Codes: the PRN Family Grows Again © Istockphoto.Com/Agencyby ©

working papers Codes: The PRN Family Grows Again © iStockphoto.com/agencyby ©

Unlike most families that become more robust the more the members have in common with each other, pseudorandom noise families grow stronger the more different their sequences are to one another. This column introduces some new offspring from the familiar elements of GNSS and other radionavigation system codes.

Stefan Wallner, José-Ángel Ávila-Rodríguez to this very interesting field of research merit (FOM) indicating the perfor- European Space Agency is now expected in the near future. In mance of the PRN code set. A large fact, not only additional CDMA-based variety of FOMs can be imagined, and seudorandom noise (PRN) signals will appear from GNSS satellites, dedicated publications deal with this sequences are an essential ele- but a steadily growing interest in ground subject. See, for example, the paper by ment of any radionavigation sat- based, continuously transmitting pseu- F. Soualle et alia referenced in the Addi- P ellite system that is based on code dolites can also be observed. tional Resources section near the end of division multiple access (CDMA) tech- This interest will undoubtedly ani- this article. niques. Indeed, these sequences enable mate further research on high-per- The correlation function goes back a navigation receiver to distinguish one forming sets of PRN codes with certain to the mean squared error concept satellite from another. characteristics regarding code length introduced by Carl Friedrich Gauss A comprehensive introduction to and the number of codes to support (1777–1855) and is without doubt one CDMA, including its history in general design of the new signals to be provided of the most widely used and most pow- and PRN sequences in particular, can be by these systems. The New PRN Code erful means to characterize the perfor- found in the Working Papers column by Family introduced in this column could mance of PRN sequences. It can mea- G. W. Hein et alia published in the Sep- be one potential candidate for such sys- sure the communality between different tember 2006 issue of Inside GNSS. tems because it offers a number of highly sequences of length N and is defined for After the finalization of the Galileo advantageous characteristics that we will 0-hertz Doppler frequency offset as PRN codes back in 2004 and the defini- analyze in this article. tion of the GPS L1C codes in about the System designers need to select the same timeframe, additional attention best codes according to some figure of

www.insidegnss.com september/october 2011 InsideGNSS 83 working papers

where Adequate Code Lengths u(n) n-th chip of sequence u Prime Code Length v(n + l) n+l-th chip of sequence v. Prime Squared Code Length The correlation function shown by Code Length Adequate for LFSR this equation is also referred to as even correlation because it does not account for a flip of the sequence within the integration period as might be induced due to a data or secondary code bit change. 012345 This article will address the issue of odd Code Length N correlations later on. The objective of the following discussion is to introduce and FIGURE 1 Adequate code lengths for LFSR-based sequences, prime, and prime-squared length mathematically derive the New PRN sequences Code Family and characterize it in terms of correlation performance. Crosscorrelation of GPS C/A Code and Truncated GPS C/A Code 0.8 Generation of PRN GPS C/A sequences Truncated GPS C/A Regarding the generation of PRN 0.6 sequences, we can distinguish between two categories: 0.4 • PRN sequences, where the value of

