NOISE IN COMMUNCATION SYSTEM
- Introduction
- Thermal Noise
- Shot Noise
- Low Frequency or Flicker Noise
- Excess Resister Noise
- Burst or Popcorn Noise
- General Comments
- Noise Evaluation – Overview
- Analysis of Noise in
Communication Systems
-
Thermal Noise
-
Noise
Voltage Spectral Density
-
Resistors
in Series
-
Resistors
in Parallel
- Matched Communication Systems
- Signal - to – Noise
- Noise Factor – Noise Figure
- Noise Figure / Factor for
Active Elements
- Noise Temperature
- Noise Figure / Factors for
Passive Elements
- Review – Noise Factor / Figure
/ Temperature
- Cascaded Networks
- System Noise Figure
- System Noise Temperature
- Review and Application
- Algebraic Representation of
Noise
- Additive White Gaussian Noise
1. Introduction
Noise is often described as the limiting
factor in communication systems: indeed if there as no noise there would be
virtually no problem in communications.
Noise is a general term which is used to
describe an unwanted signal which affects a wanted signal. These unwanted
signals arise from a variety of sources which may be considered in one of two
main categories:-
a)
Interference, usually from a
human source (man made)
b)
Naturally occurring random
noise.
Interference arises for example, from other
communication systems (cross talk), 50 Hz supplies (hum) and harmonics,
switched mode power supplies, thyristor circuits, ignition (car spark plugs)
motors … etc. Interference can in principle be reduced or completely eliminated
by careful engineering (i.e. good design, suppression, shielding etc).
Interference is essentially deterministic (i.e. random, predictable), however
observe.
When the interference is removed, there
remains naturally occurring noise which is essentially random
(non-deterministic),. Naturally occurring noise is inherently present in
electronic communication systems from either ‘external’ sources or ‘internal’
sources.
Naturally occurring external noise sources
include atmosphere disturbance (e.g. electric storms, lighting, ionospheric
effect etc), so called ‘Sky Noise’ or Cosmic noise which includes noise from galaxy, solar noise and ‘hot spot’ due
to oxygen and water vapour resonance in the earth’s atmosphere. These sources
can seriously affect all forms of radio transmission and the design of a radio
system (i.e. radio, TV, satellite) must take these into account.
The diagram below shows noise temperature
(equivalent to noise power, we shall discuss later) as a function of frequency
for sky noise.
The upper curve represents an antenna at low
elevation (~ 5o above horizon), the lower curve represents an
antenna pointing at the zenith (i.e. 90o elevation).
Contributions to the above diagram are from
galactic noise and atmospheric noise as shown below.
Note that sky noise is least over the band –
1 GHz to 10 GHz. This is referred to as a low noise ‘window’ or region and is
the main reason why satellite links operate at frequencies in this band (e.g. 4
GHz, 6GHz, 8GHz). Since signals received from satellites are so small it is
important to keep the background noise to a minimum.
Naturally occurring internal noise or
circuit noise is due to active and passive electronic devices (e.g. resistors, transistors ...etc) found in communication systems. There
are various mechanism which produce noise in devices; some of which will be
discussed in the following sections.
2. Thermal
Noise (Johnson Noise)
This type of noise is generated by all
resistances (e.g. a resistor, semiconductor, the resistance of a resonant
circuit, i.e. the real part of the impedance, cable etc).
Free electrons are in contact random motion
for any temperature above absolute zero (0 degree K, ~ -273 degree C). As the
temperature increases, the random motion increases, hence thermal noise, and
since moving electron constitute a current, although there is no net current
flow, the motion can be measured as a mean square noise value across the
resistance.
Experimental results (by Johnson) and
theoretical studies (by Nyquist) give the mean square noise voltage as 
Where
k = Boltzmann’s constant = 1.38 x 10-23 Joules per K
T
= absolute temperature
B = bandwidth noise measured in (Hz)
R
= resistance (ohms)
The law relating noise power, N, to the
temperature and bandwidth is
N
= k TB watts
These equations will be discussed further in
later section.
The equations above held for frequencies up
to > 1013 Hz (10,000 GHz) and for at least all practical
temperatures, i.e. for all practical communication systems they may be assumed
to be valid.
Thermal noise is often referred to as ‘white
noise’ because it has a uniform ‘spectral density’.
Note – noise power spectral density
is the noise power measured in a 1 Hz bandwidth i.e. watts per Hz. A uniform
spectral density means that if we measured the thermal noise in any 1 Hz
bandwidth from ~ 0Hz → 1 MHz → 1GHz …….. 10,000 GHz etc we would measure the
same amount of noise.
