NAVAL POSTGRADUATE SCHOOL THESIS

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1 NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA THESIS EXTENDING THE UNAMBIGUOUS RANGE OF CW POLYPHASE RADAR SYSTEMS USING NUMBER THEORETIC TRANSFORMS by Nattaphum Paepolshiri September 11 Thesis Co-Advisors: Phillip E. Pae David C. Jenn Approved for publi release; distribution is unlimited

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3 REPORT DOCUMENTATION PAGE Form Approved OMB no Publi reporting burden for this olletion of information is estimated to average 1 hour per response, inluding the time for reviewing instrution, searhing existing data soures, gathering and maintaining the data needed, and ompleting and reviewing the olletion of information. Send omments regarding this burden estimate or any other aspet of this olletion of information, inluding suggestions for reduing this burden, to Washington headquarters Servies, Diretorate for Information Operations and Reports, 115 Jefferson Davis Highway, Suite 14, Arlington, VA -43, and to the Offie of Management and Budget, Paperwork Redution Projet (74-188) Washington DC AGENCY USE ONLY (Leave blank). REPORT DATE September REPORT TYPE AND DATES COVERED Master s Thesis 4. TITLE AND SUBTITLE Extending the Unambiguous Range of CW Polyphase 5. FUNDING NUMBERS Radar Systems Using Number Theoreti Transforms 6. AUTHOR(S) Nattaphum Paepolshiri 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Postgraduate Shool Monterey, CA SPONSORING /MONITORING AGENCY NAME(S) AND ADDRESS(ES) N/A 8. PERFORMING ORGANIZATION REPORT NUMBER 1. SPONSORING/MONITORING AGENCY REPORT NUMBER 11. SUPPLEMENTARY NOTES The views expressed in this thesis are those of the author and do not reflet the offiial poliy or position of the Department of Defense or the U.S. Government. IRB Protool Number: N/A. 1a. DISTRIBUTION / AVAILABILITY STATEMENT Approved for publi release; distribution is unlimited 13. ABSTRACT (maximum words) i 1b. DISTRIBUTION CODE A Polyphase ontinuous waveform (CW) radar systems often use the popular Frank ode and P4 ode due to their linear time-frequeny harateristis as well as their low periodi ambiguity sidelobes. The phase relationship of the Frank ode orresponds to a sawtooth folding waveform. The phase relationship of the P4 ode is symmetrial with a paraboli distribution. The radar system s unambiguous target detetion range is limited by the number of subodes within the ode period (ode length). Inreasing the ode length to extend the unambiguous range results in a larger range-doppler orrelation matrix proessor in the reeiver, a longer ompression time and an inrease in the reeiver s bulk memory requirements. In addition, the entire ode period may not be returned from the target due to a limited time-on-target resulting in signifiant orrelation loss. To signifiantly extend the unambiguous range beyond a single ode period, this thesis explores the relationship between the polyphase odes (Frank and P4) and the number theoreti transforms (NTT) where the residues exhibit the same distribution as the polyphase values. The unambiguous range is extended from the number of subodes within a single ode period to the dynami range of the transform without requiring a large inrease in orrelation proessing. The dynami range of a NTT is defined as the greatest length of ombined phase sequenes that ontain no ambiguities or repeated paired terms. By transmitting N oprime ode periods, the unambiguous range an be extended by onsidering the paired values from eah sequene. A new Frank phase ode formulation is derived as a funtion of the residue number system (RNS) where eah residue orresponds to a phase value within the ode period (modulus) sequene. Based on the symmetrial distribution of the P4 ode, a new phase ode expression is derived using both the symmetrial number system (SNS) and the robust symmetrial number system (RSNS). Here eah phase value within the ode period orresponds to a symmetrial residue. MATLAB simulations are used to verify the new expressions for the RNS, SNS and RSNS phase odes. Implementation onsiderations of the new approah are also addressed. 14. SUBJECT TERMS CW radar, Polyphase sequene, Frank ode, P4 ode, Unambiguous range, Pulse ompression, Residue Number System, Symmetrial Number System, Robust Symmetrial Number System. 17. SECURITY CLASSIFICATION OF REPORT Unlassified 18. SECURITY CLASSIFICATION OF THIS PAGE Unlassified 19. SECURITY CLASSIFICATION OF ABSTRACT Unlassified 15. NUMBER OF PAGES PRICE CODE. LIMITATION OF ABSTRACT NSN Standard Form 98 (Rev. -89) Presribed by ANSI Std UU

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5 Approved for publi release; distribution is unlimited EXTENDING THE UNAMBIGUOUS RANGE OF CW POLYPHASE RADAR SYSTEMS USING NUMBER THEORETIC TRANSFORMS Nattaphum Paepolshiri Lieutenant, Royal Thai Navy B.S., Royal Thai Naval Aademy, 5 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN ELECTRONIC WARFARE SYSTEMS ENGINEERING and MASTER OF SCIENCE IN ELECTRICAL ENGINEERING from the NAVAL POSTGRADUATE SCHOOL September 11 Author: Nattaphum Paepolshiri Approved by: Phillip E. Pae Thesis Co-Advisor David C. Jenn Thesis Co-Advisor R. Clark Robertson Chair, Department of Eletrial and Computer Engineering Dan C. Boger Chair, Department of Information Sienes iii

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7 ABSTRACT Polyphase ontinuous waveform (CW) radar systems often use the popular Frank ode and P4 ode due to their linear time-frequeny harateristis as well as their low periodi ambiguity sidelobes. The phase relationship of the Frank ode orresponds to a sawtooth folding waveform. The phase relationship of the P4 ode is symmetrial with a paraboli distribution. The radar system s unambiguous target detetion range is limited by the number of subodes within the ode period (ode length). Inreasing the ode length to extend the unambiguous range results in a larger range-doppler orrelation matrix proessor in the reeiver, a longer ompression time and an inrease in the reeiver s bulk memory requirements. In addition, the entire ode period may not be returned from the target due to a limited time-on-target resulting in signifiant orrelation loss. To signifiantly extend the unambiguous range beyond a single ode period, this thesis explores the relationship between the polyphase odes (Frank and P4) and the number theoreti transforms (NTT) where the residues exhibit the same distribution as the polyphase values. The unambiguous range is extended from the number of subodes within a single ode period to the dynami range of the transform without requiring a large inrease in orrelation proessing. The dynami range of a NTT is defined as the greatest length of ombined phase sequenes that ontain no ambiguities or repeated paired terms. By transmitting N oprime ode periods, the unambiguous range an be extended by onsidering the paired values from eah sequene. A new Frank phase ode formulation is derived as a funtion of the residue number system (RNS) where eah residue orresponds to a phase value within the ode period (modulus) sequene. Based on the symmetrial distribution of the P4 ode, a new phase ode expression is derived using both the symmetrial number system (SNS) and the robust symmetrial number system (RSNS). Here eah phase value within the ode period orresponds to a symmetrial residue. MATLAB simulations are used to verify the new expressions for the RNS, SNS and RSNS phase odes. Implementation onsiderations of the new approah are also addressed. v

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9 TABLE OF CONTENTS I. INTRODUCTION...1 A. CONTINUOUS WAVE RADAR SYSTEMS USING POLYPHASE MODULATION...1 B. PRINCIPAL CONTRIBUTIONS...5 C. THESIS OUTLINE...6 II. III. IV. PRINCIPLES OF PHASE MODULATION RADAR...7 A. CW PHASE MODULATION CONCEPTS...7 B. CODE COMPRESSION OF RECEIVED SIGNAL...9 C. PERIODIC AUTOCORRELATION FUNCTION...14 D. PERIODIC AMBIGUITY FUNCTION...16 FRANK POLYPHASE MODULATION AND THE RESIDUE NUMBER SYSTEM...19 A. FRANK PHASE CODE...19 B. THE RESIDUE NUMBER SYSTEM... C. RESIDUE-FRANK PHASE CODE...4 D. RESOLVING RANGE AMBIGUITIES USING N RESIDUE-FRANK PHASE CODE SEQUENCES Blok Diagram of the Residue-Frank Radar System...6. Transmitted and Referene Codes for Compression Calulating the Target Range Resolving Multiple Target Range Ambiguities Pratial Considerations...36 a. Detetion of Target in Noise Using RNS Compression...36 b. Required CW Signal Power Versus Required Output SNR..41. Unambiguous Detetion Range for a Constant Output SNR...43 P4 POLYPHASE MODULATION AND THE SYMMETRICAL NUMBER SYSTEM...45 A. P4 PHASE CODE...45 B. THE SYMMETRICAL NUMBER SYSTEM...47 C. SYMMETRICAL RESIDUE-P4 PHASE CODE...5 D. RESOLVING RANGE AMBIGUITIES USING N SYMMETRICAL RESIDUE-P4 PHASE CODE SEQUENCES Blok Diagram of the Symmetrial Residue-P4 Radar System Transmitted and Referene Codes for Compression Calulating the Target Range Resolving Multiple Target Range Ambiguities Pratial Considerations...6 a. Detetion of Target in Noise Using SNS Compression...6 b. Required CW Signal Power Versus Required Output SNR..65 vii

10 . Unambiguous Detetion Range for a Constant Output SNR...66 V. P4 POLYPHASE MODULATION AND THE ROBUST SYMMETRICAL NUMBER SYSTEM...69 A. P4 PHASE CODE...69 B. THE ROBUST SYMMETRICAL NUMBER SYSTEM...69 C. ROBUST SYMMETRICAL RESIDUE-P4 PHASE CODE Example of Phase Sequene Calulation ACF, PACF, and PAF...74 D. RESOLVING RANGE AMBIGUITIES USING N ROBUST SYMMETRICAL RESIDUE-P4 PHASE CODE SEQUENCES Blok Diagram of the Robust Symmetrial Residue-P4 Radar System Transmitted and Referene Codes for Compression Calulating the Target Range Resolving Multiple Target Range Ambiguities Pratial Considerations...87 a. Detetion of Target in Noise Using RSNS Compression...87 b. Plot of Required Signal Power Versus Required SNR Output...9. Unambiguous Detetion Range for a Constant Output SNR...93 VI. CONCLUDING RESULTS AND RECOMMENDATIONS...97 A. EQUATION SUMMARY...97 B. COMPARISON OF 3-CHANNEL MODULAR NUMBER SYSTEMS WITH THE SAME DYNAMIC RANGE...99 C. COMPARISON OF 3-CHANNEL MODULAR NUMBER SYSTEMS WITH THE SAME MODULI...1 D. RANGE DETECTION ERROR CONTROL...11 E. RECOMMENDATIONS FOR FUTURE WORK...11 APPENDIX LIST OF VARIABLES...13 LIST OF REFERENCES...15 INITIAL DISTRIBUTION LIST...19 viii

11 LIST OF FIGURES Figure 1. Geometry of CW radar using phase modulation odes to sample a moving target....8 Figure. Phase oded signal with the Barker ode N 7, A 1, f 1 khz, tb 1 ms....9 Figure 3. The relationship between unompressed pulse train and phase-oded CW signal for (a) Transmitted signal, and (b) The output of the signal proessed in the reeiver....1 Figure 4. Correlation reeiver mathed to N r periods of a transmitted polyphase ode for (After [9])....1 Figure 5. Example of ode ompression in CW Barker ode N 7 and N p Figure 6. Normalized orrelation output of Barker ode sequene N 7 and N 1 for (a) ACF and (b) PACF r Figure 7. PAF of Barker ode N 7 with (a) Nr 1 and (b) Nr Figure 8. Frank ode modulation for M 8, N 64 (From [1])... Figure 9. Signal phase (radians) modulo versus k -index for phase hange of the Frank ode with M 8, N 64 (From [1])... Figure 1. ACF (a) and PACF (b) for the Frank ode with M 8, pp 1, and Nr 1 (From [1])...1 Figure 11. PAF for the Frank ode modulation with M 8, pp 1, and Nr 1 (From [1])...1 Figure 1. The RNS residue folding waveforms for m1 4 and m Figure 13. Blok diagram for the radar using N residue-frank phase ode sequenes...6 Figure 14. Illustration of transmitted signal using N 3 residue-frank phase ode sequenes for m1 3, m 4, and m Figure 15. Plot of the residue-frank phase ode sequene for m 1 3, m 4, and m3 5 ( M RNS 6) showing (a) residues, and (b) phase sequenes from ()...9 Figure 16. Illustration of ompression at the reeiver and the range bin for RNS m1 3, N 9, and Nr Figure 17. Example of ompression output for target detetion using N 3 residue- Frank phase ode sequenes with m1 3, m 4, and m Figure 18. Resolving the range ambiguity of targets using RNS for m1 3, m 4, and m Figure 19. Missed detetions of the first set of residues Figure. Residue-Frank signal without noise for m 1 3, m 4, and m3 5 with M 6, A 1, f 1 MHz, f 7 MHz, pp 1, and t 1 s RNS ix s b

12 Figure 1. Residue-Frank signal with SNR = 3 db for m 1 3, m 4, and m3 5 with M RNS 6, A 1, f 1 MHz, fs 7 MHz, pp 1, and tb 1 s Figure. Residue-Frank signal with SNR = db for m1 3, m 4, and m3 5 with M RNS 6, A 1, f 1 MHz, fs 7 MHz, pp 1, and tb 1 s Figure 3. Target detetion for SNR = 3 db showing the range bins for m1 3, m 4, and m3 5 with the two targets at 44 m and 584 m Figure 4. Target detetion for SNR = db showing the range bins for m1 3, m 4, and m3 5 with two targets at 44 m and 584 m....4 Figure 5. Illustration of detetion error from the threshold detetor (CFAR) using the residue-frank for m1 3, m 4, and m Figure 6. Average power of the CW transmitter for residue-frank with m1 3, m 4, and m Figure 7. Comparison of the maximum unambiguous range of CW radar system for SNR 13 db using the residue-frank phase ode for m1 3, m 4, and Ro 3 5 m with the radar using eah Frank ode sequene individually with the orresponding N Figure 8. P4 phase sequene for N 64 (From [1]) Figure 9. Signal phase (radians) modulo versus k -index for phase hange of the P4 ode with N 64 (From [1])...46 Figure 3. ACF (a) and PACF (b) for the P4 ode with N 64, pp 1, and Nr 1 (From [1])...47 Figure 31. PAF for the P4 ode modulation with N 64, pp 1, and Nr 1 (From [1])...47 Figure 3. The SNS residues for m1 4 and m Figure 33. Blok diagram for the radar using N symmetrial residue-p4 phase ode sequenes....5 Figure 34. Illustration of transmitted signal using N 3 symmetrial residue-p4 phase ode sequenes for m1 7, m 8, and m Figure 35. Plot of the symmetrial residue-p4 for m1 7, m 8, and m3 9 ( M ˆ SNS 37 ) showing (a) symmetrial residues, and (b) phase sequenes from (34) Figure 36. Illustration of ompression at the reeiver and the range bin for SNS m1 7, N 7, and Nr Figure 37. Illustration of target detetion by using the range bin matrix for SNS-P4 ode with m1 7, m 8, and m Figure 38. Resolving the range ambiguity of targets using SNS for m1 7, m 8, and m Figure 39. Missed detetions of the first set of symmetrial residues....6 x

