The study of performance limits of receive antenna selection for MIMO spatial multiplexing in nonstationary channel Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

THE STUDY OF PERFORMANCE LIMITS OF RECEIVE ANTENNA SELECTION FOR MIMO SPATIAL MULTIPLEXING IN NON-STATIONARY CHANNEL

Mikhail Yu. Starovoytov,

Nokia, Moscow, Russia, mikhail.starovoytov@nokia.com

Keywords: MIMO spatial multiplexing, receive antenna selection, channel prediction, non-stationary channel.

The Single User MIMO Spatial Multiplexing (SU MIMO SM) mode allows for reaching the highest spectral efficiency per transmit antenna (TX antenna) for the fixed number of receive data processing circuits (RX circuits) per user. The scheme where the subset of a number of RX antennas (installed for example on the roof of the moving vehicle) is dynamically chosen and connected to the same number of RX circuits - is known as antenna selection and switching, and it's the simplest way to get an energy efficiency gain for SU MIMO SM.

For stationary radioconditions the solution for the problem of choosing the optimal subset of RX antennas is well studied. In the non-stationary case the choosing of RX antennas shall be done for every coming slot (7 symbols, 0,5 ms) in advance; it involves the use of channel prediction on the basis of measurements in the past, which is also a well developed area in technical literature. However the combination of non-stationary radio channel and antenna selection is not covered in literature due to controversy in problem statement: to provide the circularly repeated reading of the pilot information from reference channels for all RX antennas on one hand, and not to lose the gain from applying the RX antenna selection on another hand. A new method to combine these requirements is proposed in this work, without the need to go beyond the existing LTE specifications in part of physical channels organization.

In the model experiments for MIMO 4x4 Spatial Multiplexing mode at modulation I6QAM and selection of 4 out of 8 RX antennas, the 1 dB energy efficiency performance gain is availed with the proposed method compared to the mode without antenna selection. The proposed method could also serve an efficient theoretical instrument to estimate the upper limits of energy efficiency performance of SU MIMO SM system with RX antenna selection in dynamically changing radioconditions - without dependence on a particular channel prediction method used.

Information about author:

Mikhail Yu. Starovoytov, Nokia, Manager, Moscow, Russia

Для цитирования:

Старовойтов М.Ю. Изучение пределов помехоустойчивости нестационарного канала MIMO c выбором антенн на приеме // T-Comm: Телекоммуникации и транспорт. 2017. Том 11. №8. С. 63-68.

For citation:

Starovoytov M.Yu. (2017). The study of performance limits of receive antenna selection for MIMO spatial multiplexing in non-stationary channel. T-Comm, vol. 11, no.8, рр. 63-68.

7ТЛ

introduction

For the MIMO model the known problem of RX antennas selection is formulated as follows: number of RX antennas is more than number of receiving chains; it's necessary to switch the receiving chains to the same number of RX antennas so that to get the energy efficiency gain. For the known characteristics of radio channel, which in practice could be availed for stationary radioconditions, the topic of RX antennas selection is well studied [4, 5]. The dominant criterion used in the known up to date antenna selection algorythms in M1MO systems, is the criterion of maximization of mutual information in the MIMO channel [3, 51.

The case of essentionally non-stationary radio channel that involves the techneeks of pilot-aided channel evaluation in time-frequency domain with due precision is also studied well [3, 9] — but without antenna selection option. Antenna selection requires channel prediction of certain (not very high) accuracy for the next period of time in future, and the prediction techneeks for the purpose are available [8, 14, 15]. The combination of non-stationaiy radio channel and antenna selection is not covered in literature due to the controversy in requirements: to provide the circularly repealed reading of the pilot information from reference channels for all RX antennas on one hand, and not to lose gain from RX antenna selection on another hand.

In this article the problem is stated in the way: without changing the time and frequency resources allocation structure adopted presently in LTL 3GPP standard, to find the RX antenna selection algorithm that will ensure the flow of pilot data measurements for all RX antennas at a constant rate - at the cost of losing certain share of gain compared to ideal RX antenna selection, but still providing gain compared to the case without RX antenna selection.

