Digital Radar Implementation with AmplitudePredistortion

B. Sun∗, M. Yeary∗, F. Uysal∗†, N. Goodman∗, C. Fulton∗, R. Rincon‡,∗Dept. of Electrical & Computer Engr., Advanced Radar Research Center, Univ. of Oklahoma, Norman, OK, USA

†Microwave Sensing, Systems and Signals (MS3), Delft University of Technology, Delft, Netherlands‡ NASA/Goddard Space Flight Center, Code 555, Greenbelt, MD, USA

Abstract—The modern advancements in digital electronicsallow waveforms to be easily synthesized and captured using onlydigital electronics. The synthesis of radar waveforms using onlydigital electronics, such as Digital-to-Analog Converters (DACs)and Analog-to-Digital Converters (ADCs) allows for a majorityof the analog chain to be removed from the system. In order tocreate a constant amplitude waveform, the amplitude distortionsmust be compensated for. The method chosen to compensate forthe amplitude distortions is to pre-distort the waveform so, whenit is influenced by the system, the output waveform has a nearconstant amplitude modulus. The effects of the predistortion wereobserved to be successful in both range and range-Doppler radarimplementations.

I. INTRODUCTION

The broad applications of radar cause for multiple differentconfigurations of radar. Radars need three major componentsto function, the transmit chain, the receive chain and theprocessing computer [1]. Digital radar systems have inher-ent challenges and constraints. Currently, digital radars aregrowing in popularity specifically in phased array radars.Phased array radars traditionally require each element ofthe array to have their respective excitation circuits. Mostphased array radars are designed with a calibration stagewhere the waveforms are manipulated to ensure the effects ofmutual coupling and other element distortions are mitigated[3]. The predistortion technique described in this paper isuseful in the phased array applications also. For instance,a SAR system that performs beam steering can utilize thetechnique to ensure all beams have a constant modulus. Thesystem designed and used in this paper was modeled afterthe EcoSAR system developed by NASA [4]. The EcoSARsystem utilizes independent loopback calibration tests for eachtransmit and receive element [5]. The predistortion techniquein this paper is designed to utilize the loopback tests togenerate a waveform with a constant modulus. Furthermore,the predistortion technique proposed can be inplemented onlow cost systems utilizing Field Programmable Gate Arrays(FPGAs) like those described in [6]. Moreover, multiple inputmultiple output (MIMO) radar systems, like the one describedin [7][8], could also benefit from the predistortion techniqueto design the optimal waveform.

Next, Fig. 1 depicts an example of the problem that thispaper addresses. The upper subplot shows a digitized signalthat was captured by our system operating in a loopback test.The lower subplot depicts this signal’s spectrum. The loopback

test was comprised of an FPGA, ADC, DAC, high-poweredamplifier, bandpass filters, and dummy load. From our tests inthe lab, the upper plot depicts a 2 µs pulse, where the lowerplot depicts its spectrum with 200 MHz bandwidth centeredat the expected center frequency of 145 MHz. The problemmanifests itself both in the time and frequency domains; inparticular, a non-constant envelop is obvious. The next sectionof our paper discusses our approach to digital predistortion tocorrect this far from ideal waveform.

Fig. 1. upper: time domain samples of the received signal, lower: spectrumof the signal.

II. PREDISTORTION

The concept of predistortion is to distort the original signalin a way that when influenced by the system, the distortions arereversed to produce the desired output. The system responsecan be compensated for by “predistorting” the waveformbefore it is influenced by the system. As described in [12],noise and distortion can be differentiated because distortionis a fixed effect on the signal where as noise is a statisti-cally, random process. For instance when implementing LFMwaveforms in hardware for radar transmission, the hardwareinduces an amplitude modulation. The amplitude modulationsis caused by the hardware being incapable of creating sharp,rapid changes in voltage. The amplitude modulation can bereferred to as the system response. The system response causesthe LFM to not have constant power over the radar pulsewhich then effects the range of the radar over the pulse.To maximize the range of the radar the maximum power

