Multi-sensor Doppler radar for machine tool collision detection

Adv. Radio Sci., 12, 35–41, 2014 www.adv-radio-sci.net/12/35/2014/ doi:10.5194/ars-12-35-2014 © Author(s) 2014. CC Attribution 3.0 License.

Multi-sensor Doppler radar for machine tool collision detection T. J. Wächter1 , U. Siart1 , T. F. Eibert1 , and S. Bonerz2 1 Technische 2 Ott-Jakob

Universität München, Lehrstuhl für Hochfrequenztechnik, Arcisstr. 21, 80333 Munich, Germany Spanntechnik GmbH, Industriestr. 3–7, 87663 Lengenwang, Germany

Correspondence to: T. J. Wächter ([email protected]) Received: 26 January 2014 – Revised: 11 July 2014 – Accepted: 16 July 2014 – Published: 10 November 2014

Abstract. Machine damage due to tool collisions is a widespread issue in milling production. These collisions are typically caused by human errors. A solution for this problem is proposed based on a low-complexity 24 GHz continuous wave (CW) radar system. The developed monitoring system is able to detect moving objects by evaluating the Doppler shift. It combines incoherent information from several spatially distributed Doppler sensors and estimates the distance between an object and the sensors. The specially designed compact prototype contains up to five radar sensor modules and amplifiers yet fits into the limited available space. In this first approach we concentrate on the Doppler-based positioning of a single moving target. The recorded signals are preprocessed in order to remove noise and interference from the machinery hall. We conducted and processed system measurements with this prototype. The Doppler frequency estimation and the object position obtained after signal conditioning and processing with the developed algorithm were in good agreement with the reference coordinates provided by the machine’s control unit.

1 Introduction Collisions within the operating space of machine tools have many different causes. Main causes for collisions during human operation are wrong programming (wrong part, compensation or offsets), wrong machine setup (wrong tool, clamping or raw parts) and inattention or faulty operation by the operator himself. The so called technological collisions concern crashes between tool and work piece only. The second group, the geometric collisions, involves further elements of the machine like the bearings, holder, spindle, axis or the driving shaft.

In the case of tool collisions, tremendous impact forces cause heavy damage to the work piece, the tool and to other elements of the drive train. The results are expensive damage, costs for production downtimes and high rejection rates. The goal is to guarantee high machine availability to maximize efficiency and output. Up to now, no system is known which reliably provides information in order to avoid collisions with non-moving components and to actively protect the machine components against physical damage. There are some promising sensor based systems that can decouple the inner main motor spindle from the outer spindle stock, hence protecting the components from mechanical overload to reduce damage to a minimum. Other methods, without sensors, monitor the current flux through the motor and driving chain and activate when a certain threshold is exceeded. An elaboration on available safety systems can be found in Abele et al. (2012). All of these are reactive systems which are activated after a crash. With these, mitigation of damage is possible, but not their prevention. Optical systems like laser scanners or (multi-)camera based systems are active observers. The disadvantage of these systems is that they are blinded due to cooling fluid, drizzle, dust and dirt from the milling treatment which pollute the lens or the sensor space. So their use is limited to phases where the metal processing is stopped or to other applications with non-cutting manufacture. In this paper, the implementation of a prototype for a new possible solution is presented based on an autonomous active radar sensor safety system which operates in the 24 to 24.25 GHz ISM band. The utilized sensors are commercially available low-cost modules for short-range radar (SRR) applications, containing all front-end parts of a CW radar, including complex down-conversion. Figure 1 shows the prototype concept, here equipped with four modules in a

Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.

2

2 36

W¨ chter, U. T. T. F. Eibert and S. Bonerz: Multi-Sensor Doppler Radar for Machine for Tools T.T.J.J.W¨ aachter, U.Siart, Siart, F. Eibert and S.etBonerz: Multi-Sensor Machine Too T. J. Wächter al.: Multi-sensor DopplerDoppler radar for Radar machine tools z

z sensor sensor platform

platform metal sphere

metal sphere

x

x

y

y

Fig. 1. Concept of the toroidal holder with four modules in a symmetric arrangement and the internal coordinate system. A metal Figure 1. Concept of the toroidal holder with four modules in a sphere serves as a measurement target. symmetric arrangement and the internal coordinate A metal Fig. 1. Concept of the toroidal holder with four system. modules in a symsphere serves as a measurement target. metric arrangement and the internal coordinate system. A metalFigure 2. The implemented toroidal sensor platform equipped with Fig. 2. The and implemented sensor platform with four radars integratedtoroidal amplifiers embedded in equipped an accessible, sphere serves as a measurement target. four radars and amplifiers embedded in an accessible, milling machine tool to provide complete coverage of the maneuverable testintegrated environment. Obtained signals are conducted via maneuverable Obtained signals are conducted via three-dimensional surrounding space. coaxial lines totest theenvironment. data storage and processing system. symmetrical, perpendicular arrangement. The origin of the coaxial lines to the data storage and processing system. We designed a toroidal holder with integrated low-noise

internal coordinate system is at the center point of the holder amplifiers (LNAs) which allows test scenarios with differ-