From the equation N=kTB, noise power
spectral density is 
watts per Hz.
watts per Hz.
I.e. Graphically
3. Shot
Noise
Shot noise was originally used to describe
noise due to random fluctuations in electron emission from cathodes in vacuum
tubes (called shot noise by analogy with lead shot). Shot noise also occurs in
semiconductors due to the liberation of charge carriers, which have discrete
amount of charge, in to potential barrier region such as occur in pn
junctions. The discrete amounts of charge give rise to a current which is
effectively a series of current pulses.
For pn junctions the mean square shot
noise current is
Where
is the direct current
as the pn junction (amps)
is the reverse
saturation current (amps)
is the electron
charge = 1.6 x 10-19 coulombs
B
is the effective noise bandwidth (Hz)
Shot noise is found to have a uniform
spectral density as for thermal noise.
4. Low
Frequency or Flicker Noise
Active devices, integrated circuit, diodes,
transistors etc also exhibits a low frequency noise, which is frequency
dependent (i.e. non uniform) known as flicker noise or ‘one – over – f’ noise.
The mean square value is found to be proportional to 
where f is the
frequency and n= 1. Thus the noise at higher frequencies is less than at lower
frequencies. Flicker noise is due to impurities in the material which in turn
cause charge carrier fluctuations.
where f is the
frequency and n= 1. Thus the noise at higher frequencies is less than at lower
frequencies. Flicker noise is due to impurities in the material which in turn
cause charge carrier fluctuations.
5. Excess Resistor Noise
Thermal noise in resistors does not vary
with frequency, as previously noted, by many resistors also generates as
additional frequency dependent noise referred to as excess noise. This noise
also exhibits a (1/f) characteristic, similar to flicker noise.
Carbon resistor generally generates most
excess noise whereas were wound resistors usually generates negligible amount
of excess noise. However the inductance of wire wound resistor limits their
frequency and metal film resistor are usually the best choices for high
frequency communication circuit where low noise and constant resistance are
required.
6. Burst Noise or Popcorn Noise
Some semiconductors also produce burst or
popcorn noise with a spectral density which is proportional to
.
.
7. General Comments
The diagram below illustrates the variation
of noise with frequency.
For frequencies below a few KHz (low
frequency systems), flicker and popcorn noise are the most significant, but
these may be ignored at higher frequencies where ‘white’ noise predominates.
Thermal noise is always presents in
electronic systems. Shot noise is more or less significant depending upon the
specific devices used for example as FET with an insulated gate avoids junction
shot noise. As noted in the preceding discussion, all transistors generate
other types of ‘non-white’ noise which may or may not be significant depending
on the specific device and application. Of all these types of noise source,
white noise is generally assumed to be the most significant and system analysis
is based on the assumption of thermal noise. This assumption is reasonably
valid for radio systems which operates at frequencies where non-white noise is greatly
reduced and which have low noise ‘front ends’ which, as shall be discussed,
contribute most of the internal (circuit) noise in a receiver system. At radio
frequencies the sky noise contribution is significant and is also (usually)
taken into account.
Obviously, analysis and calculations only
gives an indication of system performance. Measurements of the noise or
signal-to-noise ratio in a system include all the noise, from whatever source,
present at the time of measurement and within the constraints of the
measurements or system bandwidth.
Before
discussing some of these aspects further an overview of noise evaluation as
applicable to communication systems will first be presented.
8. Noise Evaluation
Overview
It has been stated that noise is an unwanted
signal that accompanies a wanted signal, and, as discussed, the most common
form is random (non-deterministic) thermal noise.
The essence of calculations and measurements
is to determine the signal power to Noise power ratio, i.e. the (S/N) ratio or
(S/N) expression in dB.
i.e.
Let S= signal power (mW)
N
= noise power (mW)
Powers are usually measured in dBm (or dBw)
in communications systems. The equation 
is often the most
useful.
is often the most
useful.
The 
at various stages in a
communication system gives an indication of system quality and performance in
terms of error rate in digital data communication systems and ‘fidelity’ in
case of analogue communication systems. (Obviously, the larger the , the better the system will be).
at various stages in a
communication system gives an indication of system quality and performance in
terms of error rate in digital data communication systems and ‘fidelity’ in
case of analogue communication systems. (Obviously, the larger the , the better the system will be).
