Theory Communication

CHAPTER 4: NOISE
Prepared by:
DR NOORSALIZA BINTI ABDULLAH
DEPARTMENT OF COMMUNICATION ENGINEERING
FACULTY OF ELECTRICAL AND ELECTRONIC ENGINEERING

DEFINATION OF RANDOM VARIABLES
A real random is mapping from the sample space Ω (or S) to the
set of real numbers.
A schematic diagram representing a random variable is given
below
Ω

ω1 ω2
ω3

X (ω1 )

X (ω2 )

ω4

X (ω3 ) X (ω4 )

Figure 4.1 : Random variables as a mapping from Ω to R
Dept. Of Communication Engineering,
Faculty Of Electrical And Electronics,
Universiti Tun Hussein Onn Malaysia

A random variable, usually written X, is a variable whose
possible values are numerical outcomes of a random
phenomenon, etc.; individuals values of the random variable X
are X(ω).
There are two types of random variables, which is Discrete
Random Variables and Continuous Random Variables.


Discrete Random Variables
A sample space is discrete if the number of its elements are
finite or countable infinite, i.e., a discrete random variable is
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).

Continuous Random Variables
A continuous random variable is one which takes an infinite
number of possible values. Continuous random variables are
usually measurements.
Examples include height, weight, the amount of sugar in an
orange, the time required to run a mile.
A continuous random variable is not defined at specific values.
Instead, it is defined over an interval of values, and is
represented by the area under a curve (in advanced
mathematics, 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 the
random variable is infinite.


Figure 4.2 : Random variables (a) continuous (b) discrete.

Example 4.1
Which of the following random variables are discrete and which are
continuous?

c)

X = Number of houses sold by real estate developer per week?
X = Number of heads in ten tosses of a coin?
X = Weight of a child at birth?

d)

X = Time required to run 100 yards?

a)
b)

Answer:
(a) Discrete (b) Discrete (c) Continuous (d) Continuous

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.


RANDOM SIGNALS
Random signals cannot be characterized by a simple, welldefined mathematical equation and their future values cannot
be predicted.
Rather, we must use probability and statistics to analyze their
behavior.
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.


Deterministic Signal

Figure 4.3: An example of a deterministic signal, the sine wave.

Random Signal

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.

RANDOM PROCESSES
As mentioned before, in order to study random signals, we
want 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
Model Random Signal and Random Noise


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 space
mapped into a number, whereas for a random process it is
mapped into a function of time.


Figure 4.5: Example of random process represent the temperature of a city at 20
hours.

POWER SPECTRAL DENSITY
Random process is a collection of signals, and the spectral
characteristics of these signals determine the spectral
characteristic 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 concentrated
at high frequencies.

Power spectral density determines the power distribution (or
power spectrum) of the random process.
PSD of a random process X(t) is denoted by SX(f), denotes the
strength of power in the random process as a function of
frequency.
Units for PSD is Watts/Hz.

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.
However there always remain inescapable random signals, that
present a fundamental limit to systems performance.


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 voltage
produced by random movements of
mobile electrons.


GAUSSIAN PROCESS
Gaussian process is important in
communication systems.
The main reason is that thermal
noise in electrical devices produced
by movement of electrons due to
thermal agitation is closely modeled
by a Gaussian process.
Due to the movements of electrons,
sum of small currents of a very large
number of sources was introduced.
Since majority sources are
independent, hence the total current
is sum of large number of random
variables.
Therefore the total currents has
Gaussian distribution.

Figure 4.7 Histogram of some noise voltage
measurements


Definition
A random process X(t) is a Gaussion process if for all n and all
(t1,t2,…,tn) the random variable {X(ti)}ni=1 have a jointly Gaussian
density function.
Gaussian or Normal Random Variables
( x − m )2
−
where m = mean
1
2σ 2
(4.1)
f X ( x) =
e
σ = standard deviation
2πσ
σ2 = variance
A Gaussian random variable with mean m and variance σ2 is denoted
by Ν(m, σ2)
Assuming X is a standard normal random variable, we defined the function
Q(x) as P(X > x). The Q function is given by relation
2

Q( x) = P( X > x) = ∫

∞

x

1 − t2
e dt
2π

(4.2)

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:
(4.3a)
Q(-x) = 1 – Q(x)
(4.3b)
Q(0) = ½
Q(∞) = 0
(4.3c)
Table 3.1 gives the value of this function for various value of x.
For Ν(m, σ2) random variable, a simple change of variable in the integral
that computes P(X > x) results in P(X > x) = Q[(x – m)/σ].
tail probability in Gaussian random variable.


