Analytical estimation of capacity of the communicational channel with the frequency response of the resonance filter in the presence of ISI and the use of PSK-n-signal Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

ANALYTICAL ESTIMATION OF CAPACITY OF THE COMMUNICATIONAL CHANNEL WITH THE FREQUENCY RESPONSE OF THE RESONANCE FILTER IN THE PRESENCE OF ISI AND THE USE OF PSK-N-SIGNAL

Ilya M. Lerner,

Kazan National Research Technical University Keywords: communicational channel

named after A.N. Tupolev-KAI, Kazan, Russia, capacity, resonance filter, PSK-n-signals.

aviap@mail.ru

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 the data transmission in the presence of intersymbol interference produced by selective systems presented at radio engineering data transmission systems. Despite the attractiveness of this approach, its technical implementation is associated with a number of 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.

Since the increase in the alphabet of channel symbols leads to the need to increase the signal-to-noise ratio (SNR) in order to achieve a given signal reception quality, it is advisable to consider receiving algorithms based on utilizing frequency filtering to increase SNR and consequent use of multithreshold decision device at situation of strong ISI in com-municational channel.

In this paper, we consider capacity of such channel with the frequency response of the resonance filter and using PSK-n-signals. Capacity estimation was performed by analytical proving. New properties for such channel were revealed.

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. 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. Том 11. №9. С. 65-73

For citation:

Lerner I.M. (2017). Analytical estimation of capacity of the communicational channel with the frequency response of the resonance filter in the presence of ISI and the use of PSK-n-signal. T-Comm, vol. 11, no.9, рр. 65-73.

Introduction

Volume of transmitted information exponentially increasing from year lo year is a current trend of modem 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 (RDTS), which currently operate under conditions of limited frequency resources and constantly increasing requirements to an effective frequency resource usage [I j.

One of the most effective approaches lo solving this problem is the conversion to the data transmission in the presence of intersymbol interference f 1ST> produced by selective systems presented at RDTS. Despite the attractiveness of this approach, its technical implementation is associated with a number of 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.

So widely used methods: zero-forcing lifter (ZFF), zero-forcing equalizer (ZFE), linear minimum squared estimation (LMSE) equalizer and decision feedback equalizer (DFE) have a significant loss in noise immunity as opposed to optimal reception algorithms using the maximum likelihood (ML) criterion f 1,2]. in some cases the first three methods of mentioned above cannot be realized for real channels 111 and as it was mentioned in [2] these methods are inadequate as compensators for the ISI on channels with spectral nulls, which may he encountered in radio transmission. The last one has the same problem as first three methods for channels, which may be encountered in radio transmission but with less effect. It should be noted that all of them do not lead to complete Suppression of ISI and they provide good results only with minor ISI when amplitude frequency response has slow variation on magnitude 11.2],

As for ML algorithms (since they were constructed on Viterbi algorithm) their main disadvantage is the complexity of implementation, depending exponentially on the transmission speed and memory channel, which leads to the impossibility of their usage with a large alphabet of channel symbols. In the literature, various approaches to reducing the complexity of the Viterbi decoder for this case are considered: a decrease in channel memory by means of a preliminary linear correction, slightly distorting amplitude frequency response of the channel [3, 4], a shortened search of the most probable paths along the lattice |5,6J; combination of two methods mentioned above [7,8 j. However, the complexity of the receiver in this case decreases insignificantly, still remaining much more complex than a linear equalizer. One of the latest works in this direction shows that obtaining suboptimal algorithms based on ML algorithms leads to considerable computational complexity described by exponent law when the number of channel symbols exceeds 4 and/or in a case of high transmission rate 11].

In this case, one of the most interesting approaches to getting high speeds on radio communication channel is the usage ofpre-distortion signals and receiving algorithms concerted with them.

Since the increase in the alphabet of channel symbols leads to the need to increase the signal-lo-noise ratio (SNR) in order to achieve a given signal reception quality, it is advisable to consider receiving algorithms based on utilizing frequency filtering to increase SNR and consequent use of multithreshold decision device at situation of strong ISI in communications! channel. Their effi-

ciency is expedient to estimate the ultimate capacity of linear selective systems (capacity of channel in this particular case) in the absence of noise when the decision device is a multithreshold device. The last condition determines the efficiency limit for such RDTS.

