To the matter of optimization of the amplitude-frequency responses of communication channels with PSK-n-signals based on the theory of resolution time Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

TO THE MATTER OF OPTIMIZATION OF THE AMPLITUDE-FREQUENCY RESPONSES OF COMMUNICATION CHANNELS WITH PSK-N-SIGNALS BASED ON THE THEORY OF RESOLUTION TIME

DOI 10.24411/2072-8735-2018-10308

Ilya M. Lerner,

Kazan National Research Technical University named after A.N. Tupolev-KAI, Kazan, Russia, aviap@mail.ru

Keywords: Capacity estimation, ISI, PSK-n-signals, bandpass filters, optimization problem.

Currently, the annual increase in the amount of transmitted information is an essential feature of modern society. This fact, subject to its mobility, leads to the need to search for new approaches that can significantly increase the capacity of radio engineering data transmission systems. Such approaches include the method that realize the developing of wireless communication systems based on the resolution time theory, which allows quite efficient operation in the communication channel with memory in the presence of intersymbol interference. In this paper, based on the mathematical modeling of such communication systems which use multiposition phase-shifted signals unity ultimate capacity estimations are obtained. Criteria for optimization the shape of the amplitude-frequency responses of the communication channel and the configuration of signal constellations are formulated. The results obtained in the work confirm the fundamental possibility of achieving unity capacity of at least 6 bit / s * Hz in the absence of frequency detuning and at least 5.83 bit / s * Hz in the it presence.

Information about author:

Ilya M. Lerner, associated professor, candidate of physico-mathematical sciences, Kazan National Research Technical University named after A.N. Tupolev-KAI, Department of Radioelectronic and Quantum Devices, Kazan, Russia

Для цитирования:

Лернер И.М. К вопросу оптимизации амплитудно-частотных характеристик каналов связи с ФИН-^сигналами, построенных на основе теории разрешающего времени // T-Comm: Телекоммуникации и транспорт. 2019. Том 13. №9. С. 36-49.

For citation:

Lerner I.M. (2019). To the matter of optimization of the amplitude-frequency responses of communication channels with PSK-n-signals based on the theory of resolution time. T-Comm, vol. 13, no.9, рр. 36-49. (in Russian)

Introduction

Volume of information transmitted by radio engineering data transmission systems {ReDTS) exponentially increasing from year to year is the immanent feature of modern society. This leads to considerable technical problem of creating new ReDTS since spectral efficiency requirements are also increasing year by year [1,2].

One of the most effective approaches to solving this problem is the conversion to transmitting, receiving and processing the signals in the presence of strong intersymbol interference (1SI) produced by frequency selective channel, including selective systems of ReDTS radio path. However, the implementation of this approach is associated with several difficulties leading to the considerable increase of complexity of the receiver itself along with an increase in the number of interfering symbols. Quite often this ultimately arises the issue about the practical feasibility of the entire system [1-8].

So computational complexity of the optimal algorithms of signal receiving in the presence of IS! - algorithms using the maximum likelihood criterion, increase exponentially with the increase in the number of interfering symbols due to the increase of transmission rate as well as with the increase of the volume of the alphabet of channel symbols [1,2].

Using the suboptimal algorithms of receiving which initially reduce IS! level {by using zero-forcing filter, or linear minimum squared estimation equalizer, or decision feedback equalizer) and then perform the optima) signal processing in the absence of communication channel memory [1-6] is also not without drawbacks. So according to papers [1,2,81 using such suboptimal algorithms results in: 1) significant loss in noise immunity as compared to optimal algorithms of receiving; 2) dependence of their noise immunity on the shape of amplitude frequency response (AFR) of communication channel, particularly when it is subjected to considerable variations on magnitude. The last feature is the most pronounced in radio channels and doesn't allow achieving high noise immunity of radio channels [1,2,5,8].

One of the most promising ways of improving the performance of ReDTS in a difficult signal-noise environment is to use a new method in which the symbol duration is selected taking into account the resolution time [1,2,4,5,7,8], while the IS1 is not compensated in receiver.