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transmitted for a LFM pulse should have a constant modulusideally at the maximum voltage of the waveform. In orderto achieve a constant modulus, the system response must beremoved. The technique used to removed the system responsestarts with the input wave form being predistorted by theopposite of the system response so that when influenced bythe system a constant modulus waveform is produced. Thepredistortion technique used requires the system response tobe only a distortion in amplitude and not contain a distortionin phase. In addition the concept of predistortion requires thesystem response must be deterministic, meaning the systemresponse does not change from pulse to pulse. The techniqueused is applied in the calibration stage of the radar thus,the technique will be applied once every time the system ispowered on and the predistortion calculated will be applied toall pulses in that data collection section. If the system responseis not deterministic the predistortion will not be applicablefor every pulse. In addition if the system response is notdeterministic, the system response would be different by thetime the predistortion is applied. As for phase, the techniqueused is designed to remove amplitude distortions and doesnot effect phase whether the phase is distorted or not. If thesystem response introduces unwanted phase then the proposedtechnique will not remove the undesired phase. The techniquefor predistorting signals with phase distortions can be foundin these papers [15][16]. The response will vary from systemto system and any components added or removed from thesystem will create a different system response. Furthermore,the response can even change from system run to system run.For example, simply power cycling the system could cause thesystem to produce a slightly different system response. If thesystem is power cycled, the technique proposed is requiredto be applied again to acquire a new predistortion capableof correcting the new changes in the system. If the systemresponse is only amplitude and deterministic then the predis-tortion applied to the input waveform can be characterized asthe inverse of the system response envelope.

Mathematically, the amplitude distortion is a constant influ-encing the waveform with time varying phase θ(t), as shownin (1). The amplitude modulation can be characterized as atime varying component a(t), the effects of the componentcan easily be removed by doing the inverse operation shownin (2). x(ejθ(t)) is the signal with amplitude distortion andxdes(e

jθ(t)) is defined as the desired signal.x(ejθ(t)) = a(t)ejθ(t) (1)

xdes(ejθ(t)) =

1

a(t)x(ejθ(t)) (2)

Pre-distorting the input signal with the inverse of the systemresponse envelope, requires the system response to be obtainedand finding the effects of the system on the input referencewaveform after the system. The effects of the system on theinput signal can only be obtained by sending the a referenceLFM signal through the system and observing the differencesbetween the received signal distorted by the system and theinput reference signal. The distorted waveform is used toestimate the amplitude distortion provided by the system re-

sponse. Assuming the distortion from the system is amplitudedistortion only, the distortion can be modeled by the envelopeof the signal. The envelope is then inversed and multiplied bythe reference waveform to obtain the predistorted waveform.The predistorted waveform is then re-transmitted and acquiredvia the analog to digital converter (ADC). The resulting signalshould have significantly less amplitude modulation if any atall. The estimation method used utilizes the Hilbert transformto derive an estimation of the envelope of the signal withequation (4) and equation (3) [2].

Xre(ejw) =

1

2π

∫ π

−πXim(ejv)cot(

w − v

2)dv + x[0] (3)

Xim(ejw) = − 1

2π

∫ π

−πXre(e

jv)cot(w − v

2)dv (4)

In equations (4) and (3), the variables v and w are phase(2πf )and phase(2πf ) transformed respectively. In addition, Xim

and Xre are the imaginary part of signal and real partof signal respectively. Lastly, x[0] is the frequency offset.The envelope is estimated by taking the magnitude of thecomplex waveform and applying a smoothing median filter.The smoothing median filter can be configured to have moreor less taps to control how close the envelope estimation is tothe magnitude of the signal. For instance, a smoothing filterwith a smaller amount of taps will produce an envelope thatis closer to the magnitude of the analytic signal compared toa filter with more taps creates an envelope that will go frompeak to peak in a sinusoidal signal producing an envelopeestimation without the oscillations. The result is a envelopethat captures the amplitude distortions of the system ratherthan the amplitude variations of the waveform.

An example of the envelope estimation with differentamounts of filter taps is shown in Fig. 2.

Fig. 2. Envelope Estimation

The estimated envelope is now defined as the systemresponse a(t) and the pre-distortion needed to reverse theeffects cause by the system is simply the inverse of the systemresponse. The waveform initially transmitted is then multipliedby the pre-distortion and a Tukey window to minimize anyover compensations the system might induce due to thebeginning and ending rapid changes in the waveform. The

Tukey window is a combination of a cosine and series ofconstant values. The equation for a Tukey window is shownin (5) [13][14]. For the system designed a Tukey window withr = 0.1 was used.

