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On Application of Exact Mathematical Representation in Time-Frequency Plane to Pulse Doppler Radar Signal Processing


G.T. Avanesyan, A.A. Hakhoumian, A.H.Muzhikyan

Simultaneous precise determination of coordinates and velocity of interesting target is impossible with use of a single rectangular pulse, due to the uncertainty principle. Range (time delay) and velocity (Doppler frequency shift) of the target are determined by independent processing procedures. Typically, these are correlation to determine the delay, and Fourier transform to determine the frequency shift. In the present paper the possibility of simultaneous precise determination of these two parameters is shown, using a new asymptotically exact signal processing method in the time-frequency plane (TFP). The method is based on a unique waveform with strong exponential localization property in the TFP. It coincides with its own Fourier spectrum. The basis is generated by anti-commutative subgroups of Weyl-Heisenberg group. This approach overcomes the difficulties posed by the Balian-Low theorem. In pulse Doppler radar systems the received signal carries two types of changes in comparison with transmitted: it is delayed in time and shifted by frequency. In terms of TFP, the shifts in the plane correspond to the simultaneous transmission and modulation (the Weyl-Heisenberg group). An example for Gaussian signal with shift in the TFP is considered, which corresponds to actual shifts in time and frequency Δt = a/8 and Δf = b/6. Their product is far below the limit of the uncertainty principle. The problem of determining the shift of the signal in the TFP, corresponding to Δt and Δf, and, therefore, the range and velocity, is reduced to a simpler «recognition» problem on small number of lattice sites. A sample «recognition» method is to compare the expansion coefficients of received signal with ‘reference’ expansions, pre-calculated for the shifts of received signal in the TFP. The criterion of comparison for normalized signals is mean-square error (MSE). The ‘reference’ expansion with lowest MSE corresponds to the target range and velocity. Lattice dimensions, within which signal is processed, are taken to include the space of TFP area, where received signal can be located. This approach doesn’t require use of windowing as an additional tool. The number of expansion coefficients is easy to find from the required accuracy of Δt and Δf.
May 29, 2020

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