Since the capacity estimations for the considered class of ReDTS with PSK-n signals, was previously carried out for specific implementations of the amplitude-frequency responses (AFR), this did not allow formulating the criteria for their optimization with the goal of maximizing capacity, as well as formulating the requirements for the configuration of the signal constellation. Therefore, the aim of this work is to solve this particular problem,

I. Problem Statement

To solve this problem, we use the communication channel model, which was earlier developed by the author within resolution time theory [7]. It should be noted that the linear selective system (LSS) in the presented math model determines the IS! caused by the frequency-selective properties of the real channel, including those that are caused by the selective systems ReDTS. Since such ReDTS imply high data transfer rate, so we assume that channel parameters remain unchanged during transmission session.

Within of considered channel model, the capacity for the number of discrete states n in phase in the signal constellation is defined as [7J:

where

(1)

a resolution time, in the

general represents a set of symbol durations which are detined taking into account the restrictions introduced by the subsystems of the receiver. At these durations, the values of information parameter of the received signal distorted by the IS1 (a slow ly varying phase in the considered case) at moments of infonnation retrieval will differ from its true values in the worst case (the initial phase of the transmitted pulse of the PSK-n signal in considered case) by the amount of permissible error a, for any of the possible transmitted information sequences of unlimited dimension. Here X a]1d S are the number of the "transparency window" and

their total number, respectively; tw sl and /w cnd arc the symbol durations at which the "transparency Window" begins and ends, respectively; tb is a boundary time. In the absence of

"transparency windows", the resolution time is determined only by the boundary time.

Thus, the expression {1) can be represented as:

(2)

C. =C ;C =C. ;c:=c.

where C. is estimation of potential capacity without utilizing

' wo.w

"transparency windows"; Cy and Cy are the upper and lower boundary estimations for "transparency window", respectively.

In this work, when solving this problem, we confine ourselves to the most common in practice AFR of communication channels, which we implement due to LSS that satisfy the following criteria: I) (0„ /2AQ(I >15, where 2AQ() is the resultant bandwidth of LSS system and C0() is the average frequency of the LSS system; 2) its amplitude and phase frequency responses possess an even and odd symmetries, respectively, with respect to to0,

At the same time each symbol of the transmitted information sequence arriving at the input of the transmitter randomly takes equiprobably any of the n phase values of the signal constellation

(A-+0.5^sign{||0.5/7|¡-0.5/7--0.25}--jj0.5/;||)A(psl. (3)

Here k = \,n; || . || is rounding operation to the nearest integer; sign ( . } is signuin function; A(pst = 2iT / H is step between adjacent phase values of the signal constellation; c, — 1 for case, when n is even number, otherwise = 0, and this denotes the initial phase shift of the signal constellation (p . = / n .

The PSK.-/7 signal generated hy the transmitter at the earner

frequency M0 + Ato based on transmitted information sequence

arriving at its input with a symbol duration equal to the resolution time is determined by the following expression [7]:

(4)

(5)

(') = (')| = \z(t) / ic0 (y[ö)0 + Ä0)])|; Vcom (') = arg [Z(i)/k0 (y [<Ö0 + A<ö])],

(6)

where HC(m (/) and H'com (0 are the envelope and slowly varying phase of PSK-n signal, which must be measured after compensation for its static changes, introduced by LSS.

After that, the receiver at frequency cofi + Aco measures the

parameters of the PSK-« signal subjected to the compensation operation. At the same time, measurement errors are introduced into its slowly varying phase, as follows

K (0 = exp(y[K + Aft))/]){[! -l(r)]exp(y<p5C)+

r-\

xexp(7yr) + l(i-(/-l)rres)exp(yy;)) where / £ N is the number symbols in transmitted information

r

sequence; l(r) is Heaviside step function; yr - ^ + tpsc is

<ri

the initial phase ofr'h radio pulse ofPSK-n signal at the output of transmitter, where 0? is a phase jump caused by transmission of

</-th information symbol.

