Technical Report #USC 02-0511
Electronic Noise Characterization −
Part I: System Concepts and Theory
Dr. John Choma
Professor of Electrical Engineering
Scholar in Residence, Raytheon Space and Airborne Systems Electronics Center
University of Southern California
Ming Hsieh Department of Electrical Engineering
University Park: Mail Code: 0271
Los Angeles, California 90089–0271
213–740–4692 [USC Office]
626–915–0944 [Home Fax]
818–384–1552 [Cell]
johnc@usc.edu
ABSTRACT:
This report provides detailed discussions of the system concepts that mathematically
underpin a characterization of electronic noise phenomena in electronic systems and
circuits. Noise generated by thermal effects in two-terminal resistances, shot effects in
active elements, and flicker noise phenomena exuding a generally troublesome inverse
frequency dependence are studied at some depth. The analytical means for assessing
the impact that these parasitic noise sources exert on the idealized (noiseless) properties of commonly encountered active networks receive considerable attention. This
attention is synergistic with the fundamental goal of forging generalized, but
mathematically tractable, circuit design strategies that mitigate, or at least minimize,
noise-induced degradation of observable circuit performance.
July 2011
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1.0.
University of Southern California Viterbi School of Engineering
Choma
INTRODUCTION
The study of electronic noise in high performance analog integrated circuits requires a
consideration of at least three issues. The first of these issues is the identification of the principle
noise sources in an electronic circuit. To this end, we shall address the three most common
sources of electrical noise. The first of these noise sources is thermal noise −often called
Johnson noise or Nyquist noise−[1]-[2], which materializes in virtually all passive conductors.
Unlike most other noise sources, thermal noise does not require that the afflicted device conduct
current conduction. A second noise source is shot noise, which is evidenced whenever a device
conducts a nonzero static current and presents a potential barrier that transported charges must
overcome to sustain the directed current. Finally, flicker noise is principally observed at relatively low frequencies. Like shot noise, flicker noise requires a directed current flow and in general, the amount of flicker noise increases with increases in current level. An especially troublesome aspect of flicker phenomena is that the “low” frequency passband over which it can be a
significant, if not dominant, source of electrical noise in an electronic system can extend to
distressingly higher frequencies as device unity gain frequency capabilities expand. Popcorn
noise (also termed burst noise), which seemingly derives from metal ions that contaminate a
semiconductor body, is not addressed in this report owing to the unavailability of satisfying
analytical formulae that relate burst phenomena to relevant physical constraints or measureable
electrical parameters. Fortunately, popcorn noise, which has been observed in gold-doped bipolar junction transistors, is generally manifested only sporadically and only at frequencies that are
far below the signal frequencies associated with broadband and radio frequency (RF) bandpass
amplifiers. Finally, a consideration of excess noise, which is a kind of flicker noise effect exhibited by carbon composition resistors, is similarly spurned in subsequent discussions[3].
The second noteworthy issue implicit to this study of electrical noise is the designoriented analysis promoting the phenomenological understanding of noise properties that underpins prudent low noise design. At possible risk of oversimplification at this early juncture of our
work, the dominantly undesirable effect of electrical noise phenomena is that noise voltages and
currents coalesce with applied signals to the point that signal amplitudes can become comparable
to, or even dwarfed by, noise. The analyses undertaken herewith therefore focus on theoretic
methods and practical design strategies that preclude signal obscurity in the face of electrical
noise, along with ensuring the reliable detection and processing of signals immersed in inherently noisy environments. We shall demonstrate that this signal detection and processing requires judicious biasing and other optimal design strategies that minimize the effective input
noise voltages and currents applied to an electronic system. To this end, we shall introduce
generalized design strategies that promote lowering the noise floor of a network to a level that
ensures reliable observability and efficient processing of small input signals.
The final issue is addressed by the second part of this report. This vital engineering issue is the documentation and application of the circuit level noise models that underpin our
analytical endeavors. To this end, we shall concentrate on noise models for the passive resistor,
the PN junction diode, and the MOSFET. We shall then exploit these models to assess the noise
characteristics and properties of a few commonly encountered amplifiers.
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TIME DOMAIN REPRESENTATION[4]
2.0.
Electrical noise is a random voltage or current waveform. The amplitudes of these
noise voltages and currents typically subscribe to a Gaussian distribution and therefore, they cannot be quantified at specific times. Because a clarion analytical representation of noise in the
time domain is not practicable, we have few engineering options but to resort to such statistical
characterization metrics as the mean square value of noise and its companion, the root mean
square (RMS) value. We shall also deem it profitable to exploit such additional engineering
performance metrics as the signal -to- noise ratio (SNR), the noise figure or noise factor, and
the correlation coefficient associated with the algebraic combination of noise waveforms. In the
subsections that follow, we examine these metrics and their implications for the special, but commonly encountered, case of a noise voltage or current having zero average value. We proffer
that a zero average noise value is common because if a waveform is inherently random in nature,
the subject waveform has equal probabilities of being positive or negative over time.
2.1.
ROOT MEAN SQUARE VALUE OF NOISE
The root mean square (RMS) value, Erms, of a noise voltage waveform, symbolized abstractly in the time domain as en(t), is
T
E
rms

1
e 2 (t)dt 
T n

0
e 2 (t) ,
(1)
n
where e 2 (t) , which denotes the mean-squared value of en(t), is nonzero even if en(t) projects zero
n
average value. We point out in the interest of clarity that throughout this report, we shall invoke
the symbol, “E,” to designate the RMS value of a random voltage waveform. Moreover, “v”
and “i” designate time deterministic voltage and current, respectively, while “e” and “j” are
their respective random, or noise, counterparts. We note from (1) that the square of an RMS voltage returns the mean square value of the subject voltage. In (1), T is a sufficiently long averaging period. If the random process underlying en(t) is ergodic1, the value, T, selected for the
integration process is unimportant. In actual laboratory practice, however, large T generally
produces more meaningful mathematical results than does a small averaging interval. In the
interest of completeness, a noise current waveform, in(t), has an RMS current value, Jrms, given
by
J
rms

1
T
T
2
 jn (t)dt
0

j 2 (t) .
(2)
n
Once again in the interest of analytical clarity, symbol “J” designates the RMS value of noise
current, whence the square of the RMS current value is the mean square noise current.
For all practical purposes, the RMS voltage or current stipulates the noise power associated with a random voltage or current. Indeed, the square of the RMS voltage developed
across a 1-ohm resistance and the square of the RMS current conducted by the same 1-ohm resis-
1
Ergodicity refers to a random process in which every sequence or sample of the time domain variable is equally
representative of the entire time domain waveform.
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tance can be thought of as the normalized noise power, Pn, dissipated in the resistance. In
particular, we note that
P 
E
n
2
rms 
1
or very simply,
P  E
n
2
 1   ,
 J rms
(3)
2
2
 J
,
rms
rms
(4)
where it is understood that Pn is in units of watts despite being numerically equal to the square of
either RMS noise voltage or RMS noise current. When expressed in terms of the square of the
RMS voltage developed across a resistive branch, the normalized noise power scales in proportion to branch conductance. On the other hand, Pn scales with branch resistance when power is
expressed in terms of the squared RMS current flowing through the resistive branch.
2.2.
SIGNAL TO NOISE RATIO AND NOISE FIGURE
The signal to noise ratio, SNR, at a stipulated circuit port is commonly (but not universally) expressed in units of decibels (dB) and is given by
 2 
V 2 
V

 v (t) 
 rms   20 log  rms  ,
(5)
SNR  10 log 
10
log


 E 2 
E

 e 2 (t) 
 rms 
 rms 
 n 
where the port of interest is understood to support a time-deterministic signal, v(t), which is immersed in a noise voltage. The normalized power of the time deterministic signal is V 2 , while
the normalized noise power implicit to the signal prevailing at the port of interest is
rms
E2 .
rms
It is
intuitively clear that we wish the SNR to be as large as possible. For example if the RMS signal
voltage established at a given network port is 5 mV and the corresponding RMS noise voltage is
1 μV, the SNR is a laudable 74 dB. But if Vrms = Erms, SNR = 0 dB, which suggests the
improbability of discriminating between signal and noise voltages because neither signal nor
noise is dominant at the subject port. Of course, if the SNR is not expressed in units of decibels,
it assumes a simple numerical value, and (5) becomes
2
V

V2
rms
  rms ,
SNR  
2
E

E
 rms 
rms
which is effectively a ratio of signal power -to- noise power evidenced at the input port.
(6)
The SNR is a first cousin to the noise factor, F of a network. A demonstration of the
relationship between signal -to- noise ratio and noise figure best derives from a consideration of
the linear network abstraction in Figure (1). In this diagram, a signal source applied to the linear
network manifests an RMS signal voltage at the input port of Virms and a resultant signal response
at the output port of Vorms. If the linear network is an active circuit, input/output (I/O) gain is
plausible so that Vorms > Virms. But all networks are unavoidably contaminated by noise, which
means, among other things, that the reliable detection and processing of the applied input signal
is rendered potentially problematic. If, in the present case, the RMS noise voltage measured at
the input port and due exclusively to noise associated with the applied input signal is Eirms, the
resultant signal -to- noise ratio evidenced at this input port is
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Input Signal
Source
Virms
Eirms
*
Output
Port
Linear
Network
*
Eorms
Vorms
Load
Input
Port
Figure (1). Symbolic depiction of a linear network to which an input signal of RMS
value Virms is applied to the input port and a corresponding signal of RMS
value Vorms results at the output port. Both input and output ports are
contaminated by noise voltages (abstracted here as “*” within the voltage
generator symbol) whose respective RMS values are Eirms and Eorms.
2
V
irms
SNR 
,
i
2
E
irms
(7)
where we hope that SNRi is substantively larger than unity; that is, we wish the signal to dwarf
the noise at the input port, as opposed to noise blurring the signal identified for processing. In
response to the source-derived input noise measured by voltage Eirms and noise generated within
the network undergoing investigation, an RMS output port noise voltage of Eorms is observed,
thereby presenting an output port signal -to- noise ratio of
2
orms
SNR 
.
o
2
E
orms
V
(8)
Then, the noise factor of the linear network undergoing investigation is defined numerically as
SNR
i ,
F 
(9)
SNR
o
or in decibel units (whereupon it becomes known as the noise figure) as
 SNR 
i .
(10)
F  10 log 
 SNR 
o

