A STATISTICAL MODEL FOR FLICKER NOISE

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A STATISTICAL MODEL FOR FLICKER NOISE

V. Reinhardt

To cite this version:

V. Reinhardt. A STATISTICAL MODEL FOR FLICKER NOISE. Journal de Physique Colloques,

1981, 42 (C8), pp.C8-211-C8-216. �10.1051/jphyscol:1981825�. �jpa-00221720�

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JOURNAL DE PHYSIQUE

Colloque C8, suppl&nent au n012, Tome 42, de'cembre 1981 page C8-211

A STATISTICAL MODEL FOR FLICKER NOISE

V, S . Reinhardt

NASA Goddard Space F l i g h t Center, Greenbelt, MD 20771, U.S.A.

Recently J.J. Gagnepain, et. all have proposed that flicker of frequency noise in crystal oscillators is due to a phonon-phonon interaction in the piezoelectric crystal

resonator causing fluctuations in the relaxation time on Q of the crystal. They have developed a theoretical model which predicts a Q - ~ dependence of the spectral density from this process and have experimentally verified this Q - ~

dependence for flicker of frequency noise in crystal

resonators. Gagnepain, et. a l l have not yet derived the f-l behavior in the spectral density, but have indicated that P.

Handel has derived a flicker spectrum for fluctuations in the cross section of a scattering process using quantum theory. Using quantum electrodynamics, P. Handel has, in fact, derived a flicker spectrum for the current and phase fluctations of a DC and AC current of discrete carriers

( 2 r 3 r 4). Unfortunately P. Handel's derivations are difficult to follow for those who are not familiar with quantum electrodynamics and quantum noise theory. Based on Gagnepain's et. al's idea of flicker noise coming from a random fluctuation in the relaxation time of an excitation process, a straight forward statistical model which

generates a flicker fluctuation spectrum from a white noise fluctuation input can be derived. This model has the important feature of allowing the derivation of an

autocorrelation function a s well as the spectral density.

Consider some variable, Y, which is excited by a white noise process and which relaxes with a rate constant

.

The differential equation which describes this process is:

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981825

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JOURNAL DE PHYSIQUE

where n C i > is a random white noise function (Langevin function). From this equation it can be shown5 that:

where S&~o)and S,(w)are the double sideband spectral densities of the autocorrelation functions of Y and n respectively. If n is a random white noise function:

and:

~ a k i n g the inverse fourier transform to obtain, R, ( y ) , the autocorrelation function of Y yields:

Now consider a process which is the sum of a set o f independant such processes with

X:

ranging from ~5 to L where C is very small and L is very large. For

independent processes, the spectral density of this sum process is given by

':

substituting ( 3 ) for

Sa ((L*JJ

^yields:

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Changing variables, this becomes:

Performing the integration and the limiting processes, one obtains:

Thus,SCwj has a flicker spectrum. Equation (5) shows that C and L d o not have t o (go t o 0 and infinity) respectively to obtain (6) a s long as:

It is instructive to use this model to derive an

autocorrelation function for flicker noise. Using ( 4 ) , the autocorrelation function for the sum process becomes:

Allowing

e

to go to zero and, in the case of Z = 0 ,

allowing t to go to infinity, of course, will produce a logarithmic divergence in the integral so the function does not strictly exist. One can use ( 8 )

,

however, to define c a s a distribution similar to the Dirac

$

"function." This can prove useful in solving problems involving flicker noise where a time domain approach is simpler or more appropriate.

To show that R ( " c ) can be used, consider calculating the zero d e a d time Allan Variance G

,

^{o y ' c} ^{Z ]}

,

for flicker of phase noise. Let R(r)be the autocorrelation function for

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C8-2 14 JOURNAL DE PHYSIQUE

x = ? / a

,

the clock reading error. The Allance Variance c a n b e written in terms of R'( T ) as:

~ h u s , for flicker o f phase noise:

Now the 6-> 0 limit causes no problems because the integrand behaves a s (1/2)

r

for very small 2/

.

^The

L * c u limit still causes problems. It is well known, however

,

for flicker o f phase noise, that the spectral bandwidth must be limited for

6

to exist s o a bandwidth limitation must be introduced into (10).

In the time domain, there are two convenient ways to introduce a bandwidth limitation, o n e i s to associate L with an upper spectral bandwidth W M

.

The reasoning behind this is that the averaging process involved in bandwidth limitation will average to zero the effects of the noise processes for which y

> .

In this case, the Allan Variance is:

The integral, in general, is difficult to s o l v e analytically, but for u!, T 4 <: /

,

can be reduced to:

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A second much more mathematically fruitful way of treating the effects of bandwidth limitation i s to rewrite

( 1 0 ) as:

- 3 Y - 2

" D - ~ / L J H ~ - 4 , y.

e

KC e J Y

^{( 1 3 1}

0,'tr3 =pj

0

4 Y F C 2

This form reduces the effects noise processes Eor

x > >

^&^J^,

in a smooth way. It also has the advantage that the integral is analytically solvable in the general case.

Using :

one can reduce (13) to:

P O

r

^{b ~ ,}^'Y < < I

,

⁽¹⁹⁾^reduce%

to (12). For w, % > 3 I

,

^{( 1 5 )}^becomes:

(16) differs from previously published equationsC; by ( 9 / 8 ) ~ .

This may be due to the differing models used for bandwidth limitation effects, but, in any case, the discrepancy should be further investigated.

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JOURNAL DE PHYSIQUE

R e f e r e n c e s

1. J.J. G a g n e p a i n , e t . a l , " R e l a t i o n b e t w e e n '/f n o i s e and Q - f a c t o r i n q u a r t z r e s o n a t o r s a t room a n d low t e m p e r a t u r e s , f i r s t t h e o r e t i c a l i n t e r p r e t a t i o n , ' ' 3 5 t h a n n u a l Symposium o n F r e q u e n c y C o n t r o l ( P h i l a d e l p h i a , 1 9 8 1 ) .

2. P.H. H a n d e l , "Quantum t h e o r y o f l / f n o i s e , " P h y s i c s L e t t e r s , 53A, # 6 , 4 3 8 , ( 1 9 7 5 ) .

3 . P.H. H a n d e l , " l / f n o i s e

-

An I n f r a r e d Phenomenon," P h y s . Rev. L e t t e r s , 3 4 , R24, 1 4 9 2 , ( 1 9 7 5 ) .

4 . P.H. H a n d e l , " N a t u r e o f l / f P h a s e Noise,'' Phys. Rev.

L e t t e r s , 3 4 , # 2 4 , 1 4 9 5 , ( 1 9 7 5 ) .

5. W.B. D a v e n p o r t and W.L. R o o t , An I n t r o d u c t i o n t o t h e T h e o r y o f Random S i g n a l s and N o i s e , c h 6. McGraw-Hill,

(New Y o r k , 1 9 5 8 ) .

6 . B. E. BLAIR, e d , Time a n d F r e q u e n c y : T h e o r y and F u n d a m e n t a l s , c h 8 , NBS Monograph 1 4 0 , U.S. Government P r i n t i n g O f f i c e C a t a l o g #C13.44:140 ( W a s h i n g t o n , D.C., 1 9 7 4 ) .

7. R. C. W e a s t , e d , CRC Handbook o f C h e m i s t r y a n d P h y s i c s , 5 5 t h E d i t i o n , A-155, CRC p r e s s ( C l e v e l a n d , 1 9 7 4 ) .