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Conditional bias of QL estimate of initial phase as a function of SNR

Estimation of the initial phase of narrowband radio signal with unknown amplitude and duration

Andrey P. Trifonov, Yu. E. Korchagin, M. V. Trifonov, K. S. Kalashnikov


The quasi-likelihood and maximum likelihood algorithms for estimating the initial phase of radio signal with arbitrary shaped envelope and unknown duration and amplitude have been synthesized. Schematic block diagrams of initial phase meters are proposed. Characteristics of the synthesized algorithms were determined, and errors of the initial phase estimates were compared. The structure of quasi-likelihood estimate of the initial phase is shown to be invariant with respect to the lack of knowledge of radio signal amplitude. However, a mismatch between the expected value of signal duration and its true value can lead to a marked increase of dispersion of the initial phase quasi-likelihood estimate. The loss in accuracy of the initial phase estimate due to a priori lack of knowledge of signal duration has been determined. Analytical expressions for statistical characteristics of maximum likelihood estimate of initial phase at large signal-to-noise ratios are shown to coincide asymptotically with the characteristics of maximum likelihood estimate of the initial phase of radio signal with a priori known amplitude and duration. Therefore, a priori unknown signal duration (asymptotically with an increase of signal-to-noise ratio) does not affect the accuracy of the maximum likelihood estimate of initial phase. The computer methods of statistical simulation were used to determine the applicability limits of asymptotic expressions for characteristics of the maximum likelihood estimate in respect of signals with linear and exponential envelopes.


maximum likelihood estimate; quasi-likelihood estimate; initial phase; duration; amplitude; bias; dispersion

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KULIKOV, E.I.; TRIFONOV, A.P. Estimation of Signal Parameters against the Background of Interferences [in Russian]. Moscow: Sov. Radio, 1978.

TIKHONOV, V.I. Optimal Signal Reception [in Russian]. Moscow: Radio i Svyaz’, 1983.

IVASCHENKO, P.V.; PEREKRESTOV, I.S. “Optimal estimates of phase carrier during demodulation of digital modulation signals,” Tsifrovye Tekhnologii, n.6, p.40-47, 2009. URI:

HARISOV, V.N.; BULAVSKY, N.T.; LUPINA, M.V. “Algorithm of delay estimation based on phase processing of multi-frequencies signals,” Radiotekhnika, n.7, p.80-83, 2006. URI:

ZAKHAROV, A.V. “Efficiency of estimating the radio signal phase in the presence of fast fading,” Vestnik VGU. Ser. Fizika, Matematika, n.2, p.221-228, 2010. URI:

GLADKIKH, N.D. “Assessing the impact of the expansion of the Doppler bandwidth for error estimation phase hydroacoustic (HA) signal,” Electronics and Communications, n.6, p.85-93, 2012. URI:

NOELS, N.; STEENDAM, H.; MOENECLAEY, M.; BRUNEEL, H. “Carrier phase and frequency estimation for pilot-symbol assisted transmission: bounds and algorithms,” IEEE Trans. Signal Process., v.53, n.12, p.4578-4587, 2005. DOI:

YANG, G.; WANG, J.; ZHANG, G.; SHAO, Q.; LI, S. “Joint estimation of timing and carrier phase offsets for MSK signals in alpha-stable noise,” IEEE Commun. Lett., v.22, n.1, p.89-92, 2018. DOI:

IVANOV, S.I.; LIOKUMOVICH, L.B.; MEDVEDEV, A.V. “Estimation of the parameters of the phase modulated signal in presence of the background noise using complete sufficient statistics,” Proc. of XX IEEE Int. Conf. on Soft Computing and Measurements, SCM, 24-26 May 2017, St. Petersburg, Russia. IEEE, 2017, p.11-13. DOI:

CHERNOYAROV, O.V.; GLUSHKOV, A.N.; LITVINENKO, V.P.; LITVINENKO, Y.V.; MATVEEV, B.V. “Fast digital algorithms for the coherent demodulation of the phase-shift keyed signals,” Proc. of Conf. on Dynamics of Systems, Mechanisms and Machines, Dynamics, 14-16 Nov. 2017, Omsk, Russia. IEEE, 2017, p.1-5. DOI:

TRIFONOV, A.P.; SHINAKOV, Y.S. Joint Discrimination of Signals and Estimation of their Parameters against the Background of Interferences [in Russian]. Moscow: Radio i Svyaz’, 1986.

TIKHONOV, V.I. Statistical Radio Engineering [in Russian]. Moscow: Radio i Svyaz’, 1982.

GRYAZNOV, M.I.; GUREVICH, M.L.; RYABININ, Y.A. Measurement of Pulse Parameters [in Russian]. Moscow: Radio i Svyaz’, 1991.

KORCHAGIN, Y.E. “Estimation duration of signal with unknown amplitude and phase,” Radiotekhnika, n.9, p.11-19, 2013. URI:

TRIFONOV, A.P.; KORCHAGIN, Y.E.; TRIFONOV, M.V.; CHERNOYAROV, O.V.; ARTEMENKO, A.A. “Amplitude estimate of the radio signal with unknown duration and initial phase,” Appl. Math. Sci., v.8, n.111, p.5517-5528, 2014. DOI:

TRIFONOV, A.P.; BUTEIKO, V.K. “Characteristics of joint estimates of signal parameters in conditions of partial regularity violations,” Radiotekh. Elektron., v.36, n.2, p.319-327, 1991.

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