The results of signal peculiarities' investigation of an autodyne short-range radar system with linear frequency modulation for the moving reflecting object are described. The autodyne signal plots are obtained for the cases when its period duration is much longer than delay time of the reflected emission as well as for the cases when this inequality is not satisfied. Experimental data are obtained on the 8mm-range Gunn-diode oscillator with frequency tuning by the varicap.
A system of linearized (in the vicinity of a steady-state) differential equations for small relative variations of the amplitude and the frequency for a single-circuit oscillator being under impact of the proper reflected emission has a ( ) = 0 + Δ ( ) = 0[1 + ( ) = 0[1 + ( )] = 0[1 + are the loaded Q-factor and the cavity frequency, relatively; , , and are dimensionless parameters defining the oscillator increment slope, non-isodromity and non-isochronity, relatively; Γ( , ) = Γ0[ ( , )/ ( )], ( , ) = Ψ( ) − Ψ( , ) are the modulus and the phase of instantaneous reflection factor presented to the oscillator output; Γ0 is the emission damping factor in amplitude at its propagation to the object and back; ( , ), Ψ( , ) are the oscillation amplitude and phase from the system pre-history ( − ); = / , are efficiency and the external Q-factor of the oscillating system; ( ), Ψ( ) are the current amplitude and phase at t moment; = 2 / , where l is a distance to the reflecting object, and c is the velocity of emission propagation.
At FM of the oscillator by the variation of the vericap bias voltage, both frequency and the “patrasitic” amplitude modulation (PAM) if emission occur: where relatively; Δ oscillation modulation; = Δ / 0 and = Δ
/ 0 are coefficients of frequency (FM) and amplitude (AM) modulation, , Δ are maximal deviations of amplitude and phase from theirк steady-state values 0 and 0 due to
( ) is the normalized modulating function. The quasi-periodic solution of the first
approximation [20] of equation system (
[ ( , ) − ]
( , ) = 0{1 −
relatively.
where , are coefficients of autodyne amplification and the frequency deviation, relatively;
=
( ), =
( ) are angles of phase shift; = / , = / are coefficients of oscillator non-isodromity and non-isochronity,
To obtain expressions describing an autodyne response in general case of arbitrary ratio of delay time of reflected
emission and the period , we use the known approach to analysis of retarded systems, which was developed for the
case of signal description of usual autodynes (without FM) in [18]. Its essence consists in expansion of functions ( , )
and Ψ( , ) of delayed impact in the Tailor series over small parameter – the delay time with respect to the current time
. St that, we assume an absence of functions (
Γ( , ) = Γ0{1 + 2 }Γ ∑
=0(−1) ( )
, ) − + Θ ( )]
(
, Θ ( ) = − ( +1)
Then, we take into consideration some features of FM SRR functioning. At frequency control with the help of the varicap, the PAM deep is usually small. In addition, we may neglect autodyne amplitude variations Γ0 : Γ0 ≪ 1.
Therefore, in expression (
= 0. The account of series terms of higher orders in these expressions allows, as shown below, taking into consideration also the phase variation (
, ) dynamics of reflected emission.
In this connection, expressions for normalized (with respect to maximal values) variations of amplitude and frequency in the form of autodyne amplitude (AAC) ( , ) and frequency (AFC) (
, ) have a view: ( , ) = 2 + where = / is the parameter of normalized (with respect to autodyne signal period) distance to the reflecting object;
= Δ is the parameter of “FM base”, which defines a number of signal periods keeping within a period of the
modulating function at fixed reflecting object; Δ
= Δ
/2 ;
is the feedback parameter of the autodyne
system; = Ω
/2 is the normalized time of the modulating function; Ω
is the modulation frequency; ( ),
Θ ( ) are amplitude values and phase shifts of series m-th terms (
= 0,1, … ) in expressions (
, ) = ( , ) = ( , )
Γ0 ( , ) Γ0 = = −
, ) − ]
( ).
