Optimized NLFM Pulse Compression Waveforms forHigh-Sensitivity Radar Observations

James M. Kurdzo∗†, Boon Leng Cheong∗‡, Robert D. Palmer∗†‡ and Guifu Zhang∗†∗Advanced Radar Research Center, University of Oklahoma, Norman, Oklahoma, USA

†School of Meteorology, University of Oklahoma, Norman, Oklahoma, USA‡School of Electrical and Computer Engineering, University of Oklahoma, Norman, Oklahoma, USA

120 David L. Boren Blvd., Ste. 4600, Norman, OK 73072 USA, kurdzo@ou.edu

Abstract—In order to provide adequate sensitivity in low-power radar systems, pulse compression has been used fordecades in military applications, and more recently, weatherradar. Due to the distributed nature and relatively low returnpower of hydrometeors, power efficiency of pulse compressionwaveforms is of great importance. The Advanced Radar ResearchCenter at the University of Oklahoma has been developingnovel waveform design techniques for weather radar platformswhich provide excellent sidelobe performance while maintainingoperational processing gains as high as 0.95 due to limited useof amplitude modulation. While directly applicable to weatherobservations, such waveforms are capable of lowering price pointson all types of radar systems, ranging from military uses and air-craft detection to SAR applications. These waveforms have beenimplemented on the Advanced Radar Research Center’s PX-1000transportable, solid-state, polarimetric X-band weather radar,which operates at 100 Watts on each channel. Due to the very lowpeak transmit power, as well as a fully customizable waveformimplementation and real-time signal processing architecture,PX-1000 serves as an excellent testbed for waveform researchand development. An overview of the technical design methodfor optimized nonlinear waveforms is presented in detail, withcomparisons to other popular nonlinear and heavily-windowedtechniques. A description of implementation in a real system(PX-1000) is presented, including the need for, and implementa-tion of, system-specific pre-distortion. A preliminary explorationof recent progress on Doppler tolerance correction for distributedweather targets is shown. Finally, data from the 20 May 2013Moore, Oklahoma EF-5 tornado are shown, with a discussionof the technical implementation of optimized waveforms onPX-1000, including blind-range mitigation, multi-lag calculationof polarization moments, and the assumptions required for usingcurrent-generation pulse compression techniques in observationsof extreme weather.

I. INTRODUCTION AND MOTIVATION

Radar systems are constantly in need of higher sensitivityand lower costs in order to remain a viable tool in military,weather, airborne, and various other disciplines. Because returnpower is directly related to pulse length, sending a longer pulsewill result in greater sensitivity. However, a significant issuein utilizing a longer pulse is the associated loss in range reso-lution. This issue is solvable by using a frequency-modulatedpulse instead of a constant-frequency pulse, and match filteringthe received signal with a copy of the transmitted pulse. Insteadof a direct relation between pulse length and range resolution,this dependence is decoupled, allowing for significantly higherrange resolution and sensitivity. This technique, known aspulse compression, has been in use for decades with varyingresults, based on the method used for waveform design.

While pulse compression has many possible positive impli-cations on radar design and implementation, there are a numberof issues that arise in practice. Of principal concern is theappearance of range sidelobes. Unlike a constant-frequencypulse in traditional radar designs, frequency-modulated pulsesincoherently interfere with their matched filter, resulting ina compressed pulse which has peaks of false return powerthroughout the length of the pulse in range. This phenomenonhas been well studied, and numerous attempts to mitigate rangesidelobes have been made.

One of the most common methods for lowering rangesidelobes is to taper the waveform, either during transmit orin the filter (or both). This amplitude modulation results in apotentially significant loss in power in the transmitted pulse, orif applied in the matched filter, a loss in signal-to-noise ratio(SNR). This loss is known as the two-way processing gain (1),with 1.00 corresponding to a rectangular pulse (no amplitudemodulation), and lower values corresponding to increasinglyaggressive windows. Amplitude modulation can be applied atvarying degrees of aggressiveness to a simple, linear frequencymodulated waveform in order to lower sidelobes. Amplitudetapering with processing gains as low as 0.50 have been usedon transmit and/or receive with pulse compression waveforms.In some cases, amplitude modulation is seen somewhat as anafterthought in order to correct for non-ideal waveform perfor-mance, and in many instances, is not detailed in final studiesand results due to its “necessity.” However, a Blackman-Harriswindow (for example) applied on receive only (0.50 two-wayprocessing gain), results in a loss of 3.00 dB compared to arectangular pulse. This loss in power translates to a drasticincrease in overall system cost.

