Effects, Estimation, and Compensation of Frequency Sweep Nonlinearity in FMCW* Ranging Systems

Committee membersApplied Physics Prof. dr. A.P. Mosk (COPS), ir. R. Vinke (Thales), prof. dr.

W.L. Vos (COPS), ir. H.T. Griffioen (Thales)

Applied Mathematics Dr. G. Meinsma (MSCT), prof. dr. A.A. Stoorvogel (MSCT), dr. A. Zagaris (AAMP)

* Frequency-Modulated Continuous-Wave

Contents• Introduction • Digital chirp generation and its effect on the

performance of a FMCW radar• Compensation of frequency sweep

nonlinearity by digital post-processing• Applications of FMCW to optics• Conclusions

Radar• Radio Detection And Ranging• “To see and not be seen”

RAF Chain Home radar site

German U-boat surrendering (depth charge in profile)

Heinkel HE-111 bombers

Pulsed radar

Intercept receivers• Jamming• Direction finding (DF)• Anti-radiation missiles (ARMs)

Prowler armed with HARM high-speed anti-radiation missiles

DRS ZA-4501 shipboard DF antenna array

LPI radar

pulse with high peak power

continuous wave with low peak power

time

power

• Low probability of intercept

Thales Smart-Lpower megaWatt

Thales Scout Mk2power milliWatt

FMCW radar• Frequency-modulated continuous-wave

time

time

frequency

amplitude

bandwidth = 50 MHz

sweep period = 500 µs

carrier frequency = 10 GHz

chirp (𝑡 )=cos [2𝜋 ( 𝑓 𝑐𝑡+ 12𝛼𝑡 2)] , where𝛼=𝐵𝑇

Principle of FMCW rangingtransmitted linear

chirp

received echoes

frequency difference

frequency

time

time

target ‘beat’ frequencies

FMCW transceiver

chirp generator

spectrum analyzer

time

coupler

mixer

transmit antenna

receive antenna

target

RF

LO

IF

frequency

power

frequency

Frequency sweep nonlinearity

transmitted non-linear

chirp

received target echoes

beat frequency

frequency

time

time

“Ghost” targets

beat frequency

frequency

time

time

power

frequency

transmitted non-linear

chirp

received target echo

“ghost” targets

target

Analog chirp generation• YIG (Yttrium, Iron, and Garnet)-tuned oscillator

A.G. Stove, Measurement of Spectra of Microwave FMCW Radars, Thales Aerospace UK, working paper (2006).

Digital chirp generation• Direct digital synthesizer (DDS)

address generator

RAM or ROM

D/A converter

low-pass filter

clockto transmitter

• Clock speed 1 GSPS• Integrated 14-bit DAC

Output of a AD9910 sweeping from 180 MHz to 210 MHz

Source: J. Ledford, Master’s Thesis, University of Kansas (2008).

Quantization of phase

0000…0

1111…1

‘jump’ size

sine look-up table (ROM)

‘phase accumulator’

Δ𝜙

Δ𝜙=2𝜋2𝑊radians

𝑊=number of bits of the phase accumulator

AD9910 synthesizer

clock

Worst-case “ghost” target

SFDR=20 log10 (2 Δ𝜙 )≈ 92dB

• ‘Spurious-free dynamic range’

• “Ghost” targets practically negligiblepower

frequency

SFDR = 92 dB

Compensation of phase errors

• Burgos-Garcia et al., Digital on-line compensation of errors induced by linear distortion in broadband FM radars, Electron. Lett. 39(1), 16 (2002).

• Meta et al., Range non-linearities correction in FMCW SAR, IEEE Conf. on Geoscience and Remote Sensing 2006, 403 (2006).

