1

In order to reduce errors, the measurement object and the

measurement system should be matched not only in terms of

output and input impedances, but also in terms of noise.

The purpose of noise matching is to let the measurement

system add as little noise as possible to the measurand.

We will treat the subject of noise matching in Section 5.4.

Before that, we have to describe the most fundamental types

of noise and its characteristics (Sections 5.2 and 5.3).

Reference: [1]

5.2. Noise types

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

2

Reference: [1]

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

5.2.1. Thermal noise

Thermal noise is observed in any system having thermal losses

and is caused by thermal agitation of charge carriers.

Thermal noise is also called Johnson-Nyquist noise. (Johnson,

Nyquist: 1928, Schottky: 1918).

An example of thermal noise can be thermal noise in resistors.

3

vn(t)

tf(vn)

vn(t)

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

R V

6

Vn rms

Example: Resistor thermal noise

Normal distribution according to thecentral limit theorem

T 0

2R()

0

White (uncorrelated) noise

en2

f0

4

C enC

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

To calculate the thermal noise power density, en2( f ), of a

resistor, which is in thermal equilibrium with its surrounding, we

temporarily connect a capacitor to the resistor.

R

Ideal, noiseless resistor

Noise source

Real resistor

A. Noise description based on the principles of

thermodynamics and statistical mechanics (Nyquist, 1828)

From the point of view of thermodynamics, the resistor and the

capacitor interchange energy:

en

55. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Illustration: The law of equipartition of energy

m v

2

2

Each particle has three degrees of freedom

mivi 2

2

mi vi 2

2=

m v 2

2= 3

k T2

In thermal equilibrium:

65. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

C V 2

2=

k T2

In thermal equilibrium:

Illustration: Resistor thermal noise pumps energy into the capacitor

Each particle has three degrees of freedom

CV

2

2

mivi 2

2

75. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Since the obtained dynamic first-order circuit has a single

degree of freedom, its average energy is kT/2.

This energy will be stored in the capacitor:

R

Ideal, noiseless resistor

Noise source

Real resistor

C enC

H( f )= enC ( f )

enR ( f )

enR

C V 2

2=

k T2

In thermal equilibrium:

8

kTC

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

= =nC 2 =

C nC 2

2

kT

2

kT

C

C vnC (t) 2

2

According to the Wiener–Khinchin theorem (1934), Einstein

(1914),

enR 2( f ) H(j2 f)2

e j 2 f d f

nC 2 RnC () =

1 d f

enR2( f )

1) +2 f RC(2

enR2( f )

4 RC

enR2( f ) = 4 k T R [V2/Hz].

Power spectral density of resistor noise:

9

SHF EHF IR R

10 GHz 100 GHz 1 THz 10 THz 100 THz 1 GHz

1

0.2

0.4

0.6

0.8

enR P( f )2

enR( f )2

f

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

enR P 2( f ) = 4 R [V2/Hz] .

B. Noise description based on Planck’s law for blackbody

radiation (Nyquist, 1828)

h f

eh f /k T 1

A comparison between the two Nyquist equations:

R = 50 ,C f 0.04 = F

= R 50 ,C = 0.04 f F

105. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

The Nyquist equation was extended to a general class of

dissipative systems other than merely electrical systems.

eqn2( f ) = 4 R + [V2/Hz] .

h f

eh f /k T 1

Zero-point energy f(t)

h f

2

C. Noise description based on quantum mechanics

(Callen and Welton, 1951)

eqn ( f )2

enR ( f )2

SHF EHF IR R

10 GHz 100 GHz 1 THz 10 THz 100 THz 1 GHz f0

2

4

6

8

Quantum noise

11

The ratio of the temperature dependent and temperature

independent parts of the Callen-Welton equation shows that at 0

K there still exists some noise compared to the Nyquist noise level

at Tstrd = 290 K (standard temperature: k Tstrd = 4.001021)

10 Log dB. 2

eh f /k T 1

f, Hz

Ratio, dB

102 104 106 108 1010 1012

20

40

60

80

100

120

0

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

12

An equivalent noise bandwidth, B , is defined as the bandwidth

of an equivalent-gain ideal rectangular filter that would pass as

much noise power as the filter in question.

By this definition, the B of an ideal filter is its actual bandwidth.