SNRloss = 10log10

(

N∑n=1

wtwr

)2

N

(N∑n=1

(wtwr)2) (1)

In order to increase power efficiency and utilize lessaggressive windowing, the concept of a non-linear frequencymodulated (NLFM) waveform was suggested in the early1960s [1], and became common use by the 1990s [2]. Atfirst, stepped non-linear waveforms were used, meaning thattwo different frequency modulation rates were used throughoutthe pulse; a sharper change at the ends, and a more moderatechange throughout the middle. This increase in modulation

978-1-4799-4195-7/14/$31.00 ©2014 IEEE

rate on the edges introduced an effective windowing, whichlowered sidelobes without the need for as much amplitudemodulation. The technique resulted in lower sidelobes thana rectangular linear frequency modulated pulse, but still wasnot sufficient, meaning aggressive windows were still needed.

Soon thereafter, continuous non-linear frequency modu-lated (NLFM) waveforms were devised, with the most popularmethod being developed by De Witte and Griffiths (2004)[3]. For the following decade, nearly all forms of pulsecompression used methods either derived from [3], or from thestationary phase principle, with various types of windowing.The best solutions thus far have lacked flexibility to providevery low sidelobes without significant windowing, as well asthe ability to be designed for specific hardware implemen-tations. While waveforms could be designed with somewhatacceptable processing gain and theoretically low sidelobes,the implementation through actual systems often results insignificantly degraded performance.

This paper presents a technique that utilizes a highlyflexible optimization algorithm for designing pulse compres-sion waveforms. Instead of using previously attained formulaefor the frequency modulation function of a NLFM wave-form, a significantly more flexible approach is presented. Thistechnique allows for globally optimized theoretical waveformdesign, with or without amplitude modulation. Additionally,the technique is capable of optimizing waveforms based onthe measured transfer function of a given system. This meansthat when used in an actual system, the waveform is tailoredto the hardware being used, and the resulting performance canoften be significantly closer to theoretical values. The proposedmethod has the potential to be utilized within all types ofhardware, ranging from weather radar to military solutionssuch as airborne and satellite-based radar systems.

II. THEORETICAL DESIGN METHODOLOGY

The most critical component to any nonlinear waveform de-sign technique is the frequency function. Previous techniqueshave attempted to design flexible waveforms via stepped fre-quencies, semi-linear chirps, and polynomial-based frequencyfunctions. However, in order to allow for very low amplitudemodulation, an effective windowing can be applied to a rectan-gular waveform which allows for sharper changes in frequencynear the edges of the function. This is achieved in our methodvia the use of Bezier curves, a common application similar tospline curves commonly used in graphical design applications[4].

Bezier curves, depending on the bounds set by their gov-erning algorithm, are customizable and flexible to significantlyhigher degrees than allowable by even very high-order polyno-mial functions. Bezier curves can be implemented in softwarevia a series of anchor points which pull an otherwise linearfrequency function into any combination of shapes. Thesepull vectors represent the mathematical construct of the Beziercurve in a computationally simple format, allowing for directimplementation within an optimization framework. The com-bination of computational simplicity and high-order flexibilitymakes for efficient processing and design of waveforms forany type of radar platform.

Fig. 1. Application of Bezier curve generated via pull vectors from originallylinear function. Note 15 distributed anchor points, with the ends and centerpoint locked, and each half of the function forced into symmetry.

