AD-AO98 306 WEAPONS SYSTEMS RESEARCH LAB ADELAIDE (AUSTRALIA) FIG 20/1AN EXTENSION OF A PREVIOUS RESULT ON THE AN4ALYSIS OF A METHOD 0--ETChl

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DEPARTMENT OF DEFENCEDEFENCE SCIENCE AND TECHNOLOGY ORGANISATION

WEAPONS SYSTEMS RESEARCH LABORATORY

X DEFENCE RESEARCH CENTRE SALISBURY

M SOUTH AUSTRALIA

Cz TECHNICAL MEMORANDUMWSRL-0106-TM

AN EXTENSION OF A PREVIOUS RESULT ON THE ANALYSIS OF

A METHOD OF MEASURING THE SIGNAL TO NOISE RATIO OF

A SINUSOID IN NOISE

A.P. CLARKE DTICELECTEM

APR291981I TN U 7T - S1 AT E2 A T IO N A L d

--C:<!"C'AL INFORMATION SERVICE WPcVWO uCE ANO SELL THIS REPORT

LAM.Technical Memoranda are of a tentative nature, representing the views of theauthor(s), and do not necessarily carry the authority of this Laboratory.

Approved for Public Release.

COPY No. 24 SEPTEMBER 1960

81 4 29 I

J

The official documents produced by the Laboratories of the Defence Research Centre Salisburyare issued in one of five categories: Reports, Technical Reports, Technical Memoranda, Manuals andSpecifications. The purpose of the latter two categories is self-evident, with the other three categoriesbeing used for the following purposes:

Reports : documents prepared for managerial purposes.

Technical records of scientific and technical work of a permanent value intended for otherReports scientists and technologists working in the field.

Technical : intended primarily for disseminating informatinn within the DSTO. They areMemoranda usually tentative in nature and reflect the personal views of the author.

Ko

UJNCLASSIFIED

AR-002-031DEPARTMENT OF DEFENCE

DEFENCE SCIENCE AND TECHNOLOGY ORGANISATION

WEAPONS SYSTEMS RESEARCH LABORATORY

P 1 ?'lPZ

(~j EHNICAL MEMO.

(I AN EXTENSION OF A EVIOUS RESULT ON THE ANALYSIS OF AMETHOD OF MEASURING THE_5.IGNAL-TO-NOISE RATIO OF A

SINUSOID IN NOISE.

o"//~~A.P./Clarke /

SUMMARY

This paper presents the derivation of closed form expressionsfor the probability density function and the associated meanand variance when estimating tonal signal-to-noise ratio froma power spectrum obtained as the average of a set of powerspectra. Computer programs are described that can aid inexperimental design and some possible lines of examinationare indicated.

POSTAL ADDRESS: Chief Superintendent, Weapons Systems Research Laboratory,Box 2151, GPO, Adelaide, South Australia, 5001.

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Other c. Summary inNumbers: Isolation: Unclassified

3 ITITLE JAN EXTENSION OF A PREVIOUS RESULT ON THE ANALYSIS OF A METHOD OF MEASURINGTHE SIGNAL TO NOISE RATIO OF A SINUSOID IN NOISE

4 PERSONAL AUTHOR(S): 5 DOCUMENT DATE:

CSeptember 1980

A.P. 6.1 TOTAL NUMBER

OF PAGES 19

6.2 NUMBER OFREFERENCES: 4

71 7.1 CORPORATE AUTHOR(S): 8-RNCENUMBE

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13 r ANNOUNCEMENT UMITATIONS (of the information on these itges):

No limitation

14 ERPOS Sga to noise ratio Variance (statistics) 15 COSATi CODES:Analyzing Power spectra

a. EJC Thesaurus TransmissionTerms Computer programs 2001

Density functionsMean

b. Non-ThesaurusTerms

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17 SUMMARY OR ABSTRACT:(if this is security classified, the announcement of this report will be similarly classified)

This paper presents the derivation of closed form expressions for theprobability density function and the associated mean and variance whenestimating tonal signal-to-noise ratio from a power spectrum obtainedas the average of a set of power spectra. Computer programs aredescribed that can aid in experimental design and some possible linesof examination are indicated.

