1 School of Electrical, Electronics and Computer Engineering University of Newcastle-upon-Tyne Noise in Communication Systems Noise in Communication Systems

1

School of Electrical, Electronics andComputer Engineering

University of Newcastle-upon-Tyne

Noise in Communication Noise in Communication SystemsSystems

Prof. Rolando CarrascoProf. Rolando Carrasco

Lecture Notes Newcastle University

2008/2009

2

Noise in Communication SystemsNoise in Communication Systems

1. Introduction2. Thermal Noise3. Shot Noise4. Low Frequency or Flicker Noise5. Excess Resister Noise6. Burst or Popcorn Noise7. General Comments8. Noise Evaluation – Overview9. Analysis of Noise in Communication

Systems• Thermal Noise• Noise Voltage Spectral Density• Resistors in Series• Resistors in Parallel

10.Matched Communication Systems

11. Signal - to – Noise12. Noise Factor – Noise Figure13. Noise Figure / Factor for Active

Elements14. Noise Temperature15. Noise Figure / Factors for Passive

Elements16. Review – Noise Factor / Figure /

Temperature17. Cascaded Networks18. System Noise Figure19. System Noise Temperature20. Algebraic Representation of Noise21. Additive White Gaussian Noise

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1. Introduction. Introduction

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:-

•Interference, usually from a human source (man made)•Naturally occurring random noise

Interference

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.

4

1. Introduction (Cont’d). Introduction (Cont’d)

Natural Noise

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.

5

2. Thermal Noise (Johnson Noise)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).

Experimental results (by Johnson) and theoretical studies (by Nyquist) give the mean square noise voltage as

)(4 22_

voltTBRkV

Where k = Boltzmann’s constant = 1.38 x 10-23 Joules per KT = absolute temperature

B = bandwidth noise measured in (Hz)R = resistance (ohms)

6

2. Thermal Noise (Johnson Noise) (Cont’d)2. Thermal Noise (Johnson Noise) (Cont’d)The law relating noise power, N, to the temperature and bandwidth is

N = k TB wattsN = k TB watts

Thermal noise is often referred to as ‘white noise’ because it has a uniform ‘spectral density’.

7

3. Shot Noise3. 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. • 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 coulombsB is the effective noise bandwidth (Hz)

• Shot noise is found to have a uniform spectral density as for thermal noise

22 )(22 ampsBqIII eoDCn

8

4. Low Frequency or Flicker Noise4. 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.

5. Excess Resistor Noise5. 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.

6. Burst Noise or Popcorn Noise6. Burst Noise or Popcorn NoiseSome semiconductors also produce burst or popcorn noise with a spectral density which is proportional to

2

1

f

9

7. General Comments7. General Comments

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.

10

8. Noise Evaluation8. Noise Evaluation

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.

dBmdBmdB

dB

dBm

dBm

dB

ratio

NSN

S

NSN

Sei

mW

mWNNand

mW

mWSS

thatrecallAlso

N

S

N

S

N

S

N

S

1010

10

10

10

log10log10..

1

)(log10

1

)(log10

log10

11

8. Noise Evaluation (Cont’d)8. Noise Evaluation (Cont’d)

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.

12

Noise may be quantified in terms of noise power spectral density, po watts per Hz, from which Noise power N may be expressed as

N= po Bn watts

8. Noise Evaluation (Cont’d)8. Noise Evaluation (Cont’d)

Ideal low pass filter Bandwidth B Hz = Bn

N= po Bn wattsPractical 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.

13

9. Analysis of Noise In Communication Systems9. Analysis of Noise In Communication Systems

Thermal Noise (Johnson noise)Thermal Noise (Johnson noise)

This thermal noise may be represented by an equivalent circuit as shown below

)(4 2____

2 voltTBRkV

____2V nVkTBR 2

(mean square value , power)then VRMS =

i.e. Vn is the RMS noise voltage.