The proposed method utilizes a genetic algorithm as thecore optimization framework for waveform design. While itis important to note that the Bezier curve format allows forimplementation within any reliable optimization technique,genetic algorithms were chosen for their simplicity, flexibility,and speed. The input to the genetic algorithm contains 12changeable variables with pre-defined bounds based on thedesign specifications (chirp bandwidth, pulse length, and soon). Each of these variables represents a component of a pullvector for the originally straight frequency function. It wasfound that 15 evenly-spaced anchor points, each with X andY components that combine to form a pull vector, providedsufficiently high-order flexibility for our design needs. Dueto locked end and center points, as well as the desire forwaveform symmetry (for improved Doppler tolerance [5]),only six of these 15 anchor points are part of the optimization.Since each of the six remaining anchor points makes use of twocomponents in order to create a vector, 12 total optimizablevariables are fed as inputs to each generation of the geneticalgorithm. When taking into account the potential pull vectorresolution and search space (in the case presented, 2,001 pos-sible values for each pull vector component were used; more(less) can be used for increased (decreased) performance),a total of 4.21 x 1039 possible frequency function solutionsare effectively searched during an optimization run. Fig. 1demonstrates the progression from a linear function to a highlynonlinear function using pull vectors and the Bezier techniquewithin an optimization algorithm [6].

The genetic algorithm operates via a fitness function, whichdetermines the viability of a frequency function at each gener-ation (or iteration). While it is typical, via recent literature,to design waveforms based on peak sidelobe level (PSL),integrated sidelobe level (ISL), and 3-dB range resolution, wefound that focusing on PSL and null-to-null mainlobe width(MLW) provided the most tangible results. When MLW is nottaken into account in the optimization framework, a distinctbroadening of the mainlobe occurs due to the desire to lowerPSL. While a null may not occur in this situation, the widened

MLW effectively acts as a raised PSL not easily detectableby the fitness function. Therefore, a simple fitness functionwhich takes into account only PSL and null-to-null MLW wasdeveloped for use in the optimization:

F =PSL

MLW(2)

At each iteration, the genetic algorithm attempts to mini-mize the fitness function by decreasing PSL and/or decreasingMLW. These results are made possible by changing the original12 optimizable parameters, which in turn alter the frequencyfunction (Fig. 1) used to calculate an autocorrelation function(ACF) for analysis at each generation. Typically, stoppingcriteria based on necessary design specifications are imple-mented in order to finish the optimization within a reasonableamount of time. Using standard COTS computing hardware, atypical theoretical optimization for an individual system willtake under 2-3 hours to converge on an acceptable solution. Aflowchart for the genetic algorithm is provided in Fig. 2a.

Initial ConditionsCandidate

Frequency FunctionsGenerate Waveforms

Performance EvaluationPerformance

Plateau?

Parameter

Adjustments

Optimization CompleteYES

NO

Initial ConditionsCandidate

Frequency FunctionsGenerate Waveforms

Performance EvaluationPerformance

Plateau?

Parameter

Adjustments

Optimization CompleteYES

NO

Amplitude and

Phase Pre-Distortion

a)

b)

Fig. 2. (a) Flowchart for the genetic algorithm. (b) Same as (a), but withpre-distortion taken into account.

III. THEORETICAL COMPARISONS

The method described in Section II can be applied toany combination of bandwidth, pulse lengths, and amplitudemodulation specifications. As a primer for comparison betweenthis method and previous pulse compression techniques, someof the most commonly-utilized and prevalent methods in theliterature have been chosen as methods for comparison. Acrossthese chosen waveforms, a common analysis point of time-bandwidth product (TB) equal to 270 was chosen due tothe value’s prevalence in previous literature. This TB wasachieved via a 200 µs, 1.35 MHz bandwidth pulse for eachwaveform. The comparison waveforms all utilized mismatchedfiltering with amplitude modulation applied on receive only.The values of two-way processing gain for these waveformswere 0.50 [7], 0.73 [8], 0.73 [2], and 0.51 [3]. These fourwaveforms were compared with four equal TB (and equal

Fig. 3. Example 0.02 ROF comparison waveform ACF at 270 TB.

mainlobe width/resolution) optimized frequency modulated(OFM) waveforms using the proposed technique, with a raisedcosine matched filter with roll off factors (ROF) of 0.10, 0.02,0.01, and 0.00 (no windowing). The two-way processing gainvalues for the designed waveforms were 0.95, 0.99, 0.99, and1.00, respectively.

