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DIGITAL SPREAD SPECTRUM SYSTEMS

DIGITAL SPREAD SPECTRUM SYSTEMS. ENG-737. Wright State University James P. Stephens. AUTOCORRELATION OF PN SEQUENCES. A very important property of PN sequences is the periodic auto correlation function Recall that autocorrelation is defined as:

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DIGITAL SPREAD SPECTRUM SYSTEMS

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  1. DIGITAL SPREAD SPECTRUM SYSTEMS ENG-737 Wright State University James P. Stephens

  2. AUTOCORRELATION OF PN SEQUENCES • A very important property of PN sequences is the periodic auto correlation function • Recall that autocorrelation is defined as: • Simply stated, it is a comparison of a sequence to a time-shifted version of itself • The equation asks “For an offset of m chips, how similar is the sequence to inself?”

  3. PN SEQUENCE AUTOCORRELATION FUNCTION Normalized Rbb() NTc 1.0 1/N  Tc

  4. Sc(f) N + 1 sinc2(f Tc) N2 1/NTc f 2/Tc -1/Tc 1/Tc -2/Tc POWER SPECTRAL DENSITY FOR M-SEQUENCE Chip Duration Tc and Period NTc

  5. + DISCRETE AUTOCORRELATION Example n = 3 N = 2n – 1 = 7 Initial Fill = 0 0 1 The periodic autocorrelation: . . . 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 1 1 . . 1 -1 -1 1 -1 1 1 m=0 1 1 1 1 1 1 1 = 7 1 -1 -1 1 -1 1 1 m=1 -1 1 -1-1 -1 1 1 = -1 and so on . . . . . . . Sequence  1 0 0 1 0 1 1 Convert to {-1, 1}  1 -1 -1 1 -1 1 1

  6. DISCRETE AUTOCORRELATION Example Discrete Time Plots In the limit as m   Approaches that of White Noise

  7. BINARY SYMBOL CONVERSION • Consider an m-sequence [ck] Where, ck {0, 1} for all k • Now define the m-sequence [bk]  {-1, 1} • By letting bk = 2ck -1, this translates 0  -1 and 1  1 • This is a useful MatLab conversion necessary for many MatLab computations: Note: In GF(2) arithmetic, multiplication of 1 and -1 is the same as addition of 0 and 1. Example: Exclusive OR Multiplication

  8. MATLAB IMPLEMENTATION • The following files are found on my web site at http://www.cs.wright.edu/~jstephen/ • pnf.m • oct2bin.m • masseyf.m • Example: • pn1=pnf(6,103,1,1500,1,1); • pn2=pnf(6,103,1,63,1,1); • pn1=(pn1*2)-1; • pn2=(pn2*2)-1; • auto=xcorr(pn2,pn1); • plot(auto)

  9. MATLAB IMPLEMENTATION (Cont.) • auto=auto(100:400); • plot(auto)

  10. M-SEQUENCE SECURITY ISSUES • How secure are m-sequences? Even if they do not repeat for years? • Massey’s Algorithm can determine the generating polynomial, at most, in ‘2 times n’ error free chips • Example: • Generate m-sequence where N=7, i.e. n=3 • Fill up matrix using only 2*n = 6 chips • Obtain system of equations that can be solved simultaneously • Thus, a code that uses 89 stages that will not repeat for 73,000 years (1 Mcps chip rate) can be determined by only processing 2 * 89 = 178 chips

  11. CODE SEQUENCE DETERMINATION • Berlekamp-Massey linear feedback shift register prediction algorithm • Gold’s Frequency Agile Predictor • B-M requires (at most) 2n error-free chips to determine generating polynomial • Gold’s based on relative times of arrival of single frequency • Demonstrated on Workstation with MATLAB

  12. MASSEY ALGORITHM EXAMPLE a(k) Code sequence of period N=7 (n=3): a = a(0), a(1), a(2), a(3), a(4), a(5), . . . . a = 1 1 1 0 1 0 0 1 1 1 0 1 0 0 (14 chips) a(k) = C1 a(k-1) + C2 a(k-2) + C3 a(k-3) a(3) a(2) a(1) a(0) = 0 1 1 1 a(4) a(3) a(2) a(1) = 1 0 1 1 a(5) a(4) a(3) a(2) = 0 1 0 1 We obtain the system of equations: (1) 0 = 1 C1 + 1 C2 + 1 C3 (2) 1 = 0 C1 + 1 C2 + 1 C3 (3) 0 = 1 C1 + 0 C2 + 1 C3

