chapter 4 noise

7/28/2019 Chapter 4 Noise

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Dept. Of Communication Engineering,

Faculty Of Electrical And Electronics,Universiti Tun Hussein Onn Malaysia

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.
http://cnx.org/content/m10656/latest/http://cnx.org/content/m10656/latest/


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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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Faculty Of Electrical And Electronics,

Universiti Tun Hussein Onn Malaysia

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

chapter 4 noise

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