Noise, which accompanies the signal is
usually considered to be additive (in terms of powers) and its often described
as Additive White Gaussian Noise, AWGN, noise. Noise and signals may also be
multiplicative and in some systems at some levels of , this may be more significant then AWGN.
, this may be more significant then AWGN.
In order to evaluate noise various
mathematical models and techniques have to be used, particularly concepts from
statistics and probability theory, the major starting point being that random
noise is assumed to have a Gaussian or Normal distribution.
We may relate the concept of white noise
with a Gaussian distribution as follows:
Gaussian distribution – ‘graph’ shows
Probability of noise voltage vs voltage – i.e. most probable noise voltage is 0
volts (zero mean). There is a small probability of very large +ve or –ve noise
voltages.
White noise – uniform noise power from ‘DC’
to very high frequencies.
Although not strictly consistence, we may
relate these two characteristics of thermal noise as follows:
The probability of amplitude of noise at any
frequency or in any band of frequencies (e.g. 1 Hz, 10Hz… 100 KHz .etc) is a
Gaussian distribution.
Noise may be quantified in terms of noise
power spectral density, p0 watts per Hz, from which Noise power N
may be expressed as
N=
p0 Bn watts
Where Bn is the equivalent noise
bandwidth, the equation assumes p0 is
constant across the band (i.e. White Noise).
Note - Bn is not the 3dB bandwidth,
it is the bandwidth which when multiplied by p0
Gives the actual output noise power N. This
is illustrated further below.
Ideal low pass filter
Bandwidth
B Hz = Bn
N= p0 Bn watts
Practical LPF
3
dB bandwidth shown, but noise does not suddenly cease at B3dB
Therefore, Bn > B3dB, Bn
depends on actual filter.
N=
p0 Bn
In general the equivalent noise bandwidth is
> B3dB.
Alternatively, noise may be quantified in
terms of ‘mean square noise’ i.e. 
, which is effectively a power. From this a ‘Root mean square
(RMS)’ value for the noise voltage may be determined.
, which is effectively a power. From this a ‘Root mean square
(RMS)’ value for the noise voltage may be determined.
i.e.
RMS = 
In order to ease analysis, models based on
the above quantities are used. For example, if we imagine noise in a very narrow
bandwidth, 
, as 
, the noise approaches a sine wave (with frequency ‘centred’
in df).
, as , the noise approaches a sine wave (with frequency ‘centred’
in df).
Since an RMS noise voltage can be
determined, a ‘peak’ value of the noise may be invented since for a sine wave
RMS
= 
Note – the peak value is entirely fictious
since in theory the noise with a Gaussian distribution could have a peak value
of +
or - volts.
or - volts.
Hence we may relate
Mean square 
RMS 
(RMS) Peak noise voltage
(invented for convenience)
RMS (RMS) Peak noise voltage
(invented for convenience)
Problems arising from noise are manifested
at the receiving end of a system and hence most of the analysis relates to the
receiver / demodulator with transmission path loss and external noise sources
(e.g. sky noise) if appropriate, taken into account.
The transmitter is generally assumed to
transmit a signal with zero noise (i.e (S/N) at the Tx ∞.
∞.
General communication system block diagrams
to illustrate these points are shown below.
Transmission Line
R = repeater (Analogue) or Regenerators
(digital)
These systems may facilitate analogue or
digital data transfer. The diagram below characterize these typical systems in
terms of the main parameters.
PT represents the output power at
the transmitter.
GT represents the Tx aerial gain.
Path loss represents the signal attenuation
due to inverse square law and absorption e.g. in atmosphere.
G represents repeater gains.
PR represents receiver inout
signal power
NR represents the received
external noise (e.g. sky noise)
GR represents the receiving
aerial gain.
represents the at the input to the
demodulator 
represents the quality of the output signal for analogue
systems
DATA ERROR RATE represents the quality of
the output (probability of error) for digital data systems
9. Analysis of Noise In Communication Systems
Thermal
Noise (Johnson noise)
It has been discussed that the thermal noise
in a resistance R has a mean square value given by
Where
k = Boltzmann’s constant = 1.38 x 10-23 Joules per K
T
= absolute temperature
B = bandwidth noise measured in (Hz)
R
= resistance (ohms)
This is found to hold for large bandwidth
(>1013 Hz) and large range in temperature.
This thermal noise may be represented by an
equivalent circuit as shown below.
i.e. equivalent to the ideal noise free
resistor (with same resistance R) in series with a voltage source with voltage
Vn.
Since
(mean square value ,
power)
then VRMS = = 
in above
= in above
i.e. Vn is the RMS noise voltage.
