chapter 4 noise
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DEFINATION OF RANDOM VARIABLES
A real random is mapping from the sample space (orS) to theset of real numbers.
A schematic diagram representing a random variable is given
below
1 2
34
R)( 1X )( 2X )( 3X )( 4X
Figure 4.1 : Random variables as a mapping from to R
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A random variable, usually writtenX, is a variable whosepossible values are numerical outcomes of a random
phenomenon, etc.; individuals values of the random variable X
areX().
There are two types of random variables, which is Discrete
Random Variablesand Continuous Random Var iables.
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Discrete Random Variables
A sample space is discrete if the number of its elements arefinite orcountable infinite, i.e., a discrete random variableis
one which may take on only a countable number of distinct
values such as 0,1,2,3,4,........
Examples of discrete random variables include the number of
children in a family, the Friday night attendance at a cinema,
the number of patients in a doctor's surgery, the number of
defective light bulbs in a box of ten.
A non-discrete sample space happens when the sample space
of the random experiment is infinite and uncountable.Example of non-discrete sample space is randomly chosen
number from 0 to 1 (continuous random variables).
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Continuous Random Variables
A continuous random variableis one which takes an infinitenumber of possible values. Continuous random variables areusually measurements.
Examples include height, weight, the amount of sugar in anorange, the time required to run a mile.
A continuous random variable is not defined at specific values.Instead, it is defined over an intervalof values, and isrepresented by the area under a curve(in advancedmathematics, this is known as an integral).
The probability of observing any single value is equal to 0,since the number of values which may be assumed by therandom variable is infinite.
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Figure 4.2 : Random variables (a) continuous (b) discrete.
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Example 4.1Which of the following random variables are discrete and which are
continuous?
a) X = Number of houses sold by real estate developer per week?b) X = Number of heads in ten tosses of a coin?
c) X = Weight of a child at birth?
d) X = Time required to run 100 yards?
Answer:
(a) Discrete (b) Discrete (c) Continuous (d) Continuous
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SIGNALS: DETERMINISTIC VS. STOCHASTIC
DETERMINISTIC SIGNALS Most introductions to signals and systems deal strictly with
deterministic signals as shown in Figure 4.3. Each value of
these signals are fixed and can be determined by a
mathematical expression, rule, or table.
Because of this, future values of any deterministic signal can
be calculated from past values. For this reason, these signals
are relatively easy to analyze as they do not change, and we
can make accurate assumptions about their past and future
behavior.
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RANDOM SIGNALS Random signals cannot be characterized by a simple, well-
defined mathematical equation and their future values cannot
be predicted.
Rather, we must useprobability and statistics to analyze theirbehavior.
Also, because of their randomness as shown in Figure 4.4,
average values from a collection of signals are usually studied
rather than analyzing one individual signal.
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Deterministic Signal
Random Signal
Figure 4.3: An example of a deterministic signal, the sine wave.
Figure 4.4: We have taken the above sine wave and added random noise to it to come up with a
noisy, or random, signal. These are the types of signals that we wish to learn how to deal with so
that we can recover the original sine wave.
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RANDOM PROCESSES
As mentioned before, in order to study random signals, wewant to look at a collection of these signals rather than just
one instance of that signal. This collection of signals is
called a random process.
Is an extension of random variables Also known as Stochastic Process
ModelRandom Signaland Random Noise
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Outcome of a random experiment is a function
An indexed set of random variables
Typically the index is time in communications
The difference between random variable and random process
is that for a random variable, an outcome is the sample spacemapped into a number, whereas for a random process it is
mapped into a function of time.
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Figure 4.5: Example of random process represent the temperature of a city at 20
hours.
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POWER SPECTRAL DENSITY
Random process is a collection of signals, and the spectralcharacteristics of these signals determine the spectralcharacteristic of the random process. Slow varying signals (of a random process) have power concentrated at
low frequencies.
Fast changing signals (of a random process) have power concentratedat high frequencies.
Power spectral density determines the power distribution (orpower spectrum) of the random process.
PSD of a random processX(t) is denoted by SX(f), denotes the
strength of power in the random process as a function offrequency.
