1
Chapter 10S Johnson-Nyquist noise Masatsugu Sei Suzuki
Department of Physics, SUNY at Binghamton (Date: January 02, 2011)
Johnson noise Johnson-Nyquist theorem Boltzmann constant Parseval relation Correlation function Spectral density Wiener-Khinchin (or Khintchine) Flicker noise Shot noise Poisson distribution Brownian motion Fluctuation-dissipation theorem Langevin function ___________________________________________________________________________ John Bertrand "Bert" Johnson (October 2, 1887–November 27, 1970) was a Swedish-born American electrical engineer and physicist. He first explained in detail a fundamental source of random interference with information traveling on wires. In 1928, while at Bell Telephone Laboratories he published the journal paper "Thermal Agitation of Electricity in Conductors". In telecommunication or other systems, thermal noise (or Johnson noise) is the noise generated by thermal agitation of electrons in a conductor. Johnson's papers showed a statistical fluctuation of electric charge occur in all electrical conductors, producing random variation of potential between the conductor ends (such as in vacuum tube amplifiers and thermocouples). Thermal noise power, per hertz, is equal throughout the frequency spectrum. Johnson deduced that thermal noise is intrinsic to all resistors and is not a sign of poor design or manufacture, although resistors may also have excess noise.
http://en.wikipedia.org/wiki/John_B._Johnson
2
____________________________________________________________________________ Harry Nyquist (February 7, 1889 – April 4, 1976) was an important contributor to information theory.
http://en.wikipedia.org/wiki/Harry_Nyquist ___________________________________________________________________________ 10S.1 Histrory
In 1926, experimental physicist John Johnson working in the physics division at Bell Labs was researching noise in electronic circuits. He discovered random fluctuations in the voltages across electrical resistors, whose power was proportional to temperature. Harry Nyquist, a theorist in that division, got interested in the phenomenon and developed an elegant explanation based on fundamental physics. hence this type of noise is called Johnson noise, Nyquist noise, or thermal noise. 10S.2 Spectral density
Suppose we measure a physical quantity x(t) as a function of time t. We define the Fourier transform of )(tx as
)(2
1)( txdteX ti
Note that
3
)()(2
1)(*
XtxdteX ti .
since x(t) is real. The inverse Fourier transform is given by
)(2
1)(
Xedtx ti .
(a) Parseval relation.
0
2
2
*
)'(*
*'*
)(2
)(
)'()'(')(22
1
)'(')(2
1
)'('2
1)(
2
1)()(
Xd
Xd
XdXd
dteXdXd
XedXeddtdttxtx
ti
titi
since
2*2
)()()()()()( XXXXXX .
(b) We use a truncated signal.
It is sometimes necessary, in order to avoid convergence problem, to approach the definition of the integral as a limiting process. Using the Parseval relation, we get
0
22/
2/
22 )(2
lim)(1
lim)( XdT
dttxT
txT
T
TT
.
We define the spectral density as
4
)()(2
lim2 GX
TT
, (1)
which is the ensemble average of 2
)(X . Then we have
0
2 )()( Gdtx . (2)
((Note)) We assume that the average of process parameter over time and the average over the statistical ensemble are the same (ergodicity). 10S.3 Correlation function
The limiting process required to calculate G() is sometimes awkward to use in practice. Here we show that correlation function can be closely related to the spectral density. The correlation function associated with a stationary random function of time t, x(t), is defined by
txtx ()( . (3)
which means that Eq.(3) is independent of t. The Fourier transform of )( tx is given by
)()'(2
1)(
2
1 '
Xetxdteetxdte itiiti
.
Correspondingly, the inverse Fourier transform is given by
)(2
1)( )(
Xedtx ti .
We need to calculate the product given by
)'()('2
1)()( ')(
XXeeddtxtx iti .
It is sometimes necessary, in order to avoid convergence problem, to approach the definition of Eq.(3) as a limiting process. The integral of this product is
5
0
2
2
'
2/
2/
')'(2/
2/
)()cos(2
)(
)()(
)'()()'('22
1
)'()('2
1)()(
Xd
Xed
XXed
XXedd
XXedtedddttxtx
i
i
i
T
T
itiT
T
Note that
)'(22/
2/
)'(
T
T
tidte .
