communications formulas and concepts v3.pdf

8/20/2019 COMMUNICATIONS Formulas and Concepts V3.pdf

1/24

AMPLITUDE MODULATION

• AM Wavet

V t

V t V t V mc

mmc

mcc )cos(2

)cos(2

sin)( ωωωωω +−−+=where: Vc = maximum voltage of the carrier signal

Vm = maximum voltage of the original

modulating signalc = 2! f c = frequency of the carrier signalm = 2! f m = frequency of the modulating

signal

minmax

minmax

V V V V

V V

mc

m

+−==

where: m = modulation indexVmax = maximum peak-to-peak voltage swingof AM waveVmin = minimum peak-to-peak voltage swingof AM wave

AM wave equation in terms of modulation index

t mV

t mV

t V t V mcc

mcc

cc )cos(2)cos(

2sin)( ωωωωω +−−+=

• AM Bandwidth

m f BW 2= where: BW = bandwidth

f m = modulating signal frequency

• AM Power and Current

RV

RV

RV

P USB LSBcarr t 222

++= R

V P cc 2

2

=

488

2222 m P

RV

RV m

P P cmc

USB LSB ====

21

2m P P

c

t += 2

122 m

I I

c

t +=

where: Pt = total transmitted powerPc = unmodulated carrier powerIt = total transmitted currentIc = unmodulated carrier currentm = modulation index

Note: The voltage should be in rms

Amplitude Modulation with Multiple Signals

+=

21

2t

ct

m P P

......2322

21 mmmmt ++=

where: mt = total modulation indexm1, m2, m3 = modulation index of signalhaving index 1, 2, 3 respectively

Power Savingsa. Single Sideband (SSB)

t

C USB LSB

P P P PS += /

b. Single Sideband full carrier (SSBFC)

t

USB LSB

P P

PS /=

c. Two independent Sidebands

t

C

P P

PS =

• Tuned Radio-Frequency (TRF)AM ReceiverTRF Design formulas

LC f r

π21=

BW f

Q r = where: Q = quality factor

f r = frequencyBW = Bandwidth

• Superheterodyne Receiveri s si f f f 2+=

where: f si = image frequencyf s = signal frequencyf i = intermediate frequency

221 ρα Q+=

image

RF

RF

image

si

s

s

si

f f

f

f

f f

f f −=−=ρ

where:" = image-frequency rejection ratio(IFRR)Q = quality factor of the circuit

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2/24

TELEVISION

Details of Horizontal BlankingPeriod Time, sec

Total line (H) 63.5H blanking 0.15H-0.18H or 9.5-11.5H sync pulse 0.08H, or 4.75 ± 0.5Front porch 0.02H, or 1.27Back porch 0.06H, 3.81Visible line time 52-54

Details of Vertical BlankingPeriod Time

Total field (V) 1/60s 0.0167sV blanking 0.05V-0.08V or 9.5-11.5Each V sync pulse 27.35#sTotal of 6 V sync pulse 3H = 190.5#sEach equalizing pulse 0.04H = 2.54#s

Each serration 0.07H = 4.4#sVisible field time 0.92V-0.95V, 0.015-0.016s

Picture Information Encoding

BG RQ

BG R I

BG RY

31.052.021.032.028.060.011.059.030.0

+−=−−=++=

Relative amplitude for the AM RF picture signalTip of sync = 100%Blanking level = 75%Black setup = 67.5%Maximum white = 10 to 15% or 12.5% (typical)

NAVIGATIONAL AIDS

Directional Gain

θφπ4=dir G L

λθ =

where:$ = horizontal beam-width (radians)% = the wavelength of the radar

L = the dimension of the antenna in thedirection of interest (i.e. width or height)φ = vertical beam-width (radians)

RADAR Pulse (Waveform) RT PW PRT +=

PRT PRF

1= PRT PW

DR =

DR P P PEAK AV ×= PW PRT P

P AV PEAK = where: PRT = Pulse Repetition Time

PW = Pulse Width (#s)RT = Rest Time (#s)PRF = Pulse Repetition FrequencyDR = Duty Cycle or Duty RatioPAV = Average PowerPPEAK = Peak Power

Maximum unambiguous range

2 PRT

c Runamb =

Minimum displayed range

2min PW

c R = where: c = speed of light (3×108 m/s)