It should be noted that using radio pass band filters in the radio path of RDTS leads to additional iSI which influence is considered. So it would be better to obtain the solution of this problem for radio frequency signals.

Due lo the fact that RDTS using multi-position phase-shift keying signals with n discrete stales (PSK-tt-signals) arc the most widely spread nowadays, we will solve (his problem for RDTS of this class.

In this paper, we present an analytical proving of capacity estimation Tor such channels with amplitude frequency response identical to resonant filter.

I. Problem statement

In this case, capacity estimation is considered for a communication channel (Fig. 1), which consists of a transmitter, a linear time invariant (LTI} system, a receiver and a decision device (DD).

In this work, we assume that LTI system is a tuned resonant filter that possesses the following property: ojlt / 2A£J > 15 where 2AQ - the resultant bandwidth; ~ resonant frequency of considered filler. We also assume that receiver is an ideal phase detector similar to DD that docs not introduce additional ISI.

Trammilkr

LTlSysliM SmW Rcwh-er »J 8 DD

in."J

Fig, 1, The considered communication channel

Wc assume that each symbol of' primary signal thai impacts on the input of transmitter takes cquiprobably one of the n phase values of the signal constellation

V, =(k +0.5 v sign (||0.5«||-a S/i-0.25)-||0.5»|)Aip„- A) Flere k = 1, n; ||. |j — rounding operation to the nearest integer;

in accordance with the requirements to the PSK-n signal generated by the transmitter V takes zero for the PSK-i? signal with initial shift of the signal constellation q> = () and V = 1 when

<pn=x/n'i sign( . ) - si gnu m function: -2k tn ~ step between adjacent phase values of the signal constellation.

The PSk-n signal generated by the transmitter from the moment t = 0 can be represented as follows

= I[IiH'' -"-v,)]J("V + ¥«1)U^%))

H ,1

w here / e M - the number Of transmitted radio pulses forming the PSK-n signal, - resolution time is the minimum transmission

time of each of pulses when 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; ](/) - Heaviside step function: q _y* +(p !

i

(0a - the carrier frequency of ihe PSK-n signal equal to the average frequency of the tuned LTI system: <pn - constant phase shift at the frequency CO() introduced by tuned LTI systetn; 28r = y/ ((r- i)fm }- y ((r- 2) ) ~ phase jump caused by the transmission of r-th radio pulse where y/((r—) = 2(1, + 3, and 2 ) = 20 r , + &, , - initial phases of r"1 and (,--|)'h

radio pulses of PSK-n signal at the output of transmitter; y/Ur—2)tr ) = <p if r— 1 because we assume thai before the

start of transmission, until the moment I = 0, LT! system is in a steady state and harmonic oscillation exp( /{uy+ <p -<p0)) ac,s

on its input.

For considered ease the resolution lime t,, must correspond to the minimum lime when the following condition is true

x,m a<*„, (rtm) -\v„ )-»//((/--I K, <3)

Mere # - the permissible phase settling error at the output

of LTI system, which is related to the fiducial permissible settling

error by the relation a.=a iAa> : a (rt \ ~ measured set* I) №) • SI /ft* \ rtf* I

tling phase error at the output of LTI system for the r,h radio pulse when it ends; y/ (rtr } - the measured value of slowly vary ing

phase of the r"1 radio pulse al the output of LTI system obtained by a receiver at moments ri

In liiis case PSK-n signal at the output of a LTI system starting from the time / = 0 according to Ihe superposition principle and results of [9] can be represented in the form ■■%„(/) = ¿('J exp () = k (iw,) exp (j (ii)„t

= - (4)

.< -i L

>c[exp(y(t). .))-exp(y(-e, +.£))]]. where & =(!-#,, (r)) ifr = l:

if r = 2J and B„{f) -set-

tling function [9]; Z(t) - complex envelope of the transient

process caused by the passage of the PSK-n signal through LTI system; k(ja0) = A(¡ioa)exp(/<pfl) - complex transmission coefficient of a tuned LTI system, where we assume & (¡i,ja) = I:

<r-t

The signal at the output of the receiver, based on equation (4), is defined as

MU'

The recovery of the primary signal is made by DD based on the decision rule

where /(*',/■) = mjo\wm(>%„)-<«,„„-

a/m < 0.5Ai/?ir ■ The last of these conditions is necessary to ensure

the uniqueness of the DD solution and it is obvious that in the absence of noise aim ->0.5A<pw ■

Analyzing results of j 1()| we can conclude liiat for considered channel capacity determination may be formulated as the maximum information transmission rate on the communication channel with equivocation produced by IS! where maximum is taken over all possible sources of information used as input to the channel.