For practical filters, B is greater than their 3-dB bandwidth. For

example, an RC filter has B = 0.5 fc, which is about 50%

greater than its 3-dB bandwidth.

As the filter becomes more selective (sharper cutoff

characteristic), its equivalent noise bandwidth, B,

approaches the 3-dB bandwidth.

D. Equivalent noise bandwidth, B

Reference: [4]

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

13

R

C

en o( f )

fc = = f3dB 1

2 RC

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

en in

=en in2 0.5 fc

Vn o rms2 = en o

2( f ) d f 0

= en in2H( f )2 d f

0

Example: Equivalent noise bandwidth of an RC filter

=en in2

1

1) + f / fc (2d f

0

Vn o rms2 = en in

2 B

14

fc

2 4 6 8 10

1

Equal areas

1

0.5

f /fc

B = 0.5 fc 1.57 fc R

C

en o

0

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

en in

en o2

en in2

0.01 0.1 1 10 100

0.1

1

f /fc

fc

B

0.5 Equal areas

en o2

en in2

fc = = f3dB 1

2 RC

Example: Equivalent noise bandwidth of an RC filter

15

Two first-order independent stages B = 1.22 fc.

Butterworth filters:

H( f )2= 1

1 ) +f / fc (2n

Example: Equivalent noise bandwidth of higher-order filters

First-order RC low-pass filterB = 1.57 fc.

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

second order B = 1.11 fc.

third order B = 1.05 fc.

fourth order B = 1.025 fc.

16

Amplitude spectral density of noise:

en = 4 k T R [V/Hz].

Noise voltage:

Vn rms = 4 k T R fn [V].

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

en = 0.13R [nV/Hz].

At room temperature:

17

Vn rms = 4 k T 1k 1Hz 4 nV

Vn rms = 4 k T 50 1Hz 0.9 nV

Vn rms = 4 k T 1M 1MHz 128 V

Examples:

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

18

1) First-order filtering of the Gaussian white noise.

Input noise pdf Input and output noise spectra

Output noise pdf Input and output noise vs. time

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

E. Normalization of the noise pdf by dynamic networks

19

Input noise pdf Input noise autocorrelation

Output noise pdf Output noise autocorrelation

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

1) First-order filtering of the Gaussian white noise.

20

Input noise pdf Input and output noise spectra

Output noise pdf Input and output noise vs. time

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

2) First-order filtering of the uniform white noise.

21

Input noise pdf Input noise autocorrelation

Output noise pdf Output noise autocorrelation

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

2) First-order filtering of the uniform white noise.

22

Different units can be chosen to describe the spectral density of

noise: mean square voltage (for the equivalent Thévenin noise

source), mean square current (for the equivalent Norton noise

source), and available power.

en2 = 4 k T R [V2/Hz],

in2 = 4 k T/ R [I2/Hz],

na = k T [W/Hz] . en

2

4 R

F. Noise temperature, Tn

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

23

Any thermal noise source has available power spectral density

na( f ) k T , where T is defined as the noise temperature, T = Tn.

It is a common practice to characterize other, nonthermal

sources of noise, having available power that is unrelated to a

physical temperature, in terms of an equivalent noise

temperature Tn:

Tn ( f ) .

na2( f )

k

Then, given a source's noise temperature Tn,

na2( f ) kTn ( f ) .

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

24

Example: Noise temperatures of nonthermal noise sources

Cosmic noise: Tn= 1 … 10 000 K.

Environmental noise: Tn(1 MHz) = 3108 K.

T

Vn2( f ) = 320 2(l/) k T = 4 k T R

l

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

25

Ideal capacitors and inductors do not dissipate power and then

do not generate thermal noise.

For example, the following circuit can only be in thermal

equilibrium if enC = 0.

G. Thermal noise in capacitors and inductors

R C

Reference: [2], pp. 230-231

enR enC

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

26

Reference: [2], p. 230

In thermal equilibrium, the average power that the resistor

delivers to the capacitor, PRC, must equal the average power that

the capacitor delivers to the resistor, PCR. Otherwise, the

temperature of one component increases and the temperature of

the other component decreases.

PRC is zero, since the capacitor cannot dissipate power. Hence,

PCR should also be zero: PCR [enC( f ) HCR( f ) ]2/R where

HCR( f ) R /(1/j2f+R). Since HCR( f ) , enC ( f ) .

