AMPLITUDE AND PHASE ADAPTIVE NULLING WITH A

Transcription

1 AMPLITUDE AND PHASE ADAPTIVE NULLING WITH A GENETIC ALGORITHM Y. C. Chung Electrical Engineering Dept. University of Nevada Reno, NV USA R. L. Haupt Electrical and Computer Engineering Dept. Utah State University Logan, UT USA Abstract-This paper introduces amplitude and phase nulling with a genetic algorithm with the intent of finding the optimum number of LSB of control needed to place multiple nulls while not significantly distorting the far field pattern. Many interference scenarios were modeled and individual runs were averaged to conclude that five LSB of amplitude and three LSB of phase control out of a total of eight bits are best for the linear array modeled. Representative samples of some of the interference scenarios modeled are shown. 1. INTRODUCTION Genetic Algorithms (GAs) have been applied to many adaptive antenna array and electromagnetics applications [1-16]. Adaptive phaseonly nulling with a genetic algorithm was introduced in [1]. Our adaptive nulling approach with a GA implementation uses the total output power from the array, rather than the signal at each element, as feedback for placing the null. As a result, it works with minimal hardware requirements [1-3]. The adaptive genetic algorithm is superior to an adaptive gradient algorithm. Calculating the gradient in every generation requires more power measurements, and it slows convergence speed of the algorithm

2 632 compared to the GA [4, 5]. In addition, the gradient algorithm gets stuck in a local minimum while the GA is less likely to get stuck in local minimum. A least mean square (LMS) algorithm updates weights based on a least mean square (LMS) error and a convergence speed constant, A I multiplied by covariance matrix. It is hard to choose a proper value of dynamically since it should be decided on the basis of the covariance matrix, and the power ratio between desired signal and interference signal [17]. The results of the LMS error and the derivatives of the gradient algorithm are a continuous numbers so that the gradient and the LMS algorithms require continuous value for the amplitude and phase weights while a GA does not. A GA can use continuous and discrete parameters, and it can control and update digital phase shifters and amplitude weights directly without generating quantization error in ev- ery generation. The genetic algorithms have been compared with the gradient algorithm and the minimization of mean square error method, and the superiority of GAs has been proved in many references [1, 4-6, 13, 15]. The digitally controlled phase-only adaptive nulling was introduced by Baird [18], and the adaptive nulling with amplitude control was shown in [19]. In addition, adaptive nulling by controlling element position and elevation were shown in [20, 21], and the adaptive nulling by controlling phase and element position was introduced by Lio [15]. The element location and elevation of a fixed element array cannot be controlled, and it is also easier to control mainbeam direction by controlling phase instead of controlling amplitude only. Therefore, the phase-only adaptive nulling has been popular. Nevertheless, the phase-only nulling has a common problem; It cannot generate nulls at symmetric locations about the mainbeam with small values of phase, the phase-only nulling places a null at one an- gle, causes the sidelobe at the symmetric location to rise [1, 5, 7, 22]. One way to generate nulls at the symmetric locations with phase-only control was introduced by Shore [22], but the method has significant pattern distortion. Phase-only adaptive nulling with a genetic algorithm was introduced by Haupt [1, 2], but the phase-only nulling with a GA still has the common phase-only nulling problem. The amplitude & phase adaptive nulling with a GA overcomes the problem of phase-only adaptive nulling, and it has faster convergence speed and generates deeper

3 n 633 nulls than phase-only adaptive nulling [7, 8]. The amplitude & phase adaptive nulling with a GA controls some of the least significant bits (LSB) of the phase shifters and amplitude weights to minimize the total output power, while it maintains the mainbeam and the gain, and minimizes the pattern distortion [1, 5, 7]. The convergence speed and null depths of the amplitude & phase adaptive nulling with a GA are compared with those of the phase-only adaptive nulling with a GA when symmetric and non-symmetric power interference about the mainbeam is incident at various locations-single, adjacent, symmetric and non-symmetric about the mainbeam. Furthermore, the mainbeam reduction and the sidelobe distortion of the adapted patterns are considered based on the number of bits for control of the phase-only and the amplitude & phase adaptive nulling with a GA. 2. AMPLITUDE & PHASE ADAPTIVE ARRAY AND FORMULATION A adaptive linear array having digital attenuators and phase shifters is shown in Figure 1. The amplitude weights and the phase shifters are controlled by a GA to place nulls in the directions of interference. The mathematical model for a linear array of point sources with sin element patters is given by

