Optimized NLFM Pulse Compression Waveforms for High-Sensitivity Radar Observations

Transcription

1 Optimized NLFM Pulse Compression Waveforms for High-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 USA, Abstract In order to provide adequate sensitivity in lowpower radar systems, pulse compression has been used for decades in military applications, and more recently, weather radar. Due to the distributed nature and relatively low return power of hydrometeors, power efficiency of pulse compression waveforms is of great importance. The Advanced Radar Research Center at the University of Oklahoma has been developing novel waveform design techniques for weather radar platforms which provide excellent sidelobe performance while maintaining operational processing gains as high as 0.95 due to limited use of amplitude modulation. While directly applicable to weather observations, such waveforms are capable of lowering price points on all types of radar systems, ranging from military uses and aircraft detection to SAR applications. These waveforms have been implemented on the Advanced Radar Research Center s PX-1000 transportable, solid-state, polarimetric X-band weather radar, which operates at 100 Watts on each channel. Due to the very low peak transmit power, as well as a fully customizable waveform implementation and real-time signal processing architecture, PX-1000 serves as an excellent testbed for waveform research and development. An overview of the technical design method for optimized nonlinear waveforms is presented in detail, with comparisons to other popular nonlinear and heavily-windowed techniques. A description of implementation in a real system (PX-1000) is presented, including the need for, and implementation of, system-specific pre-distortion. A preliminary exploration of recent progress on Doppler tolerance correction for distributed weather targets is shown. Finally, data from the 20 May 2013 Moore, Oklahoma EF-5 tornado are shown, with a discussion of the technical implementation of optimized waveforms on PX-1000, including blind-range mitigation, multi-lag calculation of polarization moments, and the assumptions required for using current-generation pulse compression techniques in observations of extreme weather. I. INTRODUCTION AND MOTIVATION Radar systems are constantly in need of higher sensitivity and lower costs in order to remain a viable tool in military, weather, airborne, and various other disciplines. Because return power is directly related to pulse length, sending a longer pulse will result in greater sensitivity. However, a significant issue in utilizing a longer pulse is the associated loss in range resolution. This issue is solvable by using a frequency-modulated pulse instead of a constant-frequency pulse, and match filtering the received signal with a copy of the transmitted pulse. Instead of a direct relation between pulse length and range resolution, this dependence is decoupled, allowing for significantly higher range resolution and sensitivity. This technique, known as pulse compression, has been in use for decades with varying results, based on the method used for waveform design. While pulse compression has many possible positive implications on radar design and implementation, there are a number of issues that arise in practice. Of principal concern is the appearance of range sidelobes. Unlike a constant-frequency pulse in traditional radar designs, frequency-modulated pulses incoherently interfere with their matched filter, resulting in a compressed pulse which has peaks of false return power throughout the length of the pulse in range. This phenomenon has been well studied, and numerous attempts to mitigate range sidelobes have been made. One of the most common methods for lowering range sidelobes is to taper the waveform, either during transmit or in the filter (or both). This amplitude modulation results in a potentially significant loss in power in the transmitted pulse, or if 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 amplitude modulation), and lower values corresponding to increasingly aggressive windows. Amplitude modulation can be applied at varying degrees of aggressiveness to a simple, linear frequency modulated waveform in order to lower sidelobes. Amplitude tapering with processing gains as low as 0.50 have been used on transmit and/or receive with pulse compression waveforms. In some cases, amplitude modulation is seen somewhat as an afterthought in order to correct for non-ideal waveform performance, and in many instances, is not detailed in final studies and results due to its necessity. However, a Blackman-Harris window (for example) applied on receive only (0.50 two-way processing gain), results in a loss of 3.00 db compared to a rectangular pulse. This loss in power translates to a drastic increase in overall system cost. ( N ) 2 w t w r n=1 SNR loss = 10log 10 ( N ) N (w t w r ) 2 n=1 In order to increase power efficiency and utilize less aggressive windowing, the concept of a non-linear frequency modulated (NLFM) waveform was suggested in the early 1960s [1], and became common use by the 1990s [2]. At first, stepped non-linear waveforms were used, meaning that two different frequency modulation rates were used throughout the pulse; a sharper change at the ends, and a more moderate change throughout the middle. This increase in modulation (1) /14/$ IEEE