At the LSS output, the PSK-/J signal from the moment of lime t = 0 in accordance with 17] has the form

s™, (0 = ¿(Oexp(./[K + Aw)i]) =

= *«[,/K + AM)]exp(/[(w0 + Ara)r])x

x ([l - Ba {/, j Ato)] exp(y*({>3C) + /-i

r=l

where 2, (/) is complex envelope of the PSK-/i signal at the output

of LSS; B0(t,/A(a)

is settling function for LSS [9];

K (4®o + A0)]) = K (,/[wo + Aco])exp(y'(p0) and tp0 are

complex transmission coefficient of LSS and constant phase shift

introduced by it at a frequency ro0 + Acq , respectively.

Before measuring the slowly changing phase of the PSK-/? signal, the receiver compensates for static signal changes introduced by LSS

where ^ ,,,„(/) is the measured slowly varying phase; A'{!) is a

stationary random proecss corresponding to measurement errors of a slowly varying phase. Each of its sections is a random variable with a uniform distribution law, whose distribution density is determined as follows

fx¥)=

2e„

x e

0, xé[-sv;sv] ,

(8)

where S(|1 - the absolute value of the ultimate measurement error

of the slowly varying phase introduced by the receiver. Its fiducial value is defined as li — £ /A(p .

"ly l|J Tsl

The decision device recovers each symbol in received information sequence when the following condition is met jVon, (dt^ ) > A-/Ihn;s ( Mthres is radio signal threshold amplitude) according the following rule

(9)

where

At I.'.' 1 I

Hereinafter d-\,l is number of received symbol, 11. Problem Solution.

To achieve the aim of this paper, a software package was created; its basis is a numerical method for capacity estimation for considered channel model, published in [7J.

The solution to the optimization problem is to determine the

conditions under which the ultimate capacity Cu| is achieved, as

well as study the properties of this solution for considered channel model. Considered solution for given values of fiducial permissible phase error a0 = <Xpm / A<ps[ (au < 0.5 - j and

threshold amplitude of the radio signal Mis made in accordance with the expression:

Cul - max{Cwo w;Cw},

(it))

where C = max C, , C = max C are estimates of the

wo.w I, y

highest capacity in the absence of "transparency windows" utilizing and with their use, respectively; T] is parameter affecting the shape of the APR of LSS.

The properties of the resulting solution arc studied by analyzing the behavior of the following estimations for the given parameter values T] • , a(), Mth(CS (for brevity, the last two parameters will be omitted in the future):

1) maximum capacity Cmax without utilizing «transparency windows» and corresponding number of discrete slates nmax nec-essaty for its implementation; the highest estimations of the upper

Cn and lower CM| capacity boundaries utilizing the ^lnax ,h

"transparency window'7 and the number of discrete states «'max necessary for their implementation. These estimations are defined as follows

'Vs : Cmas (%v) = f(nm^ ) = max jc^ |T| = t^},

^.JU :Cnu [%,) = ) = max {cj n = r|siv}, (|1)

cn, (v) = Cx|w = *,W'X = XraE!-

2) the local minimum capacity estimation Cmin without utilizing "transparency windows" and corresponding number of discrete states >7min necessary for its implementation; the local smallest estimations of the upper Cs and lower Cs capacity boundaries utilizing the "transparency window" and the

number of discrete states n'min necessary for their implementation. These estimations are defined as follows

"min ■ Cm.n ( V ) = /{»ma ) ~ m;11 {ct.„. I n = 5 } •

»'«ta (v)=/(n'......(12)

Cs, (v) = Q|« = «'„,i„,x=z,„1Lv

with the following conditions ^ — K : —Unin and

« ™n ->■ 111111 are meet.

3) the mean values of the upper CV] and lower CV1 ^ boundaries of capacity utilizing yuma:,"transparency window" and the number of discrete states n 'M necessary for their implementation; the mean capacity CM without utilizing "transparency windows" and the number of discrete states nM necessary for its implementation . These estimations are defined as follows

single-stage filter on coupled circuits of the second type [9, p. 143]. As a parameter rj for the above LSS we will use the degree of coupling between the coupled circuits p, since its change affects on the shape of frequency response of these filters.

