overview of radio links - university of california, santa ... · title: title of talk author: mark...

Copyright Mark Rodwell, 2016

Overview of Radio Links

Mark Rodwell, University of California, Santa Barbara

ece145c lecture notes


Radio Waves,

Propagation,

and Antennas


The Radio Spectrum

109

1010

1011

1012

1013

1014

1015

Frequency (Hz)

microwave

SHF*

3-30 GHz

10-1 cm

mm-wave

EHF*

30-300 GHz

10-1 mm

optic

al

385-7

90 T

Hz

sub-mm-wave

THF*

0.3-3THz

1-0.1 mm

far-IR: 0.3-6 THz

near-IR

100

-385

TH

z

3-0

.78

m

mid-IR

6-100 THz

50-3 m

104

105

106

107

108

109

1010

Frequency (Hz)

MF*

0.3-3 MHz

1000-100 m

HF*

3-30 MHz

100-10 m

VHF*

30-300 MHz

10-1 m

UHF*

0.3-3 GHz

1 m-10 cm

microwave

SHF*

3-30 GHz

10-1 cm

10-1

100

101

102

103

104

Wavelength (meters)

10-6

10-5

10-4

10-3

10-2

10-1

Wavelength (meters)

*ITU band designations ** IR bands as per ISO 20473


Frequencies of a Few Services (Rough #s)

109

1010

1011

1012

1013

1014

1015

Frequency (Hz)

radar sate

llite T

V

sate

llite lin

ks

wire

less H

DM

I

imag

ing

104

105

106

107

108

109

1010

Frequency (Hz)

AM

rad

io

FM

rad

io

TV

TV

cell phones

WiFi

10-1

100

101

102

103

104

Wavelength (meters)

10-6

10-5

10-4

10-3

10-2

10-1

Wavelength (meters)


Choice of Transmission Frequency

earth theof Curvature

beam. blocking of Ease

size. antenna Required

bad. and weather good :nattenuatio cAtmospheri

.allocationfrequency Government


Atmospheric Attenuation in Bad Weather

very heavy fog

Liebe, Manabe, Hufford, IEEE Trans Antennas and Propagation, Dec. 1989

propagation in bad weather favors use of frequencies below ~10 GHz

rain

heavy rain

tropical deluge

Olsen, Rogers, Hodge, IEEE Trans Antennas &

Propagation Mar 1978

Hurricane


Curvature of the earth, reflection from the ionosphere

earth. theof curvature by the limited

ision transmissradio range-Long

ion transmissrange-long enables

ionosphere thefrom Reflection

ionosphere in the MHz 10order of

frequency, resonance plasmaelectron the

below sfrequencie requires This

(Marconi) carriersfrequency -lowor

satellites requires radio nentalInterconti


Fresnel Zone: Where the Radio Beam Carriers The Signal

Zone.Fresnel (first) thecalled is This

.2/ areaan in carried is antennasbetween beam The

2/

2/2/2/

)()2/1()2/1(

)(/1/1)(

and assume usLet

.4/)(

.4/ than less is lengthspath in difference if phase-in (almost) add Paths

1

2

1

1

2

2

2

1

2

21

2

2

2

2

2

1

2

1

21

2

2

2

2

2

1

2

121

22

1

22

1

121

21

22

1

22

1

RHA

RH

RHRHRH

RRRHRRHR

RRRHRRHRRRHRHR

RHRR

RRHRHR


Fresnel Zone: Where the Radio Beam Carries The Signal

blocked. be likely to less are

signals frequency)-(lowerelength Longer wav

beam. in thepower theofmost block will

.2/ area hence 2/ diameter ofobject an

, and assuming Still

11

121

RARH

RHRR


Nearly Isotropic Antennas

/2~ islenght y when effectivel radiates :antenna Dipole

antenna" wave-quarter"

plane. ground wideusingby

length in 1:2 reduced becan Antenna

plane lin vertica )cos(~ as variesand

plane, lin vertica isotropic ispattern radiation

http://en.wikipedia.org/wiki/File:Elem-doub-rad-pat-pers.jpg


Directional Antennas

Uda-Yagi

http://en.wikipedia.org/wiki/File:Two_meter_yagi.jpg http://en.wikipedia.org/wiki/Log-periodic_antenna