13 Figure 4. Symmetrial residue-p4 signal without noise for m 1 7, m 8, and m3 9 with Mˆ SNS 37, A 1, f 1 MHz, fs 7 MHz, pp 1, and 1 s tb Figure 41. Symmetrial residue-p4 signal with SNR = 3 db for m1 7, m 8, and m3 9 with Mˆ SNS 37, A 1, f 1 MHz, fs 7 MHz, pp 1, and 1 s tb Figure 4. Symmetrial residue-p4 signal with SNR = db for m1 7, m 8, and Figure 43. Figure 44. Figure 45. Figure 46. Figure 47. m3 9 with Mˆ SNS 37, A 1, f 1 MHz, fs 7 MHz, pp 1, and tb 1 s....6 Target detetion for SNR = 3 db showing the range bins for m1 7, m 8, and m3 9 with two targets at 89 m and 99 m Target detetion for SNR = db showing the range bins for m1 7, m 8, and m3 9 with two targets at 89 m and 99 m Illustration of detetion error from the threshold detetor (CFAR) using the symmetrial residue-p4 for m1 7, m 8, and m Average power of the CW transmitter for symmetrial residue-p4 with m1 7, m 8, and m Comparison of the maximum unambiguous range of a CW radar system for SNR 13 db using symmetrial residue-p4 phase ode for m1 7, Ro m 8, and m3 9 with the radar using eah P4 ode individually with the orresponding N Figure 48. RSNS residues RS mk, for modulus m16, N, and N RSNS Figure 49. The robust symmetrial residue-p4 phase ode sequene using m 16, N, N RSNS 64 is shown with the relationship with the P4 ode using N Figure 5. Power spetral density of the robust symmetrial residue-p4 signal using m 16, N, and N 64 with f 1 khz, f 7 khz, and pp RSNS Figure 51. The ACF and PACF of RSNS phase-oded signal ( m16, N, N RSNS 64) at the reeiver with Nr 1, f 1 khz, fs 7 khz, and pp Figure 5. The PAF of the signal for (a) the robust symmetrial residue-p4 phase ode sequene using m 16, N, N RSNS 64, and t b 1 ms, and (b) the P4 ode using N 3, and tb ms Figure 53. Blok diagram for the radar using N robust symmetrial residue-p4 phase ode sequenes Figure 54. Illustration of transmitted signal using N 3 robust symmetrial residue- P4 phase ode sequenes for m1 3, m 4, and m xi s

14 Figure 55. Plot of the RSNS using m 1 3, m 4, and m3 5 with Mˆ RSNS 43 for (a) folding waveforms, and (b) their phase sequenes....8 Figure 56. Illustration of ompression at the reeiver and the range bin for RSNS m1 3, N 18, and Nr Figure 57. Illustration of target detetion by using the range bin matrix for RSNS-P4 ode with m1 3, m 4, and m Figure 58. Resolving the range ambiguity of targets using RSNS for m1 3, m 4, and m Figure 59. Robust symmetrial residue-p4 signal without noise for m1 3, m 4, and m3 5 with Mˆ RSNS 43, A 1, f 1 MHz, fs 9 MHz, pp 1, and t b.33 s Figure 6. Robust symmetrial residue-p4 signal with SNR = 3 db for m1 3, m 4, and m3 5 with Mˆ RSNS 43, A 1, f 1 MHz, fs 9 MHz, pp 1, and t b.33 s Figure 61. Robust symmetrial residue-p4 signal with SNR = db for m1 3, m 4, and m3 5 with Mˆ RSNS 43, A 1, f 1 MHz, fs 9 MHz, pp 1, and t b.33 s Figure 6. Target detetion for SNR = 3 db showing the range bins for m1 3, m 4, and m3 5 with two targets at 14 m and 14 m....9 Figure 63. Target detetion for SNR = db showing the range bins for m1 3, m 4, and m3 5 with two targets at 14 m and 14 m Figure 64. Illustration of detetion error from the threshold detetor (CFAR) using the robust symmetrial residue-p4 for m1 3, m 4, and m Figure 65. Average power of the CW transmitter for robust symmetrial residue-p4 with m1 3, m 4, and m Figure 66. Comparison of the maximum unambiguous range of CW radar system for SNR 13 db using robust symmetrial residue-p4 ode for m 1 3, Ro m 4, and m3 5 with the radar using eah P4 ode individually for the orresponding N N / N...94 RSNS xii

15 LIST OF TABLES Table 1. Finding the RNS dynami range for m1 4 and m Table. Comparison of the residue-frank phase ode sequene ( m 4 ) with the Frank phase ode sequene ( M 4, N 16 )...5 Table 3. Bin matrix and range intervals of 3 hannel RNS with m1 3, m 4, and m Table 4. Finding the SNS dynami range for m1 4 and m Table 5. Comparison of symmetrial residue-p4 phase ode sequene ( m 8, N 8 ) with the P4 phase ode sequene ( N 8 ) Table 6. Bin matrix and range intervals of 3 hannelsns for m1 7, m 8, and m Table 7. The RSNS folding waveforms for m 1 3 ( s 1 ), m 4 ( s 1), and m3 5 ( s3 ) (After [])....7 Table 8. Comparison of the robust symmetrial residue-p4 phase ode sequene ( m 4, N, N RSNS 16 ) with the P4 phase ode ( N = 8) Table 9. Bin matrix and range intervals of 3 hannel RSNS for m1 3, m 4, and m Table 1. Summary of equations for eah modular phase ode Table 11. Summary of equations for N hannel modular number system Table 1. Comparison of eah modular phase ode in whih eah has the same dynami range = 6, and tb, t b 1 s Table 13. Comparison of eah modular number system in whih eah has the same moduli [3 4 5], and tb, t b 1 s...1 Table 14. List of variables xiii

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17 EXECUTIVE SUMMARY The Frank phase ode and the P4 phase ode are often used in ontinuous wave (CW) polyphase modulation radar systems to detet the target s range due to their linear timefrequeny harateristis as well as their low periodi ambiguity sidelobes. Both phase odes limit the unambiguous range R u to within a single ode period T Ntb, where N is the number of subodes and t b is the subode period. To extend the unambiguous detetion range, the radar might simply inrease the ode period of the polyphase modulation signal. However, the longer ode period requires a larger range-doppler orrelation matrix proessor, an inrease in the ompression time, as well as an inrease in the bulk memory in the reeiver. Moreover, a signifiant amount of loss is inurred if the total number of ode periods returned from the target is less than the number of referene ode periods used in the ompression proess due, for example, to a limited time-on-target. A new set of waveform arhitetures that an solve these limitations is proposed in this thesis. Number theoreti transforms (NTT) based on residue number systems are used in many digital signal proessing appliations to inrease the amount of information available from various folding waveforms. With a set of N moduli, the paired integer residue terms from the transform sequenes must be distint (no ambiguities) throughout the entire dynami range. The dynami range of a NTT is defined as the greatest length of ombined sequenes of integers that ontain no ambiguities or repeated paired terms. Consequently, the NTT residues within the dynami range an be used to onstrut a low-power CW polyphase waveform that is useful for deteting targets beyond the single ode period unambiguous range. To signifiantly extend the unambiguous range beyond a single ode period, this thesis explores the relationship between the polyphase odes (Frank and P4) and the NTT where the residues exhibit the same distribution as the polyphase values. The xv

18 unambiguous range is extended from the number of subodes within a single ode period to the dynami range of the transform without requiring a large inrease in orrelation proessing. A new Frank phase ode formulation is derived as a funtion of the residue number system (RNS) where eah residue orresponds to a phase value within the ode period (modulus) sequene. Based on the symmetrial distribution of the P4 ode, a new phase ode expression is derived using both the symmetrial number system (SNS) and the robust symmetrial number system (RSNS). Here eah phase value within the ode period orresponds to a symmetrial residue. MATLAB simulations are used to verify the new expressions for the RNS, SNS and RSNS phase odes. Implementation onsiderations of the new approah are also addressed. The relationship between the Frank phase ode and the RNS is developed based on their similar saw-tooth waveform distribution. A new equation using the RNS to derive the Frank phase ode is formulated. This new phase ode sequene is alled the residue-frank phase ode. The periodi autoorrelation funtion (PACF) and the periodi ambiguity funtion (PAF) show that the new residue-frank phase ode formula gives exatly the same results as using the formula in the Frank phase ode. Next, a set of N 3 RNS oprime moduli [3 4 5] (with dynami range M 6 ) is hosen to examine the idea of extending the unambiguous range. A MATLAB program is used to plot the results for deteting two targets that lie in an ambiguous detetion range bin if a single modulus is used. To solve the unambiguous range, the ombined residues from N sequenes are used. The result shows that the paired terms from both targets are different, meaning that both targets have different orresponding ranges. Thus, the residue-frank phase ode sequenes an be used to extend the unambiguous detetion range from R T / t N / to R, t M / where is the speed of light in free spae. u b u RNS b RNS A blok diagram of a polyphase CW radar system that uses the residue-frank phase ode is shown below in Figure 1. RNS xvi

19 Figure 1. phase ode. Blok diagram of a CW polyphase radar system that uses a residue-frank The relationship between the P4 phase ode and the SNS is developed based on their similar paraboli waveform distribution. A new equation using the SNS to derive the P4 phase ode is formulated. This new phase ode sequene is alled the symmetrial residue-p4 phase ode. The PACF and the PAF show that the symmetrial residue-p4 phase ode formula gives the same results as using the formula in the P4 phase ode. Then, the symmetrial residue-p4 phase ode sequenes using a set of N 3 SNS oprime moduli [7 8 9] (with dynami range Mˆ SNS 37 ) are examined for the two targets loated in an ambiguous range bin by using MATLAB programming. Similar to the residue-frank phase ode sequenes, the result shows that the symmetrial residue-p4 phase ode sequenes an be used to extend the unambiguous detetion range from Ru T / tbn / to R, t Mˆ /. u SNS b SNS Reonstrution of the target s range from the paired RNS and SNS phase terms an ontain inauraies when the inoming target range straddles two range bins. As a result, the reovered range of the target has a large error. To prevent the target range error, the RSNS is used. The paired terms from all N sequenes in the RSNS when onsidered together hange one at a time at eah range bin subode transition (integer Gray ode property). This property makes the RSNS partiularly attrative for ontrolling xvii

20 the target range error. In this thesis, the relationship between the P4 phase ode and the RSNS is also developed. A new equation using the RSNS to derive the P4 phase ode is formulated. This new phase ode sequene is alled the robust symmetrial residue-p4 phase ode. Within eah P4 subode, the phase shifts in the robust symmetrial residue- P4 phase sequenes are repeated N times (equal to the number of moduli), resulting in the broader mainlobe for the PACF and the redution of signal bandwidth as a funtion of N. The PACF and the PAF show that the robust symmetrial residue-p4 phase ode formula gives similar results as using the formula for the P4 phase ode. A MATLAB program is used to examine the abilities to extend the unambiguous range and to ontrol the range bin errors by using a set of N 3 RSNS oprime moduli [3 4 5] (with dynami range Mˆ RSNS 43 ). The results verify that the unambiguous range an be extended from Ru T / tbn / to R, tmˆ / where t b is the subode period in the ursns b RSNS RSNS phase ode, and the range bin has an error of only one range bin. In summary, all three NTT are investigated to show that they an be used to extend the unambiguous range. Also, the target range error from oding an be prevented by using the robust symmetrial residue-p4 phase ode sequenes. The use of these new derived phase sequenes an be applied to CW radar systems in order to have a low probability of interept harateristi given that the set of moduli and the radar signal parameters are hosen properly. The next step in this researh is to investigate the performane of using various sets of oprime moduli, and the hardware implementation and testing of an atual system and is being proposed for future work. xviii

21 ACKNOWLEDGMENTS First and foremost, I would like to speially thank my parents and girlfriend, Ying, for their love, enouragement, and understanding. Their ontinued supports help me overome any diffiulties throughout my life. This thesis, one of the most hallenging tasks I have enountered, would not be ompleted without them. I would like to thank my advisors, Professor Phillip E. Pae and Professor David C. Jenn for their wise guidane through this proess. Their inredible patiene and assistane as I struggled through onepts and problems was valued and respeted greatly. I have learned so muh throughout my time at NPS. Also, I would like to thank Mr. Matthias Wiht and his wife, Cdr. Peerapong and his family, and Mr. Paul D. Buzynski, who spent time sharing their valuable life experienes with me. Without them, I would be ompletely overwhelmed by the raziness of shool. Finally, I would like to thank the Royal Thai Navy for providing me an exellent opportunity to get a master s degree from NPS. My dream has been ompleted for studying at the great institution. xix

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23 I. INTRODUCTION A. CONTINUOUS WAVE RADAR SYSTEMS USING POLYPHASE MODULATION Continuous wave (CW) radar systems have a superior low probability of interept (LPI) performane over pulse train radar systems due to their low average power transmitted (e.g., 1 W) and their use of pulse ompression tehniques. Note we extend the onept of pulse ompression to unpulsed CW waveforms sine the tehniques are similar and the objetives are the same. Power management is also a benefit when a solid state phased array antenna is used. This enables the target return signal-to-noise ratio (SNR) to remain at a onstant value within the radar reeiver, thus lowering the transmitted power further as the range-to-target dereases. An LPI radar system is defined as a radar system that uses a speial emitted waveform intended to detet targets as well as to prevent a non-ooperative interept reeiver from interepting and deteting its emission [1]. On the other hand, pulsed radar systems require a relatively high peak power (e.g., 1 kw) to be transmitted to obtain the same probability of target detetion. Consequently, the non-ooperative interept range is signifiantly longer for the pulsed radars. Sine pure CW waveforms annot resolve the target s range, periodi modulation tehniques are used, suh as frequeny modulated CW (FMCW), frequeny-shift keying (FSK), noise modulation, phase-shift keying (PSK), as well as hybrids of these tehniques. The first step to designing CW radar systems that use periodi modulations for ompression is to deide on the range resolution that is required. This in turn sets the transmitted bandwidth of the waveform for the above tehniques (exept for FSK where the range resolution is dependent on the duration of eah frequeny). Due to the advanes in high-speed proessing and diret digital synthesis modules [], the use of PSK tehniques in CW radar is highly advantageous. CW radars that transmit and reeive PSK signals an result in LPI radar systems with small range resolution ells and are ideally suited for many sensor appliations for situational awareness, inluding multiple-input multiple-output (MIMO) onfigurations. 1