Ava il C9*

Security classification of this page: UNCLASSIFIED

WSRL-0169-TM

TABLE OF CONTENTS

Page No.

1. INTRODUCTION I - 2

2. THE CHARACTERISTIC FUNCTION OF SIGNAL PLUS NOISE POWER 2 - 3

3. THE PROBABILITY DENSITY FUNCTION OF SIGNAL PLUS NOISE POWER 3 - 4

4. THE PROBABILITY DENSITY FUNCTION OF THE NOISE POWER 4

5. THE PROBABILITY DENSITY FUNCTION FOR THE SIGNAL-TO-NOISE RATIOESTIMATOR 4 - 6

6. THE MEAN AND VARIANCE OF THE SIGNAL-TO-NOISE RATIO ESTIMATOR 7 - 8

7. AN ALTERNATIVE DERIVATION 9 - 10

8. SOME PRACTICAL RESULTS 11

9. CONCLUSIONS 11

10. ACKNOWLEDGEMENT 11

REFERENCES 12

LIST OF APPENDICES

I MATHEMATICAL DETAILS 13 - 14

II COMPUTER PROGRAMS 1s

LIST OF FIGURES

1. Variation in the probability density function for the estimate ofsignal-to-noise ratio from an average of N spectra using 10 noise binsat an input signal-to-noise ratio of 5 dB

2. Variation in the probability density function for the estimate ofsignal-to-noise ratio from an average of N spectra using 10 noise binsat an input signal-to-noise ratio of -5 dB

3. Deviation of the mean when using N spectra to estimate signal-to-noiseratio for a 1 Hz bin width and using 10 noise bins for estimating noisepower

4. A demonstration of the effect of using N spectra to estimate signal-to-noise ratio for a 1 Hz bin width and using 10 noise bins for estimatingnoise power

-1- WSRL-0169-TI

1. INTRODUCTION

The statistical properties of estimators of signal-to-noise ratio of acoustictones are needed in ongoing studies of acoustic signal processing (Task No.DST79/069 - Signal Processing for Underwater Detection) in areas as widely separatedas measurement of acoustic transmission loss and signal detection theory. In aprevious paper by the author(ref. 1) an expression was derived for the probabil-ity density function of an estimate of the signal-to-noise ratio of a sine wavein Gaussian noie. The estimate was obtained from the power spectrum of asampled data sequence. The analytic results in reference 1 have been used toobtain functional forms for the coefficient of variation of signal-to-noise ratioestimates derived from data obtained from several sea-going experiments. Theagreement between analysis and experiment was found to be excellent.

Reference 1 did not address the situation where the spectrum used for estimationwas obtained as the average of a number of spectra derived from consecutivesequences of data samples. This present paper gives a solution to this problemby using a characteriszic function approach with the aid of some integrals presentedin Appendix I. The problem is to find the distribution of z

N

1 z xi

z = 1, -1 (1)r p N

p N

j=1 i=1

where x. is the power in the signal bin for the i-th spectral estimate and yij

is the power in the j-th noise bin for the i-th spectral estimate.

The noise bin must not include the signal bin or contain extraneous signalresidues.

N is the number of spectra from which an average spectrum is obtained.p is the number of noise bins in the average spectrum used to estimate noisepower

1- is the bin width in Hertz.r

Equation (1) can be rewritten as

NSxi

1 (2)

p N

jlij

j =I i=l

WSRL-0169-TM -2-

Before proceeding with the development of the solution, some results of a standardnature are summarised from reference 2 in order to aid in reading this paper, viz:

(1) The characteristic function of a random variable x with probabilitydensity function f(x) is given by

00

(t) = dx exp(jtx) f(x) (3)

(2) The characteristic function of the sum S of N random variables whose

individual characteristic functions are 0(t) is

0s(t) 0 (t) N (4)

(3) If 0(t) is absolutely integrable over the range (-m, o) then0o

f(x) - 1 dt exp(-jtx) 0(t) (5)

2. THE CHARACTERISTIC FUNCTION OF SIGNAL PLUS NOISE POWER

For a single spectral estimate the probability density function of the signal +noise power is given by equation (7) of reference 1. Note that the nomenclatureis the same, ie:

f(x) = A2 exp I-A 2 (x + K2 )I 10 (2A2 iv/x), x > 0 (6)