http://en.wikipedia.org/wiki/File:SuperDISH121.jpg

periodic-log

reflector

parabolic

patternradiation

ofplot Typical

http://en.wikipedia.org/wiki/Radiation_pattern


Antenna Directivity

antenna isotropic ofintensity radiated

radiatedintensity peak ydirectivit

http://en.wikipedia.org/wiki/Radiation_pattern

steradiansin angle) (solidbeamwidth angular

4

steradiansin angle) (solidbeamwidth angular

sphere a of angle solidydirectivit

1.64. ofy directivit a have antennas dipole wave-half Simple


Received

Signal Strenght


Friis Transmission Formula

Rrt

dtransmitte

received eR

DD

P

P

16 2

2

2

nattenuatio catmospheri

rangeion transmiss

area aperture effective antenna

ydirectivit antenna

R

A

D

n...propagatiomultipath

path beam in the objects

planet theof curvature

terrainfrom scattering

:factorsmany Ignores

2

4

AD

? from come ipsrelationsh thesedo Where


Plane Wave Incident on a Barrier

?pattern field-far theis....what

:aperturean th barrier wi aon incident wavePlane


Angular beamwidth of an aperture

)/(sinbeamin Null

half.lower thefrom phase-of-out180 radiate will

H,height of aperture, theof half top then the,sin/ If

1

o

H

H

null

H

H

H

dB

null

null

/beamwidth power -halfat guessRough

,/22 nullsat Beamwidth

/ ,sin :ionapproximat angle Small

3


Directivity vs Area

H

H

verticaldB /beamwidth power -half verticalthen the

isheight aperture theif that determinedjust veWe'

,3

W

W

horizontaldB /beamwidth power -half *horizontal* then the

is *width* aperture theif Similarly,

,3

2

,3,3dilluminate areaan

rectangle) (red illuminate willaperture the, distance aAt

RA

R

verticaldBhorizontaldB


Directivity vs Area

area aperature theis where/4/4 :So

/ ,/ But,

radians)in (angles /4

/ydirectivit Antenna

4 : radius of sphere a of Area

22

,3,3

,3,3

dilluminate

2

AAWHD

WH

D

AAD

RAR

horizontaldBverticaldB


sphere

sphere


Transmit vs receive antenna

HW

AD

DAWH

DAWH


horizontaldBhorizontaldB

RRRR

tTTT

/ ,/

radians)in (angles /4/4

:eitherFor

y directivit , area , width ,height :apertureReceiver

y directivit , area , width ,height :aperturer Transmitte

,3,3

,3,3

2


Received power→ Friis Transmission Formula

2

2

222

22

22

16:So

/4 and /4But

4/4area dilluminate

areareceiver

antenna receiving of area :rectance Blue

tterby transmi dilluminate , radiusat area, :rectangle Red

R

DD

R

AA

P

P

ADAD

R

DA

DR

A

P

P

R

rttR

trasmitted

received

rrtt

tR

t

R

trasmitted

received


To summarize

2

2

222 16area dilluminate

areareceiver

R

DD

R

AA

P

P rttR

trasmitted

received

HW

D

AD




/ ,/

radians)in (anglesdegrees 000,41

radians)in (angles 44

,3,3

,3,3

2

,3,3

2

blockage beam

loss scatteringterrain

loss catmospheri

:Ignores


Phased Arrays


An array of antennas:

.by size overall

: themofarray large a Make

.by size

:antenna small a Take

WH

wh

*antennaarray *an is This

receive.on or t,on transmi

phasein themDrive

HverticaldB /beamwidth power -half Vertical ,3

WhorizontaldB /beamwidth power -half Horizontal ,3


An array of antennas:

examplean of Picture

steering *mechanical*by aimed

isarray that theNote

possible. also is

ngbeamsteeri *Electronic*


Beamwidth of antenna array: element phases

./2shift phase relative Electrical

sin)/( differencedelay Time

sin differencelength Path

. separationelement Vertical

. angle Physical

:phase-in antennas theDrive

l

cD

Dl

D

v

v

v

hH /not ,/ isbeamwidth angular why is This

phase.-in addnot do signals ,0 If


Electronic Beamsteering, a.k.a. phased array

sin differencelength Path

./2shift phase relative Electrical

. angle physical

at phase intoback signals bring

:shifters-phase Add

vDl

l

array phasedReceiver


Why phased arrays ?