R C

enR enC

f f

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

27

Ideal capacitors and inductors do not generate any thermal

noise. However, they do accumulate noise generated by

other sources.

For example, the noise power at a capacitor that is connected to

an arbitrary resistor value equals kT/C:

Reference: [5], p. 202

R

C VnC

enR

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

H. Noise power at a capacitor

VnC rms 2

= enR2H( f )2 d f

0

4 k T RB

4 k T R 0.5 1

2 RC

VnC rms 2

kT

C

28

The rms voltage across the capacitor does not depend on the

value of the resistor because small resistances have less noise

spectral density but result in a wide bandwidth, compared to

large resistances, which have reduced bandwidth but larger

noise spectral density.

To lower the rms noise level across a capacitors, either

capacitor value should be increased or temperature should be

decreased.

Reference: [5], p. 203

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

VnC rms 2

kT

C

R

C VnC

enR

29

Some feedback circuits can make the noise across a capacitor

smaller than kT/C, but this also lowers signal levels.

Compare for example the noise value Vn rms in the following

circuit against kT/C. How do you account for the difference?

(The operational amplifier is assumed ideal and noiseless.)

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Home exercise:

1nF

1k

1k 1pF

Vn rmsVn o rms

vs

30

Shot noise (Schottky, 1918) results

from the fact that the current is not a

continuous flow but the sum of

discrete pulses, each corresponding

to the transfer of an electron through

the conductor. Its spectral density is

proportional to the average current

and is characterized by a white

noise spectrum up to a certain

frequency, which is related to the

time taken for an electron to travel

through the conductor.

In contrast to thermal noise, shot

noise cannot be reduced by lowering

the temperature.

Reference: Physics World, August 1996, page 22

5.2.2. Shot noise

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

R

I

ii

www.discountcutlery.net

31

Reference: [1]

R

I

t

i

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

Illustration: Shot noise in a conductor

32

Reference: [1]

I

t

iR

Illustration: Shot noise in a conductor

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

I

33

We start from defining n as the average number of electrons

passing a cross-section of a conductor during one second,

hence, the average electron current I = q n.

We assume then that the probability of passing through the

cross-section two or more electrons simultaneously is negligibly

small. This allows us to define the probability that an electron

passes the cross-section in the time interval dt = (t, t + d t) as

P1(d t) = n d t.

Next, we derive the probability that no electrons pass the cross-

section in the time interval (0, t + d t):

P0(t + d t ) = P0(t) P0(d t) = P0(t) (1 n d t).

A. Statistical description of shot noise

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

34

This yields

with the obvious initiate state P1(0) = 0.

This yields

with the obvious initiate state P0(0) = 1.

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

= n P0

d P0

d t

The probability that exactly one electrons pass the cross-

section in the time interval (0, t + d t)

P1(t + d t ) = P1(t) P0(d t) + P0(t) P1(d t)

= P1(t) (1 n d t) + P0(t) n d t .

= n P1 + n P0

d P1

d t

35

In the same way, one can obtain the probability of passing the

cross section electrons, exactly:

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

= n PN + n PN 1

d PN

d t

PN (0) = 0

.

which corresponds to the Poisson probability distribution.

PN (t) = e n t ,)n t( N

N !

By substitution, one can verify that

36

N = 10

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

Illustration: Poisson probability distribution

10 20 30 40 50

0.02

0.04

0.06

0.08

0.1

0.12

PN (t) = e 1 t)1 t( N

N !

t

N = 20 N = 30

0

37

The average number of electrons passing the cross-section

during a time interval can be found as follows

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

N = e n = n e n = n,)n( N

N !

and the squared average number can be found as follows:

n = 0

)n( N 1

)N 1( !n = 1

N 2

= N 2 e n = [N (N 1) + N ] e n

)n( N

N !n = 0

n = 0

)n( N

N !

=nne n = nn. n = 2

)n ( N 2

)N 2( !

38

We now can find the average current of the electrons, I, and its

variance, irms2:

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

I = i = (q /N= q n,

irms2 = (q /N

2= (q /.

The variance of the number of electrons passing the cross-

section during a time interval can be found as follows

N2 = N

2 N)2 = n.

39

Hence, the spectral density of the shot noise

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

In2 = 2 q fs.

in( f ) = 2 q .

