Применение полиномиальной аппроксимации для решения задачи оценивания искажений сигнала в системе с MIMO в условиях априорной неопределенности Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

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POLYNOMIAL APPROXIMATION APPLICATION FOR SOLVING THE SIGNAL DISTORTION ESTIMATION PROBLEM IN A SYSTEM WITH MIMO UNDER APRIOR UNCERTAINTY

DOI 10.24411/2072-8735-2018-10133

Natalia E. Poborchaya,

MTUCI, Moscow, Russia, Keywords: MIMO, channel estimation least square method, apriori

n.poborchaya@mail.ru uncertainty, IQ imbalance, DC drift, frequency shift, Rayleigh fading.

The most important trend in the development of communication technology is an increase information transfer speed. This problem can be solved by applying MIMO technology, which consists in the transmission and reception of signals by several antennas. In this paper, two algorithms for joint channel estimation are considered, taking into account distortions such as the amplitude-phase imbalance of the quadrature components of the M-QAM signal, the constant components drift and the frequency shift remaining from the demodulation procedure. The estimation algorithms operate under uncorrelated Rayleigh fading and a priori uncertainty with respect to phase noise and observation noise. Synthesis of these algorithms is done by the method of least squares with the use of polynomial approximation. The difference between the algorithms is as follows. First algorithm estimates the channel factors first by the test sequence without determining the frequency shift, then extrapolates them for the data receiving interval and then recalculates them according to the received data sequence. This can be repeated several times during a communication session. In another algorithm, after estimating the channel factors by test symbols, an estimate of the frequency is made, which allows to reduce the number of test intervals and, thus, to increase the time for data transmission, as well as to reduce the number of time intervals over which the channel is re-evaluated according to the received data sequence. This approach simplifies the algorithm and increases noise immunity. But it can be applied if the channel multipliers almost do not change during the evaluation and detection time. This is observed at low user's transceiver movement speed or, if the communication channel is stationary.

A computational experiment was carried out for four transmitting and receiving antennas, which showed the operability of both algorithms when receiving the 4-QAM signal. The algorithm with frequency estimation allows to extrapolate the communication channel more accurately and has an energy gain of up to 5 dB in comparison with the algorithm where the frequency shift estimation is not taken into account.

Information about author:

Natalia E. Poborchaya, docent, Cand. Tech. Sciences, MTUCI, general theory of communication chair, Moscow, Russia

Для цитирования:

Поборчая Н.Е Применение полиномиальной аппроксимации для решения задачи оценивания искажений сигнала в системе с MIMO в условиях априорной неопределенности // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №8. С. 63-68.

For citation:

Poborchaya N.E. (2018). Polynomial approximation application for solving the signal distortion estimation problem in a system with MIMO under aprior uncertainty. T-Comm, vol. 12, no.8, pр. 63-68.

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Problem formulation

A lot of work has been devoted to channel estimation in M1MO systems [1-4]. There are, as a rule, the correlation function of the channel is known or the channel matrix is constant. Distortions made by the direct conversion receiver are not taken into account or evaluated separately. Consider a model of the MIMO system with N transmit and N receive antennas after demodulation (carrier removal). Observations complex vector has the form Y. = Ycj + y'Yf, where j - мнимая единица, imaginary unit,

= + + Y„ = Hs,0, + Bn +iv (1)

therein ©,=(4 - INt Ju - M-QAMdata

symbols or test sequence symbols vector,

Be/= - bcNlfm, K={btu - bm)TNxl - slowly

varying constant components vectors, p "' '

П ,=(u, ■ ■■ a V' — noise with an unknown distribution f*,vi V"1'-1 r*Ji.iVJvxl

law, mean values vectors ) - £((I J = 0, and covariance matrix Efcjil) - = >

w(o -адо

H

H -

X.l«

-ВиЛ 0

",„,»('■) Щм,(0

—И,. ,,v{/)

flWO

^■VM л-\

№ÜN

(2) (3)

Problem solution

Let the following restrictions be satisfied. During the data transmission and reception do not change I) channel factors Kj k (f) " Kj k > Kl k (0 " Ki t > 2) constant components bcU = £prf, btll = bsl; 3) amplitude and phase imbalance y., Atpt; 4) frequency Afr l,k = \,2,....,N. The evaluation interval

length for the test sequence of the is ¡ess than the value — , so

¥i

we cannot use classical DFT-based spectral analysis to estimate the frequency shift. Symbol synchronization is performed.