Figure 4.8: The Q-function as the area under the tail of a standard normal random variable.

Table 4.1 Table of the Q function


Example 4.2
X is a Gaussian random variable with mean 1 and variance 4. Find the
probability X between 5 and 7.
Ans.
We have m = 1 and σ = √4 = 2. Thus,
P( 5 < X < 7)
= P (X > 5) – P(X > 7)
= Q ((5 – 1)/2) – Q((7 – 1)/2)
= Q(2) – Q(3)
≈ 0.0214


WHITE NOISE
There are many ways to characterize different noise sources, one is to
consider the 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 spectra P(f) is
proportional to 1 / fβ for β ≥ 0.
When β = 0 the noise is referred to white noise, when β = 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 process X(t) is called a white process if it has a flat spectral
density,i.e., if SX(f) is constant for all f.


White Noise, β = 0
β =0

β =2

Brownian noise

white noise

β =1

β =3

1/f noise


White Noise
Spectral density of white
noise is a constant, N0/2
N0
SX ( f ) =
2

SX (f)

N0
2

f

0

(3.4)

White noise power spectrum
-

Where N0 = kT

RXX ( )

Autocorrelation function:

N0
2

⎛N ⎞
RXX (τ ) = F −1 ⎜ 0 ⎟
⎝ 2 ⎠
∞

=

N 0 j 2π ft
∫ 2 e df
−∞

N
= 0 δ (τ )
2
k = Boltzmann’s constant = 1.38 x

0

(3.5)
10-23

White noise autocorrelation
-

Figure 4.9: White noise (a) power spectrum
(b) autocorrelation

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 that power spectral density of thermal noise increase
with increasing the ambient temperature, therefore, keeping
electric circuit cool makes their noise level low.


TYPE OF NOISE
Noise can be divided into :
2 general categories
Correlated noise – implies relationship between the signal and the noise,
exist only when signal is present
Uncorrelated noise – present 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


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 Earth’s
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

(b) Extraterrestrial Noise
Consists of electrical signals that originate from outside earth
atmosphere, deep-space noise
Divide further into two
(i) Solar noise – generated directly from sun’s heat. There are 2
parts to solar noise:Quite condition when constant radiation intensity exist and
high intensity
Sporadic disturbance caused by sun spot activities and solar
flare-ups which occur every 11 years
(ii) Cosmic noise – continuously distributed throughout the
galaxies, small noise intensity because the sources of galactic
noise are located much further away from sun. It's also often
called as black-body noise.


(c) Man-made noise
Source – spark-producing mechanism such as from commutators in
electric motors, automobile ignition etc
Impulsive in nature, contains wide range of frequency that
propagate through space the same manner as radio waves
Most intense in populated metropolitan and industrial areas and is
therefore sometimes called industrial noise.


(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


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 effect
transistor 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

(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.

(c) Thermal noise
Due to rapid and random movement of electrons within a conductor
due 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 it
increases 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.


Noise Spectral Density
In communications, noise spectral density No is the noise
power per unit of bandwidth; that is, it is the power spectral
density of the noise.
It has units of watts/hertz, which is equivalent to watt-seconds
or joules.
If the noise is white, i.e., constant with frequency, then the
total noise power N in a bandwidth B is BNo.
This is utilized in Signal-to-noise ratio calculations.
The thermal noise density is given by No = kT, where k is
Boltzmann's constant in joules per kelvin, and T is the receiver
system noise temperature in kelvin.
No is commonly used in link budgets as the denominator of the
important figure-of-merit ratios Eb/No and Es/No.

