Phase-Only Adaptive Nulling with a Genetic Algorithm

Transcription

1 Phase-Only Adaptive Nulling with a Genetic Algorithm Randy L. Haupt HQ USAFADFBE 2354 Fairchild Dr, Suite 2F6 USAF Academy, CO hauptrl.dfee@ usafa.af.mil Sue Ellen Haupt HQ USAFA/DFP 2354 Fairchild Dr USAF Academy, CO hauptse.dfp@usafa.af.mil Abstract-This paper describes a new approach to adaptive phase-only nulling with phased arrays. A genetic algorithm adjusts some of the least significant bits of the beam steering phase shifters in order to minimize the total output power. Using a few bits for nulling speeds convergence of the algorithm and limits pattern distortions. Various results are presented to show the advantages and limitations of this approach. TABLE OF CONTENTS 1. INTRODUCTION 2. PROBLEM FORMULATION 3. THE ADAPTIVE ALGORITHM 4. RESULTS 5. CONCLUSIONS 1. INTRODUCTION Low sidelobes don't guarantee adequate reception of a desired signal in the presence of interfering sources. Adaptive nulling complements the low sidelobe antenna by placing nulls in a few low sidelobes to reject the strongest interfering sources. An ideal adaptive algorithm for a phased array antenna has the following desirable characteristics: Places multiple deep nulls in the directions of interference, Rejects interference over the bandwidth of the antenna, Places the nulls very quickly, Complements existing phased array technology, and Minimizes pattern perturbations. An adaptive algorithm possesses some of these characteristics, but no adaptive algorithm meets all the characteristics. Selection of the adaptive algorithm, hence the desirable characteristics, depends upon the antenna, the cost, the performance requirements, and the interference environment. An off-the-shelf adaptive algorithm is not usually suitable for use with an off-the-shelf phased array antenna. Many adaptive antenna array algorithms originate from the generic signal processing literature [ 13 and require modification of the algorithm and the antenna in order to have a working adaptive array. An adaptive algorithm developed for data transmission requires hardware and software modification in order for it to successfully place nulls in the far field pattern of an array /97/$ IEEE 151

2 Most adaptive antenna algorithms multiply the quiescent weights by the inverse of the sampled covariance matrix to get the adapted weights. The resulting complex weights place nulls in the far field pattern in the directions of interference. A sampled covariance matrix is formed from the complex signals received at each element in the array. Although mathematically elegant and fast these methods have two impractical hardware requirements on the antenna array. First, the array must have an expensive receiver or correlation at each element. Most arrays have a single receiver at the output of the summer, so the antenna must be designed especially for the algorithm. Not only are multiple receivers expensive, but the receivers require a sophisticated method for calibration [2]. Second, the array must have variable analog amplitude and phase weights at each element. Usually, a phased array has only digital beam steering phase shifters at the elements. The feed network determines amplitude weights. There are two problems fkom an algorithmic standpoint as well. First, digital phase shifters only approximate the phase calculated by the adaptive algorithms. The weight quantization error limits null placement. Second, these algorithms get stuck in local minima [3]. As a result, they do not find the optimum weights to reject the interference at hand. Some common adaptive algorithms include Least Mean Square Algorithm and Howells-Applebaum Adaptive Processor, and examples can be found in references [3] and [4]. These methods are very fast but the difficulties mentioned prohibit their wide-spread use, particularly for arrays with more than a handful of elements. Another class of algorithms adjusts the phase shifter settings in order to reduce the total output power from the array [5], [61, [71. These algorithms are cheap to implement because they use the existing array architecture without expensive additions, such as adjustable amplitude weights or correlators. Their drawbacks include slow convergence and possibly high pattern distortions. This class has four approaches, the last of which is the topic of this paper. The first approach is the random search algorithm [3]. Random search algorithms randomly sample a small fraction of all possible phase settings in search of the minimum output power. The search space for the current algorithm iteration can be narrowed around the regions of the best weights of the previous iteration. This approach is usually too slow for beam steering and radar applications. It is less likely to get stuck in a local minimum and does not require an expensive receiver at each element. A second approach forms an approximate numerical gradient and uses a steepest descent algorithm to find the minimum output power [8]. This approach has been implemented experimentally but is slow and gets trapped in local minima. As a result, the best phase settings to achieve appropriate nulls are usually not found. The third approach is a beam space algorithm that assumes the location of the interference is known. This algorithm forms a cancellation beam in the direction of the interference. The height of the cancellation beam is adjusted to cancel the sidelobes and place a null in the interference direction. This approach is fast but requires knowledge of the interference locations and a reasonably accurate estimate of the amplitude and phase weights at each element. Serious drawbacks to current adaptive algorithms include: 1. They require an expensive receiver at each element - makes array impractical to build; 2. They get trapped in local minima - don't use full potential of the antenna to reject interference; 152