2 rate on the edges introduced an effective windowing, which lowered sidelobes without the need for as much amplitude modulation. The technique resulted in lower sidelobes than a rectangular linear frequency modulated pulse, but still was not sufficient, meaning aggressive windows were still needed. Soon thereafter, continuous non-linear frequency modulated (NLFM) waveforms were devised, with the most popular method being developed by De Witte and Griffiths (2004) [3]. For the following decade, nearly all forms of pulse compression used methods either derived from [3], or from the stationary phase principle, with various types of windowing. The best solutions thus far have lacked flexibility to provide very low sidelobes without significant windowing, as well as the ability to be designed for specific hardware implementations. While waveforms could be designed with somewhat acceptable processing gain and theoretically low sidelobes, the implementation through actual systems often results in significantly degraded performance. This paper presents a technique that utilizes a highly flexible optimization algorithm for designing pulse compression waveforms. Instead of using previously attained formulae for the frequency modulation function of a NLFM waveform, a significantly more flexible approach is presented. This technique allows for globally optimized theoretical waveform design, with or without amplitude modulation. Additionally, the technique is capable of optimizing waveforms based on the measured transfer function of a given system. This means that when used in an actual system, the waveform is tailored to the hardware being used, and the resulting performance can often be significantly closer to theoretical values. The proposed method has the potential to be utilized within all types of hardware, ranging from weather radar to military solutions such as airborne and satellite-based radar systems. II. THEORETICAL DESIGN METHODOLOGY The most critical component to any nonlinear waveform design technique is the frequency function. Previous techniques have attempted to design flexible waveforms via stepped frequencies, semi-linear chirps, and polynomial-based frequency functions. However, in order to allow for very low amplitude modulation, an effective windowing can be applied to a rectangular waveform which allows for sharper changes in frequency near the edges of the function. This is achieved in our method via the use of Bézier curves, a common application similar to spline curves commonly used in graphical design applications [4]. Bézier curves, depending on the bounds set by their governing algorithm, are customizable and flexible to significantly higher degrees than allowable by even very high-order polynomial functions. Bézier curves can be implemented in software via a series of anchor points which pull an otherwise linear frequency function into any combination of shapes. These pull vectors represent the mathematical construct of the Bézier curve in a computationally simple format, allowing for direct implementation within an optimization framework. The combination of computational simplicity and high-order flexibility makes for efficient processing and design of waveforms for any type of radar platform. Fig. 1. Application of Bézier curve generated via pull vectors from originally linear function. Note 15 distributed anchor points, with the ends and center point locked, and each half of the function forced into symmetry. The proposed method utilizes a genetic algorithm as the core optimization framework for waveform design. While it is important to note that the Bézier curve format allows for implementation within any reliable optimization technique, genetic algorithms were chosen for their simplicity, flexibility, and speed. The input to the genetic algorithm contains 12 changeable variables with pre-defined bounds based on the design specifications (chirp bandwidth, pulse length, and so on). Each of these variables represents a component of a pull vector for the originally straight frequency function. It was found that 15 evenly-spaced anchor points, each with X and Y components that combine to form a pull