4 HANDEL'S QUANTUM 1/f NOISE AND ITS APPLICATIONS TO
COUNTING STATISTICS
Irrespective of the type of phenomenon being studied, most commonly the
radioactive decay statistics provide necessary insight of 1/f noise [2J. Counting
techniques that are employed can be adopted as such for other mechanisms,
provided these are parallel, in reference 1 these techniques are adopted as such.
However the two phenomena are not compared. The most commonly employed
theoritical formalism [68] is breifly discussed in this chapter. The parallelism
beteween the vehicular traffic and the radio-active decay statistics Is discussed in
Chapter 5
Me Donald's theorem, as cited In Ref 68, provides a link between counting
statistics and particle current noise.
However the usefulness of this theorem is
limited to shot noise for which the statistics are purely poissonian. Therefore Van
Vliet and Handel have derived a transform which links a parameter called the Allan
variance which is a two sample variance' to the noise spectral density.
This
transform called the Allan variance transform can handle 1/f spectra and also in
general spectra of the form
S(u>) = C cert1
for
-1< o> <3
(4.1).
4 1 DERIVATION OF THE ALLAN VARIANCE THEOREM [68J
If q(t) represents the number of events counted in a time interval (-»,t) and
Aq(v) denotes the autocorrelation function of q(t) then from Weinner Khintchine
theorem, Aq(v) can be written as
Aq(v) = <q(t)q(t+v)> =/Sq(o)) cos cov dco/2*
(4.2)
According to the definition of Allan variance, if m T 0) is the average counting rate
in (t,t+T) and m T (2> m is the average counting rate in (t+T,t+2T) then the Allan
variance
is given by
=(1/2) \(mT<2> . m jU))2|
The Allan variance
aMA2 =(1/2)
T
£2
(MT<2>
(4.3)
for the total number of events is given by
- MrU))2
(4.4)
<L\
where again
denotes the total number of events detected in (t,t+T) and°
Mj!2) denotes t|i|^otal nember of events detected in (t,t+2T).
Mt«)=
Jk m(t') dt*
= T mj<t)
(4.4 a)
wiiere m T<t) is the average count rate of events in (t,t+2T)
Therefore l(MT<1> - MT(2>)2|B T2\(mT(2) - mT(1>)2|- Eq.(4.4) now becomes
=(1/2) T2l(mT^ - mTW)2) * T o;
'W.
Since q(t) denotes the total number of arrivals in (q(t) = ]*m<f) dt’
Mt0) ='t/T [q (t + T) - q (t )]
£
Mt(2) = 1/T [q (t + 2T) - q (t + T )]
J
(4.5)
,t), It can be written as
(4.6)
(4.7)
From Eqns (4.4), (4.5) and (4.7)
=(1/T2)omA2
= (1/ 2T2 H[q (t + T) - (t)] - [q (t + 2T) - q (t + T )]}2
(4.8)
In accordance with Wiener Khintchine theorem amplitudes Aq (0), Aq(1) and A^(2)
are defined
Aq (0)= <q(t) q (t)> =<q(t +T) q (t+T)> * <q(t +2T) q (t+2T)>
Aq (T) = <q(t) q <t+T)> =<q(t +T)
(4.9)
q (t+2T)>
Aq (2T)= <q(t) q (t+2T)>
On expanding Eqn (4.8) and substituting (4.9), and simplifying , we get,
A2 = (1 /2T2) ( 6Aq (0) - 8Aq (T) + 2Aq (2T)j
On using the defined values of the theorem
6Aq (0) =Lim^ j Sm(to)/a>2.6 COS (0)
dco/2x
8Aq (1) =
*/Sm(to)/a>2.8 COS (coT) dro/2n
2Aq (1) =Lim
/
Sm(co)/co2.2
COS
(4.10)
(4.11)
(2a>T) doi/2n
By substituting Eqn (4.11) into (4.10), one arrives at
to
am A2 = 1/2T2Lim
fsm(co)/co2 [6-8 COS (a>T) + 2 cos (2a>T) ]da>/2;i
T
t-fO l
(4.12)
On simplification of Eqn (4.12) we obtain,
omA2 = 4/nT2 Lim
Sm(<o) Sin 4 (coT/2) dco/to2
ajvj^2 -Ain
L^m ySm(co) sin 4 (coT/2) dco/co2
(4.13)
(4.14)
11
Eqs (4 13) and (4 14) prove the Allan variance theorem.For the particular case of
1/f noise, Sm (co) has the form c/f, where c is the constant, for which am M turns
T
out to be
am
= 4/7tT2 J(cl to ) sin * (coT/2) dco/co2
Let { <0^/2)=
om M = 2c
T
solthat
eo
/ d$/?2 sin1 $ =2c ln(2)
(4.15)
e»
and omA2= 2cT2 ln(2)
(4.16)
T
The authors (68] have given another proof of the Allan variance theorem by
making use of the Laplace transform. This again gives
= 2c T 2 ln(2) for
1/f noise and cr^^2
T
= AT for white noise, where A is a constant.
