ANALYSIS OF THE INFLUENCE OF THE LAGRANGE MULTIPLIER ON THE OPERATION OF THE ALGORITHM FOR ESTIMATING THE SIGNAL PARAMETERS UNDER A PRIORI UNCERTAINTY Текст научной статьи по специальности «Медицинские технологии»

MSC 60G35

DOI: 10.14529/ mmp220210

ANALYSIS OF THE INFLUENCE OF THE LAGRANGE MULTIPLIER ON THE OPERATION OF THE ALGORITHM FOR ESTIMATING THE SIGNAL PARAMETERS UNDER A PRIORI UNCERTAINTY

N.E. Poborchaya1, E.M. Lobov1

1Moscow Technical University of Communications and Informatics, Moscow, Russian Federation

E-mail: n.poborchaya@mail.ru, e.m.lobov@mtuci.ru

The paper considers a recurrent regularizing algorithm for joint estimation of distortions of a M-ary quadrature amplitude modulation (M-QAM) signal obtained in a direct conversion receiver path. The algorithm is synthesized using a modified least squares method in the form of Tikhonov's functional under conditions of a priori uncertainty about the laws of noise distribution. The resulting procedure can work both on the test sequence and on information symbols after the detection procedure. We analyze the influence of the Lagrange multiplier on the accuracy of the estimation procedure and on the complexity of the algorithm. It is shown that, with the same accuracy, the regularizing algorithm requires significantly fewer iterations than the procedure without the Lagrange multiplier, and therefore has a lower computational complexity.

Keywords: regularizing algorithm; a priori uncertainty; modified least squares method; direct transform receiver.

Introduction

From the estimation theory, it is well known that the joint estimation of the unknown signal parameters has a higher accuracy than the separate estimation. But the main disadvantage of joint estimation algorithms is their complexity [1,2].

In communication technology, there exist incorrectly posed problems that require a special approach. This situation arises, for example, when the operator (linear or nonlinear) describing the observed process does not have the opposite one, or is determined with errors leading to the divergence of the computational algorithm used. Such problems are solved by introducing a regularizing parameter into the estimation procedure [3-5].

1. Problem Statement

Consider the quadrature components zj = Sj(©j) of the signal M-QAM, M = 22k, k G N, observed against the background noise with an unknown probability distribution: yj = zj + , where i = 1, 2,...m is a discrete time, Sj (■) is a nonlinear vector-function describing the quadrature components of the signal:

information amplitudes or symbols of the test sequence taking discrete values, ^ = + a is a random phase formed by the phases of the generators on the transmit-receive side and

the delay in the propagation channel, is a phase constant component, a is a phase noise, A/ is a frequency shift after demodulation, At is a sampling interval, a is a signal amplitude, 7, A^ are amplitude and phase imbalance, bc, bs are constant components of the signal quadratures (DC offset).

The problem of finding estimate © of the vector © was solved under the following conditions:

- random process ^ is stationary, E(^) = 02x1, E(^) = a2I2x2 is an observation noise covariance matrix, E(^) = 02x2 for i = j, E( ■ ) is expectation operator, I2x2 is identity matrix of the size 2 x 2,

- phase noise a is described by a 2nd order moving average model,

- sequence of symbols /¿, J is known.

2. Solution

We form a 2m sample vector of the observed process Ym = Then the observation equation can be written as Ym = S(©) + S, where S(©)

T ym

T

ym_i

T \T

У1

sm(©) sm-i(©)

sT (©)

T

S

T Mm

T

Mm-1

MT )T. The algorithm to

find the estimation © is sought in the class of recurrent procedures. Expanding the nonlinear function Sl(-) in a Taylor series up to linear term at the point ©1-1, where l is the iteration number, we get: Ym = D1-1F(©1) + S, where F(©j) — f — ( 1 N