B. Spectral density of shot noise

Assuming = 1/ 2 fs, we finally obtain the Schottky equation for

shot noise rms current

405. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

C. Shot noise in resistors and semiconductor devices

In devices such as tunnel junctions the electrons are transmitted

randomly and independently of each other. Thus the transfer of

electrons can be described by Poisson statistics. For these

devices the shot noise has its maximum value at 2 q I.

Shot noise is absent in a macroscopic, metallic resistor because

the ubiquitous inelastic electron-phonon scattering smoothes out

current fluctuations that result from the discreteness of the

electrons, leaving only thermal noise.

Shot noise may exist in mesoscopic (nm) resistors, although at

lower levels than in a tunnel junction. For these devices the

length of the conductor is short enough for the electron to

become correlated, a result of the Pauli exclusion principle. This

means that the electrons are no longer transmitted randomly, but

according to sub-Poissonian statistics.

Reference: Physics World, August 1996, page 22

41

The most general type of excess noise is 1/f or flicker noise.

This noise has approximately 1/f spectrum (equal power per

decade of frequency) and is sometimes also called pink noise.

1/f noise is usually related to the fluctuations of the devise

properties caused, for example, by electric current in resistors

and semiconductor devises. Curiously enough, 1/f noise is

present in nature in unexpected places, e.g., the speed of ocean

currents, the flow of traffic on an expressway, the loudness of a

piece of classical music versus time, and the flow of sand in an

hourglass.

Reference: [3]

5.2.3. 1/f noise

Thermal noise and shot noise are irreducible (ever present)

forms of noise. They define the minimum noise level or the

‘noise floor’. Many devises generate additional or excess noise.

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

No unifying principle has been found for all the 1/f noise sources.

42

References: [4] and [5]

In electrical and electronic devices, flicker noise occurs only

when electric current is flowing.

In semiconductors, flicker noise usually arises due to traps,

where the carriers that would normally constitute dc current flow

are held for some time and then released. Although both bipolar

and MOSFET transistors have flicker noise, it is a significant

noise source in MOS transistors, whereas it can often be

ignored in bipolar transistors.

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

43

An important parameter of 1/f noise is its corner frequency, fc,

where the power spectral density equals the white noise level.

A typical value of fc is 100 Hz to 1 kHz.

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

f, decades

in 2( f ), dB

fc

White noise

Pink noise

44

References: [4] and [5]

Flicker noise is directly proportional to the density of dc (or

average) current flowing through the device:

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

in2( f ) , a A J

2

f

where a is a constant that depends on the type of material, and

A is the cross sectional area of the devise.

This means that it is worthwhile to increase the cross section of

a devise in order to decrease its 1/f noise level.

455. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

For example, the spectral power density of 1/f noise in resistors

is in inverse proportion to their power dissipating rating. This is

so, because the resistor current density decreases with square

root of its power dissipating rating:

f, decadesfc

White noise

1 Ain 1W ( f )2 a A J

2

f

1 1 W

in 1W2( f ), dB

465. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

For example, the spectral power density of 1/f noise in resistors

is in inverse proportion to their power dissipating rating. This is

so, because the resistor current density decreases with square

root of its power dissipating rating:

1 1 W1 Ain 1W

2( f ) a A J 2

f

1 9 W

1/3 A

1/3 A

1/3 A

1 A in 9W2( f ) a A J

2

9 f

f, decades

in 1W2( f ), dB

fc

White noise

475. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

f, decadesfc

For example, the spectral power density of 1/f noise in resistors

is in inverse proportion to their power dissipating rating. This is

so, because the resistor current density decreases with square

root of its power dissipating rating:

White noise

1 A

1 9 W

1/3 A

1/3 A

1/3 A

1 A in 9W2( f ) a A J

2

9 f

in 1W2( f ) a A J

2

f

1 1 W

in 1W2( f ), dB

485. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

Example: Simulation of 1/f noise

Input Gaussian white noise Input noise PSD

Output 1/f noise Output noise PSD

495. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

Example: Simulation of 1/f noise

Filter

0 +1 ij

1kR

10kfc C

1/(2*pi*x) RC

j(2 pi i ) j(2 pi i )RC

1u

1

0

Real

0

100000

2 1

20

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