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Noise
What is Noise?
Noise is an unwanted signal, which appears in a system. It may be an impulse, a specific
frequency, and a set of harmonically related frequencies or range of frequencies. The cause
can be external (in which case, the noise is usually called interference) or internal (from
electronic components themselves).
Sources of Noise
Figure 1 is a convenient introduction to the many aspects of noise in digital
telecommunication systems. It is a generalisation of the coding, transmission, and decoding
of information from a number of sources, showing the nature and likely points of entry of
possible noise. Because it is a generalisation, most of the commonest sources of noise and
interference have been included, although not all transmission systems will suffer from all
the types of noise shown. Note that I use the term noise in a very general sense to cover any
unwanted signal degradation. This includes:

random noise generated within or outside system components (thermal noise in resistors,
for example, or 'sky' noise originating from various sources in the atmosphere or
beyond)

the unwanted influence of other signals being transmitted simultaneously (crosstalk in
switching units and multiplexers, or interference from other radio channels, for example)

signal degradation arising from the practical limitations of system components
(quantisation noise and jitter, intermodulation products due to component nonlinearities, and so on).
1
No signal is ever received precisely as it was transmitted. The received signal will not only
be distorted in general shape, but will also be accompanied by unwanted signals or noise
from various sources. In a digital system, the effect of random noise is to introduce some
probability of error in the decoded data. In analogue systems, effects include random
'speckle' on a television screen, and the (usually) faint background sound, rather like running
water, which can often be heard on an open telephone line or radio channel.
Communication systems may also suffer from short, intense 'bursts' of interference (from
electrical machinery, for example). A burst of such interference may completely corrupt a
comparatively large amount of transmitted data. It can be counteracted (at least in systems
which do not have to operate in real time) by monitoring the error rate at the receiver, and
organising repeat transmission of the affected data if system performance is degraded
beyond a certain limit.
The most important sources of random noise in telecommunications are: thermal noise (also
known as Johnson noise), including sky noise; shot noise (which occurs in active electronic
devices); and 1/f noise (also known as excess noise or flicker noise). In this section I shall
also consider crosstalk, harmonic distortion, and intermodulation distortion.
Thermal (Johnson) noise
In the absence of an applied electrical field, the electrons in a conductor are in purely
random motion, their agitation increasing as the temperature is raised. This random motion
leads to a randomly varying potential difference across the ends of the conductor. Thermal
noise approximates very closely to white noise, with a constant power spectrum.
A real resistor, in which thermal noise is generated, can be modelled by a noise voltage
generator in series with an ideal noiseless resistor of the same resistance. This Thevenin
equivalent circuit is shown in Figure 2(a). It can be shown, although I shall not give the
derivation here, that the mean square voltage density associated with a resistor R is
R
v  4kTR
2
i   4kT
R
R
n2
n
{Vn2} = 4kTR
where T is the absolute temperature in Kelvin and k is Boltzmann's constant (k = 1.38 x 1023
J K-1).
In a similar way, the Norton equivalent circuit of a real (noisy) resistor is shown in Figure
2(b). The mean square current density associated with the resistor R is = 4kT/R
To obtain the mean square voltage or current in any particular case, these expressions must
be multiplied by the bandwidth B over which the noise is measured:
2
v 2  4kTRB
i2 
4kTB
R
Thermal noise occurs in resistors and the resistive parts of other electronic devices (for
example, the base resistance of a transistor). It is also valid, for calculation purposes; to
consider that thermal noise also occurs in the Thevenin or Norton equivalent resistance of
the source from which a signal is obtained. In the case of a cable, for example, the
equivalent resistance is the resistive (real) part of the impedance presented by the cable to
the receiver. This is a most important point to remember, and will be taken up in the next
section.
Sky noise
Receiving antennas pick up sky noise. This is wideband noise with similar characteristics to
thermal noise, but originating in the atmosphere or beyond. Again, it can often be modelled
closely as white noise. In fact, the easiest way to include it in system noise calculations is to
model it as thermal noise in a fictitious resistor. In this case, the value of the resistor used is
the antenna's radiation resistance.
Radiation resistance is easiest to understand in the context of a transmitting antenna. It is