Thus, the noise figure is little more than the decibel value of the noise factor. Note, that because
the RMS signal and noise voltages appear as squared quantities in the signal -to- noise ratio
expressions of (7) and (8), these signal -to- noise ratios are given as ratios of mean squared (or
square of RMS quantities) signal -to- noise quantities.
Under ideal circumstances, which are observed only within the confines of hallowed
academic halls of ivy, the linear network under present consideration generates no noise in and
of itself. Under such a metaphysical circumstance, the only noise observed at the output port is
the noise incident with the input port, multiplied, or amplified, by the I/O transfer function, or
gain in the case of a linear active configuration. But since the transfer function multiplying the
input noise is the same as that which multiplies the input port signal, the signal -to- noise ratio,
SNRo, observed at the output port is necessarily equal to SNRi, whence F = 1 (or 0 dB).
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In contrast with the foregoing ideal consideration, if noise is generated within the linear
network, the net output noise is larger than the amplified input noise, which renders SNRo (which
is inversely proportional to noise level) smaller than SNRi. Accordingly, F exceeds 0 dB for
practical networks while for noiseless circuits and systems, noise figure F assumes its idealized
value of 0 dB. It should be clearly understood that the ideal situation for which F = 0 dB does
not imply zero noise level at the output port. Instead, it implies only that the noise contributed
by the linear network to the net noise level manifested at the network output port is negligible in
comparison to the output noise level incurred exclusively because of the amplified input noise.
2.3.
CORRELATION COEFFICIENT
A commonly encountered situation in the noise analysis of active or passive structures
is the case when two noise sources algebraically superimpose with one another. Figure (2a) illustrates this superposition issue for noise voltages en1(t) and en2(t), whose respective RMS values
are E1rms and E2rms. Figure (2b) is the counterpart depiction of two superimposed noise currents,
jn1(t) and jn2(t) possessed of respective RMS values of J1rms and J2rms. If the two noise voltages
sum to produce the resultant noise response, en(t), as suggested in Figure (2a), its RMS value,
Erms, derives from (11), where we have used (1) as the pertinent mathematical basis:
jn(t)
*
en1(t)
en(t)
jn1(t)
jn2(t)
*
en2(t)
jn(t)
(a).
(b).
Figure (2). (a). Superposition of two noise voltages. No polarity is assigned to the resultant terminal noise voltage, en(t) because of
the random nature of the two noise voltages, en1(t) and en2(t).
(b). Superposition of two noise currents. No direction is ascribed to the flow of the resultant net current, jn(t) because of
the random nature of the two noise currents, jn1(t) and jn2(t).
T
T
2
1 
1  2
2
2
2
E
 e (t) 
e (t)  e (t) dt 
e (t)  e (t)  2e (t)e (t) dt . (11)
rms
n
n1
n2
n1
n2
n1 n2 




T
T
0
0


The integral averaging of the first term on the far right hand side of this expression produces the
mean square value of noise voltage en1(t); that is,
1
2
2
e (t)  E

n1
1rms
T
T
2
 en1(t)dt .
(12)
0
Similarly, the mean square value of noise voltage en2(t) is
1
2
2
e (t)  E

n2
2rms
T
T
2
 en2 (t)dt .
(13)
0
In (11), we introduce the correlation coefficient, κ[5], such that
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T
1
e (t)e (t)dt
T n1 n2

 
0
E
.
E
(14)
1rms 2rms
It can be shown that −1 ≤ κ ≤ 1, where κ = 0 implies that the two noise signals are uncorrelated,
which means that the subject noise voltages derive from mutually independent physical
processes. On the other hand, κ = ±1 asserts maximal, or full, correlation between en1(t) and
en2(t). Equation (11) now becomes
E
2
2
2
 E
E
 2 E
E
.
rms
1rms
2rms
1rms 2rms
(15)
We see that for two uncorrelated noise voltages, for which κ = 0, the mean square value of the
noise voltage sum is simply the sum of the individual mean square noise voltages. On the other
hand, the mean square value of the sum of two fully correlated noise voltages, which engender κ
= ±1, is
E2
rms
 E2
1rms
 E2
2rms
 2E
E
1rms 2rms

 E1rms  E2rms 
2
,
(16)
where the plus sign applies to the case in which en1(t) and en2(t) correlate in phase with one
another, while the minus sign pertains to two noise signals that correlate anti-phase with one
another by.
Of course, for the case of two noise currents, as depicted in Figure (2b), the mean
square value of the noise current sum is, by analogy to (15),
J
2
2
2
 J
J
 2 J
J
,
rms
1rms
2rms
1rms 2rms
(17)
where it is understood that the correlation coefficient, κ, is now
 
3.0.
1
T
T
 jn1(t) jn2 (t)dt
0
J
J
.
(18)
1rms 2rms
FREQUENCY DOMAIN ANALYSIS[4]
Unlike periodic signals that deliver power at only their fundamental frequency and
possibly discrete other frequencies, noise voltages and currents have their power distributed
continuously over a frequency spectrum. This frequency domain distribution of noise power
encourages a characterization of a noise voltage, say en(t), by its noise voltage spectral density,
SV(f), which projects, as a function of frequency, the measured or theoretically discerned mean
square voltage (or normalized power) delivered to an arbitrary load -per- unit of frequency.
Thus, for a noise voltage, the dimensional unit of its noise spectral density is volts2/Hz (in reality, mean square volts/Hz). For a noise current, say jn(t), its noise spectral density, say SI(f), has
units of amps2/Hz ((in reality, mean square amps/Hz). The immediate sidebar to the noise spectral density is that it offers an ostensibly convenient frequency domain alternative to the time domain calculation of mean square value projected by (1) -through- (4). In particular, the mean
square value (or square of the RMS value) of noise voltage en(t), which postures a noise spectral
density of SV(f), is given by
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1
2
E

rms
T
T

0
University of Southern California Viterbi School of Engineering
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
2
2
e (t)dt  e (t) 
n
n
 SV (f)df
(19)
.
0
Of course, the current counterpart to the voltage expression in (19) is
1
2

rms
T
J
T

2
n
2
n

j (t)dt  j (t) 
0
 S I (f)df
(20)
,
0
where SI(f) is the spectral density of noise current abstracted in the time domain as jn(t). Because
of the integrations implicit to the preceding two relationships, we note that assuming finite noise
spectral densities, normalized power, for which mean square voltage or mean square current is a
measure, is not delivered by any noise voltage or noise current at a single, specific frequency.
Rather, such noise power is delivered only over a band, or range, of frequencies.
A simple demonstration of the last contention proves instructive. Assume that the noise
spectral density of a certain noise voltage is
S (f)  K e
V
f f
o
o ,
(21)
where Ko is a constant whose dimension is volts2/Hz, and fo is a constant frequency parameter.
Using (19), we see that the mean square value of this noise voltage is

2
E

rms
 K oe
f f
o df
 K f e
f f 
o
o o
0
 K f
o o
0
.
(22)
Alternatively, we may inquire as to the normalized power delivered by the noise voltage over a
frequency passband extending from f1 -to- f2, where f2 > f1. Such an inquiry is meaningful, for
example, if the noise interfaces with a sharply tuned bandpass filter. Then the normalized noise
power, say Pn, delivered in the frequency interval from f1 -to- f2 is
f
P 
n
2

f
K e
f f
o
o df
 K f e
o o
f f
o
f
f
1
2
1
f f 
 f f
 K f e 1 o  e 2 o 
o o

(23)
f f 
  f1 f o
2 o ,
e
e


rms 

where we have applied (22) to our band limited noise power calculation. Observe in this example that Pn = 0 if f2 = f1; that is the noise power delivered at a single frequency is zero.
 E2
3.1.
INPUT/OUTPUT NOISE PROCESSING
Consider the case in Figure (3a) of a noise voltage, eni(t), applied to the input port of a
linear network whose (I/O) transfer function is H(s). We assume that the linear network itself is
noiseless or, more pragmatically, that we wish to examine only the filtering effect that the considered linear network has on the applied input noise. If the input noise voltage has a noise spectral density of SVi(f) and the resultantly generated output noise, eno(t), displays a noise spectral
density of SVo(f), we proffer
S
Vo
(f)  H  j2πf 
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(24)
S (f) ,
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which exploits several basic principles. The first of these nuances is that since the spectral density of a noise voltage projects mean square volts/Hz, the multiplier linking the input noise spectral density to its resultant output noise spectral density is the squared magnitude of the transfer
function. This requirement follows from the fact that a mean square voltage response can never
be negative, nor can it ever be a complex number. Using the magnitude of a transfer function, as
opposed to a mere transfer function in the complex frequency plane, certainly obviates negative
and complex numerical offerings in (24). Squaring the transfer function magnitude follows
immediately from the fact that the noise spectral density is focused on mean square voltage -perhertz. Embodied within this first principle is the second principle, which is that noise spectral
density observations apply only to steady state operation. Assuming that the considered linear
network is stable, this means that the complex frequency argument, s, in the transfer function
representation of Figure (3a) must be replaced by its steady state counterpart, s = jω = j2πf. Finally, we note that (24) is divorced of any I/O phase information. This circumstance reflects our
focus on purely random phenomena that renders a practical time domain representation intractable, thereby obfuscating time domain shifts between applied noise input and resultant noise output response. The immediate ramification of (24) is a mean square output noise given by2
eni(t)
H(s)
eno(t)
en1(t)
H1(s)
en2(t)
H2(s)
en3(t)
H3(s)
(a).
+
eno(t)
(b).
Figure (3).(a). An input noise voltage, eni(t), applied to a linear network (possibly a filter) whose
I/O transfer function is H(s). The input noise voltage has a noise spectral density of
SVi(f). (b). The application of three uncorrelated noise voltages to three individual
subcircuits whose noise output responses are algebraically summed to produce the
indicated output noise voltage, eno(t).
T