Fig. 2. The implemented toroidal sensor platform equipped wi

(sensor platform). Several modulescomplete are arranged around the fourthe radars integrated amplifiers in an accessibl milling machine tool to provide coverage of thethermore, large and operating temperature rangeembedded and the rapid ent antenna configurations involving up to five radar moddistance information. Alsoenvironment. the use of the 24 GHz wideband milling machine surrounding tool to provide complete coverage of the maneuverable test Obtained signals are conducted v three-dimensional space. fluctuation of temperature must be dealt with by the equipules with variable radial distance, angle and positions in xtechnology with 5 GHz bandwidth as inlinearity automotive SRR system. was three-dimensional surrounding space. coaxial lines to the data storage and processing ment to meet the high demands on of frequency Wedirection. designed a toroidal holder with integrated Figure 2 shows the implementation with fourlow-noise radar ceasedinasfrequency of June 2013 (StrohmCW et al., 2005). The necessary We designed a toroidal holder with integrated low-noise modulated radar (FMCW) or pulsed modules(LNAs) and the corresponding integrated LNAs embedded amplifiers which allows test scenarios with differ-ramps bandwidth for a specific range resolution ∆r can be derived amplifiers (LNAs) which allows test scenarios with differradar to provide proper distance information. Also the use of in an open test environment. Some important five properties of ent antenna configurations radar from24∆r = c /(2∆f ), where c is the speed of light and ∆f ent antenna configurations involving involving upupto to five radar mod-mod-the 0 0 GHz wideband technology with 5 GHz bandwidth as distance information. Also the use of the 24 GHz wideban this additional system and the CW concept are its indepenis the absolute bandwidth. With the automotive SRR, a suffiules ules with variable angle and positions with variableradial radialdistance, distance, angle and positions in xin x-in automotive SRR was ceased as of June 2013 (Strohm et dence from the acting machine tool, robustness against envitechnology with 5 GHz bandwidth as in automotive SRR wa cient resolution of 3 cm isbandwidth achievablefor if the full bandwidth of showsthe the implementation with fourfour radarradaral., direction. Figure direction. Figure 2 2shows implementation with 2005). The necessary a specific range resronmental influences and reliability due to its simplicity. as of June 2013 (Strohm et al., 2005). The necessar 5 GHzceased is1r becan used. modules corresponding integrated LNAs embedded be derived from 1r = c0 /(21f ), where c0 is modules andand thethe corresponding LNAs embeddedolution Microwave sensorics is more integrated suited for this application bandwidth for a specific range resolution ∆r A second figure of1f merit is the accuracy δr which is dein an open test environment. Some important properties of the speed of light and is the absolute bandwidth. With thecan be derive cameras and optical sensors. is due to the su- of in anthan open test environment. SomeThis important properties fined as (e.g., Skolnik, 1960; Barton, 1964), this additional system and the CW concept are its indepenfrom ∆r = c0 /(2∆fresolution ), whereofc03 cm is the speed of light and ∆ is achievable perior properties of microwaves withconcept the above-mentioned this dence additional system and the tool, CW are its enviindepen-automotive SRR, a sufficient from the acting machine robustness against is the absolute bandwidth. With the automotive SRR, a suffi c if the full bandwidth of 5 GHz is be used. problems with dirt and moisturetool, in the operating space. Mi-envi-δr = τ √ 0 , dence from the acting machine robustness against (1) p ronmental influences and reliability due to its simplicity. A second figure of merit is the accuracy δr which is decient resolution of 3 cm is achievable if the full bandwidth o crowave sensorics is also more suited than acoustical and 2SNR ronmental influences and isreliability due(300 to this its simplicity. Microwave sensorics more suited for application fined as (e.g., Skolnik, 1960; Barton, 1964), ultrasonic sensors. The speed of sound mm/ms) causes 5 GHz used. with the pulse is risebetime τp and the signal-to-noise ratio SNR. than cameras and optical sensors. This is due to the suMicrowave sensorics is more suited for this application long signal delays compared to electromagnetic wave propac0second A figure of meritdepends is the accuracy δr which is d Just like the resolution, the accuracy on the bandperior properties of microwaves with the above-mentioned , (1) than gation. cameras and optical sensors. This is due to the su-δr = τp √ asdetermines (e.g., Skolnik, 1960; Barton, 1964), width fined which τp ≈ 0.35/∆f . Both conditions, 2SNR problems with dirt and moisture in the operating space. Miinside the enclosure ofwith the machine and processperior Operation properties of microwaves the above-mentioned sufficient range resolution and accuracy, cannot be met in the crowave sensorics is also morefrom suited thanisacoustical and uling thewith diverse radar returns there