defined as that value of load which, when connected in place of the antenna, would dissipate
the same power as that radiated by the antenna. When the same antenna is used for
reception, it can be modelled b' a Thevenin or Norton equivalent circuit with a resistor equal
to the radiation resistance. The sky noise is then modelled as though it were thermal noise,
that is noise with a mean square voltage density
V   4kT R
2
n
a
r
where Rr is the radiation resistance and Ta is known as the antenna noise temperature. The
antenna noise temperature is not necessarily the same as ii physical temperature, just as the
radiation resistance is not the electrical resistance of its material of construction. The noise
temperature can be thought of as the effective temperature of the region at which the
antenna is pointing. For example, if the antenna is used for terrestrial communication, and
its beam has only a small inclination, then its noise temperature is often close to the physical
temperature of the earth or lower atmosphere. This is commonly taken as 290K, for ease of
calculation, since kT then becomes very nearly 4 x 10-21 J. This comparatively 'warm' noise
temperature is a reflection of the comparatively 'noisy' environment in the earth's lower
atmosphere. If however, the antenna is pointing straight up into the sky (but well away from
the extremely noisy sun), then noise temperatures can be as low as a few Kelvin, reflecting
the much 'quieter' background noise of outer space.
It is important to remember that radiation resistance and effective noise temperature are
modelling tools, not physical quantities. Introducing the concepts allows the techniques of
circuit analysis to be applied to receiving or transmitting antennas. You will see in a later
section that an equivalent noise temperature can also be used to describe the noise
performance of a receiver or amplifier; again the noise temperature may bear very little
relationship to the actual physical temperature of the electronic devices involved.
3
Shot noise
Shot noise, which occurs in the active devices used in amplifiers and receivers, also behaves
as white noise over any bandwidth of practical interest. It arises from the discrete nature of
the charge carriers, which constitute a flow of current. When such charge carriers cross a
potential barrier - at a semiconductor junction, for example - slight variations in the kinetic
energy of individual carriers cause random fluctuations in the precise rate of flow of carriers
across the barrier. Even a 'constant' direct current will therefore always fluctuate slightly.
The effect can be modelled as a shot noise current source with a mean square spectral
density
I n  2qI DC B
where
In is the current shot noise
q is the electron charge (1.6 x 10-19 Coulombs)
IDC is the dc bias current
B is the bandwidth
Flicker Noise or 1/f Noise
At sufficiently low frequencies, 1/f noise, also known as flicker noise or excess noise,
predominates. As its name suggests, its power density over a given frequency range of
interest is proportional to 1/f It is therefore not white noise, since its power density is
strongly frequency dependent, increasing as the frequency falls. There is no single physical
explanation for 1/f noise, which appears to be the result of a number of different
phenomena, but it is observed in a wide variety of situations. Ultimately, at extremely low
frequencies, it may be helpful to think of 1/f noise as a 'drift' phenomenon. All physical
devices eventually drift away from their operating points in the absence of recalibration, and
this effect may be thought of as 1/f 'noise' whose frequency components have extremely
long periods of weeks or months!
Burst (popcorn) noise
This is another type of low frequency noise which is known to but which is related to the
purity of the semiconductor manufacturing process. Measurements on a device prone to
popcorn I would show a sudden, short-lived shift in the bias current, which quickly returned
to its original state. The popping bur noise gave rise to the term popcorn noise. It increases
with current and is proportional to the inverse of the square of frequency:
Burst noise 
1
f2
Problems
4
Other causes of signal degradation
Random noise is not the only source of signal degradation in a telecommunication system. I
want to conclude this section by looking briefly at two other important phenomena:
crosstalk and non-linear distortion (harmonic and intermodulation distortion).
Crosstalk
Crosstalk is the name given to the unwanted interference by a signal in one
telecommunications channel with a signal in another. For example, in analogue telephony, it
is sometimes possible to hear a faint, but irritating, second conversation in the background,
often as a result of capacitive or inductive pick-up between physically adjacent metallic
conductors. Such pick-up can occur either during transmission, when many conductors are
often closely packed in a single cable, or during switching or multiplexing.
Capacitive (electrostatic) coupling
This occurs when a signal line runs close to another line of high impedance. When the
voltage changes rapidly in a transmitter line, the victim line follows to some extent. The
cure involves:



Moving the wires apart, if possible. This reduces the capacitance between the wires.
Reducing the impedance of the victim circuit. The impedance determines the amount of
pickup between the wires.
Introducing a screen around one or both of the wires. The screen could be a metal
enclosure or a mesh as is used in coaxial cables. Not quite as effective, but of some
worth, is the technique of moving both emitter and victim wires close to a ground plane.
This is why, on some printed circuit boards; you can see large areas of conductor.
Magnetic (inductive) coupling
The magnetic field is generated when the current changes rapidly in the transmitting wire.
The victim circuit will pick this up quite easily. The cure involves:





Separating the circuits. The magnetic coupling will be reduced.
Reducing the loop area of the victim circuit. The induced signal is largely dependent
on the area of the coil since it acts like the secondary coil of a transformer.
Increasing the circuit impedance. This reduces the amount of induced current.
Using differential amplifiers in the victim circuit to cancel common-mode effects.
Amplifiers will pick up the interference as a common mode signal. Using differential
techniques will eliminate these.
Using magnetically shielded wires or screens. These screens must be made from
special mu-metal sheets, which have a high permeability to magnetic fields.
In general, a magnetic screen will also act as an electrostatic screen, but not vice versa!
5
Coaxial cable screens
Care must he taken when dealing with any coaxial screen, i.e. where and how to ground it.
The signal is usually the core wire of the cable, with the screen providing some protection
around it. The screen can be grounded as shown in Figure 3.
However, if the ground points have a small resistance between them, then earth currents can
flow, and a small voltage will be introduced. The intrusion of this signal depends on the
amplitude of the wanted signal between the equipment. One solution to this is to use a
screened twisted pair cable to interconnect the two pieces of equipment. The second wire of
the pair is connected to the screen at the transmitter end as shown in Figure 4.
The differential amplifier at the receiver then amplifies the wanted signal and rejects the
common mode signal. This form of the differential amplifier would usually be inadequate to
reject all of the common mode noise, and it would be better to see a dedicated
instrumentation amplifier.
In really bad cases of noise pickup via the connecting lead, perhaps made worse by earth
loop currents, then there are devices called isolation amplifiers. These are amplifiers, which
require two sets of power supplies - one for the input (transmitter) stage and one for the
output (receiver) stage. They exhibit complete electrical isolation from the input to the
output. Figure 5 shows the schematic layout. Transmission from the input to the output is
either by light (LED and photodiode) or magnetism (coupling transformer).
-
Signal
out
+
Signal
in
Signal
in
+
-
-
+
+
Signal
out
6
Do you think crosstalk is a problem in systems using optical fibre?
Clearly, neither capacitive nor inductive pick-up can occur between fibres, so there is no
crosstalk from one fibre to another within a cable. Crosstalk can occur, however, in other
parts of the system, where the signals are processed in electrical form (repeaters, switches
and multiplexers, for example).
One advantage of the increasing use of digital techniques is that the effects of crosstalk, like
those of random noise, can often be kept very small. As I mentioned above, crosstalk in an
analogue telephone system can result in the appearance of an intelligible second voice signal
in a channel. When digital signals are involved, crosstalk from one channel may still distort
the precise shape of a signal in another, but unless the interference is extreme, it should be
possible to regenerate an error-free signal later. And even if errors are introduced, they will
not result in an intelligible background crosstalk signal.
Summary
Random thermal noise in resistors can be modelled as either a Thevenin equivalent circuit
with a voltage generator of mean square voltage density {Vn2} = 4kTR, or as a Norton
equivalent circuit with mean square current density {In2} = 4kT/R. In each case R is the
resistance, T is the absolute temperature in Kelvin, and k is Boltzmann's constant. These
results also apply to sky noise, where R becomes the antenna's radiation resistance and T its
effective noise temperature.
Shot noise can be modelled as a random current source with zero mean, superimposed on a
direct current I such that the noise mean square current density is I n  2qI DC B , where q is
the charge on each charge carrier.
Systems also suffer from 1/f noise, which is a predominantly low-frequency effect resulting
from a number of different causes.
Crosstalk and non-linear distortion can also cause signal degradation and errors. Crosstalk is
interference from a signal in another channel, while non-linear distortion can be summed up
as follows. In the case of a single input frequency, device non-linearity leads to harmonic
distortion, that is, the presence of higher-order harmonics in the output. If there is more than
one input frequency, then the resulting intermodulation products may include the sum and
difference of any pair of input frequencies or their harmonics. All amplifiers, when driven
towards saturation, exhibit harmonic and intermodulation distortion.
7
Noise in Circuits and Systems
This section is concerned with the way the overall noise performance of a
telecommunication system is affected by the characteristics of the individual subsystems.
The analysis of noise in electronic circuits and systems is an extensive topic, and this
section cannot do more than introduce you to the most important concepts and techniques
involved. My major aim is to explain the underlying ideas, so that, if you need to study the
subject in more depth later, you will have a base from which to start.
To begin with, look at Figure 6, which represents part of the ground station of a satellite
communication system. At the antenna, there will be a certain signal-to-noise ratio,
determined by the power in the original transmitted signal, the antennas and transponders
used in the link, and the characteristics of the channel over which the signal was transmitted.
Low noise
amplifier
antenna
Second preamplifier
Rest of
receiver
Cable or wave guide
What we are most interested in, however, is the signal-to-noise ratio after amplification; it is
this, which will determine the error rate in a digital system or the final signal quality in an
analogue system. How is the final signal-to-noise ratio influenced by the various subsystems
of the ground station - the cable (or waveguide), the two amplifiers, and the receiver itself?
At the input to the cable, the signal is accompanied by noise. After passing down the cable,
both the signal and the input noise are attenuated by the same factor over the bandwidth of
the cable. But the resistive parts of the cable contribute extra, thermal noise, which appears
at the output of the cable. So, at the input of the low-noise amplifier, the signal-to-noise
ratio has already worsened.
Next, consider the effect of the low-noise amplifier itself. Both the signal and the input
noise are amplified by the same factor, within the amplifier's frequency band. But even a
low-noise amplifier adds some extra thermal and shot noise generated internally. So once
again the output signal-to-noise ratio must be poorer - that is, lower - than the input signalto-noise ratio.
Every part of the system, active or passive, causes deterioration in quality. The question is
how do we arrange things in order to minimise this deterioration? This is the essential
subject matter of this section.
Basic concepts
In analysing noise it is very easy to fall into conceptual traps. It is therefore most important
to define very clearly the concepts, conventions and symbols used. Fortunately, most of the
standard techniques of circuit analysis can be applied to noise, with one or two slight
differences. In particular, Thevenin or Norton equivalent circuits can be used to derive
values of noise voltages or currents in circuits that contain a number of noisy components.
8
As a simple illustration, consider two noisy resistors connected in series, as shown in Figure
7(a). Each noisy resistor is represented as previously described, that is, by an ideal noiseless
resistor together with a voltage source in series with it. The voltage generators are labelled
in mean square terms to simplify the notation. The Thevenin equivalent circuit is shown in
part (b) of the figure. As usual, the two resistors have been replaced by their series
combination, but what is the value of the equivalent noise voltage source?
To answer this question, I need to introduce the concept of independent random sources.
Provided that the two noise sources are independent with zero means, as is the case with
noise from two separate resistors, then the mean square values of the two sources can be
added to give the equivalent combined mean square voltage. Hence the equivalent circuit of
Figure 7(b) has a mean square voltage source equal to the sum of the sources associated
with the individual resistors:
Veq  V1  V2  4kTB(R 1  R 2 )
2
2
2
This is just the noise one would expect from a single resistor of value R1 + R2.
R2
R1 + R2
v 2  4kTBR2
2
?