1
2
2
2
2

E
e (t)dt  e (t)  S (f)df 
H  j2πf  S (f) df .
orms
no
Vo
Vi
T no
0
0
0



(25)
In simple street talk, the output noise spectral density of the system illustrated in Figure (3a) is
the input noise spectral density scaled by the squared magnitude of the frequency domain I/O
transfer function. And since the mean square noise voltage is the noise spectral density inte2
We just as easily could have formulated the following conclusion in terms of input noise current and indeed, any
combination of input and output noise voltages and currents.
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grated over all positive real frequencies, the output mean square noise voltage is the frequency
integral over all non-negative frequencies of the product of the input noise spectral density and
the squared magnitude of the system transfer relationship.
Consider now the more complicated system shown in Figure (3b) wherein three
uncorrelated noise voltages, en1(t), en2(t), and en3(t), are applied independently to three linear
subcircuits whose respective transfer functions are H1(s), H2(s), and H3(s). The output responses
of these three subcircuits are algebraically summed to forge the output noise voltage, eno(t). Because the considered noise sources are presumed uncorrelated, the mean square value of the
superposition of their noise output responses is simply the sum of the individual mean square
output responses. Thus, the noise spectral density, SVo(f), of the output noise voltage response in
Figure (3b) derives straightforwardly as
S
Vo
(f) 

H
i
j2πf 
i
2
S (f) ,
(26)
Vi
where we have borrowed from (25) and (15) and have identified Svi(f) as the noise spectral density of the ith input noise voltage, eni(t), in the diagram of Figure (3b). We have also allowed an
open summation over running index i in (26) since it is clear that any number of input noise voltages can interact with the corresponding number of subcircuits, which effectively act as noise
filters. It follows that the net mean square output noise voltage (or square of the output RMS
noise voltage) is, by way of (25),
2
E

orms
3.2.


 SVo (f)df


0 i
0
H
j2πf 
i
2
S (f) df .
Vi
(27)
NOISE EQUIVALENT BANDWIDTH
As we have demonstrated in the preceding section of material, the frequency domain
computation of the mean square noise voltage developed across the output port of a linear network derives from integrating over all non-negative frequencies the output noise spectral density,
SVo(f). Since this output spectral density embodies the product of the input noise spectral density,
SVi(f), and the squared magnitude, |H(j2πf)|2, of the applicable network transfer function, the
requisite integration over frequency is likely to be challenging, particularly if the transfer function embodies a number of significant poles and/or zeros. This cumbersome computation motivates a search for simpler computational methods that apply to most, if indeed not all, noise
analysis undertakings.
To the foregoing end, consider the diagram in Figure (4a), which depicts a linear network to which a noise voltage, eni(t), is applied to the network input port. If the noise spectral
density of eni(t) is SVi(f), as is suggested in the diagram, its exclusive contribution to the resultant
noise spectral density, SVo(f), manifested at the network output port is
S
Vo
(f)  H  j2πf 
2
(28)
S (f) ,
Vi
where, of course, H(j2πf) represents the I/O network transfer function in the steady state. By
concentrating exclusively on the input noise, as opposed to including the effects of noise sources
generated within the network, SVo(f) in (28) effectively invokes the simplifying presumption of a
noiseless network. Equivalently, we can state that the foregoing simplification corresponds to a
network boasting 0 dB noise figure (or unity noise factor). It therefore follows that the mean
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square value of the generated output port noise in the system of Figure (4a) remains pinned by
(25).
Input
Port
SVi (f)
eni (t)
Linear
Network,
H(j2f)
Output
Port
SVo (f)
Input
Port
SVi (f)
eno (t) eni (t)
|H(j2f)|
Brick Wall
Network,
HB(j2f)
Output
Port
SVoeff (f)
enoeff (t)
|HB(j2f)|
Hm
Hm
0.707Hm
f
B
f
(a).
f
(b).
Figure (4). (a). A linear network that generates an output noise voltage response, eno(t), which has a noise
spectral density of SVo(f), to an applied input noise voltage, eni(t), whose noise spectral density
is SVi(f). The subject linear network, whose noise is tacitly ignored, is presumed to be a lowpass structure having a 3-dB bandwidth of B (in Hz). (b). The input noise voltage of (a) applied to an ideal, noiseless, “brick wall” filter that, over a frequency passband of Δf (in Hz),
delivers constant gain magnitude Hm that is identical to that of the lowpass structure in (a).
We now replace the actual network used in Figure (4a) by the mathematical −and notoriously idealistic and physically unrealizable− “brick wall” system in Figure (4b), whose transfer
function, HB(j2πf), evokes a magnitude defined by
H , for 0  f  Δf
m
(29)
H  j2πf  
,
B
0 , for f  Δf
where Hm is identically the zero, or low, frequency gain magnitude of the original network in
Figure (4a). If the input noise applied to the brick wall system is the same as that applied to the
original network, the noise spectral density, SVoeff(f), manifested at the brick wall output port follows as
(f)  H
B
j2πf 
2
S (f) 
H 2 S (f), for 0  f  Δf
m Vi
.
0 , for f  Δf
Thus, the mean square value of the brick wall output port noise voltage, vnoeff(t), is
S
Voeff
e2
(t)  E 2

noeff
oeffrms
Vi


0
S
(f)df  H 2
Voeff
m
(30)
Δf

0
S (f) df ,
Vi
(31)
where we have made use of (30) and the fact that gain parameter Hm is a frequency invariant
constant.
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Now, let us choose the foregoing Δf metric so that the net mean square noise voltage at
the output port of the brick wall filter is identical to the mean square output noise voltage generated in the original system; that is
f


2
2
H
S (f) df  S
(f)df 
H  j2πf  S (f)df ,
(32)
m


Vi
0
o
Voeff

o
which produces
2
f

H  j2πf 
S (f) df 
S (f)df .
Vi
Vi
H


0
Vi
(33)
m
o
When frequency metric Δf is chosen in accordance with (33), it is termed the noise equivalent
bandwidth of the original system earmarked for an electrical noise characterization. The
relationship in (33) postulates that frequency metric Δf can be selected so that the two filters in
Figure (4) deliver the same mean square output noise voltage for the same amount of applied input noise. Specifically, output noise equality results if the area enclosed by the input noise spectral density, SVi(f), over a frequency range spanning from zero -to- Δf, is equal to the area enclosed over all frequencies (0 -to- ∞) by the product of SVi(f) and the squared magnitude of the
original network transfer function normalized to its zero frequency gain value. We note with
more than passing interest that the noise equivalent bandwidth, Δf, is the width of brick wall filter passband; that is Δf is the passband of the brick wall filter.
Although (33) postures elegant mathematics, its pragmatic execution for a generalized
transfer relationship and for any generalized input noise spectral density is potentially cumbersome. It is inevitably a daunting undertaking unless we elect to assume a constant noise spectral
density. Constant noise spectral density, which suggests that the mean square noise -per- unit of
frequency is independent of frequency, typifies a type of electrical noise that is called white
noise. As it turns out, thermal noise in traditional integrated circuit resistances and shot noise in
semiconductor devices very closely approximate sources of white noise. And since thermal and
shot noise phenomena dominate the noise landscape of active circuits, especially at moderately
high -to- high signal frequencies, constraining SVi(f) to a constant reflects sensible circuit design
engineering. With SVi(f) constant, (33) collapses to the considerably simpler expression,

f 

o
H  j2πf 
H
2
df .
(34)
m
This relationship stipulates the constraint under which frequency metric Δf can be selected so
that the two filters in Figure (4) deliver the same mean square output noise voltage for the same
amount of applied input white noise.
Numerous lowpass, high performance, electronic circuits and systems are designed to
deliver maximal 3-dB bandwidth, subject to the constraint of no, or at least minimal, peaking of
the frequency response over the network passband. When no response peaking is evident, such
broadband systems are said to deliver a maximally flat magnitude (MFM) response. To this
end, a common design target is the Butterworth lowpass system whose I/O transfer function subscribes to the relationship,
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H  j2πf 
2

University of Southern California Viterbi School of Engineering
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H2
m
(35)
,
2n
 f 
1  
B
where we remind that Hm is the low frequency magnitude of the transfer relationship, while n,
which is the order (number of poles) of the considered network, is an integer that is at least as
large as one. For any value of n, B in (35) represents the applicable 3-db bandwidth. If we
substitute (35) into (34) in an attempt to gain a basic perspective as to the relationship between
the noise equivalent bandwidth and the actual 3-dB bandwidth for signals, we find that for the
Butterworth MFM filter
Δf
 2n
(36)

.
B
sin  2n 
For n = 1, which corresponds to a dominant pole, or ideally single pole, circuit, Δf/B = π/2,
which indicates that the noise bandwidth is more than 50% larger than the signal 3-dB bandwidth. The fact that Δf exceeds B substantially is not surprising in that a single pole network has
a relaxed, 20 dB/decade magnitude response roll off at frequencies in the neighborhood, and in
excess, of the network 3-dB bandwidth, B. Consequently, noise at frequencies larger than the
network 3-dB bandwidth can be amplified to some degree to deliver a potentially large mean
square output noise over wide high frequency passbands. In contrast, no noise is delivered by
the brick wall network for frequencies larger than Δf. Accordingly, Δf must be suitably larger
than B to account for noise delivered at high signal frequencies to the output port of the practical
single pole system. The fact that Δf always exceeds B, supplemented by the fact that since (31)
confirms a mean square output noise that is directly proportional to Δf when the applied noise is
white, discourages us from exercising the design heroics to achieve circuit bandwidths that
dramatically exceed the required system bandwidth. In particular, since Δf is proportional to B,
larger bandwidths produce progressively larger normalized output noise power or mean square
output noise voltage or current. While (36) is strictly applicable to only a Butterworth MFM circuit or system, it generally serves as a reasonable design guideline for choosing a circuit or system bandwidth, B, which suitably restricts the total mean square noise (voltage or current) developed at the network output port. Indeed, the case of N = 1, which we have just considered,
applies to any dominant pole circuit or system that exudes no finite frequency zeros.
In the interest of completeness, N = 2 in (36) establishes Δf/B ≈ 1.11, which suggests a
noise equivalent bandwidth that is 11% larger than the signal bandwidth. For 3 ≤ N ≤ 9, less
than about 4% error accrues by approximating (36) with Δf/B ≈ π/3. In a word, the noise equivalent bandwidth of most practical circuits and systems abide by the design-oriented constraint,

Δf

, for 2  N  9 .


