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2.8 NOISE
2.8.1 The Nyquist Noise Theorem
We now want to turn our attention to noise. We will start with the basic
definition of noise as used in radar theory and then discuss noise figure. The type of
noise of interest in radar theory is termed thermal noise and is generated by the random
motion of charges in resistive types of devices. One of the early attempts to characterize
thermal noise was performed by Nyquist, and one of his theorems concerning thermal
noise is that the mean-square voltage appearing across the terminals of a resistor of R
ohms at a temperature T degrees Kelvin, in a frequency band B Hertz, is given by1
2
vrms
 4kTRB V 2
(2-93)
where k  1.38  1023 w-s K is Boltzman’s constant. The noise power associated with
the resistor is
2
P  vrms
R  4kTB w .
(2-94)
2.8.2 Thevenin Equivalent Circuit of a Noisy Resistor
The Thevenin equivalent circuit of a noisy resistor is as shown in Figure 2-18. It consists
of a noise source with a voltage defined by Equation (2-93) and a noiseless resistor with a
value of R .
Figure 2-18 – Thevenin Equivalent Circuit of a Noisy Resistor
If we connect the noisy resistor, R , to a noiseless resistor, RL , we can find the
power delivered to RL by using the equivalent circuit of Figure 2-18, computing the
voltage across RL and then using this voltage to find the power delivered to RL . The
resulting circuit is shown in Figure 2-19 and the voltage across RL is given by
vRL  vrms
RL
.
RL  R
(2-95)
The power delivered to RL is
1
From “Principles of Communications” by Ziemer and Tranter, Fourth Edition.
©2011 M. C. Budge, Jr
34
PL 
vR2L
RL

2
vrms
RL
 RL  R 
2

4kTRB
 RL  R 
2
.
(2-96)
If the load is matched to the source resistance, that is if RL  R , we have
PL  kTB
(2-97)
Which is the familiar form used in the radar range equation.
Figure 2-19 – Diagram for Computing the Power Delivered to a Load
2.8.3 How to Handle Multiple Noisy Resistors
If we have a network that consists of multiple noisy resistors we find the
Thevenin equivalent circuit by using a modified version of superposition. To see this we
consider the example of Figure 2-20. In the figure, the left schematic shows two,
parallel, noisy resistors and the center schematic shows the equivalent circuit based on
Figure 2-18. The right schematic is the overall Thevenin equivalent circuit for the pair of
resistors. To find vo we first consider one voltage source at a time and short all other
sources. Thus, with only source v1 we would get
vo1  v1
R2
R1  R2
(2-98)
and with only source v2 we would get
vo 2  v2
R1
.
R1  R2
(2-99)
Figure 2-20 – Schematic Diagrams for the Two-resistor Problem
©2011 M. C. Budge, Jr
35
To get the total Thevenin equivalent voltage we must consider that v1 and v2 are noises.
As such, we must add their squares. With this, we get
vo2  vo21  vo22  v12
R22
 R1  R2 
2
 v22
R12
 R1  R2 
2
.
(2-100)
If we use v12  4kTR1B and v22  4kTR2 B we find that
vo2  4kTB
R1R2
 4kTBR .
R1  R2
(2-101)
We find the Thevenin equivalent resistance by the standard means of shorting all voltage
sources and finding the equivalent resistance. The result of this is
R  R1 R2 
R1R2
.
R1  R2
(2-102)
This leads to the Thevenin equivalent circuit represented by the right schematic of Figure
2-20.
2.8.4 White Noise Assumption
Although not stated above, one of the assumptions we place on the noisy resistor
is that its noise power density is constant over the bandwidth of B . That is,
N R  kT w Hz over B .
(2-103)
In fact, although not realistic, we assume that the noise power density is constant for all
frequencies. In other words, we assume that the noise is white. This is a good
assumption in practice because radars are generally designed so that the noise spectrum
into a device is flat over the bandwidth of the device. This is specifically done to assure
that the white noise assumption can be invoked.
2.8.5 Effective Noise Temperature for Active Devices
For passive devices such as resistive attenuators we can find the noise power
delivered to a load by an extension of the technique used in the above example. For
active devices this is not possible. For these devices the only way to determine the noise
power delivered to a load is through measurement. In general, the noise power delivered
to the load will depend upon the input noise power to the device and the internally
generated noise. The standard method of representing this is to write the noise power
delivered to the load as
Pnout  GPnin  Pnint  GkTB  GkTe B
(2-104)
where


G is the gain of the device,
kTB is the input noise power (in a bandwidth of B ),
©2011 M. C. Budge, Jr
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