Nntational conventions in this article:

Vector values and matrices will be written in bold font, scalar values in normal. Upper-"//" designates Mermitian transform.

• X(3,4) — matrix X element in row 3 and column 4;

• X(:,2) - column vector built of all elements of matrix X in column 2;

• X(V,2 )- column vector built of all elements of matrix X in column 2 and rows, defined by elements of vector v (for all

elements in v positive integer);

• v=[q ; p] - the vertically slacked column vector built of

elements of column vectors q and p, example

"f ' 7 "

4 6,5

[q ; p]-

1 4

7

6,5

= [l; 4; 7; 6,5]'

vec(A) =

AC, 2)

stacked consequitive column vectors of matrix A, Example:

A =

1 1 + i 2-i 0

vec(A) =

1

2-i 1+1 0 0 3

• dei(\) - determinant of square matrix;

• v = [l:i] ~ row vector with whole number elements

from 1 to R in increasing order;

• B \ v - row vector with elements of vector B, with exception of elements of vector v (with repeated elements deleted), example: B = [l 7 7 9 4 3],v = [7 4]:B'\v = [l 9 3];

• u,e - unity of sets and belonging to set signs;

• £•««*_ t)le family of matrices of size M x N with all elements complex numbers;

• log2{.) - base-2 logarythm;

• E(.) - mathematical expectation;

1 System model SU MIMOA/x N and RX antenna selection criterion

Let's consider the mode Single User MIMO with Spatial Multiplexing (SU MIMO SM) [1, 3]. Each of M receive (RX) circuits in the user terminal is connected to one of PN RX antennas, and N pairs of transmitter (TX) circuit - transmitter antenna are active on the other side of the radio link. The RX radio channel information is not available on the TX side , so the signal of equal mean power is fed into all N TX antennas, no precoding. The case of selective switching of PA' RX antennas to M RX circuits for m = 4, PN =8 (P = 2, N = 4) is shown on

Picture 1.

M=4 n|i)

PN-8

RX circuit 3

"switching vector" (/=[2 3 S 8)

■ Itt— all-ones on diagonal and the rest zero elements matrix of size K x K ;

A(:,1)~

- column vector composed of

He tu re 1

Let's introduce designations: d(e) efl : PAr],ee[l: Ai] — "switching vector" containing all RX

antenna's numbers connected over the switch to M RX circuits;

p(vec{H)) - probability density of the joint distribution of

elements of column vector of all columns of matrix H;

Q - the family of random matrices of Rayleigh fading channel, with all elements distributed independently and identically by a normal law with zero mean and dispersion 1, meaning HeC"'": Hefi is identical to [8]: />{vee(H)) = normaI(0M,NilM.N), where c„,v is a column vector

of size M" N with all zeros. Channel model [1-3]:

y = J$NR H(d) x + n (1)

Where:

H s - normed M1MO channel matrix. We will consider HeQ.

xeC 1 - normed column vector of transmitted symbols, with components random independent values that can equiprobably take values from a limited discrete set ("alphabet") in accordance with the used method of modulation QPSK., 16QAM.....E(x■%") = I,f, here the expectation is taken over all

allowed values in the set 3.

The set E ¡s defined in a standard way [11, for 16QAM: 3=1/yfSNR-[l+i I-i -1 + ; -I-/ 2 + i 2-i -2+i -2-i

1 + 2/ 1-2i -1 + 2i -1-2i 2 + 2; 2-2i -2 + 2/ -2-2/] neC™ - normed column vector of noise:, p(n) = normal(0M;Iti), where 0W is a column vector of size M

with all zeros.

E _ tiie observed column vector.

Scalar factor %jSNR in (1) carries the traditional interpretation of Signal-to-Noise ratio in relation to one TX antenna. Wc will consider the exact value of SNR always known at RX side.