NOISE POWER
Noise power is given as
N0
df
−B 2
= N0 B

PN = ∫

B

and can be written as
PN = kTB [W]

(3.6)

(3.7)

where
PN = noise power,
k = Boltzmann’s constant (1.38x10-23 J/K)
B = bandwidth,
T = absolute temperature (Kelvin)(17oC or 290K)


NOISE VOLTAGE
Figure 4.10 shows the equivalent
circuit for a thermal noise source.
Internal resistance RI in series
with the rms noise voltage VN.
For the worst condition, the load
resistance R = RI , noise voltage
dropped across R = half the noise
source (VR=VN/2) and
From equation 4.5 the noise
power PN , developed across the
load resistor = kTB

Figure 4.10 : Noise source equivalent
circuit

The mathematical expression :

PN

(V
= kTB = N

/ 2)
V N2
=
R
4R
2

(4.8a)

V N2 = 4 RkTB
VN =

4 RkTB

(4.8b)

OTHER NOISE SOURCES
1.

2.

3.

There are 3 other noise mechanisms that contribute to internally generated
noise in electronic devices.
Generation-Recombination Noise - The result of free carriers being
generated and recombining in semiconductor material. Can consider these
generation and recombination events to be random. This noise process can
be treated as shot noise process.
Temperature-Fluctuation Noise – The result of the fluctuating heat
exchange between a small body, such as transistor, and it’s environment
due to the fluctuations in the radiation and heat-conduction processes. If a
liquid or gas is flowing past the small body, fluctuation in heat convection
also occurs.
Flicker Noise – It is characterized by a spectral density that increases with
decreasing frequency. The dependence on spectral density on frequency is
often found to be proportional to the inverse first power of the frequency.
Sometimes referred as one-over-f noise.


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 Ω.
Ans
a) Thermal noise power

b) Noise voltage

N = kTB

VN =

= 1.38 × 10 − 23 × 290 × 1×10 6

= 4 × 50 × 4 × 10
= 0 . 895 μ V

= 4 × 10 −15W

4 RkTB
− 15


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 Ω internal resistance and 100
Ω load resistance.
Ans.
N = 1.38 ×10 −23 × 290 × 10 ×103 b) V = 4 RkTB
a)
N
= 4.002 × 10 −17 W

⎛ 4 × 10−17 ⎞
N (dBm ) = 10 log⎜
⎟
⎜ 1 × 10− 3 ⎟
⎠
⎝
= −134dBm

= 4 × 100 × 4 ×10 −17
= 0.127 μV (rms )


Example 4.5
Two resistor of 20 kΩ and 50 kΩ 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


Ans.
a)
VN 2 = 4 R2 kTB

VN 1 = 4 R1kTB

− 23
3
3
=
　　 4 × 20 × 103 × 1.38 × 10 − 23 × 290 × 100 × 103 　　 4 × 50 ×10 ×1.38 ×10 × 290 ×100 ×10
=
=
　8.95 ×10 −6 V
　　 5.66 × 10 − 6V
=

b) RT= 20 × 10 + 50 × 10 = 70 × 10 Ω
3

V Ntotal =

3

3

4 RT kTB

　　 4 × 70 × 10 3 × 1 .38 × 10 − 23 × 290 × 100 × 10 3
=
　　 1 .06 × 10 − 5 V
=
( 20 + 50 )10 3
c) RT=
= 14 .28 kΩ
3
3
20 × 50 × 10 × 10
V Ntotal = 4 RT kTB

　　 4 × 14 .29 k × 1.38 × 10 − 23 × 290 × 100 × 10 3
=
　　4.78 μV
=

CORRELATED NOISE

1.

Mutually related to the signal, not present if there is no signal
Produced by nonlinear amplification, and include nonlinear
distortion such as harmonic and intermodulation distortion
Harmonic Distortion (HD)
Harmonic distortion – unwanted harmonics of a signal produced
through nonlinear amplification (nonlinear mixing). Harmonics are
integer 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 rms
amplitude 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.