(
∑ =0(−1) ( )
[ (
, )(
∑ =0(−1) ( ) [ ( , )( ) + θ + where the approximation order is denoted by indices in parenthesis near terms ( , ); ( , )(0,1,… ) = 2 + Zero approximation, when we take into account in (12) the first term only ( , )0 of the sum, corresponds to the linear phase characteristics, which is typical for the homodyne SRR. The next approximations insert nonlinearity in this function, which is the attribute of the autodyne systems caused by autodyne frequency variations. Therefore, the main attention in fulfilled research will be paid to revealing signal features of FM ASRR under conditions when the parameter is commensurable with the unity.
Then, taking into consideration the movement of reflecting object in autodyne characteristics (
, )(0), we use substitution: = 0 + ( ). As a result, we obtain: ( )(0) = 2 0 + 2 ( ) + where 0 = 0 0/2 is the initial value of normalized distance; = (Ω /Ω ) is a parameter of “Doppler signal base” showing how much periods of the Doppler signal is kept within the modulation period; Ω id the Doppler frequency; = (Δ
/ 0) is the relative value of frequency deviation under modulation; ( ) is the alternate component of the normalized distance caused by the reflecting object movement.
Now, functions (
Signals of FM ASRR are usually registered in the power source circuit of the UHF oscillator (the auto-detection signal)
or by detection of oscillation amplitude variations. Both oscillation amplitude variations in the form of AAC ( ) and
the current autodyne frequency variations in the form of AFC ( ) make a contribution into these signal formation. For
clarification of autodyne signal formation features from the moving object, we perform calculations and an analysis of
FM ASRR signals for the non-symmetric saw-tooth law. Mathematical expression for this law takes a form:
(14)
(15)
The time diagrams of APC ( ), their time-derivatives
(curves 2) calculated according (
( ) = (1000/2 )sin(2 /1000)
Diagrams presented in Figures 1 and 2 here and later, if there is no special dtupulation, are obtained at = 1, = 0.2, = 5, = 0; = 0.8, =
= 50. Hereinafter, we shall call the segments of normalized distance , which
are multiplies to integers, as corresponding to the “operation zones” starting from the first where 0 ≤ ≤ 1. We note,
that numbers M of series terms in (
≤ 0.98.
From comparison of temporal and spectral diagrams (see Figures 1 and 2) we see that the sign changing of the radial velocity causes the signal frequency variation only by the Doppler shift value. For the arriving object, the signal frequency in this case decreases while for departing it increases.
From the analysis of temporal and spectral diagrams in Figures 1,а,e and 2,, we see that results obtained for the case = 0 correspond to research results of FM ASRR signal investigation, which were resulted in publications [10, 12]. In these publications, the autodyne response formation features were considered in first approximation only, when inequality ≪ is satisfied. At that, it was shown that the phase incursion ( ) irregularity of the reflected wave is caused by frequency variations ( ).
Anharmonic distortions of the signal characteristics manifesting in “wave slope” variations depend on the parameter value, the character (increase or decrease) of modulating function, and on the oscillator internal parameters. The speed of phase incursion changing
( )/ , characterizing as the instantaneous frequency difference of radiate and reflected oscillations, has, at that, the oscillating character with the peak formation of instantaneous frequency (see Figures 1,a,e). The peak height increases with the growth of parameter, which, in turn, depends also on the oscillator internal parameters and on the level of the reflected emission level. It is proved that average values of the instantaneous frequency during modulation period are equal to signal frequency in homodyne FM ASRR under condition of inequality < 1 fulfillment. c) g) d) h) a) e) a) b) f) b)
In the considered case (arbitrary ratio of and ), with the growth of normalized distance , as it is seen from curves in Figures 1,b–d and 1,f–h, the shape of the temporal APC ( ) approaches to the linear function. At that, the peak height of normalized derivative ( )/ , the degree of anharmoniс distortions of AAC and AFC diagrams essentially decreases. Especially, this tendency is clear in the first operation zone where 0 ≤ ≤ 1. Calculations of characteristics for other values of normalized distance show that in the case of values, which are multiplies to the integer number ( = 1,2, … ), AFC and AAC have practically sine view.
