Generation & Analysis of BPSK from Truncated PRN

Sequence

*Manisha Sharma & Neeru Agarwal

Department of ECE, ASET,

Amity University, Noida

India

*Email:

Abstract— In this paper the pseudo random noise sequence is

generated in Lab View software using 9 bit LFSR and then these

are truncated and then with different seed values, different

truncation bits, the change in the properties of the sequence are

also observed with mathematical and graphical analysis. Also

with both the normal and truncated PN sequences obtained

BPSK is also simulated and its power spectral density is also

obtained.

Keywords— PRN Sequence, Truncated PRN Sequence, peak

side lobes(rms), bpsk, power spectral density, seed value, taps,

LFSR, LabView software.

I. INTRODUCTION

Linear Feedback Shift Registers (LFSR) with one or more feedbacks from the output are used to generate the PRN sequences. For a n stage shift registers a sequence will be generated which will repeat itself after a length of L= 2^n-1. Performance can be affected by truncating last few bits of the normal PN sequence but sometimes it can be beneficial in terms of acquisition time and some applications. As length of 9

th stage PN sequence is 511 and that of 10

th stage is 1023, so

there is huge difference between these selections. Hence experiments are being conducted by selecting some length in between the large gap such that the properties of the resulting truncated sequence are preserved along with the acquisition time being reduced [1].

In this paper the normal p-n sequence along with the truncated sequence is being generated in LabView software using 4 stage and 9 stage LFSR . Also mathematical studies are conducted to compare their resulting autocorrelation and peak side lobe value (RMS) for different seeds. Truncated PRN sequences can show properties near to that of normal sequence for a particular seed value, which has many benefits as it can be used in communication and also 11 bit truncation from 511 bit sequence resulting in length of 500 will be much easier to handle for calculation purposes.

LFSRs are very much important for the generation of the PRN sequences, hence their models are also being extensively studied which can provide transition states of different bits of LFSR and is also capable to switch to any possible feedback connections i.e. polynomial [2]. Many fields in communication require pseudo random sequences like error

detection, direct sequence spread spectrum (DSSS), and these sequences are being tested for many applications like in the analysis of optical DPSK transmissions modeling [3]. The PRN sequences can be generated by numerous ways like it can be generated using algebraic feedback shift registers [4], series-parallel method to generate sequence at high speeds with low-speed devices, which interests hardware designers [5]. In mathematical terms it is the generator polynomial (primitive) of variable x that represents any LFSR to produce a maximal length sequence.

A. M-sequence :

A LINEAR SHIFT-REGISTER BINARY SEQUENCE WHOSE LENGTH

IS N= 2 M − 1, WHERE M IS THE DEGREE OF THE GENERATOR

POLYNOMIAL.

B. Primitive Polynomial :

It is the generator polynomial of m-sequence. If g(x) is a

primitive polynomial of degree m and if the smallest integer n

for which g(x) divides x^n + 1 is n = 2^m − 1.

g (x) = x^5 + x^4 + x^2 + x + 1 is a primitive. But

g( x ) = x^5 + x^ 4 + x^3 + x^ 2 + x + 1 is not primitive as

x^6 + 1 = ( x + 1 )( x ^5 + x^ 4 + x^ 3 + x^ 2 + x + 1 ) ,

& hence least value of n is 6.

II. GENERATION OF NORMAL & TRUNCATED PRN SEQUENCE

A. Generation of Normal PRN Sequence:

In this paper simulation model is created in LabView to

generate the sequences. Following is the block diagram of 4

stage PRN sequence.

Figure 1: Block Diagram of 4 stage PRN sequence Generator.

This resulted PN code is shown in figure 2 & resulted

waveform is depicted in figure3.

Figure 2: Resulted PN code

Figure 3: Resulted Waveform

In this generation, for a given length of shift register, the mode

to generate pseudorandom binary sequences can be done

either by using EXOR gates or EXNOR gates. Here we have

implemented this using EXOR gates on the block diagram of

the virtual instrumentation. The front panel is representing the

code and the waveform is generated respectively. The parallel

output can be observed either on LED indicators or in

addition, a pseudo-random sequence of ones and zeros can be

produced at Serial Out. Similarly a 511 length PN sequence

can be generated using 9 stage shift register [6]. In this a nine-

element shift register is placed on a While Loop. An EXOR

gate is used whose inputs have been wired to Q5 and Q9. The

loop index keeps track of the count of loop cycle, and it stops

when the output becomes equal to the initial value. An initial

seed is set at starting of the process and each shift registers on

the loop are initialized [6]. Following is the resulting

waveform of 511 length PRN sequence.