24 Binary Barker phase oded sequenes (1, 1) are one of the most popular PSK tehniques in the design of pulse ompression radar signals due to the aperiodi autoorrelation oeffiients or time sidelobes being limited to 1/ relative to a mainlobe level of one, where longest ode length is N = 13 (or N (at zero Doppler) N is the number of subodes used. The N =169 for a ompound ode) [3]. These odes are not used in CW radar systems for LPI appliations sine they are very sensitive to Doppler shifts and an be easily deteted by an interept reeiver that uses frequeny doubling detetion, whih onsists of multiplying the signal by itself and proessing the result through an envelope detetor. Polyphase sequenes are also of finite length with N subodes and onsist of disrete time omplex values with onstant amplitude but with a variable phase. Polyphase oding refers to phase modulation of the CW arrier with a polyphase sequene onsisting of a number of disrete phase states with eah phase orresponding to a subode. The number of subodes is taken from an alphabet of size N, whih is also the proessing gain of the radar exluding any post detetion integration. Inreasing the number of phase values in the sequene allows the onstrution of longer sequenes, resulting in a greater proessing gain or ompression ratio in the reeiver (or equivalently a larger SNR). The ode period is T Ntb where the size of eah subode t b determines the 3 db bandwidth of the waveform B 1/ t b and the range resolution Rt b /. The unambiguous range is limited by the number of subodes in the sequene as R N R. With a CW polyphase waveform, the mathed filter in the reeiver is a oherent, range-doppler matrix orrelation proessor that performs a ross orrelation between the reeived signal and a referene signal whose envelope is the omplex onjugate of ode periods of the transmitted polyphase signal. The ross orrelation output values are added together to redue the ambiguity sidelobe levels. When the polyphase signals are returned and have an impressed Doppler shift, the orrelation proess used in the reeiver ompression operation is not perfet, resulting in a ertain amount of orrelation loss due to the phase shifts aross the ode period T. u N r

25 Two of the most important PSK odes that are used in CW radar systems inlude the Frank ode [4] and the P4 ode [5]. The Frank ode is a polyphase ode that is derived from a step approximation to a linear FMCW waveform using M frequeny steps and M samples per frequeny, giving the number of subodes or proessing gain as N M. The phase steps of the Frank ode exhibit a saw-tooth folding waveform distribution. The peak sidelobe level (PSL) of a single ompressed Frank ode period is PSL log (1/ M ). 1 The P4 ode onsists of disrete phases of a linear hirp waveform taken at speifi time intervals. It is derived by onverting a linear FMCW to baseband using a loal osillator on one end of the frequeny sweep and sampling the inphase and quadrature video at the Nyquist rate. The P4 phase steps exhibit a symmetrial paraboli distribution. The PSL of a single P4 ompressed ode period is PSL log1 / ( N ). Although polyphase CW modulations an serve as a transmission waveform for a LPI radar system, there are several limiting fators. Most signifiantly, the unambiguous detetion range of the waveform is limited by the number of subodes (ode length) as R N R. By inreasing the number of subodes to extend the unambiguous range, a u larger range-doppler orrelation matrix proessor in the radar reeiver is needed to provide the ode ompression. To lower the peak sidelobes, are used to ompress the 3 N r opies of the phase ode N p returned ode periods from the target. Consequently, the inreased ode length or proessing gain requires a larger ode ompression time as well as an inrease in the bulk memory requirements in the reeiver. Further, a signifiant amount of orrelation loss is inurred if the total number of ode periods returned from the target N p is less than the number of ode periods proessor, as would be the ase due to a limited time-on-target. N r used in the orrelation In this thesis, a novel relationship between the Frank ode and the residue number system (RNS) is developed in order to signifiantly extend the unambiguous range beyond a single ode period T N t while not having to inrease the number of b subodes. In addition, due to the symmetrial distribution of the P4 polyphase ode, the

26 unambiguous range is extended by developing a phase relationship based on symmetrial number systems; both the symmetrial number system (SNS) and the robust symmetrial number system (RSNS) are investigated. The phase relationships are onstruted using these number theoreti transforms to extend the unambiguous range of the radar system by assoiating eah phase subode within the ode period T with an integer sequene within a oprime modulus from the set m1, m,..., m N. The paired integer terms from eah polyphase sequene are unambiguous within the transform s dynami range. The dynami range is defined as the greatest length of ombined phase sequenes that ontain no ambiguities or repeated paired terms. The dynami range for the RNS is denoted M RNS (Frank ode). The dynami range for the SNS is M ˆ SNS (P4 ode). The dynami range for the RSNS is M ˆ RSNS (P4 ode). For eah ase, the length of the dynami range ontains multiple ode periods. In the radar reeiver, the integer values within eah modulus sequene orrespond to the phase values from a phase detetor [6]. The RNS phase sequene ontains multiple Frank odes and extends the unambiguous target detetion range from R T / t N / to u b R, t ( m)/ t M /. The SNS phase sequene ontains multiple P4 odes urns b i b RNS and extends the unambiguous target detetion range beyond a single ode period to R t Mˆ /. u, SNS b SNS It is noted that the SNS also has a diret relationship with the disrete Fourier transform (DFT) and has been used to resolve frequeny ambiguities in undersampling digital reeivers [7]. Similarly, the unambiguous range using the RSNS is extended to R, tmˆ / where t b is the subode period in the RSNS phase ursns b RSNS ode. Reonstrution of the target s range from the paired RNS and SNS phase terms an ontain inauraies when the inoming target range straddles two range bins or subodes. For example, if the target is straddling a range bin and a onstant false alarm rate (CFAR) proessor detets the target in the wrong subode within a modulus sequene, the reovered range of the target will have a large error. This is shown to be the ase for both the RNS and the SNS. Additional signal proessing an be used to eliminate 4

27 these errors but adds additional omplexity to the detetion proess. A more effiient method to eliminate the target range bin errors uses the RSNS. The RSNS is also a modular sheme; however, the paired terms from all N sequenes, when onsidered together, hange one at a time at eah range bin subode transition (integer Gray ode property). Although the dynami range of the RSNS is less than the SNS ( M ˆ M ˆ RSNS SNS ), the RSNS integer Gray ode property makes it partiularly attrative for ontrolling the target range error. In summary, all three number theoreti transforms are investigated to determine the feasibility of extending the unambiguous range of the CW polyphase emitter. B. PRINCIPAL CONTRIBUTIONS The first step in this thesis was to understand the priniples of CW phase modulation radar systems. This inluded the omplex envelope representation of the transmitted signal and the related modulation parameters suh as the ode period, the subode period, the number of subodes within the ode period, and the number of yles of the arrier frequeny per subode. Their relationship to the signal bandwidth, the range bin size, the unambiguous detetion range and the proessing gain were also studied. Upon reeption, the CW signal is ompressed, and several ode ompression tehniques were studied and modeled. To understand the reeiver performane when a partiular CW phase ode is used, the periodi autoorrelation funtion and the periodi ambiguity funtion were examined. A MATLAB ode was written to generate the CW phase oded signals and to all the periodi ambiguity funtion subroutines. The next step in the thesis was to study the Frank polyphase ode formulation and the phase distribution within a ode period. Periodi ambiguity analysis was also performed to quantify the sidelobe levels within the reeiver s ambiguity spae (range offset Doppler offset). Next the RNS was studied as a means to generate the Frank ode sine they both have sawtooth folding waveform representations. By using the RNS to reate the Frank polyphase modulation, the unambiguous range of the ode an be extended signifiantly beyond a single ode period. A new expression to generate the Frank polyphase ode was developed as a funtion of the oprime RNS moduli hosen. 5

28 By using two or more Frank ode sequenes that are relatively prime, the deteted target residue from eah sequene is ombined within a system of ongruenes whih an be solved in a straightforward manner using the Chinese Remainder Theorem (CRT). The solution is the target s unambiguous range. A MATLAB model was developed to simulate the new Frank polyphase modulation in order to verify the expression and determine the differenes in phase values (if any) that might our. Next a new CW radar signal proessing algorithm was developed to transmit and reeive the oprime Frank sequenes to demonstrate the extension of the unambiguous range. The algorithm also addressed multiple targets within the field of view. Several examples were given to demonstrate the onept. To examine the pratial onsiderations, a simulation of the transmit power versus required ompression output SNR was performed in MATLAB for several values of oprime moduli. Next, the P4 polyphase ode was examined the relationships to the SNS and the RSNS. A new formulation of the P4 polyphase ode as a funtion of the SNS and RSNS oprime moduli was developed. Analysis similar to that for the Frank ode desribed above was performed to verify the new P4 expression. Finally, a summary of the different relationships that were developed is given and a omparison was performed. C. THESIS OUTLINE In Chapter II, the fundamentals of radar systems that use polyphase modulation are reviewed. In Chapter III, the Frank ode and RNS are examined to extend the unambiguous range. In Chapter IV, the relationships between the P4 and the SNS are developed. In Chapter V, the relationships between the P4 and the RSNS are investigated to redue the possibility of a range bin enoding error. Conluding remarks are offered in Chapter VI. 6

29 II. PRINCIPLES OF PHASE MODULATION RADAR The idea of CW radar using phase modulation is reviewed in this hapter. First, the fundamentals and use of phase modulation in CW radar systems are explained. Next, the onepts of signal analysis and the tehnique used in the reeiver s signal proessing are presented. Also, the system equations for a phase modulation in CW radar are given in order to examine the signal s properties. These equations will be used to determine the new phase modulation ode s performane in the following hapters. A. CW PHASE MODULATION CONCEPTS In Figure 1, the basi CW radar geometry is illustrated. First, the CW radar transmits the phase oded signal st () to the target through the transmitting antenna. When the signal propagates to the target at a range R, it reflets bak to the CW radar with different harateristis. For example, the signal s amplitude is less and the frequeny is shifted in time in aordane with the relative movement between platforms along the line-of-sight ( V os in Figure 1) or Doppler shift. The return signal x() t is then aptured by the reeiving antenna. In the radar system s reeiver and digital signal proessor, the ompression proessing output is used to detet and extrat the target s harateristis. In phase oded CW radar systems, the phase shifting operation is performed in the radar s transmitter, with the timing information generated from the reeiver-exiter. The transmitted omplex signal an be written as [1] st () k j( ft ( t)) Ae (1) where A is the signal amplitude, f is the arrier frequeny, and k ( t ) is the time dependent phase modulation ode. The inphase I and quadrature Q representations of the omplex signal from the transmitter an be represented as and I() t Aos( f t ()) t () 7 k Qt () Asin( ft ()). t (3) k

30 Figure 1. Geometry of CW radar using phase modulation odes to sample a moving target. For one ode period T, the CW signal is shifted in phase every subode with eah ode period onsisting of N subodes. Eah subode with phase k whih is the subode period. For a speifi ode sequene, the ode period is has a duration of t b T N. tb (4) The transmitted signal an be expressed as for t T N T k b k1 u u [ t ( k 1) t ] (5) and zero elsewhere. The omplex envelope uk is uk j k e. (6) If pp is the number of yles of the arrier frequeny per subode, the bandwidth B of the transmitted signal is f 1 B. (7) pp t b 8

31 To understand the onepts of phase modulation, a binary Barker phase ode is examined [3]. In Figure, an example of one ode period with N 7 is shown. Here, the arrier frequeny is f 1 khz, the subode period t 1 ms, and from (7) pp 1. b Figure. Phase oded signal with the Barker ode N 7, A 1, f 1 khz, t 1 ms. b The reeived return signal x() t from the target an be written as xt j{ ( f)( t) k( t)} () Ae (8) where is the amplitude attenuation oeffiient of the reeived signal due to target sattering and propagation attenuation. The two-way roundtrip delay for a monostati radar system is R/, where is the speed of light in free spae. The Doppler frequeny due to the target motion is V os( )/ where is the signal wavelength and V is the target veloity. B. CODE COMPRESSION OF RECEIVED SIGNAL Code ompression (CC) is a signal proessing tehnique designed mainly to minimize the range resolution ell and inrease the SNR of the returned signal. The idea of CC is similar to that in signal orrelation in pulse train radar. As shown in Figure 3a, the CW radar repeats the phase sequene every ode period T. A similar pulse train 9

32 signal would also have a pulse repetition interval PRI = T. Referring to Figure 3b, if the pulse width pw of the pulse train signal is equal to the width of the subode period t b in the CW phase oded signal, the output s mainlobe from both orrelator and ompressor would have the same duration ( t ). These CC peaks represent the targets if the pw b peaks are higher than the threshold desired in a onstant false alarm rate (CFAR) proess. (a) (b) Figure 3. The relationship between unompressed pulse train and phase-oded CW signal for (a) Transmitted signal, and (b) The output of the signal proessed in the reeiver. The phase-oded signal is transmitted to detet the target at a range R. When reahing the target, the refleted signal returns to the radar. At the radar reeiver, the phase oded return signal is orrelated with the referene phase ode. This is ahieved by onvolving the inoming signal s phase ode with a onjugated and time-reversed version of the transmitted signal s phase ode. This operation an be done either in the 1

33 digital or analog domain. Many digital signal proessing arhitetures have been proposed for fast orrelation using fast Fourier transforms. If no ompression loss is present, the orrelated output will give a very narrow pulse width at the ompression output for the deteted target. The orrelation output is then sent to a target detetion proess suh as a onstant false alarm rate proessor. If there is any alteration of the return signal s phase (for example from a Doppler shift or a less than adequate bandwidth in the reeiver B 1/ t b ), a ertain degree of orrelation loss ours. The reeived waveform from the target is digitized and orrelated using a phase ode ompressor. The range-doppler orrelation matrix an ontain a asade of of N r sets N referene oeffiients to improve the sidelobe struture. In Figure 4, the range- Doppler orrelation matrix reeiver for zero Doppler offset is illustrated. Here, the reeived signal onsists of N p ode periods, eah of whih has N subodes. The return signal is first proessed by a filter mathed to a retangular subode of length t b, followed by a phase detetor that sends forward the phase values. The deteted output signal is then sent through a tapped delay line where eah delay D is tb. As the signal progresses through the tapped delay line, it is multiplied by the referene signal. At eah step, the multipliation for eah delay is summed separately for eah of the ode periods. To redue the sidelobe struture, a weighting funtion N r referene C i an also be added [8]. To ompute the entire range-doppler ambiguity funtion, the orrelation outputs from eah subode are multiplied by q e k j k tb for where k ranges from to NN 1. For, the output of the previous operation are again multiplied r by q e k j k tb and so on for the entire ambiguity spae [9]. 11