(37)
2.83
B
3
We note that Δf converges toward bandwidth B for progressively higher order networks. In light
of our previous frequency response observations, this situation is to be expected since high order
networks exude robust frequency rates of magnitude attenuation at high frequencies, thereby
leaving only an anemic gain magnitude with which to process input noise at high frequencies.
3.3.
NOISE SOURCES IN THE FREQUENCY DOMAIN
Thermal noise generated in resistors, or conductors in general, as well as shot noise in
electronic devices are very good approximations of commonly encountered white noise. On the
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other hand, flicker noise, which is often called pink noise, is pervasive of many active
semiconductor devices. It is an excellent example of a commonly encountered noise source
whose noise spectral density is not frequency invariant.
Noise In Resistors
3.3.1.
Noisy
R
R
*
enR(t)
[SVR(f)]
(a).
R
(b).
Noiseless
Noiseless
Practical resistances, such as the resistance of value R (in ohms) given in Figure (5a),
produce an open circuit noise voltage, enR(t), regardless of whether or not the resistors are connected in a circuit branch that conducts current. We depict this open circuit, or effective
Thévenin, noise voltage in the noise equivalent model of Figure (5b), wherein enR(t) appears in
series with the noiseless equivalent of resistance R. Of particular significance is the fact that the
noise voltage spectral density, SVR(f), corresponding to enR(t) is given by
jnR(t)
[SIR(f)]
(c).
Figure (5). (a). A practical two-terminal resistor generating white thermal
noise. (b). Thévenin type noise equivalent circuit of the noisy
resistor in (a). (c). Norton type noise equivalent circuit of the
noisy resistor in (a).
S
VR
(f)  4 kTR ,
(38)
where k is Boltzmann’s constant [1.38(10−23) joules/°K] and T is the temperature (in Kelvin degrees) of the subject resistor. To the extent that (38) correctly defines the noise voltage spectral
density of resistor noise, which is often termed thermal noise because of the direct dependence
of its noise spectral density on temperature T and the invariance of this spectral density with
current level, resistor noise is seen as an example of white noise. This means that if resistance R
is embedded in a circuit having a noise equivalent bandwidth of Δf, (19) and (31) suggest that the
net mean square voltage that resistance R delivers to the circuit branch in which it is incident is
e 2 (t) 
nR
Δf

 SVR (f)df
 4kTR
0
 df
 4kTRΔf .
(39)
0
Thus, for example, a 1 KΩ resistance functioning at 27 °C (300.16 °K), generates noise voltage
in the RMS amount of
2
f 
nR
e
4kTR = 4.07 nV/ Hz (read as “4.07 nanovolts -per- root
hertz”). Over a 1 GHz noise bandwidth, this computation implies a net RMS noise level of
almost 130 μV!
Instead of representing a practical resistor as a series interconnection of its noiseless
counterpart and a noise voltage whose mean square value3 is given by (39), we can replace the
3
Remember that a mean square voltage is the square of its corresponding root mean square (RMS) voltage. Analogously, a mean square current is the square of its corresponding RMS current value.
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Thévenin topology in Figure (5b) by the Norton noise equivalent circuit shown in Figure (5c). In
the latter representation, the noise current, inR(t), has a mean square value of
2
e (t)
4kTRΔf
2
nR
j (t) 

 4kTGΔf ,
nR
2
2
(40)
R
R
where G is obviously the conductance equivalent of resistance R. Equation (40) also implies a
noise current spectral density, SIR(f), given by
S (f)  4 kTG ,
(41)
IR
whose frequency domain nature remains white.
Noise In Junction Diodes
3.3.2.
Unlike the passive resistor, the PN junction diode depicted in Figure (6a) generates an
open circuit noise voltage, enD(t), only when it conducts a static, or quiescent, current, say IDQ,
which naturally implies forward biasing of the device. When noise is functionally dependent on
static current, it is termed shot noise, as opposed to resistive thermal noise, which is devoid of
any current dependence. The Thévenin noise equivalent circuit of the diode is offered in Figure
(6b), while Figure (6c) gives the Norton alternative to the Thévenin noise model of the diode. In
Figures (6b) and (6c), rD, the small signal terminal resistance of the diode is given by
IDQ
Noisy
rD
(a).
jnD(t)
[SID(f)]
rD
enD(t)
[SVD(f)]
*
(b).
(c).
Figure (6). (a). A forward biased diode generating noise that is functionally
dependent on the quiescent current, IDQ, which it conducts. (b).
Thévenin type noise equivalent circuit of the noisy diode in (a). (c).
Norton type noise equivalent circuit of the noisy diode in (a). In both
(b) and (c), rD represents the small signal resistance of the diode.
r
D

n V
j T
I
,
(42)
DQ
where
kT
(43)
T
q
is Boltzmann’s voltage, k is the previously introduced Boltzmann’s constant, T is the absolute
temperature of the diode junction, and q is the magnitude of electron charge, which is 1.6(10−19)
coulombs. Finally, parameter nj is the junction injection coefficient; it is typically very slightly
greater than one but unless explicitly specified, its default value is routinely taken to be one. The
diode noise voltage has a noise spectral density, SVD(f), of
S (f)  2n kT r .
(44)
V
VD

j
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Since this spectral density is independent of signal frequency, diode shot noise comprises a
second example of white noise that is commonly encountered in electronic circuits. It follows
from (44) that the total mean square shot noise developed by a diode embedded in a network
having a noise equivalent bandwidth of Δf is
e 2 (t)  2n kT r Δf ,
nD
j
(45)
D
which is roughly one-half of the mean square voltage associated with a traditional passive resistance. Assuming room temperature operation of a semiconductor junction diode that is conducting 1 mA of quiescent current, rD in (42) is, assuming nj = 1, rD = 25.89 Ω. If the applicable
noise equivalent bandwidth is 1 GHz, (45) gives a net RMS diode noise voltage of EDrms = 14.64
μV. Although this noise voltage is small, it is directly proportional to absolute diode junction
temperature, which rises with increasing current densities at the diode junction.
As in the case of the previously considered noisy resistor, the Thévenin noise model of
the PN junction diode can be transformed into the Norton alternative displayed in Figure (6c).
The Norton noise current, jnD(t), establishes a mean square value of
2
2n kT r Δf
e (t)
j
D
2
nD
j (t) 

 2qI
Δf ,
nD
DQ
2
2
r
r
D
D
(46)
where we have adopted (45) and (42). For the numerical disclosures invoked in the diode noise
voltage calculation, JDrms = 567.7 nA, which is of the same order of magnitude as the drain currents that typify a MOSFET biased in its subthreshold regime. However, resistance rD is typically of the order of only the low tens of ohms. Thus, the amount of noise current that a PN
junction diode can supply to a load imposed across the terminals of the diode is small, assuming,
of course, the likely circumstance of a load resistance that is significantly larger than rD.
3.3.3.
Noise In MOSFETs
In a MOSFET, such as the n-channel device appearing in Figure (7a) or its topologically identical p-channel counterpart in Figure (8a), three noise sources contribute to its overall
noise response. These sources intertwine with the small signal model of the MOSFET to produce the noise and signal responses of the circuit into which MOS devices are embedded. The
complete small signal model, with noise sources included, is not shown, but it will be addressed
in a subsequent report when the noise characteristics of MOSFET technology amplifiers are studied.
The first of the device noise sources is drain current noise, jnd(t)[6]. This noise is an
amalgam of thermal noise manifested by that portion of the drain-source channel that is inverted
and therefore acts as a tapered resistor and shot noise within the depleted region of the drainsource channel that is established when the MOSFET enters its saturation regime of operation.
The second source of MOSFET noise is gate noise[7], which is caused by the thermal agitation of
free carriers within the drain-source channel and, to a lesser extent, thermally noisy resistive gate
material. Gate noise is most efficiently handled by incorporating into the gate lead of a
MOSFET a series interconnection of an appropriate noise voltage eng(t), in series with a resistance, rg, While drain current and gate noises comprise reasonable approximations of white
noise sources, the third noise component indigenous to a MOSFET is drain flicker noise[8]. This
noise, which we model as a current, jnf(t), in shunt with the drain-source terminals of the transistor, is pink noise in that its mean square noise current varies inversely with frequency. Figure
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(7b) offers the pertinent noise macromodel of the n-channel MOSFET. The p-channel MOSFET
noise model is identical to that of the n-channel device and is offered in Figure (8b). In the
subsections that follow, attention is focused almost exclusively on NMOS. However, the results
disclosed apply equally well to PMOS with but minor and obvious notational changes.
eng(t)

Vds
Vgs
IdQ
rg
Noiseless

*

Noisy
IdQ

jnds (t)
jnd (t)
jnf (t)
Vds
Vgs




(a).
(b).
eng(t)
jnds (t)
Noiseless
*

IdQ
rg
jnd s(t)