While the comparison list is hardly exhaustive, Table Ishows significant differences between previous methods andthe OFM waveforms. The best peak sidelobe level achievedin the comparison studies was -71.5 dB [3], however, suchperformance came with a heavy cost of 2.96 dB loss. The bestpeak sidelobe level achieved via the proposed technique was-65.0 dB, but with a much more reasonable loss of 0.24 dB dueto the 0.10 ROF raised cosine matched filter. A nearly identicalresult of -64.8 dB was achievable with a 0.02 ROF filter, whichyields a processing gain of 0.99 and a loss of 0.05 dB. Formany radar systems, range sidelobe levels of -64.8 dB areconsidered quite reasonable, especially given typical antennasidelobe levels of both dish and array platforms. The criticaldifference becomes a sensitivity gain of 2.72 dB, enough tovastly alter the cost of a radar system. As an example, the 0.02ROF raised cosine example ACF is shown in Fig. 3.

IV. SYSTEM IMPLEMENTATION

The system being used for research, development, andtesting of waveform techniques at the University of Oklahomais the PX-1000 transportable, solid-state, polarimetric X-bandradar [9]. While this system exists primarily as a weatherobservation tool, it is important to note that it is serving asa general proving ground for waveform development. Eachstep shown throughout this section is applicable to any radarhardware capable of custom waveform designs and pulsecompression. PX-1000 operates via dual 100-W solid-statepower amplifiers, and has a mission of collecting, at times,very low power returns from distributed hydrometeors. The

TABLE I. COMPARISON OF COMMON WAVEFORM TECHNIQUES

Waveform Frequency Modulation Amplitude Modulation TB Product Processing Gain / Power Loss Peak Sidelobe LevelKlauder et al. 1960 [7] LFM Blackman-Harris Mismatch 270 0.50 / 3.00 dB -50.4 dB

Cook and Paolillo 1964 [8] Stepped LFM Hamming Mismatch 270 0.73 / 1.35 dB -58.2 dBGriffiths and Vinagre 1994 [2] Stepped LFM Hamming Mismatch 270 0.73 / 1.35 dB -62.8 dB

De Witte and Griffiths 2004 [3] NLFM Nuttall Mismatch 270 0.51 / 2.96 dB -71.5 dB0.00 ROF OFM NLFM Raised Cosine (0.00 ROF) Match 270 1.00 / 0.00 dB -56.0 dB0.01 ROF OFM NLFM Raised Cosine (0.01 ROF) Match 270 0.99 / 0.03 dB -59.9 dB0.02 ROF OFM NLFM Raised Cosine (0.02 ROF) Match 270 0.99 / 0.05 dB -64.8 dB0.10 ROF OFM NLFM Raised Cosine (0.10 ROF) Match 270 0.95 / 0.24 dB -65.0 dB

Fig. 4. Optimized frequency function for the PX-1000 waveform.

combination of PX-1000 hardware and custom in-house signalprocessing and display/control software makes PX-1000 anideal platform for pulse compression studies. The PX-1000specifications are listed in Table II.

PX-1000 allows for up to 5 MHz of chirp bandwidth, andup to a 70 µs pulse length. While these are the values thatwould typically be used within the optimization framework aswaveform specifications, we have chosen to use only 2.2 MHzof chirp bandwidth, and a 67 µs pulse length. The reasonfor this decision is due to the use of a fill pulse for blind-range mitigation, which is explained later in this section. Inaddition to these specifications, a slight amplitude modulationis added to the pulse. In theory, the optimization framework iscapable of designing high-performance waveforms with higherprocessing gains. In a real system, however, switching noiseand transmitter imperfections can corrupt the edges of a nearly-rectangular pulse. Therefore, a processing gain of 0.95 (araised cosine taper on both ends of the waveform with aROF of 0.10) was chosen for the theoretical design of thePX-1000 waveform. Using the revised specifications, a theoret-ical optimized waveform can be designed using the previouslymentioned optimization framework. The optimization results ina waveform with peak sidelobes of -59 dB, integrated sidelobesof -37 dB, and 3 dB range resolution of 120 m. The finalfrequency function is shown in Fig. 4, and the resulting ACFcan be seen in Fig. 6a.