Remember this?

time

time

intermediate frequency (IF)

frequency

Compensation algorithm𝑓 𝑏

𝑓 𝑏

𝑓 𝑏

𝑓 𝑏

collected non-linear deramped data

transmitted non-linearties removal

range deskew

non-linearities compensation

linear deramped data

time

time

time

time

Implementation

𝑄−𝛼( 𝑓 )𝑠𝐼𝐹 3𝑠𝐼𝐹 2

deskew filter𝑠𝐼𝐹 𝑠𝐼𝐹 4

𝑠𝜖∗ (𝑡 ) “Peek”

“Meta”

“Burgos-Garcia”

chirp (𝑡 )=cos [2𝜋 ( 𝑓 𝑐𝑡+ 12𝛼𝑡 2+𝜖 (𝑡 ))] 𝑠𝜖 (𝑡 )=exp [ 𝑗2𝜋𝜖 (𝑡 ) ]

phase error𝑄−𝛼 ( 𝑓 )=exp ( 𝑗 𝜋𝛼 𝑓 2)

Sinusoidal phase error (low frequency)

14.94 14.96 14.98 15 15.02 15.04 15.06-80

-70

-60

-50

-40

-30

-20

-10

0

Range (km)

Pow

er s

pect

rum

(dB

)

uncompensated

compensated (narrowband)compensated (wideband)

ideal

2𝜋𝜖 (𝑡 )=𝐴𝑠𝑙 sin (2𝜋 𝑓 𝑠𝑙𝑡 ) , 𝑓 𝑠𝑙≪√𝛼

Parameter Value Unit

10 GHz

50 MHz

500 μs

15 km

0.1 Rad

4 kHz

Sinusoidal phase error (high frequency)

14.9 14.95 15 15.05 15.1-80

-70

-60

-50

-40

-30

-20

-10

0

Range (km)

Pow

er s

pect

rum

(dB

)

uncompensated

compensated (narrowband)compensated (wideband)

ideal

2𝜋𝜖 (𝑡 )=𝐴𝑠𝑙 sin (2𝜋 𝑓 𝑠𝑙𝑡 ) , 𝑓 𝑠𝑙 √𝛼

Parameter Value Unit

10 GHz

50 MHz

500 μs

15 km

0.1 Rad

63 kHz

Cubic phase error2𝜋𝜖 (𝑡 )=𝑘3𝑡

3

Parameter Value Unit

10 GHz

50 MHz

500 μs

15 km

4 × 1011 Hz/s2

14.94 14.96 14.98 15 15.02 15.04 15.06-80

-70

-60

-50

-40

-30

-20

-10

0

Range (km)

Pow

er s

pect

rum

(dB

)

uncompensated

compensated (narrowband)compensated (wideband)

ideal

Quartic phase error2𝜋𝜖 (𝑡 )=𝑘4 𝑡

4

Parameter Value Unit

10 GHz

50 MHz

500 μs

15 km

4 × 1011 Hz/s2

14.94 14.96 14.98 15 15.02 15.04 15.06-80

-70

-60

-50

-40

-30

-20

-10

0

Range (km)

Pow

er s

pect

rum

(dB

)

uncompensated

compensated (narrowband)compensated (wideband)

ideal

FCMW in optics

• Swept-Source Optical Coherence Tomography

• Compensation algorithm not in the literature!

3D image of a frog tadpole using a Thorlabs OCS1300SS OCT microscope system.

Conclusions

• Phase quantization effects in digital chirp synthesizers have negligible effect on performance

• Frequency sweep nonlinearity can be compensated by digital post-processing of the beat signal

• Algorithm is also applicable to optics, but not mentioned in optics literature

Thank you for your attention!

Questions?

Extra slides

Effect on Doppler processing

• Systematic phase errors have negligible effect on Doppler processing

Sinusoidal phase error, 3 cycles per sweep, amplitude 0.1 radian

Sinusoidal phase error, 3.1 cycles per sweep, amplitude 0.1 radian

Spectrum of the complex exponential

‘signal’

‘replicas’

𝜃𝑚=[0,1 ,…,7 ]

8radians

Spectrum of the analytic signal

‘signal replica’

‘main’ signal

‘image replica’

Observed beat signal

‘signal × image replica’

‘signal × signal replica’

‘image replica × image replica’

‘signal ×signal’