Figure 4.11(a) show the input and
output frequency spectrums for a
nonlinear device with a single input
frequency f1.
Mathematically, THD is

%THD =

vhigher
vfundamental

x100

Where,
%THD = percent total harmonic
distortion
vhigher = quadratic sum of the rms
2
2
2
voltages,
v2 + v3 + vn
vfundamental = rms voltage of the
fundamental frequency

(4.9)

Figure 4.11: Correlated noise:
(4.10)

(a) Harmonic distortion
(b) Intermodulation distortion

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 a
nonlinear device with two input frequencies (f1 and f2).
Mathematically, the sum and difference frequencies are
(4.11)
Cross products =mf1 ± nf2
Where f1 and f2 = fundamental frequencies, f1 > f2
m and n = positive integers between one and infinity


Example 4.6
Determine
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.
a)


Ans
a)

b)

2nd harmonic = 2×fundamental freq. = 2×1 kHz =2 kHz
3rd harmonic = 3×fundamental freq. = 3×1 kHz =3 kHz
12th harmonic = 12×fundamental freq. = 12×1 kHz =12 kHz
%

2nd

V2
0.2
× 100 = 2.5%
order = × 100 =
V1
8

% 3rd order =

V3
0.1
× 100 =
×100 = 1.25%
V1
8

0.2 2 + 0.12
× 100% = 2.795%
% THD =
8

Example 4.7
For 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.


Ans f1 = 8 kHz, f2 = 3 kHz
a)

For freqin =3kHz
1st harmonic = original signal freq. = 3 kHz
2nd harmonic = 2× original signal freq. = 2×3 kHz =6 kHz
3rd harmonic = 3× original signal freq. = 3×3 kHz =9 kHz
For freqin =8kHz
1st harmonic = original signal freq. = 8 kHz
2nd harmonic = 2× original signal freq. = 2×8 kHz =16 kHz
3rd harmonic = 3× original signal freq. = 3×8 kHz =24 kHz
b)

m
1

n
1

8±3

Cross Product
5kHz and 11kHz

1
2
2

2
1
2

8±6
16±3
16±6

2kHz and 14kHz
13kHz and 19kHz
10kHz and 22kHz

Table 4.2 Electrical Noise Source Summary


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,

S
P
= S
N PN
where,

(4.12)

PS = signal power (watts)
PN = noise power (watts)

In dB

S
PS
(dB) = 10log
N
PN

(4.13)


If the input and output resistances of the amplifier, receiver, or
network being evaluated are equal
⎛V ⎞
⎛V ⎞
S
( dB ) = 10 log ⎜ s 2 ⎟ = 10 log ⎜ s ⎟
N
⎝ Vn ⎠
⎝ Vn ⎠
2

⎛ Vs ⎞
= 20 log ⎜ ⎟
⎝ Vn ⎠
where

2

(4.14)

Vs = signal voltage (volts)
Vn = noise voltage (volts)


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
S/N =

10
= 1000 　　
[unitless ]
0.01

S / N ( dB ) = 10 log 1000 = 30 [ dB ]

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 Ω, determine the S/N.
Ans
Vs
S/N =

VN

2

R =
2

42
= 640000　　
[unitless]
0.0052

S / N ( dB ) = 10 log 640000 = 58 [ dB ]

R

NOISE FACTOR (F) & NOISE FIGURE (NF)
Noise factor and noise figure are figures of merit to indicate how much a
signal deteriorate when it pass through a circuit or a series of circuits
Noise factor

F=
Noise figure

input signal-to-noise ratio
[unitless]
output signal-to-noise ratio

input signal-to-noise ratio [dB]
NF = 10log
output signal-to-noise ratio
= 10log F

(4.15)

(4.16)

For perfect noiseless circuit, F = 1, NF = 0 dB


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 simply APSi, and the output noise level is APNi.

Ap Si
Sout
Si
=
=
N out Ap N i N i

[unitless]

(4.17)

Figure 4.12 (b) shows a nonideal amplifier that generates an internal noise
Nd

Ap Si
Sout
Si
=
=
N out Ap N i + N d N i + N d Ap

[unitless]

(4.18)


Figure 4.12 Noise Figure: (a) ideal, noiseless device (b) amplifier with
internally generated noise

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
Fn − 1
F2 − 1 F3 − 1
[unitless] (4.19)

FT = F1 +

A1

+

A1 A2

+

A1 A2 ..... An −1

Figure 4.13 Noise figure of cascaded amplifiers

Where
FT = total noise factor for n cascaded amplifiers
F1, F2, F3…n = noise factor, amplifier 1,2,3…n
A1, A2…. An = power gain, amplifier 1,2,…..n
Notification remarks
Change unit of all noise factors F and power gains A from [dB]
to [unitless] before insert its into Friss formula equation


Example 4.10
The input signal to a telecommunications receiver consists of 100 μW of
signal power and 1 μW of noise power. The receiver contributes an
additional 80 μW of noise, ND, and has a power gain of 20 dB. Compute
the input SNR, the output SNR and the receiver’s noise figure.

