TRANSMIT BEAMFORMING AND WAVEFORMS FOR RANDOM, SPARSE ARRAY RADAR John K. Schindler*, Hans Steyskal† *ARCON Corporation, 260 Bear Hill Road, Waltham MA 02154 USA †AFRL/SNHA, 80 Scott Circle, Hanscom AFB, MA 01731 USA Fax: 781 377 8984; E-Mail:
[email protected] Keywords: Space-Based Radar, Space-Time Processing, Sparse Arrays, Displaced Phase Center Arrays.
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Abstract We address the design of radar waveforms, transmit and receive beamforming, and signal processing in moving, sparse array antennas with randomly positioned elements. We are motivated by a concept for less expensive space based radar consisting of a formation of small, autonomous, identical satellites in nearly the same low earth orbits. The satellites operate collaboratively as a coherent, large aperture radar to detect and locate slowly moving targets near the earth surface. Limited fuel mandates that the satellites navigate only to avoid collision within array aperture limits and not necessarily to maintain any ideal array antenna configuration. Sensitive, coherent cancellation of ground reflections implies that the transmit and receive beamforming and waveforms create highly correlated space-time samples of the reflected ground clutter. Using an electromagnetic model for the ground reflected signals, we determine general, displaced phase center conditions on the transmit waveform and transmit/receive beamforming to create the highly correlated samples. The random array element positions are assumed to be known from independent measurement. We then describe a novel, iterative approach to the design of transmit and receive beamforming weights that maximizes the ratio of the output target signal to interference when the waveform conditions for signal correlation are satisfied. A constraint on the total radiated energy is implicit in the design. Results of the optimization for specific random array realizations reveal interesting conclusions: • The displaced phase center condition requires that the phase center of the transmit and/or receive arrays translate backwards, against the direction of array motion, with nearly identical patterns between successive pulses. When applied to the random, sparse array, we find that we may use only the complete array (without translation) and/or individual array elements, with backward translation between the elements dictated by the inter-pulse time and radar speed along the array; • The inter-pulse times are selected based on the random but known transmit and/or receive inter-element spacings. Criteria for selection include (1) emphasis on longer inter-pulse times to increase radar sensitivity to slowly moving targets and (2) use of multiple inter-pulse times to minimize blind speeds in the beamforming; processing; work not protected by U.S. Copyright U.S. Government
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Iterative optimization of the transmit/receive beamforming with total energy constraint results in (1) nearly uniform transmit array illumination and (2) emphasis on individual receive array elements satisfying the displaced phase center condition and forming two pulse cancellers of the ground clutter. The canceller outputs are combined linearly to provide the detection statistic. Negligible loss (< 0.2 dB) results when optimization is constrained by the use of uniform, maximum gain transmit weights and only those receive elements satisfying the displaced center condition; Losses due to the lack of ground clutter cancellation in the displaced phase center elements are caused in part by random (and unknown) timing and element position errors. This suggests using overlapped sub-arrays in the physical element (satellite) arrays to compensate adaptively for these errors. Monte Carlo simulations reveal that with half wavelength separation between the sub-arrays, position errors on the order of one wavelength are allowable by appropriate linear combination of sub-array outputs.
1 Introduction We discuss the design of transmit waveforms and time varying transmit beamforming for the space-based radar detection of slowly moving ground targets with large, spatially thinned, random array antennas. Detection with the moving, space-based platform is based on differences in the spatial and Doppler frequency properties of the signal scattered from the moving target and ground scattering or clutter. Conventionally, the nearly monochromatic Doppler frequency of the slowing moving vehicle must be outside the Doppler bandwidth of the ground clutter as shaped by the transmit and receive antenna power patterns of the radar. Improved detectability requires reducing the power spectral density of the ground clutter within the bandwidth of the slowly moving vehicle as determined by vehicle acceleration, scatter fluctuations and the coherent dwell time of the radar. Displaced phase center and other more general receive array processing approaches based on the space-time properties of the received signals offer sensitive slow speed target detection. However they require substantial antenna pattern control, adaptive processing, and restrictions on the precision of the radar pulse repetition frequency (PRF) that are all the more difficult to implement with large, sparse, random arrays. We describe gains to be made in space-time receive processing performance with design of transmit waveforms and transmit array patterns in sparse, random arrays.