one of−1the chalc0 τp and the signal-to-noise ratio SNR. the to pulse rise time problems dirt and moisture in the operating space. Mi-with 24 GHz trasonic sensors. The speed of sound (300 mm ms ) causes , (1 δr the =24.25 τresolution, p √GHz ISM-band. lenges for the signal processing. Multiple reflexions at the Just like crowave sensorics also more suited than wave acoustical 2SNRthe accuracy depends on the bandlong delaysis compared to electromagnetic propa- and wallssignal at arbitrary distances cause clutter effects and increase ultrasonic sensors. The speed of sound mm/ms) causeswidth which determines τp ≈ 0.35/1f . Both conditions, gation. the noise level. Further challenges are (300 the non-stationary sufficient range resolution accuracy, cannot be met in the ratio SNR 2 Designed Doppler Radar with theMulti-Sensor pulse riseand time τp and theNetwork signal-to-noise Operation inside the shapes enclosure ofgeometries the machineofand processlongenvironment, signal delays compared toand electromagnetic propa-24 to 24.25 GHz ISM-band. varying thewave milling Just like the resolution, the accuracy depends on the band ing from there is operating one of thezone. chalgation. toolsthe as diverse well as radar swarf returns from treatment in the From the viewpoint of hardware one of the limiting factors in width which determines τp ≈ 0.35/∆f . Both condition lenges for the signal processing. Multiple reflexions at the The system development concentrated on using CW space in the maOperation inside the enclosure of the machine andradar process-this application is the available installation walls at arbitrary distances cause clutter effects and increase sufficient range resolution accuracy, cannot Designed multi-sensor Doppler radar network only.diverse This, onradar the one hand, is due there to the robustness tool. A block diagram of a singleand sensor module and the be met in th ing the returns from is non-stationary one of of thethechal-2chine the noise level. Further challenges the GHz tocomponents 24.25 GHzisISM-band. technology and the low cost argumentsare of the electronic hardsignal24 processing given in Fig. 3 showing the lenges for the signal processing. Multiple reflexions at theFrom the viewpoint of hardware one of the limiting factors in environment, varying shapes and geometries themachine milling ware. On the other hand, the available space inofany RF front-end, which includes the 24 GHz voltage controlled wallstools at arbitrary distances cause clutter effects and increase as well as swarf from treatment in the operating zone. this application is athe installation space in the(Tx) matool is limited. This only permits additional components of oscillator (VCO), 90°available hybrid coupler, the transmitting In this first approach the system development concentrated chine tool. A block diagram of a single sensor module andNetwork the noise level. challenges are the small size and Further low complexity. Furthermore, thenon-stationary large operand receiving (Rx) 4-patch antenna array and the IQ mixing 2 Designed Multi-Sensor Doppler Radar on using CW radar only. This, on the one hand, is due to the signal processing components is given in Fig. 3 showing environment, varying shapes andrapid geometries millingelement. Due to the homodyne mixing process the output of ating temperature range and the fluctuationof of the temperthe robustness of thewith technology and the lowtocost arguments the RF front-end, which includescomplex the 24 GHz voltage conature must be dealt by the equipment meet the high the viewpoint downconverted A tools as well as swarf from treatment in the operating zone. the module Fromisthe of hardwareDoppler one of signal. the limiting factors ◦ hybrid of the electronic hardware. On the other hand, the available trolled oscillator (VCO), a 90 coupler, the transmitdemands on linearity of frequency ramps in frequency modfrequency gap of 30 MHz between the carrier frequencies is The system development concentrated on using CW radar this application is the available installation space in the m space anyradar machine tool is This to only permits adting (Tx) and 4-patch antenna arraybetween and the ulatedinCW (FMCW) or limited. pulsed radar provide proper introduced by receiving tuning the(Rx) VCOs to avoid crosstalk only.ditional This, components on the oneofhand, due thecomplexity. robustness of theIQ mixing chine tool. ADue block diagram of amixing singleprocess sensorthemodule and th small is size andtolow Furelement. to the homodyne technology and the low cost arguments of the electronic hardsignal processing components is given in Fig. 3 showing th ware.Adv. OnRadio the other hand, the 2014 available space in any machine RF front-end, which includes the 24 GHz voltage controlle Sci., 12, 35–41, www.adv-radio-sci.net/12/35/2014/ tool is limited. This only permits additional components of oscillator (VCO), a 90° hybrid coupler, the transmitting (Tx small size and low complexity. Furthermore, the large operand receiving (Rx) 4-patch antenna array and the IQ mixin ating temperature range and the rapid fluctuation of temperelement. Due to the homodyne mixing process the output o