Then we have
0
2
2/
2/
)()cos(2
lim
)()(1
lim()(
XdT
dttxtxT
txtx
T
T
TT
The correlation function C() is obtained as the ensemble average of txtx ()( ,
)(2
1
)()cos(
)()cos(2
lim
()()(
0
0
2
Ged
Gd
XdT
txtxC
i
T
, (4)
where we use the relation(Eq.(1)). The inverse Fourier transform of G() is given by
6
2
)(2)(
2
1 CGed i
.
The corresponding Fourier transform is
0
)()cos(2
4
)(2
2
2
)(2
2
1)(
Cd
Ced
CedG
i
i
. (5)
Equation (5) is the Wiener-Khinchin (WK) theorem. G() and C() are a Fourier transform pair;
G(): spectral density
C(): correlation function 10S.4 Properties of the correlation function
The correlation function txtxC ()()( is an even function of .
)()()()()(()()( CtxtxtxtxtxtxC .
We also have
0)]()([ 2 txtx ,
or
0)()(2)()( 22 txtxtxtx ,
or
)()()()()0( 2 CtxtxtxC .
10S.5 Johnson-Nyquist theorem
7
At any non-zero temperature we can think of the moving charges as a sort of electron gas trapped inside the resistor box. The electrons move about in a randomised way — similar to Brownian motion — bouncing and scattering off one another and the atoms. At any particular instant there may be more electrons near one end of the box than the other.
We consider a resistance R which has a length l, a cross section A. The relaxation time of the
carriers (electrons) is e. There are n electrons per unit volume (n is called the electron density). The total number of electrons is N;
nAlN . From the Ohm's law, the open circuit voltage is
)( uneRARJAIRV ,
where V is a voltage, I is a current, J is the current density, A is the conductor cross section area, e is the absolute value of the charge of electron (e>0), and u is the average velocity of electrons. Since
N
iiu
Nu
1
1,
we have
N
ii
N
ii
N
ii Vu
l
eRu
Ne
Al
NRAV
111
1)( ,
where
ii ul
eRV .
Suppose that the correlation function is expressed by
00 22
2()()(
eu
l
eReVtVtVC iiii .
The spectral density is given by
8
20
022
0
22
0
)(12
4
)cos(2
4
)()cos(2
4)(
0
i
i
ul
Re
edul
Re
CdG
(Lorentz-type spectral density)
In general, the spectral density G() has a Lorentz-type when the correlation function C() is
described by a exponential decay; exp(-/t0). For the voltage signals, the spectral density G() is customary to use units of [V2 s] = [V2 Hz−1].
We note that
Tkum Bi 2
1
2
1 2 , (equipartition theorem)
where m is the mass of electron and kB is the Boltzmann constant. In metal at room temperature,
e is very small; e<10-12 s. For the DC and microwave regions, 12 cf . Then we get
0
2Re
2
4)(
m
Tk
lG B
, (white noise)
which is independent of . We get
0
2
0
2
4
)(2
)()(
m
Tk
l
Ref
Gf
GGdV
B
i
,
since = 2f. We have
fTRkml
NAeRfTRkVNV BBi 4)(4
20
222
.
Here we note that the resistance is define by
9
02
2
02
1
NAe
ml
Am
eAlN
l
A
lR ,
and
12
02
02
2
20
2
ml
NAe
NAe
ml
ml
NAeR
.
Experimentally it is predicted that the plot of 2V vs R may exhibits a straight line. The slope
will give some estimation for the value of the Boltzmann constant kB.
10S.6 Evaluation of 2V
The Boltzmann constant is given by
kB = 1.380650410 x 10-23 J/K.
fTRkV B 42 ,
where T is the temperature (K), R is the resistance, and f is the frequency range.
(i) T = 293.15 K, f = 106 Hz, R = 1 M
VV 238.1272 .