Units for PSD is Watts/Hz.
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RELATIONSHIP OF RANDOM PROCESS
AND NOISE
Unwanted electric signals come from variety of sources,
generally classified as human interference or naturally
occurring noise.
Human interference comes from other communication systems
and the effects of many unwanted signals can be reduced or
eliminated completely.
Howeverthere always remain inescapable random signals, that
present a fundamental limit to systems performance.
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THERMAL NOISE
Thermal noise is the noise
resulting from the random motion
of electrons in a conducting
medium.
Thermal noise arises from both the
photodetector and the load resistor.
Amplifier noise also contributes to
thermal noise.
A reduction in thermal noise is
possible by increasing the value of
the load resistor.
However, increasing the value of
the load resistor to reduce thermal
noise reduces the receiver
bandwidth.
Figure 4.6 Fluctuating voltageproduced by random movements of
mobile electrons.
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GAUSSIAN PROCESS
Gaussian process is important in
communication systems. The main reason is that thermal
noise in electrical devices producedby movement of electrons due tothermal agitation is closely modeled
by a Gaussian process.
Due to the movements of electrons,sum of small currents of a very largenumber of sources was introduced.
Since majority sources areindependent, hence the total currentis sum of large number of random
variables. Therefore the total currents has
Gaussian distribution.Figure 4.7 Histogram of some noise voltage
measurements
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Definition
A random processX(t) is a Gaussion process if forall n and all
(t1,t2,,tn)the random variable {X(ti)}ni=1have a jointly Gaussian
density function.
Gaussian or Normal Random Variables
where m = mean = standard deviation
2 = variance
A Gaussian random variable with mean m and variance 2 is denoted
by N(m, 2)
AssumingXis a standard normal random variable, we defined the functionQ(x) asP(X > x). The Q function is given by relation
2
2
( )
21( )2
x m
Xf x e
2
21
( ) ( )2
t
xQ x P X x e dt
(4.1)
(4.2)
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The Q function represent the area under the tail of a standard random
variable.
It is well tabulated and used in analyzing the performance of
communication system.
Q(x) satisfy the following relations:
Q(-x) = 1Q(x)Q(0) =
Q() = 0
Table 3.1 gives the value of this function for various value ofx.
ForN(m, 2) random variable, a simple change of variable in the integral
that computesP(X > x) results inP(X > x) = Q[(xm)/].
tailprobability in Gaussian random variable.
(4.3a)(4.3b)
(4.3c)
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Figure 4.8: The Q-function as the area under the tail of a standard normal random variable.
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Table 4.1 Table of the Q function
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Example 4.2
Xis a Gaussian random variable with mean 1 and variance 4. Find the
probabilityXbetween 5 and 7.
Ans.
We have m = 1 and = 4 = 2. Thus,P( 5 5)P(X > 7)
= Q ((51)/2)Q((71)/2)
= Q(2)Q(3)
0.0214
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WHITE NOISE
There are many ways to characterize different noise sources, one is to
considerthe spectral density, that is, the mean square fluctuation at any
particular frequency and how that varies with frequency.
In what follows, noise will be generated that has spectral densities that vary
as powers of inverse frequency, more precisely, the power spectraP(f) is
proportional to 1 /ffor 0.
When = 0 the noise is referred to white noise, when = 2, it is referred
to as Brownian noise, and when it is 1 it normally referred to simply as 1/f
noise which occurs very often in processes found in nature.
White process is a process in which all frequency component appear with
equal power, i.e. power spectral density is constant for all frequencies.
A processX(t) is called a white process if it has a flat spectral
density,i.e., ifSX(f) is constant for allf.
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White Noise, = 0
1 3
0 2
Brownian noisewhite noise
1/f noise
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Spectral density of white
noise is a constant,N0/2
Autocorrelation function:
White Noise
0( )2
X
NS f
1 0
20
0
( )2
2
( )2
XX
j ft
NR F
N
e df
N
WhereN0
= kT
k= Boltzmanns constant = 1.38 x 10-23Figure 4.9: White noise (a) power spectrum
(b) autocorrelation
(3.4)
(3.5)
f
SX(f)
White noise- power spectrum
0
White noise- autocorrelation
)(RXX
0
N02
N0
2
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Properties of Thermal Noise
Thermal noise is a stationary process
Thermal noise is a zero-mean process
Thermal noise is a Gaussian process
Thermal noise is a white noise with power spectral density
SX(f)=kT/2=Sn(f)=N0/2.
