Homodyne detection of short-range Doppler radar using a forced oscillator model

This article presents the homodyne detection in a self-oscillation system, which represented by a short-range radar (SRR) circuit, that is analysed using a multi-time forced oscillator (MTFO) model. The MTFO model is based on a forced oscillation perspective with the signal and system theory, a second-order differential equation, and the multiple time variable technique. This model can also apply to analyse the homodyne phenomenon in a difference kind of the oscillation system under same method such as the self-oscillation system, and the natural oscillation system with external forced. In a free oscillation system, which forced by the external source is represented by a pendulum with an oscillating support experiment, and a modified Colpitts oscillator circuit in the UHF band with input as a Doppler signal is a representative of self-oscillation system. The MTFO model is verified with the experimental result, which well in line with the theoretical analysis.


Result
MTFO model. For in-depth understanding, a forced-input oscillation shall be explained using this signal and system first with a conventional analysis and then with the MTFO model.
In general, the explanation of an oscillator circuit under a single tone sinusoidal forced input situation can be accomplished using a second order differential equation to find a natural response x n (t) in Eq. (1). While a forced response is named a sinusoidal steady state response x f (t) shown in Eq. (2). A complete response x o (t) is composed of both natural and forced responses according to Eq. (3).
where A n is the natural response's amplitude, which depends upon the initial condition of the system X 0 and the forced response at initial time x f (t 0 ); ω n is the natural frequency generated from the oscillator; Y 0 is the magnitude of the input fed into the system; ω f is the input frequency; |H(ω f )| is the magnitude response of the transfer function of an oscillator H(ω f ); and ∠ H(ω f ) is the phase response of the oscillator's transfer function. From the complete response in Eq. (3), the amplitude of the natural response is affected by the forced response at the initial time only, which is in contrast with experimental results. Recently, T. Maneechukate et al. used a second-order differential equation with the multiple time variable technique to solve a variety of forced oscillator problems using a single model 47,48,[52][53][54][55][56][57] , the MTFO model. The results of their experiment confirmed that the forced response had an effect on the amplitude of the natural response at any time. To simplify analysis, we set t 0 = 0, the amplitude of the natural response in MTFO model can be shown as Eq. (4).
Scientific RepoRts | 7:43680 | DOI: 10.1038/srep43680 Then the complete response based on MTFO model can be shown as Eq. (5) where τ is an arbitrary time variable, we can also interpret τ ≡ t − t 0 as a time delay, with t 0 is the initial time of the forced oscillation system. We can note that the term of τ ω x t ( )cos( ) f n from (5), is the occurrence for nonlinearity of the system. Homodyne detection in a forced oscillation system. From our perspective, the Doppler shift signal reflected from a moving target as the input into the oscillator is shown in Eq. (6), and the resulting forced response of the second-order oscillator system can be found as shown in Eq. (7). Substituting this value into Eq. (5), one will obtain an absolute response of the second-order oscillator as shown in Eq. (8).
where ρ 0 is magnitude of the reflected Doppler, and ω d is the Doppler frequency.
To make it easy to understand, it is therefore assumed that t 0 = 0, and then τ ≡ t. When expanding Eq. (8), one can rewrite a new complete response as shown in Eq. (9).
It can be observed that the first term of Eq. (9) is the explicit Doppler frequency. By passing this absolute response through a low-pass filter, one can recover the Doppler signal without using an AM demodulator.
From the above, a simple mathematical analysis with an obvious homodyne detection in the arbitrary forced oscillation system has been shown, and the MTFO model was confirmed by a different structure of the forced oscillation system in T. Maneechukate et al.'s articles. The interesting experiment from their article that has inspired us was the pendulum with an oscillating support experiment 54 , where homodyne behaviour in the mechanical oscillation system occurred which based on the MTFO model. We redrew the experiment and results from their article in Fig. 1. While the forced was switched-off, the pendulum system was a free oscillation system, which their natural response ϕ n (t) can be explained by Eq. (11). By define g is the gravity acceleration constant, and l is the length of light rod. The natural frequency of pendulum system can be obtained as ω by ω = g l 0 , their amplitude would decay respect to time, due to damping constant β d . We can note that this system is not a self-oscillation system, due to it can oscillate without the external source. After we apply the external forced ϕ τ ωτ = B ( ) cos( ) i i 0 , the forced response φ f (τ) was found as Eq. (12).