veq  4kTB( R1  R2 )
2
R1
v1  4kTBR1
2
This aspect of ‘independence’ can be made mathematically rigorous by introducing the
concept of correlation. Independent noise sources, such as in the present example, are
uncorrelated. Do not let me give you the impression that two or more separate noise signals
are necessarily uncorrelated. For example, two noise signals might originate from the same
source, follow different paths through a system, and then combine at some later stage. Then
the two noise signals are not independent, and the combined noise power is not equal to the
sum of the individual powers.
The total power of a signal consisting of a number of independent (uncorrelated)
components is equal to the sum of the powers of the individual components.
Consider the following apparent paradox. Two equal resistors are connected in parallel.
From the previous section we know that each resistor generates a mean square thermal noise
voltage 4kTRB. The noise is uncorrelated, so the two resistors will again generate twice
that, or 8kTRB. But the combined resistance of two equal resistors in parallel is R/2, so the
9
mean square noise voltage can only be 2kTRB. Which expression (if either) is correct, and
what is the fallacy?
The basic mistake here is to jump to conclusions without properly modelling the situation!
From the previous discussion we know that a noisy resistor can be represented by either a
Thevenin or a Norton equivalent circuit. For two resistors in parallel, then, two possible
starting points are shown in Figure 8a, in which I am using mean square voltage and current
generators to simplify the notation.
R
R
in 
2
vn
2
v n  4kTBR
4kTB
R
R
in
2
R
2
To evaluate the total noise voltage or current, one or other of these circuits must be
manipulated into a suitable overall equivalent, according to the usual rules of circuit
analysis. According to these rules, the voltage generators in part (a) of the figure do not
simply add to give 8kTRB, as they did in the series combination. In fact, it is easier to start
with part (b) of the figure, because two current sources in parallel do add. The Norton
equivalent circuit is therefore as shown in Figure 9(a) and the Thevenin equivalent as in (b).
Both voltage and current sources then have the mean square values expected with a single
resistor R/2 replacing the parallel combination.

in 
2
8kTB
R
R
R
2
R
vn  in  
2
2
2
2
v n  2kTBR
2
Equivalent input noise
At the beginning of this section, I made the point that any physical component introduces
noise, and thus tends to worsen the signal-to-noise ratio. The general situation is shown in
Figure 10a for a noisy amplifier. At the output of the amplifier, the input thermal noise will
have been amplified, and it will be accompanied by additional noise generated internally by
the amplifier itself.
10
(a)
Extra internal
noise
Thermal noise
from a source
resistance Rs
ideal
+
+
Amplified thermal
noise + internally
generated noise
(b)
Extra internal
noise
Thermal noise
from a source
resistance Rs
+
+
ideal
Amplified thermal
noise + internal
noise
In each of the above figures a noisy amplifier has been modelled by an ideal noiseless
amplifier, together with an additional source of noise. In part (a) this extra, internally
generated, noise is shown adding after the ideal amplification of the signal plus input noise.
For modelling purposes, however, it is often more convenient to draw the system in the
equivalent form shown in part (b). Here again an ideal noiseless amplifier is shown, but this
time with an additional, fictitious, input source of noise. This additional input noise, when
amplified ideally, is exactly equal to the noise generated internally by the noisy amplifier.
This procedure is known as referring the noise to the input, and is an extremely useful
technique in noise analysis. However, before I develop it any further I want to look in more
detail at the relationship between input and output power in a system. In particular, I want to
introduce the concept of 'available power'.
Available power and matching
Figure 6 showed a number of individual subsystems cascaded together. A useful model of
any one such subsystem is shown in Figure 11. It is a quite general model, and can be used,
in certain circumstances, to represent a wide variety of components such as amplifiers or
attenuators. The input to the subsystem is represented by the source VS with its source
resistance RS. The input resistance of the subsystem is Rin. On the output side, the subsystem
is represented by the voltage source Vout, and output resistance Rout together with the load
resistance RL. For the time being I am going to ignore noise, and assume that VS and Vout
represent signals only.
In many communication systems involving interconnected elements -particularly those
operating at microwave frequencies, like the satellite communication system of Figure 6 the
individual subsystems are designed to be matched. 'Matching' in this context means that the
output resistance of one stage is made equal to the input resistance of the next. Under such
11
conditions no reflections occur at the interfaces between the subsystems, a feature which is
often highly desirable. It so happens that maximum power transfer takes place from one
stage to the next under matched conditions, but this is usually a secondary consideration in
telecommunications: avoiding reflections is normally the prime motivation for matching.
R
Rout
Rin
v s 
RL
vout 
So how do we model power transfer through a subsystem like that of Figure 11? A
convenient way is by means of the concept of available power, which is defined as the
maximum power that a source can transfer to a load: in other words, the power transferred
to a load under matched conditions.
Maximum power is transferred when the source resistance equals the load resistance.
Figure 12 shows a source with resistance Rs matched to an equal load. The instantaneous
voltage across the load resistor is vS/2, so the mean power transferred - by definition the
available power - is
Power =
v2
R
Rs
Rs
2
 vS 
2
 2

  vS
Rs
4 Rs
vs/2
v s 
An important point to understand is that available power is defined as a property' of the
source alone: it is the maximum theoretical available power, which could be delivered
under ideal conditions. The definition (and numerical value) remains the same even if
conditions are not matched, and all the theoretically available power is not actually
delivered to the load.
What is the available thermal noise power of a resistor R  at a temperature T K, measured
over a bandwidth B Hz?
The mean square thermal noise voltage under these conditions is 4kTRB, so the available
power is 4kTRB/4R = kTB. Note that this expression is in watts, and represents the
maximum actual power that could be transferred to a matched resistor. Note also that it does
not depend on the value of the resistor itself. A smaller resistor generates a smaller noise
12
voltage, but is matched to a lower load resistance: the available power is therefore
unchanged.
Returning to Figure 12, then, the available input power is
2
vS
4 Rs
whatever the value of Rin. Similarly, the available output power is
2
vout
4 Rout
whatever the value of RL. This leads us to the important concept of available power gain,
defined as
G = available output power
available input power
v 4R