(ii) T = 293.15 K, f = 1.9980 x 104 Hz, R = 200
VnVV 254.0349.2542 .
(iii) T = 293.15 K, f = 1.0 x 104 Hz, R = 1000
VnVV 402.0362.4022 .
10S.7 Equivalent circuit (Thevinin theorem and Norton theorem)
10
Noise volatge source (the left of the above figure) and noise current source (the right of the above figure);
RG
1
fTRkV B 42
.
)(4
422
2
2 fTGkR
fTRk
R
VI B
B
.
In Fig. we show the equivalent circuits for the noise voltage source and noise current source. These circuits are equivalent according to the Thevinin's theorem and Norton's theorem. ________________________________________________________________________ 10S.8 Measurement of Johnson noise
11
The above figure shows an equivalent circuit for the Johnson noise produced by a resistor. n is an invisible random voltage generator, connected in series with an ideal (noise free) resistor R. The voltage fluctuations are amplified and passed through a band-pass filter to a voltmeter. The filter only allows through frequencies in some range (the bandwidth),
minmax fff , (Hz)
Rin is the input resistance of the amplifier. The voltage due to the Johnson noise is
fTRkVe Bn 42 .
From the Ohm's law, the current flowing in the input resistance of the amplifier is
in
n
RR
ei
.
The voltage seen at the amplifier's input across Rin is
in
n
RR
ReRiv
.
The noise power entering in the amplifier is
22
in
n
RR
ReivP
.
The maximum noise power is obtained as
12
)(44
22
max fTkR
V
R
eP B
n ,
when Rin = R (impedance matching). The unit of Pmax is Watt (J/s) 10S.9 Shot noise
Shot noise is distinct from current fluctuations in thermal equilibrium, which happen without any applied voltage and without any average current flowing. These thermal equilibrium current fluctuations are known as Johnson-Nyquist noise or thermal noise. Shot noise is a Poisson process and the charge carriers which make up the current will follow a Poisson distribution. We define the Fourier transform of the current,
)(2
1)( tIdteY ti
.
Then we have
0
22/
2/
22 )(2
lim)(1
lim YdT
dttIT
IT
T
TT
.
We define the spectral density as
)()(2
lim2 SY
TT
, (1)
which is the ensemble average of 2
)(Y . Then we have
0
2 )(SdI .
The correlation function of current is defined by
13
)(2
1
)()cos(
)()cos(2
lim
()()(
0
0
2
Sed
Sd
YdT
tItIC
i
T
or
2
)(2)(
2
1 CSed i
.
The Wiener-Khinchin (or Khintchine) states that the noise spectrum is the Fourier transform of the correlation function (the spectral density)
0
)()cos(2
4
)(2
2
2
)(2
2
1)(
Cd
Ced
CedS
i
i
((Note))
S(): spectral density
C(): correlation function The delta-function current pulses are given by
k
kttetI )()( , k
kttetI )()( .
Then the correlation function is obtained as
14
)(
)(lim
)()(lim)(
''
2
'
2/
2/
'
2
Ie
ttT
e
dtttttT
eC
k kkk
T
k k
T
T
kkT
where
e
I
T
N ,
and I is the DC current. When the summation indices are k = k', it means that the arrival times
are equal tk = tk'. Then we just have (). If there are N values of tk such that -T/2<tk<T/2, these
terms will contribute N() to the correlation function. For tk ≠ tk' the delta functions will occur
at randomly distributed, nonzero values of . The contributions from these delta functions to the
C() will vanish. Taking the Fourier transform, we find
IeedIeS i 22
1)(
2
2)(
The spectrum is uniform and extends to all frequencies. Such a spectrum is called white. Then we have
fIeIef
IedSdI
222
22
2
1)(
00
2
.