The U.S. Air Force has considered a multi-function spacebased sensor concept that employs a cluster of identical, freefloating, semi-autonomous and self-organizing microsatellites to form a large space aperture [1]. Proposed missions include surveillance radar detection and location of moving vehicles, synthetic aperture mapping, emitter location, and both local area and wide area communications. Potentially, the use of small, free-floating satellites can accomplish the multiple missions of the sensor at reduced cost, while allowing for gradual performance enhancement and sensor maintenance using residual space launch capabilities over time. We consider perhaps the most demanding mission for this sensor -- wide area detection of slowly moving vehicles. We envision each satellite as having nearly identical, 2m x 2m arrays. Each satellite transmits its own, possibly orthogonal signal and receives the reflected signals from all transmitting satellites, similar to the French RIAS ground based system [2]. The satellites are envisioned to operate coherently at X-band (3 cm wavelength). The cluster of satellites forms a large (∼100m), sparse, multielement, time varying phased array with narrow beamwidth and concomitant grating or random sidelobes that introduce significant ground clutter into the received signal. Previous work described approaches to distributed space-based radar employing satellites in a single orbital plane [3,6,7,8,11,17]. In our previous work [9,10,12,13,14,15], the satellites were assumed to be in multiple, nearly circular, low Earth orbits with a common orbital plane and thus form a sparse, periodic, two dimensional transmit-receive array with limited along track dimension operating at one of a carefully selected set of PRFs. The PRFs are selected to exploit the nearly periodic array pattern structure maintained throughout the Earth orbit to achieve nearly optimum, displaced phase center cancellation of ground clutter. The performance of the optimum receive signal space-time processing with maximum gain transmit arrays for the detection of slowly moving ground targets was presented and compared with simpler, but sub-optimum, separable or sequential receive array-pulse processing architectures. Kepler’s laws of orbital mechanics tend to preserve the periodic array structure found necessary for optimum ground clutter suppression. However, second order effects such as differential satellite drag and gravitational perturbations, as well as limited deployment precision, make it difficult to create and preserve the necessary array periodicity for long time periods with limited satellite fuel consumption. Therefore, in this paper we study the detection performance of an array composed of satellites randomly distributed within the same limited aperture to determine gains in performance by control of the transmit waveforms and time-varying transmit array beamforming. The satellites are assumed to perform autonomous station-keeping with knowledge of the other satellite positions and velocities. They are required only to maintain position within the aperture boundaries and avoid collisions. It is assumed that global knowledge of the satellite positions and velocities is available and used for waveform and satellite scheduling and coherent transmit-receive signal
processing. Only the case of a single orbit, linear random array is considered here.
2 Waveform Design Space-time array design is based on maximization of the signal to interference ratio (SIR) given by the ratio of signal power to clutter and receiver noise power considered as a function of the column vector of complex space-time weights of the receive aperture, R . The SIR is proportional to H s R
R
(1) , H c R NCRI R where, s and c represent the signal and clutter covariance matrices respectively, implicitly a function of the transmit signal space-time weights and the pulse repetition frequency. Also NCR is the element noise to clutter ratio and R adH represents the conjugate transpose of the receive aperture weights. Optimum receive space-time processing weights follow from generalized eigen-analysis of this Rayleigh quotient with known covariance properties [4]. When the signal to interference ratio is considered as a function of the transmit weights, T , for a given receive spacetime processing, the SIR becomes proportional to H
T
H
T sT 2 , c T NCR R
(2)
where now s and c are the signal and clutter covariance matrices found with assumed known receive space-time weights. Here the SIR is no longer a Rayleigh quotient of quadratic forms in the transmit weights and thus the generalized eigen-analysis of this SIR does not apply. However, this difficulty may be resolved with a natural, physical constraint on the total radiated energy, given by H (3) To T I T Here To is the total space-time waveform energy specified to insure sufficient receive signal to noise after coherent integration. With this constraint, the signal to interference can be written as T H sT T
H
c
NCR R 2 To
.