Radar Range

4

min2)4( R

O P T

P SA A P

Rπ

=

2

4λπ O

P

A A =

4

min2

2

4 ROT

P S A P

Rπλ

=

where: R = Radar Range

PT = Transmitted PowerAP = antenna gainS = cross-sectional area of the targetA0 = captured area of an antennaPRmin = detected signal level in W

Doppler Effect

λθcos2v

F D = where: FD = frequency change between transmitter

and reflected signal

v = relative velocity between RADAR andtarget% = wavelength of the transmitted wave$ = angle between target direction andRADAR system




3/24

TRANSMISSION LINES

• Electrical CharacteristicsCharacteristic Impedance

Y Z

Z O = & where: Z = R +j L &/m

Y = G +j C S/m

G R

Z O = at low frequency,&

C L

Z O = at high frequency,&

Also,

OC SC O Z Z Z = & where: ZSC = short circuit impedance

ZOC = open circuit impedance

For Parallel-wire line:

d D

L2ln

πµ=

m H

d D

C 2ln

πε= m F

where: L = InductanceC = CapacitanceD = Separation between center to center

d = diameter of the wire

Alternate formulas:310016.1 −×= r O Z L ε #H/ft

310016.1 −×=O

r

Z C

#F/ft

Characteristic Impedance, Z 0

C L

Z O =

d D

Z r

O

2ln120ε

=

d D

Z r

O

2log276ε

=

Note: 150& ' Z0 ' 600&

Resistance, R

a

f R 81034.8 −×=

mΩ

d

f R

5=

ft −Ω

100

where: a = radius (m)f = frequency (MHz)d = diameter (inches)

For coaxial line:

d D

L ln2πµ=

m H

d D

C ln

2πε= m F

where: D = diameter of the outer conductor

d = diameter of the inner conductorAlternate formulas:

310016.1 −×= r O Z L ε #H/ft310016.1 −×=

O

r

Z C

#F/ft

Characteristic Impedance, Z 0

d D

Z r

O ln60ε

=

d D

Z r

O log138

ε=

Note: 40& ' Z0 ' 150&

Resistance, R

+×= −

d D f R

111034.8 8mΩ

where: D = diameter of the outer conductor (m)d = diameter of the inner conductor (m)

f = frequency (MHz)

+=

d D f R

111.0 ft −

Ω100

where: D = diameter of the outer conductor (inched = diameter of the inner conductor (inchesf = frequency (MHz)


air = 1Teflon = 2.1Polyethelene = 2.25Polysterene = 2.7Mylar = 3.1

Porcelain = 6

Mica = 6Paper = 7

Nylon = 8Silicon = 11.68Water = 80 @20 ‘C

Er:

Open Line :



4/24

Complex Propagation constant, ! ZY j =+= βαγ

where:" = attenuation constant or coefficient(Nepers/length)( = phase constant or coefficient(Radians/length)

= O Z R343.4α dB/length

λπωωβ 2===

P V LC radians/length

LC V P

1= m/s

where: V p = propagation velocity

• Loading Conditions Note: The zero reference is at the load not on thegenerator.

1. Z L = Z 0 (match load)with ZL = Z0 then Zin = Z0

LS R e I I

γ −= L RS e I I γ = L

S R eV V γ −= L RS eV V γ =

LS R e P P

γ 2−= L RS e P P γ 2=where: IR , IS, VR , VS = receiving and sending end

current and voltages respectivelyPR , PS = power at the receiving and sendingend) = complex propagation constantL = length of the transmission loneZin = input impedanceZL = load impedance

2. Z L " Z 0 (Mismatch)

L

Oin Z

Z Z

2

= for % /4 line

++=

L Z Z

L Z Z Z Z

LO

O L

Oin γ

γ

tanh

tanh for L >% /4

where: Zin = the equivalent impedance representingthe entire line terminated by the load

Load boundary characteristicsd jd j eV eV d V ββ −−+ +=)(

( )d jd jO

eV eV Z

d I ββ −−+ −= 1)(

Loss-less transmission line

d jd j eV eV d V γ γ −−+ +=)(

( )d jd jO

eV eV Z

d I γ γ −−+ −= 1)(

Loss-less transmission linewhere: V(d) = line voltage at point d

I(d) = line current at point dZ0 = characteristic impedance of the line

V+

= incident voltageV – = reflected voltage) = complex propagation constant for lossyline( = complex propagation constant for loss-less lined = distance from the load