En this case, equivocation produced by ISI is easy to consider using resolution time in the following equation form for capacity channel estimation

(8)

(10)

where the second multiplier in equation is maximum entropy of source and /( is taken at the condition am 0.5Af/\( •

Thus, it is necessary to determine the resolution time / of

LTI system to estimate the capacity of considered channel, which requires an analysis of the occurring transient process caused by the PSK-n signals transmission.

II. Problem solution

According to [9], the settling function lor a resonant lilter is defined by following expression

5 = c" (9)

A ' I 0 al l<0.

And the following properties are true

where T = const

We transform (4) taking account of (10), lor this we IIrst consider the transmission of the second ;ind third radio pulses through the considered LTI system. We introduce the following substitutions Mj = 12 (dt^ )| - = are Z () - (29, + &,),

which are the values of the envelope and the phase settling error ot'lhe slowly varying phase of the PSK-r; signal at the output of considered LT! system at the end of the d-Xh. radio pulse (i/-0] transmission. In this case Mn = \maa = 0-

So at ihe time / = /„,+/'j where f'e[0;i,...] № the transmission of the second radio pulse, (4) taking into account (5) and (10) takes the form

¿K+O =

The equation (4) lor the transmission of the third radio pulse can also be simplified by considering at the lime r = 2i, +/*,

where f'e[0;f l.We simplify it using properties {10) for the

considered LTI system, equation (5) and the following substitutions for d -2

0,= 0,-0.5 a,,

(12)

and vvc assume 2/ +t' = /, +('. As a result. (4) according to 1111 lakes the form

- (l- B, (/'}) XL exp(./{ -6, + £ +«,))+

Analyzing expressions (Ml and (13), we conclude that (4) for considered LTI system can he represented as

xcxp (/(fc+fl,))]]. where M 1, r = !,/ follows from an analysis of the results of [UJ; M, , =1. when ar ,= 0-

It follows from (14) that the PSK-« signal at the output of the considered LTI system is a sequence of transient processes, and each of them can be one of two types: the first type is caused by the phase jump in the harmonic oscillation when M = 1. a, = 0 if r - I, /, and the second type is caused by a simultaneous jump in the amplitude and phase when A/,. ,<L a, , if r -2J.

Thus, to determine ¡ki . it is necessary to establish the type of

the transient process with the greatest settling time for a given permissible phase settling error. The determination of the type is made based on a comparison of the values of the settling function at these moments. This is because (9) increases monotoni-cally with increasing time. In this ease, we are primarily interested in the dynamic component of the transient process, so we use equivalent values of phase jumps for the PSK-n signals

Iff , =

(15)

f (g-0.5«-1) Afsg. g=l,n+i if n=2z: zeN; [ ^f0.5Hj)ApriJ g=U; if n = 2r+l,zeN; that are obtained by transformation

w _ | -In + 20 if 3fie(<2;r); ** \+2ff+20 if 20 e(-/r:-2>r).

(16)

where 20 - original values of phase jumps.

Let us find equations for both types of transient processes that will allow to estimate the settling time. Thus, the equation allowing lo estimate the settling time i (based on the Value

BqIi, ,)) under given permissible phase settling error tor the first

type of a transient process is obtained from (14), for this we take

/'- 2, / = ,m E [0;/J, Z(,wl) = M, cxp{y(i, + 0, +«;„„)) ■ 2ft = 0

and after a series of simplest transformations we obtain the following equation

<l7>

Mere urm = ±av„,. where "+" for € (-/n0) and for 20, e (0:/r] that follows from the results of [111. Transforming (17) according to Ba (iw )t we obtain

Ba{tM)=-

sin(2^ +a'lll„) sin(2f?,+«;,m)-sin«/,

(18)

From the foregoing ii follows that (18) has the parity properly. Therefore, we will consider only the case when 20t e(0;;r].