A) System BW = B Hz N= Constant B (watts) = KBB) System BW N= Constant 2B (watts) = K2B

For A, KB

S

N

S For B,

BK

S

N

S

2

14

9. Analysis of Noise In Communication Systems (Cont’d)9. Analysis of Noise In Communication Systems (Cont’d)

22

___2

1

_______2

nnn VVV

11

____2

1 4 RBTkVn

22

____2

2 4 RBTkVn

)(4 2211

____2 RTRTBkVn

)(4 21

____2 RRBkTVn

Assume that R1 at

temperature T1 and R2 at

temperature T2, then

i.e. The resistor in series at same temperature behave as a single resistor

Resistors in SeriesResistors in Series

15

9. Analysis of Noise In Communication Systems (Cont’d)9. Analysis of Noise In Communication Systems (Cont’d)

Resistance in ParallelResistance in Parallel

21

211 RR

RVV no

21

122 RR

RVV no

22

___2

1

_______2

oon VVV

____

2nV

21

2122

2111

222

21

4

RR

RRRTRRTR

RR

kB

221

221121_____

2 )(4

RR

RTRTRRkBVn

21

21_____

2 4RR

RRkTBVn

16

10. 10. Matched Communication Systems 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:-

OrOr

An equivalent circuit, when the line is connected to the receiver is shown below.

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10. 10. Matched Communication Systems (Cont’d) Matched Communication Systems (Cont’d)

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11. 11. Signal to NoiseSignal to Noise

PowerNoise

PowerSignal

N

S

N

S

N

SdB 10log10

The signal to noise ratio is given by

The signal to noise in dB is expressed by

dBmdBmdB NSN

S

for S and N measured in mW.

12. 12. NoiseNoise Factor- Noise Figure Consider the network shown below,

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12. 12. NoiseNoise Factor- Noise Figure (Cont’d)

• The amount of noise added by the network is embodied in the Noise Factor F, which is defined by

Noise factor F =

OUT

IN

NS

NS

• F equals to 1 for noiseless network and in general F > 1. The noise figure in the noise factor quoted in dBi.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.

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13. 13. Noise Figure – Noise Factor for Active ElementsNoise Figure – Noise Factor for Active Elements

OUT

IN

NS

NS

OUT

OUT

IN

IN

S

N

N

SOUTS INSG

IN

OUT

IN

IN

SG

N

N

SF

IN

OUT

NG

N

For active elements with power gain G>1, we have

F = = But

Therefore

Since in general F v> 1 , then OUTN 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

21

13. 13. Noise Figure – Noise Factor for Active Elements (Cont’d)Noise Figure – Noise Factor for Active Elements (Cont’d)

Ne is extra noise due to active elements referred to the input; the element is thus effectively noiseless.

22

14. 14. NoiseNoise Temperature

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15. 15. Noise Figure – Noise Factor for Passive ElementsNoise Figure – Noise Factor for Passive Elements

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16. Review of Noise Factor – Noise Figure –Temperature

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17. 17. Cascaded NetworkCascaded Network

A receiver systems usually consists of a number of passive or active elements connected in series. 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.

eT eN or is the noise referred to the input.

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18. System 18. System NoiseNoise Figure

Assume that a system comprises the elements shown below,

Assume that these are now cascaded and connected to an aerial at the input, with aeIN NN

from the aerial.

Now , 333 eINOUT NNGN

ININ NFNG 1333 Since ININeININ NFNGNNGN 12222223

similarly INaeIN NFNGN 1112

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18. System 18. System NoiseNoise Figure (Cont’d)

INININaeOUT NFGNFGNFGNGGGN 111 332211123

The overall system Noise Factor is

ae

OUT

IN

OUTsys NGGG

N

GN

NF

321

ae

IN

ae

IN

ae

IN

N

N

GG

F

N

N

G

F

N

NF

21

3

1

21

1111

121321

4

21

3

1

21 ..........

1...........