T. J.et W¨ achter, U. Siart, T. F. Doppler Eibert andradar S. Bonerz: Doppler Radar for Machine Tools T. J. Wächter al.: Multi-sensor for Multi-Sensor machine tools Analog Baseband Processing

RF Front-End

Tx/Rx Patch-Ant

3

37

Algorithm for Distance Estimation

Digital Signal Processing Stereo-Audio Codec

LO active

90° Hybrid

LPF RF

balanced IQ-Mixer

fc = 20 kHz

24 GHz

A

including digital LPF, Downsampling (M ), Windowing,

D fs = 48 kHz

Radix-2-FFT, Spectral Estimation, Distance Estimation, etc.

VCO

Figure 3. Block diagram of the radar system and signal processing components of a single module. Fig. 3. Block diagram of the radar system and signal processing components of a single module. Table 1.of Overview several system parameters of the prototype. Table 1. Overview severalofsystem parameters of the prototype.

system parameter system parameter value value supply voltage +5 V +5 V supply voltage power consumption 4× power consumption ∼ 4 × ∼0.6 W0.6 W center frequency 24.125 GHz center frequency 24.1258.6 GHz antenna gain dBi antenna gain transmit power 8.6 dBi+15 dBm (EIRP) half-power beamwidth 80° (Azimuth) transmit power +15 dBm (EIRP) 34° (Elevation) half-power beamwidth 80◦ (Azimuth) IF bandwidth 3 Hz to 20 kHz 34◦ (Elevation) IF gain 40 dB IF bandwidth 3 Hz to 20 dBm kHz receiver sensitivity −80 IF gain 40 dB (B = 20 kHz, RIF = 1 kΩ) Doppler shift 160 Hz/(m/s) receiver sensitivity −80 dBm sampling frequency 48 kHz (B = 20 kHz, R = 1k) ADC resolution 16 bit −1 IF Doppler shift 160 Hz/(m s ) sampling frequency 48 kHz ADC resolution 16 bit

the single modules of the radar network. Interfering downconversion is then filtered out. The analog input stage consists of an active anti-aliasing low-pass filter (LPF) with a 3 dB cutoff frequency of 20 kHz, based on a low-noise operamplifier. input impedancecomplex is 1 kΩ. This configoutput of theational module is theThe downconverted Doppler uration a receiver sensitivity of −80 dBm. A standard stereosignal. A frequency of 30 between thequadrature carrier freaudio codecgap is used for MHz converting the analog sigquencies is introduced by tuning the VCOs to avoid crosstalk nal to a digital IQ data stream at a sampling frequency of Some system of the Interfering radar module between thefsingle modules of the parameters radar network. s = 48 kHz. and theisinput are given Table 1, compare down-conversion thenstage filtered out.inThe analog inputRFbeam stage Microwave GmbH (2011). consists of an As active anti-aliasing low-pass filter (LPF) with a the radar system acts in severe environmental condi3 dB cutoff tions frequency of 20 kHz, based on a low-noise operwith fluid, drizzle, heat etc., hermetically sealed modules areThe necessary. enclosuresisof1the radar front-end ational amplifier. input The impedance k. This config-in Fig.a2receiver are made from polyethylene and dBm. are closed and sealed uration allows sensitivity of −80 A standard the final application. Additionally, they increase the flexstereo-audioforcodec is used for converting the analog quadraibility for positioning and tilting of the modules. Coverage