= 0, x< 0

where 1o(.) is the modified Bessel function of zero order. The characteristic

function of this density function is then, after a simple rearrangement, given by

*(t) = A' exp(-A2 K2 ) f dx exp -(A2 - it) x Io(2A2 Kv ) (7)

0

By making the elementary substitutions

a = A2 exp(-A2 K2)

= A2 -jt(8)

y = 2j A2 K

-3- WSRL-0169-TM

and using the relation 10 (p) = Jo0(j) the characteristic function can be written

as

00

@(t) = 2a f dP P exp(-0A 2 ) jo0(71A) (9)

0

Applying the formula 1.1 from the Appendix I gives

¢(t) = exp(- ) (10)

ie

@(t) = A 2 + t exp(-A2 K2 ) exp -A A K2 A2 +1jtl)IA

4 + t 2

3. THE PROBABILITY DENSITY FUNCTION OF SIGNAL PLUS NOISE POWER

The density function derived in this section is for the summation in the numeratorof equation (2).

For the sum of N terms each distributed as in equation (6)

€sum = (t) N = N exp (12)

.... from equation (10)

Hence the probability density function for the sum is

_ ~ ,a N _MY2 \f(x) - 1 j dt exp(-jt x) exp 4p (13)

.... from equation (5)

As P is a function of t, viz j = A2 - it, the integral can be rewritten in termsof ] to give, after some rearrangement:

A 2 _j00

2 2f (x) = Aa-. exp (-V. x) j 8 exp~xBA (14)

A2 + jo

Using the formula 1.2 in the Appendix I gives, after some algebra:

f(x) A x .(N-l)/2 exp -A2 (x + NK2 ) JN_(2jA2 IA"i).N-1 NK2)

WSRL-0169-TM - 4 -

The Bessel function of the first kind of order n is related to the modifiedBessel function of order n by

In (X) = (_j)n jn (jx)

Hence

f(x) = A2( 1 2 (N- 1 )/ 2 exp I-A 2 (x + NK2 ) INI(2A2 K Vii (15)

4. THE PROBABILITY DENSITY FUNCTION OF THE NOISE POWER

From reference 1, equation (8), each term yij in the denominator of equation (2)

is distributed as X22 1/A *

Hence, using the well-known addition theorem(ref.21 for X2 variates, the doublesumation term in equation (2) is distributed as X 2pN, /A - ie :

if P N

j=l i=1

then

A2pN. 2 PN. ypN-1 exp(-A2y)

fU(y ) = 2 pN. r(pN)

(16)

SA2pN. ypN-1. exp(-A2 y)

r(pN)

5. THE PROBABILITY DENSITY FUNCTION FOR THE SIGNAL-TO-NOISERATIO ESTIMATOR

The probability density function for the ratio of the summation terms inequation (2) can now be obtained by applying the classical relation given inParzen(ref.2) for the density of the ratio of two independent positive randomvariables, viz:

f= I x, X()fy(x) dx (17)

0

S - WSRL-0169-TM

ie

( (N-)/22f dx A' exp(-A2 (yx+NK

(18)

eNrm (2A2K V ) A 2p N xpN =l exp(-A 2x)x r(pN)

Gathering terms independent of x gives

A2pN+ 2 (N-I)/2 exp(-NA2 K2) (19)r(pN) (p

The integral rearranges to

00

f dx xpN+(N -I) / 2 exp(-A2 x(y+l)) IN_1(2A 2IfA ) (20)

0

As stated previously a simple relation holds between the modified Bessel functions

and the Bessel functions of the first kind, ie:

-kIk(x) = JkOiX) (21)

Hence the integral becomes

00

dx xp N + ( N =I ) / 2 exp(-A 2x(y+l)) 3 JN_1(2A2Kj Vf) (22)

0

With the substitution pL2 = x the integral becomes

00

2j "N+I f d P u2PN+N exp(-A2 (y+l) p2) JN-I(2A K ,/y ,A)

0

00

2j - N + I d P mal exp(-_p 2 p2) jn- l (y A) (23)

0

where the substitutions

WSRL-0169-TM - 6-

a = 2pN+N+l

13 = A(y+l) 2

y = 2A' KjVR-

are distinct from the use previously made in manipulating equation (7).

