CC BY
0physical and mathematical sciences
Sytnik Oleg
DOI: 10.24412/2520-6990-2023-21180-20-25 O.Ya. Usikov Institute for Radio Physics and Electronics, National Academy of Sciences of Ukraine, 12
Academician Proskura St., Kharkov 61085, Ukraine.
COMPARISON THE EFFICIENCY OF THE TECHNIQUES OF SEQUENTIAL AND PARALLEL DETECTION OF MERSENNE CODE IN RADAR FOR RESCUERS
Abstract
The subject and purpose of the work. The subject of the study are sequential and parallel systems for detecting and identifying elements of Mersenne pseudo-noise code sequences, which are used as a modulating function for the rescue radar signal. The purpose of the work is to analyze the potential capabilities of sequential and parallel detection systems and to estimate the upper error limits of the probability of correctly identifying the symbols of the modulating sequence under conditions ofmultipath propagation and dispersion of the medium and with signal amplitude fluctuations.
Method and methodology. The main method is based on the fact that a generally recognized measure of the quality of work of both radar and communication systems is the probability of an error in identifying the information parameters of signals. The methodology for analyzing the efficiency of sequential andparallel systems is based on a comparison of the upper bounds of the error probabilitiesfor identifying symbols of a Mersenne code sequence with identical noise and characteristics of the signal propagation medium.
Results. Man-made or natural catastrophes like collapsed buildings caused by gas explosion or earthquake, mines or quarries accidents, snow avalanches, mudflows, sand and soil landslides, cause numerous human casualties. People under the rubble often remain alive for a period of several hours to several days. Therefore, it is very important to create portable devices for fast remote detection ofpeople behind optically opaque barriers. The class of short-range radars is one of the varieties of such devices. But effectiveness of detection depends on the methods of signal processing. The detailed comparative analysis of the sequential andparallel modes of processing is made. The upper bounds of the error probabilitiesfor identifying symbols of a Mersenne code sequence with identical noise and characteristics of the signal propagation medium has been found. It was shown that the symbol-by-symbol detection and code identification technique in terms of error probability is not inferior to parallel one.
Conclusion. It has been shown that the error probability in the suboptimal symbol-by-symbol detection scheme with decisions based on all received signals leads to a probability of error very close to that found for sequential systems. The formulas for error probability estimate has been derived that allows defining the energy gain likely to be given by sequential systems using the symbol-by-symbol detection scheme in conjunction with decisions based on all received signals, in comparison with parallel systems. The analysis of suboptimal signal processing systems has shown that the energy loss to an incomplete utilization of the energy spread between the signals in the various paths can be appreciable in channels where the signals in the paths experience an insignificantfluctuation in amplitude.
Keywords: Upper Error Bound, Doppler Radar, Detection, Error Probability, Mersenne Code, Sequential Systems, Parallel Systems, Pseudo Noise Code.
1. INTRODUCTION.
Man-made or natural catastrophes like collapsed buildings caused by gas explosion or earthquake, mines or quarries accidents, snow avalanches, mudflows, sand and soil landslides, cause numerous human casualties. People under the rubble often remain alive for a period of several hours to several days. Therefore, it is very important to create portable devices for fast remote detection of people behind optically opaque barriers. The class of short-range radars [1-5] is one of the varieties of such devices. The target for such a kind radar is the human body which is covered by optically opaque obstacles. Information signs that distinguish a living person from surrounding objects are the movement of the chest during breathing and heartbeat. It is these movements that lead to the appearance of phase-frequency modulation in the reflected signal of the radar, due to the Doppler effect. The Doppler frequency shift is related to the frequency of the carrier oscillation by the known dependence [6]
Jd ~ q Jn, (1)
where Vr - is the radial, with respect to the phase center of the radar antenna system, component of the velocity of the object; C - is the speed of propagation of the probing signal in the medium; f - is the carrier frequency.