Figure 4: Resulting Waveform of 511 length PRN sequence.

This sequence satisfies all the properties of a normal PN

sequence like balance, run and autocorrelation properties.

B. Generation of Truncated PRN Sequence:

A truncated sequence of 500 bits length can be generated by

removing last 11 bits from the above sequence which in this

simulation is achieved by using a 'delete from array' block. In

this block we can delete any number of last elements of the

initial array.

Figure 5: Truncated Sequence

III. MATHEMATICAL ANALYSIS

Observations are being made by varying the seed values and seeing their effect on the different amount of truncation of bits. Example: 11, 31, 51, 101, 151, 201, 301 etc. This is shown in table 1. This analysis shows how the root mean square value or the peak side lobes generation is being affected as we change seed values for different number of bits being truncated from the end of the normal sequence .

In the second table observations are carried out such that as the truncation is increased with respect to the normal PRN sequence the performance is affected i.e. The RMS values with respect to some seed values taken into consideration (it gives the nutshell of the previous analysis). As it is observed that for different seed values there is not much variation in the slope of the different truncation with respect to the normal 511 length sequence. Also the dB plot is shown below using MATLAB tool. Figure 6 is showing Matlab plot that with increase in truncation with respect to the normal sequence RMS values increases for each seed value but there in not much variation in slope as seed changes. Fig. 7 showing the dB plot of the same observation.

0 50 100 150 200 250 300 350

0.04

0.045

0.05

0.055

0.06

0.065

0.07

Truncation

RM

S V

alu

e

000001010

000010100

000011110

000101000

000110010

000111100

001000110

001010000

001011010

001100100

FIGURE 6: Matlab Plot

0 0.5 1 1.5 2 2.5 3 3.5 4

0.04

0.045

0.05

0.055

0.06

0.065

0.07

TRuncation in dB

RM

S V

alu

e

000001010

000010100

000011110

000101000

000110010

000111100

001000110

001010000

001011010

001100100

Figure 7: dB Plot

IV. GENERATION OF BPSK

From both the normal and truncated PRN sequences we

simulated the BPSK signal and observed their respective

power spectral densities.

FIGURE 8: Block for BPSK Generation

The above block diagram the BPSK is simulated as, phase of a

carrier (a selected signal from waveform generator) is

converted to two values according to the binary signal level.

The information of the stream is contained at the point where

phase changes occur in the transmitted signal.

V. RESULTS & DISCUSSIONS

One random data stream (A) is selected from square wave

generator and then the normal PN sequence is multiplied with

that data stream resulting in a sequence in (D). Finally BPSK

is obtained from this sequence and one sinusoidal carrier

signal with changing phase at the transitions (E). Power

spectral density for the resulting BPSK is plotted in graph (F).

Same procedure is followed with the 11 bit truncated PRN

sequence and its PSD is also plotted (I). It can be clearly seen

from both the spectral densities that as bits are truncated the

spectral performance goes poor resulting in more side lobes.

A. Selected Data Stream:

B. Sinusoidal Carrier Signal:

C. Normal PN sequence of length 511 bits:

D. Sequence Generated on multipling PN sequence with data:

E. Generated BPSK:

F. Power SpectralDensity (PSD):

G. 11 Bit Trancated PN Sequence:

H. Generated BPSK:

I. Power Spectral Density (PSD) of Trancated PN sequence:

.

VI. CONCLUSION

In virtual instrumentation simulation environment the pseudo

random noise sequences are simulated along with the

truncation by different bits. This made us to observe the

comparison between the amount of truncation increases the

peak side lobe level also increases but does not vary much for

different amount of truncation of bits. Then with the both

sequences BPSK signal is generated and its respective power

spectral densities are also plotted and it is observed that as we

truncate the sequence the PSD expands and side lobe levels are

also increased leading to change in system performance.

ACKNOWLEDGMENT

Manisha Sharma, is highly thankful to Prof. (Dr) P Bannerjee & Prof (Dr) M.K.Dutta, Department of ECE, ASET, Amity University Noida, India, for their valuable support.

