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TO A QUESTION OF CAPACITY ESTIMATION OF REAL COMMUNICATIONAL CHANNELS WITH PSK-N-SIGNALS IN THE PRESENCE OF ISI AND INSTABILITIES IN THE RECEIVER
DOI 10.24411/2072-8735-2018-10132
The reported study was funded by RFBR according to the research project № 18-37-00440
Ilya M. Lerner,
Kazan National Research Technical University
named after A.N. Tupolev-KAI, Kazan, Russia, Keywords: capacity estimation, ISI, PSK-n-signals, bandpass filters,
aviap@mail.ru increasing of spectral efficiency.
Volume of transmitted information exponentially increasing from year to year is a current trend of modern society that leads to the need for increasing the transmission speed of data transmission systems. This trend is the most pronounced among radio engineering data transmission systems, which currently operate under conditions of limited frequency resources and constantly increasing requirements to an effective frequency resource usage. One of the most effective approaches to solving this problem is the conversion to data transmission in the presence of intersymbol interference at radio engineering data transmission systems produced by their selective systems. Despite the attractiveness of this approach, its technical implementation is associated with the difficulties, that can lead to an increase of complexity of the receiver itself along with an increase in the number of interfering symbols. This ultimately arises the issue not only about the expediency of its implementation, but also about its practical feasibility. The alternative approach which allows to create the radio data transmission systems that function in the conditions of strong intersymbol interference caused by linear selective systems of radio path, in the absence of their compensation, is the appropriate choice of the channel symbol duration time, with regard to the resolution time of linear selective systems. In this paper, capacity estimation such channel is made when using a PSK-n signal, when the decision device is a comparator, and there is no need to take into account channel memory when making a decision. To solve this problem, a new numerical method for estimating capacity has been developed, it can be used to estimate the potential capacity in the absence of noise, but under conditions of various instabilities such as frequency detuning, measurement error, and signal level requirement.
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, KNRTU-KAI, REQD Department, Kazan, Russia
Для цитирования:
Лернер И.М. К вопросу оценки пропускной способности в канале с ФИН^-сигналами и с памятью, вызванной межсимвольными искажениями, при наличии нестабильностей в приемном устройстве // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №8. С. 52-62.
For citation:
Lerner I.M. (2018). To a question of capacity estimation of real communicational channels with PSK"n"signals in the presence of ISI and instabilities in the receiver. T-Comm, vol. 12, no.8, рр. 52-62.
Introduction
The annual exponential growth of transmitted information Volume by existing radio communication channels is a distinctive feature, which characterizes the modern development of digital communication systems. As a result, a significant complication of signal to noise environment of communication channel takes place in modem radio-technical data transmission systems (RTDS).
Thus, the solution of one of the most important problem of radio engineering, which implies a development of new approaches to the maximum possible reduction of die frequency band used for information signal transmission, becomes particularly significant 11-4J.
At the moment the efficiency of information transfer in real communication systems is increased by using of multiposition signals, noise immunity coding and reduction of protective frequency intervals [1,2]. At the same time, the transmission is always carried out by independent (not interfering) symbols. The requirement of absence of symbol interference restricts a choice of channel symbol duration [ 11. However, such approach seems to be inefficient for existing signal-noise situation 111.
The inost effective approach to overcome this situation is to use signals with mutual symbol interference or in other words signals with intersymbol interference (1SI). However, the use of optimal signal processing algorithms (using the maximum likelihood criterion) or approaches close to them becomes impractical even with a small number of interfering symbols [1 ].
At the same time, the use of suboptimal algorithms, which initially transform the channel with memory into a channel without memory due to widely used approaches (zero forcing filter, equalizer, realizing a minimum mean square error, or equalizer with feedback), and then ensure the use of optimal signal processing methods, also have a number of significant shortcomings [1-3]. These include: 1) a large loss in noise immunity compared to optimal signal processing methods; 2) the dependence of noise immunity on the form of the amplitude frequency characteristic (AFR) of the communication channel, which is especially important for radio channels.
The alternative approach, which allows to create RDTS that function the strong 1SI conditions, caused by linear selective systems (LSS) of a radio path, in the absence of their compensation, is the appropriate choice of the channel symbol duration time, with regard to the resolution time of LSS [4-101. In this ease, each symbol can be restored without using channel memory. Thus, this method has a low computational complexity and is devoid of those drawbacks of the intrinsic ones that were given above. The most expedient is to create such systems where phase shift keying signals with n discrete states (PSK-w- signal) are used 11 [.
The methods that allow us to analytically estimate efficiency of such systems in the absence of noise and unfavorable factors (frequency detuning, errors introduced by the receiver and etc.) are presented in papers [4,5,8| for PSK-w- and amplitude phase shift keying signals with L discrete states.
In this paper, we will examine the effect ofa number of unfavorable factors (frequency detuning, errors introduced by the receiver) and signal strength requirement on the effectiveness of such systems, by creating and then applying a numerical method for capacity estimation.