Wc make a substitution 20{ for a variable 20e|[« iii|;,tj that lakes continuous values, where ar„ e {-20:0I ■

Optimum analysis of (18) based on the solution ^"(V)

J20

- 0

shows that the extremum is achieved when

(IS) lake

and ( IS) lakes the value

I \_

2

I w

I +sin\a

I+tgJ

(19)

(20)

Taking into account that for limit values of the phase jump 16 -n and 2(9 —> la I equation (ISftakes the values

(21)

then ii follows from a comparison of(20) and (21) that the equation (IS) reaches its maximum at (19).

Under constraints e i 0; ^ j ?

\aj[<\2d[ ''lt; equation (18) is symmetric relatively (19). This

can be proved by transforming (18) taking into account the substitution 20 to 20 as 2t) = 20 where A2£?>(), as a

'■ I»

result, we get

cos(A20)

(22)

cos (¿20)+sin

At the same time ^^ ^ ^ 0.5. Equation (22) and the results obtained above imply the symmetry of (18) under the specified restrictions.

The equation allowing to estimate the settling time t (based

on the value (/[.,)) under a given permissible phase settling

error for the second type of transient process will be obtained by considering the PSK-« signal at the output of considered LTI system during the lime ¡n the/J-th =

radio pulse transmission.

In this case, (14) is represented in the following form, assuming thal, = (p-l),n.i+,^(e[«;ij-

2 ({ P -1) w+CH1 (C)- i(C ->*.))[(1 - (C)) *

(2j)

* exp (j (§„., - i>,„,+«,,)) + B0 (lm - ) x

We transform (23) taking into account that s„ (/) = (), when /<0 and using the substitutions (12) where c/ = /n, ¿{(P~>)'« + C)~M, exp(y{£ + + ap„,)), here a„ = ±a,m, where "+" for 20r e(-/r:0) and "-" for 2$ @(0;*j that follows from the results of [ 111. As a result, we obtain

'' /,' J r> / 1 \ n' i m i n t ' \\ /i1 *

m

where is the effective phase jump due to the transmission of

theyMli radio pulse.

We transform the last equation with respect to J and

obtain

(25)

where L as follows from the problem statement in ihe

definition / ; ho' I > a ~ this can be seen from the analysis of

(25), since if this condition is violated, it takes negative values, which means that time is not required to establish a given value of the phase jump under a given permissible phase settling error. From the Foregoing, it follows that (25) has the parity properly.

Let us analyze (25) in the same way as it was done with (IS). Taking the parity property into account, we can do ibis only when 20fi

Optimum analysis oF(25) based on the solution

d 2&.

■ = 0

shows that the extremum is achieved when

and (25) lakes the value

. . . _

('i.w ) ^ . , ,

S'fi I | +MII _I

Taking into account thai for limit values of the phase jump 20', = tt and 26', ->!«'„, I (25) takes values

(26)

(27)

M

p' i

Kf « \ + M.

A')-,

= 0,

(28)

then it follows From a comparison of (27) and (28) that the equation (25) reaches its maximum at (26).

We proceed directly to the problem of determining the type of the transient process with the longest settling lime for a given permissible phase settling error. We compare #„(/,,,) and

hy solving lhe lnecluallt>' Bn(/,,,„ )> Bn(t„p, ) according lo M | using (20) and (27). Alter a number of simplest transformations, inequality and its solution take lhe form

>i«WH e[0;l),

(29)

On tbe basis of (14), the solution (29) and the restrictions used for (27) and (20). taking into account lhe symmetry properties for (IS) and (25), we conclude that the inequality Bn (ivpl) > Bn (t,rl) is always satisfied when , e[0;1) and goes

to the equality when \f = \ and a will be zero. That is. the

fi-t /i i

longest settling lime of the iirst type transient process is lhe limit case of the longest settling time of the second type transient process when the solution is sought among the continuous values of phase jumps.

The above results were obtained for continuous values of

phase jumps, however, in determining tm by the first type of the

transient process, it should be taken into account that phase jumps 20. take discrete values. Fience, this affects the determination of the longest settling time. While for the second lype of transient process, the effective phase jumps take continuous values. In this connection, the follow ing question arises - whether it will he fair lo choose the first type of transient process in determining the resolution time in the case of discrete values of phase jumps.