With a fixed number of M=N RX circuits in the mobile terminal and with the channel matrix HtQ, the SU MIMO SM NX- N mode reaches the upper limit of spectral efficiency per one TX antenna in the downlink [1, 2, 3]. Therefore further on in this article we will concentrate on the case M=N.

The mutual information Ml for MIMO channel model (1) is defined as follows [3,4J:

Aff(5JVR,H(d,:)) = /o^3(dW{JSiVR-H(df:)wH(d,:)+I^)) (2)

Maximization of the expression in the right side of formula (2) makes the essense of the criterion of maximization of mutual information, that we will use in this work.

In practice on RX side we have to operate not with the true

channel matrix H, but with its estimation H, that is got after processing the pilot information. Criterion (2) applied to the classic antenna selection task for the MIMO model (1) is formulated as follows:

d = argmax | mi[ snr,h(s,)

(3)

The maximum here is searched over vectors s of size N running over all A'-element subsets of the list of indexes h : /w] ■

2 The structure of physical resources allocation in frequency and time domain in LTE according to 3GPP

To describe the principles of getting the channel matrix

estimation H we will consider the frame structure and organization of resource allocation in LTE OFDM na according to 3GPP [12], see Picture 2.

the current fpme marA

* 1 ~ pilot symbols lor one "ni aolEona

Picture 2

Resources are mapped on the time-frequency plane ("resource plane"), with time on the horizontal and frequency on the vertical scale.

In time domain each frame is composed of 10 subframes I ms each, and each subframe is split into two slots 7 symbols each (in the variant of Normal Cyclic Prefix, [ 12]). In frequency domain the range is spanned with subcarriers with inter-center distance between neighbors 15 kHz. The cell in the section of a particular lime symbol and a particular frequency subcarrier is the "resource elemcnl", the minimum unit of assigning resources in LTE.

For estimating the channel the pilot information is transmitted on the reference resource elements, which are distributed on the resource plane in the nodes of a regular grid w ith nodes assigned on a certain set of frequency subcarriers. On Picture 2 the positions of reference symbols on the resource plane for one TX antenna are marked with 1-s. Reference symbols for another TX antenna are assigned on the resource plane to another grid of the similar shape which is moved against the shown one by a fixed vector. After all reference elements assigned, the rest elements on the resource plane are used for pay load data.

Additional designations:

Ts= (t, + tcp) - the length of one symbol, equal to the sum of ts and cyclic prefix tCP (Cyclic Prefix - CP, is added to suppress the intersymbol interference). In LTE with "Normal Cyclic Prefix": t;=I/14MC.

A/ = 7. Ts ~ the time span equal to one slot, where one

reference symbol is included for each TX antenna on certain frequency subcarrier assigned for reference symbols for this particular TX antenna.

In this work it is assumed that channel matrix estimation H in each resource element is derived on the basis of values of the channel matrix in the reference elements, interpolated to the

whole resource plane [9, 13]. Alter getting for some RX antenna the channel estimation in reference elements on the grid corresponding to some TX antenna, the channel estimation for this RX-TX pair for the rest of resource plain is accomplished in two steps:

1) Interpolation in time domain, shown by thin horizontal lines on Picture 2, defines the channel estimation for certain frequency subcarriers for all resource elements placed continuously on horizontal lines.

2) Interpolation in time domain, shown by wide vertical arrows on Picture 2, defines the channel estimation for all resource elements placed on the whole resource plane.

Not only the interpolation, but also the prediction of the channel in places on resource plane to the right of the current time mark on Picture 2 is done based on pilot data in reference elements to the left of the current time mark. Channel prediction could be done by a collection of different means: spectral methods (ESPRIT. root-MUSIC) [14, 15], Kalman filtering [3, 7], regression [8], combination of extended Kalman filter with regression [9], and others.

3 The mode of reading the pilot information in reference elements with respect to antenna selection regardless of channel prediction method

The specific feature about the RX antenna selection N out of PN is: during one slot (containing 7 symbols) of duration ¿t ms, each one RX circuit could be connected to only

one RX antenna.