I. Problem Statement
In this case, capacity estimation is considered for a communication channel, which consists ofa transmitter, a linear time invariant (LT1) system, a receiver and a decision device (DD). We also assume that there is no noise in communication channel.
In this work, we assume that LTl system is a set ofa bandpass filters. The first type (Case A) of LTl system realization possesses the following properties [11 [: 1) eo0/2Ai\>15, where 2Aii„ is
the resultant bandwidth of LTl system and £&, is the average
frequency of the LTl system; 2) its AFR and phase frequency response possess an even and odd symmetries, respectively, with respect to CO0. The second type (Case B) of LTl system realization possesses the following properties 112J: considered bandpass filters have polynomial approximation AFR with ©¡/a; >1.21,
where co2, <9 are the upper and lower frequency of filter bandwidth, respectively.
Receiver provides the following function:
1) It detects a slowly varying phase of PSK-»-signal on the output of LTl system at © + Am frequency, where Aa is a frequency detuning, with a random measurement error CT(/j.
2) It detects an envelope of PSK-n-signal on the output of LTl system that needed for appropriate squelch subsystem functioning of DD.
3) It compensates for the unevenness of the transmission coefficient of LTl system on frequency© +Ae> an(J constant
phase shift (pn at the frequency a)()+Aco introduced by LTl system.
The measurement error „(/) distributed according to a uniform law, the density of which is given by the expression
where b = -a = sr, where er is the maximum measurement error
taken in absolute value introduced by receiver, which is related to its fiducial maximum measurement error by the relation
Sa = srf Aft, ■
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
V* = {* + O.S>7 sign (||0.5«||- 0.5« - 0.25)- ||0.5n||) Ao„ . (2) Here ;] = l for ease, when n is even number, otherwise // = 0. and this denotes the initial phase shift of the signal constellation (p - ¡rrj in; k=\,n\ II, II is rounding operation to the nearest
integer; sign(.) is signum function; Aft, =2n!n is step between adjacent phase values of the signal constellation.
The PSK-h signal on the transmitter output from the moment of transmission start (from the moment t = 0 ) can be represented as follows
4 (')+A®)'))* +1 ('"('" lk,>*p(/M)
where / e N* is the number of transmitted radio pulses forming the PSK.-« signal; tr is resolution time |4, 5, 8]; ](V] is ileavi-
side step function; ma + Am is the carrier frequency of the PSK-» signal; y ~ V© +(p is the initial phase of r01 radio pulse of
PSK-w signal at the output of transmitter, where @ is a phase
jump caused by transmission of 4-th radio pulse. We assume that before the start of transmission, until the moment / = 0, LT1 system is in a steady state mode and harmonic oscillation exp(/((ffl0+Aiu)i + (plr)) acts on its input.
In this case PSK-« signal at the output of a LTI system starting from the time / — 0 according to the Superposition principle and results of [8] can be represented in the form
+E[A 1K"jM>) ' iA®)]exp(/n)+
r=l
+Bn (/-{/-1) t„, jAco) exp{jy,)) where Z(t ) - complex envelope of the transient process caused by the passage of the PSK-« signal through LTI system; *o (/(««+ A®)) = K{j(«>o + A®))exp(y<p«) - complex transmission coefficient of LTI system at frequency a>0 +Am; Bn(l,jAoj)
-settling function for LTI system [11].
For case B settling function B0 (/, ¡Aw) is replaced by function
//„(/) = //(i)/k(j(co„ + Ad})), which has the same meaning and
properties, differs only in a designation |12|. In this paper under (/, jA/a\, we mean both designations.
Signal for a slowly varying phase on the output of the receiver, based on equation (4) and {1), is defined as
where \|/Hii ) = arg Z (t) is slowly varying phase of PSK-«
signal on the output of LTI system
Decision device (DD) is a multi threshold comparator with a squelch subsystem and it does not introduce additional I SI and measurement errors, which are caused by receiver and LTI system.
The recovery of the primary signal (symbols of transmitted information sequence) is made by DD based on the decision rule for each symbol:
where k^\7i:f{k\d) = nm\¥lt{dtKs)-¥k\<aim+et\ d=M is number of considered symbol; a is the permissible phase
settling error at the output of LT! system, which is related to its fiducial permissible settling error by the relation - a^t A<pa!
and only in the case when envelope of PSK-H-signal on the output of LTI system is equal or exceeds the predetermined normalized threshold signal level , analytically it can be presented
where M....., = 1-A ,
A is the difference between amplitude of
PSK-n-signal on the input of LTI system and normalized threshold signal level. Condition required to ensure the uniqueness of the DD solution is the following # <o.5A<psl ■
Per considered case, the resolution time must correspond
to the time, when one of the following inequalities becomes an equality and the other one is true.