Four Cases (loss-less transmission line)1. Z L # 0 (short circuit)

)sin(2)( d jV d V β+=

0

)cos(2)( Z

d V d I β+=

)tan()()()( 0 d jZ d I

d V d Z β==

1−=Γ R

2. Z L # $ (open circuit))cos(2)( d V d V β+=

0

)sin(2)( Z

d jV d I

β+=

)cot()()()( 0 d jZ d I

d V d Z β−==

1=Γ R

3. Z L = Z 0 (matched load)d jeV d V β+=)(

0

)( Z

eV d I

d jβ+=

0)( Z d Z =0

0 L

=Γ R

4. Z L = jX (pure reactance)- Reactive impedance can be realiz

with transmission lines terminated ba short or by an open circuit.

)tan( jZ Z in β=




5/24

- Reflection coefficient has a unitarymagnitude, as in the case of short andopen circuit load.

Shorted Transmission Line – Fixed Frequency0= L 0=in Z Series

Resonance

40 λ

in Z Inductance

4λ= L ∞→in Z ParallelResonance

24λλ


6/24

D P ERP rad = where: G = power gain (unitless)

Pin = power delivered to the feedpoint

For an isotropic antenna: rad T P P = But for a unidirectional antenna: ERP P T =

2 I P R rad rad =

where: R rad = radiation resistancePrad = power radiated by the antennaI = current at the feedpoint

Radiation resistance for l not in excess of & /82

790

=λl

Rrad

rad ind P P P −= where: R rad = radiation resistancePrad = power radiated by the antenna

T

rad

R R=η

inrad P P η= DG η=

where:* = antenna efficiency (1 for lossless ant.)R rad = antenna radiation resistanceR T = antenna radiation resistance

= R rad and R d (ohmic resistance)D = directivity (maximum directive gain)

Q f

BW r =

Dλφ 70=

where: BW = bandwidthf r = antenna resonant frequency

Q = antenna quality factorφ = beamwidth

B

F

B

F FB P

P A

ξξlog20log10 ==

where: AFB = front-to-back ratio (dB)PF = power output in the most optimumdirectionPB = power output in the opposite direction

+F = field strength in the most optimumdirection+B = field strength in the opposite direction

r

P

r I L T e 30sin60 == θ

λπ

ξ

where:+F = magnitude of field strength

r = distanceLe = antenna lengthI = current amplitude$ = the angle of the axis of the wire and the point of maximum radiation

Isotropic AntennaGain over isotropic = 0 dBBeamwidth = 360º

• Types of Antenna

A. Dipole Antennaa. Half-wave dipoleGain over isotropic = 2.14 dBBeamwidth = 55º

b. Folded half-wave dipoleGain over isotropic = 5.64 dBBeamwidth = 45º

B. Beam Antennaa. Yagi-Uda Antenna

Gain over isotropic = 7.14 dBBeamwidth = 25º

b. Rhombic AntennaGain over isotropic = 5.14 dB

C. Loop AntennaGain over isotropic = 3.14 dBBeamwidth = 200º

BAN f k V )2( π=

where: V = voltage induced in a loop antennak = physical proportional factorB = field strength flux, V/mA = loop area, m2 N = number of turns

D. Antenna with parabolic reflector2

2

===λ

π Dk AkA

A

AG

iso

s

iso

eff




7/24

πλ4

2

=iso A 42 D

A sπ=

where: Aeff = effective aperture or antenna captureareaAiso = isotropic areak = illumination factorD = diameter of parabolic reflector

2

6

=λ

DG with k = 0.65

Parabolic dipole: 2

5λ= D

Horn Antenna (Pyramidal)

Elevation Pattern: –3dBh

beamwidth λ56=

Azimuth Pattern: –3dBw

beamwidth λ70=

2

5.7

=λ

DG

E. Helical Antenna

λλ NS D

G2

15

=

λλπφ NS D

52=

where: G = Power gainφ = beamwidthD = helix diameter N = number of turnsS = pitch between turns% = wavelengthL = center-line axis length, NS

Note: If pitch is not given S =% /4F. Log-Periodic AntennaDesign factor formulas:

3

4

2

3

1

2

l l

l l

l l

r === 3

4

2

3

1

2

d d

d d

d d

r ===

n H

c f

λ=

1λc

f L =

where:21

1λ=l = the length of the longest element

d1 = the distance between the longest elemeand the second elementr = design factor which is between 0.7 and0.98

Antenna HeightFor a straight vertical antenna with h' % /4

λπ

λππ

λ hh

he2sin

2sin=

where: he = effective heighth = actual height

Note: he the antenna effective height is ½ to- of theactual height.