П) — f i —

(1 ©T

©l = Lf l, L = y 07x1 I7x7 J. The model of the estimated vector has the form: ©l = ©1-1 + where is a shaping noise with zero expectation and the covariance matrix E(£i£f) = a|I7x7, a| ^ 0, I7x7 is identity matrix. We calculate the value of the function F(-) from the left and right sides of this model and expand it in a Taylor series up to the first approximation at the point ©l-1. As a result, we obtain a model linearized with respect to the variable fl-1: fl = fl-1 + Wl-1£l, where Wl-1 = F'(©l-1) is the first derivative of F(-) at the point ©l-1. Further, the problem of finding the estimate is sought by the modified least squares method in the form of the Tikhonov functional:

*(f i)

lYm - Di_ifi||q-i + \i

fi- f

i_i

,1 — 1, 2,

Mn

P-

(2)

Here Al is a regularizing Lagrange multiplier, Euclidean norms are determined taking into account the weight matrices Q = a2I2mx2m and Pl: ||Ym — Dl-1fl||Q-i = (Ym —

Di_ifi; Q-1 (Ym - Di_ifi))

fi- f i_i

(fz- f 1-1; P- (fz- f 1-1)), (•; •) is a scalar

Ym-S(©;)

product. By minimizing (2) over f 1 under the constraint conditions —2(m-i) obtain expressions for the estimates:

a,,, we

©1—©i_i +LKi(Ym -S(©i_i)), I — 1, 2

Mn

where K

(a, I + A1P1 DT_iDi_i)_iAiPi DT_i,

Pi

ri_i + WWTaç2, ri

(3) (I -

KiDi-i)Pi(I - KlD1-1 )T + KlQKT + (I - KlDi-i)Pi-i;/MKT + KPT-/(I - KDi-i)T, Pi-i/ = E(f) = Ki-iQ + (I - K|-iD|-2)F>i-2;/M, I is identity matrix of the size 8 x 8, the initial conditions are P0,/M = 0gx2m, r0, ©0 are found from a priori information.

2

i

2

-i

P

2

Вестник ^ЭУрГУ. Серия «Математическое моделирование

и программирование» (Вестник ЮУрГУ ММП). 2022. Т. 15, № 2. С. 118-124

The Lagrange multiplier is calculated by the formula:

V2m

On

\l

Ym-S(©;_i)

- o,

|diag(Di-iPiDT-i)|

(4)

An approximate expression is obtained for the number of arithmetic operations for two variants of algorithm (3) taking into account the calculation of Lagrange multiplier (4): N(3)(4) = (1018m + 1588)M0. If the iterative update of the Lagrange multiplier value is not performed, i.e. A\ = 1, then the number of arithmetic operations can be estimated in the form N(3) = (984m + 1585)M0.

3. Computational Experiment

A computational experiment was carried out with the following data: M = 64, y =

0, 5, A/ = 180.7 Hz, At = 0, 25 ^s, <^0 = n/12, A^ = n/18, bc = 1, 3,bs = 2, a = 3,

m = 500, detection was carried out using an information sequence of 2000 symbols, the

phase noise is stationary, the standard deviation of the phase noise is one degree, the

additive noise is Gaussian, the number of realizations is 100. Initial values are taken as

^ /1 0

follows: ©0= (1, 0, 0,1, 0, 0, 0)T, To = n 1 0ix7

\ 07x1 ^^7x7

Figures 1, 2 show the dependence of the RMSE (root mean square error) of the estimation of some parameters on the number of iterations l for procedure (3) without the Lagrange multiplier (A^ = 1) and regularizing algorithm (3), (4) with ratio of signal to noise equal to 27 dB for in-phase component.