immunitymutipath and ceinterferen

signal received strong

beamwidth narrow

y,directivithigh

aperture, Large

ng)beamsteeri c(electroni

beam. narrow aimly mechanical

toneed no But,

https://en.wikipedia.org/wiki/Phased_array


Antenna & array basics

heightelement steering vertical

(radians) dthelement wi

steering horizontal

.range ngbeamsteeri maximum setselement Individual

2

areaarray 4ty)(directiviGain

heightarray beamwidth vertical

(radians) harray widt

beamwidth horizontal

gain and beamwidth setsarray Overall

derive.-remust Again, .beamdwidth increasing ,diminishes areaarray projected angle, steering largeAt

derive.-remust wespaces, are thereif other;each touch elementsarray that assume Formulas


Beamsteering electronics: Architectures

shifting-phase RF

shifting-phase LO


Beamsteering electronics: Architectures

multiple independent beamseach carrying different dataeach independently aimed# beams = # array elements

Hardware: In effect, a separate phased array for each signalSignal processing: matrix operations at baseband


MultiPath Propagaion


Multipath Propagation

n"propagatiopath -multi" :paths signalMany

pattern. beam antennain objectsMany

antennas)y directivit-(lowbeamwidth angular largeGiven

reflection ofStrenght

antennas ofy Directivit

strenght signaldifferent haspath Each

shift. phase possible :conditionboundary surface Reflecting

delay.different length,different haspath Each


Multipath Propagation: Delay Spread

2

2

1

2

2

2

1

2

2

2

2

22

1

2

1

22

2

22

1

21

22

1spreadDelay

2221

21

ion.approximat angle-small use angles, small :, Large

above. ,Pythagorus use angles, wide:, Low

:path (NLOS) LOS-Non

:path (LOS)sight of Line

R

H

R

H

c

R

H

R

HR

R

HR

R

HRD

DD

DD

HRHRD

RRR

rt

rt


Fading vs Intersymbol interference

periods symbollonger :ODFM useor

receiverin equalizer adapitive need

another with interfers periodbit One

ceinterferen lIntersymboperiod) Symbol spread(Delay

separation eappropriatat antennas receiving two:fix

signalery weak possibly vceinterferenphase ofout are Carriers

aligned~ periods symbol with arrive signals NLOS and LOS

Fadingperiod) Symbol spread(Delay


Signals to be

Transmitted


Signals we might want to transmit by radio

kb/s 320-32 :audio compressed MP3

Mb/s 100 ~ :WiFi

Mb/s 15about : HDTV Compressed

Gb/sec 1.5about :HDTV edUncompress

20kHz-20Hz :c)(optimisti hearinghuman of Range

kHz 4-0.4 :ion transmisssoundquality -Voice

waves.radioby them transmit toare weif

frequency in d translatebemust signals These


Generation of Digital Data Streams

.resolution bits of # some with digitized bemust then Samples

n.compressioby reduced becan rates data

,redundancy contains stream digital theIf

on theory)(informati

MP3,... MPG, JPG,


Recall the Sampling Theorem

.2 rate aat sampled bemust ),( to

limitedbandwidth of signal a aliasing, spectral avoid To

. sigsamplesigsig ffff

image shifted by +fsampleimage shifted by -fsample


Bandlimiting of digital data streams (1)

bandwidth signal thereduces and waveform

therounds signal digital a Filtering

spectrumflat a has impules ofA train

)2//()2/sin( spectrum

has train pulser rectangulaA

bitbit



),2/1,2/1(bandwidth

tofiltered wall-brick is stream data theIf

bitbit ff

)./()//sin( shape a has pulse dataeach then bitbit tt

stream. data in the bits sucessivebetween

(ISI) ceinterferen lintersymbo zero is thereso

.,etc,4,3,2for zero

is )/()//sin(function The

bitbitbit

bitbit

t

tt



draw. tohard is This pulses. theseof

sum a is waveformdigital Overall

like. looks waveform what theis Here

sequences. data possible allfor

repeatedly draw is Trajectory

pattern. eyean is This



zero. be willceinterferen lintersymbo then the

point, indicated about theration degree 180 a with symmetric is spectrum theIf

rate.bit the(1/2) -/an greater thsomewhat bandwidth required typical

broader, are filters Typical

rate.bit the(1/2) -/ the toequal isbandwidth totalminimum

)2/1,2/1( isbandwidth Minimum

bitbit

: theorem)sampling his (NOT theoremsideband vestigalsNyquist'


Pulses with Raised Cosine Spectra : From Wikipedia.