34 Figure 4. Correlation reeiver mathed to N r periods of a transmitted polyphase ode for (After [9]). In Figure 5, an illustration of the ode ompression for CW binary Barker ode with N 7 is shown. The radar reeives the returned Barker sequene with the number of ode periods N p 3. The number of referene ode periods used in the reeiver is Nr 1. The reeiver starts ompressing the first returned ode T 1 at, giving the orrelation output peak at a normalized delay of / t b representing the deteted target. At eah step in the ompression, the returned signal is shifted one subode period t b. When the returned ode is shifted by one ode period T, the orrelation output gives another peak value due to the seond ode T being ompressed at a normalized delay of / t b 7. The last peak from the third ode T 3 ompressed is also shown at a normalized delay offset of / t b 14. That is, the peak value for the deteted target 1

35 repeats every ode period T. Generally, the width of the PACF mainlobe is t. To be able to distinguish between two targets lose to one another, the pulses have to be separated by at least t period (whih is defined as the range resolution ell). b b Figure 5. Example of ode ompression in CW Barker ode N 7 and N p 3. In summary, the unambiguous range of a speifi CW phase-oded sequene is T Ntb Ru (9) and the range resolution for one transmitted polyphase signal is t R b. (1) Another important parameter in pulse ompression is the pulse ompression ratio (PCR) or time-bandwidth produt. This is also referred to as the radar s proessing gain 13

36 (PG) and is the preferred term in this thesis. In general, the higher the PG, the higher the SNR improvement in the signal proessing step. The radar signal proessing input SNR is SNR Ri and the radar signal proessing output SNR is related to the proessing gain of the signal proessor by SNR Ro (see Figure 1). They are SNRRo PG. (11) SNR For a polyphase ode, the PG is equal to the time-bandwidth produt or the number of subodes as frequeny Ri PG T (1 / t ) ( N t ) / t N. (1) b b b In the Barker ode N 7 example, the proessing gain is 7. For a arrier f, narrowing the subode period t b enables the transmitted signal to be spread over a larger bandwidth. Also, inreasing the number of subodes N, the orrelation output sidelobes an be dereased signifiantly while providing a larger PG. This omes at the expense of a larger range-doppler orrelation matrix proessor in the reeiver. In addition, a longer ompression time and an inrease in the size of the bulk memory is required. The signal proessing funtion used for the ompression is the autoorrelation funtion (ACF). For CW radars, the ompression is aomplished with a periodi autoorrelation funtion (PACF). Also, the periodi ambiguity funtion (PAF) is used to show the delay-offset versus frequeny-offset in the reeiver and shows the resulting sidelobe struture. This is espeially useful for monitoring the ambiguity spae for seondary targets that might hide in the sidelobe struture. Note that the zero Doppler ut of the PAF is the PACF [1]. C. PERIODIC AUTOCORRELATION FUNCTION It is important that in order to ahieve high range resolution for a polyphase ode sequene, the oded signal has to have a noise-like PACF property, lose to an impulse funtion. The ACF is used for signals of finite energy. For a CW polyphase ode 14

37 sequene with the number of phase odes T N t, a periodi omplex envelope ut () is given by [1] b N, subode duration t b, and a ode period ut () ut ( nt) (13) for n, 1,, 3,... The values of the PACF as a funtion of the delay r (whih are the multiples of t b ) are given by N 1 R( rtb) u( n) u ( n r). (14) N n1 Ideally, a perfet periodi autoorrelation funtion (zero-sidelobe) is desired where Rrt ( b) 1 when r (mod N ), and Rrt ( b) when r (mod N ). Sine the CW signal is ontinuous, a perfet periodi autoorrelation funtion an be ahieved. In Figure 6a, the ACF output for a N 7 Barker ode is plotted (in db) with no noise using f 1 khz, f 7 khz, pp 1 and N 1 using the MATLAB m-file s (amfbn7.m) [11]. The highest peak mainlobe is at the zero time delay. The width of the mainlobe is equal to the length of the subode tb. r For the Barker ode, it is found that the PSL is equal to log 1( N ) 17 db. The PACF in Figure 6b shows that the peak is repeated every ode period T N t at the value of 7. Also note from (14) that the CW b binary Barker ode is not a perfet ode. (a) (b) Figure 6. Normalized orrelation output of Barker ode sequene N 7 and Nr 1 for (a) ACF and (b) PACF. 15

38 Today, many studies on PSK show that the PSL of the phase-oded sequenes an be redued by hoosing a proper phase oding. Moreover, there are CW polyphase odes able to ahieve a zero sidelobe (perfet odes). D. PERIODIC AMBIGUITY FUNCTION In a realisti environment, the returned signal will be shifted in frequeny if the target is moving relative to the radar s antenna beam axis (Doppler effet). The PAF is an important tool to assess the response of a orrelation reeiver. In pulsed radar analysis, the ambiguity funtion (, ) is a two-dimensional funtion of time delay offset and Doppler frequeny offset showing the orrelation between the returned signal and the reeiver mathed filter. For the finite duration signal, the ambiguity funtion (AF) is determined by the properties of the pulse and the mathed filter. If ut () is the omplex envelope of both the transmitted signal and reeived signal, the AF is given by [1] * (, ) () ( ) j t utu t e dt (15) where * denotes the omplex onjugate. A target further from the radar than the referene ( ) position will orrespond to positive. A positive implies a target moving toward the radar. A more onise way of representing the ambiguity funtion onsists of examining the one-dimensional zero-delay and zero-doppler uts; that is, (, ) and (,), respetively. The mathed filter output as a funtion of a time (the signal one would observe in a radar system) is a delay ut, with onstant frequeny given by the target s Doppler shift (, ). Ideally, the ambiguity diagram should show the diagonal ridge entered at the origin and zero elsewhere, no ambiguity. The PAF, introdued by Levanon and Freedman [11], desribes the response of a orrelation reeiver to a CW signal modulated by a periodi waveform with period T, when the referene signal is onstruted from an integral number transmitted signal (oherent proessor length time) p 16 N r of periods of the NT). r The target illumination time (dwell NT must be longer than NT. As long as the delay is shorter than the r

39 differene between the dwell time and the length of the referene signal ( N N ) T, the illumination time an be onsidered infinitely long and the p r reeiver response an be desribed by the PAF given as [8] NT r r 1 j t N, (, ) ( ) ( ). r T ut u te dt r NT r r (16) (a) (b) Figure 7. PAF of Barker ode N 7 with (a) Nr 1 and (b) Nr 4. 17

40 In Figure 7a, the PAF of the Barker ode N 7 with Nr 1 is plotted using MATLAB m-file (amfbn7.m) from [1]. Also, the PAF of the Barker ode N 7 with Nr 4 is plotted in Figure 7b. By inreasing N r, the Doppler sidelobes have been signifiantly lowered. In the next hapter, the PACF and PAF are used to haraterize the CW Frank ode. The residue number system is introdued and a relationship to the Frank ode is developed. Sine there are many variables used throughout this thesis, all of important variables are listed in the Appendix to provide unambiguous meanings. 18

41 III. FRANK POLYPHASE MODULATION AND THE RESIDUE NUMBER SYSTEM In this hapter, the Frank phase ode and the residue number system (RNS) are introdued. Their relationship is presented. A. FRANK PHASE CODE The Frank ode is derived from a step approximation to a linear frequeny modulation waveform using M frequeny steps and M samples per frequeny [4]. If i is the sample number in a given frequeny and j is the frequeny number, the phase of the i th sample of the j th frequeny is given by k i, j ( i 1)( j 1) (17) M where k is the time index for a single phase hange per subode and it ranges from 1 to M, and i, j 1,,..., M. The phase values and the phase values modulo of the Frank ode for M 8, N 64 are shown in Figures 8 and 9, respetively. The arrier frequeny is f 1 khz, the sampling frequeny is f 7 khz, and the number of arrier yles per s subode is pp 1. The ACF and the PACF are shown in Figure 1 where N r is the number of referene ode periods in the reeiver. The delay axis is normalized by the subode period t b. Note from Equation (14) and Figure 1b that the Frank ode is a perfet ode. The PSL is PSL log 1/ ( M ) or 8 db. The PAF in Figure 11 shows the delay-offset and Doppler-offset sidelobes. Note that the Doppler tolerane is refleted in the way the sidelobes are arranged about the mainlobe. The study on Doppler tolerane is presented in [5] and [1]. 19

42 Frank phase shift (rad) k - index for phase hange Figure 8. Frank ode modulation for M 8, N 64 (From [1]). 6 5 Signal phase (rad) k - index for phase hange Figure 9. Signal phase (radians) modulo versus k -index for phase hange of the Frank ode with M 8, N 64 (From [1]).

43 (a) (b) Figure 1. ACF (a) and PACF (b) for the Frank ode with M 8, pp 1, and Nr 1 (From [1]). Figure 11. PAF for the Frank ode modulation with M 8, pp 1, and Nr 1 (From [1]). 1

44 B. THE RESIDUE NUMBER SYSTEM The RNS was introdued by Szabo [13]. It has been established as an important tool in parallel proessing appliations and high speed omputations [14] [16]. The RNS an serve as a soure for extending the unambiguous range by deomposing the unambiguous range into a number of parallel sub-ranges (moduli) that have a smaller number of subodes. Eah sub-range for a different oprime modulus requires only a ode period in aordane with that modulus. A muh larger unambiguous range is ahieved after the results of these smaller sub-range operations are reombined. Thus, by developing a relationship between the RNS, the polyphase distributions within the Frank ode period, and the target detetion proessing, the extension of the unambiguous range beyond a single ode period is feasible. The RNS residue distribution is based on a saw-tooth folding waveform, where the disrete residue values rise gradually and fall quikly as the input value rises gradually (i.e., in mod 5 the suession of disrete values for an inreasing input is, 1,, 3, 4,, 1,, 3, ). A single RNS sequene an be generated as a R (mod ) m m (18) where Rm m, m is the modulus, and R m is the residue of a modulo m. The integer values Rm within eah modulus sequene are Rm [,1,..., m 1,] (19) and are representative of the saw-tooth folding waveform. The sequene is repeated in both diretions forming a periodi sequene with the period PRNS m. () Given a set of RNS oprime moduli, a omplete residue number system is formed. A set of integers y, y 1,, ym 1 (mod m) form a omplete residue system if y g(mod m) for g,1,, m1. The dynami range M RNS of the RNS is the number g of paired terms from the sequene of values that ontain no ambiguities and is

45 where M RNS N m (1) i1 mi is the i th oprime modulus and N is the number of moduli, 1 i N. In Figure 1, an RNS example for N moduli, m1 4 and m 5, stair step relationship of the residues. i is shown. Note the 3 m 1 = 4 residue R m input(a) 4 m = 5 residue R m input(a) Figure 1. The RNS residue folding waveforms for m1 4 and m 5. From Equation (1), for m1 4 and m 5, the dynami range is M. In Table 1, the two sequenes are shown along with the input a. Starting at a, the total number of distint paired terms is, a 19. The index value l a 1 in Table 1 is shown to ount the integer number of distint paired terms within the dynami range. RNS Table 1. Finding the RNS dynami range for m1 4 and m 5. Input( a ) m1 4 m 5 l

46 C. RESIDUE-FRANK PHASE CODE The RNS residues Rmk, an be folded into the Frank phase ode sequene by assoiating R mk, with the Frank subode phases. The phase index k starts from 1 to N, where N is the number of subodes within one ode period. The residue-frank phase ode sequene from modulo m is given by where k k 1 R m m mk, mk, 1,,3,..., m is the phase index for eah phase hange, and x () indiates the greatest integer less than or equal to x. In Equation (), the number of subodes within one ode period is N m. For example, if we hoose m = 4, N = 16, the RNS folding waveform beomes Rm 4, k 1,,...,16 {, 1,, 3,, 1,, 3,, 1,, 3,, 1,, 3}. Next, the phase sequenes from equations (17) and () are ompared. For Equation (), the parameters used to alulate the residue-frank phase ode sequene are m 4, m 16 with m 4, k 1,,...,16 R {, 1,, 3,, 1,, 3,, 1,, 3,, 1,, 3}. For the Frank ode from Equation (17), the parameters are M 4, N 16 with i, j 1,,3,4. In Table, the phase ode sequenes are shown for omparison. The phase ode sequenes of both waveforms are idential. The ACF, PACF, and PAF of the CW waveform generated by Equation () with m 8, N 64 are also idential to those shown in Figures 1 and 11, whih were generated by Equation (17) with M 8, N 64. 4

47 Table. Comparison of the residue-frank phase ode sequene ( m 4 ) with the Frank phase ode sequene ( M 4, N 16 ). Time index Residue-Frank phase ode () Frank phase ode (17) ( k ) R k 1 m 4, k m (rad) i j m 4, k (rad) k 1,1,1 3,1 4,1 1, i, j , , , ,3 4, ,3 4,3 1,4, ,4 4,

48 D. RESOLVING RANGE AMBIGUITIES USING N RESIDUE-FRANK PHASE CODE SEQUENCES 1. Blok Diagram of the Residue-Frank Radar System To extend the unambiguous detetion range, N oprime moduli are used with Equation () to generate the residue-frank phase ode sequenes neessary to modulate the CW frequeny. The blok diagram of the radar using the residue-frank phase ode modulation is shown in Figure 13. In the transmitter, the first step is to generate and store the sequene values R mk, for eah modulus. Then, a diret digital synthesizer uses eah residue sequene R mk, to generate the N m subodes mk, aording to Equation () for eah modulus. Eah subode orresponding to R mk, has a length of t b. Together the N subodes have a ode period of T. To detet the target s unambiguous range, eah phase ode sequene must have transmitted in suession. M RNS subodes. Eah sequene is amplified and Figure 13. Blok diagram for the radar using N residue-frank phase ode sequenes. Upon reeption, the signal is amplified and downonverted to an intermediate frequeny ompatible with an available analog-to-digital (ADC) onverter tehnology. The output of the ADC is proessed by a moving target indiation (Ellipti) filter to remove the lutter, and then a phase detetor is used to determine the phase and 6

49 magnitude of eah subode. The output of the phase detetor is proessed by a phase ode ompressor whih is followed by a nonoherent post-detetion integration (PDI) proess. The output of the PDI is sent to a onstant false alarm rate proessor to detet and save the range bin of the target for that modulus sequene. This series of steps is performed for eah residue-frank phase ode sequene, and the deteted target s range bin R m is sent to a RNS-to-deimal algorithm to resolve the target s true range after all N residues R m are obtained.. Transmitted and Referene Codes for Compression In this setion, the transmission of the set of N residue-frank phase ode sequenes is disussed. The use of eah phase ode sequene as a referene sequene for ode ompression is disussed. An example using N 3 moduli with m 1 3, m 4, and m3 5 is illustrated. The dynami range from Equation (1) for these RNS moduli is M RNS The transmitted sequene for eah modulus requires the number of subodes equal to the dynami range M RNS. The transmitted signal is shown in Figure 14. For modulus m1 3, there are three residues within a ode period (, 1, ). For the residue- Frank phase ode period, the sequene has N m To over a dynami range of M RNS 6 subodes, the number of ode periods required is ˆ M RNS N. pm, (3) m For m1 3, Nˆ,3 6 / ode periods. The last 3 subodes (976 3) are not p used sine they are ambiguous. For m 4 and m3 5, the number of subodes N for eah ode period are m 4 16 and m3 5 5, respetively. Also, the maximum number of transmitted and reeived ode periods are Nˆ, (the last 4 subodes are not used) and p 7