To answer this question, it is necessary to compare (18) and (25) for Continuous values of phase jumps 20 = 20 = 20 under

the following restrictions

20e(O;Tr); 2i?j, E (0;/r);

(30)

using inequality

(3D

if this inequality has a solution under iliese constraints, then this ii has be taken into account when determining the resolution time. In this case, il is necessary lo determine a combination of the effective phase jump, the envelope and the permissible phase settling error when the sctlling time of the second type transient process is the greatest. Then it is necessary to compare it with the longest settling time of lhe Iirst type transient process with discrete values of phase jumps and a given value of the permissible phase settling error. In lhe absence of a solution of this inequality, the resolution time is determined from an analysis of the first type transient process for discrete values of phase jumps.

We oblam the solution of (31) with respect lo A/(i , by the

simplest transformations

M2e'K\)

sin(20;,-|«;„])' The solution of (32) is represented as

1

Sill

M, , - A

(32)

(33)

where A>\ is a dimcnsionless coefficient.

We substitute (33) into (25), taking into account the constraints (30), as a result we obtain the equivalent value of die settling function /î. (y _ J

(34)

sin la

-.4sin(2i?-|«;„„[)'

From the analysis of (34), since il is similar lo (25), it follows that this dependence has the parity property and reaches its maximum value when the phase jump is determined according to

(26) when 20 's replaced by 20 - Then the greatest value

Vu)of<34)*

sinlcr i + ,l

il" k

(35)

As was shown above, the relation /^J/ J > Bu[i r) is satis-

lied under the constraints used for (27) and (20), it is obvious that it is also satisfied under constraints (30). Therefore, the relation uu f ) > ga j must also be satisfied, and by substituting (20) and (35) it lakes the form I A

sin a.....+ /1

->0.

20e(-;r:O); 2$f e(-*;0);

0 * a<—\

pn n *

(37)

20&(x:2x)- 2^e(jr,2»); 20-\aU(x;2x); 20],-\aj e '

(38)

--

■y fm

20 e (-2/r;-;r); 20p e (-2;r;—;r);

20 + W I g { 20p + \a\ e ( -2M ) ;

(39)

0 < a^y

where the above conclusion for (37) is due to the parity properly of (18) and (25), and for (38) and (39) by applying transforms^ lion (16) resulting its reducing lo (37) and (30) respecfively.

Let us consider in addition a number of special cases, which can also arise in the transmission of the/Mil radio pulse. The iirst of them arises when a binary phase-shift keyed signal (n = 2) affects on the considered LTI system, and the jumps in phase are only to k and the permissible phase settling error ajm = 0 as

follows from (14) and the results of [11 ].

In addition, it follows that the resolution lime must be determined on the basis of the longest settling lime of the first type transient process for discrete jump values.

We estimate the resolution time for this particular case from the value of the settling function using (18) and assuming 20. - 71, a = 0- In this case, (18) is

R I, r HH^I)

Inn —,-j—-7C-—.-

sm ( ^r -I} + SI n |orfM

- am

sin a..

2sin a.

s\n\al!m\ + \

Solving it according to A , we obtain ]>^>0. (36)

The solution (36) contradicts the initial condition (33) imposed in the solution of this problem, and (31) is not satisfied. Consequently, (18) when replacing 20t with the variable 20

taking continuous values in accordance with constraints (30) is the envelope of the dependencies (25) family under constraints (30). Thus, the determination of the resolution time for the case of discrete values of phase jumps should be made according to the first type of the transient process, thai is, on the basis of analysis (18) and taking into account only those constraints from (30) where there is 20, while replacing 20 by the variable 2ft,

which takes discrete values.

This conclusion also extends to the following restrictions

Since in the last equation the ratio represents the uncertainty (0/0), then, applying the LTIospital rule, we obtain

Jim = <40>

2 cos

The second special case occurs when 20' = ±x - We consider only for 20r - ii due to the parity property of (IS) and (25). The phase jump 2On = !i satisfies the inequality since

this jump is possible with two options: I) 20' = 20r = x (considered above in the 1st special case); 2) 20v = 29 -ar , -:r when ar | & 0 that is possible only if any of the restrictions (30), (37) - (39) is satisfied for 2il , and uy ,.