4.2 APPLICATION TO COUNTING STASTICS[68]
Table - 4.1 gives the T dependance and the results obtainable from Sm(to) via
the Allan transform theorem and the spectral densities for the various forms of
noise in this table me is the average count rate and k is called the super-poisson
factor, B and C are constants depending on the model explaining the noise
phenomene and
Lorentaan noise
a = 1/t
where t
is the time constant (rate constant) of
Supposing that noise is composed of white noise, 1/f noise and Lorentaan
noise, the total noise spectral density can be written as
Sm(o>) = 2km0 + 2n (I/®) + 4 (a/(a? +o>2))
(4.17).
Another useful parameter called the relative Allan variance R(T) has also been
introduced which Is defined as
R(T) = oM A2 / <Mt>2
(4.18).
T
Here <M j> is the mean count and is equal to me T for a stationary process.
From Table 4 1, the total Allan variance can be written as
aMA2= km0T * 2cT2
In (2) + (B/a2) (4e-aT-e-2aT+2aT-3]
(4.19)
23
Using Eqn (4.18) the Relative AAiian Variance is given by
R(T) = (k/m0T + 2c' in (2) + (BWt2) [4e-«T~e-2aT+2aT-3]
where
c* = c/m02
and
B' = B/m02
(4.20)
(4.21)
Table 4.1 - Dfferent Types of Noise - Expressions for Allan Variance
and Noise Spectral Density.
(Source: C M Van Vtlet 1982)168]
°MA2
Type of Noise
Poissonian shot noise or
full shot noise
sm(®)
<Mj>=m0T
2m0
General shot noise or
non-Poissonian shot noise
k meT
1 /f noise
2cT2 In (2)
Lorentzian flicker noise
2(A my2)T
(B/a2) [4e-“T-e-2aT+2aT-3]
2nC/a>
4Boo(a2+CD2)
if the counting time interval T is very much greater than the trapping time ie., for
T»a -1 we can write:
R(T) * (k/m0T)+ 2c’ in (2) + (B7a2T2)
From Eq. (4.22) it can be seen that for T —►
.
(4.22)
,
R(T)
—► 2 c* ln(2). This
limiting value of the relative Allan variance called the 'flicker floor* from whose
value the strength of 1/f noise can be determined. Therefore,
F = 2c* ln(2)
(4.23)
If the spectral density of the counting rate Sm(co) contains shot noise, 1/f noise
and Lorentzian noise then
Sm({0) = 2m0 + 2nq(co\ + 4Rtt/(a~+o>2)
(4.24)
with ail the symbols having their usual meanings and described earlier. Using the
Allan variance theorem, the Allan variance for Eq.(4.24) is found to be
nM£2 =
m0T + 2CT2
m (2) + (B/a2) [4e^_e-2aT+2aT-3]
(4.25)
The relative Allan vanance is
R(T) = 1/ m0T + 2c' In (2) + (B/ m02T2a2) [4e-«T-e-2liT+2aT* 3]
(4.26)
As described before, for short time intervals, the term 1/m T due to to possonian
statistics domainates ans R(T) is propotional to 1/T. When T is long enough and
there is no Lorentzian noise, thenR(T) approaches 2C'ln(2), a constant called the
flicker floor. However, when T is large enough so that noise is negligible
Lorentzian noise dominates
In such cases, the relative Allan variance does not
show a flicker floor but increases beyond the flicker floor.
For aT » 1, from Eqn (4.26)
RfO * 1/m«>T f 2c’ In (2) + 2B/Ta
(4.27)
Fig 4.1 shows the relative Allan variance versus 1/T as expected for shot noise
plus Lorentzian noise, shot noise plus 1/f noise and shot noise only.
4 3 ALLAN VARIANCE THEOREM AND HANDEL'S MODEL[68J
According to Handel's theory, the spectral density of fluctuations Sm(w) in the
average number of counts <m>=m0 is given by
<m>2 sm(o) = 2 «A/f
(4.28)
A corrective factor called the coherence factor has been introduced in Eq.(4.28)
which takes into account the correlations between the various particles emitted
during the emission process or between the emerging particles after a scattered
proces
Thus,
Sm(co) = (2 aA/f m0”2 )
t,
(4.29)
s,
A value of
=1 indicates that there is complete coherence while a value $ <1
indicates incoherence.
For 1/f noise, the spectral pattern is of the form
equating,
Sm(ct>)=c/f.
Therefore
c/f = (2 aA/f) m02 l
(4.30)
from which, c = 2 aA m02 $ combining with Eqn 4.21 we get c-2 aA
The presence of non white noise or more precisely 1/f noise in counting statistics
can be detected from a measurement of the relative Allan variance R(T) as a
function of 1/T
Such a plot should show the presence of a flicker floor of
magnitude given by Eq.(4.3i) for large counting time intervals:
F = 4 z 2ct A 5 ln<2)
(4.31)
4 4 CORRECTIONS TO THE EXPRESSION FOR ALLAN VARIANCE
DUE TO NON ADJACENT SAMPLING [68]
Adjacent sampliing being difficult because of the dead time of the instruement
recording the counts, corrections to the Allan variance have been incorporated. If
t is the dead time between samples then thje corrected Allan variance can also
be applied
This factor do not generally enter the sound recordings due to
vehicular traffic since the response time of the microphone Is very small of the
order of a fraction of milli second. This needs no discussion even though it was
dealt in Ref 1.