vector, provided sufficiently high-order flexibility for our design needs. Due to locked end and center points, as well as the desire for waveform 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 two components in order to create a vector, 12 total optimizable variables are fed as inputs to each generation of the genetic algorithm. When taking into account the potential pull vector resolution and search space (in the case presented, 2,001 possible values for each pull vector component were used; more (less) can be used for increased (decreased) performance), a total of 4.21 x possible frequency function solutions are effectively searched during an optimization run. Fig. 1 demonstrates the progression from a linear function to a highly nonlinear function using pull vectors and the Bézier technique within an optimization algorithm [6]. The genetic algorithm operates via a fitness function, which determines the viability of a frequency function at each generation (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, we found that focusing on PSL and null-to-null mainlobe width (MLW) provided the most tangible results. When MLW is not taken into account in the optimization framework, a distinct broadening of the mainlobe occurs due to the desire to lower PSL. While a null may not occur in this situation, the widened

3 MLW effectively acts as a raised PSL not easily detectable by the fitness function. Therefore, a simple fitness function which takes into account only PSL and null-to-null MLW was developed for use in the optimization: F = P SL MLW At each iteration, the genetic algorithm attempts to minimize the fitness function by decreasing PSL and/or decreasing MLW. These results are made possible by changing the original 12 optimizable parameters, which in turn alter the frequency function (Fig. 1) used to calculate an autocorrelation function (ACF) for analysis at each generation. Typically, stopping criteria based on necessary design specifications are implemented in order to finish the optimization within a reasonable amount of time. Using standard COTS computing hardware, a typical theoretical optimization for an individual system will take under 2-3 hours to converge on an acceptable solution. A flowchart for the genetic algorithm is provided in Fig. 2a. a) (2) Initial Conditions Candidate Frequency Functions Generate Waveforms Fig. 3. Example 0.02 ROF comparison waveform ACF at 270 TB. b) Optimization Complete Initial Conditions Optimization Complete YES YES Parameter Adjustments NO Performance Plateau? Candidate Frequency Functions Parameter Adjustments NO Performance Plateau? Performance Evaluation Generate Waveforms Amplitude and Phase Pre-Distortion Performance Evaluation Fig. 2. (a) Flowchart for the genetic algorithm. (b) Same as (a), but with pre-distortion taken into account. III. THEORETICAL COMPARISONS The method described in Section II can be applied to any combination of bandwidth, pulse lengths, and amplitude modulation specifications. As a primer for comparison between this method and previous pulse compression techniques, some of the most commonly-utilized and prevalent methods in the literature have been chosen as methods for comparison. Across these chosen waveforms, a common analysis point of timebandwidth product (TB) equal to 270 was chosen due to the value s prevalence in previous literature. This TB was achieved via a 200 µs, 1.35 MHz bandwidth pulse for each waveform. The comparison waveforms all utilized mismatched filtering with amplitude modulation applied on receive only. The values of two-way processing gain for these waveforms were 0.50 [7], 0.73 [8], 0.73 [2], and 0.51 [3]. These four waveforms were compared with four equal TB (and equal mainlobe width/resolution) optimized frequency modulated (OFM) waveforms using the proposed technique, with a raised cosine matched filter with roll off factors (ROF) of 0.10, 0.02, 0.01, and 0.00 (no windowing). The two-way processing gain values for the designed waveforms were 0.95, 0.99, 0.99, and 1.00, respectively. While the comparison list is hardly exhaustive, Table I shows significant differences between previous methods and the OFM waveforms. The best peak sidelobe level achieved in the comparison studies was db [3], however, such performance came with a heavy cost of 2.96 db loss. The best peak sidelobe level achieved via the proposed technique was db, but with a much more reasonable loss of 0.24 db due to the 0.10 ROF raised cosine matched filter. A nearly identical result of db was achievable with a 0.02 ROF filter, which yields a processing gain of 0.99 and a loss of 0.05 db. For many radar systems, range sidelobe levels of db are considered quite reasonable, especially given typical antenna sidelobe levels of both dish and array