In a first embodiment of the invention the fractional value k_Fmc is determined according to equation (6), whereas in a second embodiment of the invention the fractional value k_Frac' is determined according to equation (11) . In a third embodiment the fractional value k_Frac" is determined by using a linearization technique according to equation (17), whereby quotient values /(k) and /(k + l) according to the first embodiment in equation (3) or alternatively quotient values y(k)' and /(k + Y)' according to the second embodiment in equation (8) can be used. The calculation of the fractional value k_Frac"' according to equation (18) in a fourth embodiment represents a further alternative.

The influence of different fractional values k_Fmc

determined on the basis of estimated transmission

coefficients of the transmission matrix H on the error- vector-magnitude (EVM) of a received symbol in the

constellation diagram for different signal-to-noise-ratio values is shown in Fig. 9. The error-vector-magnitude reduces for a higher probability value.

The integer-valued rank rank_Int , rank_Int' and the fractional value k_Frac , k_Frac' , k_Frac" , k_Fr " can be summed to an effective rank rank_Eff , rank_Eff " as outlined exemplary in equation (7) resp. (12) and displayed in a display unit 3. In Fig. 6A, 6B, 6C and 6D exemplary types of displays for an effective rank rank_Eff , rank_Eff " are shown.

The invention is not limited to the disclosed embodiments. The invention comprises all combinations of all features claimed in the claims, of all features disclosed in the description and of all features drawn in the figures of the drawings .

Claims

1. Method for determining a number of uncorrelated transmission channels in a multiple-input-multiple-output transmission system comprising the following method steps:

• determining the transmission coefficients

{ _γ, ₂,...^_ΝΜ) of a transmission matrix (H) corresponding to the multiple-input-multiple-output transmission system,

• determining a number N of eigenvalues of the

transmission matrix (H) ,

wherein the number of uncorrelated transmission channels is determined by a rank k of the transmission matrix (H) , for which a quotient (y(k) ; /(k)') between the sum of the k biggest eigenvalues and at least the next smaller eigenvalue of all N—k smaller eigenvalues is below a threshold value ( p_Th ) .

2. Method according to claim 1,

characterized in

that the quotient (y(k); /(k)') between the sum of the k biggest eigenvalues and at least the next smaller

eigenvalue is the quotient (/(k)) between the sum of the k biggest eigenvalues and the sum of the N—k smallest eigenvalues .

3. Method according to claim 1,

characterized in

eigenvalue is the quotient (/(k)') between the sum of the k biggest eigenvalues and the N-k^th smallest eigenvalue.

4. Method according to any of claims 1 to 3,

characterized in

that the number of uncorrelated transmission channels is 1, if the k biggest eigenvalues equals zero, for which a quotient (y(k); /(k)') is below a threshold value ( p_Th ) .

5. Method according to claims 2 or 4,

characterized in

that a probability value (k_Fmc) indicating a correctness of the value k determined for the rank of the transmission matrix (H) is determined by the quotient (/(k)) between the sum of the N—k smallest eigenvalues multiplied with the threshold ( p_Th ) and the sum of the k biggest

eigenvalues .

6. Method according to claims 3 or 4,

characterized in

that a probability value (k_Frac') indicating the correctness of the value k determined for the rank of the

transmission matrix (H) is determined by the quotient iy{k)') between the sum of the N— k smallest eigenvalues multiplied with the threshold and the N-k^th smallest eigenvalue .

7. Method according to any of claims 1 to 6,

characterized in

that a probability value (k_Frac") indicating the correctness of the value k for the rank of the transmission matrix (H) is determined by calculating a fractional value

(k_Frac") corresponding to the threshold value ( p_Th ) on a linearized curve of quotient values ( y(k) , /(k + l) ; y(k)' ,

/(k + l)') between the quotient ( /(k + l) ; /(k + l)' ) corresponding to the value k + l and the quotient (y(k) ; y(k)^r)

corresponding to the value k .

8. Method according to any of claims 1 to 7,

characterized in

that the probability value ( k_Fmc ; k_Fm ; k_Frac" ) is zero, if the determined rank k equals N .

9. Device for determining a number of uncorrelated transmission channels in a multiple-input-multiple-output transmission system with a measurement unit (1) for determining the transmission coefficients { _γ, ₂,...^_ΝΜ) of a transmission matrix (H) corresponding to the multiple- input-multiple-output transmission system, a processing unit (2) for determining the eigenvalues of the

transmission matrix (H) and the number of uncorrelated transmission channels on the basis of a rank of the transmission matrix (H) depending on the eigenvalues and a display unit (3) for graphical displaying the number of uncorrelated transmission channels.

10. Device according to claim 9,

characterized in

that the processing unit (2) determines the number of uncorrelated transmission channels by a rank k of the transmission matrix (H), for which a quotient (y(k);y(k)^r) between the sum of the k biggest eigenvalues and at least the next smaller eigenvalue of all N-k smaller

eigenvalues is below a threshold value ( p_Th ) .

11. Digital storage medium with electronically readable control signals, which work together with a programmable computer in a way that all method steps of claims 1 to 8 can be performed.

12. Computer program with program-code-means, which performs all method steps of claims 1 to 8, if the program is performed on a programmable computer and/or a digital signal processor.