Ans.
a) Input SNR =

Si
100 × 10 -6
=
= 100 [ unitless ]
-6
Ni
1 × 10

Input SNR(dB) = 10 log 100 = 20 [ dB ]


b) The output noise power = internal noise + amplified input noise

N out = N D + Ap N i = 80 μW + (100 × 1× 10 −6 W )
　　　1.8 × 10 − 4 [W ]
=
The output signal power = amplified input signal

S out = Ap Si = 100 × 100 × 10 −6
　　　× 10 − 2 [W ]
=1
S out
1× 10 -2
Output SNR=
=
= 55.56[unitless ]
-4
N out 1.8 × 10
Output SNR(dB) =

10 log 55 .56 = 17 .45[ dB ]


c) Noise Figure NF = 10 log

input SNR[unitless ]
100
= 10 log
output　
SNR[unitless ]
55.56

= 2 . 55 [ dB ]


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.
b.
c.

Input S/N ratio (dB)
Output S/N ratio (dB)
Noise factor and noise figure


Ans
a) Input SNR

S i 2 × 10 -10
=
= 1 × 10 8 [unitless ]
N i 2 × 10 -18

10
Input SNR(dB) = log 100000000 = 80 [ dB ]
b) The output noise power Nout = ND + Ap Ni = 6×10−12 + (1×106 × 2×10−18)

　　　 ×10−12[W ]
=8

The output signal power S out = Ap Si = 1×106 × 2 ×10 −10
　　　 2 ×10 − 4 [W ]
=
2 × 10 -4
= 74 [ dB ]
Output SNR(dB) = 10 log
- 12
8 × 10

c)
Noise factor F =

input SNR [ unitless ] 100000000
=
= 4[ unitless ]
output 　SNR [unitless ]
25000000

Noise figure NF = 10 log 4 = 6.02[ dB ]


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]
10
Power gain A = A = A = 10 10 = 10[unitless ]
1

2

3

3
10

Noise Factor F1 = F2 = F3 = 10 = 2[unitless ]
Using Friss formula ,
F − 1 F3 − 1
+
[unitless ]
Total noise factor FT = F1 + 2
A1
A1 A2
2 −1 2 −1
= 2+
+
10 10 × 10
[unitless ]
= 2.11　　　　　　　

Total noise figure NFT = 10 log 2 . 11 = 3 . 24 [ dB ]

EQUIVALENT NOISE TEMPERATURE (Te)
The noise produced from thermal agitation is directly proportional to
temperature, thermal noise can be expressed in degrees as well as watts or
dBm.
Mathematically,

N
T=
KB

(4.20)

where T = environmental temperature (kelvin)
N = noise power (watts)
K = Boltzmann’s constant (1.38 x 10-23 J/K)
B = bandwidth (hertz)


Te is a hypothetical value that cannot be directly measured
Convenient parameter often used . It’s also indicates reduction in the
signal-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 for Te , range from (20 K – 1000 K) for noisy receivers.
Mathematically,
(4.21)

Te = T (F − 1)

Where

Te =equivalent noise temperature (kelvin)
T = environmental temperature (290 K)
F = noise factor (unitless)
Conversely, F can be represented as a function of Te :

Te
F =1+
T

(4.22)


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 F = 1 + Te = 1 + 75 = 1 .258 [unitless ]
290
T
Noise figure NF = 10 log 1 . 258 = 1[ dB ]
b) Noise factor F = anti log(

NF
6
) = anti log( ) = 4[unitless]
10
10

Equivalent noise temperature Te = T ( F − 1) = 290 ( 4 − 1)

　　 870[ K ]
=

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