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5. SOURCES OF ERRORS. 5.2. Noise types
1
5.2. Noise types
x+D x
Measurement
Object
Matching
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.
Measurement
System
Influence
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 in Sections 5.2 and 5.3 the
most fundamental types of noise and its characteristics.
MEASUREMENT THEORY FUNDAMENTALS. Contents
5. Sources of errors
5.1. Impedance matching
5.4.1.
5.4.2.
5.4.3.
5.4.4.
5.2.
Anenergetic matching
Energic matching
Non-reflective matching
To match or not to match?
Noise types
5.2.1. Thermal noise
5.2.2. Shot noise
5.2.3. 1/f noise
5.3.
Noise characteristics
5.3.1. Signal-to-noise ratio, SNR
5.3.2. Noise factor, F, and noise figure, NF
5.3.3. Calculating SNR and input noise voltage from NF
5.3.4. Vn-In noise model
5.4.
Noise matching
5.4.1. Optimum source resistance
5.4.2. Methods for the increasing of SNR
5.4.3. SNR of cascaded noisy amplifiers
2
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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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.
Reference: [1]
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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Example: Resistor thermal noise
vn(t)
T0
vn(t)
6s
Vn
R
V
t
f (vn)
2s
Normal distribution
according to the
central limit theorem
2R(t)
en2
White (uncorrelated)
noise
0
f
0
t
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
5
A. Noise description based on the principles of
thermodynamics and statistical mechanics (Nyquist, 1828)
To calculate the thermal noise power density, enR2( f ), of a
resistor, which is in thermal equilibrium with its surrounding, we
temporarily connect a capacitor to the resistor.
Real resistor
R
Ideal, noiseless resistor
enC
C
enR
T
T
Noise source
From the point of view of thermodynamics, the resistor and the
capacitor interchange energy:
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
6
Illustration: The law of the equipartition of energy
z
Each particle has
three degrees
of freedom
mivi 2
2
m v2
2
y
x
In thermal equilibrium:
mi vi 2 m v 2
kT
=
=
3
2
2
2
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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Illustration: Resistor thermal noise pumps energy into the capacitor
z
Each
particle
(mechanical
equivalents of
electrons in
the resistor)
has three
degrees
of freedom
mivi 2
2
CV 2
2
The
particle
(a mechanical
equivalent of
the capacitor)
has a single
degree of
freedom
y
x
In thermal equilibrium:
C VC 2 k T
2 = 2
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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H( f ) =
Real resistor
R
Ideal, noiseless resistor
enC
enC ( f )
enR ( f )
C
enR
T
T
Noise source
In thermal equilibrium:
C VC 2 k T
2 = 2
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:
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
C VC 2 C vnC (t) 2
C snC 2
kT
=
=
2 =
2
2
2
9
 snC
2
kT
=
.
C
According to the Wiener–Khinchin theorem (1934), Einstein
(1914),

snC 2 = RnC (0) =  enR 2( f ) H(j2p f)2 e j 2p f t d f
0
= enR

2( f )
kT
enR2( f )
1
d
f
=
=
.
 1+ (2p f RC)2
C
4 RC
0
Power spectral density of resistor noise:
enR2( f ) = 4 k T R [V2/Hz].
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
10
B. Noise description based on Planck’s law for blackbody
radiation (Nyquist, 1828)
enR P
2( f ) =
4R
hf
eh f /k T - 1
[V2/Hz].
A comparison between the two Nyquist equations:
enR P( f )2
enR( f )2
1
R = 50 W,
C = 0.04 f F
0.8
0.6
0.4
0.2
1 GHz
10 GHz
SHF
100 GHz
EHF
1 THz
10 THz
IR
100 THz
R
f
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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C. Noise description based on quantum mechanics
(Callen and Welton, 1951)
The Nyquist equation was extended to a general class of
dissipative systems other than merely electrical systems:
eqn
2( f ) =
4R
hf
eh f / k T
hf
[V2/Hz]
+
-1 2
Zero-point energy
eqn ( f )2
enR ( f )2
8
6
4
Quantum noise
2
0
1 GHz
10 GHz
SHF
100 GHz
EHF
1 THz
10 THz
IR
100 THz
R
f
 f(T)
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
12
The ratio of the temperature dependent and temperature
independent parts of the Callen-Welton equation shows that
at 0 K and f  0 there still exists some noise compared to the
Nyquist noise level at T0 = 290 K (standard temperature:
k
T0 = 4.0010-21)
10 Log
0
-20
Remnant noise
at 0 K, dB
-40
-60
-80
-100
-120
100
103
2
eh f / k Tstd
-1
106
[dB]
109
f, Hz
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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D. Equivalent noise bandwidth, B
An equivalent noise bandwidth, B , is defined as the bandwidth
of an equivalent-gain ideal rectangular filter that would pass as
much power of white noise as the filter in question:

B
0
IA( f )I2
IAmax
I2
df .
IA( f )I2
IAmax I2
B
Lowpass
B
Bandpass
1
Equal areas
0.5
Equal areas
f
0
Df
Df
linear scale
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
14
Example: Equivalent noise bandwidth of an RC filter
R
enR
en o( f )
C

Vn o 2 =  en o2( f ) d f
0

=  en in2H( f )2 d f
0
1
fc =
= D f3dB
2p RC

= en in
2
1
 1 + ( f / f )2 d f
c
0
= en in2 0.5 p fc
Vn o 2 = en in2 B
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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Example: Equivalent noise bandwidth of an RC filter
R
en o
en o2
en in2
B = 0.5 p fc = 1.57 fc
1
C
en in
Equal areas
0.5
fc
fc =
1
= D f3dB
2p RC
0
en o2
en in12
1
2
4
6
8
10
f /fc
B
Equal areas
0.5
0.1
fc
0.01
0.1
1
10
100
f /fc
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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Example: Equivalent noise bandwidth of higher-order filters
First-order RC low-pass filter
B = 1.57 fc.
Two first-order independent stages
B = 1.22 fc.
Butterworth filters:
H( f
)2
=
1
second order
B = 1.11 fc.
third order
B = 1.05 fc.
fourth order
B = 1.025 fc.
1+ ( f / fc )2n
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
Amplitude spectral density of noise, rms/Hz0.5:
en =  4 k T R [V/Hz].
At room temperature:
en = 0.13 R [nV/Hz].
Noise voltage, rms:
Vn =  4 k T R B [V].
17
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
Examples:
Vn =  4 k T 1MW 1MHz = 128 mV
Vn =  4 k T 1kW 1Hz = 4 nV
Vn =  4 k T 50W 1Hz = 0.9 nV
18
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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E. Normalization of the noise pdf by dynamic networks
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
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1) First-order filtering of the Gaussian white noise.
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
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2) First-order filtering of the uniform 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
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2) First-order filtering of the uniform white noise.
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
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F. Noise temperature, Tn
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.
R
en2 = 4 k T R [V2/Hz],
en ( f )
in ( f )
in2 = 4 k T/ R [A2/Hz],
R
R
na( f )
en ( f )
en2
na 
= k T [W/Hz].
4R
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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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:
na( f )
na ( f )
Tn ( f ) 
.
k
en( f )
Nonthermal sources of noise
Then, given a source's noise temperature Tn,
n a ( f )  k Tn ( f ) .
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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Example A: Noise temperatures of nonthermal noise sources
1. Environmental noise: Tn(1 MHz) can be as great as 3108 K.
2. Antenna noise temperature:
T
l
vn2( f )
= 320 p2(l/l)2 k T [W J / Hz]= 4 k Ta RS [V2/Hz]
l << l
RS = 80 p2(l/l)2 [W] is the radiation resistance.
Reference: S. I. Baskakov.
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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Example B: Antenna noise temperature, Ta (sky contribution only)
104
Galactic noise limit
Antenna noise temperature, Ta (K)
TG  100 l2.4
103
O2
300
H2O
102
101
Quantum noise limit
TQ = h f / k
100
100
101
102
103
104
105
106
107
108
109
Frequency, MHz
S. Okwit, “An historical view of the evolution of low-noise concepts and techniques,” IEEE Trans. MT&T, vol. 32, pp. 1068-1082, 1984.
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
27
Example C: Noise performance of any antenna/receiving system, Top
Antenna
Ta
Receiver
RS
G
l
vS
Ta + Te
Noiseless
Ta
is the antenna noise temperature, K,
Te
is the effective input receiver temperature, K,
Top= Ta+Te is the operating noise temperature, K.
Compare: a 75 K receiver versus a 80 K receiver, vis-à-vis
a 0.999-dB receiver versus a 1.058-dB receiver.
S. Okwit, “An historical view of the evolution of low-noise concepts and techniques,” IEEE Trans. MT&T, vol. 32, pp. 1068-1082, 1984.
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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G. Thermal noise in capacitors and inductors
We will show in this section that in thermal equilibrium any
system that dissipates power generates thermal noise; and
vice versa, any system that does not dissipate power does not
generate thermal noise.
For example, ideal capacitors and inductors do not dissipate
power and then do not generate thermal noise.
To prove the above, we will show that the following circuit can
only be in thermal equilibrium if enC = 0.
R
enR
C
enC
T
T
Reference: [2], pp. 230-231
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
R
29
PRC
C
PCR
enR
enC
T
T
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 = 0, where
HCR( f ) = R /(1/j2pf+R). Since HCR( f )  0, enC ( f ) = 0.
f>0
f>0
Reference: [2], p. 230
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
30
H. Noise power at a capacitor
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:

