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GENERATION AND VALIDATION OF GAUSSIAN NOISE USING RANDOM SEQUENCE

Thottempudi Pardhu Usha Rani Nelakuditi Suresh Pampana

Vignan University,[email protected] Vignan University,Vadlamudi [email protected] Research Centre IMARAT [email protected]

Abstract -In general many real time problems arerepresented with random variables. But a randomvariable is characterized by a Probability DensityFunction(PDF). As per the Central Limit Theorem anystochastic process handled with many random variablescan be converged as a normal distribution. Hencenormal distribution plays a key role in modeling realtime problems contains many random variables. Thispaper dealt with the generation of Gaussian PDF usinguniform random sequence. The generated PDF isvalidated in LABVIEW.

Key words Gaussian noise, PDF, RandomSequence, LABVIEW.

I. INTRODUCTIONGaussian Random Numbers(GRN) plays a criticalrole in simulating problems related tocommunications and financial modeling etc. Recentadvances in computing, demands simulationenvironments for the timely examination of problemson hand. Modern processors programmed toimplement a computational process can often reach arate of 10 8 outputs per second. The requirement togenerate extremely large numbers of gaussianrandom numbers elevates the importance of the

quality of the GRNG. GRNGs can be classified into four basic categories,cumulative density function (CDF) inversion,transformation, rejection, and recursive methods. TheCDF inversion method simply inverts the CDF to

produce a random number from a desireddistribution. Transformation methods involve thedirect transformation of uniform random numbers toa gaussian distribution. The third category rejectionagain starts with uniform random numbers and atransformation but has the additional step ofconditionally rejecting some of the transformedvalues. Recursion utilizes linear combinations of

previously generated gaussian numbers to producenew outputs. Another alternative classification is exact or

gaussian random numbers if implemented in an ideal environment. For example, the Box-Mullermethod subjects uniform numbers to varioustransformations in order to produce Gaussian outputs.If a perfect and infinitely precise uniform RNG wereused, and if the functions themselves were evaluatedwith infinite precision, perfect Gaussian randomnumbers would be produced. Approximate methods,

on the other hand, will produce outputs that areapproximately gaussian even if the arithmetic used is perfect. An example of this is the central limittheorem, which is only exact when an infinite numberof uniform random numbers are combined and somust be approximate in any practical implementation[2][3][4].

Any kind of noise such as gaussian noise,rayleigh noise, exponential noise and chi-squarednoise can be generated from uniform randomvariables. Uniform random variables are in turngenerated from random number sequences. Randomnumbers can be generated with the followingconditions.

1. The pseudo Random Number (PRN)sequence has a finite length N.

2. The PRN x(i), i = 1, 2, 3, . . . N, occurs onlyonce in the total population N.

3. The PRN sequence must be contiguous inthe range, i.e. there shall be no missing norduplicated number in N.

4. The sequence must not have periodicity inthe range.[5][6]

A random number sequence that a computer programgenerates that meets the conditions above is called aPRN sequence. Some commercially available

programs claim to generate random numbers but failto meet the conditions, especially condition(3),thecontiguity. Among many techniques of generatingPRNs, we choose the mixed congruential method.The mixed congruential method is compact, portable,andeasy-to-understand.Themixedcongruentialmethod is described by

equation1. xi+1=a.x i+c(modulo m)approximateExact methods would produce perfect where a: Multiplierconstant

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xi: Initial seed; c: Increment constant; m:Modulus constant, the total population of thesequence N, must be a prime number. The mixed congruential method is a versatile . In thismethod independent PRN sequences can begenerated by changing the constants a and/or cand the initial seed x i. A large number of independentPRN sequences would be obtained with the same

population. Mixed congruential method is employedin generating PRNs. The primary reason for selectingthis method is that it is a simple algorithm, portableand versatile. Random numbers can be generatedwith a contiguity and without duplication by usingthe steps given below.

1. Select a prime number m immediately larger than thedesired N.

2. Generate a PRN sequence by the mixed congruentialmethod with modulo m and the proper choice ofa, c and the initial seed x i.xi+1 = a.x i + c(offset by one) xi+1= (a.x i + c) - 1 (offset by zero)

3. Search and find the missing numbers and pair ofduplicated numbers, and correct PRN sequence.

4. Discard the highest number(s) and form the N random sequence that is contiguous in the range.

White Gaussian noise is generated from a pair of independent unit uniform random variables shown in equation 2 .

Gin(i) = 2 1 .co s2 . 2 --------------(2) Gqd(i) = 2 1 .sin 2 . 2

G in(i)and G qd(i) are the in-phase and quadrature phase Gaussian noise components respectively. The white implies that the noise is uniform across thefrequency spectrum. Gaussian or sometimesnormal refers to the amplitude distribution, the

probability density function of the random variables,or the frequency of occurrence in the noise stream.The noise power of the Gaussian distribution is given

by, noise power = mean-squared + variance . Gaussian random variables with a nonzero mean anda specified variance other than unity are generated

by, G(mean, var) = mean + . G(0.0, 1.0) The in- phase Gaussian noise component is shown inFigure1. The probability density function in the bargraph is attached to the right. We expect the bargraph to approach to a normal when the number ofsamples is increased.