The settling function for the Is1 type filter has the form [9]:

4(*,p) = l-exp(-(l+-/p)jc)^cosfiv+-1 sinpx^, (14)

Here and below .v — ATi/ = AQr/ / a v - non-dimension

Aco Acq

time; p =-= a.-

AQ A AÎ1

— aAp0 is a generalized detuning

[9], p0 is the resultant generalized detuning; aA > 0 is the value of the generalized frequency detuning at which the level ripple of amplitude frequency response of the filter is

^^/¿(ajwi, (55)

where jfe is the largest value of the shortened transmission

coefficient (AFR) of filter [9]. The value parameter aA for this

filter is determined by solving equation (15) using expression (5.1.1) presented in [9]. For this filter, the critical degree of coupling between the circuits is pcr = 1, and the greatest value is P = 2.41 [9].

The settling function for the 2lui type filter with humps on one level has the form [9]:

exp(-(2 +

l±Jpf l + f}:

(16)

X ^ - jpcosP-V + [ P + ^ sin p.vj

S

CM, f Hgiv ) :

N=nm

r I r, ) -^HglV ) —

tl — H ■ + 1 'max "mm 1

Z ( ^ ill]

n — H ■ + I

max nun

(13)

argmin (A')|,

"m= argmin |cMii

As LSS we will use the following types of bandpass filters with the same resulting bandwidths: 1st type - a single-stage bandpass filter on coupled circuits of the first type [9, p. 111J; 2" type — three-circuit bandpass filter [9, p. 162J; 3rd type - a

For this type of filter the value parameter a^ is determined

by solving equation (15) using expression (7.1.2) presented in [9], For this filter, the critical degree of coupling between the circuits

is p„ , and the greatest value is p = 6 [9].

The settling function for the 3rd type filter has the form [9]:

B0 (p) = 1 - exp(■-{1 + /p) a-) cos px - ~~ sin p.v j. (17)

For this type of filter the value parameter a v is determined

by solving equation (15) using expression (6.1.1) presented in [9]. For this filter, the critical degree of coupling between the circuits ¡s ptr - 0.486 , and the greatest value is ¡3 — 1.15 [9].

The following parameters were used for calculations utilizing numerical method for capacity estimation [9'|:

• common for all types of filters: accuracy of calculating the symbol error rate (SER) caused by ISI [9] is ^ = 10"; dimensionless time step with which SF.R is calculated is AQ/CS =10 5 xa,; the values of the fiducial permissible phase

Fig, 2, Dependencies of unity capacity estimations without utilizing "transparency windows" (see sub-figure a. h. c) and the number of discrete slates (sub-figure d. e. f) required tor their implementation on fiducial degree of coupling between circuits. Resultant generalized detuning pfl= 0,05. Type of LSS: type filter (curves with a

diamond marker); 21|J lype filler (curves without murker); 311' type fitter (curves with asterisk marker). Unity capacity estimation type: C nl A (solid line); ( (dash dotted

tine); CM (dashed tine). Number of'discrete states estimation type: jj (solidline); (j (dash dotted line); /jm (dashed line). Parameters: A-/||lr = 0, (X,, = 0,499

(see sub-figure a. d)\ Mlh = 0.9, ct0 = 0.499 (see sub-figure h, e); = 0.9, a0 = 0.25 (see sub-figure c,f)

Fig. 3. Dependencies of unity capacity estimations without utilizing "transparency windows" (see sub-figure a, h, t) and the number of discrete states (s Lib-figure d, e. f) required for their implementation on fiducial degree of coupling between circuits. Resultant generalized detuning po = 0.1. Type of LSS: Is1 type filter (curves with a

diamond marker); 2'"1 type filter (curves without marker); 3"1 type filter (curves with asterisk marker). Unity capacity estimation type: C (solid line); Cmm (dash dotted

line); CM (dashed line). Number of discrete states estimation type: n (solid line); n in (dash dotted line); tiM (dashed line). Parameters: A/lht =0, Ol0= 0,499