Vds
Vgs


(c).
Figure (7). (a). A conducting n-channel MOSFET that generates noise because of three distinct sources
of internally generated noise. Biasing supportive of the quiescent drain current, IdQ, is not
shown. Proper biasing requires a gate-source voltage, Vgs, larger than the threshold voltage,
Vh and drain-source voltage Vds ≥ (Vgs − Vh). (b). Noise macromodel accounting for three
internal noise sources for the device in (a). (c). The noise model of (b) with the two drain
circuit noise components combined into a single drain noise current, jnds(t).
Vsd
Vsg
eng(t)

rg
IdQ
Noiseless

*

Noisy
IdQ
jnds (t)
jnd (t)

jnf (t) Vsd
Vsg




(a).
(b).
Figure (8). (a). A conducting p-channel MOSFET that generates noise because of sources of internally
generated noise. Biasing supportive of the quiescent drain current, IdQ, is not shown.
Proper biasing requires a source-gate voltage, Vsg, larger than the threshold voltage, Vh and
source-drain voltage Vsd ≥ (Vsg − Vh). (b). Noise macromodel that accounts for three internal noise sources for the device in (a).
3.3.3.1. Drain Current Noise
The drain current noise, jnd(t), is largely a thermal noise component when the MOSFET
operates in its ohmic regime. But as operation draws toward pinch off and thence into saturation,
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this current coalesces thermal noise phenomena with a modicum of shot noise. The mean square
value of the drain current noise evidenced in a conducting MOSFET operated in its saturation regime is taken to be[6]
j
2
(t)  4kT  g Δf ,
nd
m
(47)
where gm represents the forward transconductance of the transistor in saturation. It also happens
to be the channel conductance, say gdo, in the ohmic regime of operation under the condition of a
drain-source voltage, Vds, of zero. Specifically,


W 
W 
(48)
V  V  2 C   I
,


m
c ox  L  gs
h
c ox  L  dQ
where μc, in units of cm2/volt-sec, is the free carrier mobility in the channel (electron mobility,
μn, for n-channel transistors and hole mobility, μp, for p-channel devices), while Cox is the density
(in units of farads/cm2) of the gate oxide capacitance. Moreover, W/L is the gate width -tochannel length gate aspect ratio of the transistor, Vgs is the applied gate-source voltage, Vh denotes the threshold voltage, which is modulated somewhat by bulk-source voltage[9], and finally,
IdQ represents the quiescent drain current flowing in the device In (47),
2
(49)
   1,
3
with the understanding that the upper limit for γ applies to the ohmic domain of MOSFET operation, while the lower limit is suitable for the saturation regime. It should be noted that (47) effectively defines the mean square noise current of a simple passive resistance whose conductance
value happens to be γgm. We should also point out that since hole mobility can be up to a factor
of three or so smaller than electron mobility, conductance gdo, and hence the mean square value
of the drain noise current, is smaller for PMOS than it is for a comparable NMOS transistor.
g
  C
3.3.3.2. Gate Noise
The mean square value of the gate noise voltage, eng(t), deployed in the gate circuits of
the MOS transistors in Figures (7b) and (8b), is[7]
e 2 (t)  4kT  r Δf ,
ng
(50)
g
where
r

1
5g

1
W 
5 C   V  V
c ox  L  gs
h

1
(51)
,



m
W 
50   ox    I
c  T   L  dQ
 x 
εox is the dielectric constant of silicon dioxide [345 fF/cm], and Tox is the thickness of the gate
oxide. Parameter δ, the gate noise coefficient, is typically assigned a value approaching δ = 4/3.
In other words, δ ≈ 2γ when the considered MOSFET is biased in saturation. We observe that
resistance rg, and hence the mean square gate voltage in (50), increases with longer channel
lengths, narrower gate widths, and thicker gate oxides. It also increases with smaller drain currents.
g


3.3.3.3. Drain Flicker Noise
As we noted earlier, flicker noise, unlike white thermal or shot noises, produces a mean
square voltage or current that varies as a nominal inverse function of frequency. Although
flicker noise is known to occur in resistors, bipolar junction transistors, and even in vacuum
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tubes, it is most prominent in electronic devices whose conduction mechanisms are principally
determined by surface mechanisms and the properties of interfacial imperfections. Accordingly,
MOSFETs, whose drain currents rely on the drift of free charge carriers in a thin, strongly inverted, drain -to- source channel that is induced at the surface interface between the gate oxide
and semiconductor bulk, is vulnerable to flicker effects. This observation encourages the putative conclusion that flicker noise in MOSFETs is caused by imperfect semiconductor surfaces,
charge entrapment at oxide-semiconductor interfaces, foreign impurities in oxide and bulk
semiconductor volumes, and related other interfacial shortfalls that occur during routine device
processing.
The mean square value of the flicker noise current, jnf (t), in Figure (7b) is
 K f  g 2 
2
m  Δf ,

(52)
j (t)  
nf
 f   WLC 2 


ox 
where, as incorporated into (48), the saturation regime value of forward transconductance gm
is[10]
g  2  C W L  I
,
(53)
m
c ox
dQ
with the understanding that carrier mobility μc assumes either its electron mobility value, μn, or
its hole mobility value, μp, depending on whether the considered transistor is NMOS or PMOS,
respectively. For NMOS, the flicker noise coefficient, Kf, is of the order of 5(10−27) coul/m2. In
contrast, PMOS produces Kf in the neighborhood of (10−28) coul/m2; i.e. the flicker noise coefficient for PMOS is about 50-times smaller than that of NMOS. We recall that hole mobility is
about three-times smaller than electron mobility. Thus, a PMOS transistor produces a flicker
mean square noise that is as much as 150-times smaller than the flicker noise incurred by a
comparably sized NMOS transistor conducting a drain bias current that is identical to that conducted by its PMOS counterpart.
There are at least three other interesting sidebars to (52). The first of these luminescent
observations derives from substituting the forward transconductance expression of (53) into (52)
to arrive at the alternative mean square flicker current relationship of
 2 c K f   I dQ 
 Δf .

j 2 (t)  
(54)
nf
2


  L C 
f


ox 
This result advances ominous undertones to channel length downsizing, as well as to large quiescent drain currents. In particular, we see that the mean square value of the drain flicker current is
inversely proportional to the square of channel length and directly proportional to the static drain
current. Thus, with all else being equal, a 45 nM MOSFET can exude as much as more than 8.3times the mean square value of drain flicker noise current than does a 130 nM transistor.
A second interesting point is revealed by exploiting the facts that (1) the gate-source
capacitance, Cgs, of a MOSFET operated in its saturated domain is
2
C
 WLC ,
(55)
gs
ox
3
and (2) the unity gain frequency, fT, of a MOSFET derives from
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f
T


2 C
g
University of Southern California Viterbi School of Engineering
m
gs
C
gd


k g
3k g
m m 
m m ,
2 C
4 WLC
gs
ox
Choma
(56)
where Cgd is the gate-drain capacitance of the transistor, we have adopted (55), and we have
introduced parameter km to designate the capacitance divider,
C
gs
(57)
k 
.
m
C C
gs
gd
Since the gate-drain capacitance in saturation collapses to only the capacitance associated with
the gate oxide overlap with the drain implant, km is very nearly unity for a self-aligned gate
MOSFET biased to operate in its saturation regime. If we now combine (56) and (52), we see
that
2
 4 f   Δf 
2
T

j (t)  WL K 
.
nf
f  3k   f 


 m 
(58)
Thus, the mean square value of the drain flicker noise current is directly proportional to the gate
area, which is WL. This behavior seems reasonable in view of the fact that larger gate areas enclose progressively larger volumes that can serve as sanctuaries for large amounts of trapped
charges and greater volumes of ionic impurities. We also see that higher bandwidth devices, in
the sense of enhanced fT, promote larger mean square flicker currents. Since increases in fT are
generally realized through decreases in channel length L, this observation appears to resonate
with the channel length conclusions we have drawn from (54). Moreover, since fT is nominally
inversely dependent on gate-source capacitance, it can be reduced artificially by appending circuit capacitance across the gate-source terminals of the transistor. Obviously, this artificial
reduction of the transistor unity gain frequency is sensible in only those applications that do not
mandate large bandwidths.
Clearly, the mean square flicker noise defined by (52), (54), or (58) is a monotone
decreasing function of frequency, decreasing from very large values at low frequencies to ultimately inconsequentially small values at high frequencies. Because the flicker noise component,
jnf(t), of drain current is shunted by, and therefore algebraically superimposes with, the thermal
noise component, jnd(t), of the drain current noise, it is of interest to find the frequency at which
the flicker noise component reduces to the thermal noise component. This frequency, say fc, is
called the flicker noise corner frequency. It can be found simply by equating the mean square
value of the flicker current in (58) to the mean square drain noise current in (47). The result is
 f
4WLK  T
f  3k
 m
f 
c
kT  g
2



 .
(59)
m
Since the drain noise and drain flicker currents are uncorrelated, the mean square value of the net
drain-source noise current, jnds(t), which we show in Figures (7c) and (8b), is the sum of the
individual mean square values of the drain noise current and the flicker noise current. From (59),
(58), and (47), this net mean square current is therefore expressible as
f 

j 2 (t)  4kT  g  1  c  Δf .
(60)
nds
m

f


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In the sense that potentially large flicker noise dominates the drain noise for frequencies f satisfying f < fc, (59) and its companion equation, (60), comprises arguably bad news for high performance device enthusiasts. In particular, wideband devices, which naturally boast large fT, have,
by (59), large corner frequencies, fc. This state of affairs implies by virtue of (60), that signal
frequencies f must be progressively larger to render f >> fc, and hence, a drain noise that is both
reasonably small and nominally independent of frequency.
The last point is highlighted graphically in Figure (9). In this figure, we plot the
normalized mean square value of the drain current noise in (47), the normalized mean square
value of the flicker noise component that derives from (58) and (59), and the normalized mean
square value of the net drain-source current in (60). The drain current noise and flicker noise are
assumed uncorrelated, and the normalization current factor for all three curves is (4kTγgmΔf).
Note that in the figure before us, signal frequency f must be at least three-fold the corner frequency, fc, in order for flicker effects to be deemed inconsequential.
Normalized Mean Square Noise Current
12
10
Net Drain-Source Noise
8
6
Drain Current Noise
4
Drain Flicker Noise
2
0
0.10
0.32
1.00
3.16
10.00
31.62
100.00
Normalized Frequency, f/fc
Figure (9). The noise currents in the drain-source port of the MOSFETs modeled in Figures
(7b) and (8b). Corner frequency fc is defined such that at this frequency, the flicker
and drain current noise components are identical. The drain and flicker noise currents are presumed uncorrelated. The normalization factor for the mean square
noise current is (4kTγgmΔf).
A simplification, as it were, is to refer the net drain-source noise current, jnds(t), in the
noise macromodel of Figure (10a), to the gate circuit as a voltage, ends(t), which we delineate in
Figure (10b). Since gate-source signal voltages are multiplied by forward transconductance to
produce drain signal currents, we argue that the mean square value of this reflected noise voltage, ends(t), is
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

