Peculiarities of signal formation of the autodyne short-range radar with linear frequency modulation

Research results of signal formation peculiarities in autodyne short-range radars with linear frequency modulation by non-symmetric saw-tooth law are presented. Calculation expressions for autodyne signals are obtained in the general case of arbitrary delay time of reflected emission. Temporal and spectral diagrams of autodyne signals are defined in the cases when its period duration is much more than the delay time of reflected emission, as well as for cases when this inequality is not satisfied. Experimental investigations are performed using the autodyne oscillator on the Gunn diode of 8mm-range, which is electronically controlled in frequency by the varicap.


Introduction
Autodyne short range radars with frequency modulation (FM ASRR) have found wide application as guard sensors, anti-collision sensors for transport vehicles, measuring devices for car motion in hump yards, detectors of point-work occupation and railway crossings and many others [1][2][3][4][5] due to construction simplicity and low cost of the transceiver device. Operation principle of these devices is based on the so-called "autodyne effect", which consists in oscillator parameter variations under influence of reflected emission [5]. These variations registration in the form of autodyne signals and their further processing provide a possibility to obtain information about reflecting object and its relative motion parameters.
A large number of publications is devoted to peculiarities of FM ASRR autodyne signal formation, in which different mathematical models and impact function representations for proper reflected emission are used.
In earlier publications, which were mainly issued in the period of autodyne theory coming into being, the impact function was represented by variable load, which varies with a frequency of the signal formed [6]. In later publications, this function was described by the equivalent current (or voltage) source having the relative frequency offset caused by emission FM and by reflected radio-signal delay [5,[7][8][9]. Within the limits of described approaches, the oscillation frequency and amplitude are changing with the differential frequency and the output signals become harmonic as a result of these variation detection.
In publications of the recent time, the presence of anharmonic distortions [10,11]  Then, on the base of this solution, we are going to 1 For example, in 8mm-range, in typical situation of 500 MHz frequency deviation [4], the frequency modulation by non-symmetric saw-tooth law equaled to 10 kHz and the distance to the object 75 m, the delay time is 0.5 · 10 −6 sec, and the autodyne signal period is equal to 0.4 · 10 -6 sec. examine peculiarities of autodyne response formation at different FM parameters and distances to the radar object for the non-symmetric saw-tooth FM law. Partially, this task were discussed in reports [12][13][14]. 1 Initial expressions for signal analysis A system of differential equations linearized in the vicinity of steady-state mode for small relative variations of the oscillation amplitude and frequency for singlecircuit oscillator, which is under impact of the proper reflected (from the radar object) emission, has a view [12,15]:   ( ), Θ ( ) are amplitude values and phase shifts ofth terms ( = 0, 1, . . . , ) of series in equations (7) and (8): Autodyne variations of oscillation amplitude Γ 0 are usually very small: Γ 0 << 1. Therefore, we can neglect by the second term in large parenthesis of (7). Such an approximation in the autodyne system mathematical model supposes an account the phase delay only of the reflected emission. At that, we note that known solutions for the autodyne response following from (7) - (9), are obtained in the first approximations assuming = 0. Account of series terms of higher order in these expressions, as it will shown later, allows taking into consideration also the dynamics of phase variation ( , ) at any ratio on delay time of reflected emission and the autodyne signal period а .
The zero approximation, when we take into consideration the first term only of the sum in (12) ( , ), corresponds to the linear phase characteristic, which is typical for homodyne SRR. The further approximations insert nonlinearity in the function, which is the autodyne system attribute. Therefore, the main attention below in research fulfilled will be focused on revealing of ASRR signal peculiarities under conditions, when the parameter is commensurable with 1. ( ) [11] for the non-symmetrical modulation law: We note that the number of series terms in (7), (8) and approximations of -order in (12), which were taken in calculations, ensure the convergence of calculation results in the ranges ≤ 5 and ≤ 0.98. Plots of AFC ( ) and AAC ( ) are presented in Figure 6 for the case of = 2.5 and = 5. As we see from plots in Figure 6, in spite of the fact that > 1, the autodyne response in higher operation zones of FM ASRR is smooth and has not jumps and breaks. At that, the distortion level of quasi-harmonic characteristics equals to 31%.
Results obtained here seem to be contradictory to the habitual representations [10,11] and have quite explainable physical sense. For its understanding, it is enough to take the simplified model of representation of interaction process of the autodyne with the proper reflected emission, which can be described by the steps method [17].