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The frequency caused by the breathing process is within 0.1...0.2 Hz and heartbeat gives Doppler frequency shifts approximately 0.9...1.2 Hz [7, 8]. Moreover, the radar can detect alive human beings at a very short distance approximately from 0.1 meter to long distance 10 meters. At the same time, between the radar's antenna and target can be suited to obstacles like brick or concrete walls, doors and some other. The obstacle material absorbs the energy of the probing signal, and due to dispersion, the speed of the high-frequency components of the signal spectrum differs from the speed of the low-frequency components, which makes it difficult to use pulsed and video pulse radar's techniques.
The technical requirements for such radars are quite stringent [9, 10]. At the same time, radars with continuous pseudo noise modulation are capable of solving the detection problem It is efficient to use sequences based on the Mersenne code [11, 12] as a modulating function. Continuous modulation with pseudo-noise signals such as Mersenne codes, although it allows you to build radars for rescuers, is also not without drawbacks, which are caused by the appearance of errors in the identification of code symbols of the sequence caused by noise and interference. Errors in the code structure during the signal receiving, which are caused by interference, destroy the correlation function, and as a result, reduce the characteristics of the radar in detecting targets.
It is well known that optimal detection on the basis of symbol-by-symbol estimation [13] over an observation interval in presence of inter symbol interference secures a nearby ultimate error probability performance for sequential systems in most cases of practical interest. But error probability performance of sequential systems, like pseudo noise codes [14], using in account the inter symbol interference cannot be achieved through classical equalizers, it can only be done on the basis of optimal receivers. This can significantly complicate signal processing. The symbol-by-symbol technique has been discussed in [13]. But it is interesting to compare the efficiency of the sequential and parallel techniques for detecting code parcels under noise for rescuer radar purposes. The main focus of this work is aimed at trying to find an upper bound on error probability codes parcels detection in sequential and parallel digital processing probing signals modulated by Mersenne code.
2. CRITERIA EFFECTIVENESS OF SYSTEM PERFORMANCE.
A commonly accepted measure of performance for any radar system is error probability [15]. To find the upper bound on error probability need to determine the error of detection of the average elementary pulse of code sequentially, using various techniques in multipath fading medium First of all, need to compare binary synchronous systems of the sequential and parallel types for error probability performance under following conditions :
a) the elementary pulses of Mersenne code are followed at a constant signaling rate:
V = 1/T = n / Tpar, Tpar = nT , (2)
where T and Tpar - are the sampling intervals in sequential and parallel systems, respectively, and n - is a number of frequency channels in parallel system;
b) the average transmitter power P0 remains unchanged in both types of systems;
c) both systems use a two-path channel in which the individual channels are independent and slowly fading and have generalized Gaussian statistics, with random phase shift between the regular path code components;
d) the time delay At spread between the paths is such that
At< T. (3)
The relations existing at At = T characterize the error probability performance of sequential systems, and at At > T , under optimal signal detection. If (3) is satisfied, it sets the limit of intersymbol interference in sequential systems at two adjacent pulses. For rescuer radar systems, using Mersenne code modulation of probing signals, the time delay spread between the path is usually sufficiently small. So as for parallel systems
At<< Tpar;
e) the channel is subject to additive normal white noise with an energy spectrum Gn ;
f) the systems under comparison use the same length Mersenne code parcels and optimal according to criterion maximum likelihood receivers.
The generalized form of the transmitted signal is
^ (t ) = A (t) cos (a0t + t)) h (t), (4)
the received signal may be expressed as
t) = /j A (t) cos [®01 + t ) + ^] h (t ) + y2 A (t -At) cos [®01 + p(t -At) + ^2] h (t -At) ,(5)
where
f 1 for 0 < x < T h(t) = \ w |0 forT < x < 0 ,
Yk - is gain of the k-th path deemed constant during the observation interval, (pk - is phase shift associated with k -th path, also deemed constant during the observation interval.