The channel matrix estimation algorithm, in which the frequency shift is not determined, and the extrapolation is performed by the estimated approximating coefficients is described in [5-7]. Bringing it here again does not make sense. The advantage of this approach is the possibility of using it w ith time-varying factors hcli(i), fljktf) with complete a priori uncertainty about the channel. This algorithm will be called "algorithm 1" further.

Let us consider in more detail the second approach. Channel matrixes H H^ estimation using test sequences is also carried

out as in "algorithm 1".

I. Approximation. Let us represent (2), (3) as a polynomial of order p within the time window A - (1 2 • • • m), m - test sequence length [7,8]:

%i# ^ *i HUJ k (0 - d,Xw,; (4)

= (5)

i = 1,2,,.../?;;

/ = 1,2,......a - discrete time, £(•) - mathematical expectation

operator, I - NxN unit matrix, «7"»-operation transpose sign.

Elements of channel matrices Hf(, Hn are defined by expressions

H\cj k CO = Kj k (0 cos(%) - Kt k (0 sin(-% )>

H 1 sj k (0 4.1 k (OsinUv,) + htJ k (/) cos(%),

H2cj *(0 =■ r, (Kj i (0COS(%) - Kj k (0 sini^a)),

H2,J k (0 = Tl (hi *(') SÍnC*7* ) + Kjk (') C05^)) ' therein I - receive antenna number, k - transceiver antenna number, / = 1,2.....,N , £ = 1,2,.....N , hc,k{i),Kn(i) -

channel factors, xw = IttA^TJ + <p,., xVi=xw+A<pr y,, Atp, - amplitude and phase imbalance respectively, Aft - frequency, which remained after the demodulation procedure, <ph - phase, defined as p,, = <ptv + Qu, C,¡ = b„su+blsIJ_t + b2£u„2 - phase noise, is given by the second-order moving average model, . -white noise with E(s,,) = 0; E(ef¡) = <J~,, b0,bt,b2 - constant coefficients, Tc - IJki symbols duration.

It is required from the observations (1) to find the estimation II , H of the channel matrices H 11 to forecast them for data receiving interval.

a\c.Uk

"\sS)A ^i.i.Ui a\s,pjk . f ■ '(P+IM '

а2с.ПЛ a2c,\Л ' '' a2c.pJk

a2s.QJk a2 s.Uk a2s.pjk f ■

2. Estimation by test sequence. The problem will be solved by the least squares method, processing the whole sample of the signal with the volume m. Let us consider the model (1), where

Xi=(Xf.t -K1.2 ) V;(| ' X, = (>'«,1 ys/.2 "' y'si.N )Vi.!"

Form the observed quadrature vectors for /-oil receiving antenna

as XJ ={y^u y^u ■■■ XuL' X^&W

Then, taking (4), (5) into account, we have

Y^DAz + n,,. V^DZ^+T^.

therein

(6)

1) =

d„/t,„ ■■■ d„/,,„ -dj*,

''m-lAw-l

<u, ■4-íi

ч^. Л

-d,„ Алы 1

'^(i.Vf/H-iHD

T

D, =

d.J,.

d/„

rX ^ -^wi

\CJN

b.

V».