50 Nˆ.4 (the last 15 subodes are not used), respetively. In Figure 14, the ode period p,5 orresponding to eah modulus is denoted as T, where p represents ode period number. Note that the number of reeived ode periods proessed in the range-doppler matrix N should be less than the number of ode periods reeived N. That is r mi p N N Nˆ. Otherwise, a ertain amount of orrelation loss ours. r p p p Figure 14. Illustration of transmitted signal using N 3 residue-frank phase ode sequenes for m1 3, m 4, and m3 5. In Figure 15a, the residues R mk, for m1 3, m 4, and m3 5 ( M RNS 6) are shown with Rmk, R, where mk nm n, 1,,.... The residue-frank phase sequenes mk, from eah modulus are shown in Figure 15b. Eah phase ode sequene has a length N m, giving mk,. mk, nm If the reeiver uses one ode period as a referene ( Nr 1), the referene ode sequene for ompression has a phase sequene from index k 1 to m. 8

51 Residue R 3 Residue R 4 Residue R m 1 =3 m =4 m 3 = time-index (a) phase(rad) phase(rad) phase(rad) T 3 =9 m 1 =3 T 4 =16 m =4 T 5 =5 m 3 = time-index (b) Figure 15. Plot of the residue-frank phase ode sequene for m1 3, m 4, and m ( M 6) showing (a) residues, and (b) phase sequenes from (). 3 5 RNS 1 3 In Figure 16, the ompression and range bin detetion proess is illustrated for m and N 1. The residue-frank phase ode sequene is returned to the radar r reeiver with delay and then proessed by a filter mathed to a retangular pulse of 9

52 duration t b. The phase detetor is used to alulate the phase of the signal whih is passed to the ompressor. The output from the ompression is then proessed by the threshold detetion to see whether the amplitude reahes the desired minimum value. The ompressor proesses a single ompression delay within a sampling period t s. If the threshold detetor detets the pulse, the presene of the target will be saved and assoiated with residue R m. Eah delay D l has a duration of t b (e.g. D 1 is for the time delay between to t b, D is for the time delay between t b to t b and so on). For m 4 and m3 5, the proess is similar, exept the phase referene values have to orrespond to the transmitted phase ode sequene that is being reeived. Figure 16. Illustration of ompression at the reeiver and the range bin for RNS m1 3, N 9, and N 1. r 3

53 3. Calulating the Target Range After the returned signal is ompressed, the first sample appears in the reeiver s range bin that orresponds to a duration. For eah modular sequene, the target detetion orresponds to a residue R m. By ombining the paired residue from eah sequene, the target s range bin an be alulated. Continuing the example in the previous setion, we show the residues for m1 3, m 4, and m3 5 and the orresponding target range in Table 3. Here, the signal bandwidth B 1 MHz, and the subode period t 1 s. The unambiguous range using residue-frank phase ode is M RNStb RuRNS,. (4) For this example, the maximum unambiguous range is RuRNS, 9 km ( M RNS 6) and is onsiderably longer than that of any single ode period. The range resolution is Rt b / 15 m. The target s range now has to be R R urns, in order to be deteted with no range ambiguity. The range interval for eah range bin of this example is also given in the Table 3. b Table 3. Bin matrix and range intervals of 3 hannel RNS with m1 3, m 4, Bin a m1 3 m 4 m3 5 Range(m) 31 m3 5. Bin a m1 3 m 4 m3 5 and Range(m)

54 In Figure 17, an example of the target detetion proess using m1 3, m 4, and m3 5 is illustrated. The residue-frank phase ode signal is transmitted to the target and the target s return is in range bin a 5 or 77 m. In the ode ompression step, the returned signal is ompressed orresponding to eah modulus in onseutive order. Shown in eah graph are residues R mk, (stair-step waveform), the range bin index a, and 3

55 a heliopter target detetion. The range bin index a goes from to 59 ( M RNS 6 ). Sine the phase ode sequene is periodi, the residues is also periodi with R R For modulus m1 3, the CC peak is repeated with mk,. mk, nm T m1 N t b mt 1 b 9 tb. The threshold detetor detets the orrelated signal output, and the range bin residue R 3 is stored. Here, the range bin residue orresponding to the orrelated output for hannels m1 3 is at R3. The proesses are the same for hannels and 3 using the moduli m 4 and m3 5, respetively. The deteted target range bin residues are R4 1 and R5. The residues are paired as [ 1 ], and a RNS-to-deimal algorithm is used to alulate the target s true range R. From Table 3, the paired values [ 1 ] orrespond to the range bin a 5, and the alulated target range is between 75 m and 9 m. Figure 17. Example of ompression output for target detetion using N 3 residue- Frank phase ode sequenes with m1 3, m 4, and m

56 Although the look up table in Table 3 an be used to find the target s range from the orresponding residues, for a pratial implementation the table would be very large. Alternatively, the system of simultaneous ongruenes of the form a R (mod m ) an be solved with a straightforward appliation of the Chinese remainder theorem (CRT). The system to be solved for this example is mi a (mod3), a 1(mod 4), a (mod 5). This CRT algorithm only requires the target range bin residues R m i. The CRT solution of the system is a M N RNS bjrm (5) j j1 m j where a is the range bin index. If we onsider this number modulo m i, we find that M RNS a br i m R mod. i m m i i The b j values an be found with the Eulidean algorithm, m i whih is a repeated appliation of the division algorithm [17]. We note that the b j values are also hardwired into the system and depend only on the moduli and not at all on the range bin residues. 4. Resolving Multiple Target Range Ambiguities In this setion, the detetion of multiple targets is onsidered. As illustrated in Figure 18, the residue for the two targets are periodi as Rmk, R., If there are no mk nm missed targets in the detetion proess, the sequene of residues within eah modulus an be paired diretly for the solution of the target s range. To solve for the range of eah target, the residues are paired as [ ] for the first target and [ ] for the seond target. Using the CRT or lookup table, we find the range of the first target is 15 R1 3 m ( a ) and the range of the seond target is 3 R 315 m ( a ). i 34

57 Figure 18. Resolving the range ambiguity of targets using RNS for m1 3, m 4, and m3 5. In Figure 19, if the first set of residues for the two targets are not deteted, the residue-frank phase ode waveform still allows the two target range values to be determined. Note the residues are determined but there is no information available to pair them up orretly. Additional information must be used in order to pair them orretly. For example, for m 1, we have R3 {, }, for m, we have R4 {, }, and for m 3, we have R5 {, }. To pair them up orretly, the magnitude information assoiated with eah subode (from the phase detetor) an be used. Figure 19. Missed detetions of the first set of residues. A more onvenient method is to extend or fill in the residues aording to their ode period, whih an be identified through examination of the residues throughout the unambiguous range. This result gives the same residue onfiguration as shown in Figure 18. Note that this method works even with the target residues overlapping. 35

58 5. Pratial Considerations a. Detetion of Target in Noise Using RNS Compression In this setion, the performane of the N 3 residue-frank for m 1 3, m 4, and m3 5 with M RNS 6 is examined by using MATLAB to plot the returned signals from two targets at ranges of 44 m and 584 m. The transmitted signal has an amplitude A 1, arrier frequeny f 1 MHz, sampling frequeny f 7 MHz, pp 1, and t 1 s. For a pratial ase, the signals are reeived with noise. The noise b hanges the returned phase sequenes and affets the ompression output value. Here, two different SNRs are examined. In Figure, the transmitted signal without noise for eah hannel is plotted. In Figure 1, the signal with SNR = 3 db is plotted. The signal power is high ompared to the noise power. Consequently, the noise signal does not signifiantly affet the phase sequenes. In Figure, the SNR is redued to db. The phase sequenes are hanged signifiantly. s Amplitude Amplitude Amplitude m 1 = time(s) x 1-5 m = time(s) x 1-5 m 3 = time(s) x 1-5 Figure. Residue-Frank signal without noise for m1 3, m 4, and m3 5 with M 6, A 1, f 1 MHz, f 7 MHz, pp 1, and t 1 s. RNS s b 36

59 Amplitude Amplitude Amplitude m 1 = time(s) x 1-5 m = time(s) x 1-5 m 3 = time(s) x 1-5 Figure 1. Residue-Frank signal with SNR = 3 db for m1 3, m 4, and m3 5 with M RNS 6, A 1, f 1 MHz, fs 7 MHz, pp 1, and tb 1 s. 5 m 1 = 3 Amplitude time(s) x 1-5 m = 4 5 Amplitude time(s) x 1-5 m 3 = 5 5 Amplitude time(s) x 1-5 Figure. Residue-Frank signal with SNR = db for m1 3, m 4, and m3 5 with M 6, A 1, f 1 MHz, f 7 MHz, pp 1, and t 1 s. RNS s b 37

60 In Figures 3 and 4, the range bin profiles for eah hannel after threshold detetion of the ompression output for the two targets are shown with SNR = 3 db and SNR = db, respetively. The first target appears at a range bin a with the paired residues [ ] (range 3 45 m). The seond target appears at range bin a 38 with the paired residues [ 3] (range m). Note that the targets straddle the range bins sine the first sample reeived from the targets lies in the orret range bin. Also shown is the noise signal that is ompressed before the signals are returned to the reeiver. Shown in Figure 4, the amplitudes of ompressed pulses degrade when the SNR is dereased to db. Also, the sidelobes from the ompression output are higher when the noise power is inreased. Another important point is that the average sidelobes from the higher modulus (e.g. m3 5 ) are lower than those from the lower modulus (e.g., m1 3 and m 4 ). This is due to the different proessing gain or N subodes that eah sequene provides. For m1 3, m 4, and m3 5, the proessing gains are m1 9, m 16, m3 5, respetively. 38

61 Noise only present Note the ambiguities present if only one modulus is used (e.g. m 1 ). Signal and noise RNS bin Target 1 ACF from m 1 =3 ACF from m =4 ACF from m 3 = Target 4 sample Target 1 at [ ] Target at [ 3] Figure 3. Target detetion for SNR = 3 db showing the range bins for m1 3, m 4, and m3 5 with the two targets at 44 m and 584 m. 39

62 Degradation of amplitude due to the high noise signal Average sidelobes are lower with higher modulus (provides higher PG) RNS bin Target 1 Target 1 ACF from m 1 = ACF from m = ACF from m 3 =5.5 4 sample Figure 4. Target detetion for SNR = db showing the range bins for m1 3, m 4, and m3 5 with two targets at 44 m and 584 m. 4

63 One situation that may our is that the threshold detetor might read the wrong range bin beause the amplitudes of the ompressed pulses flutuate due to the noise present. This results in a target range error. In Figure 5, the first target in hannel m 4 an be seen with the ompressed pulse deteted by a CFAR (threshold level at.8) at residue R4 3 instead of R4. So the paired terms [ 3 ] (range 75 7 m) are omputed instead of the true range at [ ] (range 3 45 m). This is a large range error. CFAR might read residue 3 instead of, resulting in a new paired terms [ 3 ] bin a = 47 instead of bin a = [ ] Figure 5. Illustration of detetion error from the threshold detetor (CFAR) using the residue-frank for m1 3, m 4, and m3 5. b. Required CW Signal Power Versus Required Output SNR In this setion, the CW power required when using the residue-frank phase ode sequenes to detet the target at the maximum unambiguous range is presented. Given the fixed unambiguous range Ru 9 m, the CW radar power an be hanged aordingly as a funtion of the relationship between the required as [1] SNR Ro at the output of ompressor. The SNR Ro and the average CW power ( P CW ) is expressed 41

64 P CW R (4 ) ktfb L L ( SNR / PG) (6) 4 3 u B R Ri RT RR Ro GG t rtl where R u is the unambiguous range, 3 k B joule/k (Boltzman s onstant), T is the ambient noise temperature ( T 9 K ), F R is the reeiver noise fator, radar reeiver s input bandwidth in Hz, G t B Ri is the and G are the transmitting and reeiving antenna gains, T is the target s radar ross setion in m, L is the two-way atmospheri transmission fator, L RT is the loss between the radar s transmitter and antenna, and L RR is the loss between the radar s antenna and reeiver. signal-to-noise ratio r In Figure 6, the CW signal power as a funtion of the required output SNR Ro for m1 3, m 4, and m3 5 is plotted with BRi 1 MHz, tb 1 s, G t Gr 3 db, f 3 GHz,.1 m, 1 m, FR 5 db, L LRT T L 1, and R M t / 9 m (maximum detetion range without ambiguity). RR u RNS b R u =9 m PG=N =9 (m 1 ) PG=N =16 (m ) PG=N =5 (m 3 ) 1 P CW (W) SNR Ro (db) Figure 6. Average power of the CW transmitter for residue-frank with m1 3, m 4, and m3 5. 4

65 Note the differene in the average power required in eah hannel due to the different PG. The hannel m3 5, giving PG N m 5, requires less power i than the other hannels, m 1 3 ( PG 9 ) and m 4 ( PG 16 ), to detet the targets within the same unambiguous detetion range. For example, at SNR 3 db, the Ro required CW power P CW for m 3 5, m 4, and m 1 3 are 7, 1.5, and 18 W, respetively. Note that this helps ontribute to the LPI harateristi of the emitter.. Unambiguous Detetion Range for a Constant Output SNR It is important to see how superior the residue-frank phase ode is ompared to a single Frank ode with the unambiguous range R T / t N / and u b limited to only one ode period. Here, the relationship between the average signal power P CW and the maximum detetion range R max is PCWGG t rtl Rmax. (7) 3 4 ktfb B R Ri ( SNRRo / PGL ) RT LRR However, the resolvable detetion range is limited by the unambiguous range. The radar, however, an detet the targets beyond R u but there are ambiguities. 1 4 In Figure 7, the detetion range R R is plotted as a funtion of the max required CW power with onstant SNR 13 db, B 1 MHz, t 1 s, G Ro b t G 3 db, r f 3 GHz,.1 m, 1 m, T 9 K, T F 5 db, and L L L 1. Also, R RT RR the performane of eah Frank ode using the orresponding N is plotted. For the Frank ode (17), M1 3, N 9 and M 4, N 16, and M3 5, N 5. Shown in the graph, the single Frank ode has a limited unambiguous range, where M1 3 gives R u M gives / 135 m, 4 R u / 4 m, and M 3 5 gives R 31 u / 375 m. These values are less than using the residue-frank phase ode, giving R M t / 9 m. urns RNS b 43