A simple application of the formula given by equation I.1 in the Appendix Ienables the integral (22) to be evaluated to be evaluated to give

2 jN+l 1.a+n-l. N-1 a+N-I 2

______2 40)_ MC_ N, - (24)

r(N) 2N 9a+N-1

Substitution for a, 0, y gives, after some rearrangement:

r(pN+N) KN- I N(N- )/ 2 y (N-1)/2 M(pN+N,N, NA2K2 (yl)) (25)y+l

r(N) A2pN+2 (y+l)pN+N

Forming the product with (19) gives, after some algebra:

N-1= exp(-NA2 K2 ) y- M(pN+N,N,NA2 K2 (-)) (26)f (pN,N) (y+l)pN+N

where 9(...) is the bivariate f-function.

The distribution of z as defined by equation (2) then follows from

fz(z) rfrz+l (27)

where z = p y - I from equation (2).r r

The above is simply derived from the classic relation(ref.2)

f (Y) (-y--) (28)aX+B a X a

r ppN e-NA2K 2 (rz+l) N-1 M(pN+N,N,NA (rz+l)) (29)Sie f z(Z) = p N

1(pN,N) (rz+p+l)pN+N

A simple consideration of equation (2) indicates that z will only take values in

1 <z<r

-7 - WSRL-0169-TM

6. THE MEAN AND VARIANCE OF THE SIGNAL-TO-NOISE RATIO ESTIMATOR

The mean and variance of z follow readily from the moments of y, where y is theratio of the summation terms appearing in equation (2). The moments of y can bederived from the density equation (26), ie:

n+(N-l)

Efynl - exp(-NA2 K2 ) y 2 K2 (30

P(pNN) d (Y l)pN+N M(pN+N,NNA (30)

0

A se'ies expansion for M(.,.,.) written in the form00

M(a,b,z) = ab) r(a+r) zr (31)=ra rb r- r!1

r=O

can be substituted in the above equation to give, after reversing the order of

integration and summation and using the F-function form of the 0-function:_ = n+N-l4 r

EI, exp(-NA2 K2 ) 00 (pN+N+r) (NA2K2)r f y (32)

E[ - (pN) L r(N+r) rl f dy (y+I)PN N+r

r=0 0

As shown in the derivation 1.3 in the appendix the integral is simply the9-function

0((n+N+r), (pN-n))

This 0-function can be written in terms of appropriate F-functions and after arearrangement of terms the n-th moment of y can be written as

E[yn] = exp(-NA2K2 ) P(2N-n,N+n) M(N+n,N,NA2 K2 ) (33)P(pN, N)

Hence

E[ y] = N exp(-NA2K2) M(N I,N,NA2 K2 ) (34)pN-1

A simple application of the formulae 13.4.1 and 13.6.12 in Abromawitz and Stegun(rcf.3) gives

El y = N (I+A2 K2 ) (35)pN-1

A similar application of the same formulae enables the second moment of y to bewritten

NE[ y' I = (pN-l) (pN-2) (NA2 K2 +N+2) (l+A2 K2 ) -i1 (36)

~I

WSRL-0169-TM - 8 -

From equation (2)

E[z] = E[y] - 1 (37)

r r

and

var(z) = -5 E[.I - E2[y1J (38)

Hence precise expressions can be written for the mean and variance of z, ie:

E[z] = pN(1+A2 K2) 1 (39)= r (p-) -r

and

2 14+2N pN+N-l)A2K2 + (pN2+N2-N) (0var(z) r4 L! 4 (pN-1)2 (pN-2) (40)

And, the coefficient of variation defined as

C = a(41)E[ z] C1

where a is the standard deviation can be written down immediately.

Substituting N=1 in equations (39) and (40) gives

E[z] - p(1 A2K2 ) - (42)r(p-1) r

var(z) = -P,2 A4K* + 2pA2 (4p-2 (43)

and after some algebra

p A4 K4 + 2pA2K2 + p

C = pA2 K2 + I p-2 -3 (44)

These are precisely the respective equations developed in reference 1 and hence

equations (42), (43) and (44) supply a partial validation of the analysiscontained in this memorandum.