4kT   1  c 
2

j (t)
f 


e 2 (t)  nds
Δf .
nds
2
g
g
m
m
f
(61)
An important design-oriented motivation underlies the foregoing transformation. In particular,
and within the constraint of uncorrelated noise sources, the mean square value of the reflected
noise, as per (61), simply adds to the mean square value of the gate noise, which (50) defines
analytically. As a result, the two gate circuit noise voltages in Figure (10b) can be replaced by
the equivalent single noise voltage, en(t), shown in Figure (10c), with the understanding that the
mean square value of en(t) is
Noiseless

IdQ
rg
*
eng(t)
jnds (t)
ind (t)

jnf (t) Vds
Vgs


rg
IdQ

Noiseless
eng(t)
*

ends(t)
*
(a).
Vds
Vgs



rg
*
en(t)
IdQ

Noiseless
(b).
Vds
Vgs


(c).
Figure (10). (a). The MOSFET noise macromodel of Figure (7c). (b). Noise
currents in the drain-source circuit of the noise model in (a) referred
to the gate circuit as a noise voltage, ends(t). The two gate circuit
noise voltages are uncorrelated. (c). Superposition of the two noise
voltage sources in the gate circuit of the model in (b) into an equivalent, single noise voltage, en(t), in the gate circuit.
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e 2 (t)  e 2 (t)  e 2 (t)
n
nds
ng
 
f 

 
f 

   1  c 

   1  c    5  
(62)
f 
f 
 

 

 4kT 
  r  Δf  4kT 
 Δf ,
g
g
g
m
m












where we have exploited (51). Aside from simplifying the noise macromodel of Figure (10a) to
the single noise generator architecture depicted in Figure (10c), (62) serves the important purpose of effectively defining the maximum tolerable input noise, or noise floor, of a simple common source amplifier. In such an amplifier, the signal to be processed is a voltage placed in series with the gate terminal of the transistor, which is to say that this signal voltage appears in
series with the equivalent input noise voltage, vn(t). It follows that if the amplifier is to detect
reliably the signal voltage, the applied signal must overcome, in addition to its own internally
generated noise, the RMS value, Enrms, of the equivalent input noise voltage. At a minimum,
therefore, this RMS noise voltage must be smaller (and preferably, substantially smaller) than the
RMS value, Vsrms, of the applied signal; namely,
 
f 

   1  c    5  
f 


E
e 2 (t)  4kT  
(63)

 Δf  Vsrms .
nrms
n
g
m






The result at hand implies the need for large transconductance and small flicker corner frequency
if the noise inherently produced by a MOSFET is precluded from significantly contaminating the
input signal identified for appropriate processing.
3.3.3.4. Another Source of MOSFET Noise
In addition to the drain noise we have elucidated in Section (3.3.3.1), thermal noise
associated with the substrate resistance can superimpose with the drain noise current, jnd(t), in
Figure (10a)[7]. The mean square value of this substrate thermal noise, which is oft referred to
as epitaxial noise, is


2


4kTR
g
2
sub mb
 Δf ,
(64)
j (t)  
sub
2

 1  2πf RsubC gb 
where Rsub is the average value of the bulk substrate spreading resistance, Cgb represents gatebulk capacitance, and gmb designates the bulk transconductance (partial derivative of drain signal
current with respect to bulk-source signal voltage). For most, but assuredly not all, analog
MOSFET applications, substrate noise is likely to be inconsequential for three reasons. First, the
bulk transconductance, gmb, is ideally zero and likely to at least be a very small transconductance,
especially if the MOSFET is fabricated in a monolithic process that features very thin gate
oxides[9]-[10]. Second, substrate resistance Rsub, can be reduced by prudent layout strategies
entailing the liberal use of substrate electrical contacts that are returned to circuit ground. Third,
at high frequencies, where capacitance Cgb effectively bypasses the substrate resistance, the

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
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denominator on the right hand side of (68) becomes large, thereby rendering the mean square
value of the substrate noise current inconsequential.
4.0.
TWO PORT NOISE ANALYSIS
The majority of analog MOSFET amplifiers behave electrically as linear two port networks. As such, their I/O gains are independent of signal amplitudes and noise levels, thereby
allowing the convenience of referring all network noise sources to any system port and in
particular, to the system input port. When we speak of an “input port referred noise,” or more
simply, “input referred noise,” we are talking about one or perhaps two effective noise sources.
The mean square values of these effective noise generators are judiciously chosen so that when
they activate the input port of the noiseless equivalent of the network we are studying, they deliver the same net output noise as observed in the original network, which is doubtlessly permeated by numerous internal sources of electrical noise. Thus, the action of referring the net
noise level at the output port of a network to the network input port serves to simplify circuit
analysis in that potentially many noise sources intrinsic to the two port network are supplanted
by a mere one or two input port noise generators. Additionally, once we have succeeded in
referring all network noise sources to the input port, we can straightforwardly compare the
strength of an applied signal to the noise source (or sources). Aside from ascertaining the noise
floor, the careful examination of the nature of the effective input noise sources is likely to forge
meaningful design strategies aimed toward minimizing the deleterious effects of noise. Such
minimization is tantamount to reducing the noise floor.
Figure (11) abstracts the foregoing declarations. In Figure(11a), we show a linear,
noisy, two-port network whose applied input signal is represented by a Thévenin equivalent circuit comprised of voltage source Vs, and impedance Zs. Additionally, parasitic signal noise in the
form of random voltage ens(t), whose mean square value is e 2 (t) , superimposes with, and therens
fore blurs, the time deterministic signal, Vs. The output port of the network is terminated in the
load impedance, Zl. In response to the input signal, the indicated signal noise, noise sources
generated within the linear two-port network, and any noise generated by the load termination,
the net input port voltage consists of signal voltage Vi superimposed with noise voltage eni(t).
Analogously, the net output port voltage response superimposes a signal component, Vo, and a
noise voltage, eno(t).
In Figure (11b), we examine only the noise responses of the system diagrammed in Figure (11a) by setting the input time deterministic signal, Vs, to zero. Additionally, we set all noise
sources within the two-port configuration and any noise implicit to the load termination to zero.
As a result, the indicated two-port is the noiseless counterpart of the original two-port network in
Figure (11a) and, of course, Zl is the noiseless equivalent of the original load termination. In
place of these zeroed noise sources, we advance two noise sources, an equivalent input noise
voltage, en(t), and an equivalent input noise current, in(t), at the input port, as shown in the diagram. These “equivalent” noise sources are selected so that the output noise response, vno(t), remains identical to the output port noise voltage observed in the original system of Figure (11a).
The determination of the equivalent input noise voltage and current first requires the
computation or measurement of the mean square value, e 2 (t) , of the net output noise voltage,
no
eno(t). Such a computation must judiciously consider the noise associated with the applied signal
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Vs
Zin

*
ens(t)
Zl
Potentially
Noisy

Noisy, Linear
Two-Port Network
source, the noise sources implicit to the two-port network and the terminating load impedance,
any frequency dependencies of all of these noise sources, and finally, correlation factors evidenced between any two noise sources. Then, if the input port -to- output port voltage gain is
Av(j2πf), which, because of linearity, can be deduced from signal considerations alone as
Vi + eni (t)
Vo + eno (t)
Zs
ens(t)
*
eno (t)
Noiseless
jn (t)
Zin
Zs
*
eni (t)
Zl
en (t)
Noiseless, Linear
Two-Port Network
(a).
(b).
Figure (11). (a). A noisy, linear two-port network whose applied input excitation is the
superposition of a time deterministic signal, Vs, and a noise voltage component, ens(t). (b). The noiseless equivalent of the noisy two-port network in (a).
The equivalent noise voltage and noise current sources, en(t) and jn(t), respectively, are selected such that in conjunction with the input signal noise, ens(t),
the total output port noise voltage, eno(t), is identical to the output noise response observed in (a).
A (j2πf) 
V
o ,
V
i
v
(65)
the mean square value (or square of the corresponding RMS value) of the input port noise voltage, vni(t), corresponding to output noise voltage, vno(t), is
e 2 (t) 
ni
e 2 (t)
no
A (j2πf)
2
.
(66)
v
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But we observe in Figure (11b) that the input port noise voltage, eni(t) is identically equal to
noise voltage en(t) under the conditions of zero signal noise and zero source impedance. Hence,
we arrive at the mean square value of the equivalent input noise voltage by computing the mean
square value, e 2 (t) , of the equivalent input noise voltage, en(t), in accordance with
n
e 2 (t)  e 2 (t) v (t)0 
n
ni
ns
Z 0
e 2 (t)
no
2
A (j2πf)
v
s
.
(67)
v (t) 0
ns
Z 0
s
We also observe in Figure (11b) that if the source impedance, Zs, is infinitely large, the
input port noise voltage is necessarily produced exclusively by the flow of noise current jn(t) into
the input port of the network. Thus, with j 2 (t) representing the mean square value of the equivan
lent input noise current, jn(t),
e 2 (t)
ni
 Z (j2πf)
in
Z 
2 2
j (t) ,
n
(68)
s
whence,
e 2 (t)
ni
j 2 (t) 
n
Z 
s
Z (j2πf)
e 2 (t)
no
2

2
Z (j2πf)
in
in
A (j2πf)
v
.
2
(69)
Z 
s
Equations (68) and (69) establish the mathematical basis for identifying the mean square values
(or the square of the RMS values) of the equivalent input noise voltage and the equivalent input
noise current that act to emulate the original output port noise response of the considered twoport network.
4.1.
NOISE FLOOR
In Figure (11b), let the RMS values of signal source noise voltage ens(t), equivalent input noise voltage en(t), equivalent input noise current jn(t), and input port noise voltage eni(t), be
denoted by Ensrms, Enrms, Jnrms, and Enirms, respectively. By superposition,
 Z