«,,„, + £ !«,„, {4№ )| = (dt„s) ■- yd 11 (g)
¿«./^ A™ Clu}=)| J
where ,)> A., {dt„,) ¡s measured settling phase and am-
plitude error at the output of LTI system for the ith radio pulse when it ends, respectively; y/^ [chr ) is measured values of slowly varying phase of the dh radio pulse at the output of LTI system obtained by a receiver at moment dtns •
Analyzing results of 14, 5, 8, 131 we can conclude that capacity determination for considered channel may be formulated as the maximum information transmission rate on the communication channel with equivocation produced by 1S1 in the absence of errors during recovery by DD, where maximum is taken over all possible sources of information used as input to the channel. In this ease, the distribution of primary symbols in information sequence should be equiprobable, since other distribution will not change the maximum information transmission rate [8].
In this case, equivocation produced by IS] is easy to consider using resolution time in the following equation form for capacity channel estimation
= —log2n»
res
where the second multiplier in equation is the maximum entropy of source, u =\lt is maximum transfer rate.
II-. res
In the case of presence of "transparency windows", we should talk about a number of estimates of channel capacity |8J. Obviously, upper Cm and lower bounds C,^ of channel capacity in this ease, according to (9) can be found as
,(f| botmd
(10)
of channel capacity
UPPer Cup.», and loWer boLlnds
when /:th "transparency window" is used have the forms
C„P.w, -QwL ,
Here 1
'end.
'si, 'etkI,
_ (Ii)
is the start an end time of f = 1, K (K is the
amount of "transparency windows") "transparency windows", respectively; the boundary time f is the greatest time, from
which any primary symbol from any transmitted sequence can be reliably reconstructed by DD from received channel symbol at the output of the LTI system at given playback quality.
Thus, it is necessary to determine the resolution time / of
LTI system to estimate the capacity of the considered channel,
which requires the analysis of the occurring transient process caused by the PSK-h signals transmission.
II. Problem Solution. Theoretical Aspects
From expressions (5) - (8) it follows that in order to gel estimation of / ^ first of all it is required to determine resolution time
for slowly varying phase tn,v and for envelope tns , considering
the possible presence of "transparency windows" for given values of permissible phase settling error a and normalized threshold
signal level Jl/ ,, respectively. In this case, the resolution time / will he determined as follows: 1) in the absence of "transparency" windows im = max{i„,S(i;/fW }. where , are resolution time for envelope and slow varying phase, respectively; 2) in the presence of "transparency windows" t is determined
by the following expression
i/K„> 0,
if Km =0, K„> 0
n) ÇC'».«.. », ];Ц[ч 9Ut,}['*H fo H f; H vUj K>Q П Ü^v^X^HtaH
if к, > о, кг= о
(12)
where / , t are the start and the end time of к -th "transparency window" for envelope, respectively; / ^ ,/ ; are the
start and the end time of k,-th "transparency window" for slowly varying phase, respectively; t ,t. are the boundary times for
i «V
envelope and slowly varying phase, respectively; Kw, К are
amount of "transparency windows" for envelope and slowly varying phase, respectively.
To estimate resolution time t, tlic following is required to
determine conditions for each d-th {d = j7/j information symbol, which for given values of « and д ensure that: 1) the d-th
° fwt '-'лиI '
symbol greatest settling times for envelope and slowly varying phase on the LTI output in the absence of "transparency windows" and 2) in the presence of "transparency" windows the greatest number of windows with the smallest time duration (the greatest settling time for each window determines its start and the smallest settling time determines its end) and the greatest boundary time for the d-lh symbol is provided. This involves calculation of "effective memory" G, which is the number of symbols preceding the received one, which is required to evaluate resolution time with a given absolute tolerance с. Next, in this paper, problem of determining these conditions will be called the settling time problem.
In contrast to the methods that are presented in papers [4,5,8J in this study we will solve this problem using probabilistic approach that is based on cyclostationary behavior of PSK-«-signal slowly varying phase and its envelope on the output of LTI system. This approach is based on validity of transposition principle for LT! system.
In this case, "effective memory" G is the number of the first symbols of the information sequence, transmitted at a given
transmission rate u , which are needed for transformation of
mux
the PSK-« signal on the output of LTI system from nonstationary to cyclostationary process and resolution time is a period of cyclostationary process.
A. Algorithm of Capacity Estimation In this case, the algorithm that solves this problem has the following structure:
On the first step, the greatest settling time / ( of the first
information sequence symbol of PSK-// signal on the output of LTI system (d= 1) for a given value of & is estimated.
— On the second step, the "effective memory" (j for a given value of absolute tolerance c and / ( is evaluated.
— On the third step, for a given value of X the least number of realizations N n is evaluated when the greatest difference, taken module, between two cumulative distribution functions of measured settling phase error a (lG +lW 1 for different
numbers of realizations of random information sequence is less or equal X.
— On the fourth step, using results of previous steps dependences of Symbol error rate (SER) caused by IS! produced by LTI system on symbol duration time r > t v. are estimated for
two types of symbols in transmitted information sequence: a) once for any of the symbols \>>G +1 and b) for each symbols
set,
v < G - Then based on obtained results the estimations of the reso-
v — Kl|
lution time / and "effective memory" G are made taking into
account function, analog of SER, for envelope to ensure the second inequality of system (8). Then the capacity estimation is made. Let us consider each step of the algorithm more in detail.
