Copyright © 2011 by Wayne Stegall
Created January 26, 2011. See Document History at end for
details.
Flicker Noise Calculations
Introduction
As a result of expanding the functionality of the Op Amp Noise Calculator, I thought it useful
to do another article on noise.
For most noise calculations pertaining to electronics, it suffices
to consider white noise only. However account of another
type of noise does sometimes merit concern. That is flicker or
1/f noise. The power of this noise of unknown origin is inversely
proportional to the frequencies of interest, distributing equal power
to every octave. This pink noise profile declines 10dB/decade or
3dB/per octave, a rate half that of the stopband rolloff of a
firstorder RC network. Added rootsumsquare to the white noise
level also present creates a noise response break frequency below
which the flicker
noise rises at its characteristic rate. This noise corner frequency is the
frequency where the flicker noise is equal to the white noise
specification in V/√Hz (see figure 1 below).
If you are new to noise calculations you may want to examine Simple Noise Calculations
before proceeding.
Figure
1:
Noise
plot
shows
1/f
noise
rise
below
10Hz
noise
corner
from
normalized
white
noise
level


Mathematical Analysis
The specification of (f) or (f_{1},f_{2})
next to power
or voltage in these analyses indicate that these variables are
functions of frequency or a range of frequencies and does not indicate
a multiplication. Multiplication will be indicated by another
level of
parentheses or an ×.
Because noise is always specified relative to frequency in Hz rather
than radian frequency, it is well to derive the math based on
f rather than ω.
The flicker power function (in W/Hz) is inversely proportional to the
frequency.
The constant k
is arbitrary at this point.
Because the mean square voltage
^{1} and
current are proportional to their
corresponding power, that is v
_{n}(f)
^{2} = p
_{n}(f)R,
then
Because it is common to relate flicker noise to the characteristic
white noise at a frequency where both are equal called the noise corner
frequency (f
_{nc}),
if we set
(3)

k = 
v_{wn}^{2}(f) × f_{nc}
R 
the equation (2) becomes:
(4)

v_{fn}^{2}(f) = v_{wn}^{2}(f)

f_{nc}
f

v
_{wn}(f) here represents the semiconductor white noise
specified V/√Hz in datasheets; v
_{fn}(f) is specified in units
of V/√Hz as well.
Integrating the 1/f function through the
range of frequencies representing bandwidth gives the corresponding
total mean square flicker
noise
(5)

v_{fntotal}^{2}(f_{1},f_{2})
=

∫ 

v_{wn}^{2}(f) 
f_{nc}^{ }
f

df^{ }

Knowing that ∫1/f df = log
_{e}(f) = ln(f) the integral becomes
(6)

v_{fntotal}^{2}(f_{1},f_{2})
=
v_{wn}^{2}(f) × f_{nc} × (ln(f_{2})ln(f_{1}))

That subtraction of logarithms is equivalent to division allows a more
convenient representation.
(7)

v_{fntotal}^{2}(f_{1},f_{2})
=
v_{wn}^{2}(f) × f_{nc} × ln(f_{2/}f_{1})

Take the square root of total mean squared voltage noise to get total
flicker noise:
(8)

v_{fntotal}(f_{1},f_{2})
=
v_{wn}(f) 

Adding It Up
After calculating flicker noise it is well to add up all noise
sources. Flicker noise adds rootsumsquare as does white noise.
Total white noise specified as a function of voltage noise per √Hz and
bandwidth is:
(9)

v_{wntotal}(f_{1},f_{2})
=
v_{wn}(f)√B
=
v_{wn}(f) 

Square total white noise and flicker noise and add them to get total
mean square noise:
(10)

v_{ntotal}^{2}(f_{1},f_{2})
=
(v_{wn}^{2}(f) × (f_{2}  f_{1}))
+ (v_{wn}^{2}(f) × f_{nc} × ln(f_{2/}f_{1}))

Take square root of total mean squared voltage noise to get total noise:
(11)

v_{ntotal}(f_{1},f_{2})
=
v_{wn}(f) 

(f_{2}  f_{1})
+
f_{nc}·ln(f_{2/}f_{1}) 

A Statistical Diversion
The statistical chance
that noise
voltage level will take a certain
instantaneous level follows the pattern of a Gaussian
distribution. This is the
familiar bell curve. The RMS noise value corresponds to one
standard deviation (written 1σ). This is because the standard
deviation
calculation is itself an RMS calculation of the variance of the
changing variable relative to the mean (0V or 0A in the case of
noise). A peak to peak ±1σ contains 68.3% of statistically
possible values, a value that differs from the 70.7% value we expect
from a RMS calculation of a sine wave. This is because the sine
wave, not being random, does not have a Gaussian distribution.
That ±3σ contains 99.9% of all possible levels represents a
peaktopeak signal excursion of 6σ, leaves the peak value to deviate
only
3σ. These standard deviation factors become conversion factors to
convert between RMS, peak, and peaktopeak noise specifications:
v_{n(rms)}

= 
v_{n(peak)} 
÷3 
= 
v_{n(pp)} 
÷6 
v_{n(peak)} 
= 
v_{n(rms)} 
×3 
= 
v_{n(pp)} 
÷2 
v_{n(pp)} 
= 
v_{n(rms)} 
×6 
= 
v_{n(peak)} 
×2 


Figure 2: Plot of
noise Gaussian showing relationship to different signal designations. 