4 634 Figure 1. Diagram of an amplitude & phase adaptive linear array with a genetic algorithm. When the amplitude weights have even symmetry and the phase weights have odd symmetry, then the mathematical model in equation (1) simplifies to 3. AMPLITUDE & PHASE ADAPTIVE NULLING WITH A GENETIC ALGORITHM The amplitude & phase adaptive genetic algorithm controls the digitized amplitude weights and phase shifters to minimize the total out- put power of the array. The mainbeam degradation is limited by using only some of the least significant bits (LSB) of the amplitude weights and phase shifters. The flow chart of the amplitude & phase adaptive

5 635 nulling with a genetic algorithm is shown in Figure 2, and it is almost identical to that of the phase-only nulling with a GA [1, 2]. The al- gorithm begins with creating initial population and measuring output powers of the initial population. The cost function of the adaptive genetic algorithm evaluates the power corresponding to the amplitude weights and phase shifter settings. Figure 2. Flow chart of amplitude and phase adaptive nulling with a genetic algorithm. Figure 3 shows M rows of the initial population of the amplitude weights and phase shifter settings with corresponding powers using 4 LSB of amplitude weights and 3 LSB of phase shifters for control. Each row of the matrix (chromosome) has information about the amplitude weights and the phase shifter settings of each element placed side-byside. There are * (A+P) N columns and M rows where the number

6 636 Figure 3. A population of amplitude and phase settings with corresponding output powers. There are 4 LSB of amplitude and 3 LSB phase control. of elements is 2N, and the numbers of the amplitude and the phase shifter bits for control are A and P, respectively. The size of M should be large enough to keep the algorithm out of local minima [1, 2, 7], and the size of A and P also proper to generate nulls fast with a small mainbeam degradation. The initial population of chromosomes are evaluated by an adaptive array, and they are ranked from the best fit to the worst fit. The bottom 50% of the initial population is discarded, and the remain- ing top half is called the good population or parents. Two partners are randomly selected from the good population, and they mate at a randomly selected crossover point. Figure 4 shows a mating between two parents: the bits to the right of the random crossover point are swapped to generate two new offspring. The total number of new offspring is the same as the number of discarded population members. A small percentage of bits of new offspring and their parents are ran- domly switched from a 1 to a 0 or visa versa. This process is called mutation and a high percentage of mutation gives more freedom to search new areas of amplitude and phase settings. The mutated bits are also shown with italic letters in Figure 4. There is no mutation of the best chromosome on each generation. The mutated offspring are evaluated, and these processes are repeated until the solution is satisfied or the process reaches a given generation.

7 637 Figure 4. Randomly selected two partners generate offspring and their mutation. 4. RESULTS The array is a half wavelength spaced 32 element linear array, lying along the x-axis with a -30 db Chebychev amplitude taper and zero phase shifter settings initially. An 8 bit amplitude weight and an 8 bit phase shifter are used for each array element. A GA controls only some of the least significant bits (LSB) of the amplitude weights and phase shifters to minimize the total output power. The 3, 4, and 5 least significant bits (LSB) of phase shifters out of the 8 bits are used for the phase-only control adaptive array with a GA since the 2 LSB of the phase shifter control is not enough to generate a null, and more than 5 LSB of the phase shifter control gives large mainbeam and sidelobe distortions. The combinations of 3, 4 and 5 LSB out of the 8 bit amplitude weights and phase shifters are used for the amplitude & phase control adaptive nulling with a GA. The results are evaluated and compared when interference is incident at various locations (single, adjacent, symmetric and non-symmetric) about the mainbeam. GA parameters are a population of 24, discard rate of 0.5, and single random crossover with 5% mutation rate. The results of the phase-only adaptive nulling with a GA and the amplitude & phase adaptive nulling with a GA on each scenario are averaged over 30 runs

8 638 Figure 5. Null depth of phase-only and amplitude & phase adaptive nulling with a GA using various bits for control: single interference. after 25 generations. The results of various scenarios follows. The first scenario has a single interference with signal-to-noise-ratio (S/N=10000) incident on a sidelobe. One of the sample results shown here has an interference location at 0 = 141. The averaged results of other single interference locations are almost identical to the results shown here. The average null depth of the amplitude & phase nulling and the phase-only nulling vs. generation are compared in Figure 5 based on the various numbers of bits for control. Figure 5 shows that the amplitude & phase nulling with a GA converges faster than phase- only nulling with a GA, and the adapted null depths of the amplitude & phase adaptive nulling with a GA are around -5 db lower than those of the phase-only adaptive nulling with a GA. In addition, the 5 LSB of the amplitude weight with the 3 LSB of the phase shifter (5,. 3 bit) converges faster than any other number of bits control of the phase-only nulling and the amplitude & phase nulling. An example of one of the adapted patterns of the both amplitude & phase control (5, 3 bit: solid line) and phase-only control (3 bit: dashed line) after 25 generations are displayed with the quiescent array pattern (dotted line) in Figure 6. Figure 6 shows that the adapted sidelobe distortion of the