It is clear thatpower spectral density of thermal noise increase
with increasingthe ambient temperature, therefore, keeping
electric circuit cool makes their noise level low.
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TYPE OF NOISE
Noise can be divided into :
2 general categories
Correlated noiseimplies relationship between the signal and the noise,exist only when signal is present
Uncorrelated noisepresent at all time, whether there is signal or not.Under this category there are two broad categories which are:-
i) Internal noise
ii) External noise
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UNCORRELATED NOISE
Can be divided into 2 categories
1. External noise
Generated outside the device or circuit
Three primary sources are atmospheric, extraterrestrial and man made
(a) Atmospheric Noise Naturally occurring electrical disturbance originate within Earths
atmosphere
Commonly called static electricity
Most static electricity is naturally occurring electrical conditions,
such as lighting In the form of impulse, spread energy through wide range of
frequency
Insignificant at frequency above 30 MHz
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(b) Extraterrestrial Noise
Consists of electrical signals that originate from outside earthatmosphere, deep-space noise
Divide further into two(i) Solar noisegenerated directly from suns heat. There are 2
parts to solar noise:- Quite condition when constant radiation intensity exist and
high intensity Sporadic disturbance caused bysun spotactivities andsolar
flare-ups which occur every 11 years
(ii) Cosmic noisecontinuously distributed throughout thegalaxies, small noise intensity because the sources of galacticnoise are located much further away from sun. It's also oftencalled asblack-body noise.
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(c) Man-made noise
Sourcespark-producing mechanism such as from commutators inelectric motors, automobile ignition etc
Impulsive in nature, contains wide range of frequency thatpropagate through space the same manner as radio waves
Most intense in populated metropolitan and industrial areas and is
therefore sometimes called industrial noise.
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(d) Impulse noise
High amplitude peaks of short duration in the total noise spectrum. Consists of sudden burst of irregularly shaped pulses.
More devastating on digital data,
Produce from electromechanical switches, electric motor etc.
(e) Interference External noise
Signal from one source interfere with another signal.
It occurs when harmonics or cross product frequencies from one
source fall into the passband of the neighboring channel.
Usually occurs in radio-frequency spectrum
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2. Internal noise
Generated within a device or circuit.
3 primary kinds, shot noise, transit-time noise and thermal noise
(a) Shot noise
Caused by random arrival of carriers (hole and electron) at the
output element of an electronic device such as diode, field effecttransistor or bipolar transistor.
The currents carriers (ac and dc) are not moving in a continuous,
steady flow, as the distance they travel varies because of their
random paths of motion.
Shot noise randomly varying and is superimposed onto any signal
present.
When amplified, shot noise sounds similar to metal pellets falling
on a tin roof.
Sometimes called transistor noise
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(b) Transit-time noise (Ttn)
Any modification to a stream of carriers as they pass from the input
to the output of a device produce irregular, random variation
(emitter to the collector in transistor).
Time it takes for a carrier to propagate through a device is an
appreciable part of the time of one cycle of the signal , the noise
become noticeable.
Ttn is transistors is determined by carrier mobility, bias voltage, and
transistor construction.
Carriers traveling from emitter to collector suffer from emitter
delay, base Ttn
,and collector recombination-time and propagation
time delays.
If transmit delays are excessive at high frequencies, the device may
add more noise than amplification of the signal.
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(c) Thermal noise
Due to rapid and random movement of electrons within a conductordue to thermal agitation
Present in all electronic components and communication system.
Uniformly distributed across the entire electromagnetic frequency
spectrum, often referred as white noise.
Form of additive noise, meaning that it cannot be eliminated , and itincreases in intensity with the number of devices and circuit length.