v 4R
2
out
out
2
s
s
2
vout  Rs 

 2 
v s  Rout 
In this section I am going to consider matched systems only. In such cases it is usual for all
cables, amplifiers, etc. to have identical input and output impedances, so that
RS = Rin = Rout = RL
Then the available power gain is simply
G
vout
vs
2
2
which is equal to the square of the voltage gain, assuming a flat frequency response.
You may well be wondering why have I gone to so much effort to define precisely the term
'available power gain' when the final result is simply voltage gain squared. The answer is
that, although I shall consider only matched systems here, the techniques introduced can
also be applied to other cases. In cases where all input and output resistances are not
identical it is vital to define and measure power gains in the appropriate way. For the
analysis of noise in cascaded subsystems, available power gain turns out to be the most
appropriate measure. Some of the results derived later in this section hold in general,
13
unmatched conditions, but only when the gains are defined as above. If I were to present the
material as though power gain were always simply the square of the voltage gain, I would
be oversimplifying matters to the point of misleading you.
Noise temperature
Applying the ideas just introduced to noisy systems turns out to be fairly straightforward.
First of all, I shall return to the noisy amplifier, modelled as in Figure 10, supposing as
before that the input noise is thermal noise only from a source resistance RS at temperature
Ts K.
What is the available noise power density of the source resistor RS?
It was shown earlier that the available power is kTSB (in watts). The noise is white, so the
available power density is simply kTS (in watts per hertz).
The available noise power density at the output due to the thermal noise of the source
resistor is therefore kTSG, where G is the available power gain of the amplifier. However,
there is also the additional noise generated by the amplifier itself, conveniently modelled as
an extra noise source at the input. We can express this fictitious source as an additional
power density kTe at the input, where Te is known as the noise temperature (or, more
strictly, the effective input noise temperature) of the amplifier. It may be considered as the
temperature of a fictitious resistor, equal in magnitude to the source resistance, which would
generate the same noise power (after amplification) as the device itself.
What is the total available noise power density at the output of the amplifier, assuming that
the internally generated noise is uncorrelated with the input noise?
The input noise contributes kTSG, while the amplifier contributes kTSG. Uncorrelated
powers (or power densities) add, so the total noise power density at the output is kG(TS + Te).
The noise temperature of a device or subsystem is an important practical parameter. It can
be measured by applying standard noise sources to the input of the device and measuring the
total output noise, which will include the extra noise generated by the device itself. The
measured output can therefore be used to derive a value of Te.
In general, the noise temperature of a device depends on the source resistance Rs for two
distinct reasons. The first has been mentioned already, namely, the fact that the noise
temperature is the temperature of a fictitious resistor of magnitude Rs. The value of Rs must
therefore also affect the value of the noise temperature. Second, however, the actual noise
generated internally also turns out to depend on the source resistance of a device. For
example, amplifiers generate least internal noise when their inputs are short-circuited, that
is, with Rs = 0. For these reasons, noise temperatures must always be quoted under specified
source impedance conditions.
14
Noise temperature and signal-to-noise ratio
To illustrate how the notion of noise temperature can help in making design decisions,
consider the simple system of Figure 13, which shows an antenna with a noise temperature
Ta connected directly to a receiver with a noise temperature of Tr = 150K. Suppose that the
output signal-to-noise ratio is barely acceptable, and that the designer needs either to
increase transmitter power or to improve the receiver performance by upgrading its
amplifier, thereby reducing its noise temperature to 50 K, say. Both options will involve
extra expense, but which will be more cost effective?
antenna
Receiver G
Ta
Tr
Consider first a situation in which the background noise received by the antenna is fairly
high (Ta = 350 K), as might be the case for noisy terrestrial communication. Then the total
available noise power density at the output in the existing system is equal to
kG(Ta + Tr) = kG(350 + 150) = kG(500)
If the better amplifier is used, the total output noise power density becomes
kG(350 + 50) = kG(400)
Assuming that the signal strength remains the same, the improvement in signal-to-noise ratio
brought about by upgrading the amplifier is equal to the ratio of these noise power densities,
that is
500kG = 1.25
400kG
or
10 log 1.25 = 0.97dB
By what factor would the transmitter power have to be increased to bring about the same
improvement in the signal-to-noise ratio?
Increasing the transmitter power should increase the received signal power by the same
factor. Assuming that the noise level is unaffected by the increase in transmitter power, then
the signal-to-noise ratio will also increase by the same factor. In this case, then, increasing
the transmitter power by 25% will have the same effect as upgrading the low-noise
amplifier.
15
SAQ 1 Suppose that in a different application the antenna picks up much less background
noise than before, so that Ta = 50. By what factor would the transmitter power now have to
be increased to bring about the same improvement in signal-to-noise ratio as upgrading the
amplifier?
Precisely which approach the designer should choose in this example depends on various
factors, including the relative costs of upgrading the amplifier and increasing the transmitter
power, and also whether it is possible to increase power by the necessary amount to bring
about a useful increase in the final signal-to-noise ratio. The example dramatically illustrates
the importance of the noise characteristics of the 'front end' of a receiving system, in this
case, the noise picked up by the antenna itself. This is quite reasonable: if the received
signal at the antenna is already greatly degraded by noise, then improving the performance
of subsequent subsystems may result in comparatively little improvement. (I shall look at
this phenomenon in more detail shortly.)
The concept of noise temperature has become important in system design precisely because
of the ease of carrying out calculations such as these. It is also easy to relate the noise
temperature of a system to the actual signal-to-noise ratio at its output. The output signal-tonoise ratio is equal to S0/N0 where S0 and N0 are available output signal and noise powers
respectively. But
So
SG
 i
No NiG
where Si is the available input signal power, G is the available power gain of the system, and
Ni is the available input noise power including the equivalent input noise power introduced
by the system itself. That is, the noise is all referred to the input. Ni is therefore equal to k(TS
+ Te) B where Te is the noise temperature of the system and B is the bandwidth of interest.
Hence
So
Si G

