International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 9- September 2013
To study, analysis and simulation of (Spread Spectrum) FH system based on Quadratic Prime Code
using BPSK and FSK
kapil Bhardwaj1;Sandeep Agrawal 2
M.E student, digital communication ,mahakal institute of technology ujjain ,RGPV Bhopal
2
Associate Prof. mahakal institute of technology ujjain , RGPV Bhopal
1
Abstract
Performance of a FH communication system can be improved if QPC are used as a frequency generator instead of
PN sequence. Quadratic prime codes are new types of frequency hopping (FH) sequences constructed by expanding
the construct idea of prime codes to finite extension fields.
The linear complexity and characteristics of frequency interval of quadratic prime codes are analyzed and compared
with those of prime codes.
Keywords: frequency hopping, spread spectrum, quadratic prime
code , prime code, Galois field
Introduction
With excellent anti-jamming, anti multi-path fading and
multiple access networking performance, frequency-hopping
(FH) technique with Quadratic Prime codes has been widely
used not only in military communication but also in civil
mobile communication . FH sequences have decisive impact
on the performance of FH communication systems, and how
to seek and design FH sequences with ideal performance is
one of the important topics in FH communication systems
research. Quadratic Prime codes are FH sequences with ideal
Hamming correlation properties, and the only FH sequences
that achieve the five theoretical limits described. Quadratic
prime codes are constructed by expanding the construct idea
of prime codes to extension Galois fields GF (P²)[2]. Compared with prime codes, the sequence period of quadratic
prime codes is increased from P to P², and sequence number
increased from P-1 to P²-1with ideal Hamming autocorrelation property and crosscorrelation no greater than 2. After
classification, the sequences in the same group maintain
ideal Hamming auto and cross-correlation properties. More
important, quadratic prime codes can provide P (P-1)/2 families of FH sequences rather than one family for prime codes,
which is preferable to improve the security of communication systems. The linear complexity of FH sequences is an
important index for the secrecy intensity of FH communication system. In order to obtain maximum security intensity,
FH system requires that the FH sequences it uses have the
largest possible linear complexity; therefore, how to design
FH sequence with large linear complexity is a research focus
[3]. In addition, FH communication is vulnerable to various
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kinds of jam, such as tracking jam, multipath jam, some frequency band jam, monophonic collimation jam, wideband
jam, and so on. FH sequences with wide frequency interval
could help improve the anti-jamming capability of FH communication [1]. This paper investigates the linear complexity
and characteristics of wide frequency interval of quadratic
prime codes, and new FH sequences with wide frequency
interval are proposed based on the combination of prime
codes and quadratic prime codes.
Quadratic Prime Code
Quadratic prime code is based on finite field GF (P^2) in the
form of x² -y-1 ( y is the second non-surplus of prime number
P) ,and it is the result of domain multiplication of mode of quadratic polynomial. Some good feathers of quadratic prime codes
are : the maximum of QPM(Quadratic Prime Code)’s autocorrelation side lobe is 0, the good Hamming cross-correlation
distance feature, the uniformly frequency distribution characteristics. Therefore, quadratic prime code has become a new type
frequency hopping sequences. According to finite field theory
[6], we use a two dimensional vector in GF (P) to express the
parameters of quadratic prime code. The polynomials are:
L(x)=ix+j, m(x)=ax+b, k(x)=k1x+k0
i,j,a,b,k1,k0 Є GF(P) and (a,b) ≠( 0,0)
(1)
According to the above agreements, using
f(x)=x2-y
(2)
as irreducible polynomial using below formula to achieve the
multiplication of GF(P²) domain:
K[m(x),l(x)] =m(x)l(x) mod f(x)
=(ax+b)(ix+j)mod(x2-y)
= [aix2 +(aj+bi)x +bj] mod (x2-y)
=(aj
bi)x +(bj yai)
means mod P adder and (aj
bi)x +(bj yai)
corresponding frequency gap
Each code group contains the number quadratic prime
codes’ is P²-1, it means the quadratic prime sequence’ period
P². The hamming autocorrelation side lobe of this quadratic
rime code is 0, the maxim hamming cross correlation value
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 9- September 2013
is 2. The linear complexity of the quadratic prime code sequence of FH has direct impact on the anti-decipher ability
and the ideal linear complexity of FH sequence is about half
the length of code sequence. Assuming the length of quadratic prime code is N, then the linear complexity of the sequence is (N +1) / 2 Therefore, the quadratic prime code has
ideal characteristics of linear complexity.