FIBER OPTICS

• Nature of Lighthf E P =

where: E p = energy of a photon; Joules (J)h = Planck’s constant, 6.625×10-34 J-sf = frequency, Hz

frequency of red light = 4.4×1014 Hzfrequency of violet light = 7×1014 Hz

• Snell’s Law2211 sinsin θθ nn =

where: n1 = refractive index of material 1n2 = refractive index of material 2$1 = angle of incidence$2 = angle of refraction

Note: 1 Å = 10 –10 m1 micron = 10 –6 mnair = 1.0003, 1

vc

n = where: n = refractive index

c = speed of lightv = velocity of light at material withrefractive index of n

Note: Angle of incidence and refraction aremeasured from normal




8/24

1

2sinnn

C =θ

where:$C = critical angle

• Propagation of Light Through a Fiber$1 < $C 1 light is refracted$1 > $C 1 light is reflected

$1 = $C 1 reflected or refracted

C in n θθ cossin 1(max) = where:$in(max)= acceptance angle

= acceptance cone half angle

22

21 nn NA −=

Where: NA = Numerical Aperture

• Mode of Propagation2

21V N =

NAd

nnd

V OO λ

πλ

π =−= 2221

1

21

nnn −=∆

where: N = number of modesV = V numberd = diameter% = wavelength

NA = numerical Aperturen1 = refractive index of coren2 = refractive index of cladding2 = fractional index difference

• Optical Fiber System DesignMathematical AnalysisThe power budget is the basis of the design of anoptical fiber link.

Total gain – Total losses 0

Therefore(P t + P r ) – ( ! f + ! c + ! s + f m ) 0

Thus, L = P t – P r = ( ! f + ! c + ! s + f m )

where: Pt = transmitted powerPr = receiver sensitivity (minimum received power)" f = fiber attenuation

" c = connector attenuation" s = total splice lossesf m = fiber marginL = distance between repeaters

t B Z ∆=51

where: Z = system lengthB = maximum bit rate2 t = total fiber dispersion

RADIO WAVE PROPAGATION

• The Electromagnetic WaveVelocity of propagation

µε1= pV m/s

0µµµ r = 0εεε r = where:# = permeability of the medium (H/m)

3 = permittivity of the medium (F/m)

The Power Density

24 r G P

A ERP T T

π==℘ W/m2

The Electric Field Intensity or Strength

r

G P H T T

30==αξ V/m

where:" = characteristic impedance of free space,&H = rms value of magnetic field intensity ostrength (A/m)

The characteristic impedance of a medium

εµ

α = &

Characteristic impedance in free space67 1026.1104 −− ×=×= πµo H/m

129 10854.83610 −− ×==

πε o F/m

Ω=Ω=×= −−

377120

3610

1049

7

π

π

πα




9/24

The Attenuation of Power Density and ElectricField Intensity

1

2

2

1 log20log10)(r r

dB A =℘

℘=℘

1

2

2

1 log20log10)(r r

dB A ==ξξ

ξ

• The effects of environment to propagation ofradio waves

Refractive indices of different materialsH2O 1.33Glass 1.50

Quartz Crystal 1.54Glycerin 1.47Diamond 2.42

Snell’s Law

2

1

1

2

2

1

2

1sinsin k k V V nn === θθ

where:$2 = angle of refraction$1 = angle of incidenceV2 = refracted wave velocity in medium 2V1 = incident wave velocity in medium 1k 1 = dielectric constant of medium 1k 2 = dielectric constant of medium 2n1 = refractive index of medium 1n2 = refractive index of medium 2

k V cn p

==

where: n = refractive indexc = velocity of light in free spaceV p = velocity of light in a given medium

Resultant field strength between waves travelingin different (direct and reflected paths)

λδπξξ2

2sin2 d r = V/m

d hh ar at 2=δ

where:+d = direct radio wave field strength (V/m)4 = the geometrical length difference between the direct and reflected pathshat and har = the heights of transmitting andreceiving antenna above the reflecting planetangent to the effective earth

• The Propagation ModesThe Radio Frequency Spectrum

Band Name Frequency (MHz) PropagationVLF 0.01 – 0.03 Ground WavLF 0.03 – 0.3 Ground WavMF 0.3 – 3.0 Ground WavHF 3.0 – 30 Sky Wave