each chip of the sequence is based Occurrence on a mathematical, closed form algo- 0.2 rithm, and • PRN sequences that result from a numerical optimization or selection 0 process. -150 -100 -50 0 50 100 150 For the latter category, the individual Crosscorrelation [nat. units] chips of the PRN codes cannot be determined by applying a closed mathemati- FIGURE 2 Degradation of correlation performance after truncation cal formula, while specific algebraic formulas can be given for the first category rate and its corresponding symbol rate. enabling the construction of prime of sequences that result immediately in For PRN sequences that are derived length sequences significantly exceed the the PRN sequences. from closed analytical expressions, any code lengths from which to choose when Well-known PRN sequences that deviation from the preset code length setting up LFSR-based sequences. This belong to this first category include immediately results in a significant is further underlined in Figure 1, which • Gold codes degradation in terms of increased cross- shows that in a mathematical sense the • Kasami codes correlation and out-of-phase autocorre- prime numbers lie much denser in the • Weil codes lation. For instance, any codes based on natural numbers IN than the numbers • Bent-function sequences linear feedback shift registers (LFSRs), 2n − 1 for integers n do. • No such as Gold codes and Kasami codes, For 0 < N < 50,000, a total of 5,133 • Gong/Paterson, and can be set up for any length N = 2n − 1 prime numbers exist but only 48 prime • Z4 linear Family I and II. for integer values n that respect mod(n, squares, and LFSR-based sequences can Articles and papers referenced in the 4) ≠ 0. only be set up for 15 different sequence Additional Resources section discuss In contrast to LFSR-based sequences, lengths. most of these in further detail. Weil codes can be constructed for any As already mentioned, any trunca- These codes offer almost ideal auto- prime code length. This is a benefit tion or elongation of PRN sequences and cross-correlation properties for zero because there are more prime numbers immediately results in a significant Doppler frequency offset. Unfortunately, than lengths N = 2n − 1, thus allowing increase in auto- and cross-correlation they face the restriction that they can for additional flexibility. Paterson or as shown in the Figure 2. In this example, only be constructed for specific code Gong sequences need to comply with a just the very last chip of 1,023-chip Gold lengths. code length being the square of a prime codes — as they are used for the GPS The PRN code length results gen- number. C/A code signal — was deleted, lead- erally from dividing the signal’s chip Obviously, the potential code lengths ing to truncated Gold codes of length

84 InsideGNSS september/october 2011 www.insidegnss.com 1,022. While the cross-correlation for N that follow the form, N = p − 1, where new PRN code family. From here on, Gold codes of order 10 is bounded by p refers to any prime number larger this column will refer to these as SLCE -65 and 63, the correlation values of the than 7. So, any sequence contained in sequences. truncated version spread out to -120 and the new code family results in an even SLCE sequences are based on primi- 132 (indicated by the red ellipses in the length. This is of particular interest tive root elements. A short definition of figure), and are thus significantly worse. because most of the code families that primitive roots is as follows: Obviously, in many cases the require- are generally known in the literature If m is a positive integer, the con- ments of the PRN code length as driven are constructed from closed formulas gruence classes coprime to m form by the signal’s chip rate and its symbol that include a sufficiently large number a group with the multiplication rate are not compatible with the avail- of codes with the property of having modulo m as the internal operation; able, analytically defined options for an odd length. This is indeed the case it is denoted by and is called the PRN code generation. In order to close for LFSR-based sequences as well as for group of units (mod m) or the group this gap, some signal designs make use Weil-codes. of primitive classes (mod m). A gen- of truncation or elongation to come to The new code family can be derived erator of this cyclic group is called a the desired code lengths. from a single sequence in contrast to, for primitive root modulo m. For instance a head/tail concept could example, Gold codes and Kasami codes. To demonstrate how this works, we be applied as it was under early consid- Indeed, these require two appropriately take for illustration purposes the task eration for the Galileo E5 PRN codes. In selected maximum length sequences of finding a primitive root for m = 14. order to achieve the required code length (M-sequences) that form the basis from Consequently, the group of all co-prime a full run of a LFSR-based sequence (rep- which to derive the full code family. numbers with respect to m = 14 is denot- resenting the head) needs to be concat- M-sequences can be generated using ed as and is shown to be formed by enated with a truncated LFSR-based maximum LFSR, and they produce every = {1,3,5,9,11,13}. For an integer num- sequence (representing the tail). Similar binary sequence that an LFSR can cycle ber pr to be primitive root modulo of 14, concepts can be also be found at GPS through except the all-zero state. In this we now need to prove that the function L1C, where Weil sequences with a length way an n-stage LFSR is capable of gener- of 10,223 chips are complemented by a ating a binary sequence of length 2n − 1, well chosen 7-chip pad (inserted into the if the feedback taps are chosen properly, is surjectiv onto , i.e., all elements of Weil sequence at a specific index), and in and the resulting M-sequence shows a can to be “obtained” by the func- Compass signal designs. spectrally flat autocorrelation function. tion g. As an example, we will next test In order to allow for maximum flex- Two M-sequences showing specific whether 3 or 9 is a primitive root mod- ibility regarding the PRN code length, cross-correlation properties are referred ulo 14. This is depicted inT able 1, where numerical generation and optimization to as a preferred pair and form the basis all results are taken modulo 14. methods have been identified. Genetic from which to derive Gold or Kasami As we can see from the table, the algorithms can be used to construct ran- Codes (A good overview on M-Sequenc- integer powers of 3 modulo 14 generate dom codes of any desired length, opti- es, Gold Codes, and Kasami Codes can all elements of while, for instance, 5 mized for any potential FOM imagin- be found in the previously referenced is not obtained by any power of 9 mod- able, and implemented during the design Working Papers column by G. W. Hein ulo 14. Thus we can conclude that 3 is a of the PRN sequences. (For further et alia and in Spread Spectrum Systems primitive root modulo 14 while 9 is no details, see the patent application by J. for GNSS and Wireless Communications, primitive root modulo 14. Further anal- Winkel listed in Additional Resources.) authored by J. K. Holmes.) ysis shows that only the numbers 3 and 5 Alternatively, signal designers can As with the new family of PRN codes are primitive root elements modulo 14. make use of chaotic algorithms to set up proposed here, the Weil codes are just For the generation of the following the PRN sequences displaying properties based on a single generative sequence. sequences, it is interesting to know how that are as close as possible to random A method to generate binary sequences many different primitive root elements sequences, as described in the patent appli- has been proposed independently by can be determined for a specific integer cation publication by M. Hadef et alia. V. M. Sidelnikov and A. Lempel et alia m. In order to do so, we recall the totient The new code family that we shall (see Additional Resources). The binary function φ(m) that goes back to the Swiss define next belongs to the first category, sequence derived from their methods mathematician Leonhard Euler (1707– with its sequences constructed based on will serve as a generative one for our 1783) plays an important role. Euler’s toti- closed-form algebraic formulas. pr g (pr,1) g (pr,2) g (pr,3) g (pr,4) g (pr,5) g (pr,6) g (pr,7) Generation of New Family 3 3 9 13 11 5 1 3 with Even Code Length 9 9 11 1 9 11 1 9 We introduce next a new code family TABLE 1. Test for primitive root modulo 14 that can be constructed for code lengths www.insidegnss.com september/october 2011 InsideGNSS 85 working papers