platforms. The critical difference becomes a sensitivity gain of 2.72 db, enough to vastly alter the cost of a radar system. As an example, the 0.02 ROF raised cosine example ACF is shown in Fig. 3. IV. SYSTEM IMPLEMENTATION The system being used for research, development, and testing of waveform techniques at the University of Oklahoma is the PX-1000 transportable, solid-state, polarimetric X-band radar [9]. While this system exists primarily as a weather observation tool, it is important to note that it is serving as a general proving ground for waveform development. Each step shown throughout this section is applicable to any radar hardware capable of custom waveform designs and pulse compression. PX-1000 operates via dual 100-W solid-state power amplifiers, and has a mission of collecting, at times, very low power returns from distributed hydrometeors. The

4 TABLE I. COMPARISON OF COMMON WAVEFORM TECHNIQUES Waveform Frequency Modulation Amplitude Modulation TB Product Processing Gain / Power Loss Peak Sidelobe Level Klauder et al [7] LFM Blackman-Harris Mismatch / 3.00 db db Cook and Paolillo 1964 [8] Stepped LFM Hamming Mismatch / 1.35 db db Griffiths and Vinagre 1994 [2] Stepped LFM Hamming Mismatch / 1.35 db db De Witte and Griffiths 2004 [3] NLFM Nuttall Mismatch / 2.96 db db 0.00 ROF OFM NLFM Raised Cosine (0.00 ROF) Match / 0.00 db db 0.01 ROF OFM NLFM Raised Cosine (0.01 ROF) Match / 0.03 db db 0.02 ROF OFM NLFM Raised Cosine (0.02 ROF) Match / 0.05 db db 0.10 ROF OFM NLFM Raised Cosine (0.10 ROF) Match / 0.24 db db TABLE II. PX-1000 SYSTEM SPECIFICATIONS General Operating Frequency 9550 MHz Instantaneous Bandwidth 5 MHz Sensitivity (w/ OFM pulse compression) < km (67 µs PW) Antenna 3-dB Beamwidth 1.8 Polarization Simultaneous Dual-Linear Gain 38.5 dbi Cross-Polarization Level < -26 db Rotation Rate Up to 50 s 1 Transmitter Type Dual Solid-State Power Amplifiers Peak Power 100 W per Channel Pulse Width 1-70 µs Pulse Repetition Frequency Hz Receiver Minimum Gate Spacing 30 m Maximum Range 60 km Fig. 4. Optimized frequency function for the PX-1000 waveform. combination of PX-1000 hardware and custom in-house signal processing and display/control software makes PX-1000 an ideal platform for pulse compression studies. The PX-1000 specifications are listed in Table II. PX-1000 allows for up to 5 MHz of chirp bandwidth, and up to a 70 µs pulse length. While these are the values that would typically be used within the optimization framework as waveform specifications, we have chosen to use only 2.2 MHz of chirp bandwidth, and a 67 µs pulse length. The reason for this decision is due to the use of a fill pulse for blindrange mitigation, which is explained later in this section. In addition to these specifications, a slight amplitude modulation is added to the pulse. In theory, the optimization framework is capable of designing high-performance waveforms with higher processing gains. In a real system, however, switching noise and transmitter imperfections can corrupt the edges of a nearlyrectangular pulse. Therefore, a processing gain of 0.95 (a raised cosine taper on both ends of the waveform with a ROF of 0.10) was chosen for the theoretical design of the PX-1000 waveform. Using the revised specifications, a theoretical optimized waveform can be designed using the previously mentioned optimization framework. The optimization results in a waveform with peak sidelobes of -59 db, integrated sidelobes of -37 db, and 3 db range resolution of 120 m. The final frequency function is shown in Fig. 4, and the resulting ACF can be seen in Fig. 6a. While theoretical results using this method are promising, it is important to demonstrate the performance of the waveform when passed through the actual system. Due to inherent transmitter imperfections, the resulting waveform performance in practice with PX-1000 is significantly worse than theory. This can be seen in Fig. 6b, where the pass-through waveform ACF has degraded to a -42 db peak sidelobe level. This is due in large part to transmitter droop, which often occurs during long pulse transmissions. Fig. 5a shows the actual pass-through waveform as measured by PX The center portion of the waveform was designed to be perfectly flat, but the transmitter has deformed the pulse in amplitude, resulting in the degraded performance seen in the ACF. In order to correct for transmitter distortions, a predistortion method was applied to the waveform before being passed through the transmitter. Pre-distortion was accomplished via measuring the pass-through signal and determining a transfer function between an intended, theoretical signal and the actual, observed signal. By inverting this transfer function, applying it to the theoretical signal within the frequency domain at each step of the