T (x) =

1

2[1 + cos(

2π

r(x− r

2)] 0 ≤ x <

r

21

r

2≤ x < 1− r

21

2[1 + cos(

2π

r(x− 1 +

r

2)] 1− r

2≤ x ≤ r

2

(5)

The Tukey window was chosen for the freedom to control theunity pass band. The over compensations are caused by theenvelope estimation being near zero at the beginning and endsof the pulses. When inverted the near zero values become largecausing the waveform to be large as well. The Tukey windowalso compensates for the bounds of the predistotion envelope.An example of the Tukey window is shown in Fig. 3. Notethe window becomes the familiar cosine window when r = 1.

Fig. 3. Tukey Window

Since the predistortion envelope is the inverse of the systemresponse, the envelope can potentially be quite large since thewaveforms are small (near zero) at the beginning and endof the pulse. Taking the inverse of a positive number nearzero causes the predistortion envelope to potentially becomeundefined. Since the undefined points are at the beginningand end of the waveform a Tukey window counteracts theundefined portion of the predistortion envelope.

III. TEST SYSTEM

The test system used comprised of four evaluation modulesfrom Texas Instruments; including two Field ProgrammableGate Arrays (FPGA), a Digital to Analog Converter (DAC),and an Analog to Digital Converter (ADC). The DAC is a 16-bit 4-channel evaluation module with an on board numericallycontrolled oscillator(NCO). The DAC’s maximum aggregatesampling frequency is 1250 MSPS. Therefore, the maximumsampling frequency for each channel is a quarter of the max-imum aggregate. For instance, the maximum input samplingrate for the DAC becomes 312.5 MSPS which by Nyquisttheorem the max frequency that can be created by each channelof the DAC is limited to 156.25 MSPS. The DAC is connectedto an FPGA pattern generator that sends the waveforms to the

DAC. The waveforms are generated in Matlab R© and quan-tized to signed 16-bit words for transmission. The waveformsamples are mixed with the numerically controlled oscillatorfrequency to obtain higher frequencies. The waveforms de-signed for this system are similar to the waveforms displayedin [8]. In addition, the relationship between the FPGA and theCPU is similar to the one described in [8]. A picture of ourtest system used is shown in Fig. 4.

Fig. 4. The Digital System Used to Synthesize Radar Waveforms

The ADC for the system is a 12-bit dual channel evaluationboard from Texas Instruments. The ADC requires an externalclock to create the sampling speed for the system. In addition,the maximum sampling speed for the DAC is 800 MSPSper channel. The ADC is limited by the Nyquist theoremmaking a 400 MHz signal the maximum signal that can berepresented. The ADC is driven by a duplicate of the FPGAthat drives the DAC. An external clock signal is provided froma function generator and fed to both the DAC and ADC toprovide a synchronous clock. The waveforms designed for thesystem were created at baseband in order to maximize theachievable bandwidth for the signal. In this case a 580 MHzclock was used to produce the sampling speed for the ADC.The system was designed to operate at a carrier frequencyof 435 MHz with a 200 MHz complex bandwidth linearfrequency modulated (LFM) waveform. The LFM waveformwas chosen based off of the waveforms inherit ability to beDoppler tolerant [9]. A shorter, 2 µs pulse signal was createdat baseband with 200 MHz complex bandwidth in Matlab R©

and the NCO set to a mixing frequency of 435 MHz to mix thesignal to the desired center frequency. The difference signalwas utilized to minimize the effect of reciprocal image createdby mixing signals. The system utilizes under sampling in orderto reduce the sampling speed of the ADC and the data volumecollected. As a result of undersampling the 435 MHz signalat 580 MHz, the signal is reflected across 290 MHz and isrepresented at 145 MHz by the ADC. The undersamplingtechnique is utilized commonly in digital radar systems [10]and has worked for the team in the past [11]. The bandwidthof the signal then spans from 45 MHz to 245 MHz whichfalls beneath the 290 MHz Nyquist limit and the signal datais preserved.

IV. PREDISTORTION RESULT

In order to test the technique, a signal is first propagatedthrough the desired test system. The output of the DAC was feddirectly into a channel of the ADC and the signal acquired. Af-ter the signal is acquired the predistortion technique is appliedvia Matlab R©. A single pulse of the received data is acquiredand the envelope estimation algorithm is applied. The envelopeis then inverted and applied to the original waveform sentthrough the system. The result of the predistortion algorithmis then sent into the system once more and the results areobserved to confirm the removal of the system response. Theresults of the predistortion is shown in Fig. 5.