ent function determines the order of or, in other words, the Number of Primitive Root Elements modulo p 2500 number of coprime elements for an integer m and is defined as 2000

1500 where pk relate to the M prime factors that constitute the integer m. The following lines provide a short proof for equation (1). ϕ (p-1) 1000 For a prime number pk there exist exactly pk − 1 coprime elements. When setting the prime number to an exponent ek IN , 500 we can exactly identify the number of elements that are not coprime to . These non–co-prime elements list to pk, 2pk 3pk,..., 0 p = p , and their number is exactly . Consequently, 0 1000 2000 3000 4000 5000 k k p

FIGURE 3 Number of primitive root elements modulo p

As any natural number m can be represented as Autocorrelation of SLCE Sequences 4500 SLCE Sequence Category 1, 4000 SLCE Sequence Category 2 3500 with M prime factors pk and corresponding orders ek we finally result in 3000 2500 2000 As shown in the text, Elementary Number Theory, authored 1500 by D. M. Burton, here exist exactly φ(φ(m)) incongruent primitive root elements modulo m if any primitive root element Autocorrelation [nat. units] 1000 exists for m. If m equals a prime number p, this simplifies to 500 φ(φ(p)) = φ(p−1). Figure 3 shows the number of primitive root 0 elements that can be obtained for prime numbers p. 0 500 1000 1500 2000 2500 3000 3500 4000 The identification of a primitive root element is the first and Delay [chips] basic step for generating SLCE sequences. For the next step we need to form a set S defined as FIGURE 4 Even autocorrelation of SLCE sequences with pr primitive root modulo p p prime number Based on this finding, we can identify the two following categories of primitive root elements prCat1 and prCat2 that lead to SLCE sequences offering slightly different autocorrelation which is actually half the desired code length. properties:

The final SLCE sequence uSLCE(n) is defined for a delay n according to the following formula:

For any prime number p, the number of primitive root elements is always even, and half of them lead to SLCE sequences with Elementary Number Theory identifies the autocorrelation a Category 1 autocorrelation function while the other half results side peaks for SLCE sequences for any primitive root elements in SLCE sequences with a Category 2 autocorrelation function. to be given by SLCE sequences of Category 1 show absolute balance; thus, the number of +1s in the sequence equals the number of -1s. For SLCE sequences of Category 2 the balance depends on k, as defined earlier. For an even k, the However, depending on the selection of the primitive root number of 1s exceeds the number of -1s by 2, while for element pr, an additional category of SLCE sequences can be odd k, there are two more -1s in the sequence than +1s. identified for which the out-of-phase autocorrelation follows:

86 InsideGNSS september/october 2011 www.insidegnss.com The even autocorrelation functions of two SLCE sequences In consequence, it turns out that based on primitive root elements from set prCat1 and prCat2 of length N=4,092 are shown in Figure 4, and we can verify formula (4) as the autocorrelation side lobes adopt only values of either 0 or -4 or only either 4,0,-4 or -8. Although the SLCE sequences show excellent autocorrelation performance, the cross-correlation between them proves just to be of a purely random nature. Indeed, it appears similar Thus, the balance criterion is close to be fulfilled in an ideal to the cross-correlation that would be obtained for any random way. Moreover, the balance criterion is fulfilled ideally for sub- and non-optimized sequences. sets of Category 1 and Category 2 sequences. The text in the accompanying box introduces the approach Correlation Performance. As already mentioned, the auto- that has been identified in order to derive from a single SLCE and cross-correlation performance is an essential metric by sequence a full set of PRN codes that not only show good auto- which to characterize the performance of PRN sequences as correlation performance but also are favorable with respect to they are applied for CDMA systems. their cross-correlation characteristics. We can consider the so-called Welch Lower Bound, or Welch Bound for short, to be the most applied theoretical limit The new code family of even length offering ideal correlation when talking about the best correlation performance that a performance can be constructed based on the following genera- family of PRN codes can achieve. The Welch Bound indicates tion scheme. The i-th PRN sequence of this family is given by: the minimum of the maximum achievable out-of-phase auto- and cross-correlation magnitudes. Indeed, no set of PRN sequences can result in maximum correlation magnitudes where lower than the Welch Bound for any set of PRN sequences. • uSLCE relates to the generative SLCE sequence belonging The article by L. R. Welch cited in Additional Resources either to Category 1 or Category 2, depending on the provides a mathematical introduction and definition of the selection of the primitive root element, Welch Bound. For a set of K sequences each of length N, the • to the element by element binary XOR addition and Welch bound calculates as • Ti indicates a cyclic shift of i chips.

The generation of the New PRN Code Family is depicted in Figure 5. Furthermore, we can easily see that, if only the code length Depending on the primitive root element on which the gen- N tends to infinity, the Welch bound simplifies to erative SLCE sequence uSLCE is based, two different categories of the new code family can be derived, namely, Category 1 and Category 2. Both categories of the new code family show excel- The well-known Gold codes tend to approach the limit of lent, but slightly different auto- and cross-correlation properties. the Welch bound for N → ∞. The maximum auto- and cross- correlation sidelobes for a set of Gold codes of order n calculate Properties and Performance of New Code to Family Several distinctive properties are associated with the new PRN code family, which we will characterize next. The performance Consequently, two categories of Gold codes can be identi- of the code family will also be discussed in the following section. fied, depending on whether n is odd or even, with different Balance. The balance property BAL for the n-th PRN maximum correlation magnitudes: sequence is defined by the addition of the individual chips, i.e.:

The balance of a PRN sequence follows the Golomb postu- lates for randomness with one FOM to characterize the ran- where N = 2n − 1 represents the code length. domness of any PRN sequence. In an ideal case the BAL FOM is We will use this Gold code limit in order to next demon- as close to zero as possible. For the new code family the balance strate the relative value of the new code families in terms of value of the balance property is closely related to the autocorre- correlation performance. We should note that Gold codes are lation function of the generative SLCE sequence, as the balance not available for an order of n being a multiple of 4 (i.e., mod(n, for the n-th PRN sequence calculates to 4) = 0). Coming back to the New Code Family as it was introduced in equation (7), any primitive root element can be used to derive www.insidegnss.com september/october 2011 InsideGNSS 87 working papers