While theoretical results using this method are promising,it is important to demonstrate the performance of the waveformwhen passed through the actual system. Due to inherent

TABLE II. PX-1000 SYSTEM SPECIFICATIONS

GeneralOperating Frequency 9550 MHzInstantaneous Bandwidth 5 MHzSensitivity (w/ OFM pulse compression) < 20 dBZ @ 50 km (67 µs PW)

Antenna3-dB Beamwidth 1.8◦

Polarization Simultaneous Dual-LinearGain 38.5 dBiCross-Polarization Level < -26 dBRotation Rate Up to 50◦ s−1

TransmitterType Dual Solid-State Power AmplifiersPeak Power 100 W per ChannelPulse Width 1-70 µsPulse Repetition Frequency 1-2000 Hz

ReceiverMinimum Gate Spacing 30 mMaximum Range 60 km

transmitter imperfections, the resulting waveform performancein practice with PX-1000 is significantly worse than theory.This can be seen in Fig. 6b, where the pass-through waveformACF has degraded to a -42 dB peak sidelobe level. This is duein large part to transmitter “droop,” which often occurs duringlong pulse transmissions. Fig. 5a shows the actual pass-throughwaveform as measured by PX-1000. The center portion of thewaveform was designed to be perfectly flat, but the transmitterhas deformed the pulse in amplitude, resulting in the degradedperformance seen in the ACF.

In order to correct for transmitter distortions, a pre-distortion method was applied to the waveform before beingpassed through the transmitter. Pre-distortion was accom-plished via measuring the pass-through signal and determininga transfer function between an intended, theoretical signaland the actual, observed signal. By inverting this transferfunction, applying it to the theoretical signal within the fre-quency domain at each step of the optimization (Fig. 2b), andnormalizing within the time domain, a “corrected” signal canbe designed. When this signal is then corrupted by predictabletransmitter distortions, the resulting signal has nearly perfectamplitude characteristics (Fig. 5b). The recovery of waveformperformance characteristics is evident in Fig. 6c, where thefinal transmitted waveform ACF displays operational peaksidelobes of -52 dB, a 10 dB increase in performance com-pared with not accounting for pre-distortion. While theoreticalperformance levels are not achieved with this method, anoperationally-feasible waveform with the same high processinggain of 0.95 is possible.

Some additional issues must be considered with a dual-polarization radar using a long pulse. First, the long pulseresults in a wide blind-range (on the order of 10 km in thiscase), which is unacceptable given the limited maximum rangeof the system (60 km). In order to mitigate this issue, themethod described in [9] for blind-range filling is used. The

remaining 2.8 MHz of bandwidth and 2 of the remaining 3 µsof available pulse length are used to transmit a short, windowedfill pulse. This technique uses a real-time software architectureto blend the transition zone between the short and long pulses,and despite lower sensitivity within the blind-range, does asatisfactory job of filling in areas near the radar. Also, withlow SNRs, especially within the blind-range, dual-polarizationestimates suffer from various biases. An implementation ofthe multi-lag technique described in [10] for recovery ofpolarimetric moments is used throughout this paper.

Finally, when using a long-pulse waveform for distributedtarget detection in extreme weather situations (i.e., hydromete-ors and debris traveling in excess of 100 ms−1 near a tornadicvortex), the issue of Doppler tolerance must be discussed.Numerous studies have explored Doppler tolerance correction,including in distributed targets [5], however, most of thesestudies offered hardware-based solutions, or techniques whichwere not computationally feasible for real-time implementa-tion. With the advent of increased computational power, as wellas the flexibility of the PX-1000 signal processing softwaresuite, an opportunity for new insight into Doppler toleranceexists. While no real-time Doppler tolerance correction was inplace for the data presented in Section V, archived time seriesdata allows for a simulated re-processing of moment data.

A significant issue with Doppler tolerance correction, espe-cially in extreme weather events where Doppler tolerance is amajor concern, is the lack of reliable real-time dealiasing. Withclosely-confined velocity gradients which can peak at over100 ms−1, dealiasing is usually performed in post-processing.However, assuming dealiased data is attainable, a number ofoptions for Doppler tolerance correction exist. Due to theforced symmetric nature of waveforms designed using thisoptimization technique, a significant increase in PSL occurs

Fig. 5. PX-1000 pass-through measurements of optimized waveform. (a)With no pre-distortion. (b) With pre-distortion applied.