VnC =  enR2H( f )2 d f
2
R
0
VnC
enR
T
C
= 4 kTR B
= 4 k T R 0.5 p
VnC 2 =
1
2p RC
kT
C
Reference: [5], p. 202
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
31
The rms voltage VnC 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.
R
VnC
enR
C
VnC 2 =
kT
C
T
Reference: [5], p. 203
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
33
5.2.2. Shot noise
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.
D
I
ii
www.discountcutlery.net
Reference: Physics World, August 1996, page 22
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
34
Illustration: Shot noise in a diode
D
i
I
t
Reference: [1]
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
35
Illustration: Shot noise in a diode
D
i
I
I
t
Reference: [1]
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
36
A. Statistical description of shot noise
We start from defining n as the average number of electrons
passing the p-n junction of a diode during one second, hence,
the average electron current I = q n.
We assume that the probability of passing two or more
electrons simultaneously is negligibly small, P>1(d t) = 0. This
allows us to define the probability that an electron passes the
junction in the time interval d t = (t, t + d t) as P1(d t) = n d t
(d t is approaching the time taken for an electron to travel
over the junction, < 1 ns).
vDt
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
37
Next, we derive the probability that no electrons pass the
junction in the time interval (0, t + d t):
P0(t + d t ) = P0(t) P0(d t) = P0(t) [1- P1(d t)] = P0(t) - P0(t) n d
t.
This yields:
d P0
dt
= - n P0
with the obvious initiate state P0(0) = 1.
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
38
The probability that an electron passes the junction 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 .
This yields
d P1
dt
= - n P1 + n P0
with the obvious initiate state P1(0) = 0.
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
39
In the same way, one can obtain the probability of passing the
junction N electrons:
d PN
dt
= - n PN + n PN -1
.
PN (0) = 0
By substitution, one can verify that
PN (t) =
(n t) N
N!
e- n t ,
which corresponds to the Poisson probability distribution.
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
40
Illustration: Poisson probability distribution
PN (t) =
(n t) N
N!
e- n t
0.14
n = 10
t=1
0.12
0.1
0.08
0.06
0.04
0.02
0
5
10
15
vDt
20
N
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
41
Illustration: Poisson probability distribution
PN (t) =
(n t) N
N!
e- n t
0.1
n = 10
t = 0.01
0.08
0.06
0.04
0.02
0
5
10
15
vDt
20
N
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
42
The average number of electrons passing the junction during a
time interval (0,t ) can be found as follows:

(nt ) N
N= 0
N!
SN
Nt =
e - n t = nt e - n t

(nt ) N-1
S (N -1) !
N= 1
= nt ,
= e nt
and the average squared number can be found as follows:

Nt 2 =
SN
2
(nt ) N
N!
N= 0
= (nt )2


S [N (N -1) + N ]
e- nt =
(n t ) N-2
N= 0
S (N -2) ! e
N= 2
(nt ) N
- nt
+ nt = (nt )2+ nt.
N!
e- nt
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
43
The variance of the electron flow during the time interval t can
be found as follows:
sN2 = Nt2 - ( Nt )2 = nt.
We now can find the average current of the electrons, I, and its
variance, irms2:
I=
q Nt
in rms2
t
=
= q n,
q sN
t
2
=
q 2 nt
t
2
=
qI
t
.
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
44
Illustration: The relationship between t and B
Let us suppose that we measure the shot noise at the
output of an ideal low-pass filter. Then according to the
Nyquist criterion:
IA( f )I
Noise bandwidth
f
B
The highest noise frequency waveform
i
in rms =
I=qn
The maximum measurement time
t
t = ?1/2B
in z-p
2t =? 1/B
t
qI
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
B. Spectral density of shot noise
Assuming t = 1/( 2 B), we finally obtain the Schottky equation
for shot noise rms current
in rms2 = 2 q I B.
Hence, the spectral density of the shot noise
in( f ) =  2 q I .
45
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
46
C. Shot noise in resistors and semiconductor devices
Through a p-n junction (or any other potential barrier), the
electrons are transmitted randomly and independently of each
other. Thus the transfer of electrons can be described by Poisson
statistics. In this case, the shot noise has its maximum value at
in2( f ) = 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 does exist in mesoscopic (nm) resistors, although at
lower levels than in a diode 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
47
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise
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 devices generate additional or excess noise.
The most general type of excess noise is 1/f or flicker noise.
This noise has approximately 1/f power 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 device
properties caused, for example, by electric current in resistors
and semiconductor devices.
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.
No unifying principle has been found for all the 1/f noise sources.
Reference: [3]
48
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise
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 bipolar, JFET, and MOSFET transistors have flicker
noise, it is a significant noise source in MOS transistors,
whereas it can often be ignored in bipolar transistors (and some
modern JFETs).
References: [4] and [5]
49
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise
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 ff is 100 Hz to 1 kHz (MOSFET: 100 kHz).
in ( f ), dB
-10 dB/decade
Pink noise
White noise
f, decades
ff
50
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise
Flicker noise is directly proportional to the dc (or average)
current flowing through the device:
in2( f ) =
Kf m I m
fn
where Kf is a constant that depends on the type of material,
1 < m < 3, and 1 < n < 3.
References: [4] and [5]
51
5. 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.
Example: Let us compare 1/f noise in 1 W, 1 W and 1 W, 9 W resistors
for the same 1 A dc current:
in 1W ( f ) =
1A
Kf I
f 0.5
1 W, 1 W
in 9W ( f ) = ?
1A
1 W, 9 W
52
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise
in 9W ( f ) = ?
1A
1 W, 9 W
in 1W /3
1/3 A
1A
{3[(in 1W/3)·1]2}0.5 = in 1W/30.5
3W
53
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise
{3[(in 1W/30.5)/3]2}0.5 = in 1W/(30.5·3)
in 9W ( f ) = {3[(in 1W/(30.5·3)]2}0.5 = in 1W/3
3W
1A
1 W, 9 W
in 9W 2( f ) = in 1W 2( f )/9
in 1W2( f ), dB
1 W, 1 W
Pink noise
1 W, 9 W
White noise
f, decades
ff (9 W)
ff (1 W)
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5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise
Example: A simulation of 1/f noise
Input Gaussian white noise
Input noise PSD
Output 1/f noise
Output noise PSD
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise
Example: A simulation of 1/f noise
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