(see sub-figure a d); A</lhre. = 0.9, u.(, = 0.499 (see sub-figure h. e), A-/lhrm - 0,9, atl - 0,25 (see sub-figure e.j)

1.2 1.4 1.6 I H 2 P. 2.4 1.2 1.4 l.fi 1 .H I IV 3 4

O f>

Fig. 14. Dependencies of unity capacity estimations utilizing % "transparency windows" (see sub-figure a. h, e) and the number of discrete states (sub-figure J, e,J) required for their implementation on fiducial degree of coupling between circuits Tor the 3rd type filter. Unity capacity estimation type: CH (solid line with diamond marker); ¿.. (dash dolled line with diamond marker)', C„ (solid line willi asterisk marker); r, (dash dolled line with asterisk marker); C, (solid line with

III My Mj Sy

circular marker); Cs (dash doited line with circular marker). Number of discrete states estimation type: jj^ (solid tine with diamond marker); n^ (solid line with circular marker); tl.. (solid line with asterisk marker). Parameters: M. =0, ttn = 0.499 (see sub-figure u. d); M, =0.9, a,, = 0.499 (see sub-figure 6, e);

M Uirc* trtirs * u

A/lh = 0.9, au = 0.25 (see sub-figure c.J). Resultant generalized detuning p() = 0,1 (black tine); p(| = 0.15 (red line)

l'l H | i.2 1.4 1,6 I.H 2 ' 2.4 '(Vh : 1.2 1.4 1.8 2 I' 2.4

c) (1

Fig. 15. Dependencies of unity capacity estimations utilizing "transparency windows" (see sub-figure a. h, c) and ihc number of discrete slates (sub- figure d, e.f)

required for their implementation on fiducial degree of coupling between circuits for the 3rd type filter. Unity capacity estimation type: C,, (solid line with diamond

Hu

marker); ¿H (dash dotted line with diamond marker); ¿M (solid line with asterisk marker); ¿^ (dash dolled line with asterisk marker); Cs (solid line with circular marker); Cs (dash doited line with circular marker). Number of discrete states estimation type: jj^ (solid tine with diamond marker); fjrain (solid line with circular marker), fj^ (Solid line with asterisk marker). Parameters: = 0, 0l0 = 0.499 (see sub-figure a, d); Mihr. =0.9, a,, = 0.499 (see sub- figure h. e);

Mthr. = 0.9, a„ = 0,25 (see sub-figure C. J). Rcsultanl generalized detuning p() = 0.2 (black line); p(| = 0.25 (redline)

Analyzing the dependencies shown in Figures 1-15, we can Come to the following conclusions:

1) In general, the ultimate capacity values for ReDTS with PSK-n signals, developed on the theory of resolution time, are achieved by utilizing "transparency windows". Exclusions from this rule are the following: I > in case when AFR channel is the same as 3rd type filter; 2) at low: values of frequency detuning (¡p„|e[0.05;0.l]) with utilizing the following parameters

A-/Ihi . >0 and a(l = 0,499 - The greatest value of unity ultimate

capacity estimation among all types ofLSS is cul »9 hit/fllz^s)

which is achieved at n = 4 without utilizing of "transparency windows" and frequency detuning, in the absence of measurement errors and any restrictions on the amplitude of the received PSK-n signal. The channel AFR in this case is coincide with the similar characteristic of the 3rtl type filter with parameters correspond to a single oscillatory circuit see [7,111,

2) In the case of significant restrictions (M^ =0.9, a,, =0,25)

on the amplitude of received PSK-n signal and high level measurement errors of slowly varying phase, introduced by receiver, the unity ultimate capacity value can reach values of ClM =6.5 bit/(Hz-<s) at n — 128 in the absence of frequency detuning. The presence of frequency detuning leads to decreasing in the capacity in following way: 1) unity ultimate capacity is reduced in worth case to value Cu] = 5,88 bit/(Hz>s) <« = 127 and

2'ul or 3rd filter type) at low frequency detuning values [ptl| e[0.05;0.l] and 2) at large frequency detuning values

|pJe[0,15;0.25] in the worst case - to value C, =5.839 bit/(Hzxs) (»