9 639 Figure 6. Adapted amplitude & phase (Solid line), adapted phaseonly (dashed line) and quiescent (dotted line) array pattern for single interference at 141 degrees. phase-only nulling algorithm is higher than that of the amplitude & phase nulling algorithm. The second scenario has the interference source incident on two adjacent sidelobes. The results with the locations of neighboring interference at 0 = 130 and 135 degrees are shown here. The null depth at 0 =130 degrees vs. generation for both amplitude & phase nulling and phase-only nulling with a GA are shown in Figure 7. The null depths at 0 =135 degrees in db vs. generation are not shown since they are almost identical to Figure 7. The 5 LSB of amplitude weight with the 3 LSB of phase shifter for control (5, 3 bit) converges faster than any other bits for con- trol. The averaged adapted null depth of the amplitude & phase (5, 3 bit) adaptive nulling with a GA after 25 generations is approximately - 6 db lower than that of the best phase-only (5 bit) adaptive nulling with a GA in Figure 7. Figure 8 compares the adapted patterns of the amplitude & phase (5, 3 bit) and the phase-only (5 bit) nulling after 25 generations with the quiescent pattern. The adapted pattern of the amplitude & phase control (5, 3 bit) shows -57 and -52 db

10 640 Figure 7. Null depth (at 130 degrees) of phase-only and amplitude & phase adaptive nulling with a GA using various bits for control: adjacent interference at 130 and 135 degrees. null depths at the neighboring locations at and 135 degrees while the phase-only (5 bit) nulling generates only -43 and -45 db in Figure 8. The sidelobe distortion of 5 LSB phase-only is significant in Figure 8, and the numerical peak sidelobe levels based on the num- ber of bits for control are also compared at the end of this section. The result shows that using 5 LSB of amplitude with 3 LSB of phase for control outperforms any others bit configurations. The third scenario has the multiple equal power interference sources incident on non-symmetric locations about the mainbeam. An exam- ple with the interference at 0 =59 and 0 =130 degrees is shown here. Figure 9 shows null depths at 0 =59 degrees for both amplitude & phase and phase-only nulling with a GA. The null depth at the other non-symmetric location (0=130 degrees) is almost identical to Figure 9. The convergence speed of the 5 LSB of amplitude with the 3 LSB of phase for control (5, 3 bit) is close to that of the 5 LSB of amplitude with 4 LSB of phase control (5, 4 bit). The convergence speed of the 4 LSB phase-only control is closed to the (5, 3 bit) and (5, 4 bit) amplitude & phase control. Figure 10 shows the adapted patterns of both the (5, 3 bit) amplitude & phase and the 4 LSB control

11 641 Figure 8. Adapted amplitude & phase (Solid line), adapted phase-only (dashed line) and quiescent (dotted line) array pattern for adjacent interference at 130 and 135 degrees. Figure 9. Null depth (at 59 degrees) of phase-only and amplitude & phase adaptive nulling with a GA using various bits for control and non-symmetric interference at 59 and 130 degrees.

12 642 Figure 10. Adapted amplitude & phase (Solid line), adapted phaseonly (dashed line) and quiescent (dotted line) array pattern for non- symmetric interference at 59 and 130 degrees. phase-only nulling with a GA after 25 generations with quiescent pattern. The null depth of both the 5 LSB amplitude & 3 LSB phase and the 4 LSB control generates similar null depths. The last scenario simulates symmetric interference sources with equal power incident at various symmetric locations about the mainbeam. Figure 11 and 12 show null depths of both the amplitude & phase and the phase-only adaptive nulling with a GA at the symmetric locations, =50 and 130 degrees. The 5 LSB of amplitude with the 3 LSB of phase control (5, 3 bits) works best. While 3, 4 and 5 LSB control of the phase-only control do not generate deep nulls, the 5 LSB of amplitude & 3 LSB phase control adaptive nulling with a GA generates deep nulls at the symmetric locations about the mainbeam in Figure 11 and 12. The problem of the phase-only nulling [1, 5, 7, 22] is overcome with the amplitude and phase nulling with the correct number of bits for control. Figure 11 and 12 also show the 5 LSB phase-only control generates null depths at symmetric locations around -47 db after 25 generations while the 5 LSB amplitude & 3 LSB phase control adaptive nulling generates lower than -60 db. The

13 643 Figure 11. Null depth (at 50 degrees) of phase-only and amplitude & phase adaptive nulling with a GA using various bits for control: symmetric interference at 50 and 130 degrees. Figure 12. Null depth (at 130 degrees) of phase-only and amplitude & phase adaptive nulling with a GA using various bits for control: symmetric interference at 50 and 130 degrees.