(4)
I T
Now this is a Rayleigh quotient and subject to conventional eigen-analysis for the transmit weights that maximize signal to interference, with a total radiated energy constraint. In this paper we investigate an iterative approach to the design of both transmit and receive weights, including the use of pulse stagger (variable inter-pulse times) to facilitate displaced phase center implementation with a linear array having element positions that are random but known by measurement. Initially, the transmit space-time weights are assumed to be a linear combination of maximum gain and random weights on all elements or on selected elements that satisfy the generalized displaced phase center condition. The optimum receive weights are found from analysis of Equation (1) and given these weights, the covariance matrices of
Equation (4) can be computed, yielding optimum transmit weights from eigen-analysis of Equation (4). After normalization of the transmit weights to conform to the total energy constraint Equation (3), the iterative process returns to Equation (1) to find new receive space-time weights with the covariance matrices in Equation (1) found from the latest transmit weights. This process continues, giving monotonically non-decreasing improvement in signal to interference and terminates when incremental increases are sufficiently small. This approach does not necessarily yield global maximization of signal to interference. Indeed, selecting the initial transmit weights as described above facilitates investigation of the possibility of multiple local maxima that may depend on initial condition assumptions. In this paper, the design is performed assuming that the elements of the covariance matrices are known (as opposed to measured) in order to establish (a) performance bounds on adaptive processing and (b) essential features of simplified space-time signal processing architectures suitable for adaptive processing. This iterative design provides optimum, time varying transmit and receive beamforming weights. However, implicit in the weight design are the selected inter-pulse times of the transmit waveforms that are necessary to compute the signal and clutter covariance matrices. In this design process, the inter-pulse times are selected to insure correlation between the space-time samples of the clutter signal and thus permit coherence signal processing. Following Ward [16] and Klemm [5], the elements of the clutter covariance matrices can be found using the space-time structure of the signals scattered from the spatially distributed, random ground clutter. The elements are a function of the space-time variable, ξ, where −
+
−
+
ξ
(
− τ
)−
λ
(
− τ
)
+
−
. (5)
Here dti(dtj) denotes the random position of the ith(jth) transmit element/satellite in the array at the time tp(tq) of reception of the pth(qth) pulse; drm(drn) denotes the random position of the mth(nth) receive element/satellite in the array at the time tp(tq) of reception of the pth(qth) pulse; VTi, VTj, VRm, andVRn represent the along track speed of the ith and jth transmit elements and mth and nth receive elements respectively and τd is the delay to the target/clutter zone. When all elements/satellites have a common along track speed VT and ξ is small compared to a wavelength insuring highly correlated space-time samples, the inter-pulse time δt = tq-tp must satisfy −
δ
+
−
(6)
Equation (6) expresses a general displaced phase center condition that must be satisfied. This condition implies that for each pulse, the sum of the distances to the transmit and receive element/array phase centers from a fixed spatial reference point is conserved. A consequence of this condition is that the pathlength from transmit to receive phase centers is constant for each pulse for scattering elements at all azimuth
angles from the array, facilitating wide angle clutter cancellation. To satisfy Equation (6) and insure that the interpulse time is positive, there must exist backward displacement of the transmit array phase center (dtj < dti) and/or receive array phase center (drn < drm) on the later pulse. Further, this backward displacement must be achieved in a manner that achieves nearly identical array patterns. This displaced phase center condition determines feasible array configurations and inter-pulse times for random, sparse array processing. We must use either (a) the complete sparse array with no backward displacement on transmit (receive) and individual, identical elements/satellites with backward displacement between the elements/satellites on receive (transmit) or (b) individual, identical elements/satellites with backward displacement on transmit and/or receive. Spatial displacement of sub-arrays of the random sparse array result in sub-array patterns that are different, degrading the potential for wide-angle clutter cancellation. The optimization process described above selects the appropriate transmit/receive elements or arrays provided the inter-pulse times used in the optimization satisfy Equation (6) based on the assumed known element positions.
3 Performance This design approach has been applied to a linear array with five elements (satellites) in the same circular, low earth orbit and pseudo-randomly distributed within an aperture of 1000λ. Exclusion zones of ±10λ around each element are assumed in order to avoid collision among the freely moving satellite elements. Each element (satellite) is itself a 10λ array with uniform 0.5λ element spacing. Here λ is the wavelength at the radar carrier frequency. Only element (satellite) motion along the axis of the array is allowed. The element (satellite) positions are assumed to be known for purposes of the design, although compensation for residual position uncertainty was analyzed. Equation (6) implies that the inter-pulse times must be less than the sparse array transit time – the time required for the array to move a single array length. The distribution of interpulse times is defined by the random but known inter-element spacings. Smaller inter-pulse times are more probable in the distribution [10]. The inter-pulse times for waveform transmission are selected to (1) emphasize longer inter-pulse times to increase radar sensitivity to slowly moving targets and (2) minimize blind speeds in the beamforming processing by the use of multiple inter-pulse times. Blind ranges are not a consideration here since signal delay to the target zone is greater than one array transit time, the maximum inter-pulse time. Figure 1 illustrates the evolution of the space-time transmit and receive processing weights during iterative optimization. We assumed five pulses with known inter-pulse times and a target radial speed that is Gaussian distributed with RMS target speed equivalent to 5 mph (2.2 m/s). The weights converge rapidly to nearly uniform weights on transmit and
Loss in SIR Improvement (dB)
•• •••
analyzed using Monte Carlo simulation. Cross track position and pattern errors are the subject of ongoing investigations. Gaussian position or timing errors with specified error variance were introduced and the average loss in signal to interference ratio when compared to the no error case was computed assuming that the transmit and receive processing weights are those of the no error case.