VHF 30 – 300 Space WaveUHF 300 – 3,000 Space WaveSHF 3,000 – 30,000 Space WaveEHF 30,000 – 300,000 Space Wave

A. The Ground (Surface) Wave MethodThe field strength at a distance ( ' )

r I ht

λα

ξ =

The signal receive at that distance if a receiving

antenna is in placer hV ξ=

where:" = characteristic impedance of free spaceht and hr = effective height of the transmittinand receiving antennasI = antenna currentr = distance from transmitting antenna

B. The IonosphereThe refractive index of the ionosphere

2

811sin

sin f

N n r

i

−== θθ

where: N = number of free electrons per m3

f = frequency of radio wave (Hz)

The Ionospheric LayersD Layer – average height 70 km, with an averathickness of 10 km.E Layer – existing at a height about 100 km, withthickness of 25 km.F 1 Layer – exists at a height 180 km, daytim

thickness is about 20 km.F 2 Layer – height ranges from 250 – 400 km daytime and at night it falls to a height of 300 kwhere it combines with F1 layer, approximatthickness at about 200 km.

The height of the ionospheric layer

θtan2d

h =




10/24

The critical frequency (f c)

max9cos N MUF f c == θ

The Maximum Usable Frequency (MUF)

θθ

seccos c

c f f

MUF ==

The Optimum Working Frequency (OWF) orFrequency of Optimum Transmission (FOT)

MUF FOT OWF 85.0==

C. The Space Wave PropagationThe Radio Horizon Distance

k d d

EC 21=

0kR Re =where: EC = Earth’s Curvature

R e = effective earth’s radiusR 0 = earth’s radius, 3960 mi , 6371 kmk = correction factor for relatively flat earthk = 4/3

The maximum line of sight distance betweentransmitter and receiver towers is given by

r t hhd d d 4421 +=+=where: ht and hr = in meters

d, d1 and d2 = in kilometers

r t hhd 22= +where: ht and hr = in feet

d = in miles

The correction factor (k)1005577.0 ]04665.01[ −−= s N ek

where: Ns = surface refractivity

D. Tropospheric Scatter Wave (Troposcatter)Propagation

Operates at the UHF band (between n 350 MHz to10 GHz (and used to link multi-channel telephonelinks). The common frequencies are 0.9 GHz, 2 GHzand 5 GHz.

NOISE

• Noise CalculationkTB N =

where: N = noise powerk = Boltzmann’s constantT = resistor temperatureB = bandwidth of the system

Note: 17 °C/290 K is the typical noise temperature

kTBRV n 4= in #Vwhere: Vn = noise voltage

R = resistance generating the noise

Series Resistors

...222321+++= nnnn V V V V T

Parallel Resistors

...222321+++= nnnn I I I I T

For a diode, the rms noise current BeI I Dn 2= typically in#A

where: e = charge of an electron (1.6×10-19 C)ID = direct diode currentB = bandwidth of the system

B I I e I o Dn )2(2= + where: I0 = negligible reverse saturated current

I. Addition of noise due to several sourcesT n kTBRV T 4=

II. Addition of noise due to several amplifiers incascade

'...'' 321 neq R R R R R ++++=

( ) ( ) ( ) ( ) ( )212122213

21

2

1 ......

−++++= nn

eq A A

R

A A

R

A

R R R

III. Signal-to-Noise Ratio

RS

dBS log10)( =


k = _________________ 1(1- (0.0466) e^0.005577N s Ns = surface refractivity



11/24

IV. Noise Factor (NF) or Noise Figure (F)

o

o

i

i

N S

N S

NF =

NF dB F log10)( =

For a noiseless receiver, NF = 1; F = 0 dB

V. Equivalent Noise Temperature (T e))1( −= NF T T oe

where: Te = equivalent noise temperatureTo = reference temperature = 290 K NF = noise factor

For a noiseless receiver, Te = 0 K

For attenuator elements)1( −= LT T pe

where: L = loss (absolute value)T p = physical temperature (K)

VI. Overall Noise Factor (Friis’ Formula)

132121

3

1

21 ...