We estimate the limiting permissible phase settling error a for the second of the enumerated realization options of

this spccial case when (31) has ho solution, then the resolution time is determined from the analysis of the first type of transient process. Due to the parity property of (18), (25) and the transformation (16), we estimate the limiting permissible phase settling error when Z0p = 20r+\ap\ = x, ap , <0-

Sincc the relation Ji is satisfied

when Bu(r^ ) > 0.5. and 20p ~ n is true when ffti(^ J<0.5 that

follows from (28) analysis, we can formulate the following relation assuming 20 =20 = 2ltr I = 2a

fnfi.lna

* pitl. Ilm

= Tt-\a

p-t ■

(4L)

where 1« I < a , which follows from the results obtained

| /»-11 ^ fttujftti

above.

Obviously, the least value a

/ml m

is the case when

pwJmr

, then a =£ Since a <0.5Awr and the

;".-, !;:■' —, im t m

simplest i'SK.-/; signal is the signal with /1 = 3, the corresponding resolution time is determined by the first type of transient process for all PSK-» signals.

The third special case arises when 1) 20r - x and at , > 0 or

20 ——it and ar l < 0: 2) 20r = x and ]

0 or 20p = -x

and a:,,>0 where ,|* f°r the first condition, the

analysis results arc similar to those obtained under the constraints (30) and (37).

For the second condition, since (18) and (25) are even, we make an analysis when 20 =tt + |« As before, we use inequality (31) in the analysis lo determine the type of transient process used to estimate the resolution time. We transform the initial value of the considered phase jump 2O by (16) to the

equivalent value 20p = -2x + 20r = -;r+|e(1 ,| • As a result, the inequality ailer the simplest transformations and taking into ac-

70

counl 20 = 7i lakes (he form

sin ar„, + A■/,,. ; sin (\20r | - or,,,, ) sin +sin( 20 - a)im ) Its solution has the form

>1.1.

(42)

sin

(43)

«„„ =mio(jr-|i|f(|)<0.5A(p1,-

(46)

The greater part of the change in the slowly varying phase at the output of LTl system corresponds to the first stage as compared to the third stage, the smaller value is taken by ^ ■ Thus.

this gives an additional upper limit on the choice of a as

(47)

where F > 1 is the dimension less coefficient.

Substituting (43) into (25) taking into account the equivalent value of the phase jump used in considering this condition leads the first term in (42) to the form of (34) performed earlier in this paper. Based on this it can be concluded that (42) has no solution, and therefore for the second considered condition of this particular ease, the resolution time is determined from the first type transient process analysis.

The fourth special case arises when 28 = 0 and 2i7, = 0.

When 26 = 0 the phase jump of the signa! at the input of the

considered LTl system is absent, as a result, the process of establishment from transmission of the previous radio pulse continues, and at the end of the /i-th radio pulse transmission the measured error does not exceed the permissible phase settling error due to the above results. At 20y =0 the process of establishment ends

right in the moment of start of radio pulse. Thus, litis ease also does not affect on resolution time definition as compared to results obtained before.

Based on above, we conclude that the resolution time is determined for the considered LTl system by ihc first type transient process and the permissible phase settling error must satisfy the condition

0(44)

Let us consider additional restrictions connected with the features of the considered transient process in LTl system. Thus, from the analysis of (14) it follows that when

<45)

where y/{rtKi) - the value of the initial phase of the (r +1 j -tli

radio pulse of the PSK-n signa) at the output of Lhe transmitter, stepwise change of the slowly varying phase of the signal ai the output of LTl system will be observed.

This feature will he considered in more detail to determine the associated restrictions in lhe application of the results. In this case, the establishment of a slowly varying phase at the output of LTl system occurs in three stages, what follows from the analysis of (14). At the first stage, a slowly varying phase is established

from 10 <"+" 'f te„„KJ>0 and if

i//m (>'/, ,) < 0 ), at the second there is a jump from ±x to +rr . The third stage is its establishment from ±tt to the value '/',„.(('' + !)', ,)• Therefore, the resolution lime must provide the

end of the second stage of establishment for correct operation of the DD. The smallest value am providing the end of the second

stage is called a critical errorObviously, a v is defined as

follows

It follows from (47) that lhe choice of the ^ directly affects

the critical value of the permissible phase settling error «. In

the case when a = () there is a cons!ant error when receiving

the PSK-« signal in dynamic mode. To avoid this, it is necessary to select p correctly when generating the PSK-/> signal so for

»= 2r, z^2- p = 0.5A^>; 'i"d for » = 2z+l, r/>tt=0 where

sefi.