Estimation of the greatest settling time of the first information sequence symbol.
The greatest settling time / is obtained by solution the following set of equations for a =0.499
tan
K4
(13)
-sin(ÎO^}/2) + 2Reg„(./A^,{f)!})sin({Q^)/2)
ws({©M}/2)-2M,(yA«>.k})sin( RJ'2)
produced by a set of equivalent phase jumps [5], caused by transitions between the values of phases in signal constellation (2),
K,}= _
i{g-0.5w-l)Aft. g = l,Ji + l if n = 2z; z e N; "I (g-|0.5n||) Aft,, g = En if « = 2z + l; zeN; with the respect of corresponding set of settling times |/ j. Here
\a J = a sign( {© J'J ■ Equation (13) follows from the results of papers [15].
In the presence of "transparency windows" \ =/ , where
if y.OI
/ is the start of first "transparency window" for slowly vary-
ing phase, caused by the transmission of first information symbol with value © , in the absence - / -i , where / is a settling time for slowly varying phase, caused by the transmission of first information symbol with value © ^.
Thus, the greatest settling time / of the first information
sequence symbol is t = maxIt i- The absence of consideration
"i s I «J
of the influence of the envelope behavior in / is due to the fact
that envelope is a non-informative parameter in the considered case and / ; is a start time for finding resolution time.
B. Effective Memory Estimation for LTI System To estimate the "effective memory" G for LTI system first of all we need to obtain an expression that allows us to estimate the considered measured settling errors at the end of the c/-th radio pulse at the LTI system output how it was done in paper [8]. for
this we assume in the second equality of (4): t = dtns, / = d = and tlien we
divide both sides of the equality by kt (/j(j0(( + Ao))exp(y((iao+Aia)fepo +/,,))■ As a result, we get
the following equation for ¿{dt, )
(15)
- r j nr A
T=0
[l - (dt„)]exp(./«Mi «„ exp (-./>,) >
r=0
where Zr= V 0(j; \ = ¿„((>/+1)tm,j&ioj-Bn{{d-r)^,jte>);
tj=r+\
After a series of simple transformations (15), the expression for the (dtrei )| has the form
X B, cos (^) - cos am (dt^) j
7 " \ Ml
1 —
£^+0 + *,) + (16)
Lr=o
+2|'VSr X Ö cos (^ - ) - X 5 cos (Zr + ar„„ (dr„))
V r=0 u=f +\ r=0
and the relation for the a (dt 1 can be found as
ills 1
(17)
Let us test the obtained expression (16) for the extremum for c/-ih symbol with the respect to a (dtKX) ■ Determination of conditions in which the expression reaches the extremum will be made only at stationary points. This is due to the following reasons; I) settling function is a smooth function, what we can conclude, analyzing expression (2.4.7) in paper [11, p.37]; 2) singular critical point is excluded from consideration, when ABU (dtrex) = I
and aun(dirii) is unspecified, because in this case Z(dt, )-0
and therefore DD will not be able to make the correct decision according to (6).
In this case, the solution of the problem formulated above is the solution of equation t^A^ (dit )|/ dam (dtns) = 0 ■ Partial derivative necessary in this equation can be represented in the following form using (16)
da,jdlj |AM1«J| (18)
,i _ J
COSam (dt„) ]T B, sin xr + sin amy {dtn,s) £ K eosxr
=__r=j>_
(Aflir (^«s )|
and the solution of the equation (^ )|/d«„, (dt ) = n takes the form
¡an=
ttrsnz,
(19)
XA C03^r /},, +£5Pcosjr,
r=0 r=(l
Comparing expressions (17) and (19) we conclude that they are identical and the greatest values of measured settling phase errors taken by modulo correspond to the greatest amplitude settling errors taken by modulo.
Therefore, the following algorithms can be used to estimate the size of "effective memory" G ,, for / ( . For a given value of
absolute tolerance error £, we use the following rule to estimate
0\., [8]
where p _ ß I , W > G.
(20)
The last inequality follows from majorized series theory application to equation to (15) in the following form
ikm-u) l£iKiand ln this
r=l r=! r=l r=I
case, we take into account G symbols only previous the received one.
The problem (20) is solved by numerical approach, assuming d = 1000 ■ The value d = 1000 is chosen due to |F[ (J > 10
for t =t .
re)
Futher after estimation the resolution time the final value of "effective memory" G for resolution time / is calculated using
(20) and substitutions c -G\R =R ',F=B, ■
UI, : 6 r
C. Least Number of In formation Sequence Realizations Estimation
To estimate the value of a least number of information sequence realizations Nmm for given value of tolerance X we need
to solve the following problem numerically
(21)
where ^ e N is the number of the current step; + 1 is the number of step number with which to compare;
^(M^.G^+l)» ^(fcA^.t^+l] are cumulative distribution functions of measured settling phase error
({<?*, for (G«', +,)"th radl0Pulse with lhe sPeed of
transmission l // ^ for numbers N ^, A1', of information sequence realizations, respectively; (p e [-2,t;2,t] is the threshold of measured settling phase error.