~L,

-9 - WSRL-0169-Tn

7. AN ALTERNATIVE DERIVATION

On realising (note the acknowledgement at the end of this paper) that the quotientin equation (1) is the ratio of a non-central chi-squared variate (with 2N degreesof freedom) to a chi-squared variate with 2pN degrees of freedom ie the quotienthas the distribution of a non-central F distribution, it is possible to derivethe formulas equations (29), (38) and (40) as particular cases of a more generalformula. The non-central F with P"1 degrees of freedom in the numerator andP2 degrees of freedom in the denominator has a probability density function

fy Y)= exp(-X/2) P_ 111/2 i'1/2 - 1I F~22) P200

x ~~+ Pzy(VI +P2 )/ 2 X j r7/2l+ 2+j

j=0

x (45)

where X is the non-centrality parameter. This equation can obviously be writtenas

exp(-X/2) :1 v! /2 ' If2

f (y) = rcv 2/2) UZ

xl + P, z -(vi + P2)/2 r((v, + P2)/2) xS V--2 r(,/ /2)

Mv 1 + v2 , P ZXP _ (46)2 2 2P2 + 2VI Z)

On making the substitutions

V, = 2N, V2 = 2pN, X = 2NA2 K2

and carrying out some elementary algebra

- 1 pN ' K2

f(y) exp(-N A2 K2 ) yN - I pN M(pN + NN, p y (47)

P(pN,N) (y + p)pN + N

In this alternative derivation z = 1y - 1r r

WSRL-O169-TM1 - 10 -

Hence

fz(Z) = rfy(rz + 1)

as a result of applying equation (2)

pN (N 2 K)N-I Cr z lie fz(z) exp(-NAK ) (rz + 1) M(pNN,N,NAK 2 Lrzp+lj )

0(pN,N) (rz + p + 1)pN + N

which agrees with equation (29).

The nth order moment of the non-central F-distribution to be derived fromequation (45) involves evaluating an integral

00 dy yn yvP/22' y

f PlIY) (Vi + P2)/2 Vl (48)V2 P

0

The integral can be written as

00

I dz (49)+ (1+az) C2

0

wherevI .I ' 2 Vl

C, = n + j+ - -1, C2 = 2 + j, a =P

On making the substitution az = A the integral becomes

a00 1

a C + f A I)C2

0:4

- 1

aC, + 1 P (C2 - C, -1, CI .+ 1) (SO)

On substituting this integral in the appropriate expression for E[y n] and carryingout some algebra

E y = exp(-X/2) 2.'> n I + n,2 [,/ - ) I(. ,)TI- rPv2) r( P,)

- 11 - WISRL-0169-Th

Using the substitutions v, = 2N, P2 - 2pN, X - 2NA2 K2 for values of n of 1and 2 in this expression then enables E~z] and E{z 2 ] to be derived from

1 1Z = I y - I

r r

Although not carried out in detail the results are easily shown to be identicalto equations (39) and (40).

8. SOME PRACTICAL RESULTS

In order to give some idea of how to use the formulae, two steps were taken.First, a program (program A in Appendix II) was developed to examine the variationin the form of the density function as the number of spectra used to obtain anaverage spectrum is increased. Results are shown in figure 1 for a moderatelyhigh input signal-to-noise ratio of S dB, using 10 noise bins and a bin width of1 Hz, ie p=lO, r=l. It is surprising that even at this signal level the spreadin a number of estimates can be high. This spread should be compared with theresults in figure 2 for a much lower input signal-to-noise ratio of -S dB. Thecomparison indicates a necessity for a further detailed examination.