optimization (Fig. 2b), and normalizing within the time domain, a corrected signal can be designed. When this signal is then corrupted by predictable transmitter distortions, the resulting signal has nearly perfect amplitude characteristics (Fig. 5b). The recovery of waveform performance characteristics is evident in Fig. 6c, where the final transmitted waveform ACF displays operational peak sidelobes of -52 db, a 10 db increase in performance compared with not accounting for pre-distortion. While theoretical performance levels are not achieved with this method, an operationally-feasible waveform with the same high processing gain of 0.95 is possible. Some additional issues must be considered with a dualpolarization radar using a long pulse. First, the long pulse results in a wide blind-range (on the order of 10 km in this case), which is unacceptable given the limited maximum range of the system (60 km). In order to mitigate this issue, the method described in [9] for blind-range filling is used. The

5 remaining 2.8 MHz of bandwidth and 2 of the remaining 3 µs of available pulse length are used to transmit a short, windowed fill pulse. This technique uses a real-time software architecture to blend the transition zone between the short and long pulses, and despite lower sensitivity within the blind-range, does a satisfactory job of filling in areas near the radar. Also, with low SNRs, especially within the blind-range, dual-polarization estimates suffer from various biases. An implementation of the multi-lag technique described in [10] for recovery of polarimetric moments is used throughout this paper. Finally, when using a long-pulse waveform for distributed target detection in extreme weather situations (i.e., hydrometeors and debris traveling in excess of 100 ms 1 near a tornadic vortex), the issue of Doppler tolerance must be discussed. Numerous studies have explored Doppler tolerance correction, including in distributed targets [5], however, most of these studies offered hardware-based solutions, or techniques which were not computationally feasible for real-time implementation. With the advent of increased computational power, as well as the flexibility of the PX-1000 signal processing software suite, an opportunity for new insight into Doppler tolerance exists. While no real-time Doppler tolerance correction was in place for the data presented in Section V, archived time series data allows for a simulated re-processing of moment data. A significant issue with Doppler tolerance correction, especially in extreme weather events where Doppler tolerance is a major concern, is the lack of reliable real-time dealiasing. With closely-confined velocity gradients which can peak at over 100 ms 1, dealiasing is usually performed in post-processing. However, assuming dealiased data is attainable, a number of options for Doppler tolerance correction exist. Due to the forced symmetric nature of waveforms designed using this optimization technique, a significant increase in PSL occurs Fig. 6. Measured pass through ACFs of an optimized waveform for the PX-1000 system. (a) Original waveform after optimization. (b) The theoretical waveform from (a) after it has been sent through the transmitter, with no predistortion applied. (c) Pass through waveform with pre-distortion applied. only on one side of the ACF, based on the phase direction of the moving target. Assuming the phase is known (i.e., the velocity moment is both dealiased and reliable without correction in its own right), a combination of matched filter directions can be used to lower embedded peak sidelobes. Unfortunately, this technique results in generally widened null-to-null mainlobe widths in high Doppler velocity situations. While this is certainly preferable, additional methods are possible. The authors are currently experimenting with FIR filter options for application to the matched filter, depending on the expected phase shift of distributed targets. This technique also requires the assumption that original velocity estimates are reliable without any prior correction, however, preliminary results have shown promising recovery of the reflectivity moment. This method works via an additive adjustment for phase shifts across the entire length of the pulse, while also accounting for many distributed targets along a radial. Additional work on this topic is necessary and ongoing, and both theoretical examples and comparisons using archived extreme weather events will prove useful in future studies. V. Fig. 5. PX-1000 pass-through measurements of optimized waveform. (a) With no pre-distortion. (b) With pre-distortion applied. R ESULTS As a demonstration of observational capability using the proposed waveform optimization technique, data from the 20 May 2013 Moore, Oklahoma EF-5 tornado collected with PX-1000 are presented below. Observations at approximately