1...11−

−++−+−+=n

n

GGGG NF

GG NF

G NF

NF NF

VII. Overall Noise Temperature

1321211 ......32

1−

++++=n

eeeee GGGG

T

GG

T

G

T T T n

• Information TheoryHartley Law

n BC 2log2= bpswhere: C = channel capacity

B = channel bandwidth (Hz)n = number of coding levels (2 for binary, 8for octal, 10 for decimal etc.)

Shannon-Hartley Law)/1(log2 N S BC += bps

)/1log(32.3 N S BC += bpswhere: S/N = signal-to-noise ratio (absolute value)

Note: For a practical telephone channel B = 3.1 kHz(300 – 3400 Hz).

Total information sentCt H = bits

Power required2

2

)1( −= n P P n

where: Pn = power required in the n-level code

P2 = power level required in the binary codn = number of levels in a code

• Noise Measurements UnitsdBrn (dB above reference noise)

W N

dBrn 12101log10 −×

= 90+= dBmdBrn

dBa (dB above adjusted noise)For a pure tone:

5.11101log10 −×=

N dBa

85+= dBmdBa

For F1A weighted:82+= dBmdBa

dBrnC (dB above reference noise, C-messageweighted)

90+= dBmdBrnC

pWp (picowatts, psophometrically weighted)12

2

10600

)hom( −×Ω= etricV psop

pWp

310log10 −= pWp

dBmp

Transmission level point

TLP S S

dBTLP 0

log10)( =

0dBmdBmdB S S TLP −= dBdBmdBm TLP S S −=0

dBa0 (dBa at 0 dBm level point)dBTLP dBadBa −=0

dBrnC0 (dBrnC at 0 dBm level point)dBTLP dBrnC dBrnC −=0




12/24

ANGLE MODULATION

• Angle Modulation CharacteristicsPhase Deviation/ Modulation IndexPM waveform:

mV K m 1=∆= θ where: m =2$ = modulation index or peak phase

deviation (radians)K 1 = deviation sensitivity of the PMmodulator (rad/V)Vm = peak modulating signal amplitude (V)

FM waveform:

m

m

f V K

m 2=

where: K 2 = deviation sensitivity of the FMmodulator (rad/V-s)f m = modulating signal frequency (Hz)

Frequency DeviationPM waveform:

mm f V K f 1=∆ where:2 f = peak frequency deviation of PM

waveform (Hz)

FM waveform:

mV K f 2=∆ where:2 f = peak frequency deviation of FM

waveform (Hz)Percent Modulation (FM or PM)

100mod%max

×∆∆=

f f

ulation actual

where:2 f actual = actual frequency deviation of carrierin hertz2 f max = maximum frequency deviationallowed for communication system

Deviation Ratio

(max)

max..m f f R D

∆∆=

where: D.R. = deviation ratio of an FM waveform2 f m(max) = maximum modulating frequency

Power Relations in an Angle-Modulated Wave

RV

P ct 2

2

=

where: Pt = total transmitted power in an angle-modulated waveform (modulation or nomodulation)VC = peak amplitude of the carrier signalR = load resistor

Bandwidth Requirements for Angle-ModulatedWaves Low-index modulation (narrowband FM)

m f B 2≈ High-index modulation

f B ∆≈2

Using the Bessel Table (practical bandwidth))(2 m f n B ×=

where: n = number of significant sidebands

Using Carson’s Rule (approximate bandwidth)

)(2 m f f B +∆= Noise and Angle ModulationMaximum phase deviation due to an interferingsingle-frequency sinusoid:

c

n

V V ≈∆θ radians

where:2$ = peak phase deviation due to interferinsignalVn = peak amplitude of noise voltageV

c = peak amplitude of carrier voltage

Maximum frequency deviation due to an interferinsingle-frequency sinusoid:

nc

n f V V

f

≈∆ Hertz

where:2 f = peak frequency deviation due tointerfering signalf n = noise modulating frequency

FM Noise Analysism N f Φ=δ

=Φ −S

N 1sin

N

S

N S

δδ=

where:4 N = frequency deviation of the noise5 = phase shift (radians)4S = frequency deviation of the carrier




13/24


14/24

Vmin = equal to the resolutionVmax = maximum voltage that can bedecoded by the DAC

To determine the number of bits required for aPCM code

DRn ≥−12where: n = number of PCM bits (excluding sign bit)

