Analysis of Different PN Sequences

ISSN 2249-6343 International Journal of Computer Technology and Electronics Engineering (IJCTEE) Volume 1 , Issue 2

Analysis of Different Pseudo Noise Sequences Alka Sawlikar, Manisha Sharma

 — Pseudo Pseudo noise (PN) sequences are widely used in Abstract  digital communications communications and the theory involved has been treated extensively in this paper. This paper analyses some interesting properties of PN sequence. This paper further elaborates about various methods of generation of PN sequences, few methods discussed in this paper are: series-parallel method for high-speed generation, avoiding the zero states, shift registers with feedback, chaotic pattern generation. A detailed comparison of PN sequence and chaotic sequence is also described in this along with comparative analysis of LFSRs, Gold sequence, Barker sequence, Kasami sequence.

II.

PROPERTIES OF PN SEQUENCES

A PN sequence is a bit stream of ‘1’s and

occurring randomly, with some unique properties. The sequence serves as a reference pattern with known random characteristics for  the analysis, optimization and performance measurement of  communication communication channels and systems. ‘0’s

A. B alance Property  Property 

In each period of a maximum maximum length sequence, the number  of 1s is always one more than the number of 0s.

 Index Terms —  Spread spectrum, pseudo-noise, Chaos, LFSR, Code division multiple access, pseudo-noise codes, Maximal length, Gold, Barker, Kasami, autocorrelation, cross correlation.

B. Run Pr operty  operty 

I. INTRODUCTION Pseudo random binary sequences (PRBSs), also known as  pseudo noise (PN), linear feedback shift register (LFSR) sequences or maximal length binary sequences (m sequences), are widely used in digital communications, instrumentation and measurements [Lathi(1998)]. In a truly random sequence the bit pattern never repeats. A pseudo random binary sequence is a semi-random sequence in the sense that it appears random within the sequence length, fulfilling the needs of randomness, but the entire sequence repeats indefinitely. To a casual observer the sequence appears totally random, however to a user who is aware of the way the sequence is generated all its properties should be known. PN sequences have several interesting properties, which are exploited in a variety of applications. Because of  their good autocorrelation two similar PN sequences can easily be phase synchronized, even when one of them is corrupted by noise. A PN sequence is an ideal test signal, as it simulates the random characteristics of a digital signal and can be easily generated. The following fig. 1 shows the overview of PN sequence.

Among the runs of 1s and 0s in each period of a maximum maximum length sequence, one half the runs of each kind are of length one-fourth are of length two, one eighth are of length three, and so on as long as these fractions represent meaningful numbers of runs[Hykin(2001)]. C. Correlation Pr operty  operty 

Correlation is a measure of similarity between two sequences. When the two sequences compared are different it

is the ‘cross correlation’ and when they are the same it is the ‘autocorrelation’. Mathematically, the correlation between two sequences x (k) and y(k) as a function of the time delay m is expressed as  L 1

 R(m) xy   x(k ) y (k   m) ……….(1) k  0

The correlation equation for the digital bit sequence can thus be written as

R (m) (m) = total number of ‘1’s ‘1’s / total number number of bits (2) Reference Code

b1

b2

1

2 a1

b1

b3

3 a2

b2

+

a3

 y2

bk 

4

Shift Register 

 K  a4

b3

+ +  y1

b4

ak 

b4

bk 

+  y3

Modulo-2 Adder 

+ y4

yk 



K 

 y1  a1  b1

 y1 1

 0 if  a1  b1  1 if  a1  b1

 y1  a1  b1  0 if  a  b1 Fig. 2. Correlator 1  1 if  a1  b1

Fig 1.Overview of PN Sequence

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ISSN 2249-6343 International Journal of Computer Technology and Electronics Engineering (IJCTEE) Volume 1 , Issue 2 Fig.2 shows a correlator of length K. One sequence ai is For example, the polynomial 1 + xI4 + x15 means that the shifted through a K bit shift register and the output of each outputs from stages 14 and 15 are modulo- 2 added and fed stage is applied to a set of K XNOR gate for comparison.  back to the input of first stage of a 15- stage register to get a 215 - 1 length sequence, as shown in Fig. 4. D. Shift an d add 

B. Seri es-paral lel m ethod f or h igh -speed PN gener ation 

When a PN sequence is shifted and the shifted sequence modulo-2 added to the un-shifted sequence with an exclusive-OR gate, the result is the same PN sequence with some other shift. This is illustrated in Fig. 3, where a 15 bit PN sequence, a (k)), is arbitrarily shifted by4 bits to get, a (k - 4). The two sequences when modulo-2 added give a sequence which is a 3 bit shifted version, a (k - 3), of the original sequence a (k). Only when the PN sequence is modulo-2 added to itself without shift is the result a sequence of zeros [Li and Hykin (1995)]. A direct application of this property is in the generation of two identical sequences with a known large delay between them.