= 128 and 3rd filter type); thus, ultimate capacity reduction occurs only by 10% in the worst case at |p(i = 0.253) The expediency of utilizing "transparency windows" is caused not only by the realization of the Cw > C„ ratio in the

general, but also by the fact that it allows to provide the higher values of frequency selectivity and reduce the requirements for the parameters of the symbol subsystem synchronization subsystem. Herewith in this case, an increase in the required number of discrete slates for implementation of capacity estimations can be observed compared with the case when the "transparency windows" are not utilized. In addition, when implementing this operation mode, it is necessary to take into account the following features when choosing the AFR: 1) in general values of frequency selectivity, at which c„ is realized, decrease with raising |p(jj;

2) while increasing IpJthe reducing in the range of frequency

selectivity values at which capacity estimations are equal or close to the ultimate capacity value is observed and changes in dependences of capacity values around last one due varying frequency selectivity rapid are significant. The most pronounced this fact is shown for large values of frequency detuning(|p0|>0.15); 3) in

the absence of frequency detuning realization of capacity estimation c„ 'n die case (m >0; A/lhrK >0) occurs at large values of

frequency selectivity compared with the ease (m -q; jW[lms >0)

in the absence of band ripple and at low values in their presence compared to the case ij^-0; M±res >0)■ 'n the presence of frequency detuning realization of capacity estimation cw in the ease

(u„ >0: jV/lhrcs >0) occurs at low values of frequency selectivity

compared with the case =0; ,Wlhrcs > 0) ■

4} Utilizing the "transparency windows" also makes it possible to reduce the requirement to symbol synchronization for ReDTS based on the resolution time theory with capacity estimations close to the its ultimate value. In the absence of frequency detuning and in the case of ^ = 0; M > o it is necessary to use

the AFR with ripple level in the bandwidth in the case of two-humped AFR it should not exceed 3.38%, and for the three-humped one it should not exceed 0.1% With the value of the parameters M^ = 0.9, a(l = 0.25, ripple level in the bandwidth for

a two-hump AFR should not exceed 0.5%, and for a three-hump -0.01%. In case when we want to provide a high frequency selectivity while reducing the requirements for symbol synchronization subsystems even with parameters MthK = 0.9, a„ - 0.25 we have

to utilize AFR with ripple level in the range from 19.04% to 21.25% (Is1 type filter) and 15.71% to 27% (3rd type filter) in the case of two humps in the bandwidth and for a three-hump frequency response in the range from 2% to 5,1% (2MiI type filter).

5) Utilizing operating mode without using the "transparency windows" and frequency detuning is generally advisable to reduce the required number of discrete states necessary lo achieve CM ■ So the use of mean capacity estimations instead of the highest one allows reducing the required number of discrete states ("max /"m ) 'n signal constellation in general by at least two times.

When utilizing "transparency windows" this gain is essentially smaller, so in the absence of measurement errors the gain is / n , > I 14 and in their presence »' / n > i 26. At the same

max M 1 ma.\ M

time, when using "transparency windows", the loss in capacity (c„ -C,, Vc„ x 100% in the presence of measurement errors in

the worst case reaches 9.4%, and in their absence it does not exceed 2.8%. The capacity loss (cmai - CM)/ C„,M x 100% i" the absence of

"transparency windows" in the worst case will be 11,9% at = a0 = 0.499; 15% at M^ = 0.9, a,, = 0.499 and 14.7%

at Mtm =0.9, a0 = 0.25 ■

Acknowledge

The reported study was funded by RFBR according to the research project № 18-37-00440

References

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К ВОПРОСУ ОПТИМИЗАЦИИ АМПЛИТУДНО-ЧАСТОТНЫХ ХАРАКТЕРИСТИК КАНАЛОВ СВЯЗИ С ФМН-^СИГНАЛАМИ, ПОСТРОЕННЫХ НА ОСНОВЕ ТЕОРИИ РАЗРЕШАЮЩЕГО ВРЕМЕНИ