The above equation indicates that the noise
power is proportional to bandwidth.
For a given resistance R, at a fixed
temperature T (Kelvin)
We have 
, where 
is a constant – units
watts per Hz.
, where is a constant – units
watts per Hz.
For a given system, with constant, then if we
double the bandwidth from B Hz to 2B Hz, the noise power will double (i.e
increased by 3 dB). If the bandwidth were increased by a factor of 10, the
noise power is increased by a factor of 10.
constant, then if we
double the bandwidth from B Hz to 2B Hz, the noise power will double (i.e
increased by 3 dB). If the bandwidth were increased by a factor of 10, the
noise power is increased by a factor of 10.
For this reason it is important that the
system bandwidth is only just ‘wide’ enough to allow the signal to pass to
limit the noise bandwidth to a minimum.
I.e. Signal Spectrum
Signal Power = S
A) System BW = B Hz
N= Constant B (watts) = KB
B) System BW
N= Constant 2B (watts) = K2B
For A, 
, For
B, 
, For
B,
i.e. 
for B only ½ that for
A.
for B only ½ that for
A.
Noise
Voltage Spectral Density
Since date sheets specify the noise voltage
spectral density with unit’s volts per
(volts per root Hz).
(volts per root Hz).
This is from
i.e. Vn is
proportional to 
. The quantity in bracket, i.e. 
has units of volts per. If the bandwidth B is doubled the noise voltage will
increased by. If bandwidth is increased by 10, the noise voltage will
increased by
.
i.e. Vn is
proportional to . The quantity in bracket, i.e. has units of volts per. If the bandwidth B is doubled the noise voltage will
increased by. If bandwidth is increased by 10, the noise voltage will
increased by.
Resistance
in Series
Assume that R1 at temperature T1
and R2 at temperature T2, then
(we add
noise power not noise voltage)
Mean square noise
If T1= T2 = T then 
i.e. The
resistor in series at same temperature behave as a single resistor 
.
.
Resistance
in Parallel
Since an
ideal voltage source has zero impedance, we can find noise as an output Vo1,
due to Vn1
, an output voltage V02 due to Vn2.
Assuming 
at temperature 
and 
at temperature 
at temperature and at temperature and
Hence,
and
Thus = 
= 
Since usually equals (say T)
usually equals (say T)
or
i.e. the two noisy resistors in parallel
behave as a resistance 
which is the equivalent resistance of the parallel
combination.
which is the equivalent resistance of the parallel
combination.
10. Matched Communication Systems
In communication systems we are usually
concerned with the noise (i.e. S/N) at the receiver end of the system.
The transmission path may be for example:-
a)
A transmission line (e.g. coax
cable).
Zo is the
characteristics impedance of the transmission line, i.e. the source resistance
Rs. This is connected to the receiver with an input impedance RIN .
b)
A radio path (e.g. satellite,
TV, radio – using aerial)
c)
Again Zo=Rs
the source resistance, connected to the receiver with input
resistance
Rin.
Typically Zo is 600 ohm for
audio/ telephone systems
Or
Zo is 50 ohm (for radio/TV systems)
75 ohm (radio frequency systems).
An equivalent circuit, when the line is
connected to the receiver is shown below. (Note we omit the noise due to Rin –
this is considered in the analysis of the receiver section).
The RMS voltage output, Vo
(noise) is
Similarly, the signal voltage output due to
Vsignal at input is
Vs(signal)
= 
For maximum power transfer, the input 
is matched to the
source 
, i.e. = = R (say)
is matched to the
source , i.e. = = R (say)
Then 
And
signal, 
Continuing the analysis for noise only, the
mean square noise is (RMS)2.
But 
is noise due to Rs =R,
i.e. .
is noise due to Rs =R,
i.e. .
Hence
Since average power = 
Then 
i.e. Noise Power 
watts
watts
For a matched system, N represents the
average noise power transferred from the source to the load. This may be
written as
where
is the noise power
spectral density (watts per Hz)
is the noise power
spectral density (watts per Hz)
is the noise equivalent
bandwidth (Hz)
k is
the Boltzmann’s constant
T is
the absolute temperature K.
Note that is independent of
frequency, i.e. white noise.
is independent of
frequency, i.e. white noise.
These equations indicate the need to keep
the system bandwidth to a minimum, i.e. to that required to pass only the band
of wanted signals, in order to minimize noise power, N.