It is expedient to solve the problem (21) by transforming it to the following system of inequalities and taking into account the law of large numbers
O :
2_min / | < v
(22)
where Q = N/n is a specific number of information sequence realizations.. Due to validity of the law of large numbers for practical realization of numerical solving of (22) it will be sufficient to test the second inequality of system (22) during 1000 steps {in this case, the step is Q f /£> = 10) after Qmin and the number of step corresponding to it p(Qmiri) were found. Further for calculations for solving the problem (22) we will be use the following expressions
D = max\F (^^„.C,, + lJ-F^ (p, £?,.',.,..G^ +1)|
(23)
D. Estimation of Symbol Error Rate, Resolution time, Capacity
SER caused by IS1 produced by LTI system and squelch error rate (SqER), analog of SER for envelope, are estimated for two types of symbols: a) once for any symbol v > G +1, when the
output PSK-zi signal is a cyclostationary process and h) for each symbol v < G, when the output process is non-stationary. For case a) SER Psi,r (t, ) is evaluated using the expression
p (r 1-1 " V 1 1 (24)
'» V*j / j, >
min
and SqER t> {r,) is determined using the expression
p (r 1 = 1- -*.-.1—-~
Armn ' (25)
for case b), SER p t (r(,v) is evaluated using the expression
irM^l-
<vm
(26)
and SqER P (ta,v) is determined using the expression
N.
(27)
where ^„„((G„i+l)r,4 [¿((g^ +l)rs,b)\* wm(vrs,b) and
1^(^,6)1 are the values of the measured slowly varying phase
and envelope on the output of LTI system at the end of (g +1)
and V-th radio pulse of the PSK-/J signal for the 6-th realization of the transmitted random information symbol sequence, respectively; yv and yv{b) - initial phase values of the Gx, +1
and V -th radio pulse of PSK-« signal on the input of LTI system for the 6-th realization of the transmitted random information symbol sequence, respectively.
To estimate the resolution time we need to solve numerically a set of equations, produced by equating expressions (24) - (27) to zero, with the respect of r . As the result, we obtain a set of solutions for envelope if )G+I and slowly varying phase ij .
t "'» I )„d
In this case the resolution time will be determined as follows
=H jf Q{U,]{n 'iU„}}and capacity estimation is
made using(9)- (11).
III. Algorilhm Application And Results of Capacity Estimation
Let's consider the application of the above algorilhm for capacity estimation of considered communication channel. As LTI system we consider lhe following bandpass filters with identical bandwidth: 1) a single stage filter on coupled circuits with different degrees Of coupling [11, p-111]; 2) three circuits filter with two coupled of them (see scheme [1 l,p,162]); 3) a single stage on coupled circuits with different degrees of coupling [11, p. 143].
1) A single-stage bandpass filter with coupled circuits [11, p.lll]. The settling function for a single-stage filter has the form
Here and below x = Mit = AQatIaA - non-dimension time;
Affl _ Aw _ a is dimensionless detuning, p - the h AD. 4 AH,,
resultant dimensionless detuning; p is a degree of circuit coupling. For one staged filter with degree of circuit coupling p -\
(critical value) aA = 1.414 and for p = 2.41 aA =3.11 g
2) Bandpass three-circuit filter [11, p. 162] with two coupled circuits and with the parameter y = 2- This condition ensures the location of all three humps of the filter's AFR at the same level. The settling function of such a filter has the form
"(1+jpf+f
B0(x,p)= l-
l+fif
-exp(-(2 + jp)x)~
I+ JP ß
(29)
-jp cos ßx + p +
sin px
For aA = 2 for pr = yß and aL = 6.64 for /7 = 6-
3) A single-stage bandpass filter with coupled circuits [11, p. 143]. The settling function for a single-stage filter lias the form
tf, (x, />)=1 - exp (- (1+,/p) x)icos/7*-j-j^sin0x\ ■ (3°) For as = 1.5 for pr = 0.468 and aA = 2.46 for p = l. 15 -
4»
ra
£ io
J
111 10
ti IV ■ ' 1 ! | J
lu \ 1 i. л_____\ __
^ ~ ~ ~ ' ■ -----
0 I 2 3 4 5 6 7 * 9 id II 12 13 M li Non-riLmciulorul symbol durnlioii. Al"! r
Fig. 1, Dependence of "effective memory" G(r,) on non-dimensional symbol duration AQ„ri ■ Type of LTI system: a single-stage bandpass filter with a degree of circuit coupling p - \ (black line) and /7 = 2.41 (red line)', a three-circuit filter with a degree of circuit coupling fi -yjl (green line) and /3 = 6 (blue line)', a band pass filter (3-rd type) with a degree of circuit coupling jj = 0.468(magenta line) and /7 = 1.15 (cyan line). Resulting dimensionless frequency detuning p : 0 (solid line); 0.5 (dashed line)
Table. I
The greatest settling time for first symbol on the LTI output
Number of type öf LTI system Degree of circuit coupling fj Number of discrete states n First approximation of effective LTI memory The greatest non-dimensional settling time tor first symbol on the l.TI output
8 4 1.9566
1.414 32 2 2.70437
128 1 3.114