14 644 Figure 13. Adapted amplitude & phase (Solid line), adapted phaseonly (dashed line) and quiescent (dotted line) array pattern for symmetric interference at 50 and 130 degrees. adapted patterns of the amplitude & phase control (5 LSB amplitude with the 3 LSB phase control) and the 5 LSB phase-only nulling are shown in Figure 13. It shows that the null depths generated by the 5 LSB amplitude with the 3 LSB phase control are lower than those of the phase-only control. In addition, the sidelobe distortion and the mainbeam degradation of the 5 LSB phase-only nulling are larger than that of amplitude and phase nulling when they generate nulls in Figure 13. The mainbeam degradation and the adapted peak sidelobe levels are compared based on the number of bits for control in Tables 1 and 2. The 5 LSB amplitude with 3 LSB phase control of the amplitude & phase adaptive nulling with a GA outperforms any other phase-only and amplitude & phase adaptive nulling when symmetric interference is incident on an array. Most of the possible scenarios are simulated: single interference and multiple interference cases with various interference locationsadjacent, non-symmetric and symmetric interference locations with symmetric power. In all the scenarios, the amplitude & phase nulling with a genetic algorithm generally converges faster and generates deeper nulls than the phase-only nulling with a genetic algorithm. The

15 645 Table 1. Main beam reduction of adapted pattern based on various bits for control. Table 2. The peak sidelobe level of the adapted pattern with the 2nd scenario. amplitude & phase adaptive nulling with a GA (5 LSB of amplitude with 3 LSB of phase control) outperforms others in all the scenarios. The mainbeam reduction is considered for both amplitude & phase adaptive nulling and phase-only adaptive nulling with a GA. As far as a GA controls only some of the least significant bits out of the 8 bit amplitude weights and phase shifters to minimize the total output power, there is no significant change in mainbeam direction and gain of the array. The 5 LSB of amplitude with the 3 LSB of phase for control adaptive nulling has fast convergence speed, and generates deep nulls for all scenarios nevertheless 1 db mainbeam reduction which is reasonably low compare to the 5 LSB phase-only nulling in Table 1. Table 1 also shows that the mainbeam reduction increases when the number of phase bits for control increases. The sidelobe distortions based on the number of bits for control of phase-only nulling and amplitude & phase nulling are compared in Figure 14 when the adjacent interference is incident on the arrays - the 2,d scenario. The 5 LSB amplitude with 3 LSB phase control

16 646 Figure 14. Relative sidelobe comparison based on bits for control of the phase-only nulling and the amplitude & phase nulling. adaptive nulling has low peak sidelobe level on adapted pattern as the 3 LSB phase-only nulling in Table 2. Furthermore, Table 2 and Figure 14 show that the more number of phase bits for control gives higher peak sidelobe levels on the adapted patterns. 5. CONCLUSIONS There is a tradeoff between the number of LSB in the amplitude weights and/or phase shifters used in nulling and the performance of the adaptive antenna. On the one hand, using only a few LSB creates little pattern distortions but does not create adequate nulls. On the other hand, using many LSB places great nulls but creates significant pattern distortion. Many different jamming configurations were simulated. Representative samples were shown here. The runs were averaged for each interference scenario. In addition, the interference sources were moved to various sidelobe locations. After simulating many different scenarios, we conclude that 5 LSB of amplitude control and 3 LSB of phase control works best for up to two interference sources for the linear array simulated in this paper. Keeping the number of LSB at a minimum is important in order for an adaptive antenna to reduce far field pattern degradation.