On the basis of the obtained results, the problem of determining the resolution lime reduces to determining lhe value of lhe phase jump from the set of positive values (15) that is least different from the value determined by (19). The solution of this problem is

(48)

where p £ K is die desired parameter. In this case, the initial equation where the solution is should be found takes the form 2ff<m _ Mfc, (49)

20

7T

According to the problem statement, we equate the left side of the given equation to unity and transform it relatively to fj

using hy = z£. Applying the rounding operation to the result.

we gel

171 n

= - + a, 4

(50)

In ill is ease, lhe value of the sctlling function al the resolution lime based on (50), (48) and (18) is determined as

sin 1* 17 4 Hi )

sill .2 7T „ n H | + sin 2k

(51)

The resolution time for the considered LTl system under the

PSK-« signal With (51) and (9) is

>

L

1

AH

f

In

1

L'-MUJ

AH

■In

. 2n sin — - a, n

. f2n{ n v . 2 7T

sin LTl 7+atr 4 -t-sin —a0 n

(52)

When substituting (52) into (8), the equation defining the capacity for considered channel using the PSK-/) signal takes the form

AQ

• Ist*

- ïpgjî

-In

Sill

an

+ sin ■

in

KvlHb))

«n

The dependences

ut"

the specific

(53) capacity

( C - C

/2Af[bits/(H2*s)], where 2AF=2AS272jt) on the

number of discrete states /1 under given value of the fiducial settling error was plotted using (53) (Fig, 2).

0 10 20 30 40 if) 60 70 80 n 100 Fig. 2. Dependence of the specific capacity on the number of discrete states of the PSK-n signal for a channel with a LTI system with the frequency response of the resonance filter for given fiducial permissible settling error a = 0,4999

From the dependencies shown in Fig. 2, it follows that the capacity when PSK-n signals transmit through the considered channel with a limited bandwidth at values n less than 30 is complex with an explicit maximum. For large values of n the capacity is dropped.

We determine the capacity c - Ittn C . applying the

L'llospita! rule to (53) because it represents uncertainty (oo/cc}

in this case. Due to n —mo-, ■ " ~}1 after the simplest trans-

b I 4 formations of (53), we obtain

Afi I f \ . 4tt In

C =—-lim -- ■ - I -sin—a„ +sin 1»-™ I 2 n

CC„

t4

(54)

We obtain the following applying the L'l lospital rule again to

(54)

_2tt

„ Aß ,f f 4rr „ . In \ hp, C, lim~r— cos—«„ + 2sin—a0 =—. In2fl-f « \ " a ) ln2

(55)

li follows from (55) that the capacity for the considered channel under the influence of the PSK-n signal with /; is a quantity limited in the absence of noise.

III. Conclusions

The capacity When PSK-n signals transmit through the considered channel with a limited bandwidth at values n less than 30 is complex with an explicit maximum. For large values of« the capacity is dropped. Capacity of considered channel is defined by equation (55) when the number discrete states tends to infinity

References

1. Mord vi no v. A.E. (2008). Possibility qf Increasing the frequency efficiency of communication tines due to the use of signals with mutual interference of symbols: dissertation [Issledovanie vozmozknasii povysheniia chdstotnoi effektivnosti liaii sviazi za scher ispol'zovaniia signalov s vzaimnoi interferentsiei simvolov: dis. ... kand. tech. mink}. Moscow, MEI (TU), 150 p.

2. Proakis, J.G. (2000). Digital Communications, 4th ed.. McGraw-Hill, N.Y., 92 & p.

3. Qureshi. S.. Newhall E. (1973). Adaptive Receiver for Data Transmission over Time-dispersive Channels. IEEE Trans, No. 3, pp. 448-457.

4. Falconcr, D.D., Magee, F.R. (1973). Adaptive Channel Memory' Truncation for Maximum Likelihood Sequence Estimation". BSTJ. No. 9, pp. 1541-1562.

5. Clark, A .P., Harvey, .I.D., Driseoll, J.P. (1978 ). Nearmaximum Likelihood Detection Process for Distored Digital Signals. The Radio and Electronic Engineer. No.ii, pp.301-309.

6. Clark, A.P.. Najdi, N.Y.. Fairfield. F.J. (1983). Data transmission at 19.2 kbits/s over telephone circuits. The Radio and Electronic Engineer, No. 4, pp.157-166.

7. Vachula, G.M., Hill, F.S. (1981). On Optimal Detection of Band-limited PAM Signals with Excess Bandwith. IEEE Trans, No.6, pp.886-890.