The current fluctuations have a standard deviation of
fIeI 22 ,
where e (>0) is the absolute value of the charge of electron, Δf is the bandwidth in hertz over which the noise is measured, and I is the average current through the device. For a current of 100 mA this gives a value of
15
nAI 179.02
if the noise current is filtered with a filter having a bandwidth of f = 1 Hz. The plot of 2I vs
I may exhibits a straight line. This slope will give some estimation for the value of charge e.
e = 1.602176487 x 10-19 C (from NIST Physics constant) ((Note)) Derivation of the formula by van der Ziel (Poisson distribution)
We define N as the number of carriers passing a point in a time T at a rate n(t).
T
dttnN0
)( , and TnN .
where N and n are ensemble averages and this result follows from that fact that time average equals ensemble average (the ergodicity). We assume that n follows the Poisson statistics. The average current is
neT
NeI
The spectral density S is given by
Iee
IeneneS 2222 2222
where nn 2 for the Poisson distribution. Since S is independent of the frequency f (white noise), we have
fIefSI 22
10S.10` Flicker noise (1/f noise, pink noise): DC current related noise
Flicker noise, also known as 1/f noise, is a signal or process with a frequency spectrum that falls off steadily into the higher frequencies, with a pink spectrum. It occurs in almost all electronic devices, and results from a variety of effects, though always related to a DC current.
16
Fig. Spectral density vs frequency. 1/f noise in the low frequency range (the flicker noise).
White noise in the higher frequency range. ______________________________________________________________________________ 10S.11 Brownian motion
Suppose that a force (t) is applied to a particle with a mass m along the x direction. According to the Newton's second law, The velocity )()( txtv satisfies the following
differential equation,
)()()(
ttvdt
tdvm . (1)
In a situation such that the particle moves randomly in the fluid at a constant temperature T, the
parameters and (t) no longer be regarded as independent ones. It is required that and (t) are closely related to each other. This is the key point of the Brownian motion (Einstein's relation).
Note that (t) is a force applied to the particle as a result of the collision of molecules of fluid with the particle. This force is considered to be random force (fluctuating force). We assume that
the time average of (t) is zero and exhibits a white random force (Gaussian);
0)( t , and
)'(2)'()( tttt , (2)
where is the magnitude of the random force. The first term of Eq.(1) is the effect of friction,
where is the coefficient of friction. We define the relaxation time as
17
m .
The solution of Eq.(1) is obtained as
t
t
sttt sdsem
tvetv0
0 )(1
)()( /)(0
/)( .
When )( 0tv is fixed,
)()(1
)()( 0/)(/)(
0/)( 0
0
0 tvesdsem
tvetv ttt
t
sttt ,
indicating that )(tv decays with time (relaxation time ). For simplicity, we assume that v(t0) is
finite. In the limit of t0 →-∞, we have
t
st sdsem
tv )(1
)( /)( .
Then we get
0)(1
)( /)(
t
st sdsem
tv ,
since 0)( s . The correlation function is given by
0
21212
0
12
21212
21212
2211221
)(2
)(2
)()(1
)()(1
)()(
21
2 221 11
2 221 11
2 221 11
uuttdueduem
ssdsedsem
ssdsedsem
dssedssem
tvtv
uu
t stt st
t stt st
t stt st
18
where t1 - s1 = u1 and t2 - s2 = u2. Using the Mathematica, we get
||
221
21
)()(tt
em
tvtv
. (3)
_____________________________________________________________________________ ((Mathematica))
_____________________________________________________________________ When t = t', we have
22 )(
mtv
. (4)
Using the equipartition law in the classical limit,
Tktvm B2
1)(
2
1 2 , (5)
we get the relation
22 )(
mm
Tktv B
, (6)
or
eq1
Integrate
IntegrateExpu1 u2
DiracDeltat1 t2 u1 u2,
u1, 0, , u2, 0, Simplify, t1 t2 Reals & ;
Simplifyeq1, t1 t21
2t1t2
Simplifyeq1, t1 t21
2
t1t2
19
TmkB . (7)
This is the relation of dissipation-fluctuation. The parameter is the magnitude of the fluctuating force and /m is the co-efficient of friction (dissipation). This relation is first derived by
Einstein (1905). 10S.12 Diffusion constant D
The velocity correlation function is rewritten as
||
21
21
)()(tt
B em
Tktvtv
.