The maximum PN data rate depends on the type of logic device used. Since only one gate delay (due to the XOR gate) is introduced in the feedback path the maximum PN rate can  be close to the highest operating frequency of the shift register. The operating frequency of the PN sequence can be  pushed beyond the shift register clock rate by using a high-speed multiplexer. This technique uses the

‘subsequence’ property of a PN sequence already discussed [Gupta and Kumareshan (2005)]. As we can demultiplex a PN sequence into two similar sequences at half the rate, we can also multiplex two PN sequences to obtain a sequence at double the rate. However, the two sequences should necessarily have a phase shift of half the sequence length.Fig.5 shows the circuit for getting PN sequence of  length 27 -1 using series parallel method.

24-1=15 bits

Start

D3Q3

D7 Q7

D2Q2

D6Q6

D1Q1

D5 Q5

S1

S2

Start

D4Q4

Fig. 3. Addition of PN Sequence

III. GENERATION OF PN SEQUENCES

Fig. 5 Series-parallel method for high-speed PN generation

A. Usin g Shi f t Register wi th f eedback 

A PN sequence is generated using a shift register and modulo-2 adders. Certain outputs of the shift register are modulo-2 added and the adder output is fed back to the register. An N-stage shift register can generate a maximal length sequence of 2N-1 bits. Only certain outputs, or taps, can generate a maximal length sequence. The generator  output is expressed as a polynomial in 'x'.

C. Avoi din g the Zero State 

An N-bit register can generate 2N-1 states as against the 2N states of a binary counter, as shown in the state diagram in Fig. 6. Although the counter states generate an ascending or  descending sequence, the PN generator output states are apparently random. The all-zero state is missing in the PN sequence. This state is inhibited, because the generator  remains latched to it. The modulo-2 adder in the feedback  circuit feeds only '0's to the input. An additional circuit is needed to detect the 'all zeros' state and reset the PN register  to a valid state. When the register length, N, is small, a NOR gate can

decode ‘0’ outputs of the register, forcing a ‘1’,to the feedback input, through an OR gate, as shown in Fig.7a. A modulo- N down counter may be used for decoding the ‘all Fig. 4 A PN Generators with Polynomial 1 + xI4 + x 15 

zeros’ state in the PN sequence for large N. The PN data is applied to the load input after inversion as shown in Fig.7b.

157

1110

ISSN 2249-6343 International Journal of Computer Technology and Electronics Engineering (IJCTEE) Volume 1 , Issue 2  but a preferred maximal sequences can only produce Gold codes, as shown in Fig.8. Fig 11(e) & (f) shows the encrypted 1111 1111 0000 speech signal and its spectrum using Gold sequences. 1110 0111

1101

0001

1101

0011

0010 Up counter 

Down counter 

1010

0001

0101

1000

E. Bar ker Sequences 

0011

Barker sequences are short length codes that offer good correlation properties. A Barker code is a sequence of some finite length N such that the absolute value of discrete

0100

1010 1001 1000

0100

1011

0101

0111

autocorrelation function r│ (Ʈ) │ ≤ 1 for Ʈ≠ 0. Barker 

0010

0110

0110

sequences have many advantages over other PN sequences [Kumar et al. (2008)]. These sequences have uniformly low auto-correlation side these pseudo-random or pseudo-noise (PN) properties include, among other properties, (a) balance, (b) run and (c) auto-correlation lobes (≤1), but the size of  these families is small. Fig.9 shows Barker sequence generator.

1001

1100

Fig. 6 State diagrams for (a) binary counter and (b) PN generator

X1 X5

X2

X3

X4

XOR 

 NOR 

Fig .7 (a). A PN generator with ‘all zeros’ decoder

PN generator  (2 N-1) Fig. 9 Barker Sequence Generator

F . Kasami Sequences  Output

Kasami sequences are also PN sequences of length N = 2n-1, which are defined for even values of n there are two classes of Kasami sequences: (i) small set of Kasami sequences, (ii) large set of Kasami sequences. Small set of  Kasami sequences are optimal in the sense of matching

 N

LOAD

CO

Welch’s lower bound for correlation functions. A small set of 

CE

Clock 

Kasami sequences is a set of 2n/2 binary sequences [Kumar et al. (2008)]. Fig.10 shows Kasami sequence generator. Fig.7 (b). ‘All zeros’ decoding with a counter

D. Gol d Sequences 

Gold sequences are generated by the modulo-2 operation of  two different m-sequences of same length. Any two msequences are able to generate a family of many non-maximal  product codes, a4

a3

a2

a1

a0

g(D)=45

a'4 a'3

a'2

a'1

Small set of Kasami sequences are optimal sequences and have better correlation properties compared to Gold sequences. But the set contains less number of sequences. For  the shift register of length n the number of possible sequences n/2 for the small Kasami sequence set is only 2 sequences, n whereas Gold code set contains 2 + 2 sequences. The number  of sequences can be increased by making some relaxation on the correlation values of the sequences. The resulting set of  sequences is called large set of Kasami sequences [Kumar  et  al. (2008)].