3 3. They slowly converge - often not useful for radar or scanning applications; 4. They can't be implemented on existing antennas--they require adjustable amplitude weights and receivers at every element in addition to beam steering phase shifters; 5. They cause the main beam to move from its desired pointing direction; and 6. They significantly raise the sidelobe levels of a low sidelobe array. elements \/ Y incident field This paper describes a simple technique suitable for implementation on existing phased arrays. The approach combines a genetic algorithm with the hardware limitations of the array to place nulls in the directions of interference with small perturbations to the far field pattern. Excellent nulling results are possible for most interference scenarios. 2. PROBLEM FORMUL,ATION A linear array antenna is a group of equally spaced antennas arranged along a line and whose outputs are added together to provide a single output. Figure 1 shows a diagram of such an array. Mathematically, the array far field pattern is given by [ 11 where wn = ant?" complex weight at element n 2N = number of elements in the array Y =kdu+ A k = 2dA A = wavelength d = spacing between elements U = cos$ $ = angle of incidence of electromagnetic plane wave A = beam steering phase Figure 1. Diagram of a phase-only adaptive linear array The amplitude weights are fixed. Lowering the sidelobe levels requires an even phase shift about the center of the array [9], while nulling requires an odd phase shift [lo]. Since nulling is of importance here, (1) simplifies to M(u) = N n=l cos[(n - 1)Y + A,, (2) Note that Y = kdu and no longer includes the beam steering phase. Since the beam steering phase is quantized, it differs from (n - l)a and must be represented by A,,. The steering phase, A,,, is calculated first and the beam steered to the proper angle, before the nulling phase, 6,, is found. Figure 1 is a diagram of a phase-only adaptive array with A,, 4. The digital phase shifters have B bits. B needs to be as small as possible to reduce the cost of the phase shifter but should be large enough to maintain low sidelobes over the scan angles of 1 53

4 the array. The quantized steering phase at element n is given by whereas the nulling phase is represented by P 6, = 2ax b, 2- (4) p=l where B = total number of bits in phase shifters P = number of phase bits used for nulling [ b,b,... bp] = vector containing the nulling bits representing 6, round{ *} = round * to the nearest integer rem{ *} = takes only the digits to the right of the decimal point of * In order to minimally perturb the array pattern, the adaptive algorithm assumes P<B. 3. THE ADAPTIVE ALGORITHM A phase-only adaptive algorithm modifies the quantized phase weights based on the total output power of the array. If no interference is present, then the algorithm tries to minimize the desired signal. To prevent desired signal degradation, the algorithm should only be turned on when the desired signal becomes swamped by the interference or the nulling phase shifts should be small. This potential problem is discussed in more detail in the next section. The disadvantages of the phase-only algorithms make them unlikely candidates for use with antenna arrays. This section presents a method that is as fast as the beam space algorithm, doesn t easily get stuck in local minima, and limits pattern distortion. The adaptive algorithm is based on a genetic algorithm and uses a limited number of bits of the digital phase shifters. A genetic algorithm is a computer program that finds an optimum solution by simulating evolution in nature. In this application the phase shifter settings evolve until the antenna pattern has nulls in the directions of jammers. A genetic algorithm was chosen for this application because it is an efficient method to perform a search of a very large, discrete space of phase settings to achieve the minimum output power of the array. An adaptive phase-only array has 2NB possible phase settings, many corresponding to local a in the total power output. Such a large number of phase settings and local minima make random search and gradient based algorithms impractical to use. Fill phase settings matrix with random 1 s and 0 s Figure 2. Flow chart of the genetic algorithm used for adaptive nulling. 154