For example, a resistance at a temperature
of 290 K (17 deg C),
= kT is 4 x 10-21 watts per Hz. For a noise
bandwidth Bn = 1 KHz, N is 4 x 10-18 watts (-174 dBW).
If the system bandwidth is increased to 2
KHz, N will decrease by a factor of 2 (i.e. 8 x 10-18 watts or -171
dBW) which will degrade the (S/N) by 3 dB.
Care must also be exercised when noise or
(S/N ) measurements are made, for example with a power meter or spectrum
analyser, to be clear which bandwidth the noise is measured in, i.e. system or
test equipment.
For
example, assume a system bandwidth is 1 MHz and the measurement instrument
bandwidth is 250 KHz.
In the above example, the noise measured is
band limited by the test equipment rather than the system, making the system
appears less noisy than it actually is. Clearly if the relative bandwidths are
known (they should be) the measured noise power may be corrected to give the
actual noise power in the system bandwidth.
If the system bandwidth was 250 KHz and the
test equipment was 1 MHz then the measured result now would be – 150 dBW (i.e.
the same as the actual noise) because the noise power monitored by the test
equipment has already been band limited to 250 KHz.
11. Signal – To – Noise
The signal to noise ratio is given by
The signal to noise in dB is expressed by
or 
since
= S dBm if S in mW
= S dBm if S in mW
and
= N dBm
= N dBm
then 
for S and
N measured in mW.
for S and
N measured in mW.
12. Noise Factor – Noise Figure
Consider the network shown below, in which represents the at the input and
represents the at the output.
represents the at the input and represents the at the output.
In general ≥ , i.e. the network ‘adds’ noise (thermal noise tc from the
network devices) so that the output (S/N) is generally worst than the input.
≥ , i.e. the network ‘adds’ noise (thermal noise tc from the
network devices) so that the output (S/N) is generally worst than the input.
The amount of noise added by the network is
embodied in the Noise Factor F, which is defined by
Noise
factor F = 
F equals to 1 for noiseless network and in
general F > 1.
The noise figure in the noise factor quoted
in dB
i.e. Noise Figure F dB = 10 log10 F F ≥ 0 dB
The noise figure / factor is the measure of
how much a network degrades the (S/N)IN, the lower the value of F,
the better the network.
The network may be active elements, e.g.
amplifiers, active mixers etc, i.e. elements with gain > 1 or passive
elements, e.g. passive mixers, feeders cables, attenuators i.e. elements with
gain <1.
13. Noise Figure – Noise Factor for Active
Elements
For active elements with power gain G>1,
we have
F
= = 
=
But 
= G 
= G
Therefore F
= 
F
= 
If the 
was due only to G
times 
the F would be 1 i.e.
the active element would be noise free. Since in general F v> 1 , then is increased by noise
due to the active element i.e.
was due only to G
times the F would be 1 i.e.
the active element would be noise free. Since in general F v> 1 , then is increased by noise
due to the active element i.e.
Na represents ‘added’ noise measured at the
output. This added noise may be referred to the input as extra noise, i.e. as
equivalent diagram is
Ne is extra noise due to active elements
referred to the input; the element is thus effectively noiseless.
Hence
F = = F = 
= F =
Rearranging gives,
14. Noise Temperature
is the ‘external’
noise from the source i.e. = k
is the equivalent noise temperature of the source (usually
290K).
We may also write 
= k
, where is the equivalent noise temperature of the element i.e. with
noise factor F and with source temperature.
= k, where is the equivalent noise temperature of the element i.e. with
noise factor F and with source temperature.
i.e.
k= (F-1) k
= (F-1) k
or = (F-1)
= (F-1)
The noise factor F is usually measured under
matched conditions with noise source at ambient temperature, i.e. ~ 290K is usually
assumed, this is sometimes written as ,
, i.e. ~ 290K is usually
assumed, this is sometimes written as ,
This allows us to calculate the equivalent
noise temperature of an element with noise factor F, measured at 290 K.
For example, if we have an amplifier with
noise figure FdB = 6 dB
(Noise factor F=4) and equivalent Noise temperature = 865 K.
= 865 K.
Comments:-
a)
We have introduced the idea of
referring the noise to the input of an element, this noise is not actually present
at the input, it is done for convenience in the analysis.
b)
The noise power and equivalent noise
temperature are related, N=kTB, the temperature T is not necessarily the
physical temperature, it is equivalent to the temperature of a resistance R
(the system impedance) which gives the same noise power N when measured in the
same bandwidth Bn.
c)
Noise figure (or noise factor
F) and equivalent noise temperature are related and both
indicate how much noise an element is producing.
are related and both
indicate how much noise an element is producing.