Set as upper bound on the performance of communication system.
Temperature dependent, random and continuous and occurs at all
frequencies.
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Noise Spectral Density
In communications, noise spectral densityNo is the noise
power per unit of bandwidth; that is, it is the power spectraldensity of the noise.
It has units ofwatts/hertz, which is equivalent to watt-seconds
or joules.
If the noise is white, i.e., constant with frequency, then thetotal noise powerNin a bandwidth B is BNo.
This is utilized in Signal-to-noise ratio calculations.
The thermal noise density is given by No= kT, where kis
Boltzmann's constant in joules per kelvin, and Tis the receiversystem noise temperature in kelvin.
No is commonly used in link budgets as the denominator of the
important figure-of-merit ratiosEb/No andEs/No.
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NOISE POWER
Noise power is given as
and can be written as
PN= kTB [W]where
PN= noise power,
k= Boltzmanns constant (1.38x10-23 J/K)B = bandwidth,
T= absolute temperature (Kelvin)(17o
C or 290K)
0
0
2
B
NB
NP df
N B
(3.6)
(3.7)
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NOISE VOLTAGE
Figure 4.10 shows the equivalent
circuit for a thermal noise source.
Internal resistanceRIin series
with the rms noise voltage VN.
For the worst condition, the load
resistanceR = RI , noise voltagedropped acrossR = half the noise
source (VR=VN/2) and
From equation 4.5 the noise
powerPN, developed across the
load resistor= kTB
VN/2
VN/2VN R
RI
Noise Source
The mathematical expression :
2 2
2
/ 2
4
4
4
N N
N
N
N
V V
P kTB R R
V RkTB
V RkTB
Figure 4.10 : Noise source equivalentcircuit
(4.8a)
(4.8b)
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OTHER NOISE SOURCES
There are 3 other noise mechanisms that contribute to internally generated
noise in electronic devices.1. Generation-Recombination Noise - The result of free carriers being
generated and recombining in semiconductor material. Can consider thesegeneration and recombination events to be random. This noise process can
be treated as shot noise process.
2. Temperature-Fluctuation NoiseThe result of the fluctuating heatexchange between a small body, such as transistor, and its environmentdue to the fluctuations in the radiation and heat-conduction processes. If aliquid or gas is flowing past the small body, fluctuation in heat convectionalso occurs.
3. Flicker NoiseIt is characterized by a spectral density that increases with
decreasing frequency. The dependence on spectral density on frequency isoften found to be proportional to the inverse first power of the frequency.Sometimes referred as one-over-f noise.
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Example 4.3
Calculate the thermal noise power available from any resistor at room
temperature (290 K) for a bandwidth of 1 MHz. Calculate also the
corresponding noise voltage, given that R = 50 .
Ansa) Thermal noise power b) Noise voltage
W
kTBN
15
623
104
1012901038.1
V
RkTBVN
895.0
104504
4
15
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Example 4.4
For an electronic device operating at a temperature of 17 oC
with a bandwidth of 10 kHz, determine
a) Thermal noise power in watts and dBm
b) rms noise voltage for a 100 internal resistance and 100 load resistance.
Ans.
a) b)W
N
17
323
10002.4
10102901038.1
dBm
NdBm
134
101
104log10
3
17
)(127.0
1041004
4
17
rmsV
RkTBVN
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Example 4.5
Two resistor of 20 k and 50 k are at room temperature (290
K). For a bandwidth of 100 kHz, calculate the thermal noise
voltage generated by
1. each resistor
2. the two resistor in series
3. the two resistor in parallel
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Ans.
a)
b) RT=
c) RT=
V
kTBRVN
6
3233
11
1066.5
101002901038.110204
4
V
kTBRVN
6
3233
22
1095.8
101002901038.110504
4
333 107010501020
V
kTBRV TNtotal
5
3233
1006.1
101002901038.110704
4
k28.14
10105020
10)5020(33
3
V
k
kTBRV TNtotal
78.4101002901038.129.144
4
323
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CORRELATED NOISE
Mutually related to the signal, not present if there is no signal
Produced by nonlinear amplification, and include nonlineardistortion such as harmonic and intermodulation distortion
1. Harmonic Distortion (HD)
Harmonic distortionunwanted harmonics of a signal produced
through nonlinear amplification (nonlinear mixing). Harmonics areinteger multiples of the original signal.
