Application of multi-harmonic measuring signals for control of the technical state of dynamic objects is considered. Implementation of the traditional approach to the evaluation of the amplitude-frequency spectrum characteristics of dynamic objects, based on the application of generators of sinusoidal signal, requires a considerable complexity of measurements. It is necessary to set sequentially the control frequency of the object, technical state of which is monitored. Elimination of this shortcoming can be performed by application of multi-harmonic measuring signals. It is proved that the multi-harmonic signals with complex law of modulation of pulse duration possess the greatest functional capabilities for controlling the spectral composition, when the obtaining the necessary spectrum of the signal is achieved by changing the moments of switching its levels. The purpose of the paper is to obtain an analytical relation that binds the spectrum of the amplitudes of the harmonic components of a multi-harmonic measuring signal with a set of its switching moments. The methodology of synthesis of multi-harmonic measuring signals parameters is proposed. Quadrature formulas are obtained for determining the amplitude spectrum of a multi-harmonic signal with arbitrary law of modulating its pulses duration. In order to generalize the resulting equations, we have proposed an universal formula for computing the lower limit of the summation index for even or odd number of switch points. The article presents and solves the problem of finding an analytical apparatus that binds the characteristics of the amplitude-frequency spectrum of a multi-harmonic signal inherent in an arbitrary law of pulse duration modulation, with a set of values of switching points. The final equation for computing the amplitude of a multi-harmonic measuring signal is obtained. The forms of multi-harmonic signal with an even and an odd number of switching points are given.

[1] Yu. Pavlenko, S. Slavinsky, “Questions of metrological support of spectrum analyzers”, *Ukr. Metrolog. Journ.*, iss.3, p.35-42, 1999.

[2] S. Herasimov, Yu. Shapran, M. Stakhova, “Measures of efficiency of dimensional control under technical state designation of radio-technical facilities”, *Information Processing Systems*, iss.4(152), p.148-154, 2018.

[3] V. Chinkov, Yu. Krichtin, “An analysis of the current state and perspective directions of synthesis of optimal multi-harmonic signals with a normalized spectrum for the control of the technical condition of samples of weapons and military equipment”, *Information Processing Systems*, iss.5(21), p.214-217, 2002.

[4] A. Bratslavska, S. Herasimov, H. Zubrytskyi, A. Tymochko, A. Timochko, “Theoretical basic concepts for formation of the criteria for measurement signals synthesis optimality for control of complex radio engineering systems technical status”, *Information Processing Systems*, iss.5(151), p.151-157, 2017.

[5] S. Herasimov, O. Timochko, S. Khmelevskiy, “Synthesis method of the optimum structure of the procedure for the control of the technical status of complex systems and complexes”, *Bull. Sc. Papers Kharkiv Nat. Air Force Univ.*, iss.4(53), p.148-152, 2017.

[6] E. Kudryavtsev, *Mathcad 2000 Pro*. Moscow, RF: DMC Press, 2001.

[7] I. Honorovsky, M. Demin, *Radiotechnical circuits and signals*. Moscow, RF: Radio and communications, 1994.

[8] S. Baskakov, *Radiotechnical circuits and signals*. Moscow, RF: Higher school, 2000.

[9] V. Zadiraka, *Theory of calculating the Fourier transformation*. Kyiv, Ukraine: Scientific thought, 1983.

[10] W. Gander, W. Gautschi, “Adaptive Quadrature – Revisited”, BIT, vol.40, p.84-101, 2000.

[11] I. Bronshtein, K. Semendyaev, *Handbook of Mathematics*, Moscow, RF: Science, 1981.