Fig. 5. Result of Predistortion

If the predistorted signal continues to produce an undesiredamplitude distortion, the signal is reconditioned with slightadjustments to the envelope estimation technique to produceeither a tighter or more smooth estimated envelope. Forinstance, if the predistorted signal contains a large increasein modulus at the end of the pulse, it is likely caused bythe estimated envelope containing a near zero value at thebeginning or the end of the pulse used to calculate thesystem response. When the envelope is inverted the near zerovalues become quite large and will cause an overshoot inthe predistortion applied to the waveform. As a result, thewaveform becomes over distorted and the system responsedoes not contain enough distortion to produce a constantmodulus. The smoothing filter effectively causes the estimatedenvelope to “loosen”and produce more of a gradual transitionalong the edges of the pulse, which in turn causes the ends ofthe envelope to increase. When the envelope is inverted withlarger end values of the envelope, the resulting predistortionto be applied contains more defined end points that lowerthe amount of distortion applied. Therefore, minimizing theundesired increases at the edges of the system response pulse.Variations in the modulus can also be caused by misalignmentin the pulse captured. For instance, if the pulse captured forenvelope estimation is cut off or contains too many points theenvelope estimated contains either an incomplete version ofthe system response or distortions not caused by the system.If the pulse is cut off the system response in not completely

represented and thus the predistortion calculated does notcompensate for all of the system response. Also, if the pulsecontains too many points it will capture parts where thesystem was not transmitting a pulse which causes the estimatedenvelope estimate noise as part of the system response. Bothissues can be mitigated by correctly capturing the pulse neededfor the envelope estimation technique. The signal shown in5 envelope is flat and constant, proving the amplitude pre-distortion technique works and is effective in the currentapplication. In addition, the predistorted signal will have alower amplitude, as shown in Fig. 5, than the ideal waveformdue to the nature of applying an inverse envelope. The loss inamplitude can be compensated for by adding additional gain.

V. KU-BAND OUTDOOR EXPERIMENT

The radar system originally designed operated in the UHFband, however, locating a set of antennas to test the efficiencyof the system was not feasible. The UHF band antennas wereoverly large and costly for the prototyping test in mind. Thesolution was to modify the system to include two standardgain horn antennas and a UHF to Ku-band up/down converter.The up/down converter contained an amplifier capable ofproducing the 30 dBm signal required to satisfy the designrequirements. In addition, the up/down converter allowed fortwo antennas to be connected one for transmitting and onefor receiving. The two standard gain horns could be utilizedtogether to eliminate the need for a circulator. On the receivechain, the up/down converter contained a limiter and a lownoise amplifier producing the desired power needed to safelyfeed the signal into the ADC. The Ku-band System is shownin Fig. 6.

Fig. 6. Ku Band Radar Design

The inclusion of the up/down converter effectively shiftedthe carrier frequency up to the 13 GHz. However, since theconverter reverses the conversion on the receive chain thetechnique for predistortion still applies as long as the up/downconverter is included during the predistortion calculation.In fact, with the converter the antennas become the onlycomponent that is removed from the system during the loopback scenario. In place of the removed antennas, the highpower attenuator is used to simulate the loss accrued whenthe waveform propagates through the air.

A. Radar Implementation

The system used with the Ku-band up/down converter isshown in Fig. 7. The Ku-band implementation was initially

used as a range radar to prove the functionality of the designedsystem. The system was driven by a function generator tocreate the clock used by both the DAC and ADC. The centerfrequency of the Ku-band system was 13 GHz. The desiredtarget to detect was a rectangular trailer placed at 363 metersaway from the transmit platform.

Fig. 7. System Used for Range Doppler Measurements

The radar was placed on the balcony of the Radar Innova-tions Lab at the University of Oklahoma and the target wasplaced on the other end of the field at the specified distance.The radar antennas were mounted on a tripod that allowed foradjustments in angle height and azimuth. The radar antennaswere pointed at the target using levels and laser range findersto ensure the target was well within the radar’s line of sight.The initial test of the radar showed the desired target produceda weak reflection to the radar and it was not immediately clearthe target was detected. In order to ensure the radar systemwas not the issue corner reflectors were placed and distancesmuch closer to the radar at varying distances to ensure theradar was detecting targets as expected. The initial resultsfrom the corner reflector targets showed positive. The datacollected clearly showed two targets at the correct distance.In addition, the field contained a number of other scatterersincluding a fire hydrant. All of the scatterers are believed tobe picked up by the radar. The corner reflectors were placedin two configurations and compared to the radar data withno corner reflectors present. The first configuration placed thefirst corner reflector roughly 120 meters from the radar andthe second corner reflector was placed about 20 meters beyondthe first corner reflector or 140 meters from the radar. Thesecond configuration was nearly identical to the first with theexception of placing the second corner reflector even furtherat roughly 160 meters from the radar. Each test utilized thepredistortion technique and the range results were capturedand post processed using the matched filter acquired after thepredistortion technique. In addition, the power level of theradar was varied using the gain of the NCO within the DAC.The result of the best power test is shown in Fig. 8. With theconfidence that the radar was functioning as expected the datawas recaptured a number of times using the maximum gainwith no corner reflectors present. Post processing continuously