a'0

Fig. 8. Gold Sequence Generator

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ISSN 2249-6343 International Journal of Computer Technology and Electronics Engineering (IJCTEE) Volume 1 , Issue 2 IV. COMPARATIVE ANALYSIS h( x)   x

6

x

5

LFSR satisfies all the properties of PN sequences but for  high degree of recursions it is computationally infeasible to evaluate the distance between the phase shifts. It has good autocorrelation but the sequence is not quite maximal length. With Series Parallel method, the rate of generation of PN sequence is at high speed and is used for doubling clock rate of PN generator. Performance of Gold code is good as compared to maximum length sequence and sample to sample correlation is reduced [Kumar  et al. ( 2008)].Small Kasami sequences have less autocorrelation and hence more cross correlation values but the number of sequences that can be generated are less. Thus the security provided by these sequences is less compared to Barker s equence [Kumar et al . (2008)].

1

h= [1,2, ......,0] an arbitrary nonzero vector 

h' ( x)   x6  x5  x3  x 2  1 h'  [d 0 , d 1 ,....d n1 ]

Fig.10 Kasami Sequence Generator

G. Correlation M easur es 

PN sequences of desired length are generated as described, and the MSAAC and MSACC measures are computed for the code set. Table.1 shows the correlation measures for PN sequences of length 16 bits and Table.2 is correlation measures for 32-bit PN sequences. From the results, among all PN sequences m-sequences have low MSAAC values since these sequences have single peak auto correlation function. But these sequences are not suitable for speech encryption since there is only one possible m-sequence of  given LFSR length. Gold codes have less MSAAC and MSACC values and for a given length of m-sequence one can generate more number of Gold codes so, these sequence effectively remove the intelligibility of the speech signal by de-correlating the speech samples. The MWH codes have  better auto correlation properties as compared to WH codes,  but they have poor cross correlation properties. OVSF codes with some specific repetitive sequences gives less correlation values. The MSAAC and MSACC values for OVSF codes with repetitive sequences {-1,-1,-1, 1} are less and these values are equal to the correlation values of MWH codes. Table 1: Correlation measures for PN sequences of length16 bits MSACC

Concatenated Shift Registers are computationally feasible and provides a large class of linearly concatenated shift registers to generate approximately maximally spaced phase shifts of PN sequence for use in pseudo random number  generation [Hurd and Welch (1997)]. In Chaos Based PN sequence the generation of pseudo noise is using digital signal processor which is used in secure communication [Guo and Wang (2010)]. It has good statistical properties and generates large set of PN sequence. It is used to randomize the signal over the links and provides PN sequence of higher rate. It is used to generate encryption key and its implementation is done using digital signals. Fig 11.showing time domain and its spectrum representation for 30 ms voice segment of the speech utterance using m-sequence, Gold code, WH codes, MWH codes and OVSF codes. Fig.12 to 16. shows a periodic auto-correlation function and cross-correlation function of Gold sequence, Barker-like sequence, large Kasami sequence of length 63 bits and MWH sequence, Gold sequence of length 64 bits.

Sequence

MSAAC

m-sequences

0.3467

WH codes

4.0625

0.7292

MWH codes

1.8125

0.8792

V. CONCLUSION

OVSF codes

1.8125

0.8792

We have shown the analysis of different PN sequences with different generation methods. A PN sequence generated by every method is analyzed to check if properties are satisfied. Advantages of every method of PN sequence is studied i n this  paper. PN sequences are used as spreading code. Correlation measures for PN sequences of length16 bits and 32 bits is given in Tables. We can extend analysis of PN sequences by giving different parameters which are helpful for different applications such as speech encryption and even generate PN sequence using elliptic curves over prime field.

Table 2: Correlation measures for PN sequences of length32 bits Sequence

MSAAC

MSACC

m-sequences

0.4807

WH codes

6.5938

0.7873

MWH codes

3.2188

0.8962

OVSF codes

3.2188

0.8962

Gold codes

0.6866

0.7451

Barker sequence

0.8127

1.0505

REFERENCES

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Fig 11 : Time domain and its spectrogram representation for 30 ms voice segment of the speech utterance , for original speech segment ((a)&(b)), for encrypted signal, using m  -sequences ((c)&(d)), signal using Gold codes ((e)&(f)), using WH codes ((g)&(h)), using MWH codes ((i)&(j)), and using OVSF codes ((k)&(l)).

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Fig.12: Aperiodic (a) auto-correlation function, (b) cross-correlation function, of Gold sequence of length 63 bits.

Fig.13: Aperiodic (a) auto-correlation function, (b) cross-correlation function, of Barker-like sequence of length 63 bits.

Fig 14. Aperiodic (a) auto-correlation function, (b) cross-correlation function, of large Kasami sequence of length 63 bits.

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(a) Fig.15: Aperiodic (a) auto-correlation function, (b) cross-correlation function, of MWH sequence of length 64 bits

Fig.16: Aperiodic (a) auto-correlation function, (b) cross-correlation function, of orthogonal Gold sequence of length 64 bits

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