Fig. 6. Measured pass through ACFs of an optimized waveform for thePX-1000 system. (a) Original waveform after optimization. (b) The theoreticalwaveform from (a) after it has been sent through the transmitter, with no pre-distortion applied. (c) Pass through waveform with pre-distortion applied.

only on one side of the ACF, based on the phase directionof the moving target. Assuming the phase is known (i.e.,the velocity moment is both dealiased and reliable withoutcorrection in its own right), a combination of matched filterdirections can be used to lower embedded peak sidelobes.

Unfortunately, this technique results in generally widenednull-to-null mainlobe widths in high Doppler velocity situa-tions. While this is certainly preferable, additional methodsare possible. The authors are currently experimenting with FIRfilter options for application to the matched filter, depending onthe expected phase shift of distributed targets. This techniquealso requires the assumption that original velocity estimatesare reliable without any prior correction, however, preliminaryresults have shown promising recovery of the reflectivitymoment. This method works via an additive adjustment forphase shifts across the entire length of the pulse, while alsoaccounting for many distributed targets along a radial. Addi-tional work on this topic is necessary and ongoing, and boththeoretical examples and comparisons using archived extremeweather events will prove useful in future studies.

V. RESULTS

As a demonstration of observational capability using theproposed waveform optimization technique, data from the20 May 2013 Moore, Oklahoma EF-5 tornado collected withPX-1000 are presented below. Observations at approximately

480 m above ground level at a range of between 10-12 kmat peak tornado intensity (with radial velocities approaching90 ms−1 in the core vortex) offer opportunities to exploresensitivity of measurements, the blind-range transition zone,and Doppler performance of the optimized waveform. Fig. 7shows reflectivity and dealiased velocity moment estimates at20:20:18 UTC, the time of near-peak intensity.

Fig. 7. Reflectivity (left; dBZ) and dealiased Doppler velocity (right; ms−1)moment estimates of the 20 May 2013 Moore, Oklahoma EF-5 tornado atnear-peak intensity (20:20:18 UTC).

At the same time, the core vortex of the tornadic circulationwas located just beyond the blind-range transition zone, whichcan be seen via the circle of lowered SNR approximately10 km from the radar location in the top panels in Fig. 7.In the areas of higher SNR, such as surrounding the tornadicsignature and debris to the south of the tornado, the transitionzone is not nearly as evident, as seen in the bottom panels inFig. 7. Despite no attempted correction for Doppler tolerancein the waveform or matched filter designs, the general structureof both the reflectivity and Doppler velocity fields match bothexpectations and other observations by radar platforms in thearea.

Fig. 8 shows the differential reflectivity and correlationcoefficient moment estimates using the multi-lag method from[10] at the same time as the data shown in Fig. 7. Again, thehorizontal structure of both moments agrees well with whatwould be expected from supercell dynamic theory [11]. Withinthe blind-range, where SNR values are significantly lower, theuse of multi-lag estimation offers nearly identical quality towhat is seen outside of the blind-range.

VI. CONCLUSIONS

A novel approach to optimized high-sensitivity pulse com-pression waveform design for radar systems has been pre-sented. A discussion of the optimization framework shows theworkflow of the design process, and a comparison to otherpulse compression techniques has been made. Data from asignificant tornadic weather event in Oklahoma show promisefor the combination of long-pulse waveforms, blind-range

Fig. 8. Differential reflectivity (left; dB) and correlation coefficient (right;unitless) moment estimates of the 20 May 2013 Moore, Oklahoma EF-5tornado at 20:20:18 UTC.

mitigation, multi-lag estimation for polarimetric moments, andfuture applications to Doppler tolerance correction in extremeweather and distributed target scenarios.

ACKNOWLEDGMENT

This work was supported by the National Severe StormsLaboratory (NOAA/NSSL) under the Cooperative AgreementNA08OAR4320904. The authors would like to thank JohnMeier and Redmond Kelley for discussions regarding wave-form design and system implementation, as well as DavidBodine for insight into analysis of the 20 May 2013 case. Wewould also like to thank the Toshiba Corporation for usefuldiscussions regarding waveform techniques.

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