Generation of 1st New PRN Code pair combination exists that does not adopt uSLCE the corresponding correlation magnitude. The black circles indicate the mean value T1u SLCE with which the corresponding correlation magnitude is adapted. Here the average is Generation of 2nd New PRN Code taken over all K2/2 - K/2 potential code pair

uSLCE combinations. The dashed red line in Figure 7 further 2 T uSLCE indicates the cumulative frequency of the Analogue construction of nth New PRN Code various mean values, while the green line relates to the correlation limit that could be FIGURE 5 Generation of the new PRN code family achieved for a set of Gold codes of appro- priate length. it. Depending on the selected primitive root element, two cat- As outlined previously, the new code family can be con- egories of SLCE sequences can be obtained and, consequently, structed for any code length equal to p-1, p being a prime num- two different categories of PRN code families also result, each ber. For demonstration purposes, we selected a code length of them showing slightly different, but still excellent correlation of 4,092 chips. On the one hand, this complies with the code characteristics. The two categories can also be distinguished by their cor- Max. Correlation Magnitudes of New Code Family responding maximum correlation magnitude: 500 450 400 350 300 The formula specifying the maximum correlation magni- 250 tude has been derived empirically. Figure 6 shows the maxi- 200 mum correlation for both categories. Next, Table 2 characterizes a large number of known PRN 150 code families in terms of their code length and the size of the Max. Correlation [nat. units] 100 resulting family as well as the maximum correlation magni- 50 tudes. The last two rows relate to the New PRN Code Family as 0 defined by equation (7). A comparison with the existing PRN 0 0.5 1 1.5 2 2.533.544.5 5 4 code families shows that: Code length x 10 • Only families #7, 10, and 11 allow for even code length, but with significant higher restrictions than for the categories FIGURE 6 Maximum correlation magnitudes of New Code Family of New Code Families #13 and 14. For a further discussion of this point, see also the article by J. Rushanan (2007) listed Crosscorrelation, Un-Optimized Codes, Length 4092 100 in Additional Resources. • The maximum correlation magnitude of the New Code Family is identical to #2, 4, 7, 9, and 11. 10-2 The evaluation of PRN code families regarding their auto- and cross-correlation performance is provided next in the form Figure 7 of correlation histograms. provides a sample histogram 10-4 that also includes a listing of correlation percentiles. The number of cross-correlation magnitudes resulting from an entire code family consisting of K sequences, each of length -6 Relative Frequency 10 N, sums up to (K2/2 – K/2)N, and all these correlation values form the test statistics to derive the percentiles represented in Figure 7. 10-8 The blue crosses in Figure 7 indicate the maximum/minimum relative frequency with which the corresponding corre- 0 -60 -50 -40 -30 -20 lation magnitude shows up in a specific correlation function, Correlation [dB] depending on the selection of two PRN sequences out of the full FIGURE 7 Sample correlation histogram code family. If a minimum is not indicated, at least one code

88 InsideGNSS september/october 2011 www.insidegnss.com Autocorrelation, New Code Family CAT1, Length 4092 Autocorrelation, Optimized Codes, Length 4092 100 100

10-2 10-2

10-4 10-4

10-6 10-6 Relative Frequency Relative Frequency

10-8 10-8

-70 -60 -50 -40 -30 -20 -10 0 -70 -60 -50 -40 -30 -20 -10 0 Correlation [dB] Correlation [dB] Crosscorrelation, New Code family CAT1, Length 4092 Crosscorrelation, Optimized Codes, Length 4092 100 100

10-2 10-2

10-4 10-4

10-6 10-6 Relative Frequency Relative Frequency

10-8 10-8

-70 -60 -50 -40 -30 -20 -70 -60 -50 -40 -30 -20 Correlation [dB] Correlation [dB]