pulse of TransformFourier

systemsion transmissreal ingapproximatin used are These signals. ISI-zero offamily a of

onidealizati almathematic a is waveformpulse cosine- Raised The

http://en.wikipedia.org/wiki/Raised-cosine_filter

pulse of waveformTime

1.for , ),( i.e. )/1,/1( to

, 0for , )2/,2/( i.e. , )2/1,2/1( from vary will

signal digital baseband theof rangefrequency the1, to0 from varies As

bitbitbitbit

bitbitbitbit

ff

ff


Pulses with Raised Cosine Spectra: Filters

)2/sin(/2)/()(function transfer havemust (#2)filter channel

thespectrum, /2)/()2/sin( a hasalready pulser rectangula a Since

.)( spectrum cosine-raised thehavemust which *pulse filtered* theisIt

bitbit

bitbit

fH

fH

hardware. realin present not isfilter first theso

impulses,not pulses,r rectangula produce circuits digital real course, Of


Root Raised Cosine Spectra in Transmission Links

noise.receiver limit toas soreceiver thebandlimit further must We

band.frequency allowed n thestay withi er to transmittthebandlimit must welink, radio aIn

filters. *cosine-raised-root* using edaccomplish is This

criterion.Nyquist meet must ISI; zero have alsomust signal received final The


Root Raised Cosine Spectra in Transmission Links

)2/sin(/)2/()()(/)()(

:is responsefilter er transmitt thepulses,r rectangulaby driven isit Given that

)()(

:responsefrequency cosine-raised-root have alsomust then filter receive The

)()(

:signal ed transmittfor the spectrum *cosine-raised-root* a Choose

)()(

:function cosine-raised a is )(

2

3

bitbitRCrectTF

RCF

RCT

RCfiltered

filtered

jHjHjHjH

jHjH

jHjH

jHjH

jH


Modulation and

Frequency Conversion


Frequency Conversion / Modulation

ation).(multiplic mixingby done isit ;modulation called is This

carrier.frequency -high on the signalour imposemust we this,do To

105

106

107

108

109

1010

1011

Frequency (Hz)

).-(DC to)2/-(DCbetween spectrum baseband a has stream data the,on Depending bitbit ff