111

n

nsys GGG

F

GGG

F

GG

F

G

FFF

The equation is called FRIIS Formula.

28

19. System 19. System NoiseNoise Temperature

29

20. Algebraic Representation of 20. Algebraic Representation of NoiseNoise

Phasor Representation of Signal and NoiseThe general carrier signal VcCosWct may be represented as a phasor at any instant in time as shown below:

If we now consider a carrier with a noise voltage with “peak” value superimposed we may represents this as:

Both Vn and n are random variables, the above phasor diagram represents a snapshot

at some instant in time.

30

20. Algebraic Representation of 20. Algebraic Representation of Noise (Cont’d)Noise (Cont’d)

nn CosVtx )(

nn SinVty )(

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

31


32


33


Considering the general phasor representation below:-

34


tCosVV

tSinV

nnc

nn

1tan

tCosV

V

tSinV

V

nc

n

nc

n

1tan 1

From the diagram

35

21. Additive White Gaussian 21. Additive White Gaussian Noise Noise

Additive

White

White noise = fpo = Constant

Gaussian

We generally assume that noise voltage amplitudes have a Gaussian or Normal distribution.

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

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School of Electrical, Electronics andComputer Engineering

University of Newcastle-upon-Tyne

Error Control CodingError Control Coding

Prof. Rolando CarrascoProf. Rolando Carrasco

Lecture Notes University of Newcastle-upon-Tyne

2005

37

Error Control CodingError Control Coding

• In digital communication error occurs due to noise

•Bit error rate =

•Error rates typically range from 10-1 to 10-5 or better

• In order to counteract the effect of errors Error Control Coding is used.

a) Detect Error – Error Detection

b) Correct Error – Error Correction

)(largeforbitsN

bitsNinerrorsofno NN

38

Channel Coding in CommunicationChannel Coding in Communication

39

Automatic Repeat Request (ARQ) Automatic Repeat Request (ARQ)

40

Automatic Repeat Request (ARQ) (Cont’d) Automatic Repeat Request (ARQ) (Cont’d)

41

Forward Error Correction (FEC)Forward Error Correction (FEC)

42

Block CodesBlock Codes

• A block code is a coding technique which generates C check bits for M message bits to give a stand alone block of M+C= N bits

• The code rate is given by Rate = N

M

CM

M

8

7

17

7

• A single parity bit (C=1 bit) applied to a block of 7 bits give a code rate

Rate =

43

Block Codes (Cont’d)Block Codes (Cont’d)

7

4

• A (7,4) Cyclic code has N=7, M=4

Code rate R =

A repetition-m code in which each bit or message is transmitted m times and the receiver carries out a majority vote on each bit has a code rate

mmM

M 1Rate

44

Message Transfer Message Transfer

It is required to transfer the contents of Computer A to Computer B.

COMPUTER A COMPUTER B

• The messages transferred to the Computer B, some may be rejected (lost) and some will be accepted, and will be either true (successful transfer) or false

• Obviously the requirement is for a high probability of successful transfer (ideally = 1), low probability of false transfer (ideally = 0) and a low probability of lost messages.

45

Message Transfer (Cont’d) Message Transfer (Cont’d)

Error control coding may be considered further in two main ways

1. In terms of System Performance i.e. the probabilities of successful, false and lost message transfer. We need to know error correcting /detection ability to detect and correct errors (depends on hamming distance).

2. In terms of the Error Control Code itself i.e. the structure, operation, characteristics and implementation of various types of codes.

46

System PerformanceSystem Performance

In order to determine system performance in terms of successful, false and lost message transfers it is necessary to know:

• the probability of error or b.e.r p.• the no. of bits in the message block N• the ability of the code to detect/ correct errors, usually expressed as a minimum Hamming distance, dmin for the code

RNR ppRRN

NR

1

!!

!)(

This gives the probability of R errors in an N bit block subject to a bit error rate p.

