Correction of IQ mismatch for a particle tracking radar
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RESEARCH PAPER
Correction of IQ mismatch for a particletracking radar
FELIX RECH AND KAI HUANG , For a better understanding of granular flow problems such as silo blockage, avalanche triggering, mixing and segrega-tion, it is essential to have a ‘microscopic’ view of individual particles. In order to cope with the difficulty arising fromthe opacity of granular materials, such as sands, powders and grains, a small scale bi-static radar system operating at GHz (X-band) was recently introduced to trace a sub-centimeter particle in three dimensions. Similar to a moving tar-get indicator radar, the relative movement of the tracer with respect to each of the three receiving antennae is obtainedvia comparing the phase shift of the electromagnetic wave traveling through the target area with an IQ-Mixer. From theazimuth and tilt angles of the receiving antennae obtained in the calibration, the target trajectory in a three-dimensionalCartesian system is reconstructed. Using a free-falling sphere as a test case, we discuss the accuracy of this radar systemand possible ways to enhance it by IQ mismatch corrections.
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I. INTRODUCTION
Since the beginning of last century, radar technology hasbeen continuously developed and benefiting us in many dif-ferent ways: From large scale surveillance radar systemsthat are crucial for aircraft safety and space exploration [1],to small scale systems for monitoring insects [2]. Consid-ering the limit of radar tracking technique, it is intuitiveto ask: How small can an object be accurately tracked bya radar system? Can it be as small as a tracer particlewith a size comparable to a grain of sand? If this chal-lenge can be solved to a satisfactory level, radar technologycan help us greatly in a better monitoring, understanding,and predicting granular flow problems that widely existin nature, industry and our daily lives, ranging from geo-science (e.g. landslide, debris flow), through chemical andcivil engineering (e.g. pile drilling), to space exploration(e.g. landing on an asteroid) [3, 4]. The reason behind isthat the mobility of single particles in a granular materialcan influence dramatically its collective behavior, owing toits discrete nature as well as heterogeneous distributions offorce chains inside [5].In the past decades, there have been substantial pro-gresses in imaging granular particles [6]. Due to the factthat most of the granular materials are opaque, optical Experimentalphysik V, Universität Bayreuth, 95440 Bayreuth, Germany Division of Natural and Applied Sciences, Duke Kunshan University, No. 8 DukeAvenue, Kunshan, Jiangsu, China 215316
Corresponding author:
Kai HuangEmail: [email protected] means for imaging particles in three dimensions (3D), suchas refractive index matching [7], are very limited. Instead,X-ray tomography [8] and Magnetic Resonance Imaging[9] have both been used to identify the internal structuresof granular materials. However, the limited time resolu-tion of scanning technique as well as the huge amountof data to be processed hinder the investigation of granu-lar dynamics. For the investigation of granular dynamics,alternative approaches including radar systems for tracerparticles have also been discussed [10].Recently, we introduced a small scale continuous-wave(CW) radar system working at X-band to track a spheri-cal object with a size down to mm [11]. In comparisonto other techniques, the continuous trajectory of a tracerparticle obtained by the radar system helps in decipheringgranular dynamics greatly. Here, we characterize the uncer-tainty of this system, discuss possible sources of error andways to improve the accuracy. II. EXPERIMENTAL SET-UP ANDTRACKING ALGORITHM
Figure 1 shows the block diagram of the radar system (a)together with a sketch of the set-up used to test the accuracyof reconstructed trajectories (b). The bi-static radar systemoperates at 10 GHz (X-band) with one transmission (Tx.)antenna pointing in the direction of gravity (defined as − z direction) and three receiving (Rx.) antennae mountedsymmetrically around the z axis. Polarized electromagnetic(EM) waves, after being scattered by the tracer particle, are GRANUL AR PART ICL E T RACKING WIT H RADAR
Powerdivider
Fig. 1.: Block diagram of the radar system (a) and a sketchshowing the configuration of the transmission (Tx.) andreceiving (Rx.) antennae as well as the tracer holdingdevice.captured by the Rx. antennae. With the help of an IQ mixer,which compares the phase shift between the Tx. and Rx.antennae, the change of the absolute traveling distance forthe i th antenna L i = l + l i can be obtained, where l and l i are the distance between Tx. Ant. and the target and thedistance from the target to the Rx. Ant, respectively. Subse-quently, a transformation matrix ~T is applied to the distancevector ~L = ( L , L , L ) to reconstruct the tracer trajectoryin a 3D Cartesian system. The smallest spherical objectbeing traceable by the system was found to be ∼ mm,in agreement with the prediction of radar equation. Moredetailed descriptions of the radar system can be found in[11].IQ-Mixers play an essential role in the accurate rangingof a target. The LO and RF inputs correspond to the sig-nal sent to the Tx. antenna [ a cos(2 πf t ) ] and that receivedby a Rx. antenna [ b cos(2 πf t + θ ) ], where a and b arethe magnitudes of the corresponding signals, f and f arethe transmitted and received signal frequencies. The outputsignals