From the results, the direct conversion achieved by forcing the pendulum with a sinusoidal signal is based on the MTFO model. These results implied that there is intrinsic homodyne for an arbitrary oscillation system; therefore, the MTFO model can also be used for specific structures, such as a modified Colpitts oscillator. This movable support is directly connected to the cart's pillar, and the body of the cart is connected to the centre of the speaker, which is controlled by a signal generator. Thus, the output signal from the speaker is the external input ϕ i (t) that is injected through the support of the pendulum system. The light rod's position ϕ c (t, τ) is measured by a Hall effect motion sensor, UGN3503, that is placed near the circular flat magnet; the sensor is connected through an oscilloscope, and the FFT (fast Fourier transform) function is used to display the output in the frequency domain. Moreover, the pendulum swings less than 5 degrees to avoid nonlinear behaviour. (b) A natural frequency spectrum at 1.5 Hz, with no forced input from the oscillating support. (c) Complete response spectra of the forced pendulum system, which is forced by a sinusoidal wave at 1.1 Hz.
Scientific RepoRts | 7:43680 | DOI: 10.1038/srep43680 Modified Colpitts oscillator analysis using the MTFO model. In this section, more potential for using MTFO has been shown. The complicated of many difficult equation is raised due to more complex system analysed by basic dynamic law with multi-time variable technique only, otherwise the second order within higher order differential can be shown in this section, then the MTFO model still can be applied to analysed system. Then a second-order oscillator can be used for homodyne detection, as demonstrated in this article using a modified Colpitts oscillator in Fig. 2.
To prove homodyne detection in the modified Colpitts oscillator, which is based on the MTFO model, one can analyse the system by starting from law of dynamic of the circuit such as KCL (Kirchhoff current law), when there is still no Doppler reflected from the target, and find the natural frequency response. Here, a high-frequency small-signal equivalent circuit 58 of the circuit in Fig. 2 is shown in Fig. 3, where C π , r π C μ , and r μ are intrinsic capacitor and resistor which depend on active BJT device. When using Miller's theorem 58 to separate C π and r π to be Z μi and Z μo , one can rewrite a new equivalent circuit as shown in Fig. 4. After applying KCL (Kirchhoff current law) 59,60 into each node and using Laplace transform, the following relationships can be found as: where v ic (s), v ib (s), and v ie (s) are the collector-, the base-, and the emitter-fed input node respectively. The BJT ports node for the collector, the base, and the emitter are represented as v c (s), v b (s), and v e (s). And v o (s) is the output node. And the iedance for Z 1 , Z 2 , Z 3 , Z 4 , and Z 5 can be found as:  where a 6 , a 5 , a 4 , a 3 , a 2 , a 1 , and a 0 are simplify parameters (see Method). After solving Eq. (23), the natural frequency is derived as in Eq. (24). 2  3  1  3   2  3  4  1  3  4  1 A suitable point where one should feed the reflected Doppler is not necessarily through a system feedback path, the output of the circuit. For an oscillator under forced oscillation, an analysis can be performed when an input signal is injected into either the emitter, the collector, or the base of the transistor, while the output can be measured at the Collector. In this article, an experiment and analysis had been undertaken for all cases where the reflected Doppler has been injected into the emitter, the collector and the base of the transistor.
As the reflected Doppler has been injected into the base circuit of the transistor of the oscillator, the analysis must be carried out on the transfer function of the system. By solving node equations, the relationships between the base, emitter, or collector inputs and the output are shown in Eqs (25) to (27), respectively.  After substituting the impedance value into the Eqs (25) to (27) and taking some approximation by omitting some insignificant terms, the magnitude and phase response of the system can be found as shown in Eqs (29) to (31).
( )((( ( ) (2 ))) ( (2 )) ) tan (2 ) 4 ( ( ) (2 )) 4 (2 ) Subsequently, the forced response of the system for the base-, emitter-, and collector-fed circuits can be found by substituting Eqs (29) to (31) into Eq. (6) as follows: From the above analysis, a complete response for the base-, emitter-, and collector-fed circuits has been established as shown below: From the above analysis of the forced response, it can be concluded that the reflected signal can be fed into either the collector-, the base-, or the emitter-fed circuits of a transistor. Additionally, when substituting into the absolute response, the result will be similar. Only the magnitudes and phases will differ; however, in all cases, the first term obtained will be the Doppler frequency component. From these, we can apply more than one Doppler shifted signal to circuit, and we still also can classify the point of fed circuit by magnitudes and phases response of system. Then, we can apply this advantage to improve more application in our future work, such as define multiple target in three different direction. In next section, these system analyses have been investigated by our experiment.
Short-range Doppler radar experiment. From the previous section, we verify the principle by an experiment using a modified Colpitts oscillator that was established from the schematic in Fig. 2. While there is not any Doppler reflected back, the spectrum of the output signal from the oscillator measured with a spectrum analyser set at a normalized impedance value of 50 Ω is shown in Fig. 5. The signal obtained is a sinusoidal signal with a fundamental frequency of 2.76 GHz.
