S.B. Makarov – Dr.Sc.(Eng.), Professor, 

Higher School of Applied Physics and Space Technologies of Peter The Great St. Petersburg Polytechnic University E-mail: makarov@cee.spbstu.ru

A.M. Markov – Leading Engineer, 

Peter The Great St. Petersburg Polytechnic University E-mail: markov@cee.spbstu.ru

The paper considers binary spectrally efficient signals with phase modulation for various values of phase deviation, duration and shape of the phase pulse v(t) = qsinp(πt/nT) in the interval [0, nT], where T is the transmission duration of a bit. The time and spectral characteristics of such signals sequences were analyzed, including in conditions of intersymbol interference that occurs when n = 2, 3… For the evaluation of the BER performance in the presence of additive Gaussian white noise, we use the coherent reception algorithm constructed on the basis of the Viterbi algorithm. It was performed the parametric optimization of the phase pulse parameters p and q at which the maximum BER performance is achieved in the area of error probability equals about perr ≈ 10−3 and BER curves were obtained. It is shown that using the Viterbi algorithm allows to obtain an energy gain up to 1.8 dB for n = 2 and up to 3.5 dB for n = 5 with respect to opposite signals for the area of the error probabilityperr ≈ 10−3…10−4.

The paper considers signals formed on the basis of phase pulses of the shape Sinpx. In contrast to frequency shift keying with a continuous phase (CP-FSK), an integrator is not used in the modulation of studied signals. The first part of the work is devoted to the consideration of the time and spectral characteristics of the studied signals, the second part considers the signal reception algorithm by the Viterbi algorithm, the third part considers the bit error rate of signals and analyzes it.

Various values of the phase pulse duration, the degree of the pulse and the phase deviation have been considered in the paper. If the duration of the phase pulse exceeds the bit duration, then there is intersymbol interference in the signal sequence. A feature of the studied signals is the presence, besides a continuous, a discrete component in the power spectral density. The bandwidth of studied signals becomes less than it is for classical binary phase-shifted keying with a rectangular envelope.

As a coherent reception algorithm in a channel with additive Gaussian white noise, we considered the algorithm implemented as the Viterbi algorithm. The Viterbi algorithm consists in a recursive search of the optimal bit sequence by maximum likelihood sequence estimation (MLSE). At each step of the algorithm, the number of bit combinations in first decreases by 2 times, due to the exclusion of bit combinations for which it is defined that they cannot provide the maximum value of the likelihood function, and then increases by 2 times by adding two opposite bits to each combination. Then, among all the remaining combinations, the best one is selected. By simulation it was found that with an increase in the duration of the phase pulse compared to the bit duration, the BER performance of studied signals becomes better than for opposite signals. The optimal values of the degree of the phase pulse and the coefficient proportional to the phase deviation, which provides the minimum BER have been found. If a phase pulse duration equal to the bit duration, the rectangular pulse is the optimal pulse shape and the BER performance is equal to the BER performance of opposite signals.

To explain the increase in BER performance Euclidean distance between signals differing by one bit has been analyzed. It was found that the maximum Euclidean distance is achieved for a rectangular phase pulse of duration nT. However, during the simulation, it was found that the optimal pulse is different from a rectangular shape, which is possibly associated with an increase in the probability of a double bit error with a decrease in the degree of the phase pulse.

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2. Jeremiah F. Hayes. The Vitervi Algorithm Applied to Digital Data Transmission. IEEE Communications Magazine. 1975. V. 13. № 2.
3. Makarov S.B., Markov A.M. Priem v «tselom» sluchainykh posledovatelnostei chastotno-manipulirovannykh signalov s mezhsimvolnoi interferentsiei. Radiotekhnika. 2011. № 3. S. 46−51. (in Russian)
4. Prokis Dzh. Tsifrovaya svyaz: Per. s angl.. Pod red. D.D. Klovskogo. M.: Radio i svyaz. 2000. (in Russian)
5. Antoshkin A.V., Azarov A.I. Potentsialnaya pomekhoustoichivost chastotno-modulirovannykh signalov s nepreryvnoi fazoi. Izvestiya Tulskogo GU. Tekhnicheskie nauki. 2018. № 7. S. 420−427. (in Russian)
6. Makarov S.B. Artamonov A.A., Kosukhin I.L. Spektralnye kharakteristiki sluchainoi posledovatelnosti zavisimykh FM signalov s ogibayushchei, opisyvaemoi polinomom n-i stepeni. Tekhnika sredstv svyazi. Ser. Tekhnika radiosvyazi. 1990. № 8.
7. Makarov S.B. Protsess gruppirovaniya oshibok v demodulyatorakh s obratnoi svyazyu po resheniyu. Sb. statei «Pomekhoustoichivost i effektivnost sistem peredachi informatsii». Odessa. 1986. (in Russian)
8. Makarov S.B., Kryachko M.A., Smirnov S.M. Primenenie spektralno-effektivnykh signalov na osnove atomarnykh funktsii v zadache EMSRES. Nauchno-tekhnicheskie vedomosti SPbGPU. 2008. № 2. (in Russian)