Лернер Илья Михайлович, Казанский национальный исследовательский технический университет им. А.Н. Туполева - КАИ,

Казань, Россия, aviap@mail.ru

Исследование выполнено при финансовой поддержке РФФИ в рамках научного проекта № 18-37-00440 Аннотация

В настоящее время ежегодное повышение объема передаваемой информации является неотъемлемой чертой современного общества. Это факт при условии его мобильности приводит к необходимости поиска новых подходов, позволяющих существенным образом повысить пропускную способность радиотехнических систем передачи информации. К таким подходам относиться построение беспроводных систем связи на основе теории разрешающего времени, позволяющих достаточно эффективно работать при сильных межсимвольных искажениях в канале связи с памятью. В данной работе на основе математического моделирования таких систем связи получены оценки предельной удельной пропускной способности, в которых применяются многопозиционные фазоманипулированные сигналы. Сформулированы критерии для оптимизации формы амплитудно-частотной характеристики канала связи и конфигурации сигнальных созвездий. Полученные в работе результаты подтверждают принципиальную возможность достижения удельной пропускной способности не менее 6 бит/с*Гц при отсутсвии расстройки по частоте и не менее 5,8 бит/с*Гц при её наличии, в том числе и при высокой частотной избирательности канала связи.

Ключевые слова: пропускная способность, фильтры, ФМн-n-сигналы, критерии оптимизации.

Литература

1. Лернер И.М. Метод оценки пропускной способности реальных каналов связи с многопозиционными фазоманипулированными сигналами при наличии межсимвольных искажений и его применение // Т-Сотт: Телекоммуникации и транспорт. 2017. Т. 11. № 8. С. 52-58.

2. Лернер И.М. Аналитическая оценка пропускной способности канала связи с частотной характеристикой резонансного фильтра при наличии межсимвольных искажений и использовании многопозиционного фазоманипулированного сигнала // Т-Сотт: Телекоммуникации и транспорт. 2017. Т. 11. № 9. С. 65-73.

3. Лернер И.М., Камаллетдинов Н.Н. К вопросу увеличения скорости передачи в фазовых радиотехнических системах передачи информации, работающих при сильных межсимвольных искажениях в линейном радиотракте // Наукоемкие технологии в космических исследованиях Земли. 2017. Т. 9. № 5. С. 92-104.

4. Лернер И.М., Ильин Г.И. Об одной возможности увеличения скорости передачи при наличии дестабилизирующих факторов в системах связи, использующих взаимную интерференцию символов // Физика волновых процессов и радиотехнические системы, 2017. №4. С.24-34.

5. Лернер И.М., Чернявский С.М. Оценка пропускной способности реальных каналов связи с АФМН-М-сигналами при наличии МСИ // Т-Сотт: Телекоммуникации и транспорт. 2018. Том 12. №4. С. 48-55.

6. Лернер И.М., Файзуллин Р.Р., Чернявский С.М. К вопросу повышения спектральной эффективности фазовых радиотехнических систем передачи информации, функционирующих при сильных межсимвольных искажениях // Известия высших учебных заведений. Авиационная техника. 2018. №1. С.113-118.

7. Лернер И.М., Ильин Г.И. Численный метод оценки потенциальной пропускной способности при использовании ФМн-п-сигнала в канале связи с межсимвольными искажениями // Вестник КГТУ им. А.Н. Туполева. 2018. № 4. С. 138-149.

8. Лернер И.М., Ильин Г.И., Ильин А.Г. К вопросу о циклостационарности АФМн-М-сигналов, наблюдаемых на выходе канала связи с межсимвольными искажениям // Вестник КГТУ им. А.Н. Туполева. 2018. № 3. С. 107-117.

9. Евтянов С.И. Переходные процессы в приемно-усилительных схемах, М.: Связьиздат, 1948, 221 с.

Информация об авторе:

Лернер Илья Михайлович, к.ф.-м.н., доцент кафедры РЭКУ, Казанский национальный исследовательский технический университет им. А.Н. Туполева - КАИ, Казань, Россия