From the spectrograms presented in Figure 2,c,d we can see that in the case considered, with the growth of normalized distance the higher harmonics level sharply decreases. The harmonic coefficient THD is the generalized parameter, which characterizes the distortion degree of quasi-periodic oscillations. Calculation results of these coefficients are presented in Figure 3 in the form of ( ) for different values of . We took into account in these plots calculation the amplitudes of the first ten harmonic components of the Fourier series expansion. ( ) versus normalized distance calculated for and region of small values of the first operation zone. For = 0.8, the harmonic coefficient in this zone achieves approximately 40%. With further transition to higher-order operation zones, as we mentioned above, signal distortions significantly decrease achieving minimal values when normalized distance is a multiple of the integer ( = 1,2, … ).
Returning to analysis of calculation results presented in Figure 2, it is necessary to note the presence of DC components 0( ) and 0( ) in spectrograms in Figures 2,c,b,d. The presence of DС component 0( ) in the oscillator response on frequency variation ( ) in definite operation zones indicate some offset of frequency average value under the impact of reflected signal. These components deserve special attention while studying FM ASRR signals. Their account may be necessary, for instance, at signal processing, during the analysis of system noise immunity etc.
Calculation results of relative levels of DС components 0( ) and 0( ) depending on the normalized distance values at = 0.8 are presented in Figure 4 for autodyne responses ( ) and ( ).
From obtained curves in Figure 4, one can see that in the first operation zone of FM SRR for large values of parameter , when the
~1, the DC level may have values, which are commensurable with amplitudes of the signal.
Inside the higher-order zones, we can neglect by the DC component influence. We should also note that when the inequality
≪ 1 is true, DC components in the FM ASRR output signals are practically eliminated.
( ) = (
= ±1 for different normalized distance to the reflecting object. The signal spectrograms presented in Figure 6 were calculated for the same cases and normalized distances .
From comparison of temporal and spectral diagrams of Figures 5 and 6 it follows that when changing the radial velocity sign the average value of signal frequency over modulation period does not change. In the case of arriving object, the autodyne response frequency, on the ascending part of modulation characteristic, decreases by the Doppler shift value while on the descending part – it increases by the same value. Motion direction change leads to changing of described order to the reverse one.
At that, we should note that the growth of signal frequency in one half-period and frequency reduction in the other half-period in the case of the moving reflector and change of this order at changing of the motion direction (see Figure 5) are in full correspondence with homodyne radar systems theory [9]. This phenomenon is widely used at SRR signal processing and for range information extraction, the object velocity and its motion direction.
Difference with the FM ASRR consists in presence of its anharmonic distortions under condition, when ~1, and arrival of the additional phase shift caused by dynamics of autodyne variations of oscillation frequency. At that, signal spectrum “sprawling” (see Figure 6,a–d) takes place similar to the spectrum of frequency-shift keyed oscillation [21]. Here, higher harmonics and components of their combinative interaction are present as well as low-frequency components on modulation frequency corresponding to the frequency shift from the moving reflecting object and DС component. Figure 5: Time diagrams of formation of APC ( ) their derivatives ( )/ , AAC ( ) (curves 1) and AFC ( ) (curves 2), calculated for cases of the arriving (a–d) and departing (e–h) reflecting object at different normalized distances: = 0 ( ), = 0.5 ( ), = 1 ( ) and = 1.5 ( )
From spectrograms presented in Figure 6,b–d, we see that in the case under consideration, the level of higher harmonic components essentially decreases at growth of the normalized distance . THD calculation results versus normalized distance are presented in Figure 7 in the form of ( ) for various values of .
From the obtained plots in Figure 7, one can see that the largest FM ASRR signal distortions for symmetrical law are observed in the middle part of the first operation zone. For = 0.8, the harmonic coefficient achieves here the value of 40%. With further transition to operation zones of higher order, as it was mentioned above, signal distortions essentially decrease.
Returning to spectrograms in Figure 6, it is necessary to note also the DC presence of 0( ) and 0( ). Calculation results of relative levels of DC components 0( ) and 0( ) depending on normalized distance value at = 0.8 for autodyne responses ( ) and ( ) are presented in Figure 8. From comparison of obtained curves in Figures 4 and 8, it is seen that they practically coincide.