6 480 m above ground level at a range of between km at peak tornado intensity (with radial velocities approaching 90 ms 1 in the core vortex) offer opportunities to explore sensitivity of measurements, the blind-range transition zone, and Doppler performance of the optimized waveform. Fig. 7 shows reflectivity and dealiased velocity moment estimates at 20:20:18 UTC, the time of near-peak intensity. Fig. 8. Differential reflectivity (left; db) and correlation coefficient (right; unitless) moment estimates of the 20 May 2013 Moore, Oklahoma EF-5 tornado at 20:20:18 UTC. mitigation, multi-lag estimation for polarimetric moments, and future applications to Doppler tolerance correction in extreme weather and distributed target scenarios. ACKNOWLEDGMENT Fig. 7. Reflectivity (left; dbz) and dealiased Doppler velocity (right; ms 1 ) moment estimates of the 20 May 2013 Moore, Oklahoma EF-5 tornado at near-peak intensity (20:20:18 UTC). This work was supported by the National Severe Storms Laboratory (NOAA/NSSL) under the Cooperative Agreement NA08OAR The authors would like to thank John Meier and Redmond Kelley for discussions regarding waveform design and system implementation, as well as David Bodine for insight into analysis of the 20 May 2013 case. We would also like to thank the Toshiba Corporation for useful discussions regarding waveform techniques. R EFERENCES At the same time, the core vortex of the tornadic circulation was located just beyond the blind-range transition zone, which can be seen via the circle of lowered SNR approximately 10 km from the radar location in the top panels in Fig. 7. In the areas of higher SNR, such as surrounding the tornadic signature and debris to the south of the tornado, the transition zone is not nearly as evident, as seen in the bottom panels in Fig. 7. Despite no attempted correction for Doppler tolerance in the waveform or matched filter designs, the general structure of both the reflectivity and Doppler velocity fields match both expectations and other observations by radar platforms in the area. Fig. 8 shows the differential reflectivity and correlation coefficient moment estimates using the multi-lag method from [10] at the same time as the data shown in Fig. 7. Again, the horizontal structure of both moments agrees well with what would be expected from supercell dynamic theory [11]. Within the blind-range, where SNR values are significantly lower, the use of multi-lag estimation offers nearly identical quality to what is seen outside of the blind-range. [1] [2] [3] [4] [5] [6] [7] [8] [9] VI. C ONCLUSIONS A novel approach to optimized high-sensitivity pulse compression waveform design for radar systems has been presented. A discussion of the optimization framework shows the workflow of the design process, and a comparison to other pulse compression techniques has been made. Data from a significant tornadic weather event in Oklahoma show promise for the combination of long-pulse waveforms, blind-range [10] [11] C. E. Cook and M. Bernfeld, Radar Signals. Academic Press, H. D. Griffiths and L. Vinagre, Design of low-sidelobe pulse compression waveforms, IEEE Electronics Letters, vol. 30, no. 12, pp , E. De Witte and H. D. Griffiths, Improved ultra-low range sidelobe pulse compression waveform design, IEEE Electronics Letters, vol. 40, no. 22, pp , G. E. Farin, Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide, 4th ed. Orlando, FL: Academic Press, N. J. Bucci and H. Urkowitz, Testing of Doppler tolerant range sidelobe suppression in pulse compression meteorological radar, in International Radar Conference. Moorestown, NJ: IEEE, 1993, pp J. M. Kurdzo, B. L. Cheong, R. D. Palmer, G. Zhang, and J. B. Meier, A pulse compression waveform for improved-sensitivity weather radar observations, J. Atmos. Ocean. Technol., under review. J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, The theory and design of chirp radars, Bell Syst. Technol. J., vol. XXXIX, no. 4, pp , C. E. Cook and J. Paolillo, A pulse compression predistortion function for efficient sidelobe reduction in a high-power radar, Proc. IEEE, vol. 52, pp , B. L. Cheong, R. Kelley, R. D. Palmer, Y. Zhang, M. Yeary, and T.Y. Yu, PX-1000: A solid-state polarimetric X-band radar and timefrequency multiplexed waveform for blind range mitigation, IEEE Trans. Instrum. Meas., vol. 62, no. 11, pp , L. Lei, G. Zhang, R. J. Doviak, R. Palmer, B. L. Cheong, M. Xue, Q. Cao, and Y. Li, Multilag correlation estimators for polarimetric radar measurements in the presence of noise, J. Atmos. Oceanic Technol., vol. 29, no. 6, pp , M. R. Kumjian and A. V. Ryzhkov, Polarimetric signatures in supercell thunderstorms, J. Appl. Meteor. Climatol., vol. 47, pp , 2008.