There are various degrees of harmonic distortion.
2nd order HT, ratio of the rms amplitude of the second harmonic to the
rms amplitude of the fundamental.
3rd oder HT, ratio of the rms amplitude of the third harmonic to the rmsamplitude of the fundamental.
Total harmonic distortion (THD), ratio of the quadratic sum of the rms
values of all the higher harmonics to the rms value of the fundamental.
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Figure 4.11(a) show the input and
output frequency spectrums for a
nonlinear device with a single input
frequencyf1.
Mathematically, THD is
Where,
%THD = percent total harmonic
distortion
vhigher= quadratic sum of the rmsvoltages,
vfundamental = rms voltage of the
fundamental frequency
V1 V1
V2
V3
V4Frequency
f1
f1
2f1
3f1
4f1
Input signal
Harmonicdistortion
Input frequency spectrum Output frequency spectrum
(a)
Frequency
V1
V2
f1 f2
V1 V2
f1 f2
VsumVdifference
Input signals
f1-f
2f1+f
2
Intermodulation
distortion
Input frequency spectrum Output frequency spectrum
(b)
Figure 4.11: Correlated noise:
(a) Harmonic distortion
(b) Intermodulation distortion
100THD%lfundamenta
higher
xv
v
223
22 nvvv
(4.9)
(4.10)
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2. Intermodulatin Distortion (ID)
Intermodulation distortion is the generation of unwanted sum and
difference frequency when two or more signal are amplified in a
nonlinear device such as large signal amplifier.
The sum and difference frequencies are called cross products.
Figure 4.11(b) show the input and output frequency spectrums for anonlinear device with two input frequencies (f1 andf2).
Mathematically, the sum and difference frequencies are
Cross products =mf1nf2
Wheref1 andf2 = fundamental frequencies,f1 >f2
m and n = positive integers between one and infinity
(4.11)
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Example 4.6
Determine
a) 2nd, 3rd and 12th harmonics for a 1 kHz repetitive wave.
b) Percent 2nd order, 3rd order and total harmonic distortion for a
fundamental frequency with an amplitude of 8 Vrms, a 2nd harmonic
amplitude of 0.2 Vrms and a 3rd harmonic amplitude of 0.1 Vrms.
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Ans
a) 2nd harmonic = 2fundamental freq. = 21 kHz =2 kHz
3rd harmonic = 3fundamental freq. = 31 kHz =3 kHz
12th harmonic = 12fundamental freq. = 121 kHz =12 kHz
b) % 2nd order =
% 3rd order =
% THD =
%5.21008
2.0100
1
2 V
V
%25.1100
8
1.0100
1
3
V
V
%795.2%1008
1.02.022
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Example 4.7For a nonlinear amplifier with two input frequencies, 3 kHz and 8 kHz,
determine,
a) First three harmonics present in the output for each input frequency.
b) Cross product frequencies for values of m and n of 1 and 2.