5 phase shifter r, nulling bits Y Y Y Y Y E E E E E )b) b) 6) - output power Figure 3. A population of phase shifter settings with corresponding output powers. Figure 2 shows a flowchart of the adaptive genetic algorithm. It begins with an initial population consisting of a matrix fiued with random ones and zeros. Each row of the matrix (chromosome) consists of the nulling bits for each element placed side-by-side. There are NP columns and M rows. The output power corresponding to each chromosome in the matrix is measured and placed in a vector (Figure 3). M must be large enough to adequately search the solution space and help the genetic algorithm arrive at an excellent solution. On the other hand, M needs to be small, so the algorithm is fast. The speed of the algorithm is also a function of N and P. As N and P increase in size, M needs to be larger to keep the algorithm out of local minima, and the number of iterations required for convergence increases. The output power vector and associated chromosomes are ranked with the lowest output power on top and the highest output power on the bottom. The next step discards the bottom X% of the chromosomes, because they have the greatest output power. The algorithm generates new nulling chromosomes from the chromosomes that were kept to replace those discarded (Figure 4). Two chromosomes are selected at random. Chromosomes with lower output power receive a correspondingly higher probability of selection. Next, a random point is selected and bits to the right of the random point are swapped to form two new chromosomes. These new chromosomes are placed in the matrix to replace two settings that were discarded, and their output powers are measured. When enough new chromosomes are created to replace those discarded, the chromosomes are ranked and the process repeated. A small number (less than 1%) of the nulling bits in the matrix can be randomly switched from a one to zero or visa versa. These randomly induced errors (mutations) allow the algorithm to try new areas of the search space, while it converges on a solution. Usually, the best phase setting is not randomly altered. More general descriptions of genetic algorithms can be found in [ll] and [12]. The next section shows results for determining the best values for P and M. r m cd c e, d 9 phase shifter. output settings - power Figure 4. Two partners are selected from the mating pool to produce two offspring that are put back into the population to replace those chromosomes that were discarded. 155

6 4. RESULTS The genetic algorithm used here begins with 20 chromosomes. Only 10 are kept during the 25 iterations of the simulation. Each iteration, the bottom 6 chromosomes are discarded and replaced by chromosomes generated from the top 4 chromosomes. These numbers are small enough to keep the algorithm fast but large enough to place the nulls. The first example array modeled in this paper has 40 elements and a 30 db Chebychev amplitude taper. Elements are spaced 0.5A0 apart at the center frequency fo. Phase shifters must have six bit accuracy in order to keep the quantization lobe slightly below the general sidelobe level over the scan angles of the array (+-30" from broadside). The adaptive array model was tested for five different interference scenarios and judged based on three performance criteria. Results appear in Table 2. The first performance characteristic is the sidelobe reduction at the interference angle(s). When there are two interference sources (ui=.62 and.72), the minimum sidelobe reduction of the two angles is listed in the table. The second performance characteristic is the number of power measurements required for a 10 db reduction in the sidelobe level. This number directly relates to the speed of the algorithm. The third performance characteristic is the maximum sidelobe level, which is an indication of the amount of pattern distortion. A lower value for any performance characteristic indicates better performance. Table 1. The genetic algorithm adaptive array was tested for five scenarios. The seven columns to the right list the number of nulling bits and the phase value of the most significant bit (MSB). The three performance characteristics judge the algorithm for null depth, speed, and pattern distortion. A low value for a performance characteristic is good. 1 56