ture signal to a digital IQ data stream at a sampling frequency of fs = 48 kHz. Some system parameters of the radar module and the input stage are given in Table 1, compare RFbeam Microwave GmbH (2011). As the radar system acts in severe environmental conditions with fluid, drizzle, heat etc., hermetically sealed modules are necessary. The enclosures of the radar front-end in Fig. 2 are made from polyethylene and are closed and sealed for the final application. Additionally, they increase the flexibility for positioning and tilting of the modules. Coverage simulations based on deterministic ray-tracing showed that www.adv-radio-sci.net/12/35/2014/

simulations on deterministic ray-tracing that of the sur15◦ tiltingbased of the modules gives best showed coverage 15° tilting of the modules gives best coverage of the surrounding space around the tool (Azodi et al., 2013). rounding space around the tool (Azodi et al., 2013). Theinput input stage is into builta metal into ashielding metal shielding The stage is built enclosure. Itenclosure. It supposed to protect the circuit only against electromagisissupposed to protect the circuit not onlynot against electromagnetic but also the harshthe environmental neticinterference interference but against also against harsh environmental conditions in machine tools. tools. Experiments showed that inter- that interconditions in machine Experiments showed ference from power supplies and switching electrical actuaference power supplies, switching electrical actuators tors as wellfrom as mechanical vibrations are there to effectively and mechanical vibrations are thereinterference to effectively increase increase the noise level. The high-frequency can be after data by digitalinterference filtering as de- can be supthesuppressed noise level. Theacquisition high-frequency scribed in the nextdata section. The low frequency components pressed after acquisition by digital filteringinas described the sub-kilohertz range remains as superimposed interfering in the next section. Low frequency components in the subsignals.

kilohertz range remains as superimposed interfering signals.

3 Implemented Digital Signal Processing Stage As radar modules digital are basedsignal upon aprocessing homodyne archi3 the Implemented stage tecture, the IQ mixer output appears in baseband centered around DC. The analog bandwidth of the active LPF is from the20radar modules areneeded based upon a homodyne archi3As Hz to kHz. This covers the range of relative vetecture, the IQ mixer output appears in baseband centered locities especially for slowly moving targets. After digital the bandwidth signal is buffered andactive the DCLPF is from around DC. conversion The analog of the bias is removed. NextThis a decimation stageneeded follows. range It consists 3 Hz to 20 kHz. covers the of relative veof a digital LPF and subsequent downsampling by factor of locities especially for slowly moving targets. M . Figure 4 illustrates the preprocessing steps between ADC conversion buffered andAfter spectraldigital frequency estimation.the Thesignal digitalisLPF is a 4th and the DC order type Next I filter awith 0.25 dB ripple and follows. fc = bias Chebyshev is removed. decimation stage It con160Hz cutoff frequency. equates todownsampling the Doppler sists of a digital LPFThis andcutoff subsequent by facshift due to the highest expected relative velocity of 1 m/s. tor of M. Figure 4 illustrates the preprocessing steps beThe filter process reduces the noise contribution, hence the tween ADC spectral frequency estimation. The digital sensitivity level isand decreased down to −101 dBm. Eachisobservation window has lengthtype N T Iseconds and is0.25 dB ripLPF a 4th order Chebyshev filter with weighted a window decimation. T =cutoff 1/fs equates to ple andwith fc = 160 Hzfunction cutoffafter frequency. This is the sampling period and N is the number of sampling the Doppler shift due to the highest expected relative velocity points. The−1 decimation process does not change the observa-

of 1 m s . The filter process reduces the noise contribution, hence the sensitivity level is decreased down to −101 dBm. Each observation window has length N T seconds and is weighted with a window function after decimation. T = 1/fs is the sampling period and N is the number of sampling points. The decimation process does not change the observation period. The new sampling period is TM = MT = M/fs and the number of sampling points is N/M. A Kaiser window with 80 dB sidelobe suppression has proven sufficient selectivity and low spectral leakage. The Kaiser window shape parameter value was chosen as α = 3.422. Adv. Radio Sci., 12, 35–41, 2014

4

38

T. J. W¨achter, U. Siart, T. F. Eibert and S. Bonerz: Multi-Sensor Doppler Radar for Machine Tools ui (k)

T. J. ′Wächter et al.: Multi-sensor Doppler radar for machine T. J. W¨achter, U. Siart, T. F. Eibert and S. Bonerz: Multi-Sensor Doppler Radar for Machine Tools

4

k = 0, 1, 2, .. .., N -1

u(k)

ui (k)

from ADC

fs

buffer

+

fs = 48 kHz

k = 0, 1, 2, .. .., N -1

NT u(k)

from ADC

·j fs = 48 kHz

k = 0, 1, 2, .. .., N/M-1

+

T = 1/fbuffer s

uq (k) ·j uq (k)