The energy of the signal (5) can be expressed as
E = E0 [/f + /2 + 2ÏIÏ2 C0S (ft - ft )«
(6)
where E0 = - f A2 (t) dt, 2 *
and
f A ( t ) A ( t-r) dt
a = tf (Ar) = -< 1.
V ' 2E
For a parallel system (where At << T ), the correlation coefficient is a — R (A t) ~ 1, while for sequential systems it may be very nearly zero. For a rectangle envelope of elementary code pulse, a — 1 — At / T . In the assumption that bit time T is set at 0.1 of maximum value, for a Gaussian envelope a can be written as
exp
a = ■
-0.5 (Ar/ T )2 O[ 3(1 -Ar/ T )]]
3)
(7)
where - is the probability integral [16].
For a binary sequential system utilizing the symbol-by-symbol detection scheme and an observation interval Tobs = (B +1) T , the minimum attainable probability of error (the ultimate performance) is defined as
iJKi^n)],
p = 0.5
1 -O(
(8)
Tobs
where Eeq = J ^(t) — s2 (t)] dt, Gn - is the energy spectrum of additive noise in the channel.
0
For bipolar signals such that s1 (t) = ¿2 (t) = s (t) the equation (8) can be rewritten as
p = 0.5
1 -O
1/ GH f s'2 (t) dt
J
J
(9)
or, recalling (5) and assuming the channel characteristics to be precisely known, the probability of error may be defined as
p = 0.5
1 -O
(yj 2h0 [/ +72 +2?^ cos (ft -ft )(1 -Ar/T )] )
(10)
where h = E0 / Gn = P0T / Gn, P0 - is the average signal power of the transmitter.
3. AN UPPER BOUND ON ERROR PROBABILITY.
Assuming optimal detection, a sequential system and no guard spaces between information symbols, the result (10) is obtained in conjunction with ideal decision feedback (with a probability close to unity). Given the existing SNRs and channel variation rates, practical designs come very close to receivers with such feedback. It is to be
stressed that when the average transmitter power P0 and the overall signaling speed are held constant, E0 = P0T can also define the energy ofeach transmitted signal code element in each n frequency channels in parallel system, because an increase of n times in pulse duration (in comparison with a sequential system) entails a decrease by the same factor in average signal power. Thus, equation (10) gives a measure of performance for parallel system
as well, assuming optimal coherent detection subject to the condition Ar << Tpar. Using an integral representation for the Kramp function [16]
1 œ 1
05 [1 -O( J )]=ï f 1+7 exP
f
1 +1
2
dt
(11)
equation (10) can be rewritten as 1 r 1
p = _fr^2exP
tT{ 1 +1
-h02 (1 +12 ) (/ + r22 ) - 2h2 (1 +12 ) a cos (ft -ft ) /1/2 ] dt. (12)
Applying a generalized Gaussian model, it should first examine generalized Rayleigh fading in the case where the asymmetry factor of the orthogonal components is
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2023 / PHYSICAL AND MATHEMATICAL SCIENCES 23
P2 =°ll = 1. (13)
In such a model, the phase difference = is uniformly distributed, and by averaging (12) over W,
it can be got
1 r 1 r r -n
-h02 (1 +t2)(/ +/22)I0 [2h20a(l +12)riy2 J dt, (14)
ir i
P eXP
^ I + t
where I0 [•] - is the Bessel function [16]. The joint probability density W2 (y1,Y2 ) can be written as
W2 (ylty2) = ^exp[—/ / y2f — /f / / — qf — qf][Io (2q/ / $)Io (2^2/2 / )j, (15)
where qf = (/fg k / yj/f ) (k = 1,2) - is the ratio of average powers of regular (index reg) and fluctuating
(indexf) parts of the signal in the k-th path respectively.