FIGURE 8 Auto- and cross-correlation histogram of New Code Family for a FIGURE 9 Auto- and cross-correlation histogram of optimized code family length of 4,092 chips for a length of 4,092 chips length requirement of the new code family and, on the other Wallner et alia), the correlation of code families showing ideal hand, a highly optimized code family of this length is available performance at 0-hertz Doppler offset degrades when a Dop- that we can use for comparison. pler offset is applied, converging to the performance shown by For this performance demonstration, shown in Figures 8 un-optimized code sets. and 9, the green line needs to be seen as an indicator identifying Figure 10 next shows the maximum absolute correlation a level of excellent correlation performance as can be obtained values based on a pairwise correlation evaluation. Three dif- for Gold codes. However, please note that Gold codes of length ferent maximum correlation magnitudes were observed for a 4,095 chips (which is very close to the code length of 4,092 that code set of length 4,092, and their relative distribution is shown we will analyze next) do not exist because the order of the LFSR in Table 3. must not be a multiple of 4. The corresponding LFSR order Size of Code Family. The new code family includes a number would be 12 in this case. of N/2-1 individual codes where the code length N is given by The following results should only be seen as examples; simi- N=p−1 with a prime number p. lar results can be obtained for any code length with which the new code family is built. The New PRN Code Family is based Deriving a Subset Offering Good Odd on an SLCE sequence that has been derived from the primitive Correlation root element 5 according to equation (7). The selected primitive As already indicated at the beginning of this column, even root element 5 belongs to Category 1, thus showing a slightly auto- and cross-correlation are not the only measures by which lower maximum correlation. to judge the performance of a PRN code family. Typically the We must note that the correlation performance evaluation navigation data or the secondary code bits are modulated onto shown next focuses on a 0-hertz Doppler frequency offset. the primary PRN code, applying binary phase shift keying As has been shown on several occasions (see the article by S. (BPSK). www.insidegnss.com september/october 2011 InsideGNSS 89 working papers

However, as the objective of this column is to introduce a new PRN code family, we will restrict ourselves to the even and odd correlation and ignore further evaluation of other specific properties. We should point out that any PRN code family derived by analytical formula can only offer good even correlation performance. Up to now — and this also applies to the New PRN Code family introduced in this column — no PRN code family derived from closed mathematical expres- sion is known that offers at the same time excellent even and odd correlation characteristics. The odd correlation for mathematically derived PRN code sets is generally not superior to what can be obtained from any unoptimized PRN code family. Therefore, we need to identify ways that provide a selection of codes showing — in addition to excellent even correlation — TABLE 2. Comparison of various PRN code families (Characteristics for families 1 to 12 are derived from the good odd correlation performance. The article, “The Spreading and Overlay Codes for the L1C Signal,” by J. Rushanam.) selection of a subset of codes showing also good odd correlation performance can be Correlation magnitude accomplished by either applying a bottom- 124 128 132 [natural units] up or a top-down approach. Correlation magnitude [dB] -30.37 -30.10 -29.83 The bottom-up approach starts with the identification of an initial seed sequence followed by a search for sequences that Relative occurrence 0.04% 75.03% 24.93% keep the maximum odd correlation of the resulting set under TABLE 3. Maximum cross-correlation magnitudes and their occurrence control. The code family of GPS L1C was constructed following this concept, too. Whenever the navigation or secondary code bits within the The following discussion proposes and applies a top-down integration period induce a flip of the PRN code sequence, the method. Due to the nature of the concept we refer to it as itera- resulting correlation function is referred to as an odd correla- tive sifting. tion. The difference between the even and the odd correlation The first step is the generation of the full code family and is outlined in Figure 11. evaluation of its performance of odd correlation. For each code The odd correlation for two sequences u and v of length N Maximum Pairwise Crosscorrelation [nat. units] is calculated according to the following formula: 2000

132 with 1500

1000 128 Obviously, apart from even and odd correlations, a large number of additional figures of merit exist to judge the per- Code Number formance and compare different PRN code families. These performance measures include Excess Line Weight and Excess 5000 Welch Square Distance, as well as the correlation accounting 124 for Doppler frequency offset. The interested reader is invited to consult the previously ref- 0 500 1000 1500 2000 erenced paper by F. Soualle to gain more insight into the differ- Code Number ent existing FOMs that can be applied for PRN code evaluation. FIGURE 10 Maximum pairwise cross-correlation in nat. units