47

System Performance (Cont’d)System Performance (Cont’d)

Hence, for an N bit block we can determine the probability of no errors in the block (R=0) i.e.

NN pppN

N)1(1

!0!0

!)0( 00

• An error free block

• The probability of 1 error in the block (R=1)

111 )1(1!1!1

!)1(

NN ppNpp

N

N

• The probability of 2 error in the block (R=2)

22 1!2!2

!)2(

Npp

N

N

48

Minimum Hamming distanceMinimum Hamming distance

• A parameter which indicates the worst case ability of the code to detect /correct errors.

Let dmin = minimum Hamming distance l = number of bits errors detected

t = number of bit errors corrected

dmin = l + t + 1 with t ≤ l

For example, suppose a code has a dmin = 6.

We have as options 1) 6= 5 + 0 + 1 {detect up to 5 errors , no correction}2) 6= 4 + 1 + 1 {detect up to 4 errors , correct 1 error}3) 6= 3 + 2 + 1 {detect up to 3 errors , correct 2 error}

49

Minimum Hamming distance (Cont’d)Minimum Hamming distance (Cont’d)

Messages transfers are successful if no errors occurs or if t errors occurs which are corrected.

i.e. Probability of Success =

t

i

ipp1

)()0(

Messages transfers are lost if up to l errors are detected which are not corrected, i.e

Probability of lost = p(t+1) + p(t+2)+ …. p(l)

• Fortunately, the higher the no. of errors, the less the probability they will occur for reasonable values of p.

• For option 3 for example, if 4 or more errors occurred, these would not be detected and these messages would be accepted but would be false messages.

l

ti

ip1

)( =

50

Minimum Hamming distance (Cont’d)Minimum Hamming distance (Cont’d)

Message transfers are false of l+1 or more errors occurs Probability of false = p(l+1) + p(l+2)+ …. p(N)

=

N

li

ip1

)(

Example Using dmin = 6, option 3, (t =1, l =4)

Probability of Successful transfer = p(0) + p(1)

Probability of lost messages = p(2) + p(3) + p(4)

Probability of false messages = p(5) + p(6)+ …….+ p(N)

51

Probability of Error Probability of Error

• Each bit has a probability of error p, i.e. probability that a transmitted ‘0’ is received as a ‘1’ and a transmitted ‘1’ is received as a ‘0’.

• this probability is called the single bit error rate or bit error b.e.r.

• For example, if p = 0.1 , the probability that any single bit is in error is ‘1 in 10’ or 0.1.

• If there were 5 consecutive bits in error, the probability that the 6th bit will be in error is still 0.1, i.e. it is independent of the previous bits in error.

52

Probability of Error (Cont’d) Probability of Error (Cont’d)

Consider a typical message block below.

Error Control Coding Data Information

Address bits Synchronization bit pattern

• The first requirement for the receiver/decoder is to identify the synchronization pattern (SYNC) in the received bit stream and then the address and data bits etc may be relatively easily extracted.

•Because of errors, the sync’ pattern may not be found exactly.

53


• When synchronization is achieved, the EC bits which apply to the ADD (address) and DATA bits need to be carefully chosen in order to achieve a specified performance.

• Synchronization is required for Error control coding (ECC ) to be Applied.

• To clarify the synchronization and ECC requirements, it is necessary to understand the block error rates.

• For example, what is the probability of three errors in a 16 bit block if the b.e.r is p = 10-2?

54


Let N be number of bits in a block. Consider N=3 block.

• Probability of error = p , (denote by Good , G)• Probability that a bit is not in error = (1-p), denote by Error, E• An error free block, require ,G G G i.e, Good, Good and Good.