After averaging (14) over / and /2 and some evaluations it can be get following equation for probability
-hf qf (1 +12)[ 1 + h2f (1 +12)(1 — a)2 — qfhf (1 +12)][1 + h2f (1 +12)(1 — a2)j
p = — i exp
I + 2h2 (i +12) + hf (i +12)2 (i-a2)
10 2 qqahf (i +12 )
i + 2h2 (i +12) + hf (i +12)2 (i-a2)
(i+1 2 ) i + 2h2 (i +12) + hf (i +12)2 (i-a2 )]
dt
(16)
where hf = /f h0 - is the average ratio between the energy of the fluctuating part of the signal in any path
and the noise power spectral density.
Equation (14) will follow from equation (16) as a special case under following conditions:
h2f = 2 /1 = /reg,l, /2 = /reg,2 and
E{h2} = hf (1 + qf), E{h2} = hf (1 + qf ), S2 = E{h2}/ E{hf }.
For Rayleigh fading and for S2 = 1, it can be written
dt
P =
1 iTT J i
X 0 (i +12) E{h2}(i +12)2 (i-a2) + 2E{h2}(i +12) +1
For sequential systems under condition
E{hf }Vi-a2 »i
formula (17) will have got a view
i
dt
E{hf }(1 — a2(1 +12)3 ,
where E {•} - is a mathematical expectation operator.
In a sequential systems, the overlap of waves in the various paths (a > 0 ) entails a loss of energy
r=1/V1—a
(17)
(18)
(19)
a
For parallel systems ( a = i ) under condition E {hf } >> i from (17) it follows that
2 CO
ii2W i
dt
2E{hf }xJ0 (i +12)
(20)
(21)
From comparison of (19) and (21) it can be noted that the energy gains likely to arise from sequential systems over parallel in the case of Rayleigh channel is
x
2
TJ = -
a
2V3P
(22)
A plot of r(p) defined by (22) for a = 0 is given in Fig. 1
FIG. 1: Maximum energy gain of a sequential system in comparison with a parallel system, as a function of the permissible error probability: 1 - q2 = q\ S2 = 1; 2 - q2 = ql = 0, 01 = 0; 3 -
q2 = q22 = 0, p = 1.
The gain increases as the requirement for performance become more stringent. If the fluctuation of signals in the various paths is insignificant (q1 >> 1 and q2 >> 1), integration of (14) by the Laplace method for sequential systems at E [h } + E {h2}>> 1 and a = 0 yields
E\hf }(\+52
P
fiTEft}(ïTF) ■
(23)
However, for parallel systems ( C = 1 ), for the same signal intensity in the various paths ( §2 = 1 ) and given the same conditions, the probability error is
1
P
tt^I2TE {h2} '
(24)
Under the defined conditions ( qx >> \, q2 >> \ and ß2 = \ ), the energy gain r of sequential systems is connected to the permissible probability error, p , by the relation
:^(T3/2)rln (tV4) )\
P =
(25)
A plot of r (p) defined by (25) is likewise given in Fig.1.
CONCLUSION
Thus, the analysis of error probability has shown that a combination of suboptimal symbol-by-symbol detection scheme with decisions based on all received signals (see [13]) leads to a probability of error very close to that found for sequential systems. The relations derived above fully define the energy gain likely to be given by sequential systems using the symbol-by-symbol detection scheme in conjunction with decisions based on all received signals, in comparison with parallel systems. Through a combination decisions based on all received signals which are Mersenne code combinations during the observation interval and a decision in favor of the first symbol in the sampled code combination, it is possible, under conditions of ideal feedback, to synthesize a linear correlative receiver which
secures a nearly ultimate error probability performance in sequential systems. It can clearly be seen from the comparison of (23) and (24). The energy loss to an incomplete utilization of the energy spread between the signals in the various paths can be appreciable in channels where the signals in the paths experience an insignificant fluctuation in amplitude. It is interesting to note, that even under the conditions, with the signals in the various paths having the same intensity, the sequential symbol-by-symbol receiver gives a greater energy gain in comparison with parallel systems.
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