N o k (Ts  Te ) B
16
SAQ 2 The available signal power at the receiving antenna of the system of SAQ1 is 1 pW.
What is the output signal-to-noise ratio of the upgraded system, assuming a bandwidth of 5
MHz? (Assume a value for k, Boltzmann's constant, of 1.38 x l0-23J K-1.)
Noise temperature of cascaded subsystems
I now want to consider what happens when a number of subsystems are cascaded to form a
complete system.
Consider first two matched subsystems connected together, for example, the low-noise
amplifier and second amplifier of the satellite ground station of Figure 14.
The general situation is shown in Figure 14. The first amplifier has input noise power
density kTS from a source with noise temperature TS, and its own internally generated noise
is referred to the input as an extra noise density kTe1 where Te1 is the amplifier's noise
temperature. The total available noise power density from the first stage is therefore
kG1 Ts + kG1Te1 Similarly, the second stage has a noise temperature Te2 associated with
equivalent additional noise power density kTe2 at its input.
kTe1
kTe2
+
kTs
+
G1
+
+
Te1
G2
Te2
What is the total noise power density at the output of the second stage?
The noise power density from the first stage is amplified by G2, and there is also additional
noise density from the second amplifier equal to kG2 T2 The total output noise power
density is therefore
17
kG1 G2Ts + kG1 G2Te1+ kG2Te2
again assuming uncorrelated noise sources.
To characterise the total noise of the two cascaded stages, we need an equivalent combined
model as shown in Figure 15. That is, the two amplifiers are replaced by a single block of
available power gain GT = G1G2 and overall noise temperature Te. So the total output noise
power density from the equivalent combined system is kGT Ts  kGT Te
kTe
+
kTs
+
GT = G1G2
kGT (Ts + Te)
T e= ?
Setting this expression equal to the one derived in the preceding in-text question, we have
kGTTs + kGTTe = kG1G2Ts + kG1G2Te1, + kG2Te2
Since GT = G1 G2 the first terms on each side of this equation are equal (they both represent
the final output noise density originating from the input noise at the first stage). Hence
kGTTe = kG1G2Te1 + kG2Te2
Dividing each side by kG1G2 (= kGT) gives
Te  Te1 
Te 2
G1
SAQ 3 Show that, for three subsystems in cascade, Te  Te1 
Te 2
T
 e3
G1 G1G2
18
The same argument can be extended to any number of cascaded subsystems, leading to the
general expression:
Te  Te1 
Te 2
T
Te 4
 e3 
 ......
G1 G1G2 G1G2 G3
A great deal of information is contained in this general result. The first important point to
note is that, provided that the gains of the subsystems are all high, then the overall noise
temperature is determined primarily by the noise temperature of the first element. In fact, we
can go even further. If G1 alone can be made high enough, then Te1 can be made to dominate
all the other terms in the expression.
Hence, an important general rule is to ensure that the first element in a cascaded system
generates as little noise as possible, and has a sufficiently high gain to ensure that the signal
is much greater than any noise subsequently introduced. Now, if you look back at Figure 6,
you will see that the first element in the system (after the antenna itself) is a cable or
waveguide, which is bound to attenuate the signal to some extent - it will have a gain of less
than one. I shall look at how to include attenuators in the system model later on. However,
from the preceding discussion it should be clear that, if it is necessary for a lossy element
such as a cable or waveguide to be the first element in a system (as it often is), it is
important to keep the The attenuation introduced as low as possible. Simply connecting an
amplifier after the lossy element will not do. In many cases, in fact, an amplifier would be
mounted (if possible) on the antenna itself.
Noise factor
Noise temperature is a very useful parameter and, as already seen, calculations can be
performed easily given numerical values. Nevertheless, it is sometimes more convenient to
express system performance in a way that relates even more directly to signal-to-noise
ratios. This can be done by introducing the concept of noise factor.
Like noise temperature, the noise factor (or noise figure) F of a system is a measure of the
noise introduced by the system itself. It may be defined loosely (I shall explain why in a
moment) in the following way:
noise factor F = signal-to-noise ratio at the system input
signal-to-noise ratio at the system output
This ratio is always greater than 1, and the larger the ratio, the more noisy the system. It is
commonly quoted in decibels, using the expression 10 log F. (Some writers reserve the term
noise figure for a noise factor expressed in decibels.)
A typical noise factor for a high-frequency amplifier is 10dB. At the output of the amplifier,
how much of the noise power is contributed by the amplifier itself, and how much is an
amplified version of the source input noise?
10dB is equivalent to a noise factor F = 10. This means that the input signal-to-noise ratio is
ten times the output signal-to-noise ratio. In other words, the total output noise power is ten
times the amplified input noise power. So nine-tenths of the output noise must have been
generated by the amplifier, and one-tenth is amplified source input noise.
19
Formally, the noise factor is defined in terms of input and output noise power densities,
rather than signal-to-noise ratios. I shall state the formal definition, and then look at the
circumstances under which we can use the looser idea of the ratio of S/N at input and output.
The noise factor of a system is the ratio of:
1
the total noise power density at the output when the noise temperature of the input
termination is 290K, to
2
that portion of (1) generated by the input termination at the same temperature.
Throughout this section I am going to assume white noise, and a system frequency response
that is flat over the range of frequencies of interest. In such cases, provided that the system
is at the standard temperature of 290 K, then the loose definition in terms of signal-to-noise
ratios is equivalent to the more formal definition. For a given signal level and a flat
frequency response, the ratio of input to output S/N is then equal to the total output noise
power density divided by that fraction resulting from the input noise alone. I suggest you
think about this for a moment, and refer back to the last in-text question if you feel you need
further clarification.
A most important point to note is that noise factors are defined at the standard temperature
of 290 K. Noise factors are therefore easiest to use when all system elements are at this
standard temperature. When this condition does not hold, it is often preferable to work with
noise temperatures. Like noise temperatures, noise factors are affected by the source
impedance, and measured values must be quoted under specified conditions.
The relationship between noise factor and noise temperature follows directly from the above
definition. Denoting the standard temperature of 290 K by T0, the gain of the system by G,
and its noise temperature by Te we have
total output noise power density = kG(T0 + Te)
portion due to input noise = kGT0
The noise factor F is the ratio of these two quantities:
F
kG(T0  Te )
kGT0
1 
1 
Te
T0
Te
290
Hence, for example, a noise temperature of 500 K is equivalent to a noise factor of 1 +
500/290 = 2.72. In decibels this becomes 10 log 2.72 = 4.4dB.
20
SAQ 4.4 A system has a noise factor of 6dB. What is its noise temperature, and what
fraction of the total output noise power is generated internally by the system itself?
Here
Cascaded subsystems
An expression for the overall noise factor of a system of cascaded elements can be derived
immediately from the similar expression for noise temperatures. Since
Te
To
rearranging gives
F  1
Te  F - 1To  F - 1290
The figure below shows the general situation for cascaded subsystems. We know from noise
temperature of cascaded subsystems section that
Te  Te1 
Te 2
T
Te 4
 e3 
 ......
G1 G1G2 G1G2 G3
so substituting directly for noise factors gives
(F2 - 1)T0 (F3 - 1)T0