The second step taken was to develop a program (program B in the appendix) toexamine the variation with signal-to-noise ratio of the mean, standard deviation,and coefficient of variation of the estimate for a specified number of bins usedto estimate noise power. Typical results for the mean and coefficient of vari-ation are shown in figures 3 and 4 respectively, using p=l0 and r-l as in figure 1.In figure 3 the bias (due to the assymetry of the distribution) is clear at evenhigh signal levels. At low signal levels it is obvious that it is necessary toaverage a large number of spectra to minimise the bias in the mean. In figure 4the behaviour in the coefficient of variation indicates the large number of spectrato be analysed to keep the frequency of occurrence of negative estimates ofsignal-to-noise ratio to a minimum. This need is of course going to conflictwith the non-stationarity of the statistics of any physical medium in which anexperiment is taking place such as when measuring acoustic signal transmissionproperties in the ocean.

9. CONCLUSIONS

This paper presents a rigorous mathematical analysis and a brief look at someexperimental implications of a definition and associated measurement techniqueof the signal-to-noise ratio of a sine wave in white noise. The computerprograms developed as a result of this analysis can be used to carry out adetailed examination of any proposed experimental scenario. During the develop-ment of the analysis it has appeared that the probability distribution functionfor both signal plus noise and signal-to-noise ratio are both amenable to ananalysis that extends techniques reported in this paper and it is proposed topublish results on this shortly. It is anticipated that this further work willthrow some interesting light on some of the problems of detection theory.

10. ACKNOWLEDGEMENT

I would like to register here my gratitude to Dr.D.A. Gray for pointing out theconnection between the sought-after result and the non-central F-distribution.This has enabled the analysis in Section 7 to be presented as an ideal verificationof the earlier portion of the paper.

WSRL-0169-TM - 12-

REFERENCES

No. Author Title

I Clarke, A.P. "On the Distribution of Signal-to-NoiseRatio when Estimated from a PowerSpectrum".Technical Memorandum WSRL-0139-TM, 1980

2 Parzen, E. "Modern Probability Theory and itsApplication".John Wiley and Sons, 1960

Abromowitz, M. and Handbook of Mathematical Functions

Stegun. I.A. Dover Publications, 1970

4 Watson, G.N. A Treatise on the Theory of BesselFunctions; CUP, 1944

- 13 - WSRL-0169-TM

APPENDIX I

MATHEMATICAL DETAILS

1.1 A general integral formula first due to Hankel(ref.4)

I dt t. - exp(-p2t2 ) JN( a t) r + p 2 *4a2N- M(U P_ + 1~~, T2

0 F(P + 1) 2" 1 P 2

Iarg p I < I, Re(O + v) >0

If the substitutions a =, ' = 0, A 2, p = 410 are made then

dt t exp(-13t 2 ) J (7t) I MQI_1,

0

1 _9/- exp (7)

from applying formula 13.6.12(ref.3).

1.2 An integral formula(ref.4), generally attributed to Sonine, for the ordinaryBessel function is

J (z) = NZv dt1+ exp(t -z2

JC+jOO

where the path of integration is the straight line RI(t) - C > 0

If the substitutions V+1 = N, t x3, z2 NY 2x, C - A are made, then

Jv-1i (7v NX) = ? N-1 N(N-1)/ 2 A2 N /J x(N-1)/2 f 0 eN

A2 -j--

1.3 0-functions can be written in the integral form

0(p,q) = /(y+K)P+q

0

WSRL-0169-TM - 14 -

Make the parameter substitutions

q = n+N+r

p = pN - n

Then

p + q = pN + N + r

Hencen+N-l1+r

dy Y =N- j(n+N+r,pN-n)

(y+l) pNN+r0

as i(p,q) 1f(q,p)

- IS - ISRL- 0169-TM

APPENDIX 11

COMPUTERh PROGRAMN

61636 CXSS EUATE SiNO PIp FOR ,sua:0064 C 1/101.WREOC N SPEC?" (NX.)4006 C P NOISE $INS 00C.)00070 C S SIGNL TO NOISE 07T1009)

"on3 FUNCTION CNSF(A,CZ)"too S-1."tie Ye.

