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FPGA Implementation of Pseudo Noise Sequences based on Quadratic Residue Theory

International Journal of Computer Applications
Foundation of Computer Science (FCS), NY, USA
Year of Publication: 2016
A. Rajagopal, K.L. Sudha, Dundi Ajay

A Rajagopal, K L Sudha and Dundi Ajay. Article: FPGA Implementation of Pseudo Noise Sequences based on Quadratic Residue Theory. International Journal of Computer Applications 134(9):10-14, January 2016. Published by Foundation of Computer Science (FCS), NY, USA. BibTeX

	author = {A. Rajagopal and K.L. Sudha and Dundi Ajay},
	title = {Article: FPGA Implementation of Pseudo Noise Sequences based on Quadratic Residue Theory},
	journal = {International Journal of Computer Applications},
	year = {2016},
	volume = {134},
	number = {9},
	pages = {10-14},
	month = {January},
	note = {Published by Foundation of Computer Science (FCS), NY, USA}


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 M-sequences, 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.


  1. Kenneth Ireland and Michael Rosen, A Classical Introduction to modern number theory, Springer-Verlag, 2nd Ed.1990.
  2. M. R. Schroeder, Number Theory in Science and Communications, Springer-Verlag, 2nd Ed.1997.
  3. Zhang Guohua and Zhou Quan, Pseudonoise codes constructed by Legendre sequence, Electronics Letters 38 (2001), no. 8, 376–377.
  4. K. VeerabhadraRao and V. Umapathi Reddy, "Biphase Sequence Generation with low Side-lobe Autocorrelation Function," IEEE Transactions on Aerospace and Electronic Systems, vol. 22, March 1986.
  5. J. J. Rushanan, "Weil Sequences: A Family of Binary Sequences with Good Correlation Properties," in IEEE International Symposium on Information Theory, Seattle, WA, pp. 1648 – 1652, 2006.
  6. D.V. Sarwate and M.B. Pursley, Cross correlation properties of pseudorandom and related sequences, Proc. of the IEEE, vol. 68, No. 5, May 1980, pp. 593-619.
  7. Simon Haykin, Digital communication, John Wiley & Sons, 2006.
  8. CherukuRavikumar and K.L.Sudha, “Legendre and PolyphaseSidel’nikov Sequence for Applications in Space Communication”, International Journal of Emerging Science and Engineering (IJESE) ISSN: 2319–6378, Volume-2 Issue-9, July 2014.
  9. Dundi Ajay, K.L.Sudha and A.Rajagopal, “DSP implementation of Weil and Sidelnikov binary Pseudo Random Noise Codes”, IEEE International Conference on Electrical, Computer and Communication Technologies, 2015.(ICECCT 2015), Coimbatore, India.


PN sequence, Quadratic residue, Legendre, Weil, correlation, FPGA.