The displacement correlation function can be obtained as follows.
)}1({2
)()()]0()([
/2
0
||
2
0
1
0
212
0
12
21
tB
t tttB
tt
etm
Tk
edtdtm
Tk
tvtvdtdtxtx
When t>>, we have
tTk
tm
m
Tkt
m
Tkxtxx BBB
222
)]0()([ 22 . (8)
We define the Diffusion constant D as
Tk
t
xtxD B
t
2
)]0()([lim
2
. (9)
((M athematica))
20
10S.13 Langevin function
We consider the case that the finite force F is further applied to the particle. We set up the Lagrange equation
Fttvdt
tdvm )()(
)( . (10)
Using the assumaption that 0)( t , we get
Ftvdt
tvdm )(
)( . (11)
The solution of this equation is
00
00
)0()1(
)(]1[)()(
0
)(
tttt
ttm
ttm
evem
F
etveF
tv
In the limit of t→∞ (in thermal equilibrium), )(tv becomes constant (steady state),
Fm
Ftv
)( , (12)
where is the mobility and is defined by
Clear"Global`";
f1 Expt1 t2
UnitStept1 t2
Expt1 t2
UnitStept2 t1;
eq1
IntegrateIntegratef1, t1, 0, t,
t2, 0, t Simplify, t 0, 0 &
2 t 1 t
21
m
. (13)
There is a relation between D and as
Tkm
TkD B
B , (Einstein's relation), (14)
10S.14 Fluctuation-dissipation (FD) theorem (i) First-type FD theorem
The velocity correlation function is given by
||
)0()(t
B em
Tkvtv
.
Taking the integral over time, we have
Tkm
Tkdte
m
Tkdtvtv B
Bt
B
0
|
0
)0()( . (15)
Then the mobility can be rewritten as
0
)0()(1
dtvtvTkB
. (Type-1 FD theorem) (16)
The transport co-efficient (mobility, conductivity) can be described by the time correlation of the velocity (current density). (ii) Second-type FD theorem
From the relation
)(2)0()( tt ,
We have
Tkdttdtt B 22)(2)0()(
,
22
or
Tkdttdtt B 22)0()(2)0()(0
,
or
dttTkB
)0()(1 . (17)
The co-efficient of the friction can be described by the time correlation of the fluctuating force. 10S.15 Langevin equation for electrical conductivity
We consider the motion of the i-th particle with mass m and charge e in the presence of
fluctuating electric field i(t);
eEtetvtvdt
dm iii )()](
1)([
We define the current density as
n
ii tevtJ
1
)()(
From the above equation, we have
neEtetvtvdt
dm
n
ii
n
ii
n
ii
111
)(])(1
)([
,
or
EnetnetJtJdt
dm 22 )()](
1)([
,
where n is the particle density and the fluctuating electric field is given by
23
n
ii t
nt
1
)(1
)(
which is a average fluctuating electric field applied to the n charged particles.
0)( t .
and
)'(2
)'()(2 ttTmk
ttq Bijji
.
Here we assume the independence of { )(ti }. In the steady state, we have
EEm
netJ
2
)( .
The electrical resistance R is
Anq
ml
A
l
A
lR
2
1
The fluctuating electric field (t) can be described by
)'(2)'(2)'(2
)'()(2
ttl
RATktt
ml
RATmktt
nq
Tmktt BB
B
Using the relatrion V(t) = l (t),
)'(2)'()( ttTRAlktVtV B .
or
)(2)0()( tTRAlkVtV B
or
24
0
)0()(11
)0()(2
11dtVtV
TklAdtVtV
TklAR
BB
10S.16 Fluctuation-dissipation theorem for Conductivity
Suppose that E = 0. Then we have
)()(1
)(2
tm
netJtJ
dt
d
The solution for this equation is given by
0
0
)()()( 0
)(2 ttt
t
st
etJdssem
netJ
For simplicity, we assume that J(t0) is finite. In the limit of t0 →-∞, we have
t st
sdsem
netJ )()(
)(2
.