Taking P=3 and y=3 as an example, 8 sequences with
length of 9 are constructed and listed in Table I.
Q1
Q2
Q3
Q4
Q5
Q6
Q7
Q8
{0,1,2,3,4,5,6,7,8}
{0,2,1,6,8,7,3,5,4}
{0,3,6,2,5,8,1,4,7}
{0,4,8,5,6,1,7,2,3}
{0,5,7,8,1,3,4,6,2}
{0,6,3,1,7,4,2,8,5}
{0,7,5,4,2,6,8,3,1}
{0,8,4,7,3,2,5,1,6}
Table I:Quadratic prime code for P=3
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 9- September 2013
OUTPUT DATA
BPSK
MODULATOR
FH MODULATOR
CHANNEL
BPSK
DE MODULATOR
FH DEMODULATOR
RANDOM DATA
FREQUENCY
SYNTHESIZER
FREQUENCY
SYNTHESIZER
QUADRATIC
PRIME CODE
QUADRATIC
PRIME CODE
FH SYNCHRONIZER
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 9- September 2013
Results and Analysis
Fig 2: waveform of BPSK,FSK using QPC and their bit error
rate performance
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International Journal of Engineering Trends and Technology (IJETT) – Volume 4 Issue 9- September 2013
In fig 2 various waveforms are shown. At top left waveform
for BPSK system are shown, digital data is first randomly
generated and it modulates the high frequency carrier signal
using BPSK. BPSK is more immune to noise and interference.In FH system modulated signal is hopped at different
frequencies so we have generated some frequency where the
signal is hopped on periodic basis, that signal is called frequency hopped signal. That signal is transmitted through
Gaussian channel and at the receiver the reverse operation is
performed by using frequency synchronizer and demodulator.Top right part displays the system using MFSK
.Remaining part is same for this system. Bottom left displays
the BER versus SNR for BPSK system Which shows better
results than MFSK. Bottom right displays the BER versus
SNR for MFSK system
Acknowledgments
I am very thankful to Prof Sandeep Agrawal of MITS
ujjain for the support and guidance to develop this paper.
Conclusion
FH system using QPC with BPSK gives better performance
than using MFSK. BPSK gives better bit error rate results
than MFSK. Theoretical analysis and computation results
show that, the proposed FH sequences have all advantages
of prime codes and quadratic prime codes in that they not
only have wide frequency interval and uniform frequency,
but also have good Hamming correlation and ideal linear
complexity. Quadratic prime codes have better confidentiality than prime codes. Moreover, two methods for constructing FH sequences with wide frequency interval are proposed
with the combination of prime codes and quadratic prime
codes.
[5]
[6]
[7]
Yang Yixian and Lin Xuduan, Coding cryptology.
Beijing: Posts and Telecommunications Press,
1992, pp. 553-556
Liu Qingge,Yang Dongkai, Zhang Qishan, “Concatenated prime codes and quadratic prime codes,”
11th IEEE Singapore International Conference on
Communication Systems, 2008. Singapore, 2008,
pp. 241-245
Mei Wenhua and Yang Yixian, “Families of Frequency Hopping Sequences with Given Minimum
Gap,” Journal of China Institute of Communications. China, vol.18(5), pp. 37-44, 1997
Biographies
KAPIL BHARDWAJ received the B.E. degree in Electronics Engineering from the RGPV University BHOPAL in
2004, And pursuing M.E. degree in Digital Communication
from the RGPV University of BHOPAL respectively. Currently, He is an associate Professor of Electronics Engineering at MIT UJJAIN. His teaching and research areas include
wireless communication, automation, and embedded system
design. kapil bhardwaj may be reached at
References
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Mei Wenhua, Wang Shubo, Qiu Yonghong, and Du
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Communication. Beijing: National Defence Industry Press, 1996, pp. 13-14
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