We will base on the proposition that the future is unknown, that in present time there exists no way to define the RX antennas on which measurements should be made in future lime in a preferred way with a higher frequency, and the frequency of measurements of the channel from all N TX antennas for all PN RX antennas shall be provided equal.

This proposition is not a trivial constatation: it is known there is a way to construe the ideal noil-stationary radio channel in a way that it could be measured only once in the beginning, and the prediction of its behavior for the unlimited future could be guaranteed. This way of construction is given by spectral methods of channel representation as a superposition of a Unite number of plane waves with unchanging amplitude and direction of arrival [14, 15Shortages of this construction are known [10]: in reality as the lime is passing, the number of plane waves, their amplitudes and directions of arrival are undergoing the constant and often random changes. Additionally the diffusive reflection on the objects surrounding the RX antennas leads to the impossibility of considering the number of waves finite, and even to consider those waves plane, which leads to the inapplicability of the model. Therefore in practice even if the channel is satisfactorily descried by spectral methods, the constant channel measurements to track the changes in channel parameters shall be done.

As a result from the made proposition it follows that to provide the consequent one-time reading of pilot information from N transmit antennas on ail PN RX antennas, P slot intervals of duration At^,, each shall be needed, with reading pilot

information on the new set on A' RX antennas on every next slot interval.

4 The proposed Procedure of defining the switching vectors in a double slot

For working out the Procedure further on we will assume P - 2, keeping the aim at further modelling where we will test MIMO 4x4 with dynamic selection of 4 RX antennas out of 8 (N- 4, PN=%)

Let's call "double slot" the two consequent slot intervals of Aj = 0,5 ms each. The switching vectors of size 4 with the

pointers to RX antennas enabled in the consequent slots I and 2 in one double slot will be named d and d, (with the channel

model {1) with (| = d, for all 7 symbols in the duration of slot 1

and d - <i2 for all 7 symbols in the duration of slot 2).

from the conclusion in paragraph 3 it follows that we shall ensure: it i^d, = [l-K] in each double slot.

To ensure d i^d, =[1:8], the d, and d switching vectors

used in the current double slot - shall be derived in advance and known right before the beginning of the current double slot. It means that they shall be calculated on the basis of channel prediction {by any known prediction method) for 2A/ M -147" =

I ms ahead in the end of the previous double slot.

Lei the time mark t^T, (' ~ integer) be taken at the beginning

of a new double slot; channel matrix (r + 4)-7" in the middle of

the first slot will be designated h,, lir * and in the middle of the

second slot h ,,,..

Let S be the integer matrix of the size 70 x 8, with elements from all possible selections of 4 different numbers out of the set 11:8] without difference in their order on the first 4 positions in every ¿-th row S(k,] :4) (there are 8!/ -7« such

/(41-4!)

selections), and the rest 4 numbers remaining in the nexl 4 positions in every row: s(k,5 :8) - [1: 8] \ S(k, 1:4).

In the further Procedure ¡11 the first cycle: for the given fixed value SNR and for two known matrices

and

(t+tljTi ]et's compose matrix B of size 70/3 with Ml values in the first two positions in every row, and the value minimum over the first two positions in the third position in this row. Procedu re

of choosing d,, d, and the resulting d for one double slot: ►

for u= 1:70

B(a, 1) = m(SNR. H(,+w. {S(w, 1:4),:)) B(;r,2) = Ml(SNR, Hi(+11)I- (S(H,5 :8),:)) B(w,3) = min (B(u, 1), B{if, 2)) End

% nexl tiie function addrmax finds the row number where the maximum value for the column B(«,3) is met: z = addrmax{ B(:,3})

d, =S(z,l:4), d, = S(.z,5:8) (4)

