—A compact Self-Injection Locked Oscillator based Doppler Radar (SILO-Radar) at 5 GHz has been studied, analytically and numerically. The SILO-Radar con- sists of just a cross-coupled oscillator and a Schottky diode baseband detector, with the help of the Hilbert transform at the baseband to extract the Doppler phase information. Both analytical analysis and PSpice simulation have been performed to verify the validity of the SILO-Radar. The compact, low-power and low-cost SILO-Radar has many potential consumer applications such as auto-driving radars, Unmanned Aerial Vehicle (UA V) navigation radars, remote heartbeat/respiration healthcare biomedical radars and so on. Index Terms —Self-Injection Locked Oscillator (SILO), Doppler radar, cross-coupled oscillator. I. I NTRODUCTION Making use of the remote capability of the electro- magnetic waves from the Radio-Frequency (RF) wave to the high-frequency optics wave [1]- [63], Doppler sensors such as RF/microwave/millimeter-wave radars [1]- [10] and high-frequency Lidars have been widely used in many important areas such as police car-speeding radars, auto- driving radars, Unmanned Aerial Vehicles (UA Vs) navi- gation radars and remote heartbeat/respiration healthcare biomedical radars [7]- [10] etc. In particular, the long wavelength of the RF wave makes it ideal for long-range Doppler radar applications. Conventional Doppler radars are either based on broad- band pulse time-domain technique or Frequency Modu- lated Continuous Wave (FMCW) frequency-domain tech- nique, which makes the radar architectures very complex and expensive. So it is desirable to develop a compact cost-effective consumer Doppler radars. Self-Injection Locked Oscillator based radars (SILO-Radar) are such Doppler radar architectures [2]. In this paper, we propose a compact SILO-Radar architecture at 5 GHz that consists *Shaolin Liao is the corresponding author. The work is partly supported by China Postdoctoral Science Founda- tion No. 2020M672488. This preprint was submitted to 2020 Asia-Paciﬁc Microwave Con- ference (APMC2020), which will be held in Hong Kong, SAR, P. R. China, December 8-11, 2020.of only a cross-coupled oscillator and a Schottky diode baseband detector, via the help of Hilbert transform. II. T HESILO-R ADAR Fig. 1 shows the architecture of the SILO-Radar: it con- sists of a V oltage Controlled Oscillator a cross-coupled oscillator, which launches an outgoing wave through a patch antenna towards the Doppler object (a car here); then the outgoing wave is reﬂected by the Doppler object and is injected to the cross-coupled oscillator, changing the phase of the cross-couple oscillator (s=j!)as follows [2], (s) =!LRcos (!osc D) s+!LRcos (!osc D) (1es D) D(s);(1) where D(s)is the Laplace transform of the Doppler phase; Dis the round trip delay time between the antenna and the Doppler object; !oscis the angular frequency of the free running cross coupled oscillator; and !LRis the locking range of cross coupled oscillator. The real part of the Doppler modulated phase signal r(t)is then detected by a time-delay ( ) differential operator with a Schottky diode, r(t) = cosf!osc + (t+ ) (t)g; (2) Then the detected real part signal r(t)is sampled by an Analog-to-Digital Converter (ADC); and ﬁnally, a digital processor (e.g., an FPGA) is used to perform the Hilbert transform [64] to obtain the imaginary part of the Doppler modulated phase signal, i(t) =1 t~r(t); (3) where~denotes the convolution. Now the Doppler modulated phase can be obtained from Eq. (2) and Eq. (3), !osc + (t) (t ) = arctani(t) r(t) ; (4) 1 Fig. 1. Schematic of the SILO-Radar. Fig. 2. The SILO circuit under study based on a cross coupled oscillator. which can be solved through Laplace transform as fol- lows, (s) =Ln arctann i(t) r(t)o !osc o 1es : (5) Finally, the Doppler phase can be obtained by substi- tuting Eq. (5) into Eq. (1), D(s) =H(s)L arctani(t) r(t) !osc  ; (6) H(s) =s+!LRcos (!osc D) (1es D) !LRcos (!osc D) (1es ): III. A NALYSIS AND SIMULATION OF THE SILO-R ADAR In this Section, we show the theoretical analysis of the cross coupled oscillator and the PSpice simulation of the SILO, whose circuit is shown in Fig. 2. A. The Cross-Coupled Oscillator The transfer function of the cross-coupled oscillator is given by, Hosc(s) =sgmL 1 +sL(1AL) RH+s2LC; (7) Fig. 3. The analytical transfer function of the cross-coupled oscillator: left) magnitude (dB); and right) phase (degree). Fig. 4. The PSpice simulation of the transfer function of the cross- coupled oscillator at points A and B of Fig. 2: left) magnitude (dB); and right) phase (degree). wheregmis the trans-conductance of the transistor of the cross-coupled oscillator; AL=fgmRHis the feedback amplitude, with fbeing the feedback coefﬁcient; L,C andRHare the inductance, the capacitance and the resistance of the half circuit. The two poles of the transfer function Hoscof Eq. (7) are given by, s1;2=1gmRH 2RHCjs 1 LC1AL 2RHC2 : (8) From Eq. (8), it can be seen that the start-up condition for the cross coupled oscillator is given by, gm> RH; (9) 2 a) b) Fig. 5. Doppler information recovered by the Hilbert transform: a) the recovered imaginary part sinf (t)g; and b) the recovered Doppler velocity. where 1is required and usually 3is used to ensure the start-up of the cross coupled oscillator. In particular, when = 1, the resonant frequency is calculated as follows according to Eq. (8), !1;2=!osc=r 1 LC: (10) On Fig. 3, we have plotted the magnitude and the phase of the transfer function given in Eq. (7), for the parameters of L1=L2=L= 3 nH,C1=C=20.169 pF,R4= 2RH= 10000 Ohm, andgm= 0:6mS. From Fig. 3, it can be seen that when the feedback amplitude AL=fgmRHapproaches 1, i.e., the feedback coefﬁcient fapproaches 1=(gmRH), Full Width at Half Maximum (FWHM) the resonant peak becomes narrower, meaning higher Q factor, which is given by Q= 5 GHz/FWHM. B. PSpice Simulation of the SILO We also performed the PSpice simulation of the SILO circuit shown in Fig. 2. The magnitude and phase plots for the output points of A and B of Fig. 2 have been shown in Fig. 4. Compared with the analytical transfer function of the cross coupled oscillator shown on Fig. 3, it can be seen that both the magnitude and phase behave similarly around the design frequency of 5 GHz. C. Doppler Phase Recovered by the Hilbert Transform To show the validity of Doppler phase recovery by the Hilbert transform, we have performed Hilbert transform for a car moving at the following velocity (in unit of km/hour), v(t) = 60 +t2 20+t3 6000: (11) Fig. 5a) shows the imaginary part of the Doppler phase sinf (t)gobtained from the Hilbert transform according to Eq. (3), and Fig. 5b) shows the recovered velocity. It can be seen that both the imaginary part of the Doppler phase and the recovered velocity agree well with their given original values, showing the feasibility of the Hilbert transform for Doppler information recovery.IV. C ONCLUSION In this paper, we have studied a compact 5-GHz SILO- Radar that consists of only a cross-coupled oscillator and a Schottky diode baseband detector. The Doppler phase is extracted through the Hilbert transform at the baseband. Both analytical analysis and PSpice simulation have been performed to show the feasibility of the SILO-Radar. The SILO-Radar has the beneﬁts of compactness, low-power consumption and low cost, enabling its use in many im- portant consumer applications such as autonomous cars, UA Vs, and remote heartbeat/respiration remote healthcare etc.