Next, a reflected Doppler was fed into the oscillator circuit, and all possibilities, namely the emitter, the collector, and the base, were used to receive the input. From our experimental observation, the sensitivity of this modified Collpitts oscillator, without add on processing process is − 57.13 dB with range of bandwidth 40 kHz. Thus, an imitating Doppler-shifted signal from an RF generator equivalent to a signal reflected from a target with a relative velocity of 3 Mach at a distance of 5 m is used (equivalent to the distance in an idealistic experiment of 10 m), which is approximately equal to 20 kHz. From such a distance, the free-space path loss attenuation is calculated as − 61.3 dB.
When feeding the Doppler-shifted signal into the base-, emitter-, and collector-fed circuits, the resulting output from the collector of the transistor appears in the radar frequency spectrum as an AM signal which similar to autodyne signal as shown in Fig. 6(a,c,e), respectively. The output spectrums of the Doppler are as shown in Fig. 6(b,d,f), respectively. The results presented in Fig. 6 show that the oscillator sensitivity essentially depends upon the feed-point of the Doppler-shifted signal. The more noteworthy from Fig. 6 is the low amplitude of Doppler signal, which compare with the amplitude of natural response and the Doppler shifted signal amplitude. Then the Doppler signal hard for investigate directly in time domain, but it still exists in frequency domain observation.

Discussion
This article has been presented a new method of homodyne detection by using the MTFO model which based on the forced oscillation perspective, where can applied in a Doppler SRR. The analysis is based on signal and system theory, a second-order differential equation, and the multi-time variable technique. The mathematical technique of MTFO is uncomplicated and can also be applied to self-oscillation system such as electrical oscillator or arbitrary structure of oscillator. This imply that the forced oscillating behavior is the intrinsic behavior for every kind of the oscillator, even self-oscillation system. Then we also can apply to another different homodyne detection in the future, such as optical device. Otherwise, the nonlinear model still be an appropriate choice for analyse in-depth characteristic in self-oscillation system. From results, the output of natural oscillation system with external forced still has some difference from self-oscillation result. The AM signal or autodyne signal which occur from the output in radar frequency band is not appear in a forced pendulum's result, and this autodyne signal can be obtained from SAE method only. Although, the MTFO model is not suitable for profound analysis, but it can explain the different kind of oscillation system under single principle, then the MTFO model is the other interesting choice to analyse primary phenomenon, homodyne detection, in all of oscillation system. This phenomenon has been observed in the free oscillation system which forced by external source such as pendulum with an oscillating support experiment, and same behaviour in the self-oscillation system, which has been confirmed experimentally using a RF generator feeding a simulating reflected signal into a modified Colpitts oscillator, which generates a 2.76 GHz output. The experimental results show that the Doppler-shifted signal can be fed into any inputs of a transistor and not strictly into the feedback path of circuit. However, the Doppler output will be most prominent, and hence the system response will be highest when feeding a reflected signal into the collector of the transistor, followed by base feeding and then emitter feeding. Furthermore, the result is in line with the analysis, which confirms that there is a Doppler frequency component in the lower side of the spectrum and that the forced input can be applied into any terminal of a transistor. Moreover, the difference of magnitude and phase response for each fed circuits, the directional finding in three direction can be applied in the future.
Experimental Setup. For the explicit application proposed, a real situation was examined when a proximity fuze was 5 m away from a flat rectangle target (area ≈ 0.328 m 2 ) with a relativistic velocity of 3 Mach. All loss budgets in this short-range RADAR situation were calculated using the RADAR equation (Eq. (38)). Due to the transmission frequency being close to the reflection frequency, we then approximate them to be equal. We establish the idealistic equivalent by using an RF generator with a monopole antenna of 5.19 dBi located 10 m away from the circuit to gather the transmission and reflection distance, as shown in Fig. 7. The power of the generated RF signal (or Doppler-shifted signal) equals the output power compensated by the antenna gain for − 5.19 dBc. where P t is the transmit power, P r is the receive power, R is the distance between the proximity fuze and the target, λ is the wavelength of the carrier signal, G is the antenna gain, and σ is the RADAR cross section of the target. From (34), given R is 10 m, we can calculate σ ≈ 115.14 m 2 , and the Free Space Path Loss attenuation is calculated as − 61.29 dB. In the oscillator circuit, an equivalent input with a simulated Doppler shift is fed through the antenna connected to the impedance-matching circuit, which is attached to the selected transistor port. For the other terminals, dummy loads were connected with impedance-matching circuits as shown in Fig. 8. Base feeding (c) (d)

Emitter feeding
Collector feeding (e) (f) Figure 6. Output spectrum of the modified Colpitts oscillator. Output spectrum in a natural frequency band and a Doppler frequency band when feeding Doppler-shifted signal into the base (the spectra are shown in (a,b)), the emitter (the spectra are shown in (c,d)), and the collector (the spectra are shown in (e,f)) of the transistor.