of the IQ-Mixer are I = ab π ( f − f ) t − θ ] ,Q = ab π ( f − f ) t − θ ] . (1)Subsequently, the relative movement of the tracer isobtained from the phase shift of I + Qi in a complex plane. If L i varies with a distance of one wavelength, thevector I + Qi rotates π . As IQ mixers provide analoguesignals that representing the mobility of the tracer, thetime resolution of the radar system is only limited by theanalogue-digital (AD) converter.Although distance measures rely only on the phaseinformation, the sensitivity and accuracy of the systemdepend on IQ signal strength. In order to have a sufficientsignal to noise ratio, the directions of the horn antennae(Dorado GH-90-20) are adjusted with the help of a laseralignment and range meter (Umarex GmbH, Laserliner)to face the target area. According to the specification ofthe antenna, the main lobe of its radiation pattern has anopening angle of ∼ degrees. Thus, we estimate thefield of ‘view’ of the radar system has a volume of about × × , taking into account the averageworking distance of the antennae. The distance betweeneach antenna and the center of the coordinate system is alsomeasured by the laser meter during the adjustment process.The polarization of the antennae are adjusted to maximizethe raw I and Q signals. The whole system is covered withmicrowave absorbers (Eccosorb AN-73) to reduce clutterand unwanted noises from the surrounding. In addition,the container for granular materials and the holder of thetracer are made of expanded polystyrene (EPS), which istransparent to EM waves.A metallic sphere with a diameter d =
10 mm is usedas the tracer. It is initially held by a thin thread wrappedaround and released by gently pulling the thread such thatthe initial velocity of the falling sphere is close to . Thisdesign enables a defined and reproducible initial conditionfor a comparison among various experimental runs. Theraw IQ signals from the AD converter (NI DAQPAD-6015)are recorded and further processed with a Matlab programto obtain the reconstructed trajectories. III. DATA ANALYSIS AND ERRORCORRECTION
As Fig. 2 shows, the raw IQ signals are typically not idealin the sense that the in-phase (I) and quadrature (Q) signalsare not always orthogonal with each other. This mismatchmay arise from the DC offsets of either I or Q signal, gainand phase imbalance. How to correct such kind of errorshas been extensively discussed in, for instance [12] or [13],particularly along with the development of telecommuni-cation and non-invasive motion detecting techniques [14].The distortions are typically attributed to device imperfec-tions as well as clutter. However, for the system being usedhere, there are additional errors arising from the mobilityof the tracer itself, which can not be readily corrected witha calibration of the hardware. Moreover, distortion mayalso arise from the interaction of the scattered signal fromthe tracer with that from static objects that are not com-pletely transparent to EM waves. In that case, the existenceof ‘mirrored’ particles may lead to additional uncertainty.
ADAR TRACKING FOR GRANULAR PARTICLES −1 −0.5 0 0.5 1−1−0.8−0.6−0.4−0.200.20.40.60.81 I (arb. unit) Q ( a r b . un it ) Corrected signals
Fig. 2.: Raw (continuous lines) and corrected (open sym-bols) signals representing a free-falling sphere from aheight of cm. Red (dark red), green (dark green) andblue (dark blue) curves (points) correspond to the resultsfrom channel 1, 2, and 3, respectively. For a better visibility,the offsets of the raw signals are removed.Here, we use the following approach to correct IQ mis-match arising from multiple sources of errors. It worksbest when the object moves in a distance covering multiplewavelengths. As illustrated in Fig. 3, the correction processis composed of the following steps: First, we identify thetime segment of the raw data V raw that contains the move-ment of the tracer particle via finding the start and end ofthe fluctuations. Second, the peaks (red circles) and valleys(blue circles) of individual fluctuations are determined byfinding the local extreme values in the selected data. Third,the bias error V bias [green line in (a)] is estimated as themean value of the spline fits of peaks and valleys (dashedlines). In order to avoid unrealistic extrapolations, the biaserror starts to vary only from the first peak. Fourth, thebias error is removed and the corrected signal V raw − bias is segmented by zero crossings. Finally, the data in individ-ual segments are rescaled by local maxima and minima tocorrect gain mismatch.As shown in Fig. 2, this approach can effectively findtime dependent correction factors due to tracer movement.For the corrected data of channel 3 (dark blue circles), thereexists a slight deviation from a perfect circle, indicating theexistence of a small phase error. This arises presumablyfrom the fact that perfect polarization cannot be achievedfor all three Rx. antennae. Further investigations are neededto check whether this error can be avoided by using circu-lar polarized EM waves or by correcting the phase errorbetween I and Q signals in the Matlab program.After the correction of IQ mismatch, the correspondingphase angles are obtained by φ = arctan( Q/I ) . Because φ is a modulo of π , a further correction on the phasejump is needed to obtain the continuous phase shift Φ . In −0.0200.02 V r a w ( V ) −0.04−0.0200.02 V r a w − b i a s ( V ) Time t (s) V c o rr ec t e d ( V ) a)b)c) Fig. 3.: Process for IQ mismatch correction. (a) A repre-sentative raw signal with peaks and valleys marked withred and blue