Then we get
0)()()(2
t st
sdsem
netJ ,
since 0)( s . The correlation function is given by
0
21212
0
1
2
212122
42
21212
42
22112
42
21
)(2
)(2
)()(
)()()()(
21
2 221 11
2 221 11
2 221 11
uuttdueduem
Tkne
ssdsedsene
Tmk
m
en
ssdsedsem
en
dssedssem
entJtJ
uuB
t stt stB
t stt st
t stt st
where t1 - s1 = u1 and t2 - s2 = u2. Using the Mathematica, we get
25
2121 22
21 2
2)()(
tt
B
tt
B em
Tknee
m
TknetJtJ
.
Then we get
Tkm
Tknedte
m
TknedtJtJ B
Bt
B
2
0
2
0
)0()( .
The conductivity is expressed by the time correlation of the current dnsity;
0
)0()(1
dtJtJTkB
.
_____________________________________________________________________________ REFERENCES A.C. Melissinos and J. Napolitano, Experiments in Modern Physics, 2nd edition (Academic
Press, Amsterdam, 2003). J.B. Johnson, Phys. Rev. 32, 97 (1928). H. Nyquist, Phys. Rev. 32, 110 (1928). C. Kittel, Elementary Statistical Physics (John Wiley & Sons, New York, 1958). C. Kittel and H. Kromer, 2nd edition (W.H. Freeman and Company, New York, 1980). H.S. Robertson, Statistical Thermodynamics (PTR Prentice Hall, Englewood Cliffs, NJ, 1993). Toshimitsu Musha, World of Fluctuation (Kozansha Blue Backs, Tokyo, Japan) [in Japanese]. Masuo Suzuki, Statistical Mechanics 2nd edition (Iwanami, Tokyo, 1996) [In Japanese]. Kazuo Kitahara, Nonequilibrium Statistical Physics, 2nd edition (Iwanami, Tokyo, 1998) [in
japanese]. __________________________________________________________________________ APPENDIX A.1 Poisson distribution The Poisson ditribution function is given by
!n
eP
n
n
,
with the mean
26
0nnnPn ,
and the variance
0
2222 )(n
nPnnn ,
since
)1(0
22
n
nPnn .
Fig. The Poisson distribution function with being changed as a parameter. ((Mathematica))
m=10
20
3040
5060 70 80 90 100
20 40 60 80 100 120n
0.02
0.04
0.06
0.08
0.10
0.12
Poisson
27
PDFPoissonDistribution, k k
k
MeanPoissonDistribution
VariancePoissonDistribution
f1
ListPlotTablek, PDFPoissonDistribution10, k,
k, 0, 30;
f2
GraphicsTextStyle"10", Black, 12,
10, 0.07;
Showf1, f2
m=10
5 10 15 20 25 30
0.02
0.04
0.06
0.08
0.10
0.12
28
A.2 Gaussian distribution (normal distribution)
]2
)(exp[
2
1),;( 2
2
x
xf .
dxxxfx ),,( .
22222 ),,()(
dxxfxxx .
: mean
: standard deviation
])(2
1exp[]
2
)(exp[),;(2 2
2
2
xx
xf .
s=0.1
0.2
0.30.40.50.6
-1.0 -0.5 0.5 1.0x-m
1
2
3
4
Gaussian
29
The full width at half maximum (FWHM);
35482.2
17741.12ln2 ((Mathematica))
1
2
2 ln2- 2 ln2
FWHM
-4 -2 2 4
x - m
s
0.2
0.4
0.6
0.8
1.0
f 2p s
30
PDFNormalDistribution, , x
x2
2 2
2
MeanNormalDistribution,
VarianceNormalDistribution, 2
f1 PlotPDFNormalDistribution0, 1, x,
x, 4, 4;
f2
GraphicsTextStyle"0, 1", Black, 12,
1, 0.3;
Showf1, f2
m=0, s=1
-4 -2 2 4
0.1
0.2
0.3
0.4