8 S

S 150

aï ÏÎ

8 If ^ 100

° « G

¡If 50

Ü ^ O

[

■1 N 200

a M

çr -a

Fig. 1. Dependence of the standard deviation of the estimation of the frequency and the total phase of the 64-QAM signal on the number of iterations for algorithm (3) without Lagrange multiplier

The above figures show that the algorithm with the Lagrange multiplier has a significantly shorter transient process. Therefore, it has less computational complexity. Regularizing algorithm (3), (4) converges already after 10 iterations. Algorithm (3) with

£1 23456789 10

/

Fig. 2. Dependence of the standard deviation of the estimation of the frequency and the total phase of the 64-QAM signal on the number of iterations for algorithm (3) with Lagrange multiplier

a) b)

Fig. 3. Experimental curves of noise immunity of 64-QAM signal reception during algorithm (3) operation: without Lagrange multiplier (Al = 1 ) and with a different number of iterations M0(1 - M0 = 230; 2 - M0 = 225; 3 - M0 = 220; 4 -M0 = 215) -a), with Lagrange multiplier (4), M0 = 10 - b)

Вестник !Ю"УрГ"У. Серия «Математическое моделирование

и программирование» (Вестник ЮУрГУ ММП). 2022. Т. 15, № 2. С. 118-124

Ai = 1, which is a recurrent least squares (RLS) method with weighted matrices, converges only after M0 = 230. Then, with the size of the estimated vector equal to 7 and the length of the test sequence equal to 500, the complexity of the RLS method exceeds the complexity of

regularizing algorithm (3), (4) by approximately — 22 times. Figure 3 illustrates the

experimental probabilities of error per symbol for receiving a 64-QAM signal when using algorithm (3) with Ai = 1 and a different number of iteration steps M0, and regularizing algorithm (3), (4). Figure 3 shows that the noise immunity, which regularizing algorithm (3), (4) allows for 10 iterations, is achieved using procedure (3) without the Lagrange multiplier only at 230 iterations.

A computational experiment showed that to achieve the same noise immunity, regularizing algorithm (3), (4) requires fewer iterations, and hence fewer arithmetic operations, than recurrent least squares method (3) (Ai = 1). With the length of the estimated vector equal to 7 and the length of the test sequence equal to 500 symbols, the complexity of the recurrent least squares method exceeds the complexity of regularizing algorithm (3), (4) by 22 times.

References

1. Hsu C.-J., Cheng R., Sheen W.-H. Sheen Joint Least Squares Estimation of Frequency, DC Offset, I-Q Imbalance, and Channel in MIMO Receivers. IEEE Transactions on Vehicular Technology, 2009, vol. 58, no. 5, pp. 2201-2213. DOI: 10.1109/TVT.2008.2005989

2. Weikert O. Joint Estimation of Carrier and Sampling Frequency Offset, Phase Noise, IQ Offset and MIMO Channel for LTE Advanced UL MIMO. IEEE 14th Workshop on Signal Processing Advances in Wireless Communications, 2013, Darmstadt, pp. 520-524. DOI: 10.1109/SPAWC.2013.6612104.

3. Tikhonov A.N., Leonov A.S., Yagola A.G. Nelineynye nekorrektnye zadachi [Nonlinear Incorrect Tasks]. Moskva, Nauka Fizmatlit, 1995. (in Russian)

4. Bakushinskiy A.B., Kokurin M.Yu. [Iterative Stochastic Approximation Methods for Solving Irregular Nonlinear Operator Equations]. Computational Mathematics and Mathematical Physics, 2015, vol. 55, no. 10, pp. 1637-1645. (in Russian)

5. Golubev G.K. [Concentrations of Risks of Convex Combinations of Linear Estimates]. Problemy peredachi informatsii, 2016, vol. 52, no. 4, pp.31-48. (in Russian)

6. Poborchaya N.E. [Methods for Estimating the Parameters of a Random Signal in Conditions of a Priori Uncertainty]. Elektrosvyaz', 2010, no. 3, pp. 24-26. (in Russian)

7. Poborchaya N.E. Stationary Channel Factors and Signal Disturbances in a Direct Converter Receiver in a System with MIMO Joint Estimation Algorithm. Systems of Signal Synchronization, Generating and Processing in Telecommunications, Svetlogorsk, 2020. DOI: 10.1109/SYNCHROINFO49631.2020.9166068.