Figure 1. Evolution of Transmit and Receive Weights with Iterative Optimization; 5 Element Sparse Random Array with 5 Pulses; Slow Speed Target; Transmit Energy Sufficient for Clutter Limited Case.
Figure 2 illustrates for this case the space-time locations of significant transmit (lower left) and receive (upper right) satellite elements. Each of the five satellite elements is located in space (abscissa in aperture lengths) and time (ordinate in array transit times). The significant receive elements for seven canceller pairs are highlighted, with backward displacement of the later satellite elements shown. The processing of the canceller pairs and coherent processing of their outputs is illustrated at the right. Delay (Aperture Transit Time)
7 Displaced Phase Center Cancellers
e
-j
1
• • •
• • • e
-j
Along Track Position Errors
Timing Errors
-o- Average Loss ± Standard Deviation
-o- Average Loss ± Standard Deviation
Receive Weights
Transmit Weights
Loss in SIR Improvement (dB)
paired elements on receive for each pulse. The receive weights for each pair are the negative of one another indicating a form of two pulse cancellation on receive.
7
w1 ∼ -1
wN ∼ -1
RMS Error in Along Track Position of Element (wavelengths)
Radar Speed x Timing Error (wavelengths)
Figure 3. Average Loss in Signal to Interference (dB) due to Along Track Position and Timing Errors (wavelengths). Errors in timing when multipled by the along track speed of the radar provide effective position errors.
Figure 3 shows the results. Moderate losses (∼ 3 dB) occur with errors that are 0.01λ and substantial losses (> 15 dB) occur with errors that are 0.1λ. Thus, extraordinary accuracy in the measured element positions and inter-pulse timing is required over the large, 1000λ sparse aperture. Likely, adaptive compensation for these errors will be required. To assess the potential for adaptive compensation, we investigated the source of the performance degradation illustrated in Figure 3. Performance is expressed as the improvement in signal to interference where the sources of interference are receiver noise and ground reflections or clutter. We examined the changes in average signal power, noise power and clutter power as the RMS position and timing errors increase.
Relative Average
Figure 2. Location in Space and Time of Transmit and Receive Elements with Relative Amplitude Weights Greater than -30 dB and Associated Processing of Received Space-Time Signals with Delay Line Cancellers and Coherent Processing of Canceller Outputs.
Relative Average
Element Position (Aperture Lengths)
Along Track Position
For this case, we investigated iterative maximization of the signal to interference with the following constraints: (1) on each pulse, each element of the transmit array has the same complex amplitude and the complex pulse amplitudes were optimized; (2) the total energy transmitted was limited and sufficient for clutter limited operation and (3) only the receive elements on each pulse obeying the displaced phase center condition were allowed to operate. Constrained optimization at each of a wide range of RMS target speeds revealed less than 0.2 dB degradation in the improvement in signal to interference when compared to results of the unconstrained optimization the same RMS speed. Displaced phase center beamforming is sensitive to errors in inter-pulse timing, array positions and array patterns. The performance degradation of the array configuration described above to random along track position and timing errors was
RMS Error in Along Track Position of Element (wavelengths)
Timing
o
o
o
Signal Power - S Noise Power - N Clutter Power – C
Radar Speed x Timing Error (wavelengths)
Figure 4. Relative Signal, Clutter and Noise Power as a Function of Along Track Position and Timing Errors
The results of this investigation, shown in Figure 4, indicate that the signal power and noise power are essentially independent of position and timing errors. Here the total transmit energy and noise are set to yield a 13 dB signal to noise ratio. In contrast, the clutter power is a sensitive, monotonically increasing function of the errors. A clutter to noise ratio of 0 dB results in the 3 dB degradation in signal to interference noted above. Thus, only the clutter cancellation resulting from each of the seven space-time canceller pairs illustrated in Figure 2 is sensitive to position and timing errors.
This observation suggests an approach to compensate adaptively for these errors. We suppose that each of the receive satellite array antennas in a space-time canceller pair is subdivided into overlapped subarrays. The length and spacing between the subarrays dictates the number of subarrays that fill the full satellite array. The subarray outputs are linearly combined with complex weights selected to minimize the clutter power from contiguous range cells with only clutter present.