Since, = (F-1)
= (F-1)
Then for F=1, = 0, i.e. ideal noise free active element.
= 0, i.e. ideal noise free active element.
15. Noise Figure – Noise Factor for Passive
Elements
The theoretical argument for passive
networks (e.g. feeders, passive mixers, attenuators) that is networks with a
gain < 1 is fairly abstract, and in essence shows that the noise at the
input, is attenuated by
network, but the added noise Na contributes to the noise at the output such that
=.
is attenuated by
network, but the added noise Na contributes to the noise at the output such that
=.
Thus, since F = and =.
and =.
F = 
If we let L denote the insertion loss
(ratio) of the network i.e. insertion loss
LdB
= 10 log L
Then
L
= 
and hence for passive
network
and hence for passive
network
F
= L
Also, since = (F-1)
= (F-1)
Then for passive network
= (L-1)
Where is the equivalent
noise temperature of a passive device referred to its input.
is the equivalent
noise temperature of a passive device referred to its input.
16. Review of Noise Factor – Noise Figure
–Temperature
F, dB and are related by FdB
= 10 logdB F
are related by FdB
= 10 logdB F = (F-1)290
Some corresponding values are tabulated
below:
|
F
|
FdB
(dB)
|
(degree K) |
|
1
|
0
|
0
|
|
2
|
3
|
290
|
|
4
|
6
|
870
|
|
8
|
9
|
2030
|
|
16
|
12
|
4350
|
Typical values of noise temperature, noise
figure and gain for various amplifiers and attenuators are given below:
|
Device
|
Frequency
|
(K) |
FdB
(dB)
|
Gain
(dB)
|
|
Maser
Amplifier
|
9 GHz
|
4
|
0.06
|
20
|
|
Ga
As Fet amp
|
9 GHz
|
330
|
303
|
6
|
|
Ga
As Fet amp
|
1 GHz
|
110
|
1.4
|
12
|
|
Silicon
Transistor
|
400 MHz
|
420
|
3.9
|
13
|
|
L C
Amp
|
10 MHz
|
1160
|
7.0
|
50
|
|
Type
N cable
|
1 GHz
|
|
2.0
|
2.0
|
17. Cascaded Network
A receiver systems usually consists of a
number of passive or active elements connected in series, each element is
defined separately in terms of the gain (greater than 1 or less than 1 as the
case may be), noise figure or noise temperature and bandwidth (usually the 3 dB
bandwidth). These elements are assumed to be matched.
A typical receiver block diagram is shown
below, with example
In order to determine the (S/N) at the
input, the overall receiver noise figure or noise temperature must be
determined. In order to do this all the noise must be referred to the same
point in the receiver, for example to A, the feeder input or B, the input to
the first amplifier.
The equations so far discussed refer the noise
to the input of that specific element i.e.
or is the noise referred to the input.
To refer the noise to the output we must
multiply the input noise by the gain G.
For example, for a lossy feeder, loss L, we
had
= (L-1) , noise referred to
input
Or = (L-1) - referred to the
input.
= (L-1) - referred to the
input.
Noise referred to output is gain x noise
referred to input, hence
referred to output = G
= 
(L-1)
= (1-)
)
Similarly, the equivalent noise temperature
referred to the output is
referred to output =
(1-)
These points will be clarified later; first
the system noise figure will be considered.
18. System Noise Figure
Assume that a system comprises the elements
shown below, each element defined and specified separately.
The gains may be greater or less than 1,
symbols F denote noise factor (not noise figure, i.e. not in dB).
Assume that these are now cascaded and
connected to an aerial at the input, with
from the aerial.
Note: - for each stage is
equivalent to a source at a temperature of 290 K since this is how each element
is specified. That is, for each device/
element is specified at 290 K.
for each stage is
equivalent to a source at a temperature of 290 K since this is how each element
is specified. That is, for each device/
element is specified at 290 K.
Now
, 
Since 
and similarly 
then
The overall system Noise Factor is
If we assume 
is ≈ , i.e. we would measure and specify 
under similar conditions as 
etc (i.e. at 290 K),
then for n elements in cascade.
is ≈ , i.e. we would measure and specify under similar conditions as etc (i.e. at 290 K),
then for n elements in cascade.
The equation is called FRIIS Formula.
This equation indicates that the system
noise factor depends largely on the noise factor of the first stage if the gain
of the first stage is reasonably large. This explains the desire for “low noise
front ends” or low noise most head preamplifiers for domestic TV reception.