Received August 19, 2021

УДК 51-77:001.895 DOI: 10.14529/mmp220210

АНАЛИЗ ВЛИЯНИЯ МНОЖИТЕЛЯ ЛАГРАНЖА НА РАБОТУ АЛГОРИТМА ОЦЕНИВАНИЯ ПАРАМЕТРОВ СИГНАЛА В УСЛОВИЯХ АПРИОРНОЙ НЕОПРЕДЕЛЕННОСТИ

Н.Е. Поборчая1, Е.М. Лобов1

1 Московский технический университет связи и информатики, г. Москва,

Российская Федерация

В работе рассматривается рекуррентный регуляризующий алгоритм совместной оценки искажений сигнала многопозиционной квадратурной модуляции (M-QAM), полученных в тракте приемника прямого преобразования. Алгоритм синтезирован с помощью модифицированного метода наименьших квадратов в виде функционала Тихонова в условиях априорной неопределенности относительно законов распределения шумов. Полученная процедура может работать как по тестовой последовательности, так и по информационным символам после процедуры детектирования. Проанализировано влияние множителя Лагранжа на точность процедуры оценивания и на сложность алгоритма. Показано, что при одинаковой точности регуляризующий алгоритм требует существенно меньшее количество итераций, чем процедура без множителя Лагранжа, а значит обладает более низкой вычислительной сложностью.

Ключевые слова: регуляризующий алгоритм; априорная неопределенность; модифицированный метод наименьших квадратов; приемник прямого преобразования.

Литература

1. Hsu, C.-J. Joint Least Squares Estimation of Frequency, DC Offset, I-Q Imbalance, and Channel in MIMO Receivers / C.-J. Hsu, R. Cheng, W.-H. Sheen // IEEE Transactions on Vehicular Technology. - 2009 - V. 58, № 5. - P. 2201-2213.

2. Weikert, O. Joint Estimation of Carrier and Sampling Frequency Offset, Phase Noise, IQ Offset and MIMO Channel for LTE Advanced UL MIMO / O. Weikert // 2013 IEEE 14th Workshop on Signal Processing Advances in Wireless Communications, Darmstadt, 2013. -P. 520-524.

3. Тихонов, А.Н. Нелинейные некорректные задачи / А.Н. Тихонов, А.С. Леонов, А.Г. Яго-ла. - М.: Наука Физматлит, 1995.

4. Бакушинский, А.Б. Итерационные методы стохастической аппроксимации для решения нерегулярных нелинейных операторных уравнений / А.Б. Бакушинский, М.Ю. Кокурин // Журнал вычислительной математики и математической физики. - 2015. - Т. 55, № 10. - С. 1637-1645.

5. Голубев, Г.К. Концентрации рисков выпуклых комбинаций линейных оценок / Г.К. Голубев // Проблемы передачи информации. - 2016. - Т. 52, № 4. - С. 31-48.

6. Поборчая, Н.Е. Методы совместной оценки дрейфа постоянных составляющих и амплитудно-фазового разбаланса КАМ сигнала на фоне аддитивного белого шума // Электросвязь. - 2013. - № 5. - С.24-26.

7. Poborchaya, N.E. Stationary Channel Factors and Signal Disturbances in a Direct Converter Receiver in a System with MIMO Joint Estimation Algorithm / N.E. Poborchaya // 2020 Systems of Signal Synchronization, Generating and Processing in Telecommunications, Svetlogorsk, 2020.

Вестник ЮЮУрГУ. Серия «Математическое моделирование

и программирование» (Вестник ЮУрГУ ММП). 2022. Т. 15, № 2. С. 118-124

Наталья Евгеньевна Поборчая, доктор технических наук, доцент, доцент кафедры < Общая теория связи>, Московский технический университет связи и информатики (г. Москва, Российская Федерация), n.poborchaya@mail.ru.

Евгений Михайлович Лобов, кандидат технических наук, доцент, заведующий лабораторией <НИЛ-4803>, Научно-исследовательская часть, Московский технический университет связи и информатики (г. Москва, Российская Федерация), e.m.lobov@mtuci.ru.

Поступила в редакцию 19 августа 2021 г.