 .......
G1
G 1G 2
Dividing each side by T0 leads to the expression
(FT - I) T0  (F1 - 1)T0 
FT  F1 
F2  1 F3 - 1 ()T0

 .......
G1
G 1G 2
G1
G2
G3
Te1, F1
Te2, F2
Te2, F2

GT = G1+ G2+ G3
Te, FT
21
Cables and attenuators
All that remains now is to incorporate lossy elements, such as cables and attenuators, into
the noise model. In fact, the result I shall present applies to any lossy passive network. The
word passive is used to indicate that there are no active devices, such as transistors, in the
network to introduce shot or other non-thermal noise. The noise analysis therefore needs to
consider only thermal noise in the resistive parts of the network. To keep things simple, I
shall assume that the network is resistive only, although the analysis can be applied to those
passive circuits - filters, for example - which also involve reactive components such as
inductors and capacitors.
The insertion loss L of such a network is defined as L = input power = 1
output power G
As before, I shall consider matched conditions only.
For a matched attenuator (or other passive network) it can be shown that the noise factor F
is equal to the insertion loss L. I am not going to give a detailed proof of this, which would
involve rather more network analysis than is appropriate for this course. The general thrust
of the argument can be easily illustrated, however, with the aid of Figure 4.13. The figure
shows a passive attenuator terminated on each side by a matched resistor. The entire system
is assumed to be at the standard temperature of 290 K, as is required to derive a noise factor.
From the formal definition of noise factor, we know that
F
total output noise power density
output noise power density resulting from input noise
Now, because the attenuator is a matched, passive, resistive network, from the point of view
of the load resistor in Figure 4.13 it behaves exactly like a resistor of resistance R. In other
words, whatever the precise arrangement of resistors within the attenuator (or distributed
resistance in the case of a cable), the load simply 'sees' the matched output resistance R. So
the available output noise power density from the attenuator is the usual kT0, which forms
the numerator of the noise factor expression. To fill in the denominator, we need to know
how much of this output noise arises from the input noise after attenuation. Now, the
matched source resistor on the input side also supplies a power density of kT0, so after
attenuation, the contribution of this to the total output noise density is simply kT0/L.
Hence we can write
F
kT0
L
kT0 /L
That is, the noise factor of a matched, lossy attenuator at the standard temperature is equal to
its insertion loss.

What is the equivalent noise temperature, T1, of a matched attenuator of insertion loss
L?
22
 It was shown earlier that Te = (F - 1)290. Hence in this case T1 = (L - 1)290. Note yet
again that the equivalent noise temperature is not the same as the attenuator's physical
temperature, which is assumed here to be the standard 290 K.
Use of a television pre-amplifier
I pointed out above that noise could be reduced by arranging for the 'front-end' of a
cascaded system to have a sufficiently high gain and a sufficiently low noise factor or
temperature. To illustrate this crucial point in a situation where losses, as well as gains, are
involved, I shall show the effect of a lossy coaxial downlead connecting a television aerial
to a receiver. I shall make the following assumptions about the receiving system:
1
The aerial's radiation resistance, the cable's input and output impedance, and the
receiver's input impedance are all resistive and equal over the frequency range of
interest.
2
The physical temperature of all parts of the system is 290 K. (I can therefore use
noise factors throughout the analysis.)
3
The mean square aerial noise voltage within the frequency range of interest is
5 x 10-12 V2 and the rms signal emf at the aerial is 1 mV, both measured into the
same load.
4
The cable has a loss of 6dB; the receiver noise factor is 4 (6dB); and a low-noise
pre-amplifier is available with a noise factor of 2 (3 dB) and a gain of 20dB.
Case 1
Suppose that the aerial could be connected directly to the receiver. Using the 'loose'
definition of noise factor
noise factor 
input S/N
output S/N
For the system under consideration we can therefore rewrite this expression as
final S/N 
S/N at aerial
receiver noise factor
Now, the signal-to-noise ratio at the aerial is given by
Working in dB, the final signal-to-noise ratio at the receiver is obtained by subtracting the
noise figure from the signal-to-noise ratio at the aerial, Hence the final signal-to-noise ratio
mean square signal voltage
S