There is a danger however; if the gain of the first stage is too large, large
and unwanted signals are applied to the mixer which may produce intermodulation
distortion. Some receivers apply signals from the aerial directly to the mixer
to avoid this problem. Generally a first stage amplifier is designed to have a
good noise factor and some gain to give an acceptable overall noise figure.
19. System Noise Temperature
Since = (L-1), i.e. F= 1 + 
= (L-1), i.e. F= 1 +
Then 
and
is the receiver system
equivalent noise temperature. Again, this shows that the system noise temperature
depends on the first stage to a large extent if the gain of the first stage is
reasonably large.
The equations for and 
refer the noise to the input of the first stage. This can
best be classified by examining the equation for in conjunction with the diagram below.
and refer the noise to the input of the first stage. This can
best be classified by examining the equation for in conjunction with the diagram below.
is already referred
to input of 1st stage.
is referred to input
of the 2nd stage – to refer this to the input of the 1st
stage we must divide by G1.
is referred to input of third stage, (
G1G2) to refer to input of 1st
stage, etc.
It is often more convenient to work with
noise temperature rather than noise factor.
Given a noise factor we find 
from = (F-1)290.
from = (F-1)290.
Note also that the gains (G1G2
G3 etc) may be gains > 1 or gains <1, i.e. losses L where L = .
.
See examples and tutorials for further
classifications.
20. Review And Application
It is important to realize that the previous
sections present a technique to enable a receiver performance to be calculated.
The essence of the approach is to refer all the noise contributed at various
stages in the receiver to the input and thus contrive to make all the stages
ideal, noise free.
i.e. in practice or reality : -
The noise gets worse as we proceed from the
aerial to the output. The technique outlined.
All noise referred to input and all stages
assumed noise free.
To complete the analysis consider the system
below
The overall noise temp = 
Noise
referred to A = kTB
= k (
)B watts
)B watts
Where
, k = 1.38 x 10-23 J/K
B
is the bandwidth as measured at the output, in this case from
the
IF amp.
If we let this noise by NR, i.e.
NR = k ()B then the noise at the output would be
)B then the noise at the output would be 
i.e.
the noise referred to input times the gain of each stage.
Now consider
is not the actual (real), 
ratio that would be measured at A because NR is
all referred back from later stages.
Signal power at the output would be
Hence, by receiving all the noise to the
input, and finding NR, we can find 
which is the same as 
- i.e. we do not need to know all the system gain.
which is the same as - i.e. we do not need to know all the system gain.
Consider now the diagram below
A filter in same form often follows the
receiver and we often need to know 
, the noise power spectral density.
, the noise power spectral density.
i.e. recall that N= B = kTB.
B = kTB.= kT
We may find from
from = k()
21. Algebraic Representation of Noise
General
In order for the effects of noise to be
considered in systems, for example the analysis of probability of error as a
function of Signal to noise for an FSK modulated data systems, or the
performance of analogue FM system in the presence of noise, it is necessary to
develop a model which will allow noise to be considered algebraically.
Noise may be quantified in terms of noise
power spectral density watts per Hz such that
the average noise power in a noise bandwidth Bn Hz is
watts per Hz such that
the average noise power in a noise bandwidth Bn Hz is
N=
Bn watts
Bn watts
Thus the actual noise power in a system
depends on the system bandwidth and since we are often interested in the at the input to a demodulator, Bn is the smallest
noise equivalent bandwidth before the demodulator.
at the input to a demodulator, Bn is the smallest
noise equivalent bandwidth before the demodulator.
Since average power is proportional to 
we may relate N to a
“peak” noise voltage so that
we may relate N to a
“peak” noise voltage so that
N
= 
= Bn
= Bn
i.e. 
’peak’ value of noise
’peak’ value of noise
In practice noise is a random signal with
(in theory) a Gaussian distribution and hence peak values up to 
or as otherwise limited by the system dynamic range are
possible. Hence this “peak” value for noise is a fictitious value which will
give rise to the same average noise as the actual noise.
or as otherwise limited by the system dynamic range are
possible. Hence this “peak” value for noise is a fictitious value which will
give rise to the same average noise as the actual noise.
Phasor
Representation of Signal and Noise
The general carrier signal VcCosWct may be
represented as a phasor at any instant in time as shown below:
The phasor represents a signal with peak
value Vc, rotating with angular frequencies Wc rads per sec and with an
angle 
to some reference axis at time t=0.
to some reference axis at time t=0.