GkTe B is the noise power generated by the device (in a bandwidth of B ) and
Te is the equivalent noise temperature of the device.
For resistors, the equivalent noise temperature is an actual temperature. For active
devices the equivalent noise temperature is not an actual temperature. It is the
temperature that would be necessary for a resistor to produce the same noise power as the
active device. Both G and Te can be measured. In the above equation, and in all
calculations of noise to follow, we never specifically state the value of the bandwidth.
We simply carry as it along as a required parameter.
2.8.6 Noise Figure
An alternate to using effective noise temperature to characterize the noise
properties of devices is to use noise figure. The noise figure, Fn , of a device is defined as
Fn 
noise power out of the actual device
.
noise power out of an ideal device
(2-105)
In this definition it is assumed that the noise power into the device is given by
Pnin 0  kT0 B
(2-106)
where T0  290 K .
To compute the noise out of the ideal device we assume that the device does not
generate its own noise. Thus
Pnoutideal  GPnin 0  kT0 BG .
(2-107)
From (2-104), the actual noise power out of the device, when the input noise power is
Pnin 0 , is
Pnoutactual  kT0 BG  kTe BG .
(2-108)
With this we can relate Fn to Te as
Fn 
Pnoutactual kT0 BG  kTe BG
T

 1 e .
Pnoutideal
kT0 BG
T0
(2-109)
Alternately, we can solve for Te in terms of Fn as
Te  T0  Fn  1 .
(2-110)
An important point from Equation (2-109) is that the minimum noise figure of a
device is Fn  1 . Another important point in the above is that noise figure is always
based on an assumption that the noise power into the device derives from a resistive noise
source at the standard temperature of 290 ºK.
In working radar problems some people prefer noise figure and others prefer
effective noise temperature. Most of the noise specifications of devices and radars are
©2011 M. C. Budge, Jr
37
provided in terms of noise figure. However, as we will see shortly, effective noise
temperature, and total noise temperature, are often useful when characterizing the
combined effects of external noise sources and receiver noise.
For most devices, noise figure is determined by measurement. The exception to
this is attenuators. For attenuators, the noise figure is the attenuation. Thus, if an
attenuator has an attenuation of L (a number greater than one) the noise figure is
Fn  L .
(2-111)
The rationale behind this is that if the attenuator is matched to the source and the load
impedance, which are assumed the same, the noise power out of the attenuator is equal to
the noise power input to the attenuator. There is a further, unstated, assumption that the
noise temperature of the resistive elements that make-up the attenuator are at the same
temperature as the noise source driving the attenuator.
With the above we can derive the noise figure of an attenuator as follows. If the
attenuator is considered ideal, i.e. the resistive elements that make-up the attenuator do
not generate noise, the noise power out of the attenuator is
Pnoutattenideal  Pninatten L .
(2-112)
However, for an actual attenuator we have
Pnoutattenactual  Pninatten .
(2-113)
By the definition of Equation (2-105) the noise figure of the attenuator is
Fn 
Pnoutattenactual
P
 ninatten  L .
Pnoutattenideal Pninatten L
(2-114)
2.8.7 Noise Figure of Cascaded Devices
Since a typical radar has several devices that contribute to the overall noise figure
of the radar we need a method of computing the noise figure of a cascade of components.
To this end, we consider the block diagram of Figure 2-21. In this figure, the circle to
the left is a noise source, which in a radar would be the antenna or other radar
components. For the purpose of computing noise figure, it is assumed that the noise
source has an effective noise temperature of T0 (consistent with how noise figure is
defined – see Section 2.8.6). The blocks following the noise source represent various
radar components such as amplifiers, mixers, attenuators, etc. These blocks are
represented by their gain, G , and noise figure, F . For purposes of computing noise
figure, all of the devices are assumed to have the same bandwidth of B .
©2011 M. C. Budge, Jr
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Figure 2-21 – Block Diagram for Computing System Noise Figure
To derive the equation for the overall noise figure of the N devices we will
consider first device 1, then devices 1 and 2, then devices 1, 2, and 3, and so forth. This
will allow us to develop a pattern that we can extend to N devices.
Since we have the noise figure of each device we can compute the effective noise
temperature of each device via Equation (2-110). Thus, the effective noise temperature
of device k is
Tk  T0  Fk  1 .
(2-115)
For Device 1, the input noise power is
Pnin1  kT0 B .
(2-116)
The noise power out of an ideal Device 1 is
Pnout1i  G1Pnin1  kT0 BG1 .
(2-117)
The actual noise power out of Device 1 is
Pnout1  G1Pnin1  Pint1  kT0 BG1  kT1BG1  k T0  T1  BG1 .
(2-118)
From Equation (2-105) the system noise figure from the source through Device 1 is
Fn1 
Pnout1 k T0  T1  BG1
T

 1  1  F1
Pnout1i
kT0 BG1
T0
(2-119)
where the last equality was a result of Equation (2-109).
For Device 2, the input noise power is
Pnin 2  Pnout1  k T0  T1  BG1 .
(2-120)
The noise power out of an ideal cascade of Devices 1 and 2 is
Pnout 2i  G1G2 Pnin  kT0 BG1G2 .
(2-121)
The actual noise power out of Device 2 is
Pnout 2  G2 Pnin 2  Pint2  k T0  T1  BG1G2  kT2 BG2

T 
 k  T0  T1  2  BG1G2
G1 

.
(2-122)
The system noise figure from the source through Device 2 is
©2011 M. C. Budge, Jr
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Fn 2 
Pnout 2 k T0  T1  T2 G1  BG1G2
T
1 T2
.