% next to evaluate the properties for the whole double slot 2At =

I 111s we lake the switching vector d and channel matrix on that of two slots for which the Ml value showed lo be less (or worse):

if B(z,3) = B(2,l) ci - <1, else cl = d3

end ■

From the very construction of the Procedure it's seen that the existence of at least one solution is guaranteed. Besides: if from the column of 140 numbers sorted in ascending order w = jort(vcf([B(1: 70,1); Li(l: 70,2)])) to select the 70-th value in

the middle w(70), then the minimum of two Ml values in two consequitive slots for the chosen combination d,, d, could be

limited from below: mr«(B(z,l),B(z,2))>w(70)■ It means that

the Procedure provides the combination d., d, that gives the

MI value better than a certain threshold ("good enough"),

5 Modelling

Let's make two additional assumptions that will allow lo exclude the influence on the result of a particular chosen way of modelling in time of the non-stationary Ml MO radio channel [10, 11], and to be able study the influence of prediction accuracy without choosing any particular channel prediction method. This way the conclusions that will follow from experiments will show the effect from the proposed method on very general terms.

AI: Radio channel is assumed stationary in the duration of

^^pilot (for P=2 - during two slots, or I ms), but different in the next l-ms time periods. This way it makes true: 11 = 11-, — — — 1 (j — H ■

A2: Channel prediction for the next 1 ms in future gives a constant value for all symbols. This way it makes true:

A A A A

H,, = H;;; =•••:= Hur. =H-

For describing numerically the accuracy of channel matrix prediction let's introduce the parameter K:

H = -J\-K H + *JK ■ AH

K e [0... I], AH e fi (5)

Parameter K is introduced in a way that the matrix norms for

H and H stay unchanged. In compliance with the established practice in MIMO literature [1,2], we consider: AH e Cl.

Modelling terms: the case of SU MIM04 x 4 SM with modulation 16QAM. Bnergy efficiency testing was accomplished with generation of 20000 random channel matrices for each SNR value. Maximum likelihood demodulation with hard output [ 11 was applied, channel coding was not used.

On the vertical axis on the plot Picture 3 the "SER" - Symbol Error Rate, the share of erroneously detected symbols is shown. Curves shown on Picture 3 are:

For the case of no antenna selection (N = 4, d = 11:41 = const. Legend entry: "no ant selection". For the case of RX antenna selection 4 out of Si {N = 4, PN - 8) for the proposed method, vector d is taken from the outcome of the Procedure, Two curv es are shown - for K = 0 (which is ideal channel prediction for 1 ms ahead) and for K = 0,05 which is taken from |9, 14] as a typical value for the quality of channel prediction for I ms ahead for the known methods and known models of non-stationary channel. Legend entry: "proposed ant selection

10

10

Rayleigh fading. MIM04x4 SM 16QAM 8 RX anlennas

ce

UJ (£>

10

10

....................-!.................. -no ant select ——■ proposed art on selection K=0 selection K-0,05

—"— proposed art

;

....................-i.....................i.....................

10 15 20

SNR [dB]

Picture 3

It is seen on the plot that on the level SER = 10" the proposed method gives the energy efficiency gain of 1 dB compared to basic variant without RX antenna selection. The gain is not big for the reason that the proposed method is ensuring the flow of pilot data measurements for all RX antennas at a constant rate

The closeness of curves for the proposed method for the values of K = 0 and K = 0.05 within 0,3 dB at SER = I0": -points to the low level of dependence of the proposed method of the accuracy of channel prediction. This is a general property of antenna selection algorythms in general, that makes them advantageous in practical implementation regardless of channel prediction methods used. It also shows that regardless of Al and A2 assumptions done for the sake of modelling, those assumptions don't decrease the value of the results received in defining the limits of energy efficiency gain from using the proposed method.

Conclusion

The proposed RX antenna selection algorithm ensures the flow of pilot data measurements for all RX antennas at a constant rate, and still providing the energy efficiency gain (I dB) compared lo the case without RX antenna selection. Besides, it could serve an efficient theoretical instrument to estimate the upper limits of energy efficiency performance of MIMO system with RX antenna selection - regardless of channel prediction methods and unstationary channel models used.