66 1 9 R u =9 m 8 PG=N =9 (m 1 ) 7 PG=N =16 (m ) R max (m) Residue-Frank phase ode PG=N =5 (m 3 ) Frank M1 Frank M Frank M3 3 R u (M3) R u (M) 1 One single period Frank ode R u (M1) P CW (W) Figure 7. Comparison of the maximum unambiguous range of CW radar system for SNR 13 db using the residue-frank phase ode for m1 3, m 4, and Ro 3 5 m with the radar using eah Frank ode sequene individually with the orresponding N. In onlusion, the residue-frank phase ode presented in this hapter an be used to extend the unambiguous range of the CW polyphase radar system. The issue onerned with the range bin error due to the flutuation of the ompression outputs was also introdued. In the next hapter, the relationship between the P4 ode and the symmetrial number system (SNS) is presented. As will be shown in examining the PAF, the P4 has a better property in Doppler tolerane than the Frank ode. This makes it more attrative when the emitter attempts to detet the moving targets. 44

67 IV. P4 POLYPHASE MODULATION AND THE SYMMETRICAL NUMBER SYSTEM In this hapter, the P4 phase ode and the SNS are introdued. Their relationship is presented. A. P4 PHASE CODE The P4 polyphase ode is oneptually derived from a linearly frequeny modulated waveform (LFMW). Suh odes and ompressors an be employed to obtain muh larger time-bandwidth produts. The signifiant advantages of P4 odes are low peak sidelobes and that they are more Doppler tolerant than other phase odes derived from a step approximation to an LFMW [5], [1]. In the P4 ode the loal osillator frequeny, whih is offset in the I and Q detetors, results in a oherent double sideband detetion. The phase sequene of a P4 signal is given by k k k (8) N ( 1) ( 1) where k is the time index for a single phase hange per subode and ranges from 1 to N. The phase values and phase values modulo π of a P4 ode for N 64 are shown in Figures 8 and 9, respetively. The arrier frequeny is f = 1 khz, the sampling frequeny is f s = 7 khz, and the number of arrier yles per subode is pp = 1. The ACF and PACF for Nr 1 are shown in Figure 3. The peak-to-sidelobe level is PSL or 5 db. The PACF is a perfet ode with low sidelobes. log1 / ( N ) The PAF in Figure 31 shows the delay-offset and Doppler-offset sidelobes. 45

68 -1 P4 phase shift (rad) k - index for phase hange Figure 8. P4 phase sequene for N 64 (From [1]) Signal phase (rad) k - index for phase hange Figure 9. Signal phase (radians) modulo versus k -index for phase hange of the P4 ode with N 64 (From [1]). 46

69 (a) (b) Figure 3. ACF (a) and PACF (b) for the P4 ode with N 64, pp 1, and Nr 1 (From [1]). Figure 31. PAF for the P4 ode modulation with N 64, pp 1, and Nr 1 (From [1]). B. THE SYMMETRICAL NUMBER SYSTEM The SNS is omposed of N oprime moduli. The integers within eah SNS modulus are derived from a symmetrially folded waveform [18]. The integer values within eah SNS modulus are derived from a mid-level quantization of the symmetrial 47

70 folding waveform and, therefore, inongruent modulus m (i.e., mod 5: {, 1,,, 1,, 1, }). Due to the presene of ambiguities, the set of integers within eah SNS modulus does not form a omplete residue system by themselves. It is well known that the inlusion of additional redundant moduli an effetively detet and orret errors that may our within a RNS representation of a number. The SNS formulation is based on a similar onept, whih allows the ambiguities that arise within the SNS to be resolved by using various arrangements of the SNS moduli. The vetor of integer values within a single SNS folding waveform is given as and where x S S m m m m,1,...,,,...,,1,,1, when m is odd m m,1,...,, 1,...,,1,,1, when m is even (9) (3) indiates the greatest integer less than or equal to x. An example of the SNS waveforms for m1 4 and m 5 ( N waveform equals the value of the modulus as ) is shown in Figure 3. The period of eah PSNS m. (31) m 1 = 4 residue S m input(a) m = 5 residue S m input(a) Figure 3. The SNS residues for m1 4 and m 5. 48

71 The dynami range of the SNS M ˆ SNS is ˆ j min N m M SNS m i m l i (3) l lj1 l for oprime moduli m i, with one of the moduli even, j ranges from 1 to N 1, N is the number of moduli, and mi, mi3,..., min range over all permutations {,3,, N }. The produt of j from to 1 is empty, and its value is 1. If all the oprime moduli in the system are odd, then j ˆ N 1 1 M SNS min i m m l i 1 (33) l l lj1 where j ranges from 1 to N 1 and mi 1, mi, mi3,..., min range over all permutations {1,,3,, N }. Note that the dynami range for the SNS is less than the RNS ( M ˆ SNS MRNS ). From Equation (3), for m1 4 and m 5, the dynami range is M ˆ 7. In Table 4, the two sequenes are shown along with the input a. Starting at a, the total number of distint paired terms is 7, a 6. The index value l a 1 in Table 4 is shown to ount the integer number of the distint paired terms within the dynami range. Also shaded is the region with no ambiguity. SNS Table 4. Finding the SNS dynami range for m1 4 and m 5. Input( a ) m1 4 m 5 l

72 C. SYMMETRICAL RESIDUE-P4 PHASE CODE The SNS symmetrial residues S mk, an be folded into the P4 phase sequene by assoiating S mk, with the P4 subode phases. The phase index k starts from 1 to N. The symmetrial residue-p4 phase sequene is given by m mk, Smk, m m 4 (34) where k = 1,,3,., m. The number of subodes within one ode period is N m. For example, if we hoose m 8, N 8, and the SNS sequene beomes m 8, k 1,,...,8,1,,3, 4,3,,1. S Next, the phase sequenes from equations (34) and (8) are ompared. First, Equation (34) is used to alulate the symmetrial residue-p4 phase ode sequene using m 8, 8 N with Sm 8, k 1,,...,8,1,,3, 4,3,,1. For the P4 ode in Equation (8), the parameters are N 8 and i = 1,,3,.8. In Table 5, the phase sequenes are shown for omparison. The phase ode sequenes of both waveforms are idential. The ACF, PACF, and PAF of the CW waveform generated by Equation (34) with m N 64 are also idential to those shown in Figures 3 and 31 whih were generated by Equation (8) with N 64. 5

73 Table 5. Comparison of symmetrial residue-p4 phase ode sequene ( m 8, N 8 ) with the P4 phase ode sequene ( N 8 ). Phase index ( k ) Symmetrial residue-p4 phase ode (34) P4 phase ode (8) Sm 8, k m 8, k (rad) k k (rad) (11) 1 (11) 8 (1) (1) (31) 3 (31) (41) 4 (41) (51) 5 (51) (61) 6 (61) (71) 7 (71) (81) 8 (81) D. RESOLVING RANGE AMBIGUITIES USING N SYMMETRICAL RESIDUE-P4 PHASE CODE SEQUENCES 1. Blok Diagram of the Symmetrial Residue-P4 Radar System The blok diagram of a radar system using symmetrial residue-p4 phase ode sequenes is shown in Figure 33. This diagram is similar to the residue-frank phase ode radar system in Chapter III, exept that now the symmetrial residue-p4 phase ode sequenes are used instead. Eah phase sequene with the length equal to the dynami 51

74 range M ˆ SNS is transmitted to detet the target. In the reeiver, eah ode sequene is ompressed with its orresponding onjugate of phase ode. Figure 33. Blok diagram for the radar using N symmetrial residue-p4 phase ode sequenes.. Transmitted and Referene Codes for Compression In this setion, the transmission of N symmetrial residue-p4 phase ode sequenes is explained as well as the phase ode referene for ompression. An example using N 3 moduli with m1 7, m 8, and m3 9 is illustrated. The dynami range, using Equation (3) for these SNS moduli, is Mˆ SNS 37. Eah sequene has the number of ode length equal to the dynami range with the transmitted signal as shown in Figure 34. To reate the P4 ode sequene, one ode period requires the number of subodes N m. To over a dynami range of subode, the number of ode periods required is ˆ ˆ M SNS N pm,. (35) m For m1 7 and N 7, Equation (35) gives Nˆ p,7 37 / 7 5.9, whih last 5 subodes ( ) are not inluded sine they are ambiguous. For modulus m 8 and 5

75 m3 9, the number of subodes N for one ode period are 8 and 9 respetively. Also, the maximum number of transmitted and reeived ode periods are Nˆ p, (the last 3 subodes are not used) and Nˆ p, (the last 8 subodes are not used), respetively. In Figure 34, the ode period orresponding to eah modulus is denoted as T m, pwhere p represents the ode period number. i Figure 34. Illustration of transmitted signal using N 3 symmetrial residue-p4 phase ode sequenes for m1 7, m 8, and m3 9. In Figure 35a, the symmetrial residues S mk, using m 1 7, m 8, and m3 9 ( M ˆ SNS 37 ) are shown. Sine the phase sequenes are periodi, S, S, where mk mk nm n, 1,,.... The phase sequenes from eah hannel are plotted in Figure 35b. Eah symmetrial residue-p4 phase sequene has a phase ode period equal to N m, giving. If the reeiver uses one ode period ( Nr 1), the referene ode sequene mk, mk, nm for ompression is the phase sequene from index k 1 to m. 53

76 3 m 1 =7 Residue S 7 Residue S 8 Residue S m 4 = m 4 3 = time-index (a) T 7 =7 m 1 = 7 phase(rad) phase(rad) m T = 8 8 = m T 3 = 9 9 =9 phase(rad) time-index (b) Figure 35. Plot of the symmetrial residue-p4 for m1 7, m 8, and m3 9 ( M ˆ SNS 37 ) showing (a) symmetrial residues, and (b) phase sequenes from (34). 1 7 In Figure 36, the ompression and range bin detetion proess is illustrated for m and N 1. The symmetrial residue-p4 phase ode is returned to the radar r reeiver with delay and then proessed by a filter mathed to the retangular pulse of duration t b. Then the phase detetor is used to alulate the phase of the signal and pass 54

77 it through the ompressor. The output from the ompression is then proessed by threshold detetion to see whether the amplitude reahes the desired minimum value. If the threshold detetor detets the pulse, the presene of the target will be saved with the symmetrial residue value S m. In m 8 and m3 9, the proess is similar, exept that the phase referenes have to orrespond to the transmitted phase ode that is being reeived. Figure 36. Illustration of ompression at the reeiver and the range bin for SNS m1 7, N 7, and Nr Calulating the Target Range Using all hannels, we see that the paired terms are mapped to a speifi range bin input a (similar to residue-frank ode bin detetion proess). Continuing the example in the previous setion for m 1 7, m 8, and m3 9, we show the symmetrial residue paired values and the orresponding ranges in Table 6. The signal bandwidth 55 B 1 MHz, and the subode period t 1 s. The unambiguous range using symmetrial residue-p4 is b

78 Mˆ SNStb RuSNS,. (36) For this example, the maximum unambiguous range is 555 m ( M ˆ SNS 37 ). Range resolution is 15 m ( R t b / ). Table 6. Bin matrix and range intervals of 3 hannel SNS for m1 7, m 8, and Bin a Range (m) Bin m1 7 m 8 m3 9 a m3 9. m1 7 m 8 m3 9 Range (m)

79 In Figure 37, an example of target detetion for m 1 7, m 8, and m3 9 is illustrated. The target s return is in range bin a 4 or 65 m. Shown in the graph are symmetrial residue S mk, (stair-step waveform), the range bin a, and a heliopter target detetion. The bin orresponding to the orrelated output for hannel m 1 7 is at Sm Note the repeated pulse from the ompression output due to the periodi phase ode. The proess is the same for the hannel m 8, and m3 9. The residue bins are Sm 8 4 and Sm respetively. Then the three symmetrial residues are paired as [3 4 4], and the SNS-to-deimal algorithm is used to find the target s true range R. From Table 6, the paired terms [3 4 4] orrespond to the range bin a 4, and the alulated range is between 6 m and 75 m. Figure 37. Illustration of target detetion by using the range bin matrix for SNS-P4 ode with m1 7, m 8, and m

80 In Chapter III, we solved the system of ongruenes using CRT for reovering the target range bin a ( Mˆ SNS ). Due to the definition of the SNS, a S (mod m ), m1 1 a S (mod m ),, as (mod m ). For eah of the ongruenes, either the plus or m mn N minus is orret, but we do not know whih. Thus, we have N sets of N equations. For example, if N, we have i) a S (mod m ), m1 1 a S (mod m ), m ii) a S (mod m ), m1 1 a S (mod m ), m iii) a S (mod m ), m1 1 a S (mod m ), m iv) a S (mod m ), m1 1 a S (mod m ). m The CRT guarantees that eah of these has a unique solution modulo mm, 1 and exatly one of these solutions lies within the dynami range of the system. This is the value of a. In fat, it is only neessary to solve (i) and (ii) at most beause the solutions to (iii) and (iv) are the negatives of the solutions to (i) and (ii), respetively. Reall that in the standard statement of the CRT we wish to solve for a where a Sm (mod mi), 1 i N, and the m i are pairwise relatively prime. The solution is i a M ba M ba M b a RNS 1 1 RNS RNS N N... (37) m1 m mn where M RNS N mi. Note that the values of b i depend only on i1 m i and not at all on S mi. Thus, we may assume that the onstants M b / m are known, and the SNS- 58 i RNS i i

81 to-deimal algorithm only needs to evaluate S 1 S... S modulo M RNS and m1 m N m N piks the one value that lies within the dynami range M ˆ SNS. 4. Resolving Multiple Target Range Ambiguities In this setion, the detetion of multiple targets is onsidered. In Figure 38, we see that the symmetrial residue for the two targets are periodi as Smk, Smk, nm. If there are no missed targets in the detetion proess, the sequene of symmetrial residues within eah modulus an be paired diretly for the solution of the target s range. To solve for the range of eah target, the residues are paired as [ 3 4] for the first target and [ 1] for the seond target. From the CRT or lookup table, the range of the first target is 75 R1 9 m ( a 5 ), and the range of the seond target is 39 R 45 m ( a 6 ). Figure 38. Resolving the range ambiguity of targets using SNS for m1 7, m 8, and m3 9. In Figure 39, if the first set of symmetrial residues for the two targets are not deteted, the symmetrial residue-p4 phase ode still allow the two target range values to be determined. Note the symmetrial residues are determined but there is no information available to pair them up orretly. Additional information must be used. For example, for m, we have 1 S7 {, }, for m, we have S 8 {3, }, and for m 3, we have 59