 
mean square noise voltage
 N ms
-6
10
 2 x 10 -5  (53dB)
-l2
5 x l0
23
is 53 - 6 = 47dB.
This would be just about acceptable for good-quality colour television.
Case 2
Suppose now that the receiver is connected to the aerial via the cable, as in
Figure 4.14.
Cable
Receiver
Ta = 290 K
L
1
4
G
F3 = 4
Use the cascade formula for noise factor to find the new final signal-to-noise ratio.
 For two cascaded subsystems (the cable and the receiver) the overall noise factor FT is
given by
FT  F1 
F2 - 1
G1
In this case F1 = 4 (6dB), the cable loss L is 4 and F2 (the receiver noise factor) is also 4. G1
is the gain of the cable and equals 1/L = 1/4 = 0.25. Hence,
FT  4 
4 -1
 16 (12dB)
0.25
The new signal-to-noise ratio is therefore 53 - 12 = 41dB
A value of 41 dB is unacceptable for good television reception.
Case 3
Suppose next that the aerial is connected via the pre-amplifier and cable, as shown in Figure
4.l5.
Cable
Ta = 290 K
Available signal
power =2 pW
Mast-head
Pre-amplifier
G = 100 dB
F1 = 2
1
4
G
F2  4
Receiver
L
F3 = 4
SAQ 4.5 Calculate the final signal-to-noise ratio for this case.
24
You should have found that signal-to-noise ratio is now about 50dB, which is better even
than with the receiver connected directly to the aerial.
This striking result is a classic illustration of the advantage of ensuring high gain with low
noise at the 'front-end' of a cascaded system. Because of the cascading effect, the rather poor
noise performance of the receiver itself (F = 4) is rendered negligible because of the better
noise performance (F = 2) and sufficient gain (20dB) of the pre-amplifier.
Case 4
SAQ 4.6 The pre-amplifier is moved to the bottom of the cable, that is, immediately before
the receiver. Would you expect the final signal-to-noise ratio to be higher, lower, or the
same as in case 3? Verify your answer by calculating the new output signal-to-noise ratio.
Noise factor or noise temperature?
As a result of studying this section you should be able to carry out simple calculations of the
type illustrated in the preceding in-text questions and SAQs. However, you may still be
wondering when exactly to use noise temperature and when noise factor. As I have
presented the material here, noise factor is defined only when all parts of the system are at
the standard temperature of 290 K. In fact, it is possible to extend the concept to situations
when this is not the case, although this is outside the scope of HNC. When the source
temperature is other than 290 K - for example, the antenna in a satellite communications
system - it is generally easier to work with noise temperatures. When the complete system is
at 290 K, however, noise factor is often more convenient - particularly when lossy
subsystems are involved. (This is why I did not consider the matched attenuator until I had
introduced the concept of noise factor.)
A further consideration is the 'noisiness' of the devices under discussion. Very noisy
subsystems are easier to handle in terms of noise factor, since at high noise levels Te can
become awkwardly large whereas F, particularly expressed in decibels, remains more
manageable. Conversely, components such as low-noise amplifiers are normally
characterised in terms of Te rather than F. When handled correctly, both methods will lead
to identical estimates of signal-to-noise ratios at various parts of a system.
To check that you understand how to manipulate noise factors and temperatures, try the
following SAQ, which rounds off the section by returning to the communication satellite
ground station of Figure 4.1.
Summary
All active or passive devices introduce noise. The noise temperature, Te, of a device is the
temperature of a fictitious resistor with resistance equal to the source impedance which,
when connected to the input of a noise-free equivalent of the device, would result in output
noise equal to that generated by the real, noisy device.
If several matched subsystems are cascaded, then the overall noise temperature is given by
the expression
T
T
Te 4
Te  Te1  e 2  e3 
 ......
G1 G1G2 G1G2 G3
25
The noise factor, F, of a device is defined as the ratio of the total noise power density at the
output to that portion of the output noise density resulting from the input noise, measured at
290K. This can often be taken as equivalent to
input signal - to - noise ratio
output signal - to - noise ratio
Noise factor and noise temperature are related by the expressions
F  1
Te
To
Te  F - 1To  F - 1290
The overall noise factor of a number of cascaded matched subsystems is given by
F2 - 1 F3 - l

 .........
G1
G 1G 2
The noise factor of a matched lossy element of insertion loss L = 1/G is equal to L.
F1 
When several subsystems are cascaded, then the overall output signal-to-noise ratio is
maximised by providing sufficient gain in, and minimising the noise of, the first subsystem.
In practice, other considerations such as device non-linearities and costs will place a limit on
the improvements that can be achieved in this way.
26