References

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ИЗУЧЕНИЕ ПРЕДЕЛОВ ПОМЕХОУСТОЙЧИВОСТИ НЕСТАЦИОНАРНОГО КАНАЛА MIMO

C ВЫБОРОМ АНТЕНН НА ПРИЕМЕ

Старовойтов Михаил Юрьевич, ООО Нокиа Солюшнз энд Нетворкс, Москва, mikhail.starovoytov@nokia.com Дннотация

Режим однопользовательского MIMO с пространственным мультиплексированием позволяет достичь наивысшей спектральной эффективности в расчете на одну передающую антенну для случая фиксированного количества линий обработки сигнала на приемно стороне. Исходной архитектурой на приемной (абонентской) стороне является типовая схема с антенной решеткой, установленной на крыше транспортного средства, и мобильный терминал с фиксированным количеством цепей цифровой обработки сигнала. Схема с динамическим выбором приемных антенн для подключения к цепям обработки сигнала позволяет с наименьшими издержками получить выигрыш в помехоустойчивости системы MIMO с пространственным мультиплексированием каналов. Для стационарных радиоусловий задача оптимального выбора приемных антенн хорошо изучена в литературе по MIMO. В случае нестационарного канала выбор подмножества приемных антенн на каждый следующий пакетный интервал (7 символов, 0,5 миллисекунд) должен делаться заранее; это предполагает использование предсказания канала на будущее время на основе прошлых измерений, что в технической литературе является хорошо разработанной областью. В то же время применение техники выбора антенн на приеме к условиям существенно нестационарного канала в литературе не раскрыта. Причина нам видится в противоречивых требованиях содержащихся в постановке проблемы: необходимо с одной стороны обеспечить постоянные циклически повторяющиеся чтение и обработку пилотной информации со всех приемных антенн, и с другой стороны избежать полной потери выигрыша в помехоустойчивости по сравнению с базовым случаем без выбора приемных антенн. Новый метод, предлагаемый в этой статье, представляет попытку решить эту дилемму без необходимости изменения существующей структуры назначения физических ресурсов в стандарте LTE. В модельных экспериментах показан выигрыш в помехоустойчивости 1 дБ от применения предложенного метода с выбором 4-х антенн из 8-ми по отношению к базовому MIMO с пространственным мультиплексированием каналов (без выбора антенн). Предложенный метод может также служить теоретическим инструментом для оценки достижимой помехоустойчивости режима MIMO с пространственным мультиплексированием каналов и с выбором антенн в случае динамически меняющегося радиоканала - без привязки к определенному методу предсказания канала.

Ключевые слова: MIMO spatial multiplexing, receive antenna selection, channel prediction, non-stationary channel. Литература

1. E. Biglieri, R. Calderbank, A. Constantinides, A Goldsmith, A. Paulraj, H. V. Poor. MIMO Wireless Communications. Cambridge University Press New York, NY, USA 2007. 344 p.

2. Тартаковский Г.П. Теория информационных систем. М.: Физматкнига, 2005. 304 с.

3. Emre Telatar. Capacity of multiple-antenna Gaussian channels // European Transactions on Telecommunications, vol. 10, no. 6, pp. 585-595, November-December 1999.

4. Alexei Gorokhov, Dhananjay A. Gore, Arogyaswami J. Paulraj. Receive Antenna Selection for MIMO Spatial Multiplexing: Theory and Algorithms // IEEE Transactions on Signal Processing, vol. 51, no. 11, pp. 2796-2807, November 2003.

5. Andreas F. Molisch, Moe Z. Win, Yang-Seok Choi, Jack H. Winters. Capacity of MIMO Systems With Antenna Selection // IEEE Transactions on Wireless Communications, vol. 4, no. 4, pp. 1759-1772, July 2005.

6. Крейнделин В.Б. Новые методы обработки сигналов в системах беспроводной связи. СПб.: Линк, 2009. 272 с.

7. C.K. Chui and G. Chen, Kalman. Filtering with Real Time Applications, Berlin: Springer, 1999.

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Информация об авторе: Старовойтов Михаил Юрьевич, ООО Нокиа Солюшнз энд Нетворкс, Менеджер, Москва, Россия

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