If we now consider a carrier with a noise
voltage with “peak” value superimposed we may represents this as:
In this case Vn is the peak value of the
noise and is the phase of the noise relative to the carrier. Both Vn and 
n are random variables, the above phasor diagram represents a
snapshot at some instant in time.
n are random variables, the above phasor diagram represents a
snapshot at some instant in time.
The resultant or received signal R, is the
sum of carrier plus noise. If we consider several snapshots overlaid as shown
below we can see the effects of noise accompanying the signal and how this
affects the received signal R.
Thus the received signal has amplitude and
frequency changes (which in practice occur randomly) due to noise.
We may draw, for a single instant, the
phasor with noise resolved into 2 components, which are:
a)
x(t) in phase with the carriers
b)
y(t) in quadrature with the carrier
The reason why this is done is that x(t)
represents amplitude changes in Vc (amplitude changes affect the performance of
AM systems) and y(t) represents phase (i.e. frequency) changes (phase /
frequency changes affect the performance of FM/PM systems)
We note that the resultant from x(t) and
y(t) i.e.
We can regard x(t) as a phasor which
is in phase with 
, i.e a phasor rotating at 
.
, i.e a phasor rotating at .
i.e.
x(t)Cos t
t
and by similar reasoning, y(t) in
quadrature
i.e.
y(t)Sint
t
Hence we may write
Or
– alternative approach
This equation is algebraic representation of
noise and since
= 
the peak value of x(t) is 
(i.e. when 
)
(i.e. when )
Similarly the peak value of y(t) is also (i.e. when 
)
(i.e. when )
The mean square value in general is 
and thus the mean square of x(t), i.e
also the mean square value of y(t),
i.e 
The total noise in the bandwidth, Bn is
N
= 
i.e.
NOT 
as might be expected.
as might be expected.
The reason for this is due to the 
and 
relationship in the representation e.g. when say “x(t)”
contributes 
, the “y(t)” contribution is zero, i.e. sum is always
equal to .
and relationship in the representation e.g. when say “x(t)”
contributes , the “y(t)” contribution is zero, i.e. sum is always
equal to .
The algebraic representation of noise
discussed above is quite adequate for the analysis of many systems,
particularly the performance of ASK, FSK and PSK modulated systems.
When considering AM and FM systems, assuming
a large (S/N) ratio, i.e Vc>> Vn, the following may be used.
Considering the general phasor representation
below:-
For AM systems, the signal is of the form 
where m(t) is the message or modulating signal and the
resultant for (S/N) >> 1 is
where m(t) is the message or modulating signal and the
resultant for (S/N) >> 1 is
AM
received = 
Since AM is sensitive to amplitude changes,
changes in the Resultant length are predominantly due to x(t).
For FM systems the signal is of the form 
. Noise will produce both amplitude changes (i.e. in Vc) and
frequency variations – the amplitude variations are removed by a limiter in the
FM receiver. Hence,
. Noise will produce both amplitude changes (i.e. in Vc) and
frequency variations – the amplitude variations are removed by a limiter in the
FM receiver. Hence,
The angle represents frequency /
phase variations in the received signal due to noise. From the diagram
represents frequency /
phase variations in the received signal due to noise. From the diagram 
Since 
>>
(assumed) then 
<< 1
>>(assumed) then << 1
So
{which is
also obvious from diagram}
{which is
also obvious from diagram}
Since tan = for small and is small since >>
= for small and is small since >>
Then
The above discussion for AM and FM serve to
show bow the ‘model’ may be used to describe the effects of noise.
Applications of this model to ASK, FSK and
PSK demodulation, and AM and FM demodulation are discussed elsewhere.
22.
Additive White Gaussian Noise
Noise in Communication Systems is often
assumed to be Additive White Gaussian Noise (AWGN).
Additive
Noise is usually additive in that it adds to
the information bearing signal. A model of the received signal with additive
noise is shown below.
The signal (information bearing) is at its
weakest (most vulnerable) at the receiver input. Noise at the other points
(e.g. Receiver) can also be referred to the input.
The noise is uncorrelated with the signal,
i.e. independent of the signal and we may state, for average powers
Output
Power = Signal Power + Noise Power
= (S+N)
White
As we have stated noise is assumed to have a
uniform noise power spectral density, given that the noise is not band limited
by some filter bandwidth.
We have denoted noise power spectral density
by 
.
.
White
noise = = Constant
= Constant
Also
Noise power =
Gaussian
We generally assume that noise voltage
amplitudes have a Gaussian or Normal distribution.
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