  13. MASSEY ALGORITHM EXAMPLE Add equations (1) and (2) 1 = 1 C1 + 0 C2 + 0 C3 C1 = 1 Substitute in equation (3) 0 = 1 (1) + 0 C2 + 1 C3 C3 = 1 Substitute C1 = 1, C3 = 1, into (1) gives 0 = 1(1) + 1 C2 + 1(1) C2 = 0 Solution is: X3 + X + 1

  14. Frequency Taps Feedback Taps MODEL FREQUENCY HOPPER Code-of-the-day Modulo-2 Adder Binary Register FH Output Mapper

  15. GOLDS TIME-OF-ARRIVAL ALGORITHM Code-of-the-day Code Generator Binary Output Register Output Hopping Sequence 00 > a 01 > b 10 > c 11 > d Frequency Mapper Times-of-arrival: 4,5,8,9,10,13,14,23,24,25,26,27,31

  16. PERFORMANCE • Importance of real-time implementation: • Rapidly changing battle scenario • Typically short message bursts • Ability to change code-of-the-day • Data collection time • Probability of correct synchronization • Prediction algorithm run time

  17. For 64-stage register and 128 hop frequencies a sequence of 64 X 128 = 8192 hops need to be collected • At 100K hops/sec this will take 8192/100K = 0.08192 seconds DATA COLLECTION TIME

  18. 31 correct detections of key frequency • 31-stage register • 128 hop frequencies • SNR - 12 dB • Prob correct sync = .9 PROBABILITY OF CORRECT SYNCHRONIZATION

  19. ALGORITHM TIME Based on 32-stage shift register

  20. ALGORITHM TIME VERSUS NUMBER OF HOP FREQUENCIES

  21. PARTIAL CORRELATION • The important random properties of m-sequences are the result of usually very long sequences • However, it is not possible to perform full length correlation in a receiver • Also, if the duration of a message bit is less than that of the period of the PN sequence, then to decode the message, we need to correlate over something less that a full code period • In these cases, the correlation function we just looked at does not apply • The ‘Partial Correlation’ function is written:

  22. PARTIAL CORRELATION (Cont.) • We can treat the am’s as random variables { 1} So, • The variance is:

  23. PARTIAL CORRELATION M-sequence Autocorrelation M-sequence Partial Correlation 63 chips 50 chips

  24. PARTIAL CORRELATION (Cont.) 50 chips 40 chips 30 chips 20 chips

  25. MATLAB IMPLEMENTATION OF PARTIAL CORRELATION • To perform a full autocorrelation of an m-sequence of length 63: pn1=pnf(6,103,1,1500,1,1); pn2=pnf(6,103,1,63,1,1); pn1=(pn1*2) -1; pn2=(pn2*2) -1; auto=xcorr(pn2,pn1); auto=auto(100:400); plot(auto) • To perform a partial autocorrelation for this sequence, change line 2: pn2=pnf(6,103,1,50,1,1); Try different values here ! Note: Each time you change line 2, be sure to re-enter lines 4, 5, 6, and 7 since these values must be re-computed.

  26. PERIODIC VS. APERIODIC CORRELATION 1 4 2 3 PN Code Reference Periodic 1 PN Code Reference Aperiodic

  27. APERIODIC AUTOCORRELATION OF AN M-SEQUENCE • Determine the Aperiodic autocorrelation of m-sequence [45]o n = 5, N = 31 pn1=pnf(5,45,1,31,1,1); pn2=pnf(5,45,1,31,1,1); pn1=(pn1*2)-1; pn2=(pn2*2)-1; x=xcorr(pn2,pn1); plot(x)

  28. APERIODIC AUTOCORRELATION Peak Value Approximately 15 side lobe values Index of discrimination G(x) = 45o

  29. PARTIAL APERIODIC AUTOCORRELATION G(x) = 45o 15 Chips

  30. CROSSCORRELATION • Whereas autocorrelation compares a code with itself, ‘crosscorrelation’ compares one code to another • Example: Determine the periodic and aperiodic crosscorrelation of [45]o and [75]o • Note the number of sidelobe values • Periodic sidelobes are 3-valued • Aperiodic is very random looking • Refer to the handout from Robert Gold’s publication on Linear Code Sequences

  31. CROSSCORRELATION([45]o and [75]o) Periodic Aperiodic

  32. COMPOSITE CODES • Although m-sequences are quite desirable for many applications, there is considerable research in finding new codes that offer improved performance in area • Composite codes are generated by a combination of linear maximal sequences • The following are examples of composite codes: • Gold Codes CDMA Applications • JPL Codes Rapid synchronization • Kasami Codes Length construction and CDMA Applications