GHz. 100 toMHz few perhapsbetween sfrequencie radio using signal the transmit We


Mixing = Multiplication

)cos()( tVtV BIB

)cos()( tVtV LOLOLO

0/)cos()cos()( VtVtVtV LOLOBIRF

tt

eeeeeeee

eeeeeeee

zzzzzzzz

zzzztt

zzeet

zzeet

BLOBLO

tjtjtjtjtjtjtjtj

tjtjtjtjtjtjtjtj

LOBLOBBLOLOB

LOLOBBLOB

LOLO

tjtj

LO

BB

tjtj

B

LOBBLOLOBLOB

LOBLOBBLOLOB

LOLO

BB

)(cos2)(cos2

)()(

))(()cos()cos(4

so

)cos(2

)cos(2

11111111

1111

11

11

y.efficientl functions trig. work withlearn tomust We

)sin(2 and )cos(2)sin()cos( jjjjj eejeeje


Mixing = Multiplication

)cos()( tVtV BIB

)cos()( tVtV LOLOLO

sfrequencie difference and sum

generatestion Multiplica

tVVVtVVV

VtVtVtV

BLORFIBLORFI

LORFBIRF

)(cos)2/()(cos)2/(

/)cos()cos()(

00

0

)sin()( tVtV BQB

)cos()( tVtV LOLOLO

tVVVtVVV

VtVtVtV

BLOLOQBLOLOQ

LOLOBQRF

)(sin)2/()(sin)2/(

/)cos()sin()(

00

0

n...calculatiosimilar aBy


Mixing → Convolution of Spectra

)(tVB

)(tVLO

0/)()()( VtVtVtV LObRF

)'()'()'(2

1

2

)( and )( ofn convolutio

2

)(*)()(

0

00

jdjjVjVV

V

jVjV

V

jVjVjV

LOb

LObLObRF

)()()()()( So

)(2 that Recall

)2/()2/()cos()( Suppose

LOLOLOLOLO

LO

tj

tj

LO

tj

LOLOLOLO

VVjV

e

eVeVtVtV

LO

LOLO

Spectrum of the Local Oscillator


Mixing with cosine ("I") local oscillator

)(tVB

spectrum oscillator Local

)( jVRF

parts)imaginary real, (note

spectrum signal Baseband

)cos()( Suppose tVtV LOLOLO

spectrum signal RF


Mixing with sine ("Q") local oscillator

)(tVB

spectrum oscillator Local

)( jVRF

parts)imaginary real, (note

spectrum signal Baseband

)sin()( Suppose tVtV LOLOLO

)()()()()( LOLOLOLOLO VjVjjV

sidebands) theof phase flipped the(note

spectrum signal RF


IQ Modulation

)(, tV IB

)(, tV QB

)sin()()cos()(

)sin()(

)cos()(

)(0

,

0

,

ttVttV

tV

VtVt

V

VtVtV

LOQLOI

LO

LOQB

LO

LOIB

RF

)cos( tV LOLO

)sin( tV LOLO

streams data edsynchroniz t,independen are )( and )( ,, tVtV QBIB


IQ Signal Representation

)sin()()cos()()()(Re)( ttVttVetjVtVtV LOQLOI

tj

QIRFLO

plane ain point a as drepresente is Signal


IQ Signal Representation

)sin()()cos()()( ttVttVtV LOQLOIRF

wavecosine wavesine

)cos( tLO)cos( ttLO

frequency angular

at clockwise-counter spins


Quadrature Phase Shift Keying (QPSK)

)(, tV IB

)(, tV QB

)sin()()cos()(

)sin()(

)cos()(

)(0

,

0

,

ttVttV

tV

VtVt

V

VtVtV

LOQLOI

LO

LOQB

LO

LOIB

RF

)cos( tV LOLO

)sin( tV LOLO

sequencesbinary are )( and )( ,, tVtV QBIB

QAM-4 as often) (lessknown Also


16-Quadrature Amplitude Modulation (16-QAM)

etc. QAM,-256 QAM,-64 also is There

)cos( tV LOLO

)sin( tV LOLO


Demodulation

)cos( tV LOLO )cos( tV LOLO

)sin( tV LOLO )sin( tV LOLO

)(, tV IB

)(, tV qb

)(tVri

)(tVrq)(tVS

)sin()()cos()()( 0,0, tVVtVtVVtVtV LOLOQBLOLOIBS

)2sin()()2cos(2)(2)(

)cos()sin()()(cos)(

)cos()()(

2

0,

2

0

2

,

2

0

2

,

2

0,

22

0,

0

tVVtVtVVtVVVtV

ttVVtVtVVtV

tVVtVtV

LOLOQBLOLOIBLOIB

LOLOLOQBLOLOIB

LOLOSri

LPFby removed

.2at term2)()( Similarly, LO

2

0

2

, VVtVtV LOQBrq


Receiver Sensitivity


Computing Receiver Sensitivity (1)

ed transmittbe to waveformssymbol thedefine First,

many are There ies.possibilit of couple ajust are These


Computing Receiver Sensitivity (2)

themlabel Lets examples. as theseUse

)(1,1 t

)(1,2 t

)(1,3 t

etc. #2.slot in time waveformspossiblen offirst theis )(1,2 t

etc QAM,in )( and )( :Example

periodeach in waveformsignal one than more have topossible isIt

tQtI


Energy Per Signal

)(1,1 tsym is period symbol :Correction

JoulesOhms

secondsVolts)(

1

Symbolper Energy

22

1,1

0

dttZ

Esymbol

math.in out cancel Willcorrect. physically benot Need

impedance. reference theis 0Z

slot per time waveformone have weassuming

/

power Signal

BEEP bitsymbolbitsignal


Normalization: Unit Energy Signals

)(1,1 tsignals power)unit (not *energyunit * theseMake

waveformssymbol the*all*for that Do

Joule. 1.0 isenergy symbol the

that so )( of amplitude thescaled have We

Joules0.1)(1

Symbolper Energy

1,1

2

1,1

0

t

dttZ

Esymbol


Non-interfering waveforms

period a with overlap Zeroetc. 0)()(1

periodsbetween overlap Zeroetc. 0)()(1

: want Weinterfere.not should signals base These

2,11,1

0

1,21,1

0

dtttZ

dtttZ

)(1,1 t

)(1,2 t

)(1,3 t


Easier Notation: dot products

algebra. vector likeJust products.inner are These

on so and )()(1

)()(

)()(1

)()(

:notationcompact more a is Here

2,11,1

0

2,11,1

1,21,1

0

1,21,1

dtttZ

tt

dtttZ

tt


Our Set of Unit-Energy singals:

)(1,1 t

)(1,2 t

)(1,3 t

0waveformdifferent ath product widot theTake

1. waveformsame th theproduct widot theTake

Joules 0etc.)()()()(

Joule 1 etc.)()( )()(

:notationcompact moreour Using

1,12,11,11,2

2,22,21,11,1

tttt

tttt

matrix) diagonalunit a (like

otherwise. 0

same, theare indices its if 1 is where

Joule 1 )()(

:compactly moreeven

ki,

lj,ki,lk,ji,

tt


A bit more notation and normalization:

5)9119(25.0

orm)bits/wavef (2 31/-1/3/ bemight

1(binary) 1-1/ bemight

Joule 1)()( because

period symbolper Energy signalTransmit

...)()()()(

oltageTransmit v

: two)has QAM :(careful

period symbolper waveform*one* transmit to were weIf

2

2

1,11,1

22

1,331,221,11

m

m

mm

ttmE

tmtmtmtv

tt

tttt


A bit more notation and normalization:

Joule 1)()( because

period symbolper Energy signal Received

two)has QAM :(careful

period symbolper waveformoneonly assumed veI'

...)()()()(

voltageReceived

1,11,1

22

1,331,221,11

ttmE

tmtmtmtv

rr

rrrr


Optimum Receiver (orthogonal signals, white noise)

QAMfor steps in two thisshows structure bottom the

directly, thissillustrate structure topThe

)( various with the),( multipliessimply receiver optimuman but

yet, thisprovenot usLet

ji, ttvr


Received Signal: no noise

)(1,1 t

)(1,2 t

noise. of absence in the is This

)()(

Joule 1 )()( because )()(

:smultiplier ofOutput

...)()()()()(

noise) (no Signal Received

2,11,21

lk,j,i,lk,ji,1,11,11

2,22,21,22,12,11,21,11,1

mttvQ

ttmttvI

tmtmtmtmtv

rr

rr

rrrrr

period symbolper waveformsbasis *2* assuming now are we:Note


Received Signal: with noise

)(1,1 t

)(1,2 t

ji

rr

rr

rrrrr

n

nmttvQ

nmttvI

tmtmtmtmtntv

,

2,12,11,21

1,11,11,11

2,22,21,22,12,11,21,11,1

of scovariance and variances theknow toneed We

)()(

)()(


...)()()()()()(

noise)(with Signal Received


Noise propagation through the receiver

)(1,1 t

)(1,2 t

lkji nnEnnE

ttnn

ttnn

ttnn

,,2,11,1

2,11,2

1,22,1

1,11,1

generally moreor , know toneed We

)()( :2 timeslot 1, waveformbasis

,)()( :1 timeslot 2, waveformbasis

)()( :1 timeslot 1, waveformbasis

:(s) correlator ofoutput Noise



pair. ansformFourier tr a are )2( and )( because

),()( So,

figure. noise system theis re whe

Hz)/(V )2( ofdensity spectralpower a has )(ut B

)( ofation autocorrel theis )( where

)()()(

)()()]()([

)()()()(

)()()()(

)()()()(

:(s) correlator of outputs noise of nsCorrelatio

0

2

0

,

2

0

,

2

0

,

2

0

,

1

0

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Tabulated values of the Q-function

Q(0.0) = 0.500000000Q(0.1) = 0.460172163Q(0.2) = 0.420740291Q(0.3) = 0.382088578Q(0.4) = 0.344578258Q(0.5) = 0.308537539Q(0.6) = 0.274253118Q(0.7) = 0.241963652Q(0.8) = 0.211855399Q(0.9) = 0.184060125

Q(1.0) = 0.158655254Q(1.1) = 0.135666061Q(1.2) = 0.115069670Q(1.3) = 0.096800485Q(1.4) = 0.080756659Q(1.5) = 0.066807201Q(1.6) = 0.054799292Q(1.7) = 0.044565463Q(1.8) = 0.035930319Q(1.9) = 0.028716560

Q(2.0) = 0.022750132Q(2.1) = 0.017864421Q(2.2) = 0.013903448Q(2.3) = 0.010724110Q(2.4) = 0.008197536Q(2.5) = 0.006209665Q(2.6) = 0.004661188Q(2.7) = 0.003466974Q(2.8) = 0.002555130Q(2.9) = 0.001865813

Q(3.0) = 0.001349898Q(3.1) = 0.000967603Q(3.2) = 0.000687138Q(3.3) = 0.000483424Q(3.4) = 0.000336929Q(3.5) = 0.000232629Q(3.6) = 0.000159109Q(3.7) = 0.000107800Q(3.8) = 0.000072348Q(3.9) = 0.000048096Q(4.0) = 0.000031671

Some values of the Q-functionare given below for reference.

http://en.wikipedia.org/wiki/Q-function

http://en.wikipedia.org/wiki/Q-function


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