17 647 REFERENCES 1. Haupt, R. L., "Phase-only adaptive nulling with a genetic algorithm," IEEE Trans. Antenna Propagat., Vol. AP-45, , June Haupt, R. L., and S. E. Haupt, Practical Genetic Algorithms, John Wiley & Sons Inc., New York, NY, Haupt, R. L., "Thinned arrays using genetic algorithm," IEEE Trans. Antenna Propagat., Vol. AP-42, , July Haupt, R. L., "Comparison between genetic and gradient-based optimization algorithms for solving electromagnetics problems," IEEE Trans. Magnetics, Vol. 31, , May Chung, Y. C., and R. L. Haupt, "Amplitude and phase adaptive nulling with a genetic algorithm," USNC/URSI National Radio Science Digest, 225, Atlanta, GA, June Haupt, R. L., and H. Southall, "Experimental adaptive nulling with a genetic algorithm," Microwave Journal, Vol. 42, No. 1, 78-89, Jan Chung, Y. C., and R. L. Haupt, "Optimum amplitude and phase control for an adaptive linear array using a genetic algorithm," IEEE Antennas and Propagation Soc. Int. Symp., Digest, Vol. 2, , Orlando, FL, July Chung, Y. C., and R. L. Haupt, "Adaptive nulling with spherical arrays using a genetic algorithm," IEEE Antennas and Propagation Soc. Int. Symp., Digest, Vol. 3, , Orlando, FL, July Chung, Y. C., and R. L. Haupt, "Optimizing genetic algorithm parameters for adaptive nulling," Applied Computational Electromagnetic Society Symposium Digest, , Monterey, CA, March Johnson, J. M., and Y. Rahmat-Samii, "Genetic algorithms in electromagnetics," IEEE Antennas and Propagation Soc. Int. Symp., Digest, Vol. 2, , Baltimore, MD, July Johnson, J. M., and Y. Rahmat-Samii, "A novel integration of genetic algorithms and method of moments (GA/MoM) for antenna design," Applied Computational Electromagnetic Society Symposium Digest, , Monterey, CA, March Johnson, J. M., and Y. Rahmat-Samii, "Genetic algorithms and method of moments (GA/MoM): A novel integration for antenna design," IEEE Antennas and Propagation Soc. Int. Symp., Digest, Vol. 3, , Montreal, Canada, July Jones, E. A., and W. T. Joines, "Design of yagi-uda antennas using genetic algorithms," IEEE Trans. Antenna Propagat., Vol. AP-45, , Sep

18 Johnson, M. J., and R. Samii, "Genetic algorithm optimization to antenna design," IEEE Antennas and Propagation Soc. Int. Symp., Digest, Vol. 1, , Seattle, WA, June Lio, W., and F. Chu, "Array pattern nulling by phase and position perturbations with the use of the genetic algorithm," Microwave and Optical Technology Letters, Vol. 15, No. 4, July Tennant, A., M. M. Dawoud, and A. P. Anderson, "Array pattern nulling by element position perturbations using a genetic algorithm," Electron. Lett., Vol. 30, No. 3, , Feb Steinberg, B. D., Principles of Aperture and Array System Design, Wiley-Interscience, New York, Baird, C. A., and G. G. Rassweiler, "Adaptive sidelobe nulling using digitally controlled phase-shifters," IEEE Trans., Antenna Propagation, Vol. AP-24, , Sep Vu, T. B., "Null steering by controlling current amplitude only," IEEE Antennas and Propagation Soc. Int. Symp., Digest, Vol. 2, , Ismile, T. H., and M. M. Dawoud, "Null steering in phased arrays by controlling the element positions," IEEE Trans. Antenna Propagat., Vol. AP-39, , Hejres, J. A., and J. E. Richie, "Adaptive pattern nulling utilising the elevations of the antenna array elements," IEEE Antennas and Propagation Soc. Int. Symp., Digest, , June Shore, R. A., "Nulling at symmetric pattern location with phaseonly weight control," IEEE Trans. Antenna Propagation, Vol. AP-32, , May You Chung Chung received the BS. in Electrical Engineering from Inha University, Inchon, Korea in 1990, and MSEE. degree from University of Nevada, Reno (UNR) in He is currently a Ph.D. candidate in Electrical Engineering at UNR. Since 1995, he has been employed as a teaching and research assistant in Electrical Engineering at UNR. His research interests include computational electromagnetics, optimized antenna and array design, conformal and fractal antennas, adaptive array processing, optimization techniques, and genetic algorithm. In 1996, he received an Outstanding Teaching Assistant Award from UNR. He also received an Outstanding Graduate Student Award in The NSF sponsored his 1999 IEEE AP-S paper presentation. Randy Haupt is Professor and Department Head of Electrical and Computer Engineering at Utah State University. He has a Ph.D. in Electrical Engineering from the University of Michigan, MS in Electrical Engineering from Northeastern University, MS. in Engineering

19 Management from Western New England College, and BS. in Electrical Engineering from the USAF Academy. He was a Professor of Electrical Engineering at the USAF Academy and Professor and Chair of Electrical Engineering at the University of Nevada Reno. He is co-author of the book Practical Genetic Algorithms, John Wiley & Sons, Jan