^JM-L^lflf-I d. 1

d./vi

r Y

A

%j =

/(íWy+lH )xl

p+IJ+l)

X-2C.M V rf /(2,V|;j+1)+!|KI

Au

■ I,., =

/ mxl

f \ Msin.l

Msii

Uncorrelated noise vector? T]r s, t|, , have zero mean values and covariance matrixes E{x\cjl{cj) = l:{t\ ij \) = <x;l„„m, / = 1,2,...,N -receiving antenna number. Minimizing the functional [8]

- DZ,I" = min; |jVt, -D,Z, |" = min.

we obtain expressions for estimating the approximation coefficients and the constants

Zu=(D/Dc)-ll>:;%; Z2J =(D/D,rlD/YI,, (7)

Remark. In the ease of Gaussian noise, algorithm described by (7) is the optimal estimate based on the maximum likelihood

criterion.

f X,

therein

z„ =

v./1

Li./A*

' ZJ ~

f V ^

V 4 y

(2JV(p+lH)«l

\ /(2Ar{i>+l )+])*!

/ = 1,2,..., N ■ Then the estimation of the elements of the channel matrix in the time window A will have the form

"w* (0-= d,X„ ,Jt;Hu ,,(/)=diXl, ; (8)

Hicj k(0 = d,X2i., k; H2s lk (i) ~ dfX,,_,k; (9)

4;=Z,((2tí>+l)+U); b£=±,J2N(p + \) + \,\), (10) i = 1,2,....m ,l,k = 1,2,..., N .

Using the formulas (2), (3), we express the dependence of

H2sJk(S) on HlcJM HUJk{i): H2cj k(i)=?\ cos( )HXcJ *(/)- Sin( A^, (/),

H^j k (i) = 7, cos( A^, t(í)+f, sin( , í (/).

Then, using the method of statistical averaging, we obtain expressions for estimating the amplitude and phase imbalance

/v í=, m ,=| A' - 1 f 1 f jr-f

therein (Kf>ft(0

iWÍ^WO ¿Wíof; lir =

(11) (12)

To calculate the frequency shift, we use formula

1

(13)

A1' i-1 w - ,-! l7rmJL. i Wclt(0,

+)•

^(0 - (' + mo(0 -HMk{i + rn0)HUJk(0, where A«u is the distance between two neighboring time sample (a parameter on which the quality of the frequency estimate depends), 0 < m„ <m~ 1 -

Remark. If for all receiving antennas the amplitude-phase imbalance, the frequency shift and the constant components are the same, then for their calculation, averaging over antennas:

1 N \ N

c N p d s Ntt Channel factors Hu,,k{i\tlu¡At) taking into account (2), (3), can be represented in the form

HXeJM = 4 cos(2^A/,r,i + <pH +ik{) = = Fek, cos(2rAf,Tci)~FtM s[n(2/TAJ]TJ), H\*jk(0 = At sm(2,TA/;7W + <pu + 4) = = F u cos(2.TA/;rj') + F sin(2,TA/;rj'),

h

\ "cJi y

therein ^ 4

- 4* C0S(4 ^v.« = 4 sin(4

We denote by ^(Ofí

'' cos( 2 TTÄ/j 7) /) - si n{ 2M/¡7/) 1

• _ ( F.a F¡Jk )

sin(2^A/jr,/) cos(2 7r\f,TJ)

Then the estimation of the parameters Fc , Fs tk,l,k - 1,2,..., N can be found as

Fft=q'Hw (14)

Procedure (7) - (14) will be called "algorithm 2" further.

3. Extrapolation and detection. According to earlier estimates. the matrixes Hri,Hi; are predicted for time interval

/ — w + l,w + 2,.....n by formulas

,u cos{2/zA/jl71t./) - FsM si]i(2/TA/iT;),

Huj* (0 = cos(2ttA/;7;,/) + Fc U sin(2,TA/;r /),

(15)

Y

H2Cj k (0 = Y, cos( A^ )HlcJ t (/) - y, sin( Aft )HW k (0; Hnj k (0 = ft cos(A^ )J¥W k (/)+f¡ ùn(A<f>t)HUJ t {/);

Further, using ( i 5), ( 16), we define a soft solution

therein r.=(r,. r,,)r; X =(yw y,,)r; H, =

(16)

(17)

H ^

2Nx2N

ru-(ru.i *!/./ -(Oj.l "' rJJJi):xfl'

Let 0, =(/, . J, , be a the vector of possible

I') 0 \ I-J □ /

M-QAM data symbols of the signal arriving at /-th receiving antenna, f = 1,2,,,,.,A/. Then minimizing the functional

|r^0,J|2 = mm; where fu=(riu r,uf, we obtain a

detection algorithm

&u=Qi ja=ar&nm(rUJItJo +rJUJl ja -0.5(l?Ja +jfJo)),

h

that is optimal by the maximum likelihood criterion lor Gaussian noise; 0, =f/(J , I = 1,2.....,N .