NT T = 1/fs

LPF

Chebyshev LPF n=4 fc = 160 Hz

k′ M = 0,↓1, 2, .. .., N/M-1 fs

Chebyshev n=4 fc = 160 Hz

u′ (k′ ),

u′ (k′ ),

sample M ↓rate reduction

ν = −L/2, .. .., L/2-1 ′ ′ uw (k )

fs M

ˆ.. ν = −L/2, S(ν) .., L/2-1

fs M

tools

νctr

u′w (k′ ) ˆ short time S(ν)

Kaiser window

sample rate reduction

Kaiser window

Fourier transform

short time Fourier transform

νctr

band-limited spectral centroid

band-limited spectral centroid

Fig. 4. Preprocessing of complex signal from one single path. Figure 4. Preprocessing of complex signal from one single path. Fig. 4. Preprocessing of complex signal from one single path.

Adv. Radio Sci., 12, 35–41, 2014

−50

−60 50

50 −70

0 −80

0 −50

−90

−50 −100

−100

−100 0.5

1

1.5 Time t in s

2

2.5

−110

Power Spectral Density S in dBm/Hz

−50

100 100

−150

−40

−40

−60 −70 −80 −90

(dBm/Hz) Power Spectral Density S in dBm/Hz

150 150

Frequency in Hz Frequency f(Hz) inf (Hz) f(Hz)

tion period. The new sampling period is TM = M T = M/fs tion period. The sampling period T. M M Tof = winM/fs Frequency resolution is new determined byN/M theis duration oband the number of sampling points is A=Kaiser and the number of sampling points is N/M . A Kaiser winservation. The number of sampling points for each segment dow with dow 80 dB sidelobe suppression has proven sufficient with 80 dB sidelobe suppression has proven sufficient areselectivity kept to N ∼ 2nlow for utilization ofleakage. theThe efficient Radix-2and spectral leakage. Kaiser window selectivity and low spectral The Kaiser window FFT algorithm. To extract Doppler shift and relative shape parameter value was chosen as α = 3.422. shape parameter value was chosen as α = 3.422. velocFrequency resolution is determined byfrom the duration ofobobity, the short time Fourier spectrum is derived eachof data Frequency resolution is determined by the duration servation. The number of sampling points each segment servation. Theestimator number of nsampling points for for each segment segment by the 2 for utilization of the efficient Radix-2are kept toareNkept ∼ 2ton Nfor∼ utilization of the efficient Radix-2L−1 and relative velocX FFT algorithm. To extract Doppler
shiftshift FFT algorithm. To extract Doppler relative veloc0 0 0 ˆ the)short Fourier/L spectrum derived from each data uity, exp time −j 2πνk , isand (2) S(ν) = w (k ity, the short time Fourier spectrum is derived from each data segment by the estimator k 0 =0 segment by the estimator L−1 Xand u0 (k 0 ) is the preprocessed signal. where k 0 is an ˆ integer S(ν) = u′w (k ′ )wexp (−j2πνk ′ /L) , (2) L−1  Theˆ Fourier spectrum is ′ ′ k ′=0 calculated at ′ discrete points ν, where S(ν) = uw (k ) exp (−j2πνk /L) , (2) ν = −L/2, ′. where . . , L/2 1 and L and = N/M. k′ − is an integer u′w (k ′ ) is the preprocessed signal. k =0 FrequencyThe estimation is based on computation the bandFourier spectrum is calculated at discreteof points ν, where ′ ′ where k ′ isν = an integer )L isbandwidth the preprocessed signal. −L/2, . .centroid. .,and L/2u−w1(k and = N/M . width limited spectral The for calculaspectrum is calculated at points ν,the where estimation is based on computation of bandtionThe is Fourier limitedFrequency to a window which is discrete identical to the LPF width limited spectral centroid. The bandwidth for calculaν = −L/2, . . ., L/2 − 1 and L = N/M . passband, excluding a small guard intervalisaround DC. Extion estimation is limited toisabased window identical to the LPF Frequency onwhich computation of the bandploiting the passband, sign of the received Doppler shift provided excluding a small guard interval around DC.by Exlimited spectral centroid. The bandwidth forprovided calculathewidth IQ mixer architecture integration limits can set toby ploiting the signthe of the received Doppler shift be is limited a νwindow which iswhere identical tocan thebe LPF theindices IQ to mixer architecture the integration limits set to thetion frequency