Weil Sequences: A Family of Binary Sequences with Good Correlation Properties

Rushanan, Joseph J.

doi:10.1109/isit.2006.261556

Cited by 42 publications

(34 citation statements)

References 5 publications

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“…The XOR addition of a Legendre sequence "s" of length "p" with a shifted replica of itself leads us to whole new set of family called Weil sequences W [5].…”

Section: International Journal Of Computer Applications (0975 -8887) mentioning

confidence: 99%

“…It is given by the expression, with ′τ′ as the time shift Similarly figure 5 shows the plot of even and odd autocorrelation of Weil sequence. It can be seen from the plot the even auto-correlation is not binary valued but the maximum limit of correlation side-lobe is bounded by 2√p +5 [5] where "p" is the length of the sequence. The odd cross-correlation is defined as…”

Section: -(4)mentioning

confidence: 99%

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FPGA Implementation of Pseudo Noise Sequences based on Quadratic Residue Theory

Rajagopal¹,

Sudha²,

Ajay³

2016

IJCA

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Pseudo Noise (PN) sequences are defined as a sequence of 1"s and 0"s which have randomness properties that make it appear noise-like but are generated by mathematical algorithms. PN sequences that are generated by shift registers such as Msequences, Gold sequences are known and widely used since the 1960"s for various applications. These sequences are periodic and the periodicity is always in terms of powers of 2, hence donot offer much flexibility in terms of length of the sequence. In the past decade or so, PN sequences based on Prime numbers and quadratic residue theory have been discovered and are known to exist for a greater range of permissible lengths. The properties and generation of these Prime number based sequences have not been explored fully in literature and hence this paper explains two such sequences namely Legendre and Weil sequences and simulates them to analyse their properties which test their randomness. The simulation is done using MATLAB and Verilog Hardware Description Language. Generation of these sequences is described and implementation details on the Kintex-7 FPGA device with results are brought out.

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“…The XOR addition of a Legendre sequence "s" of length "p" with a shifted replica of itself leads us to whole new set of family called Weil sequences W [5].…”

Section: International Journal Of Computer Applications (0975 -8887) mentioning

confidence: 99%

Section: -(4)mentioning

confidence: 99%

FPGA Implementation of Pseudo Noise Sequences based on Quadratic Residue Theory

Rajagopal¹,

Sudha²,

Ajay³

2016

IJCA

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“…The last line of Table 1 has the relatively new Weil sequences, which are described in the next section. The data for Table 1 is derived from [5][6][7][8]. Table 1 is the lack of variety for the lengths of the various families.…”

Section: Families Of Spreading Codesmentioning

confidence: 99%

“…The Weil sequence construction was originally proposed for primes congruent to 3 modulo 4 in [11]. We rediscovered the construction, extended it to all primes, and proved the important correlation properties in [8]. Some of the properties of Weil sequences are listed in Table 1.…”

Section: Weil Sequencesmentioning

confidence: 99%

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The Spreading and Overlay Codes for the L1C Signal

Rushanan¹

2007

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The creation of the new L1C GPS signal presented the opportunity to choose both a family of spreading codes and an associated family of overlay codes. This paper describes the rationale and construction of these families. The families were created from extensive searches with each search requiring its own fine-tuned techniques and search criteria.The L1C spreading codes comprise 210 pilot/data pairs of length-10230 sequences. The data code modulates the data message bits while the pilot code modulates the overlay code symbols, which represents a fixed repeating pattern. The codes are perfectly balanced and exhibit good auto-and crosscorrelation (both in the "odd" case, when there is a bit transition across the code boundary, and the "even" case when there is no such transition). The length 10230 precluded the immediate adaptation of wellknown spreading code families, such as Gold codes. Instead, the relatively new Weil sequence construction was adapted. Weil codes are prime length sequences constructed via shift-and-add from the well-known Legendre sequence and one of its shifts. Weil code correlation sidelobes are bounded by twice the square root of the length, which is no worse than 3 dB from commonly used Gold codes.The L1C codes were created by using Weil-codes of prime length 10223. Selected Weil-codes were padded with a fixed 7-bit pad to yield the L1C spreading code. The correlation properties of Weil-codes with pad are highly dependent on both the Weil code and the pad insertion point. Thus a search over all Weil codes and insertion points was required. The search criteria were derived from threshold bounds on the sidelobes for both auto-and cross-correlation and for both the even and odd cases. The search criteria need to be adjusted occasionally to yield more candidates codes. The overall search yielded a large set of codes from which the final set of 420 codes could be selected. Because L1C is currently considering two separate modulation schemes (BOC(1,1) and TMBOC), two separate families of codes were constructed that are optimized to the modulation The 210 L1C overlay codes are length 1800, which corresponds to a frame length of 1800 symbols. The search to construct the overlay codes used two types of criteria. First, full period even auto-and cross-correlation sidelobe bounds were specified. Second, criteria were given for the correlation sidelobes when a small subsequence of the code is correlated against the full code. For this case, lengths of 100 (1 second) and 200 (2 seconds) were used. The overlay codes are based on truncated linear feedback shift register sequences of length 2047, either m-sequences or Gold sequences. The choices for truncation points ensured good auto and crosscorrelation sidelobes while also allowing flexibility to bound the short window correlations.

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Satnav Signals

2015

Engineering Satellite‐Based Navigation and Timing

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Weil Sequences: A Family of Binary Sequences with Good Correlation Properties

Cited by 42 publications

References 5 publications

FPGA Implementation of Pseudo Noise Sequences based on Quadratic Residue Theory

FPGA Implementation of Pseudo Noise Sequences based on Quadratic Residue Theory

The Spreading and Overlay Codes for the L1C Signal

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