6 References
Inprovement Factor Gain (+) and Loss (-)
[1] Air Force Research Laboratory. “TechSat21 - Next Generation Space Capabilities” , www.vs.afrl.af.mil/TechProgs/TechSat21, (1999). [2] J. Dorey et al. “RIAS, Radar à Impulsion et Antenne Synthetique”, Int’l Conf. on Radar, (April 1989). [3] J. H. G. Ender. “Spacebased SAR/MTI using Multistatic Satellite Configurations”, 4th European Conference on Synthetic Aperture Radar (EUSAR 2002), (June 2002). [4] D. Johnson and D. Dudgeon. Array Signal Processing, Prentice Hall, (1993). Figure 5. [5] R. Klemm. Space-Time Adaptive Processing, Institution Improvement Factor Gain/Loss as of Electrical Engineers, pp 71-79, (1998). a Function of RMS [6] D. A. Leatherwood and W. L. Melvin. “Adaptive Number and Length of Sub-Arrays Position Error in Processing in a Nonstationary Spaceborne Environment”, 1, 10.0λ λ 2, 9.5λ λ Wavelengths Proceedings of the IEEE Aerospace Conference, (March 5, 8.0λ λ • 200 Trials for 10, 5.5λ λ 2003). 15, 3.0λ λ Each RMS Error [7] D. Massonnet. “Capabilities and Limitations of the • 10 λ Aperture Interferometric Cartwheel”, IEEE Transactions on • 0.5λ Between Geoscience and Remote Sensing, vol. 39, no. 3, (March Apertures 2001). [8] S. Ramongassie, et al. “Preliminary Design of the Payload for the Interferometric Cartwheel”, 4th European Conference on Synthetic Aperture Radar (EUSAR 2002), (June 2002). [9] J. K. Schindler, H. Steyskal and P. Franchi. “Pattern RMS Position Error/λ λ Synthesis for Moving Target Detection with TechSat21 - A To bound the adaptive performance, we applied random Distributed Space-Based Radar System”, Proceedings of IEE position errors with specified standard deviation in 200 Monte Radar 2002 Conference, (October 2002). Carlo trials. The average gain or loss in improvement factor [10] J. K. Schindler, H. Steyskal. “Sparse, Random Array resulting from optimum beamforming with known errors is Processing for Space-Based Radar”, Proceedings of the 36th plotted in Figure 5. It is clear from comparison of the results IEEE Southeastern Symposium on System Theory, (March at 0 dB change in improvement, that the proper use of 2004). overlapped subarrays increases permissible position errors by [11] R. J. Sedwick, T. L. Hacker, and K. Marais. 2 to 4 orders of magnitude. This results from the increased “Performance Analysis for an Interferometric Space-Based signal to noise from the multiple subarrays, each with reduced GMTI Radar System”, IEEE International Radar Conference gain, at small errors and the error compensation from the use – RADAR 2000, (May 2000). of the subarray with appropriate spacing for the errored [12] H. Steyskal, J. K. Schindler, P. Franchi and R. J. condition. Mailloux. “Pattern Synthesis for TechSat21 - A Distributed Space-Based Radar System,” IEEE Antennas and 4 Conclusions and Future Work Propagation Magazine, (August 2003). [13] H. Steyskal, J. K. Schindler. “Separable Space-Time We have described an iterative approach to the design of radar waveforms, beamforming and processing for random, Pattern Synthesis for Moving Target Detection with TechSat sparse arrays under the constraint of limited transmit energy. 21 - A Distributed Space-Based Radar System,” Proceedings Analysis of designs reveals necessary restrictions on the of the IEEE Aerospace Conference, (March 2003). waveforms and transmit beamforming and simplifications in [14] H. Steyskal, J. K. Schindler, P. Franchi, R. J. the space-time receive signal processing, when compared to Mailloux. “Pattern Synthesis for TechSat 21 - A Distributed general space-time processing. Indeed, the simplified Space Based Radar System,” Proceedings of the IEEE Conference, (March 2001). processing points to an approach to adaptive processing that Aerospace [15] H. Steyskal, J. K. Schindler. “Beamforming and Signal obviates the need for precision measurement of element Processing with Sparse, Random Arrays for Space Based positions. Radar”, Proceedings of the IEEE Aerospace Conference, Future work will include the effects of cross track motion (March 2004). resulting from earth rotation, satellite pattern errors and [16] J. Ward. “Space-Time Adaptive Processing for Airborne internal clutter motion. Radar”, MIT Lincoln Laboratory Report 1015 (ESC-TR-94109), pp 7-24, (December 1994). 5 Acknowledgement [17] M. Zatman. “Space-Time Processing using Sparse This work was sponsored by the Mathematics and Space Arrays”, Adaptive Sensor Array Processing Workshop, MIT Sciences Directorate of the U.S. Air Force Office of Scientific Lincoln Laboratory, (March 2003). Research, under the direction of Dr. Arje Nachman.
