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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 inte- gration 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 deter- mined by applying a closed mathemati- FIGURE 2 Degradation of correlation performance after truncation cal formula, while specific algebraic for- mulas 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.

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