Reading Datasheets
Datasheets often give flicker data in a form other than the most useful
noise corner frequency (f
_{nc}). A midband noise
specification will be given for the white noise. A single
specification at a lower frequency will contain flicker and white noise
from which f
_{nc} can be calculated. Alternatively, a
noise total for a range of frequencies below the white noise
specification will be given.
First, if necessary convert any peaktopeak specifications to RMS by
dividing by 6 (refer to note on conversions above), then calculate
the noise corner frequency from equations (13) or (14) below.
Derive the formula for f
_{nc} based on a single low
frequency specification.
First, solve the 1/f formula of equation (4) for f
_{nc}.
(12)

f_{nc} = 
v_{fn}^{2}(f) × f
v_{wn}^{2}(f)

Substitute the subtraction of the square of the white noise
specification from the total noise specification for the square of the
flicker noise specification (all in V/√Hz) to yield the final formula.
(13)

f_{nc} = 
(v_{ntotal}^{2}(f)

v_{wn}^{2}(f)) × f
v_{wn}^{2}(f)

Derive the formula for f
_{nc} based on a noise total for a
range of frequencies.
Solve equation (11) for f
_{nc} to get formula deriving f
_{nc}
from a noise specification through a range of frequencies.
(14) 
f_{nc} = 
(v_{ntotal}(f_{1},f_{2})/v_{wn}(f))^{2}
+ f_{1}  f_{2}
ln(f_{2}/f_{1})

Example
Take the following specifications from the
LSK170 datasheet:
 Midband white noise specification, v_{wn}(f) = 0.9nV/√Hz
at 1kHz
 Lowfrequency noise specification, v_{ntotal}(f) =
2.5nV/√Hz at 10Hz (both are labeled e_{n}^{2})
For this JFET, calculate the total white noise from 20Hz to 20kHz, the
noise corner frequency, the total flicker noise from 20Hz to 20kHz,
then the total noise from 20Hz to 20kHz.
From equation (9) calculate total white noise:
v_{wntotal}(f_{1},f_{2})
=
0.9nV/√Hz


= 0.9nV/√Hz 

v
_{wntotal}(f
_{1},f
_{2}) =
0.9nV/√Hz × 141.351√Hz =
127.216nV
From equation (13) calculate noise corner frequency.
f_{nc} = 
((2.5nV/√Hz)^{2} (0.9nV/√Hz)^{2})
×
10Hz
(0.9nV/√Hz)^{2} 
= 
(6.25a(V^{2})/Hz  0.81a(V^{2})/Hz)
×
10Hz
0.81a(V^{2})/Hz 
f_{nc} = 
(5.44a(V)^{2}/Hz) × 10Hz
0.81a(V)^{2}/Hz 
= 6.71605 10Hz = 67.1605Hz

From equation (8) calculate total flicker noise.
v_{fntotal}(f_{1},f_{2})
=
0.9nV/√Hz


67.1605Hz × ln(20kHz/20Hz) 

= 0.9nV/√Hz 




v_{fntotal}(f_{1},f_{2})
=
0.9nV/√Hz


= 0.9nV/√Hz 




v
_{fntotal}(f
_{1},f
_{2})
= 0.9nV/√Hz × 21.539√Hz =
19.3851nV
Calculate total noise from total white noise and total flicker noise by
rootsumsquare:
v
_{ntotal}^{2}(f
_{1},f
_{2}) =
(127.216nV)
^{2} + (19.3851nV)
^{2} = 16.1839f(V
^{2})
+
375.782a(V
^{2}) = 16.5597f(V
^{2})
v
_{ntotal}(f
_{1},f
_{2}) =
128.684nV
Final Thoughts
The example above illustrates an important point. The 128.684nV
total after adding flicker noise to the white noise total of 127.216nV
is only an increase of 0.1dB. It is often safe to ignore flicker
noise and calculate white noise only if the noise corner frequency is
near the lower end of the band calculated.
You can tinker with flicker noise calculations with the
Op Amp Noise Calculator.
^{1}Mean square values are squared
rms values.
^{2}E is sometimes used to
represent the voltage variable rather
than v, derived from
voltage's original name,
electromotive force or emf. The reason may lie with
combination
electromechanical systems, where v
is in conflict for use as voltage
and velocity, one of which has to be changed. E is not used in
place of Volt as a unit however. V is the preferred variable for
electronicsonly applications because e is also the base of the natural
exponential, a common function in electronics mathematics. The
similar appearance of e^{n} and e_{n}, at least in
quick reading, could create problems that that use of v for emf would
alleviate.
Document History
January 26, 2011 Created.
January 26, 2011 Changed (f1f2) to (f1,f2) to better represent
dual function arguments and added an additional comment at the end.
January 26, 2011 Corrected a name discrepancy that made equations
(11) and (14) seem inconsistent.
January 27, 2011 Made minor improvements for clarity and grammar.