NOISE
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
(100)
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 .
(101)
The Thevenin equivalent circuit of a noisy resistor is as shown in Figure 18. It
consists of a noise source with a voltage defined by (100) and a noiseless
resistor with a value of R .
Noiseless
R
vrms  4kTBR
Figure 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 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 19 and the voltage
across RL is given by
vRL  vrms
RL
.
RL  R
(102)
The power delivered to RL is
PL 
1
vR2L
RL

2
vrms
RL
 RL  R 
2

4kTRB
 RL  R 
2
.
(103)
From “Principles of Communications” by Ziemer and Tranter, Fourth Edition.
©2005 M. C. Budge, Jr
32
If the load is matched to the source resistance, that is if RL  R , we have
PL  kTB
(104)
Which is the familiar form used in the radar range equation.
Noiseless
R
vrms  4kTBR
RL
Figure 19 – Diagram for Computing the
Power Delivered to a Load
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 20. In the figure, the left schematic
shows two, parallel noisy resistors and the center schematic shows the
equivalent circuit based on Figure 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
(105)
and with only source v2 we would get
vo 2  v2
R1
R1
.
R1  R2
R2
vo
(106)
R1
R2
R
vo
v1
v2
vo  4kTBR
R  R1 R2
Figure 20 – Schematic Diagrams for Two-resistor Problem
To get the total Thevenin equivalent voltage we must consider that v1 and v2
are noises. As such, we must add their squares. Thus, with this we get
©2005 M. C. Budge, Jr
33
vo2  vo21  vo22  v12
R22
 R1  R2 
2
 v22
R12
 R1  R2 
2
.
(107)
If we use v12  4kTR1B and v22  4kTR2 B we find that
vo2  4kTB
R1R2
 4kTBR .
R1  R2
(108)
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
(109)
This leads to the Thevenin equivalent circuit represented by the right schematic
of Figure 20.
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 .
(110)
In fact, although not realistic, we assume that the noise power density is
constant for all frequencies and that the resistor is an ideal band pass filter
with a bandwidth of B . 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.
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
(111)
where G is the gain of the device, kTB is the input noise power (in a bandwidth
of B ), 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.
©2005 M. C. Budge, Jr
34
An alternate to using gain and 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
(112)
In this definition it is assumed that the noise power into the device is given by
Pnin 0  kT0 B
(113)
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 .
(114)
From (111), the actual noise power out of the device, when the input noise
power is Pnin 0 , is
Pnoutactual  kT0 BG  kTe BG .
(115)
With this we can relate Fn to Te as
Fn 
Pnoutactual kT0 BG  kTe BG
T

 1 e .
Pnoutideal
kT0 BG
T0
(116)
Alternately, we can solve for Te in terms of Fn as
Te  T0  Fn  1 .
(117)
An important point from (116) 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 provided in terms of noise figure. However, as we will see
shortly, effective noise temperature, and total noise temperature, are often of
use 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 .
(118)
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
©2005 M. C. Budge, Jr
35
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 .
(119)
However, for an actual attenuator we have
Pnoutattenactual  Pninatten .
(120)
By the definition of (112) the noise figure of the attenuator is
Fn 
Pnoutattenactual
P
 ninatten  L .
Pnoutattenideal Pninatten L
(121)
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
21. In this figure, the circle to the left is a noise source, which in a radar would
be the antenna are other radar components. For the purpose of computing
noise figure it is assumed that the noise source has an effective noise
temperature of T0 . 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 .
Source
Device 1
Device 2
Device 3
Device N
T0 , B
G1 , F1 , B
G2 , F2 , B
G3 , F3 , B
GN , FN , B
Figure 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 (117). Thus, the effective noise
temperature of device k is
Tk  T0  Fk  1 .
(122)
For Device 1, the input noise power is
©2005 M. C. Budge, Jr
36
Pnin1  kT0 B .
(123)
The noise power out of an ideal Device 1 is
Pnout1i  G1Pnin1  kT0 BG1 .
(124)
The actual noise power out of Device 1 is
Pnout1  G1Pnin1  Pint1  kT0 BG1  kT1BG1  k T0  T1  BG1 .
(125)
From (112) 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
(126)
where the last equality was a result of (116).
For Device 2, the input noise power is
Pnin 2  Pnout1  k T0  T1  BG1 .
(127)
The noise power out of an ideal cascade of Devices 1 and 2 is
Pnout 2i  G1G2 Pnin  kT0 BG1G2 .
(128)
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 

.
(129)
The system noise figure from the source through Device 2 is
Fn 2 
Pnout 2 k T0  T1  T2 G1  BG1G2
T
1 T2

 1 1 
.
Pnout 2i
kT0 BG1G2
T0 G1 T0
(130)
Or, using (116)
Fn 2  F1 
F2  1
.
G1
(131)
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 .
(132)
The actual noise power out of Device 3 is
Pnout 2  G3Pnin 3  Pint3  G3Pnout 2  Pint 3
(133)
or, substituting for Pnout 2 from (125),
©2005 M. C. Budge, Jr
37

T 
Pnout 3  k  T0  T1  2  BG1G2G3  kT3 BG3
G1 


T
T 
 k  T0  T1  2  3  BG1G2G3
G1 G1G2 

.
(134)
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
.
(135)
T
1 T2
1 T3
 1 1 

T0 G1 T0 G1G2 T0
Or, again using (116)
Fn 3  F1 
F2  1 F3  1

.
G1
G1G2
(136)
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 (133) to write the system noise figure
from the source through Device N as
FnN  F1 
F2  1 F3  1 F4  1



G1
G1G2 G1G2G3

FN  1
.
G1G2G3 GN 1
(137)
It will be left as a exercise to show that the effective noise temperature of the N
devices is
TeN  T1 
T2
T
 3 
G1 G1G2
TN
G1G2
GN 1
.
(138)
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 21
was connected to the input of Device k and we would include terms like those of
(137) 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
(139)
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 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
©2005 M. C. Budge, Jr
38
Amplifier
Attenuator
G1  100
G2  1 L  0.01
F1  4
F2  L  100
Attenuator
Amplifier
G1  1 L  0.01
G2  100
F1  L  100
F2  4
Option 1
Option 2
Figure 22 – Two Configurations Options
Fno21  F1 
F2  1
100  1
 4
 5 or 7 dB .
G1
100
(140)
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 or 26 dB!
G1
0.01
(141)
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 the 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.
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 (111) we have
Pnout  GPnin  Pnint  GkTB  GkTe B
(142)
where Pnin  kTB and T is the noise temperature of the source. If we were to
rewrite (142) using noise figure we would have
©2005 M. C. Budge, Jr
39
Pnout  GPnin  Pnint  GkTB  GkT0  Fn  1 B .
(143)
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 (138)) and Fn is the noise
figure of the N devices (see (137)).
©2005 M. C. Budge, Jr
40