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Ans f1 = 8 kHz, f2 = 3 kHza)
For freqin =3kHz
1st harmonic = original signal freq. = 3 kHz
2nd harmonic = 2 original signal freq. = 23 kHz =6 kHz
3rd harmonic = 3 original signal freq. = 33 kHz =9 kHz
For freqin =8kHz
1st harmonic = original signal freq. = 8 kHz
2nd harmonic = 2 original signal freq. = 28 kHz =16 kHz
3rd harmonic = 3 original signal freq. = 38 kHz =24 kHz
b)m n Cross Product
1 1 83 5kHz and 11kHz
1 2 86 2kHz and 14kHz
2 1 163 13kHz and 19kHz
2 2 166 10kHz and 22kHz
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NOISE
CORRELATED
NOISE
UNCORRELATED
NOISE
NONLINEAR
DISTORTION
HARMONICDISTORTION
INTERMODULATIONDISTORTION
EXTERNAL INTERNAL
SHOTTRANSIENT
TIMETHERMAL
ATMOSPHERIC EXTRATERRESTRIAL
SOLAR COSMIC
MAN-MADE IMPULSE INTERFERENCE
Table 4.2 Electrical Noise Source Summary
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SIGNAL-TO-NOISE RATIO (SNR)
Signal-to-noise power ratio (S/N) is the ratio of the signal power level to
the noise power
Mathematically,
where, PS= signal power (watts)
PN= noise power (watts)
In dB
S
N
S P
N P
( ) 10log S
N
S PdBN P
(4.12)
(4.13)
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If the input and output resistances of the amplifier, receiver, or
network being evaluated are equal
where Vs = signal voltage (volts)
Vn = noise voltage (volts)
22
2( ) 10log 10log
20log
s s
n n
s
n
S V VdB
N V V
V
V
(4.14)
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Example 4.8
For an amplifier with an output signal power of 10 W and an output noise
power of 0.01W, determine the S/N.
Ans
Example 4.9
For an amplifier with an output signal voltage of 4 V, an output noise voltage
of 0.005 V and an input and output resistance of 50 , determine the S/N.
Ans
][100001.0
10/ unitlessNS
][301000log10)(/ dBdBNS
][640000
005.0
4/
2
2
2
2
unitless
RV
RV
NS
N
s ][58640000log10)(/ dBdBNS
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NOISE FACTOR (F) & NOISE FIGURE (NF)
Noise factor and noise figure are figures of merit to indicate how much asignal deteriorate when it pass through a circuit or a series of circuits
Noise factor
[unitless]
Noise figure
[dB]
For perfect noiseless circuit,F= 1,NF= 0 dB
input signal-to-noise ratio
output signal-to-noise ratioF
input signal-to-noise ratio10log
output signal-to-noise ratio
10log
NF
F
(4.15)
(4.16)
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For ideal noiseless amplifier with a power gain (AP), an input signal power
level (Si) and an input noise power level (Ni) as shows in Figure 4.12(a).The output signal level is simplyAPSi, and the output noise level isAPNi.
[unitless]
Figure 4.12 (b) shows a nonideal amplifier that generates an internal noiseNd
[unitless]
p iout i
out p i i
A SS S
N A N N
p iout i
out p i d i d p
A SS S
N A N N N N A
(4.17)
(4.18)
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Figure 4.12 Noise Figure: (a) ideal, noiseless device (b) amplifier with
internally generated noise
Ideal noiseless
amplifierA
P= power
gain
= SiN
i
=APSi
APN
i
=Si
Ni+ N
d/ A
P
=APSi
APN
i+ N
d
(a)
Signal power out, SoutNoise power out, N
out
Signal power out, SoutNoise power out, N
out
Nonideal amplifier
AP
= power gain
Nd
= internally
generated noise
(b)
Signal power in,Noise power in,
SiN
i
Signal power in,
Noise power in,
SiN
i
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When two or more amplifiers are cascaded as shown in Figure
4.13, the total noise factor is the accumulation of the
individual noise factors.Friiss formula is used to calculate the
total noise factor of several cascaded amplifiers.