open circles. From an average of both splinefits for the peaks and valleys, the bias error (green line) forthe raw signal as a function of time is estimated. (b) Biascorrected signal time dependent rescaling factors (greenline) for the correction of gain error. (c) Corrected signalfor further analysis.this step, a threshold is introduced to determine whethera phase jump occurs or not and in which direction thejump takes place. As the phase shift of the i th channel Φ i ∝ L i , the variation of Φ with time (see the blue curvein Fig. 4) indicates that the target object moves initiallyslow and accelerates while moving away from the anten-nae. As demonstrated by a comparison between corrected Φ and uncorrected Φ uncorr phase shift, the aforementionedcorrection method can effectively reduce unrealistic fluc-tuations in the reconstructed curves. As shown in the insetof Fig. 4, the distance L obtained from Rx. antenna rep-resents exactly what is expected: The object falls freelywith a growing velocity and bounces back when reach-ing the container bottom, suggesting that the coefficientof restitution, which measures the relative rebound overimpact velocities, can be determined with the radar system.In comparison to the standard high speed imaging tech-nique [15], the radar tracking technique requires less datacollection and processing efforts. IV. VALIDATION OF RECONSTRUCTEDTRAJECTORY
From the measured distances ~L ≡ ( L , L , L ) , the recon-structed trajectory can be obtained with a coordinate trans-formation xyz = ~r − ~L~T , (2) GRANUL AR PART ICL E T RACKING WIT H RADAR
Time t (s) A ng l e / π Time t (s) L ( c m ) Uncorrected angle Φ Phase angle φ Corrected angle Φ Fig. 4.: Arc-tangential demodulation process to obtain thetraveling distance from the Tx. to a Rx. antenna. The redopen symbols correspond to the outcome from the demod-ulation and the blue curve represents the continuous phaseshift that scales with the traveling distance of an EM wave.The gray curve corresponds to the Φ without correcting IQmismatch. Inset shows an example of the variation of L over a longer time.where the vector ~r is chosen to be as it contributesonly to a constant offset to the reconstructed trajectory, thetransformation matrix reads ~T ≡ sin θ cos φ sin θ sin φ θ sin θ cos φ sin θ sin φ θ sin θ cos φ sin θ sin φ θ (3)with θ i and φ i the tilt and azimuth angles of the i th antenna,respectively. The transformation matrix is determined froma calibration process using the same tracer particle movingin a known trajectory. A detailed description of the calibra-tion and coordinate transformation processes can be foundin [11].Finally, we compare the reconstructed trajectories of afree-falling object from different initial falling heights withthe expected parabolic curve. As shown in Fig. 5, the fallingcurves agree with the expected curve well, demonstratingthat, after a proper correction of IQ mismatch, the radarsystem can be used for particle tracking. Further tests withthe Styrofoam container filled with expanded polypropy-lene (EPP) particles also show that this system can bereadily used for the investigation of granular dynamics.Further investigations will focus on particle tracking withvarious types of granular materials, particularly how to dealwith distorted signals arising from the multiple scatteringof the surrounding particles. V. CONCLUSION
To summarize, this investigation suggests that advances inradar tracking technology can be helpful in the investiga-tion of granular dynamics. Using an X-band continuous t (s) z ( c m ) /2Falling height (cm) Fig. 5.: A comparison of reconstructed free-falling curvesat various initial falling heights. The solid line correspondsto the expected free-falling curve for the largest fallingheight. Note that the curves for various H are shiftedto have the initial falling position z = 0 cm, and onlythe trajectories before the first bouncing with the con-tainer bottom are shown except for H = 22 . m. For eachcurve, one over 15 data points are shown here for a bettervisibility.wave radar system, we are able to track a centimeter sizedmetallic object in 3D, which enables, for instance, a mea-surement of the coefficient of restitution of the particle. Incomparison to other particle imaging techniques alreadybeing used for granular particles [6], continuous-waveradar tracking has the advantage of high time resolutionand low data collection and processing requirements. Withthe rapid development of radar technology, this approach isalso expected to be more cost effective and accurate.Moreover, we show that the accuracy of the radar track-ing technique depends strongly on a proper correction of IQmismatch, which arises predominately from the mobilityof the tracer itself. A practical approach has been pro-posed to correct the instantaneously changing bias as wellas gain errors in the raw IQ signals. Finally, we validatethis approach through an analysis on the reconstructedtrajectories of a free-falling sphere. ACKNOWLEDGMENT
We acknowledge Felix Ott for his preliminary work onthe experimental set-up and Klaus Oetter for technicalsupport. Helpful discussions with Valentin Dichtl, SimeonVölkel and Ingo Rehberg are gratefully acknowledged.This work is partly supported by German Research Foun-dation through Grant No. HU1939/4-1.
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Bibliographies
Felix Rech received his Bachelor of Sci-ence degree in physics from the University ofBayreuth in 2018. He is now a master student atthe technical University of Darmstadt, aimimg tofinish his studies in October 2020.