showed what was believed to be targets in the same location. Inorder to confirm the targets acquired, the concept of coherentintegration was utilized to prominently distinguish the targetsfrom the noise floor. Furthermore, the application of coherentintegration of multiple pulses allows for signals that may beburied by the noise floor to be extracted to detectable valuesas the noise floor is decreased with more integrated pulses.

Fig. 8. Range Result at Max Power

After the measurements were taken, the rectangular trailerwas targeted once again and measurements were taken andpost processed. In addition, a comparison of the fire hydranttarget with a predistorted signal vs a non predistorted signalis shown in Fig. 9. The improvement to the predistortedsignal compared with the non-predistorted signal is 0.3 dB.Considering a 3 dB loss from predistortion, the predistortedsignal still provided a slight improvement in the matched filterresponse. In addition, the SNR of the predistorted signal is alsoincreased as the peak side lobe levels are also decreased.

Fig. 9. Predistorted vs. Non Predistorted Range Results

B. Range-DopplerWhen the target was confirmed to have been detected by

the radar, the concept of producing a range-Doppler map

became a realizable goal. The basic idea of a range-Dopplermap is to collect data from a radar detecting a moving targetand processing the data in such a way that shows the rangeand Doppler shift of the target. Range-Doppler maps use theDoppler shift that changes from pulse to pulse and plots themagainst the detected range of the target. The range-Dopplermap will plot all stationary targets on a single line on theDoppler axis, or zero Doppler shift, at various ranges. In orderto create a range-Doppler map with the current setup, a targettraveling at a constant velocity of 0.3 m/s toward the radarwas targeted. The data was collected from each pulse, as thetarget progresses toward the radar, and is observed in phaseand range. A range-Doppler map is shown in Fig. 10. Thetarget scene included a pole placed about 30 meters in frontof the target. In addition, the range-Doppler map shows thetarget is indeed traveling since the Doppler shift is not zero.

Fig. 10. Range-Doppler Map of Area Around Target

Range-Doppler maps take a significant amount of process-ing as the range is extracted from the time it takes for a singlepulse to return from the target where as the phase is extractedfrom the changed in phase from each pulse. In addition, therange can be extracted from one pulse where as the phase mustbe extracted using two pulses. The data can be arranged insuch a way that allows for quick processing using the innateproperties of the discrete time Fourier transform. The datacan be organized into two different time frames, fast time andslow time. fast time is the time frame from when the transmitpulse ends to when the complete waveform is reflected back.Slow time is the time from the when the reflected pulse isreceived to when the next reflected pulse is received. The datais arranged by placing the data collected fast time sampleson the horizontal axis and placing the slow time componentsalong the vertical axis. By arranging the data with slow timeon the vertical and fast time on the horizontal allows forthe transformation from time to the frequency componentsby utilizing a two-dimensional Fourier transform. In orderto extract range, a matched filter must be applied to the fasttime samples in order to produce a single reference point forcalculating the time it took for the waveform to travel to thetarget and back. The slow time samples need to be transformed

into the frequency domain in order to resolve the Dopplerfrequency.The speed of the can be extracted from the Dopplerfrequency Fd where v = FDλ/2. Using v = FDλ/2 andthe Doppler frequency of 31 Hz, the velocity of the target iscalculated to be 0.31m/s which corresponds the the actualspeed of the target being 0.3m/s.

VI. CONCLUSION

The predistortion technique presented allow for improve-ment in the matched filter response. The range-Doppler map,created from the radar imaging a moving trailer target, accu-rately represented the target scene with the correct Dopplerfrequency at the correct range. The predistortion method pro-posed removes distortions caused by a non-constant amplitudemodulus.

ACKNOWLEDGMENT

This work was partially supported by NASA grantNNX13AD37A.

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