*9136 to RN.-1"144 vvxQ/Ms(ZN3A+RN)/(C*WI)

$41to IF(M.LE.IW.*MD:.:Jc.SL) 60 TO It

m ETURN"all END

0230 REAL N.PS,Z(32G),LDEN(32V).DNC32*)6240 Z(1)-1.

"m0 IRITE(6,100)$we7 READ(SE) NP.S

042" DO If L-t.326

001. IF(D.L.T.S.1 GO TO 26"320 D-ALOGD)00330 G 6TO2500348 20 0fl.'OGCA3S(D))66350 2r, CON11NUE01136 LGD(N( 1).ALOG(R )sPTUALOGI P )(N-1 *S003n1 ALOG(RIZII )41. )4ALGAN(PSN.Pi)-ALGARA(PVtI4S38S 1 -ALGAMA(N)-(PUN.4)$ALOG(RZ(I).P,1.IW304 1 *-fas0046 Df AT).Exp(LGDKt(I))$0410 is Z.'ZI4.*426 S.1ALOGIIS(S)

W136 ITECS,201) S"440 WITE (6.219 1

464S* DO S4 1-1,320*466 SO WNITE6.230) Z(I).DEN(I)0647. 160 FORMATI ENTER NPS (REAL) ... 100AU 200 FORMAT( S- ',F5.2)006490 210 FORMAT( z DEN'OSS* 230 FORAT(22XF.2,2XF8.Z2)"Sit STOP"sat6 END

B M2 C

09049 C CONPUTE COEPF. OF UARIATION. MIEAN, SD OF00050 C SI ESTIMATION OF TORE IN WNITE NOISE04060 C FROMI AVERAGE SPECTRA.04479 C N SPECTRA"on0 C P NOISE URNS

"I" REAL SCZ.Z1)SCi,(2.SG116 R.I.

00136 UNRTE4,10)*148 IN FORMAT( 'ENTER N.P EL..)"ISO REAO(SU Np011" DO It 141.12

01It S*10.112(X/16.)

*1m SZ I )m(P414sS6rT(Ns)(Sx82)10210 1 *l130(PSW0-.N-IMCPS00SZ"m I0 1 *(fl)N (dps-I. 3a)II(PSN-1. 116830 C(I)*sZc1)/EZ(180240 16 S5(I)*X6615 Uh!TEC6.809*1* 9"FOMAT(' SI E SZ C 1)love 00 a 1-1,12*12 111 RITE46,300) SSdE(I ) * 5Z I) * ()wew 3 FO M TCIXFS.2,3tlX. 3.3))41360 STOP*0319 ENo

WSRL-0169-TNFigure I

0.5

0.4-

N = 10f(z)

0.3-

N=3

0.2-

N= 2

0.1-

InputSNR = 3.16

0 2 4 6 8

Figure 1. Variation in the probability density function for the estimate ofsignal-to-noise ratio from an average of N spectra using 10 noisebins at an input signal-to-noise ratio of 5 dB

WSRL-0169-.tFigure 2

1.0

N = 10

0.8

f(z)

0.6 N 3

0.4-

N=2

0 2

InputSNR = 0.316

0 1 2 3 4

Figure 2. Variation in the probability density function for the estimate of

signal-to-noise ratio from an average of N spectra using 10 noise

bins at an input signal-to-noise ratio of -S dB

WSRL- 0169-TMFigure 3

MEAN OUTPUT SIGNAL TO NOISE RATIO

30

20

10

I I - I---

-35 -25 -15 -5 5 15 25 35/INPUT SIGNAL TO NOISE RATIO (dB)

N =1 ./ -10-

N I

N =10 e 5 -201

N=30 *

Ideal -30T

to.

Figure 3. Deviation of the mean when using N spectra to estimate signal-to-noise ratio for a I Hz bin width and using 10 noise bins for

estimating noise power

*

WSR L- 01(9 -'I'MFigure 4

100

S-4

COEFFICIENT

OF

VARIATION

N= 1

0.11

N = 10

N = 30

0 .0 I I 1 2

-35 --25 --15 -5 5 115 25

INPUT SIGNAL STRENGTH (dB)

Figure 4. A demonstration of the effect of using N spectra to estimate signal-

to-noise ratio for a 1 Hz bin width and using 10 noise bins forestimating noise power

WSRL-O169-T

DISTRIBUTION

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Defence Scientific and Technical Representative, London No copy

Dr D. Nunn, Department of€Electrical Engineering,University of Southampcon 1

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Farnborough 2

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WSRL-0169-TM

Copy No.

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