• Let R= the number of errors, in this case R=0. Hence we may write • Probability of error free block = Probability that R=0 or

P(R=0) = P(0) = P (Good, Good and Good)

55


• Since probability of good = (1-p) and probability are independent so,P(0)= p(G and G and G) = (1-p). (1-p). (1-p)= (1-p)3

P(0) = (1-p)3

For 1 error in any position

Probability of one error P(R=1) = P(1)

)(Pr

)(Pr

)(Pr

EandGandGobEGG

or

GandEandGobGEG

or

GandGandEobGGE

P(1) = p(1-p) (1-p) + (1-p) p (1-p) + (1-p) (1-p) p

P(1) = 3 p (1-p)2

56


For 2 errors in combination

Probability of one error P(R=2) = P(2)

)(Pr

)(Pr

)(Pr

EandEandGobEEG

or

EandGandEobEGE

or

GandEandEobGEE

P(2) = p p (1-p) + p (1-p) p + (1-p) p pP(2) = 3 p2 (1-p)

For 3 errors

)(Pr EandEandEobEEE

P(3) = p p p = p3

57


In general, it may be shown that

The probability of R errors in an N bit block subject to a bit error rate p is

RNRR

N ppCRp )1()(

orRNC !)!(

!

RRN

N

R

N

Where

is the number of ways getting R errors in N bits

RNRR

N ppCRp )1()(

Prob. of (N-R) good bits Prob. of R bits in error No. of ways getting R errors in N bits

Prob. of R errors.

58

Probability of Error (Example 1) Probability of Error (Example 1)

An N=8 bit block is received with a bit error rate p=0.1. Determine the probability of an error free block, a block with 1 error, and the probability of a block with 2 or more errors.

Prob. Of error free block,

4304672.0)0(

)9.0()1.01()1()0(

)0()0(88080

08

p

ppCp

pRp

Prob. of 1 error,

3826375.0)1(

)1.01()1.0(8)1()1(

)1()1(8181

18

p

ppCp

pRp

59

Probability of Error (Example 1) Probability of Error (Example 1) Prob. of two or more errors = P(2) + P(3) + P(4)+ …….

P(8)i.e.

8

2

)(R

Rp

It would be tedious to work this out , but since

1868952.0))3826375.04304672.0(1()2(

))1()0((1)2(..

1)2()1()0(then1)(0

p

pppei

pppRpN

R

60

Probability of Error (Example 2) Probability of Error (Example 2)

A coin is tossed to give Heads or Tails. What is the probability of 5 heads in 5 throws?

Since the probability of head, say p = 0.5 and the probability of a tail, (1-p) is also 0.5 and N=5 then

Prob. of 5 heads

25

55

5555

5

10125.3)5.0()5(

)5.0()1()5(

p

CppCp N

Similarly the probability of 3 heads in 5 throws (3 in any sequence) is

3125.0)3(

)5.0()5.0()1()3( 233

53533

5

p

CppCp

61

Synchronization Synchronization

One method of synchronization is to compare the received bits with a ‘SYNC’ pattern at the receiver decoder.

In general sense, synchronization will be •successful if the sync bits are received error free, enabling an exact match•lost if one or more errors occurs.

62

Synchronization (Cont’d) Synchronization (Cont’d)

Let S denote the number of sync bits. To illustrate let S=4 bits and let the sync pattern be 0 1 1 0

The probability of successful sync Ssucc ppP )1()0(

The probability of lost sync )0(1 pPlost

63

Error Detection and Correction Error Detection and Correction

Given that the synchronization has been successful, the message may be extracted as shown below.

Probability of successful transfer =

N

R

Rp0

)(

64

Error Detection and Correction (Cont’d)Error Detection and Correction (Cont’d)

A message, after synchronization contains N=16 bits, with a b.e.r, p= 10-2 . If the ECC can correct 1 error determine the probability of successful message transfer.

1

0

15

11611

16

16

989067.0)1()0()(

137609.0)01.01()01.0(16)1(

)1()1()1(

851458.0)01.01()1()0(

Rsucc

RNRR

N

N

ppRpp

p

ppCppCp

pp

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1 School of Electrical, Electronics and Computer Engineering University of Newcastle-upon-Tyne Noise in Communication Systems Noise in Communication Systems