REFERENCES [1] Banerjee P, Keshwala U & Kaushik M, “Study on Potentiality of

Truncated PRN Sequences for Communication”, International Conference on Communications, Devices & Intelligent Systems, 2012, pp 409-412.

[2] Ahmed A & Abri D, “Design of a Pseudo-Random Binary Code Generator via a Developed Simulation Model”, ACEEE Int. J. on Information Technology, Vol. 02, No. 01, March 2012, pp 33-36.

[3] Hadjia Badaoui, Yann Frignac & Mohammed Feham, “Pseudo Random Binary Sequences Analysis for the Modeling of Optical DPSK Transmission Systems”, International Journal of Computer Science & Communication, Vol. 1, No. 2, July-December 2010, pp. 369-372.

[4] Mark Goresky and Andrew Klapper, “Pseudo-noise Sequences based on Algebraic Feedback Shift Registers”, IEEE Transaction on Information Theory, VOL. 52, No: 4, 2006, pp 1649-1662.

[5] R.N. Mutagi, “Pseudo noise sequences for engineers”, Electronics & Communication Engineering Journal,1996, pp 79-87.

[6] Goran S. Miljković, Ivana S. Stojković & Dragan B. Denić, “Generation and Application of pseudorandom binary sequences using virtual Instrumentation”, Automatic Control and Robotics, FACTA UNIVERSITATIS, Vol. 10, No 1, 2011, pp. 51 – 58.

TABLE I

SEED

VALUES

RMS value of

11 bit

truncated PRN

seq.

RMS value of

31 bit

truncated PRN

seq.

RMS value of

51 bit

truncated PRN

seq.

RMS value of

101 bit

truncated PRN

seq.

RMS value of

151 bit

truncated PRN

seq.

RMS value of

201 bit

truncated PRN

seq.

RMS value

of 301 bit

truncated

PRN seq.

000001010 0.0370043 0.0382634 0.0394002 0.0423284 0.0458956 0.0504004 0.065412

000010100 0.0370043 0.0383088 0.0394696 0.0422886 0.0459451 0.0503523 0.064847

000011110 0.0366342 0.0377456 0.0388889 0.0421593 0.0464451 0.0513363 0.0638377

000101000 0.0370917 0.0380528 0.0393035 0.0417395 0.0457326 0.050491 0.0641826

000110010 0.036472 0.0376255 0.0387198 0.0424381 0.0456264 0.0503924 0.064827

000111100 0.0364395 0.0375851 0.0389856 0.0423833 0.046397 0.0509999 0.0640273

001000110 0.0369723 0.0379933 0.0393229 0.0426934 0.0462311 0.0502802 0.0641556

001010000 0.0369914 0.0379713 0.0392606 0.0417771 0.0457438 0.0504031 0.063858

001011010 0.036906 0.0380214 0.039337 0.0425284 0.0456424 0.0501757 0.0638987

001100100 0.0365111 0.0375552 0.0386438 0.0418813 0.0455821 0.0505496 0.0638105

001101110 0.0368408 0.0378448 0.03927 0.0418813 0.0460394 0.051276 0.0627224

001111000 0.0367302 0.0381108 0.0393009 0.0423381 0.0457495 0.0506745 0.0645458

TABLE II

seed-> 000001010 000110010 001100100 010010110 011001000 100000100 100110110 101101000 110011010

Tprn/prn

500/511 0.037004 0.03647 0.036511 0.036879 0.036549 0.03693 0.036717 0.0367 0.03653

480/511 0.03826 0.0376 0.03755 0.037797 0.037579 0.03797 0.038028 0.03815 0.03754

460/511 0.039400 0.03872 0.038644 0.038801 0.038645 0.0393 0.0392 0.03902 0.03861

410/511 0.0423 0.04244 0.0418 0.0422 0.041904 0.042422 0.042179 0.04222 0.04201

360/511 0.045895 0.04563 0.045582 0.046708 0.045886 0.045893 0..045836 0.04616 0.04602

310/511 0.050400 0.05039 0.05055 0.05024 0.050701 0.049961 0.050568 0.05036 0.05102

210/511 0.065412 0.065483 0.0638 0.0652 0.063661 0.064927 0.064767 0.06409 0.06349