90 InsideGNSS september/october 2011 www.insidegnss.com pair the maximum absolute odd corre- No Bit Flap lation value is stored in matrix form, Data Bit/Secondary Code Bit +1 Data Bit/Secondary Code Bit +1 which for the New PRN Code Family results in a square matrix of size N/2-1. Received Code After identifying the requirement regarding the size of the PRN code Code Replica family to be obtained the iterative sifting process can be initialized. The concept is based on the identification of PRN codes that lead to the maximum correlation Bit Flap magnitude within the current set. The Data Bit/Secondary Code Bit +1 Data Bit/Secondary Code Bit +1 deletion of one of the PRN codes resulting in the maximum correlation magni- Received Code tude is sufficient to produce a gain in correlation performance. In turn, this opens Code Replica the door for some random optimization. This process is continued until the code set is reduced to size and the FIGURE 11 Even (top) and Odd (bottom) Correlation result is stored. The next iteration uses the initial code family of size N and restarts the sifting process. However, due to the random nature of the process the deletion of individual PRN sequences from the overall set follows a different scheme. The two resulting code sets of size are compared and only the better performing one is maintained. Figure 12 schematically outlines the overall process. Figures 13 and 14 present the result of this selection process, using the New PRN Code family of length 4,092. The requirement regarding the mini- mal size of the PRN code set was placed FIGURE 12 Top Down Selection Process for Odd Correlation at 130 codes, which is considered sufficient for navigation applications. The that the code length directly matches Additional Resources result of the iterative sifting process is the system requirement. Consequent- [1] Burton, D. M., Elementary Number Theory, 4th indicated in Figure 14, where we can see ly, no truncation or elongation of the Edition, William C. Brown Publishers, 1989 that the maximum odd correlation mag- PRN code — which is associated with [2] Gao, G. X.,.and A. Chen, S. Lo, D. De Lorenzo, nitude is reduced from 360 to 280 (in a loss of correlation performance — is T. Walter, and P. Enge, “Compass-M1 Broadcast natural numbers), which corresponds required. Moreover, just one original Codes in E2, E5b and E6 Frequency Bands,” IEEE to 2.2 decibels’ improvement. SLCE sequence of length N is sufficient Journal of Selected Topics in Signal Processing, to derive a full set of N/2-1 PRN codes Special Issue on Advanced Signal Processing for GNSS and Robust Navigation, August, 2009 Conclusions by binary-shift-and-add logic. This alle- [3] Gold, R., “Optimal Binary Sequences for This article has presented a new family viates the need to store each chip of the Spread Spectrum Multiplexing,” IEEE Transac- of PRN codes that can be constructed PRN code in memory within the receiv- tions on Information Theory, Vol. 13, pp. 619–621, following a closed mathematical formu- er device. October 1967 la. This new code family offers excellent The code family described in this [4] Gong, G., “New Designs for Signal Sets with even correlation properties. column is large enough to serve any Low Cross-correlation, Balance Property and Large Following the approach that we have needs in the field of navigation, be it for Linear Span: GF(p) Case,” IEEE Transactions on described here, PRN codes of length satellites, pseudolites, or both. A request Information Theory, Vol. 48, pp. 2847–2867, 2002 N=p–1 (p being a prime number) can has been initiated for a patent applica- [5] Hadef, M.,,and J. Reiss and X. Chen, “Chaotic be generated, allowing for a high level tion on the New PRN Code Family. Spreading Codes and Their Generation,” Patent of fidelity regarding the code length. Application Publication, US 2010/0054225 A1, This produces a much higher likelihood March, 2010 www.insidegnss.com september/october 2011 InsideGNSS 91 working papers

Max. Odd Crosscorrelation Max. Odd Crosscorrelation 2000 280 1800 360 120 1600 320 260 1400 100

1200 80 280 240 1000

Code ID Code ID 60 800 240 220 600 40 400 200 20 200 200

00 500 1000 1500 2000 20 40 60 80 100 120 Code Number Code ID

FIGURE 13 Maximum Odd cross-correlation for full set (natural units) FIGURE 14 Maximum Odd cross-correlation after iterative sifting (natural units)