Thus, peculiarities of FM ASRR signal formation for symmetric saw-tooth law are described in the general case of arbitrary ration of the delay time of the reflected emission and the signal period . 5
To explain the character of obtained functions of normalized distance we introduce concepts of the equivalent
feedback parameter Ceq of the autodyne system “FM ASRR – the reflecting object” and an angle of dynamic
displacement of the phase ΔδDD of the autodyne response: Ceq = CFBKDF and ΔδDD = δ(rn) − δ(rn = 0). Here KDF =
∑Mm=0(−1)mXm(rn) is the “dynamic factor” of feedback included into (
From the curve KDF(rn) (see Figure 9,а) one can see that with growth of normalized distance rn in the first operation zone where 0 < rn < 1, the KDF value and relatively the value of equivalent parameter Ceq decreases almost by the order. Further, with growth of rn, the value of Ceq asymptotically damps with slight increase in the middle parts of higher operation zones where rn > 1.
Dynamic phase variations ΔδDD of signal characteristics influencing on character of their distortions, which is well seen from comparison of curves in Figures 1,а,b; 1,e,f; 5,а,b, and 5,e,f, are also small at rn increase in the first zone. At rn = 1, there is the first minimum of this value, which is determined by the θ parameter characterizing the oscillator non-isochronity (see Figure 9,b). With the further growth of rn, the asymptotic damping of “oscillations” in dynamic phase variations ΔδDD is also observed.
Results obtained here seem to be contradictory to usual insides [12] and have the entirely explainable physical sense. To understand it, it is enough to turn to the simplified model of interaction process of the autodyne oscillator with the proper reflected emission, which was considered in [22] by means of a stepping method using an example of radio-pulse, autodyne. From this model, it follows that with shortening of the relative pulse duration (which is equivalent to growth of normalized distance), a number of partial reflections decreases during radio pulse action. This leads to reduction of the equivalent feedback parameter Ceq of autodyne system and, accordingly, to decrease of the signal distortion level. At achievement of the normalized distance equaled to (rn = 1), when the reflected emission impact becomes one-partial, the formation of practically sine autodyne variations of amplitude and frequency are provided. 6
Experimental investigations of FM ASRR signal peculiarities were preformed with oscillating module on the 8mm-range Gunn diode in the autodyne sensor structure developed for monitoring of the point crossing occupation on the hump yard [4–6].
Figure 10 shows the total view of the sensor and its main components. An antenna with an UHF module and the printed board for probing signal formation, digital signal processing and coupling interface RS-485 are placed into cylindrical encapsulated case from duralumin by 92mm-diameter and 170mm-length. The dielectric lens of the antenna is simultaneously the forward wall of the case.
a) b) c)
The autodyne UHF module of the sensor is performed on the base of the Gunn diode АА727А and the varicap 3А637А-6 in the packaged fabrication. The output power is 25 mW, the central frequency is 36.5 GHz. The UHF module provides in the linear FM mode the frequency deviation in the 500 MHz band. The FM law is symmetric or nonsymmetric saw-tooth with modulation frequency of 10 kHz.
The autodyne signal is extracted in the power source circuit of the Gunn diode with the help of the wideband current transformer. After filtering and amplification, it passes to ADC of the digital signal processor (DSP) TMS320F2808 from Texas Instruments in the processing unit. In this unit, functions of modulating voltage formation, digital signal processing and the communication interface RS-485 with PC are realized with software.
The computing kernel of DSP performs the digital signal processing (filtering and the Fourier analysis), which results are transferred through the universal asynchronous transceiver to PC used as a mapping system of radar information. When the reflecting object is situated in the ASRR antenna pattern, then at output of the one of spectrum analyzer’s filters of DSP the signal will be present. Its amplitude characterizes the reflecting capability of the object under investigation, while a filter number, in which the useful signal arrives, characterizes the distance from ASRR to the object.
To exclude from the received signal spectrum of components associated with PAM and reflections from the near objects the information not interesting for us, before the Fourier transform in DSP we use signal pre-filtering. For this, the algorithm of “moving average” is used to the initial signal, and this algorithm serves as the low-pass filter. After extraction of low-frequency components by this filter, they are subtracted from the initial signal, and as a result, the “dead zone” is formed near ASRR of one-meter-order distance. The high-frequency signal components remaining at the filter output contain information about reflecting objects being in the required distance range.