Simulation

To analyze the obtained "algorithm 2", a computer experiment was performed with the following data: the number of transmitting and receiving antennas N = 4, he received 4-QAM signal the length of the test sequence, which is a 4-QAM signal with known information symbols, is m = 500, the sample size n — 7500, the order of the approximating polynomial p = 1, the standard dev iation of the phase noise of about one degree, data symbol duration T = 0.25 mkc, sampling interval A/ = T, the number of experiments is — 500. The channel factors hc lk, htJk

were formed as independent Gaussian random variables. Constant components bd,bsi, amplitude imbalance , phase imbalance A<p, and the initial random phase pl0 are uniformly dis-

7t n

tributcd over the intervals [0;2], [0.5;l],

18 18

respectively, /,A" = 1,2,...., /V, the residual frequency after the demodulation procedure is Af = 180.7 Hz, The root-mean-square errors of the signal parameters estimation obtained using the procedure "algorithm 2" for different signal-to-noise ratios q (SNR) are given in Table I.

Estimation RMS when receiving a 4-QAM signal with four transmitting and receiving antennas

Table 1

g, aG J/Q of tlie signal ■13/41 4008 33/31 30/iH 23/21 20/1X 13/11 10/8

RMS, icr4- 10"4. 10~4- IV* i Í0~3- 10"3- lo3- 10~3-

5.38 5.47 6.12 6.86 1.1 1.5 3.2 4.5

RMS,, icr* • 10H- 10"4- 10-4- itr3- 10"3- I0~3- 10"3-

4.3 4.41 5.24 6.14 1.1 1.5 3.3 4.6

RMS, 10 3 • 10"3- 10 3- 10"3. i(T3- 10"3- 10 "3 • 10

2.3 2.3 2.7 3.2 5.6 7.6 1.57 2.23

RMS*, (dCR) 0.17 0.18 0.21 0.25 0.43 0.58 1.21 1,66

(Hz) 5.51 5.58 6.03 6.5 9.46 12.29 26.47 36.77

and

Figure I shows one of the implementations of the 4-QAM signal on one of the receiving antennas with SNR q = 23/21 ,uE for the in-phase and quadrature components, respectively, without compensation and with signal compensation for the two evaluation procedures. "Algorithm 1" worked with the following settings: m - 500, n - 7500, p = I, the test sequence was used once, the length of the extrapolation and detection interval K() = 100, then the channel is re-evaluated according to the information sequence, the number of intervals K0 is equal to Q = 70 . The predicted channel for one implementation on one of the receiving antennas is illustrated in Figure 2.

Figure 3 shows the experimental error probabilities obtained after the detection procedure.

e o

i

m • C 0 • *

• m

0 rc - 0 rc

• 0 *

* • '1 w *

e o

m *

* * ¡2 o

m * * #

1 0 rc - /i

* Ë 0 * *

* « ■1 * •

■2 0 2 4 -2 0 2 A -1 0 1 -1 0 1 10 1-10 1

yc yc ft rc rc rc

a) b) c)

Kiguro 1. 4-QAM signal at the compensator input - (a), 4-QAM signal at the compensator output: using "algorithm I" - (b), using "algorithm 2" - (c)

T

т

ПРИМЕНЕНИЕ ПОЛИНОМИАЛЬНОЙ АППРОКСИМАЦИИ ДЛЯ РЕШЕНИЯ ЗАДАЧИ ОЦЕНИВАНИЯ ИСКАЖЕНИЙ СИГНАЛА В СИСТЕМЕ С MIMO В УСЛОВИЯХ АПРИОРНОЙ НЕОПРЕДЕЛЕННОСТИ