 Z

 Z Z 
in
in
E
E
(70)
E
 
 
  in s  J
,
nirms
 Z  Z  nsrms  Z  Z  nrms  Z  Z  nrms
in
s
in
s
in
s






where we have elected to add all of the three pertinent voltage components because, as we have
already observed, phase angles between truly random voltage and current components are
indeterminate. As it materializes, it is more convenient to couch (70) into the form,
E
nirms

J
nsrms

 J
nrms
Y E
s nrms
Y Y
in
,
(71)
s
where Yin is the admittance corresponding to input impedance Zin, Ys is the admittance of source
impedance Zs, and
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J
nsrms

E
University of Southern California Viterbi School of Engineering
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nsrms  Y E
s nsrms
Z
s
(72)
is the RMS Norton current equivalent of RMS signal noise voltage Ensrms.
The signal noise current, jns(t), is doubtlessly not correlated with either jn(t) or en(t).
But noise current jn(t) and noise voltage en(t) are correlated because from (67) and (69), the mean
square values of both of these noise generators are functionally dependent on the mean square
output noise voltage and the network voltage gain. Thus, the mean square noise voltage developed across the input port of the linear network is
J
2
2
e (t)  E

ni
nirms


2
2
 J
Y E
nsrms
nrms
s nrms
,
2
Y Y
in
s
(73)
where we have exploited the fact that the mean square values (or squares of RMS values) of only
uncorrelated random variables directly superimpose with one another. It is understood that both
Yin and Ys in (73) are complex functions of frequency; that is,
Y  Y (j2πf)
s
s
(74)
.
Y  Y (j2πf)
in
in
It now follows that the mean square value of the net output noise of the linear two-port network
before us is
 2
2
J
 J
Y E

2
2
nrms
s nrms 
 A (j2πf) E 2
 A (j2πf)  nsrms
e 2 (t)  E 2
 . (75)
no
norms
v
nirms
v
2


Y Y
in
s




Despite the messiness of this relationship, the mean square output noise voltage, say e 2 (t) ,
nos
attributed solely to the noise associated with the applied input signal is clearly revealed as
e 2 (t) 
nos
A (j2πf)
2
v
J
Y Y
in
2
nsrms
,
2
(76)
s
which implies that the corresponding mean square input port noise, E 2 , due exclusively to
nis
noise contamination implicit to the signal source is
e 2 (t) 
nis
e 2 (t)
nos
A (j2πf)
v
2

J2
nsrms
Y Y
in
2
.
(77)
s
In Figure (11a), we see that the signal delivered to the input port of the linear two-port
network is
 Z

I
in  V 
s
(78)
V  
,
i
Z  Z  s
Y Y
s
in
s
 in
where
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I
s

University of Southern California Viterbi School of Engineering
V
s  YV
s s
Z
s
Choma
(79)
is the Norton equivalent signal current corresponding to the applied signal voltage, Vs and its
intrinsic Thévenin impedance, Zs. In order for the two-port network to detect signal voltage Vi
faithfully and ultimately process it in accordance with the design targets stipulated for the twoport system, Vi must rise above the RMS noise voltage, Enirms, established across the network input port. Thus, we require Vi > Enirms and from (75), (76), and (83), this mandate is seen as
implying
V  E
s
nsrms

 Z J
s nrms
 E
nrms
.
(80)
The minimum detectable signal, Vsmin, which defines the noise floor of the two-port network
therefore evolves as
V
smin
 E
nsrms

 Z J
s nrms
 E
nrms
.
(81)
Thus, in addition to the self-evident observation that the input signal must overcome its own
noise level, whose RMS amplitude is Ensrms, it must also overcome the deleterious effects of the
equivalent input noise voltage and current, whose combined impact on the input port is measured
as the voltage sum, (Zs Jnrms + Enrms ).
4.2.
NOISE FACTOR
Recalling (7) -through- (9), the noise factor, F, of the two-port network displayed in
Figure (11a) is
V2
E2
i
irms

 norms 
F 
2
SNR
V
E2
o
orms
nirms
SNR
E2
1
A (j2πf)
v
2
 norms ,
E2
(82)
nirms
where we have applied (65). Before proceeding further, there may be an advantage, from the
perspective of simply expediting recollection, to expressing noise factor F in (82) in terms of
simple engineering terminology. In particular, a careful investigation of the right hand side of
(82) allows us to proffer,
Net Mean Square Output Noise Voltage (Current) Due To All Noise Sources
F 
. (83)
Net Mean Square Output Noise Voltage (Current) Due Only To Source Noise
Appealing to (75) and (77), we arrive at the relatively simple analytical expression,
F 
SNR
i
SNR
 J  Ys Enrms 
 1  nrms
o
J2
2
.
(84)
nsrms
The result at hand confirms earlier contentions to the effect that the noise factor of a network exceeds one but approaches one when the noise implicit to the network undergoing investigation is
negligible in comparison to the noise implicit to the signal source. This implicit network noise is
determined by equivalent input noise current jn(t) and equivalent input noise voltage en(t), whose
RMS values are respectively Jnrms and Enrms, as deployed in (84). Using (72), we can also cast
(84) directly in terms of the noise voltage that accompanies the input signal. Specifically,
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F 
4.3.
SNR
i 
SNR
o
University of Southern California Viterbi School of Engineering
 J  Ys Enrms 
1  nrms
Y (j2πf)
2
s
2
2
E
nsrms
 1
Z (j2πf)
s
2
Choma
 J nrms  Ys Enrms 
2
E
nsrms
2
. (85)
NOISE FACTOR OPTIMIZATION
In an attempt to formulate design strategies that ensure noise factor minimization in linear active networks, let us begin by decomposing the RMS value, Jnrms, of the equivalent input
noise current into an uncorrelated RMS component, Jurms, and a component, say Jncrms, that remains correlated to the RMS equivalent input noise voltage, Enrms. In an actual application, current component Jurms may be null, or current Jncrms may be null, independent of the status of current Jurms. We write,
J2
nrms

 Jurms  J ncrms 
2

 Jurms  Ync Enrms 
2
,
(86)
where Ync is, in general, a frequency dependent complex admittance possessed of a real, or
conductive component, Gnc, and an imaginary, or susceptive, component, Bnc, such that
Y  Y ( j2πf)  G  jB ,
(87)
nc
nc
nc
nc
If we decompose the source admittance, Ys, into its conductive and susceptive components, Gs
and Bs, respectively, such that
Y  Y ( j2πf)  G  jB ,
(88)
s
s
s
s
(86) -through- (88) inserted into (84) delivers
F  1
J
2

urms
Gnc  Gs   j  Bnc  Bs 
J
2
nsrms
2
E
2
nrms
.
(89)
In order to render (89) manageable in design environments, we shall introduce an equivalent
conductance, Gu, such that the uncorrelated component of the equivalent input noise current is
viewed as simple thermal noise generated by said conductance, Gu. This is to say that
J
2
 4kTGu Δf .
urms
(90)
Similarly for the equivalent input noise voltage,
E
2
 4kTRn Δf .
nrms
(91)
Finally, we shall presume that the noise current produced by the applied signal source can be
attributed exclusively to source conductance, Gs; that is,4
J2
nsrms
 4kTGs Δf .
(92)
Accordingly, (89) becomes
4
Attributing the signal noise to source conductance is a common engineering tack adopted in the characterization
and assessment of the noise properties of a standalone amplifier or other type of linear two-port network. But in an
actual multistage system, the signal noise is effectively the output port noise of the preceding stage, which can be
significantly larger than mere thermal noise attributed to the output resistance of the preceding stage. Of course, the
Thévenin voltage and output resistance of the preceding stage serves as the signal source for the network under
investigation.
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University of Southern California Viterbi School of Engineering

 

2
2

G   G G
 B B
R
u
nc
s
nc
s  n


F  1
.
G
Choma
(93)
s
Since the susceptive part of a physically realizable admittance can be positive or negative, noise factor minimization requires that source susceptance Bs be equal to −Bnc. The resultant noise factor expression is infinitely large for both Gs = 0 and Gs = ∞, which implies that an
optimum source conductance component, say Gsopt, exists. This optimal source conductance is
straightforwardly deduced by setting to zero the first derivative of F with respect to Gs in (93),
with Bs = −Bnc. The result is
G
G
u
(94)
G
 G
1
 G2  u ,
sopt
nc
nc R
2
R G
n
n nc
for which the corresponding minimal (or optimized) noise factor, Fmin, is found to be


G

.
u
F
 1  2R G  G
 1  2R G
1 1
min
n nc
sopt
n nc 
2 

R G 
n nc 



(95)
Design insights evolve straightforwardly from the foregoing results if we assume that
the equivalent input noise current displays no correlation with its counterpart equivalent input
noise voltage. In such an event, whose realism mandates a careful examination of the specific
application under investigation, admittance Ync in (86) and (87) is null. Obviously, the conductive component, Gnc, and susceptive component, Bnc, of admittance Ync are necessarily zero.
With Bnc = 0, noise figure minimization in (93) requires that the signal source admittance be a
purely real conductance; that is, Bs = −Bnc = 0, whence Ys in (88) is simply Gs. With Gnc = 0,
the optimum value, Gsopt, of signal source conductance Gs is, by (94), (90), and (91),
G
J
J
u  urms  nrms .
(96)
G

sopt Y 0
R
E
E
n
nc
nrms
nrms
In this expression, we have exploited the fact that if equivalent input noise current, jn(t) and
equivalent input noise voltage en(t) are uncorrelated, the RMS value, Jurms, of the uncorrelated
component of jn(t) is simply the RMS value, Jnrms, of jn(t). In short, the optimum impedance
value of a signal source applied to a noisy, linear active network displaying no correlation betwixt its equivalent input noise current and voltage is simply the ratio of the RMS values of
equivalent input noise voltage and equivalent input noise current. In the interest of completeness, we note that the minimum noise factor corresponding to (96) is, from 95),
F
min Y 0
 1  2R G
n sopt
nc