7 Two important parameters are the number of bits used for nulling and the maximum phase shift or the phase of the most significant bit (MSB). A genetic algorithm is often sensitive to the number of bits in a chromosome. Thus, comparing the performance of the algorithm against the number of bits used for nulling is important. The maximum phase shift impacts the main beam and sidelobe distortion and the null depth obtainable. Results for MSB phase values above.n/4 produced significant pattern distortions, so they are not listed in Table 1. a, t l5 N N N K INN INN I 11% N M NI I N A gradient-based algorithm using central difference approximations of the derivatives would take 80 power measurements per step in the algorithm. The genetic algorithm is a clear winner by converging in only 32 power measurements. Figure 7 is a graph of the maximum and average sidelobe reduction of the chromosomes in the population after each iteration. Iteration 0 is the quiescent level of the sidelobe. The average chromosome is of importance, because it indicates how well the antenna rejects interference during the adaptation. The average and best chromosomes are the same at iteration 11. They don't remain the same in later iterations because random mutations are introduced into the population. N NNN x -lo/ N N I -1 5 *!UN f * I element Figure 5. Adapted phase weights for the 40 element (d=0.5h) low sidelobe array with 6 bit phase shifters. Two bits were used for nulling with an MSB of d16. Interference sources appear at u=.62 and -72. Scenario 1 is a simple example of two interference sources at ui =.62 and.72. Sidelobe reduction is best for a middle-sized MSB and P. Convergence is fastest for a smaller P and MSB. As expected, sidelobe distortion increases with larger values of the phase of the MSB. This example suggests that P=2 and MSB phase = x/16 is the best combination. Figure 5 shows the adapted phase weights that produce the nulled pattern in Figure 6. The maximum phase shift to produce the nulling is ' The maximum sidelobe level increases to -27 db. B.- C " -10 E -20 B g ij - m p! U Figure 6. Adapted antenna pattern corresponding to the phase shifts in Figure

8 the size of the phase shift keeps the main beam intact. As long as P and MSB are small, then the algorithm has little effect on the main beam. An MSB phase of d16 or d32 does not require turning the algorithm off when no interference is present, because the algorithm can't attack the main beam iteration Figure 7. Genetic algorithm convergence is graphed over 25 iterations. The solid line represents the output power for the best chromosome, while the dashed line represents the average output power for all the chromosomes in the population. Scenario 2 presents the algorithm with the difficult situation where the jammers are at angles symmetric about the main beam. In Figure 6 notice the increase in the sidelobes symmetrically opposed to the nulls placed by the adaptive algorithm. This phenomena is discussed in [14]. As noted in Table 2, symmetric interference sources can only be nulled with large values of phase and many power measurements. Pattern distortions are quite high. Performance improves as the symmetry is broken. When the interference sources are one sidelobe apart, performance similar to Scenario 1 is obtained. Scenario 3 tests the bandwidth performance of the algorithm by placing two interference sources close together. In this case, the smaller P and MSB phase, the better the algorithm performance. For the most part, null depth is diminished for broadband jammers. Scenario 4 checks the algorithm when no interference is present, but the desired signal is received by the main beam. The algorithm should not attack the desired signal. Limiting Scenario 5 has the algorithm attempt to place a null in the quantization sidelobe. Results are marginal for all values of MSB phase. The quantization lobe results from the digital phase shifter quantization % 2 5-0) :._ xm *xx x m x It mx mx Ymx x 1 X!E%lK 111 i %!X * * O - X M x m x x** x x 1lx x y I x *sx element Figure 8. Adapted phase weights for the 100 element (d=0.5h) low sidelobe array with 6 bit phase shifters. Two bits were used for nulling with an MSB of n/16. Interference sources appear at u=.25 and