2.41 8 14 1.868875
32 11 2.309
128 10 2.4715
$ 8 5 2.67136
32 3 3.3374
2 128 2 3.6415
6 8 13 3.6892
32 11 4.756
128 2 10.69735
0.468 8 9 1.3516
32 4 2.3505
3 128 2 3.084
1.15 8 19 0,9246
32 15 1.3159
128 13 1.47
The dependences of "effective memory" on symbol duration for presented above LT! systems are shown on Fig. 1. These dependences were obtained by numerical simulation using (20), (28)
- (30), where ¿/=1000, ¿r=IO"!. The first approximations of effective LTI memory (JUI for considered LTI systems in absence of frequency detuning are presented at Tabl. 1, These results are obtained using expressions (13), (20), (28) - (30) for <2^ = 0.499-
Analyzing results, presented on Fig.l and Tabl. 1, we conclude the following. Decreasing the symbol duration of transmitted sequence leads to an increase by exponent law in "effective memory" of LTI system. An increase in the number of discrete states и leads to a decrease in the value of the "effective memory". The presence of frequency detuning flpj = q gj slightly affects on the value of "effective memory" in general case.
Based on the results obtained above we solve numerically the problem (22) in the absence of frequency detuning for tolerance Я — 0.01 and given values of "effective memory" G,., and xM
(Tabl. 1), using expressions (28) - (30) and (f) = arg2(<)-
The results of simulating with solutions (vertical lines) are presented on Fig. 2-7.
Analyzing results on Fig. 2-7, we conclude that for the tolerance Л = 0.01 it is necessary about 26000 different realizations of information sequence with length of G,., +1 symbols.
I Ml 1201) 1400 1601) I two 2O00
0.015
1 i 1 1 1
i i j.yj »VJ ■/ til 1 Щ' f i Ш iNMm
17 i ii V ■ 1 1 1 ■ ' V *
0 200 4011 600 8011 11100 1201) 1400 1600 1800 2000 Spccillc number :-l i i* : t ;;iir- i gAqucnuc rciiLi.'junns. Q
Fig. 2. Dependences of parameters A and D on specific number of information sequence realizations Q for a single-stage bandpass filter [11, p, 111 ] with a degree of circuit coupling j} r = ! Number of discrete
states n: S (solid line); 32 (dashed line); 128 (dash-dotted line)
0.15 § 0.1 1
j3 0.05
i i
\!
a s
i J i
i ! Si i i 'к /V ' v e £ , jfe MM ^ iyl r v
0 200 400 600 $00 1000 I21KI 1400 1600 1S00 2000 Specific ii Iitvr . f rlirri.i; г &£quenc«! raliz&lpng. Q
Fig, 3. Dependences of parameters A and D on specific number of information sequence realizations О for a single-stage bandpass filter [II, p.Ul] with a degree of circuit coupling /? = 2.41. Number of discrete states n: 8 (solidline); 32 (dashedline); 128 (dash-dottedline)
low t20ti iioo ifsoo isoo 2o«o
I 1 1 1
- i i. fjtV»
0 200 400 600 800 1000 1200 1400 1600 1300 2000 Specific number of information sequcncc rçvl I zstîoiu, Q
Fig. 4. Dependences of parameters A and D on specific number of information sequence realizations Q for a threc-circiiit filter [11, p. 162] with a degree of circuit coupling p r = . Number of discrete states n: 8 (solid line); 32 (dashed line); 128 (dash-dotted line)
I o-1
J 0.05
! \ i 1 1
\ i \ i ;
\t —---+--T----
1OOO 120(1 IJOO 1600 1800 200(1
1 i ! 1
i . Ij r j i r. 1 r* *\d iy 1 1 ' 1 1 t )
0 200 400 600 BOO 1000 1200 1400 1600 18(10 200(1 Specific number of informalion hc^Iicnuc realizations. o
Fig. 7. Dependences of parameters A and D on specific number of information sequence realizations O for a band pass filter [11, p. 143J with a degree of circuit coupling ß = 1.15 ■ Number of discrete states n: 8 (solid line)', 32 (dashed line); 128 (dash-dotted line)
0.15
o.i
005
1 1 (
\ -
V ^_ 1 -
200 400 600 BOO 10(10 1200 14(10 160(1 I BOO 2000
: 0.(105
i [ 1 1 (
• Hto i 1 ¡ju j] t. i' ,' , ..... J i \
(I 200 400 600 BOO 1000 1200 ¡ 4(10 1600 1800 2000 Specific number of information scqucnce realizations. Q
Fig. 5. Dependences of parameters A and D on specific number of information sequence realizations Q for a three-circuit filter [11, p.162] with a degree of circuit coupling p = 6 ■ Number of discrete stales n: 8 (solid line); 32 (dashed line); 128 (dasli-dotted line)
i i 1
\ i \ i
\i 1
200 4(H) 600 BOO 1000 1200 1400 1600 ISOO 2000
i j
i « ii."' ■ w 1 . !; r.AUi V , V ) 1 ki^é
0 200 400 600 8(10 (000 1200 1400 1600 1800 2000 Specific number of informaiiori sequence realizations, y
Fig. 6. Dependences of parameters A and D on specific number of information sequence realizations Q for a band pass filter [¡1, p. 143] with a degree of circuit coupling ft =0.468- Number of discrete states n: 8
(solid line); 32 (dashed line); 128 (dash-dotted line)
As a proof of the correctness of the results obtained above, dependences of SFR on symbol duration were estimated using expressions (24), (26), (28), (30) for first and third type of presented filters for a = 0,499. The choice was made due to largest
value of "effective memory" for considered number of discrete states in Tabl. I. The results of these estimations are presented on Fig. 8-50.