82 S9 {4, 1}. To pair them up orretly, the magnitude information assoiated with eah subode (from the phase detetor) an be used. Figure 39. Missed detetions of the first set of symmetrial residues. A more onvenient method is to extend or fill in the symmetrial residues aording to their ode periods, whih an be identified through examination of the symmetrial residues throughout the unambiguous range. This result gives the same symmetrial residue onfiguration as shown in Figure 38. Note that this method works even with the target symmetrial residues overlapping Pratial Considerations a. Detetion of Target in Noise Using SNS Compression In this setion, the performanes of the N 3 SNS for m1 7, m 8, and m with Mˆ 37 are examined by using MATLAB to plot the returned signal from SNS two targets at ranges of 89 m and 99 m, respetively. The transmitted signal have amplitude A 1, arrier frequeny f 1 MHz, sampling frequeny f 7 MHz, pp 1, and t 1 s. The two different SNRs are examined. In Figure 4, the transmitted signal b without noise for eah hannel is plotted. In Figure 41, the signal with SNR = 3 db is plotted. For the data shown in Figure 4, the SNR is redued to db. The phase sequenes are hanged signifiantly with the inrease in noise power. 6 s

83 Amplitude Amplitude Amplitude m 1 = time(s) x 1-5 m = time(s) x 1-5 m 3 = time(s) x 1-5 Figure 4. Symmetrial residue-p4 signal without noise for m1 7, m 8, and m with Mˆ 37, A 1, 3 9 SNS f 1 MHz, f 7 MHz, pp 1, and t 1 s. s b Amplitude Amplitude Amplitude m 1 = time(s) x 1-5 m = time(s) x 1-5 m 3 = time(s) x 1-5 Figure 41. Symmetrial residue-p4 signal with SNR = 3 db for m1 7, m 8, and m with Mˆ 37, A 1, 3 9 SNS f 1 MHz, 61 f 7 MHz, pp 1, and t 1 s. s b

84 5 m 1 = 7 Amplitude time(s) x 1-5 m = 8 5 Amplitude time(s) x 1-5 m 3 = 9 5 Amplitude time(s) x 1-5 Figure 4. Symmetrial residue-p4 signal with SNR = db for m1 7, m 8, and m with Mˆ 37, A 1, 3 9 SNS f 1 MHz, f 7 MHz, pp 1, and t 1 s. s b In Figures 43 and 44, the range bin profiles for eah hannel are shown using SNR = 3 db and SNR = db, respetively. The first target appears at range bin a 5 with the paired symmetrial residues [ 3 4] (range 75 9 m). The seond target appears at range bin a 19 with the paired symmetrial residues [ 3 1] (range 85 3 m). Also shown is the noise signal that is ompressed before the returned signals from the targets arrive at the reeiver. In Figure 44, the amplitudes of ompressed pulses degrade when the SNR is dereased to db. The effet from the noise signal degrades the ability to detet the targets similar to that disussed in the residue-frank phase ode radars. The average sidelobes from the higher modulus (e.g. m3 9 ) are lower than those from the lower modulus (e.g., m1 7 and m 8 ) due to the different proessing gain. For m1 7, m 8, and m3 9, the proessing gains are N 1 7, N 8, and N 3 9, respetively. 6

85 Noise The ambiguities present if only one modulus is used SNS bin ACF from m 1 = Target 1 Target ACF from m = ACF from m 3 = sample Target 1 at [ 3 4] Target at [ 3 1] Figure 43. Target detetion for SNR = 3 db showing the range bins for m1 7, m 8, and m3 9 with two targets at 89 m and 99 m. 63

86 Degradation of amplitude due to the high noise signal Average sidelobes are lower with higher modulus (provides higher PG) SNS bin Target 1 ACF from m 1 = 7 ACF from m = 8 ACF from m 3 = Target sample Figure 44. Target detetion for SNR = db showing the range bins for m1 7, m 8, and m3 9 with two targets at 89 m and 99 m. 64

87 Like the RNS phase ode, a situation in where the threshold detetor might read the wrong residue bin an our beause the amplitudes of the ompressed pulses flutuate due to the noise. This results in the target range error. In Figure 45, the first target in hannel m 8 is seen with the ompressed pulse deteted by CFAR (threshold level at.85) at symmetrial residue S8 instead of S8 3. So, the paired terms are now [ 4] (no range math) instead of the true range at [ 3 4] (range 75 9 m). This results in target range error. CFAR might read symmetrial residue instead of 3, resulting in a new paired terms [ 4] no mathed bin instead of bin a = 5 [ 3 4] Figure 45. Illustration of detetion error from the threshold detetor (CFAR) using the symmetrial residue-p4 for m1 7, m 8, and m3 9. b. Required CW Signal Power Versus Required Output SNR In this setion, the CW power required when using the symmetrial residue-p4 phase ode sequenes to detet the target at the fixed unambiguous range Ru 555 m is presented. The relationship between the required SNR Ro and the average CW power P CW is given by Equation (6). In Figure 46, the CW signal power as a funtion of the required output SNR Ro for m1 7, m 8, and m3 9 is plotted with BRi 1 MHz, tb 1 s, 65

88 G t G 3 db, f 3 GHz,.1 m, 1 m, F 5 db, L L L 1, and r T R RT RR R Mˆ t / 555 m (maximum detetion range without ambiguity). u SNS b R u =55 m PG=N =7 (m 1 ) PG=N =8 (m ) PG=N =9 (m 3 ) P CW (W) SNR Ro (db) Figure 46. Average power of the CW transmitter for symmetrial residue-p4 with m1 7, m 8, and m3 9. Note that the differene of the average power used in eah hannel is due to the PG. The hannel m3 9, giving PG N m 9, requires less power than the other hannels, m1 7 ( PG 7 ) and m 8 ( PG 8 ). i. Unambiguous Detetion Range for a Constant Output SNR For a single P4 ode, the unambiguous range is limited to only one ode period ( R T / t N / ). In this setion, we ompare the maximum unambiguous u b range using the symmetrial residue-p4 phase ode sequenes with the orresponding single P4 ode sequenes. The relationship between the average signal power P CW and the maximum detetion range R max is given by Equation (7). 66

89 In Figure 47, the detetion range is plotted as a funtion of the required CW power using N 3 SNS for m 1 7, m 8, and m 3 9 with SNR 13 db, B 1 MHz, tb 1 s, Gt Gr 3 db, f 3 GHz,.1 m, T 1 m, T 9 K, F 5 db, and L L L 1. For omparison, the performane of eah P4 ode R RT RR Ro using the orresponding N is plotted. For the P4 ode from Equation (8), N 1 7 and N and N 3 9. The single P4 ode has a limited unambiguous range, N 1 7 gives 8, R u N gives / 15 m, 8 R u / 1 m, and N3 9 gives R u / 135 m. These values are less than using the N 3 symmetrial residue-p4 phase ode sequenes, giving R M t / 555 m. urns SNS b 6 R u = 555 m 5 PG=7 (m 1 ) PG=8 (m ) 4 PG=9 (m 3 ) R max (m) 3 Symmetrial residue-p4 phase ode P4 N 1 P4 N P4 N 3 R u (N 1 ) R u (N ) R u (N 3 ) P CW (W) Figure 47. Comparison of the maximum unambiguous range of a CW radar system for SNR 13 db using symmetrial residue-p4 phase ode for m1 7, Ro One single period P4 ode m 8, and m3 9 with the radar using eah P4 ode individually with the orresponding N. 67

90 In onlusion, the symmetrial-p4 phase ode sequene presented in this hapter an be used to extend the unambiguous range of the CW polyphase radar systems. The issue onerned with the range bin error due to the flutuation of the ompression outputs was onsidered. In the next hapter, the relationship between the P4 ode and the robust symmetrial number system (RSNS) is presented to examine the integer Gray ode properties and how they an prevent an error in the deteted symmetrial residues. 68

91 V. P4 POLYPHASE MODULATION AND THE ROBUST SYMMETRICAL NUMBER SYSTEM As disussed in Chapters III and IV, using the RNS and the SNS may give a large error in the target s range when the threshold detetor detets the wrong residue due to the presene of noise whih an hange the phase of the reeived signal. In this hapter, the RSNS is presented, and it is shown that the range errors in the robust symmetrial residue-p4 phase ode sequene an be bounded to a small value due to the Gray ode property of the RSNS []. The P4 phase expression is presented and examples are given to illustrate the target detetion proess with this new waveform. A. P4 PHASE CODE The details of the P4 phase ode are disussed in Chapter IV. The P4 phase ode is derived from a linearly frequeny modulated waveform (LFMW). To alulate the phase sequene, Equation (8) is used. Reall the harateristis of the P4 ode sequene is its superior PAF sidelobe struture and its Doppler tolerane properties. B. THE ROBUST SYMMETRICAL NUMBER SYSTEM The RSNS is a modular system onsisting of N integer sequenes with eah sequene assoiated with a oprime modulus m i. Due to the presene of ambiguities, the set of integers within eah RSNS sequene do not form a omplete residue system. A set of integers y, y1,, y 1(mod m) form a omplete residue system if y g(mod m) for m g,1,, m1[1]. The ambiguities within eah modulus sequene are resolved by the use of additional moduli and onsidering the vetor of paired integers from all N sequenes. The RSNS is based on the sequene RS mh, [,1,,, m 1, m, m 1,,,1] (38) for an integer h suh that h m. To form the N-sequene RSNS, eah term in (38) is repeated N times in suession. The integers within one folding period of a sequene are then g 69

92 RSmh, [,,,1,1,,1,, m 1,, m 1, mm,, mm, 1,, m1,,1,,1]. This results in a periodi sequene with a period of [] That is, eah RSNS sequene ontains Pm (39) Nm. (4) m i integers with the folding period of eah sequene equal to Nm. The onstrution of the N sequenes ensures that any two i suessive RSNS vetors (paired integers from the N sequenes) differ by only one integer, resulting in an integer Gray ode property. Eah sequene orresponding to m is right (or left) shifted by s i 1 plaes. i i The hosen shift values { s 1, s,, sn } must also form a omplete residue system modulo N. Further, if eah sequene is extended periodially with period Nm as RS RS where n, 1,,..., then RS, mh, n Nm mh, ( h nnm) modulo Nm. mh is alled a symmetrial residue of The alulation of M ˆ RSNS and its loation in the sequene are a funtion of N and the hoies for m i and s i. A losed-form solution for omputing M ˆ RSNS for N moduli is reported in [], and summarized as follows: For m1 m m1 3 and 1 Mˆ 3( m m ) 6 6m 3. (41) RSNS 1 1 For 5 m1 m and m m1 Mˆ 4m m 5. (4) RSNS 1 For 5 m1 m and m m1 3 Mˆ 4m m. (43) RSNS 1 z z z A losed-form solution for N 3 moduli of the form 1,, 1 is 3 15 Mˆ RSNS m1 m1 7 (44) 7

93 where z is any integer and m1 3 [3]. An effiient algorithm for omputing M ˆ RSNS and its position for any general set of moduli is reported in [4] [5]. The shift values s i for eah sequene do not affet the RSNS dynami range but do make a differene in the loation of the beginning position h of the dynami range and the ending position hmˆ RSNS 1. After hoosing the beginning h, it is useful to know the symmetrial residues at this point (e.g., to align the symmetrial residues for eah sequene). Sine eah folding period onsists of Nm integers, the symmetrial residues are determined by first subtrating off an integer number of Nm integers as i i h ni h Nmi. Nmi This value is then used to find the symmetrial residue RS m, has [] i (45) Let RS X h mh, ni si N si ni Nmi si 1 Nm 1 Nmi si ni Nmi si 1. i N ni s i N (46) be the vetor of N paired integers from eah sequene at h where h is the position of the vetor X h within the RSNS. In Table 7, h and X h for an N = 3 RSNS system with m [3 4 5] T and right shift s [,1,] T i i are shown. For this example, the RSNS dynami range M ˆ RSNS 43, and the position begins at h 61 []. The vetor T X 61 [ 4 1]. The set of integers that lie within the dynami range M ˆ RSNS 43 ontain no ambiguities. Also note in Table 7 the integer Gray ode property where any ode transition results in just one integer hanging value by 1. Also shown is the bin index a whih runs from to 4 (total 43) that will be used to refer to the range bin detetion. 71

94 Table 7. The RSNS folding waveforms for m1 3 ( s1 ), m 4 ( s 1), and m 5 ( s ) (After []). 3 3 h m m m a C. ROBUST SYMMETRICAL RESIDUE-P4 PHASE CODE Similar to the SNS phase ode, the RSNS folding waveform an be related to the P4 ode sequene. The RSNS symmetrial residues RS, an be folded into the P4 mk phase sequene by assoiating RS, mk with the P4 subode phases. The phase index k starts from 1 to N RSNS, where N RSNS is the number of RSNS subodes within one ode period. The robust symmetrial residue-p4 phase sequene is given by and RSmk m m m mk,, N RSNS (47) mn (48) where N is the number of hannels. For example, if we hoose m 4, N, and N 16. The robust symmetrial residue-p4 phase ode sequene is RSm4, RSNS {,,1,1,,,3,3,4,4,3,3,,,1,1}. Note that the folding sequene is similar to SNS with m 8, Sm 8, k,1,,3, 4,3,,1 exept that the robust symmetrial residue-p4 phase is repeated N times; there are N phase values for eah robust symmetrial-p4 phase for every P4 phase value. That is, both k 1 and k of the RSNS orresponds to k 1 k 7

95 for the P4 phase. Consequently, the number of subodes N for the P4 ode is related to the number of RSNS subodes N RSNS as N N RSNS. (49) N Also, the PG for the robust symmetrial residue-p4 waveform is N PG RSNS RSNS N m. (5) N 1. Example of Phase Sequene Calulation In this setion, the phase ode sequenes from equations (47) and (8) are ompared. Equation (47) is used to alulate the phase sequene for m 4, N, and N 16. Using the RSNS, we find that the robust symmetrial residues are RS RSNS {,, 1, 1,,, 3, 3, 4, 4, 3, 3,,, 1, 1} where k 1,,...,16. In order to m 4, k ompare to the P4 phase sequene in Equation (8) with N N / N 16 / 8, k = 1,,3,,8, the two phase sequenes are shown in Table 8. RSNS Table 8. Comparison of the robust symmetrial residue-p4 phase ode sequene ( m 4, N, N 16 ) with the P4 phase ode ( N = 8). RSNS Phase index Robust symmetrial reside-p4 phase ode (47) P4 phase ode (8) ( k ) RS m 4, k m 4, k (rad) k k (rad) (11) 1 (11) 8 (1) (1)

96 (31) 3 (31) (41) 4 (41) (51) 5 (51) (61) 6 (61) (71) 7 (71) (81) 8 (81) ACF, PACF, and PAF In this setion, we examine the ACF, PACF, and PAF of the signal. To ompare the robust symmetrial reside-p4 phase ode sequene with the P4 phase ode sequene, the ACF, PACF, and PAF are examined for eah waveform. In Figure 48, the robust symmetrial residues using mi 16, N, and N 64 are shown. In Figure 49, the robust symmetrial residue-p4 phase ode sequene residues are shown (using Equation (47)) with f 1 khz, 74 RSNS f 7 khz, and pp. Note that pp represents the number of s