9 --lot.- C E i.- %! m - e U Figure 9. Adapted antenna pattern corresponding to the phase shifts in Figure 8. I I and the 100 element uniform array showed fast convergence, deep nulling capability, and small pattern distortions. Using only a few least significant bits and small phase values for the MSB are keys to the algorithm's performance. Its important advantage is that it is easy to implement on existing phased arrays. Disadvantages are: 1) little success at nulling interference entering a quantization sidelobe and 2) interference at symmetric angles about the main beam. Increasing the bandwidth of the interference also degrades algorithm performance. Phase-only adaptive nulling works on existing phased-array antenna designs, unlike signal processing based adaptive arrays that require receivers at every element. Its main disadvantage is slow convergence time. The genetic algorithm approach to phase-only adaptive nulling is significantly faster than the previous approaches of random search and gradient methods. Thus, it advances the stateof-the-art of phase-only adaptive nulling. I iteration Figure 10. Genetic algorithm convergence is graphed over 25 iterations. The solid line represents the output power for the best chromosome, while the dashed line represents [l] C. F. N. Cowan and P. M. Grant, Adaptive Filters, Englewood Cliffs, NJ: Prentice Hall, [2] R. Kinsey, "Array antenna self-calibration techniques," M A Workshop: Testing Phased Arrays and Diagnostics, San Jose, CA, Jun 89. [3] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Antennas, New York: Wiley, the average Output power for all the [4] R. T. Compton, Adaptive Antennas Concepts and chromosomes in the population. Performance, Englewood Cliffs, NJ: Prentice Hall, CONCLUSIONS [5] C. A. Baird and G. G. Rassweiler, "Adaptive sidelobe nulling using digitally controlled phase- The genetic performed we' for two shifters," IEEE AfJ Trans., Vol 24, No. 5, pp , jammers that were: 1) separated by an angular sep76. width of at least one sidelobe and 2) were not symmetric in angular distance about the main beam. Both the 40 element low sidelobe array 159

10 [6] M. K. Leavitt, "A phase adaptation algorithm," IEEE AP-S Trans., Vol. 24, No. 5, pp , Sep 76. [7] H. Steyskal, "Simple method for pattern nulling by phase perturbation," IEEE AP-S Trans., Vol. 31, No. 1, pp , Jan 83. [8] R. L. Haupt, "Adaptive nulling in monopulse antennas," IEEE AP-S Trans., Vol. 36, No. 2, pp , Feb 88. [9] J. F. Deford and 0. P. Gandhi, "Phase-only synthesis of minimum peak sidelobe patterns for linear and planar arrays," IEEE AP-S Trans. Vol. 36, No. 1, pp , Jan 88. [ 101 R. A. Shore, "A proof of the odd-symmetry of the phases for minimum weight perturbation phase-only null synthesis," IEEE AP-S Trans., Vol. 32, No. 5, pp , May 84. [l 11 J. H. Holland, "Genetic algorithms," Sci. Amer., pp 66-72, July [12] R. L. Haupt, "An introduction to genetic algorithms for electromagnetics," IEEE Antennas and Propagation Magazine, Vol. 37, NO. 2, pp. 7-15, Apr 95. [13] S. Lundgren and J. Sanford, "A new technique for phase-only nulling with equispaced arrays," IEEE AP-S Symposium Digest, Vol. 1, pp , Jun 95. [141 R A. Shore, "Nulling at symmetric pattern location with phase-only weight control," IEEE AP-S Trans., Vol. 32, No. 5, pp , May 84. Randy Haupt is a Lieutenant Colonel in the USAF and a Professor of Electrical Engineering at the United States Air Force Academy. He received his BS in EE from the USAF Academy in 1978, his MS in Engineering Management from Western New England College in 1981, his MS in EE from Northeastern University in 1983, and his PhD in EE from The University of Michigan in He worked as a project engineer on the OTH-B Radar and as an antenna engineer for Rome Air Development Center. Lt Col Haupt was Federal Engineer of the Year in Sue Ellen Haupt is a Research Scientist in the PAOS Program at the University of Colorado, Boulder. For the academic year, she is on a sabatical as a visiting scholar at the Physics Department at the USAF Academy. She received her BS in Meterology from Penn State in 1978, her MS in Engineering Management from Western New England College in 1981, her MS in ME from Wochester Polytechnic Institute in 1984, and her PhD in Atmospheric Science from The University of Michigan in