Table Of ContentUNITED STATES DEPARTMENT OF COMMERCE
Maurice H. Stans, Secretary
NATIONAL BUREAU OF STANDARDS 0 Lewis M. Branscomb, Director
*TECHNICAL NOTE 604
ISSUED JUNE 1971
Not. Bur. Stand. (U.S.), Tech. Note 604, 22 pages (June 1971)
CODEN: NBTNA
Efficient Numerical and Analog Modeling
of Flicker Noise Processes
J. A. Barnes and Stephen Jarvis, Jr.
Time and Frequency Division
Institute for Basic Standards
National Bureau of Standards
Boulder, Colorado 80302
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TABLE OF CONTENTS
Page
. . . . . . . . . . . . . . . . . . . .
1. Introduction. 1
. . . . . . . . . . . . . . . . .
2. The Filter Cascade 2
3. Efficient Generation of Approximate Flicker
. . . . . . . . . . . . . . . . . . .
Noise Numbers 10
. . . . . . . . . . . . . . . . . . .
4. Acknowledgment 13
. . . . . . . . . . . . . . . . . . . . .
5. References 18
iii
LIST OF FIGURES
Figure Page
1 Basic R-C filter with voltage transfer function 3
given by eq (1).
e,
2 FilteT magnitude error eq (17), relative to 6
Als'z] for A = 0.43, ~,(1)0=.0 4 seconds, and
6 = -1 over frequency range (1 Hz, 4 kHz).
9
3 Realization of impedance z1 (O), eq (19), with Ro = 8
20 k52 and Tal1) = 0.04 seconds, which approximates
fractional capacitor of order one -half over frequency
range (1 Hz, 4 kHz).
Operational amplifier system with transfer function 9
given by eq (20).
Fortran program for computing N approximate 14
flicker numbers by recursion relations, eq (36).
('IN = 102411h as been specified. )
Sample of 1024 approximate flicker numbers 15
generated from program of figure 5.
Square root of Allan variance O(7) for data sample 16
of figure 6.
Impulse response function y (4) of digital filter. 17
n
iv
Efficient Numerical and Analog Modeling
of Flicker Noise Processes
J. A. Barnes and Stephen Jarvis, Jr.
It is shown that by cascading a few simple resistor-
capacitor filters, a filter can be constructed which generates
from a white noise source a noise signal whose spectral
I
density is very nearly flicker, If -I, over several decades
of frequency f. Using difference equations modeling this
filter, recursion relations are obtained which permit very
-
efficient digital computer generation of flicker noise time
series over a similar spectral range. These analog and
digital filters may also be viewed as efficient approximations
to integrators of order one-half.
Key words: Analog noise simulation; computer noise simulation;
digital filters ; flicker nois e; fractional integra tion; recursive
digital filters.
1. INTRODUCTION
One of the most pervasive noise processes observed in electronic
I
systems is flicker noise, whose spectral density varies as If over
many decades of frequency, often beyond the low frequency limit to
which the spectral density can reasonably be measured. (See, for
example [ 11 - [6] and their bibliographies. ) While its occurrence is very
widespread, its cause, or causes, is as yet uncertain, despite the
attention of many investigators.
In order to study flicker noise and to simulate the behavior of
systems with flicker noise, several authors have presented mathematical
models which generate a discrete [i’],[ 8 1 or continuous [ 91 - [ 141 flicker
noise from a white noise process. Because of the intrinsically long
I.
correlation time associated with the flicker noise process, the models
have required large computer memories, or extensive computation, or
both. The purpose of this paper is to present an algorithm which permits
one to generate, in a very efficient manner with negligible computer
memory requirements, a sequence of numbers with a very nearly flicker
noise spectral density over several decades of frequency. It is derived
from a cascade of simple resistor -capacitor filters which, when realized,
also permits one to generate from a white noise source an analog noise
signal which has a spectral density which is very nearly flicker over
several decades of frequency.
2. THE FILTER CASCADE
Consider the filter shown in figure 1. Its voltage transfer function
is:
and w= 2nf.
Equation (1) is valid provided the loading on the output, y, of the filter
is negligible. This assumption, thus, might require the filter to be
followed by an isolation amplifier in practice. As it will be seen,
however, it is possible to construct a complete flicker filter with only
one amplifier and still not compromise the validity of eq (1).
For low frequencies (O(71 t 73) << l), one finds that g(w) M 1; and
.
for high frequencies (O 72 >> l), one finds g(w) fi: 72 If one were to
71 t Ta
have a sequence of such filters such that for the i-th filter
. . .
(p)i
71( i)= T?), i = 1,2, N, (5)
where is some constant such that 0 < 1,
2
h
LI
vx
3
and
and if these filters were cascaded with appropriate isolation amplifiers
between successive stages (to eliminate loading effects), the overall
voltage transfer function would become:
One may consider a set of n sections of this filter cascade
. . .
corresponding to i = j t 1, j t 2, j t n, where j is arbitrary.
This subset of filters would change the (magnitude) transfer function by
the factor
in the frequency interval
where
If this filter is to approximate a filter whose transfer function varies
as
IC
where A is a constant, then
1 a
Thus,
4
For a flicker filter Q = - 1 in eq (8) and, thus, eq (11) becomes
A more precise approximation to a flicker filter is obtained by increasing
fi (i. e., the ratio of the 7's given in eq (12)). However, the frequency
-N .
range of the complete filter is proportional to fl Thus, to improve
the precision without reducing the frequency range both B and the
number of sections, N, must be increased.
It is of value to take as a practical example the particular values:
N = 4 (1 3)
71") t 72(')= 372 (0)
(14)
and use the variable
The filter's transfer function then becomes (from eq (7))
1
This transfer function should be expected to have approximately an s-'
behavior over a relative frequency range of 6-4 = 9*, nearly four
decades of frequency. In fact, the magnitude error
E = -z 1,
IG(w)l -AI s (17)
1
1.1
is within f $dB relative to AIs-'~ over the range 0.25 5 5; 1000, a
relative frequency range of 4000 where A = 0.43. The location of the
approximated frequency band depends on 7 (I). For the value ~ ( =l )0 .04
2 2
seconds, the approximating band is (1 Hz, 4 kHz); the error is shown in
figure 2.
One may forget, for the time being, the heuristic arguments which
led to generating eq (16) as an approximation to the function
5