Mathematically,Friiss formula is
[unitless]
12121
3
1
21
.....111
n
nT
AAAF
AAF
AFFF
Amplifier 1
AP1NF
1
Amplifier 2
AP2NF
2
Amplifier 3
APnNF
n
Si
Ni(dB)
Input Output
So
No
Si
Ni
= + NFT
Figure 4.13 Noise figure of cascaded amplifiers
(4.19)
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Where
FT= total noise factor forn cascaded amplifiers
F1,F2,F3n= noise factor, amplifier 1,2,3n
A1,A2.An= power gain, amplifier 1,2,..n
Notification remarks
Change unit of all noise factorsFand power gainsA from [dB]
to [unitless]before insert its intoFriss formula equation
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b) The output noise power = internal noise + amplified input noise
The output signal power = amplified input signal
Output SNR=
Output SNR(dB) =
][108.1
)101100(80
4
6
W
WWNANN ipDout
][10110100100
2
6
W
SASipout
][56.55101.8
1014-
-2
unitlessN
S
out
out
][45.1756.55log10 dB
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c) Noise Figure NF =56.55
100log10
][
][log10
unitlessSNRoutput
unitlessSNRinput
][55.2 dB
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Example 4.11
For a non-ideal amplifier and the following parameters, determine
Input signal power = 2 x 10-10 W
Input noise power = 2 x 10-18 W
Power Gain = 1,000,000
Internal Noise (Nd) = 6 x 10-12 W
a. Input S/N ratio (dB)
b. Output S/N ratio (dB)
c. Noise factor and noise figure
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Ans
a) Input SNR
Input SNR(dB) =
b) The output noise power
The output signal power
Output SNR(dB)
][101102
102 818-
-10unitless
N
S
i
i
][80100000000log10 dB
][108)102101(106
12
18612
WNANN ipDout
][102
102101
4
106
W
SAS ipout
][74log10 dB
108
102
12-
-4
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c)
Noise factorF =
Noise figureNF=
][425000000
100000000
][
][unitless
unitlessSNRoutput
unitlessSNRinput
][02.64log10 dB
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Example 4.12
For three cascaded amplifier stages, each with noise figures of 3 dB and power
gains of 10 dB, determine the total noise figure.
Ans.
Change all noise figure and power gain from [dB] unit to [unitless]
Power gain
Noise Factor
UsingFriss formula ,
Total noise factor
Total noise figure NFT =
][10101010
321 unitlessAAA
][21010
3
321 unitlessFFF
][11
21
3
1
21 unitless
AA
F
A
FFFT
][11.2
101012
10122
unitless
][24.311.2log10 dB
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Te is a hypothetical value that cannot be directly measured
Convenient parameter often used . Its also indicates reduction in thesignal-to-noise ratio a signal undergoes as it propagates through a receiver.
The lower the Te, the better the quality of a receiver.
Typically values forTe , range from (20 K1000 K) for noisy receivers.
Mathematically,
Where Te=equivalent noise temperature (kelvin)
T = environmental temperature (290 K)
F= noise factor (unitless)
Conversely,Fcan be represented as a function ofTe:
1 FTTe
T
TF
e 1
(4.22)
(4.21)
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Example 4.13
Determine,
a) Noise figure for an equivalent noise temperature of 75 K.
b) Equivalent noise temperature for noise figure of 6 dB.
Ans.
a) Noise factor
Noise figure NF =
b) Noise factor
Equivalent noise temperature
][258.12907511 unitless
TTF e
][1258.1log10 dB
][4)
10
6log()
10log( unitlessanti
NFantiF
][870
)14(290)1(
K
FTTe
NOISE MEASUREMENTS
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NOISE MEASUREMENTS
To work with noise in communications systems, it must bemeasured in a meaningful way.
Noise is a random process & does not have a single valueor an equation to describe it.
The root mean square(rms) value of the noise is the mostimportant fact.
rms value is formed by taking the square root of theaverage of the individual noise voltages, which have beensquared.
NOISE MEASUREMENTS
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Consider a series of 10 noise values measured with a voltmeter
as -0.3, 1.0, 0.2, 0.5, 0.6, -0.6, 0.3, 0.1, -0.15 and 0.9 V. They are squared so that the negative values become positive, &
then these squared values are averaged.
The sum of the squares is
The average is
22222222 1.03.06.06.05.02.013.0 22 9.015.0....
20325.3 V
230325.010
0325.3V
NOISE MEASUREMENTS
NOISE MEASUREMENTS
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f i i i i
The square root of this mean is
Example 4.14
Noise values in mV as follows are measured at various times:
10, -100, 35, -57, 90, 26, 26, -10, -15 and -20. What is the rmsnoise value?
Squaring each value, we have:
100 + 10,000 + 1225 + 3249 + 8100 + 676 + 676 + 100 + 225 +
400 = 24,751 (mV)2
The average value is 24,751/10 = 2475.1 (mV)2.
The rms value = 49.75 mV.
V55.030325.0
NOISE MEASUREMENTS
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