 1 2 G R .
(97)
u n
It is enlightening to witness that (93) can be massaged into the algebraic form,
2
G
2
G G
 B  B
 s FF
,
s
sopt
s
nc
min
R
 

n


(98)
which asserts that in the source admittance plane, the contours of constant noise factor are circles
centered at (Gs, Bs) = (Gsopt, −Bnc), with radii, r(F), of
Electronic Circuit Noise, Part I_R1
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July 2011
Technical Report #02-0511_R1
r(F) 
G
University of Southern California Viterbi School of Engineering

Choma

s F F
.
min
n
R
(99)
These contours are displayed generically in Figure (12). Observe in this figure, as predicted, that
the noise factor is minimized at its optimal value of Fmin when Gs = Gsopt and simultaneously, Bs
= −Bnc. Unfortunately, Gsopt and Bnc are likely to be frequency dependent owing to flicker
phenomena implicit to the considered linear active network and the likelihood that the network
transfer function, which is pivotal to the determination of the equivalent input noise voltage and
current, is a function of frequency. Accordingly, we anticipate encountering challenges if attempts are made to achieve broadband noise factor optimization. On the other hand, such
optimization is more easily accomplished at a single frequency and perhaps even over a restricted passband.
Bs
F3
F2
F1
Gsopt
0
Bnc
Gs
Fmin
Figure (12). The contours of constant noise figure for a linear two-port network. A signal
source admittance, Ys, which equals (Gsopt − jBnc), delivers an optimal noise
figure of Fmin.
In radio frequency (RF) amplifier applications, stereotypically anemic signal strengths
encourage conjugate matching of the signal source impedance to the amplifier driving point input impedance. This matching assures maximum signal power transfer from the signal source to
the amplifier input port, thereby guarding against excessive signal power losses that derive from
inefficient signal coupling. Unfortunately, the source admittance corresponding to maximum
signal power transfer at the amplifier input port is rarely the source admittance deemed optimal
for noise factor optimization[11]. Accordingly, a design compromise for high performance RF
amplifiers is mandated to preclude both excessively large noise factor and inappropriately large
signal power losses between signal source and input port.
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5.0.
University of Southern California Viterbi School of Engineering
Choma
NOISE FACTOR OF A CASCADE
Most amplifiers are multistage units, as is underscored pictorially by the symbolic
three-stage cascade of Figure (13a). Since several noise sources are likely to appear in each
stage, the noise analysis of the entire cascade is sufficiently cumbersome to impede the formulation of relevant design insights. For this reason, the development of an efficient and mathematically tractable method of noise characterization in complex linear network architectures is an
enviable undertaking. To this end, we shall derive an expression for the overall noise factor, say
F, in terms of the individual stage noise factors, Fi, and the associated stage voltage gains, Avi. In
the subject figure, Eni denotes the RMS value of the output noise generated exclusively by the
noise sources internal to the ith amplifier stage.
Eno1
Eno2
Ens
Eno
Stage #1
Stage #2
Stage #3
{En1 , Av1 , F1 }
{En2 , Av2 , F2 }
{En3 , Av3 , F3 }
(a).
Eno1
Eno2
Stage #1
Ens
*
Stage #2
Ens
{En1 , Av1 , F1 }
Stage #3
Ens
*
*
{En2 , Av2 , F2 }
Eno3
{En3 , Av3 , F3 }
(b).
Figure (13). (a). Cascade of three linear amplifier stages. In general, the ith stage is characterized by
an RMS output noise voltage due to internal noise sources of Eni, an I/O port voltage gain
of Avi, and a noise figure of Fi. (b). The noise characterization of each of the three amplifier stages in (a). The reference input noise to each stage is identical and has an RMS
voltage of Ens, which is the same as the RMS noise voltage applied to the input port of the
cascade in (a).
In the cascade of Figure (13a), a test input noise source of RMS value Ens is applied to
the cascade input port. In order to effect a stage noise characterization that is consistent with the
noise properties deduced for the entire network cascade, we evaluate the noise performance of
each stage by applying this same noise source to the input ports of the individual stages. This
Electronic Circuit Noise, Part I_R1
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July 2011
Technical Report #02-0511_R1
University of Southern California Viterbi School of Engineering
Choma
engineering methodology is abstracted in Figure (13b). Then, using (83) as a basis for an
analytical noise factor expression and noting that En1 is the RMS value of the output noise attributed solely to the noise internal to the first stage, we have5
A
2
2
2
E
ns
n1
,
2 2
A
E
v1
ns
v1
F 
1
E
(100)
from which we deduce
E2 
n1
 F1  1 Av1
2
E2 .
(101)
ns
It is hardly surprising that the mean square internal noise generated in the first stage is zero if F1
= 1. A similar analytical tack yields for the mean square noise voltages incurred by internal
noise sources in the second and third stages,
E2

 F2  1 Av2
2
E2

 F3  1 Av3
2
n2
and
n3
E2 ,
(102)
E2 .
(103)
ns
ns
In order to evaluate the noise factor of the entire three-stage system, we need first to
find the net mean square output noise, E 2 , produced at the network output port. In arriving at
no
this metric, we note several important points in Figure (13a). First, the test noise applied at the
cascade input port is amplified by all three stages. However, the output noise caused by stage #1
internal noise sources is amplified by only the second and third stages. Similarly, the output
noise incurred by internal noise sources within the second stage is amplified by only the third
stage, while the internal third stage noise is not amplified. Accordingly, we stipulate
E2
no
 A A A
v1 v2 v3
2
E2  A A
ns
2
v2 v3
E2  A
n1
2
v3
E2  E2 .
n2
(104)
n3
Since the output mean square noise generated by only the signal source noise applied to the input
port is the first term on the right hand side of the this equation, we deduce an overall noise factor,
F, of
E2
A A A
E2  A A
ns
v2 v3
2
E2  A
n1

F 
2 2
2 2
A A A
E
A A A
E
v1 v2 v3
ns
v1 v2 v3
ns
E2
E2
E2
n1
n2
n3
 1


.
2 2
2 2
2 2
A
E
A A
E
A A A
E
v1
ns
v1 v2
ns
v1 v2 v3
ns
no
v1 v2 v3
2
v3
2
E2  E2
n2
n3
(105)
Using (101)-(103), this relationship can be written in the streamlined form
5
Metric Ens is the RMS voltage of an independent test noise source applied to the the input port of the entire cascade
in Figure (13a), which is also the noise source activating the individual test stage structures in Figure (13b). This
independent noise source can be presumed to be uncorrelated to the internal amplifier noise that generates RMS output voltage components of Eni in the ith stage.
Electronic Circuit Noise, Part I_R1
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Technical Report #02-0511_R1
F  F 
1
University of Southern California Viterbi School of Engineering
F 1
2
A
v1
2

F 1
3
A A
2
Choma
,
(106)
v1 v2
which the literature archives as Friis’ formula[12]. Obviously, (106) can be extended to any number of stages; namely,
F 1
F 1
F 1
F 1
3
4
n


 
F  F  2
, (107)
1
2
2
2
2
A
A A
A A A
A A A A
v1
v1 v2
v1 v2 v3
v1 v2 v3
v(n-1)
where, of course, n is the number of stages.
Equation (106), or its generic form in (107) is interesting from several engineering
perspectives. First, it argues that the noise factor of a cascade of linear network stages can be
determined unambiguously in terms of only the individual stage noise factors and the squared
voltage gain magnitude (or power gain) of these individual stages. Second and perhaps most
importantly, the Friis’ formula shows that if the squared voltage gain magnitude of the first stage
in the cascade is sufficiently large, the overall noise factor is largely determined by only the
noise factor of the high-gain first stage. This means, within reason, that a low noise design
essentially boils down to the low noise design of the first stage. To the latter end, note in (107)
that the noise factors of stages 2 -through- n contribute to the individual terms on the right hand
side of (107) as a noise factor less one, whereas the contribution of the first stage to overall noise
factor is the full noise factor of that stage. Finally, we observe that since the squared magnitude
of the first stage voltage gain appears in the denominator of every term beyond the first term on
the right hand side of (107), a first stage gain magnitude of less than one over the noise equivalent passband is inappropriate. Thus, for example, a source follower as the first stage of an intended low noise amplifier design is hardly prudent.
6.0.
REFERENCES
[1]. J. B. Johnson, “Thermal Agitation of Electricity in Conductors,” Physics Rev., vol. 32, pp. 97109, July 1928.
[2]. H. Nyquist, “Thermal Agitation of Electric Charge in Conductors,” Physics Rev., vol. 32, pp.
110-113, July 1928.
[3]. C. D. Motchenbacher and F. C. Fitchen, Low-Noise Electronic Design. New York: John Wiley
and Sons, Inc., 1973, pp. 172-175.
[4]. D. A. Johns and K. Martin, Analog Integrated Circuit Design. New York: John Wiley and
Sons, Inc., 1997, chap. 4.
[5]. Y. W. Lee, Statistical Theory of Communication. New York: John Wiley and Sons, Inc., 1963,
pp. 9-45.
[6]. A. van der Ziel, “Thermal Noise in Field Effect Transistors,” Proc. IEEE, pp. 1801-1812, Aug.
1962.
[7]. T. H. Lee, The Design of CMOS Radio-Frequency Integrated Circuits. United Kingdom:
Cambridge University Press, 2004, chap. 11.
[8]. K. R. Laker and W. M. C. Sansen, Design of Analog Integrated Circuits and Systems. New
York: McGraw-Hill, Inc., 1994, pp. 79-85, 161.
[9]. J. Choma, “The Metal-Oxide-Silicon Field Effect Transistor,” University of Southern
California, EE 536a Course Notes #1, July 2008, available at http://www.jcatsc.com/.
Electronic Circuit Noise, Part I_R1
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University of Southern California Viterbi School of Engineering
Choma
[10]. J. Choma, “Circuit Level Models and Basic Applications of MOS Technology Transistors,
University of Southern California, EE 536a Lecture Aid #1, 2010-2011, available at
http://www.jcatsc.com/.
[11]. D. Shaeffer and T. Lee, “A 1.5V, 1.5 GHz CMOS Low Noise Amplifier,” IEEE J. Solid-State
Circuits, May 1997.
[12]. J.D.Kraus, Radio Astronomy. New York: McGraw-Hill Book Company, 1966
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