From Fig. 8-10 we conclude that the greatest settling times for symbols with a sequence number less than or equal to the values of "effective memory" are less than the greatest settling time when the behavior of output signal becomes like a cy do stationary process. This effect fully totally coincides with theoretical results obtained in paper [8] and thus the greatest settling time when the behavior of output signal becomes cyclostationary process is the resolution time with tolerance £. Additional we can say that the estimation of resolution time corresponds G = 1 ...3.
2 2.5 3
Non-dimensional symbol duration, ASi .r^
Fig. 8. Dependences of SER on non-dimensional symbol duration AÎî„r for v symbol in transmitted information sequence of PSK.-8
signal: 1 (blue line); 4 (red line); 7 (green line); 10 (cyan line); 13 (magenta line); 15 (black line), LTt system - a single-stage bandpass filler [11, p.l II ] with a degree of circuit coupling p = 2.41
The results of unity capacity estimation Cmil—C/2AF0t
where AF„ = Afi(, / 2n for considered LT1 systems are presented
on Fig. 11-16. It is necessary to mentioned, that, in a number of cases, "transparency windows" and the dependences ofthe capacity estimations are superimposed at certain values of discrete states n.
Number oftliscrclc "
Fig. 14. Dependencies of unity capacity Cmll on the number of discrete
stales n for considered communication channel. LTI system: three-circuit filter [II, p. 162J with a degree of circuit coupling p- 6- Estimation of channel capacity for case: 1) apm - 0.499 I AMr/ = I; ea = 0; ft = 0
(black line with diamond marker and black shaded area); 2) apm = 0.499 ; A„„, = 0.1; e0 = 0; ft = 0 (green line and green
shaded area); 3) apm = 0.25 \ =0.1; £0 = 0.249; ft = 0 (red line and red shaded area); 4) apm - 0.499 i A„„, =0.1; e0 = 0 i ft = 0.5 (blue line and blue shaded area); 5) a/lln - 0.25 ; AjW = 0.1; g = 0.249 ft =0.5 (cyan line and cyan shaded area). Type of capacity estimation: low bound of capacity (solid line); upper bound (dashed line) and low bound (dash-dotted line) of capacity of "transparency window", "Transparency windows" - shadowed area
% <•
^-- t I < >~--
> ^ — ----
/
J
52 «8 IU
Number .1! discrete slat« n
Fig. 15. Dependencies of unity capacity C,(m( on the number of discrete
states n for considered communication channel. LTI system: a singlestage bandpass filter [11, p.i43] with a degree of circuit coupling 0 - 0.468 ■ Estimation of channel capacity for case:
I) a]m = 0.499 ; A„„, = 1; sa = 0; ft = 0 (black line with diamond
marker and black shaded area); 2) apm - 0.499 A,,,,, = 0. j; £u = 0
ft = 0 (green line and green shaded area); 3) apm = 0.25 I Awij = 0.1
£,. = 0.249; o. = 0 (red line and red shaded area); 4) a =0 499
0 r ll fun
A , = 0.1; ff„=0; p0 = 0.5 (blue line and blue shaded area)
5) apm = 0.25 ; A,w =0.1; sit = 0.249; ft = 0.5 (cyan line and cyan
shaded area). Type of capacity estimation; low bound of capacity (solid line); upper bound (dashed line) and low bound (dash-dotted line) of capacity of "transparency window". "Transparency windows" - shadowed area
a 2
\
- I-
4 20 J6 }1 M SI 10(1 lit 122
Number uf discrete slalcs
Fig. 16. Dependencies of unity capacity C u oil the number of discrete
states n for considered communication channel, LTI system: a singlestage bandpass filter [II, p.143] with a degree of circuit coupling P r =0.468- Estimation of channel capacity for case: 1) a n =0,499;
A Bfjt; e0 = 0: ft = 0 (black line with diamond marker and black
shaded area); 2) apm = 0.499A,rJjJ = 0.1; S0 = 0; ft = 0 (green line
and green shaded area); 3) apm = 0.25 ; A^ =0.1; £„=0.249
pa— 0 (red line and red shaded area); 4) g; = 0.499 - Awif = 0.1
S0 = (); pfi = 0.5 (blue line and blue shaded area); 5) czpm - 0.25
A«, =0.1; £„=0.249; ft = 0,5 {cyan line and cyan shaded area).