 1 1 
Pnout 2i
kT0 BG1G2
T0 G1 T0
(2-123)
Or, using Equation (2-109)
Fn 2  F1 
F2  1
.
G1
(2-124)
It is interesting to note that the noise figure of the second device is reduced by the gain of
the first device. We will examine this again in an example to be presented shortly. For
now we proceed to determine the system noise figure from the source through the third
device.
The noise power out of an ideal cascade of Devices 1, 2 and 3 is
Pnout 3i  G1G2G3Pnin  kT0 BG1G2G3 .
(2-125)
The actual noise power out of Device 3 is
Pnout 2  G3Pnin 3  Pint3  G3Pnout 2  Pint 3
(2-126)
or, substituting for Pnout 2 from Equation (2-118),

T 
Pnout 3  k  T0  T1  2  BG1G2G3  kT3 BG3
G1 


T
T 
 k  T0  T1  2  3  BG1G2G3
G1 G1G2 

.
(2-127)
The system noise figure from the source through Device 3 is
Fn 3 
Pnout 3 k T0  T1  T2 G1  T3  G1G2   BG1G2G3

Pnout 3i
kT0 BG1G2G3
.
(2-128)
T
1 T2
1 T3
 1 1 

T0 G1 T0 G1G2 T0
Or, again using Equation (2-109)
Fn 3  F1 
F2  1 F3  1

.
G1
G1G2
(2-129)
Here we note that the noise figure of Device 3 is reduced by the product of the gains of
the preceding two devices.
With some thought we can extend Equation (2-126) to write the system noise
figure from the source through Device N as
FnN  F1 
©2011 M. C. Budge, Jr
F2  1 F3  1 F4  1



G1
G1G2 G1G2G3

FN  1
.
G1G2G3 GN 1
(2-130)
40
It will be left as an exercise to show that the effective noise temperature of the N devices
is
TeN  T1 
T2
T
 3 
G1 G1G2
TN
G1G2
GN 1
.
(2-131)
In the above we found the system noise figure between the input to Device 1
through the output of Device N. If we wanted the noise figure between the input of any
other device, say Device k, to the output of some other succeeding device, say Device m,
we would assume that the source of Figure 2-21 was connected to the input of Device k
and we would include terms like those of Equation (2-130) that would carry to the output
of Device m. Thus, for example, the noise figure from the input of Device 2 to the output
of Device 4 would be
Fn24  F2 
F3
F
 4 .
G2 G2G3
(2-132)
2.8.8 An Interesting Example
We now want to consider an example that illustrates why radar designers
normally like to include an RF amplifier as the first element in a receiver. In this
example we consider the two options of Figure 2-22. In the first option we have and
amplifier followed by an attenuator and in the second option we reverse the order of the
two components. The gains and noise figures of the two devices are the same in both
configurations. For Option 1, the noise figure from the input of the first device to the
output of the second device is
Figure 2-22 – Two Configurations Options
Fno21  F1 
©2011 M. C. Budge, Jr
F2  1
100  1
 4
 5 w/w or 7 dB .
G1
100
(2-133)
41
For the second option the noise figure from the input of the first device to the output of
the second device is
Fno22  F1 
F2  1
4 1
 100 
 400 w/w or 26 dB!
G1
0.01
(2-134)
This represents a dramatic difference in noise figure of the combined devices. This
difference is due to the aforementioned property that the noise contributed to the system
by an individual device is a function of the noise figure of that device and the gains of all
devices that precede the device. In general, if the preceding devices have a net gain, the
noise contributed by a device will be reduced relative to its individual noise figure. If the
preceding devices have a net loss, the noise contributed by the device will be increased
relative to its individual noise figure.
In Option 1 of the above example, the noise figure of the two devices was close to
the noise figure of simply the amplifier. However, for the second option the noise figure
was the combined noise figures of the two devices. This is why radar designers like to
include an amplifier early in the receiver chain: it essentially sets the noise figure of the
receiver.
2.8.9 Output Noise Power When the Source Temperature is not T0
In the above, we considered a source temperature of T0 . We now want to
examine how to compute the noise power out of a device when the source temperature is
something other than T0 . From Equation (2-104) we have
Pnout  GPnin  Pnint  GkTB  GkTe B
(2-135)
where Pnin  kTB and T is the noise temperature of the source. If we were to rewrite
Equation (2-135) using noise figure we would have
Pnout  GPnin  Pnint  GkTB  GkT0  Fn  1 B .
(2-136)
If we use a cascade of N devices, G is the combined gain of the N devices, Te is the
effective noise temperature of the N devices (see Equation (2-131)) and Fn is the noise
figure of the N devices (see Equation (2-130)).
©2011 M. C. Budge, Jr
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