Figure.1:Gaussian noise, G(mean=0.0, var=1.0). Gaussian distribution is given by

1 2 =

2 2 2

where, x = Gaussian random variable G in(i) orGqd(i). = standard deviation

2 = variance m = mean The paper is organized as follows. In section II

generation of random numbers is explained. SectionIII gives the detailed explanation of generation ofGaussian noise, section IV deals with theimplementation and section V gives Results andconclusion.

II. GENERATION OF RANDOMNUMBERS

Random numbers can be generated bydifferent methods like Linear Feedback Shift Register(LFSR), Tausworthe architecture, Boxmullertechnique etc. VLSI implementation , LABVIEWimplementation of random numbers is explained inthis section by Figure.2 and Figure.3.

start HDL program

Synthesis

.bit file generation

Download into FPGA

.cdc file generation

Verification in chipscope

End Figure.2: VLSI implementation of PRNG usingTausworthe Architecture.


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Figure.5 : Block diagram of mixed congruential method logic part 2

The Pseudo random number sequence isconverted to Guassian random variable using Box-Muller transformation.The block diagram of Box-Muller transformation logic is shown below inFigure.6 and Figue.7.

Figure 6: Block diagram of Box-Muller transform

logic part 1.

Figure.7: Block diagram of Box-Muller transform logic part 2

IV. IMPLEMENTATION

The algorithm for simulating of Gaussian PDF isas follows.

1. Initially Pseudo random number sequence of(PRNA and PRNB) is generated usingmixed congruential method.

2. This Pseudo random number is converted touniform distribution by sorting.

3. By using Box-Muller transformation asmentioned in eq(5.4) is applied to

pseudorandom number sequence with unituniform distribution for generating thegaussian random variables(u and v).

4. By calculating mean,variance of Gaussianrandom variable and using Gaussiondistribution,we simulate PDF of Gaussian noise.Flow chart of implementation of Gaussian noiseis shown in Figure. 8 as shown below.

START

GENERATE RANDOM NUMBER SEQUENCE PRNA,PRNB USING MIXED

CONVERT PRN TO UNIT UNIFORM DISTRIBUTION

USING SORTING

USE BOX-MULLER TECHNIQUE TO CONVERT

PRN TO GAUSSIAN VARIABLE

CALCULATE MEAN AND VARIANCE

SIMULATE PDF OF GAUSSIAN NOISE

END

Figure.8 Flow chart for generation of Gaussiannoise


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V. RESULTS & CONCLUSION

In this paper most useful Gaussian PDFgeneration and validationis implemented usingmatlab and Lab View. Figure.9, Figure.10 andFigure.11 will show the MATLAB result ofGaussian PDF, Xilinx result of Gaussian randomvariable, LABVIEW result of generatedGaussian Noise. It is generated from the randomsequence . The result is also validated againststandard exact Box-Muller method . It is

performed in Lab-View S/W.

Figure. 9 :PDF of Gaussian Distribution

Figure.10: Simulating the Gaussian randomvariable in Xilinx

Figure.11: Simulation result of PDF ofGaussian noise

CONCLUSION

The generated normal distribution is moreapproximated to the real mathematical function. Itcan be used in many applications. In this paper mixedcongruential method is used.

REFERENCES

1. AHRENS,J.H. ANDDIETER, U. 1972.Computer methods for sampling from theexponential and normal distributions.Comm. ACM 15, 10, 873882.

2. ANDRAKA,R. ANDPHELPS, R. 1998. AnFPGA based processor yields a real timehigh fidelity radar environment simulator. InMilitary and Aerospace Applications ofProgrammable Devices and TechnologiesConference.

3. C.Wei, C.He and H.Y.Qiu, "VLSIimplementation of universal rand numbergenerator," IEEE Trans. Very Large ScaleIntegr. (VLSI) Syst., pp. 465-470, 2002.

4. D.U.Lee,J.D.Villasenor,W.Luk,P.H.W.Leong,A hardwae Gaussion noise generatorusing Boxmuller method and its erroranalysis," IEEE Trans. Com put., vol. 55,no. 6, pp. 659-671, Jun. 2006.

5. Y.Cong, 1. Z x.Gan and W R.Qi, "Real-timegenerating Gaussian random noise based onCORDIC algorithm," Journal of

NanjingUniversity of Science andTechnology, vol. 30, no. 3, pp. 336-338,351,Jun. 2006.

6. AAlimohammad, S F. Fard, B. F. Cockburn and ( Schlegel, "A Compact and Accurate Gaussian Variate Generator," IEEE Trans. Very Large Scale Integr. (VLSI) Syst, vol.16, no.5, pp. 517-527, May 2008.

7. Thottempudi Pardhu, Usha Rani Nelakuditi,P.Suresh, Novel Random SequenceGeneration And Validation Using FPGA,ICCNASP 2013,V1, PP 295-298


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