Type of capacity estimation: low bound of capacity (solid line); upper bound (dashed tine) and low bound (dash-dotted line) of capacity of "transparency window". "Transparency windows"—shadowed area
IV. Conclusion
Analyzing Fig. 11 - 16 we conclude the following:
The increase in the capacity of considered communication channel operating in the presence of strong ISI is possible due to the use of "transparency windows" including the case of frequency detuning. In the absence of unfavorable factors using "transparency windows" allows us to provide gain (r ic ) i'11
average) in the range from 1.2 to 1.9. An increase in the signal level requirement and the presence of a measurement error in the absenec of frequency detuning, which may indicate the feasibility of such RDTS, leads to a decrease the size of "transparency windows" and their appearance at small values of the number of discrete states n. In this case, upper bound of capacity estimation is reduced in average from 3 to 5%.
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К ВОПРОСУ ОЦЕНКИ ПРОПУСКНОЙ СПОСОБНОСТИ В КАНАЛЕ С ФМН^-СИГНАЛАМИ И С ПАМЯТЬЮ, ВЫЗВАННОЙ МЕЖСИМВОЛЬНЫМИ ИСКАЖЕНИЯМИ, ПРИ НАЛИЧИИ НЕСТАБИЛЬНОСТЕЙ В ПРИЕМНОМ УСТРОЙСТВЕ
Лернер Илья Михайлович, Казанский национальный исследовательский технический университет им. А.Н. Туполева - КАИ,
г. Казань, Россия, aviap@mail.ru
Исследование выполнено при финансовой поддержке РФФИ в рамках научного проекта № 18-37-00440
Аннотация
Объем передаваемой информации возрастает экспоненциально из года в год, что является тенденцией современного общества. Это приводит к необходимости увеличения скорости передачи данных систем передачи информации. Наиболее выражено это среди радиотехнических систем передачи информации, которые в настоящее время работают в условиях ограниченных частотных ресурсов и постоянно увеличивающихся требований их эффективного использования. Одним из наиболее эффективных подходов к решению этой проблемы является переход к передаче информации при наличии межсимвольных помех в радиотехнических системах передачи информации. Несмотря на привлекательность этого подхода, его техническая реализация связана с рядом трудностей, которые могут привести к увеличению сложности самого приемника при увеличении числа интерферирующих символов. В конечном итоге возникает вопрос не только о целесообразности реализации, но и о его практической осуществимости самого приёмника. Альтернативным подходом, позволяющим создавать радиотехнические системы передачи информации, которые функционируют в условиях сильных межсимвольных искажений, возникающих в линейных избирательных системах радиотракта, при отсутствии их компенсации, является соответствующий выбор длительности канального символа, осуществляемый с учетом разрешающего времени линейных избирательных систем. Производится оценка пропускной способности такого канала при использовании ФМн-п-сигнала, когда решающее устройство является компаратором, а при принятии решения отсутствует необходимость в учете памяти канала, при условии, что радиотехнические системы передачи информации работают в присутствии сильных межсимвольных искажений. Для решения этой задачи был разработан новый численный метод оценки пропускной способности, который можно использовать для оценки потенциальной пропускной способности при отсутствии шума, но в условиях различных нестабильностей, таких как расстройка по частоте, погрешность измерения, влияния уровня сигнала.
Ключевые слова: пропускная способность, фильтры, ФМн-п-сигналы, повышение спектральной эффективности. Литература
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5. Лернер И.М. Аналитическая оценка пропускной способности канала связи с частотной характеристикой резонансного фильтра при наличии межсимвольных искажений и использовании многопозиционного фазоманипулированного сигнала // T-Comm: Телекоммуникации и транспорт. 2017. Т. 11. № 9. С. 65-73.
6. 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.
7. 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.
8. Лернер И.М., Чернявский С.М. Оценка пропускной способности реальных каналов связи с АФМН-^сигналами при наличии МСИ // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №4. С. 48-55.
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Caused by the ISI // Proceedings of IEEE Conference 2018 Systems of Signal Synchronization, Generating and Processing in Telecommunications, Minsk, Jule 4-5, 2018 (в печати)
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Радиотехника и электроника. 2012, №2, С. 174-188. Информация об авторе:
Лернер Илья Михайлович, к.ф.-м.н., доцент кафедры РЭКУ, Казанский национальный исследовательский технический университет им. А.Н. Туполева — КАИ, г Казань, Россия