97 arrier yles per phase hange, and the pp value for the P4 is N times that of the robust symmetrial residue-p4 phase ode. For the P4, the subode period is pp tb. (51) f For the robust symmetrial residue-p4, the subode period is pp t b. (5) Nf From (51) and (5), we have t Nt (53) b b. Using Equation (48), we see that the phase sequene from the robust symmetrial residue-p4 for mi 16, N, and N 64 is atually the same as the P4 phase ode sequene using N 3. RSNS Robust symmetrial residue RS m phase index k' Figure 48. RSNS residues RS mk, for modulus m16, N, and N 64. RSNS 75

98 -5 For P4 ode, N 64 N RSNS 3 N -1 phase shift (rad) t' b t b k' - index for phase hange Figure 49. The robust symmetrial residue-p4 phase ode sequene using m 16, N, N 64 is shown with the relationship with the P4 ode using N 3. RSNS The power spetral density (PSD) of the signal is shown in Figure 5. The signal bandwidth B is redued by the fator of N. With pp, the bandwidth of the robust symmetrial-p4 signal is B f / pp1/ t 5 Hz (instead of B 1 khz when pp 1). In Figure 51, the ACF and PACF are plotted. The signal s harateristis are the same as the P4 phase ode sequene using N 3, output gives PSL = log 1 / ( ) b t ms. In the ACF plot, the b N = db. Note that the width of the mainlobe in the ACF plot is t b and is proportional to the number of moduli N. That is, the higher the number of moduli N, the broader the mainlobe. In Figure 5, the PAF is plotted. The PAF from the robust symmetrial residues using mi 16, N, and N 64 in Figure 5a is ompared to the PAF from the P4 ode sequene using N 3 in Figure RSNS 5b. Both PAFs are the same. 76

99 Power Spetrum Magnitude (db) Bandwidth redued to 5 Hz Frequeny Figure 5. Power spetral density of the robust symmetrial residue-p4 signal using m 16, N, and N 64 with f 1 khz, f 7 khz, and pp. RSNS s Autoorrelation [db] Periodi Autoorrelation [db] / t' b / t' b Doubled mainlobe Figure 51. The ACF and PACF of RSNS phase-oded signal ( m16, N, N RSNS 64 ) at the reeiver with N 1, f 1 khz, f 7 khz, and pp. s 77 r

100 (a) Figure 5. The PAF of the signal for (a) the robust symmetrial residue-p4 phase ode sequene using m 16, N, N RSNS 64, and t b 1 ms, and (b) the P4 ode using N 3, and tb ms. 78 (b)

101 D. RESOLVING RANGE AMBIGUITIES USING N ROBUST SYMMETRICAL RESIDUE-P4 PHASE CODE SEQUENCES 1. Blok Diagram of the Robust Symmetrial Residue-P4 Radar System To extend the unambiguous detetion range, N oprime RSNS sequenes are used. The blok diagram of the radar using N hannel RSNS phase oding is shown in Figure 53. This diagram is similar to the residue-frank phase ode radar system exept that now the RSNS moduli are used instead. Eah phase sequene with the length equal to the dynami range M ˆ RSNS is transmitted to detet the target. In the reeiver, eah ode sequene is ompressed with its orresponding referene phase ode. Figure 53. Blok diagram for the radar using N robust symmetrial residue-p4 phase ode sequenes.. Transmitted and Referene Codes for Compression In this setion, the transmission of a set of N robust symmetrial residue-p4 phase oded sequenes is explained. Also, the use of the phase ode sequenes as the referene sequene for ode ompression is disussed. An example using N 3 RSNS moduli with m1 3, m 4, and m3 5 is illustrated. The dynami range for these RSNS moduli from Equation (44) is M ˆ RSNS 43. The starting position h and ending position 79

102 hmˆ RSNS 1 are 61 and 13, respetively []. The robust symmetrial residues within the RSNS moduli are paired as shown in Table 7. The transmitted signal is shown in Figure 54. Eah sequene has a length equal to the dynami range. For m 1 3, there are four robust symmetrial residues within the sequene, RS3, 1,, and 3. To reate the P4 ode sequene, one ode period needs the number of subsodes N Nm For m 4, the robust RSNS symmetrial residues are, 1,, 3, and 4. The required number of subodes is For m3 5, the robust symmetrial residues are, 1,, 3, 4, and 5. The number of subodes is As shown in the Figure 54, the beginning of eah sequene is not neessarily the beginning of the phase ode sequene k 1 (e.g., the beginning phase sequenes for m 1 3, m 4, and m 3 5 are k 8, k 13, and k 3, respetively. The referene ode sequenes in the ompressors an be hosen to be either starting at the beginning of the sending signal waveforms or the starting of phase sequene at k 1. The importane is that the referene ode has to have a length equal to the ode period T m for eah hannel in order to orretly alulate the ode ompression output. In this example, we hoose to use the referene signals starting from the beginning of the transmitted signal ( 8 to 7 for m 1 3, 13 to 1 for m 4, and to for m 5 ). The number of sequene periods is alulated as ˆ ˆ ˆ MRSNS MRSNS N pm,. N mn (54) RSNS Sending 43 subodes, the modulus m1 3 needs Nˆ p,3 43 /18.4. For modulus m 4 and m3 5, the number of subodes N RSNS are 4 and 3, respetively. Also, the number of transmitted ode periods are Nˆ p,4 1.8 and Nˆ p,5 1.4, respetively. Here, only hannel m1 3 is sending more than one omplete ode period. So we an expet to see two ompressed pulses from hannel m 1 3 if all subodes return bak to the reeiver (there will be only one pulse in the other two hannels). 8

103 Figure 54. Illustration of transmitted signal using N 3 robust symmetrial residue- P4 phase ode sequenes for m1 3, m 4, and m3 5. In Figure 55a, the RSNS folding waveforms for m1 3, m 4, and m3 5 with Mˆ RSNS 43 are plotted. Also, the phase sequenes from eah hannel are plotted in Figure 55b. Sine RS, RS, mh n Nm periodi as mk, n Nm mk,. mh where n, 1,,, the phase sequenes are In Figure 56, the ompression and range bin detetion proess is illustrated for RSNS m1 3, and N 1. The returned phase oded signal is first proessed by a filter r mathed to the retangular pulse of duration t b. The phase detetor is used to alulate the phase of eah sample and then pass it through the ompressor. The output from the ompression is then proessed by the threshold detetion to see whether the amplitude reahes the desired minimum value. If the threshold detetor detets the pulse, the presene of the target will be saved with the robust symmetrial residue value RSm. In m 4 and m3 5, the proess is similar, exept that the phase referenes have to math with the transmitted phase ode that is being reeived. 81

104 Residue RS 3 Residue RS 4 Residue RS time-index (a) m 1 = 3 m = 4 m 3 = 5 phase(rad) - -4 T m 1 = 3 phase(rad) -5 T m = 4 phase(rad) -5 T time-index m 3 = 5 (b) Figure 55. Plot of the RSNS using m1 3, m 4, and m3 5 with Mˆ RSNS 43 for (a) folding waveforms, and (b) their phase sequenes. 8

105 Figure 56. Illustration of ompression at the reeiver and the range bin for RSNS m1 3, N 18, and Nr Calulating the Target Range Using all sequenes, we map the paired terms to a speifi range bin a (similar to the residue-frank ode bin proess in Chapter III). Continuing the example in the previous setion for m1 3, m 4, and m3 5, we show the robust symmetrial residue paired values and the orresponding ranges in Table 9. The signal bandwidth is B 1 MHz, and the subode period is t t / N.33 s. The new expression formula b b for alulating the unambiguous range using RSNS is Mˆ RSNSt b RuRSNS,. (55) For this example, the maximum unambiguous range is RuRSNS, 15 m. The range resolution is 5 m ( R t / ). b 83

106 Table 9. Bin matrix and range intervals of 3 hannel RSNS for m1 3, m 4, Bin a Range (m) Bin m1 3 m 4 m3 5 a m3 5. m1 3 m 4 m3 5 Range (m) and 84

107 In Figure 57, an example of target detetion for m 1 3, m 4, and m3 5 is illustrated. The return signal is in the range bin a 5 or 7 m. First, the phase ode signal of the modulus m1 3 (top) is ompressed. Reall that the RSNS has a width of the ompression output that overs N 3 ode sequenes at robust symmetrial residues RS3 3,, and. So the threshold detetion has to deide to take one bin, e.g., at the peak of the ompressed pulse, giving the robust symmetrial residue RS3. The proess is the same for the hannels m 4 and m3 5. The robust symmetrial residues are RS4 3 and RS5 1, respetively. The three robust symmetrial residues are paired as [ 3 1] and must be onverted to the deimal RSNS bin number. In Table 9, the orresponding range is between 5 m and 3 m (bin a 5 ). Figure 57. Illustration of target detetion by using the range bin matrix for RSNS-P4 ode with m1 3, m 4, and m

108 4. Resolving Multiple Target Range Ambiguities In Figure 58 for m1 3, the ompressed pulses are repeated as the returned ode runs through the referene ode in the ompressor. The two target returns overlap one another at range bins a 19,, and 1 and are ambiguous. Next we resolve the range ambiguity by using 3 hannels. If the threshold detetor is set to measure at the peak of the ompression output, the robust symmetrial residues from the first target are paired as [3 4 ], giving a range bin a (range 5 1 m). Similarly, the robust symmetrial residue paired terms from the seond target are [3 4], giving a range bin a (range 1, 1,5 m). The two targets have different paired terms, so the range ambiguity an be resolved. Figure 58. Resolving the range ambiguity of targets using RSNS for m1 3, m 4, and m3 5. As mentioned in Chapters III and IV, we an resolve the targets even if the first set of the residues are not deteted. Imagine we use a higher set of RSNS moduli and the dynami range is higher with the result that the ode period repeats. To find the true range of the targets that are missed, the magnitude information or the filling in the range bin aording to the ode periodiity an be used. 86

109 5. Pratial Considerations a. Detetion of Target in Noise Using RSNS Compression In this setion, the performanes of the N 3 RSNS for m1 3, m 4, and m3 5 with Mˆ 43 are examined by using MATLAB to plot the returned signal RSNS from two targets at ranges of 14 m and 14 m. The transmitted signal has amplitude A 1, arrier frequeny f 1 MHz, sampling frequeny f 9 MHz, pp 1, t 1 s, and t.33 s. In Figure 59, the transmitted signal without noise for eah hannel is b plotted. In Figure 6, the signal with SNR = 3 db is plotted. The signal power is high ompare to the noise power. As a result, the noise signal does not signifiantly affet the phase sequenes. In Figure 61, the SNR is redued to db. The phase sequenes are hanged signifiantly. s b Amplitude Amplitude Amplitude m 1 = time(s) x 1-5 m = time(s) x 1-5 m 3 = time(s) x 1-5 Figure 59. Robust symmetrial residue-p4 signal without noise for m1 3, m 4, and m3 5 with Mˆ RSNS 43, A 1, f 1 MHz, fs 9 MHz, pp 1, and t.33 s. b 87

110 Amplitude Amplitude Amplitude m 1 = time(s) x 1-5 m = time(s) x 1-5 m 3 = time(s) x 1-5 Figure 6. Robust symmetrial residue-p4 signal with SNR = 3 db for m1 3, m 4, and m3 5 with Mˆ RSNS 43, A 1, f 1 MHz, fs 9 MHz, pp 1, and t.33 s. b Amplitude Amplitude Amplitude m 1 = time(s) x 1-5 m = time(s) x 1-5 m 3 = time(s) x 1-5 Figure 61. Robust symmetrial residue-p4 signal with SNR = db for m1 3, m 4, and m3 5 with Mˆ RSNS 43, A 1, f 1 MHz, fs 9 MHz, pp 1, and t.33 s. 88 b

111 In Figures 6 and 63, the range bin profiles from two targets are shown using SNR = 3 db and SNR = db, respetively. The first target appears at range bin a with the paired robust symmetrial residues [3 4 ] (range 1 15 m). The seond target appears at range bin a, giving paired terms [3 4] (range 1 15 m). As shown in Figure 63, the amplitudes of the ompressed pulses degrade when the SNR is dereased to db. Also, the sidelobes from the ompression output are higher when the noise power is inreased. Another important point is that the average sidelobes from the higher modulus (e.g. m3 5 ) are lower than those from the lower modulus (e.g. m1 3 and m 4 ) due to the different proessing gains. For m1 3, m 4, and m3 5, the proessing gains are N 1 m1 6, N m 8, and N3 m3 1, respetively. 89

112 Noise The ambiguities present if only one modulus is used RSNS bin Target 1 1 Target ACF from m 1 = ACF from m = ACF from m 3 = sample Target 1 at [3 4 ] Target at [3 4] Figure 6. Target detetion for SNR = 3 db showing the range bins for m1 3, m 4, and m3 5 with two targets at 14 m and 14 m. 9

113 ACF from m 1 = 3 ACF from m = 4 ACF from m 3 = Degradation of amplitude due to the high noise signal Average sidelobes are lower with higher modulus (provides higher PG) RSNS bin Target 1 Target 387 sample Figure 63. Target detetion for SNR = db showing the range bins for m1 3, m 4, and m3 5 with two targets at 14 m and 14 m. 91

114 Using N robust symmetrial residue-p4 phase ode sequenes allows ontrol of the range oding error. The threshold detetor might read the wrong robust symmetrial residue bin beause the amplitudes of the ompressed pulses are flutuating. However, the paired terms from the robust symmetrial residues will hange only one position (the Gray ode property). In Figure 64, the ompressed pulse from the first target in m 4 is deteted by CFAR (threshold level at.85) at the robust symmetrial residue RS4 3 instead of RS4 4. The paired terms are now [3 3 ] (range 15 m) instead of the true range at [3 4 ] (range 1 15 m). The maximum range error is only one range bin or 5 m. CFAR might read residue 3 instead of 4, resulting in a new paired terms [3 3 ] bin a = 3 instead of bin a = [3 4 ] *error only one bin Figure 64. Illustration of detetion error from the threshold detetor (CFAR) using the robust symmetrial residue-p4 for m1 3, m 4, and m3 5. b. Plot of Required Signal Power Versus Required SNR Output In this setion, the power required when using the robust symmetrial residue-p4 phase ode sequene to detet the target at the maximum unambiguous range is presented (given the fixed unambiguous range between the required 9 Ru 15 m ). The relationship SNR Ro and the average CW power P CW is given by Equation (6). In Figure 65, the CW signal power as a funtion of the required SNR Ro for m1 3, m 4, and m 3 5 is plotted with BRi 1 MHz, tb 1 s, t b.333 s,