  33. GOLD CODES • Many details regarding Gold Codes are explained in Appendix 7 of Dixon text • To be assured of good performance in CDMA applications, a system must have: • Low crosscorrelation properties among codes • Low partial crosscorrelation properties among codes • From Gold’s report: Length 25 – 1 = 31 Fixed Polynomial [45]o Bound N Spectrum Polynomial 31 1 31(1) -1(30) 45 11 1 7(5) 3(10) 51 -1(5) -5(5) -9(5) 9 4 7(10) -1(15) 75 -9(6) 67 53 73

  34. REGARDING M-SEQUENCES FOR CDMA APPLICATIONS • There is no general bound on the cross-correlation properties of m-sequences • There is a limited number of available code for a large number of users within the family of m-sequences

  35. GOLD CODE CONSTRUCTION LRS 1 and LRA 2 must have same length Code 1 LRS 1 + Clock LRS 2 Code 2 Output = Gold Code ( Code 1 xor Code 2)

  36. + + + 1 2 3 4 5 1 2 3 4 5 GOLD CODE CONSTRUCTION x5 + x3 + 1 = [51]o Gold Code x5 + x4 + x3 + x2 + 1 = [75]o

  37. GOLD CODE CONSTRUCTION [51]o xor [75]o = ( x5 + x3 + 1)(x5 + x4 + x3 + x2 + 1) = x10 + x9 + x8 + x7 + x5 x8 + x7 + x6 + x5 + x3 + x5 + x4 + x3+ x2 + 1 = x10 + x9 + x6 + x5 + x4 + x2 + 1 = [3165]o non-maximal

  38. MAXIMUM NUMBER OF GOLD CODES • Normally, you fix one register and change the fill of the other to generate a family of Gold Codes • Since there are 2n – 1 initial fills, you can generate 2n – 1 Gold Codes from each configuration • The code length will be N = 2n – 1, where n is the length of the individual registers which must be of equal size • For the 5-stage registers shown in the previous chart, N = 31 • Since the generating polynomial for the Gold Code is degree-10, an m-sequence would be 210 – 1 = 1023 • 1023 / 31 = 33 codes • This comes from 31 initial fills plus the m-sequence generated from each separate register alone • Recall from Table 3.3 in Dixon’s text, the number of possible m-sequences given n is tabulated in the next chart

  39. MAXIMUM NUMBER OF GOLD CODES n # of m-sequences. # of Gold Codes 5 6 31 + 2 6 4 63 + 2 7 18 127 + 2 8 16 255 + 2 9 48 511 + 2 . . . . . . . . . 15 1800 32767 + 2 Not only do Gold Codes provide a larger ensemble of codes, but they provide uniformly low-cross correlation properties while exhibiting good autocorrelation properties

  40. CHOSING A SET OF ‘PREFERRED PAIRS’ • M-sequences used to generate Gold Codes must be of the same length and chosen according to a process of “preferred pairs” selection (Details are in Appendix 7 of Dixon text) • Table A7.3 says there are no preferred pairs for n=6. Not true! [103]o and [147]o are a preferred pair • Arbitrary selection of code pairs can result in very poor correlation performance • Table A7.1, page 501, shows that for n=5, a ‘preferred pair’ yields a peak value of cross correlation of 9 for a preferred pair, but 11 for a non-preferred pair • The selection of ‘preferred pairs’ will always give 3-valued autocorrelation which will bound the correlation to:

  41. Table A7.1 Performance of Preferred Pairs Compared with Worst Case Pairs

  42. PREFERRED PAIRS OF GOLD CODES

  43. CHOSING A SET OF ‘PREFERRED PAIRS’ The algorithm for choosing ‘preferred pairs’ is as follows: • Select n and a polynomial • Find k associated with the polynomial (polynomial root) • If n is odd, then calculate 2k + 1, if n is even then calculate 2(n+2)/2 + k • The number calculated is the polynomial root (k) associated with the second code that completes a ‘preferred pair’

  44. CHOSING A SET OF ‘PREFERRED PAIRS’ Example: • Let n=8, then choose 435(1) • k=1 • n is even therefore, 2(8+2)/2 + 1 = 33 None listed • Try n=8, then choose 747(11) • k=11 • N is even therefore, 2(8+2)/2 + 11 = 43  [703]o

  45. JPL RANGING CODES • JPL Codes are constructed by modulo-2 addition of 2 or more m-sequences whose lengths are relatively prime to one another • Advantages of JPL Codes: • Very long codes can be generated which are useful for unambiguous ranging over long distances • May be generated easily with a small number of short registers • Synchronization can be more easily accomplished by separate operation on each of the component codes • Relatively prime means no common factors among numbers

  46. 2m - 1 2n - 1 2p - 1 JPL CODE CONSTRUCTION a a xor b xor c b c

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