8. Weselowski, k. (1987). Ail Efficient DFE ML Suboptimal Receiver Ibr Data Transmission Over Dispersive Channels using Two-Dimensional Signal Constellation. IEEE Trans, No. 3, pp.536-339.

9. Evtyanov, S.l. (1948), Transient processes in the receiver-amplifier circuits [ Perehodnye process}- r priemno-usiliteTnyh shemah] Syjaz'izdat, Moscow. 221 p,

10. Shannon, C. E. (1948). A mathematical theory of com mini ¡cation. Beil Syst. Tech. J., No. 3, pp. .379-423.

11. Lerner, I.M, IT in, G.l. (2012). The Analysis of the Transient Process Caused by a Jump in the Amplitude and Phase of Radio Pulse at the Input of Narrowband Linear System. Journal of Communications Technology and Electronics. No.2, pp. 174-1S8.

— n

АНАЛИТИЧЕСКАЯ ОЦЕНКА ПРОПУСКНОЙ СПОСОБНОСТИ КАНАЛА СВЯЗИ С ЧАСТОТНОЙ ХАРАКТЕРИСТИКОЙ РЕЗОНАНСНОГО ФИЛЬТРА ПРИ НАЛИЧИИ МЕЖСИМВОЛЬНЫХ ИСКАЖЕНИЙ И ИСПОЛЬЗОВАНИИ МНОГОПОЗИЦИОННОГО ФАЗОМАНИПУЛИРОВАННОГО СИГНАЛА

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

Казань, Россия, aviap@mail.ru

Дннотация

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. Поскольку увеличение алфавита символов канала приводит к необходимости увеличения отношения сигнал/шум для достижения заданного качества приема сигнала, целесообразно рассмотреть возможность использование алгоритмов приема на основе использования частотной фильтрации и последующее использование многопорогового решающего устройства в ситуации сильных межсимвольных искажений, вызванных избирательными системами в канале связи. вроизведено определение пропускной такого канала связи с частотной характеристикой фильтра, соответствующей резонансному усилителю. Оценка произведена аналитическим способом путем ряда математических доказательств Выявлены новые свойства.

Ключевые слова: пропускная способность, резонансный фильтр, ФМн-п-сигналы, повышение частотной эффективности.

Литература

1. Мордвинов А.Е Исследование возможности повышения частотной эффективности линий связи за счет использования сигналов с взаимной интерференцией символов: дис. канд. техн. наук (051204) дата защ. 12.10.08, дата утв. 21.05.09. МЭИ(ТУ). 150 с.

2. Proakis J.G. Digital Communications, 4th ed., N.Y.: McGraw-Hill, 2000, 928 p.

3. Qureshi S., Newhall E. Adaptive Receiver for Data Transmission over Time-dispersive Channels // IEEE Trans, 1973, No. 3, pp. 448-457.

4. Falconer D.D., Magee F.R. Adaptive Channel Memory Truncation for Maximum Likelihood Sequence Estimation // BSTJ, 1973, No. 9, pp. 1541-1562.

5. Clark A.P., Harvey J. D., Driscoll, J.P. Nearmaximum Likelihood Detection Process for Distored Digital Signals // The Radio and Electronic Engineer, 1978, No.6, pp. 301-309.

6. Clark A.P., Najdi N.Y., Fairfield, F.J. Data transmission at 19.2 kbits/s over telephone circuits // The Radio and Electronic Engineer, 1983, No. 4, pp. 157-166.

7. Vachula G.M., Hill, F.S. On Optimal Detection of Band-limited PAM Signals with Excess Bandwith // IEEE Trans, 1981, No.6, pp. 886-890.

8. Weselowski K. An Efficient DFE ML Suboptimal Receiver for Data Transmission Over Dispersive Channels using Two-Dimensional Signal Constellation // IEEE Trans, 1987, No. 3, pp. 336-339.

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

10. Shannon C.E. A Mathematical Theory of Communication // Bell System Technical Journal, 1948, Vol. 27, pp. 379-423, 623-656.

11. Лернер И.М. Ильин Г.И. Анализ переходного процесса, вызванного скачком амплитуды и фазы радиоимпульса на входе узкополосной линейной системы // Радиотехника и электроника, 2012, №2. С. 192-206.

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