[5] Hein, G.W., and J.-A. Ávila-Rodríguez and S. Problems of Information Transmission, Vol.5, the signal structure of Galileo, RF compatibility of Wallner, “The Galileo Codes and Others,” Inside 1969, pp. 12–16 GNSS, and advanced receiver autonomous integ- GNSS, Volume 1, September, 2006 [15] Soualle, F., and M. Soellner, S. Wallner, et al., rity monitoring (ARAIM) concepts. [6] Holmes, J. K., Spread Spectrum Systems for “Spreading Code Selection Criteria for the Future José-Ángel Ávila-Rodrí- GNSS and Wireless Communications, Artech GNSS Galileo,” Proceedings of ENC 2005, Munich, guez has been since House, Boston, 2007 Germany, July 19–22, 2005 March 2010 the GNSS [7] Kasami, T., Weight Distribution Formula for [16] Wallner, S., and J. J. Rushanan, J.-A., Ávila- signal and receiver engi- some Class of Cyclic Codes, Coordinated Science Rodríguez, and G. W. Hein, “Galileo E1 OS and GPS neer of the Galileo Evo- Laboratory, University of Illinois, April 1966 L1C Pseudo Random Noise Codes — Requirements, lution Team at ESA/ ESTEC. Between 2003 [8] Lempel, A., and M. Cohn and W. L. Eastman, “A Generation, Optimization and Comparison,” Pro- and 2010 he was research associate at the Institute Class of Balanced Binary Sequences with Optimal ceedings of ION GNSS 2007, Fort Worth, Texas, of Geodesy and Navigation at the University of the Autocorrelation Properties,” IEEE Transactions USA, September 25–28, 2007 Federal Armed Forces Munich. He was awarded the on Information Theory, Vol. 23, No. 1, pp. 38–42, [17] Welch, L.R., “Lower Bounds on the Maximum Bradford Parkinson prize in 2008 and the following January 1977 Cross Correlation of Signals,” IEEE Transactions year he received the Early Achievement Award, on Information Theory, Vol. 20, pp. 397–399, [9] No, J. S., and V. Kumar, “A New Family of Binary both from the U.S. Institute of Navigation. Pseudorandom Sequences Having Optimal Peri- May 1974 Guenter W. Hein serves as odic Correlation Properties and Large Linear Span,” [18] Winkel, J., “Spreading Codes for a Satellite the editor of the Working IEEE Transactions on Information Theory, Vol. 35, Navigation System,” Patent Application Publica- Papers column. He is pp. 371–379, March 1989 tion, US 2008/0246655 A1, October, 2008 head of the Galileo Oper- [10] Olsen, J. D., and R. A. Scholtz and L. R. Welch, ations and Evolution “Bent-Function Sequences,” IEEE Transactions on Department of the Euro- Information Theory, Vol. 28, pp. 858-864, 1982 Author pean Space Agency. Pre- [11] Paterson, K.G., “Binary Sequences with Stefan Wallner studied at viously, he was a full professor and director of the Favourable Correlations from Different Sets and the Technical University Institute of Geodesy and Navigation at the Univer- MDS Codes,” IEEE Transactions on Information of Munich and graduated sity FAF Munich. In 2002 he received the presti- Theory, Vol. 44, pp. 172–180, 1998 with a Diploma in tech- gious Johannes Kepler Award from the U.S. Insti- tute of Navigation (ION) for “sustained and [12] Rushanan, J. (2006), “Weil Sequences: A no-mathematics. He was significant contributions to satellite navigation.” Family of Binary Sequences with Good Correla- research associate at the tion Properties,” IEEE International Symposium Institute of Geodesy and He is one of the CBOC inventors. on Information Theory, Seattle, Washington, Navigation at the Federal Armed Forces Germany July, 2006 in Munich from 2003 to 2010. Since 2010 he has been working as a GNSS system analysis engineer [13] Rushanan, J. (2007), “The Spreading and at the European Space Agency/ESTEC in Noord- Overlay Codes for the L1C Signal,” Journal of Navi- wijk, The Netherlands, in the field of Galileo evolu- gation, Vol. 54, No. 1, pp 43-51, 2007 tion, GNSS standardization, RNSS compatibility, [14] Sidelnikov, V. M., “Some k-valued pseudo- and future integrity provision schemes. His main random sequences and nearly equidistant codes,” topics of interests include GNSS spreading codes,

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