Поборчая Наталья Евгеньевна, Московский технический университет связи и информатики, кафедра ОТС МТУСИ,

Москва, Россия, n.poborchaya@mail.ru

Аннотация

Важнейшим направлением развития техники связи является увеличение скорости передачи информации. Эту проблему можно решить, применяя технологию MIMO, которая заключается в передаче и приеме сигналов несколькими антеннами. Рассматриваются два алгоритма совместной оценки канала с учетом искажений таких, как амплитудно-фазовый разбаланс квадратурных компонент M-QAM сигнала, дрейф постоянных составляющих и сдвиг частоты, оставшийся от процедуры демодуляции. Алгоритмы оценивания работают в условиях некоррелированных релеевскимих замираний и априорной неопределенности относительно фазовых шумов и шумов наблюдений. Синтез данных процедур проведен методом наименьших квадратов с использованием полиномиальной аппроксимации.

Отличие рассматриваемых алгоритмов состоит в том, что в одном производится оценка множителей канала без определения сдвига частоты сначала по тестовой последовательности, затем производится их экстраполяция на время приема информации и дальнейший их пересчет по информационной последовательности. Это может повторяться несколько раз в течение сеанса связи. В другом случае после оценки множителей канала по тестовым символам происходит оценка частоты, которая позволяет сократить количество тестовых интервалов и тем самым увеличить время для передачи информации, а также уменьшить число временных промежутков, на которых производится переоценка канала по информационной последовательности. Такой подход упрощает алгоритм и увеличивает помехоустойчивость. Но его можно применить, если множители канала практически не изменяются за время оценки и детектирования. Это наблюдается при низкой скорости движения абонента или, если канал связи стационарный. Проведен вычислительный эксперимент для четырех передающих и приемных антенн, который показал работоспособность обоих процедур при приеме сигнала 4-QAM. Алгоритм с оценкой частоты позволяет более точно экстраполировать канал связи и обладает энергетическим выигрышем до 5 дБ по сравнению с процедурой, где оценка сдвига частоты не учитывается.

Ключевые слова: MIMO, оценка канала, метод наименьших квадратов, априорная неопределенность, сдвиг частоты, амплитудно-фазовый разбаланс, дрейф постоянных составляющих, релеевские замирания.

Литература

1. Бакулин М.Г., Варукина Л.А., Крейнделин В.Б. Технология MIMO. Принципы и алгоритмы. М.: Горячая линия - Телеком, 2014. 244 с.

2. Нестеренко А.Н. Математическая модель MIMO-OFDM сигнала. // Интернет-журнал "Науковедение", выпуск 4 (23), 2014. С.1-12, http://naukovedenie.ru.

3. Коляденко Ю.Ю., Коляденко А.В. Математическая модель радиоканала для MIMO систем // Электронное научное специализированное издание-журнал "Проблемы телекоммуникаций", № 2(7), 2012. С. 91-109.

4. Мухин И.А. Исследование влияния погрешности оценки канальной матрицы на эффективность многоантенных систем с пространственным мультиплексированием // T-Comm: Телекоммуникации и транспорт. №9. 2012. С. 107-111.

5. Поборчая Н.Е Оценка амплитудно-фазового разбаланса и дрейфа постоянных составляющих сигнала в системе MIMO // Системы синхронизации, формирования и обработки сигналов. Т.8. №3. 2017. С. 36-41.

6. Поборчая Н.Е., Хасьянова Е.Р. Оценка и компенсация искажений сигнала в канале с доплеровским расширением спектра и релеевскими замираниями // Электросвязь. 2017. №6. С. 44-49.

7. Поборчая Н.Е., Пестряков А.В. Оценка и компенсация искажений сигнала в приемном тракте систем с MIMO // Электросвязь. №12. 2017. С. 42-48.

8. Вержбицкий В.М. Основы численных методов. М.: Высшая школа, 2005, 840 с. Информация об авторе:

Поборчая Наталья Евгеньевна, доцент, к.т.н., Московский технический университет связи и информатики, кафедра ОТС МТУСИ, Москва, Россия

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