Figure 11 shows signal spectrograms obtained from the arriving (а), fixed (b) and departing (c) reflecting object if we use in ASRR non-symmetric saw-tooth FM law. For mentioned cases Figure 12 presents the spectrograms for symmetric FM law. The electrical-mechanical imitator of the Doppler signal [23] was used as the reflecting object.
In the first group of experiments, the distance from ASRR to the imitator was 3 m (see left spectrograms in Figures 11 and 12), while in the second group – 75 м (see right spectrograms in Figures 11 and 12). In the first experiments, we attach the horn antennas to the UHF module and the imitator with the pattern width 100×100 degrees for the -3dB-level and with gain about 25 dB. In the second group we used double-mirror antennas with the elliptical small mirror with the pattern width 1.4×1.4 degrees and with gain about 42 dB. Signal amplitude (Us) in both experiments were aligned with the help of the variable attenuator inserting between the UHF module and the antenna. At that, the attenuator damping was set in such a manner to provide ≈ 0.8.
Motion direction of the reflecting object was changed by switching of the rotation direction of electric motor, which shaft is connected to the reflector in the Doppler signal imitator [23]. In the case of a fixed object, the power was switched-off from the motor and reflector was stopped.
The first experiment on the ratio of the delay time of the reflected signal and the period in both cases of used modulation laws corresponds to the beginning of the first operation zone rn ≪ 1. In the second experiment for the nonsymmetric FM law, the normalized distance is rn = 1.25, and rn = 2.5 for the symmetric law.
At utilization of the saw-tooth non-symmetric FM law, the moving reflector causes the spectrum shift towards frequency decrease when object arriving while frequency increase when departing from ASRR (see Figure 11). At utilization of the saw-tooth symmetric FM law, the moving reflector causes spectrum “splitting” into two “side” components, which are shifted from the average frequency value by Doppler shift (see Figure 12). These statements are agreed with the general theory of FM ASRR operation [9].
In the first experiment (see left diagrams in Figures 11, 12) the noticeable level of the second and third higher harmonics are seen, which is typical for anharmonic distortions of the autodyne signal. In the second experiment (see right diagrams) higher harmonic level is essentially lower than in the previous case. These results are new; they confirm the above-mentioned conclusions of the theoretical examination.
Thus, experimental data obtained confirm the adequacy of the above-developed mathematical model for analysis and calculations of signal and spectral characteristics of FM ASRRs. 7
Accordingly, the peculiarities of ASRR signal formation for linear FM obtained from the moving reflecting object are considered in this paper. The general case of arbitrary ratio of delay time of the reflected emission and the signal period is described.
Fulfilled calculations and experimental investigations of FM ASRR show that the autodyne signal frequency both for moving and fixed reflector correspond exactly to the transformed signal frequency which is obtained in the case of the homodyne system [9, 12]. Nevertheless, the presence of autodyne signal distortions and spectrum enrichment under conditions when the distance to the reflected object is small and when > 0 , the inequality ≪ is satisfied and require to take these into consideration in the devices of their processing.
Research results obtained of the autodyne characteristics for the case, when the autodyne response period is commensurable or even smaller than the delay time of the reflected emission, show that the distortion level of the autodyne signals decreases with the reflected object distance growth at hypothetical keeping of the reflected wave amplitude. From this, it follows that under condition when the feedback parameter value, which sometimes exceeds of its boundary value, in higher operation zones of FM ASRR, the stable oscillator operation with formation of quasi-harmonic signals can be ensured.
Analysis results obtained in this paper develop and expand the known research results performed in [22, 24], in account of oscillator FM as well as the results fulfilled in [10–12] in account of dynamics of reflected emission phase variation in the autodyne model at wave propagation to the object and back. Regulations revealed of signal formation in FM ASRR have the sufficiently general character and clear physical interpretation on the base of the stepping method known from the theory of retarded systems. In this connection, the results obtained, as we think, can be also used at calculation of autodyne system signals, which are made on the base of semiconductor laser modules [13–15]. Additional confirmation of developed model adequacy of FM ASRR signals is the successful functioning of some systems created on the base of 8mm-range Gunn diodes [4–6].
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