Coding Efficiency

100 _ _ . _

_ _ . _ _ ×=bitsof no Actual

bitsof no MinimumefficiencyCoding

Analog Compandinga. -law companding – used in U.S. and Japan

( )µ

µ

+

+

=1ln

1lnmax

max V V

V

V

in

out

where: Vmax = maximum uncompressed analog inputamplitudeVmin = amplitude of the input signal at a particular distant of time

# = parameter used to define amount ofcompressionVout = compressed output amplitude

b. A-law companding – used in Europe

AV

V A

V V

in

out ln1maxmax += AV V

in 10max

≤≤

A

V V

A

V V

in

out ln1

ln1max

max +

+

= AV V

Ain ≤≤

max

1

where: A = parameter used to define the amount ofcompression

ACOUSTICS

• The Sound GenerationOctave

12 −= nan f f where: f n = frequency of the nth octave

f a = fundamental frequencyn = 1, 2, 3 …

Phon)(log1040 2 sone Phon +=

The apparent loudness and loudness levels0 – 15 dB very faint15 – 30 dB faint30 – 60 dB moderate60 – 80 dB loud80 – 130 dB very loud130 dB deafening

Notes: 0 dB – threshold of hearing60 dB – average conversation120 dB – threshold of pain150 dB – permanent damage to hearing

Sound Pressure Levels of common sound sourcesSource SPL (dB)

Faintest audible sound 0Whisper 20Quiet residence 30Soft stereo in residence 40Speech range 50 – 70Cafeteria 80Pneumatic jack hammer 90Loud crowd noise 100Accelerating motorcycle 100Rock concert 120Jet engine (75 feet away) 140

• Basic FormulasSound Velocity

λ f v = Sound Velocity in Gases

O

O P vργ =

where:) = ratio of the specific heat at constantvolumePo = the steady pressure of the gas (N/m2)6o = the steady or average density of the ga(kg/m3)

In dry air (experimental)05.045.331 ±=v m/s16.042.1087 ±=v ft/s

Velocity of sound in air for a range of about 20°Celsius change on temperature

C T v 607.045.331 ±= m/s F T v 016.103.1052 ±= ft/s

where: TC = temperature in degrees Celsius




15/24

TF = temperature in degrees Fahrenheit

For TC > 20°C,

27345.331 K T v = m/s

where: TK = temperature in Kelvin

Recall: 273+= C K T T 460+= F R T T

3259 += C F T T

( )3295 −= F C T T

Sound Pressure Level

OOO I I

P P

P P

SPL log10log10log202

=

==

where: P = RMS sound pressure (N/m2)Po = reference sound pressure

= 2×10-5 N/m2 or Pascal (Pa)= 0.0002# bar= 2.089 lb/ft2

Sound Intensity

410

22 P v

P I ==

ρ W/m2

where:6 = density of air

v = velocity of sound in air6v = characteristic impedance of air to sound= 410 rayls in air

The total intensity, I T nT I I I I I ++++= ...321

The total pressure, P T 22

32

22

1 ... nT P P P P P ++++=

Sound Intensity coming from(a) a point source (isotropic) in free space

24 r W

I π

=

(b) a source at ground level

22 r W

I π

=

The Sound Intensity Level2

log10log10

==

OO L P

P I I

I

where: Io = threshold intensity (W/m2)= 10-12 W/m2

The Sound Power Level (PWL)

OW W PWL log10=

120log10 += W PWL where: W = sound power in watts

Wo = reference sound power= 10-12 W

The Relation of SPL and PWL(a) for a sound produced in free space by an

isotropic source11log20

−−= r PWLSPL

(b) for a sound produced at ground level8log20 −−= r PWLSPL

• Room AcousticsOptimum reverberation (at 500 to 1000 Hz)

RoomFunction

Reverberationtime (s)

Recording and broadcast studios 0.45 – 0.55Elementary classrooms 0.6 – 0.8

Playhouses, intimate drama production 0.9 – 1.1

Lecture and conference rooms 0.9 – 1.1Cinema 0.8 – 1.2Small Theaters 1.2 – 1.4High school auditoriums 1.5 – 1.6General purpose auditoriums 1.5 – 1.6Churches 1.4 – 3.4

Different ways in computing reverberation timesA. Stephens and Bate formula (for ideal

reverberation time computation))1070.0012.0( 360 += V r t seconds

where: V = room volume (m3)r = 4 for speech= 5 for orchestra= 6 for choir




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Optimum volume/person for various types of hallTypes of halls Optimum volume/person

(m3)Concert halls 7.1Italian-type opera houses 4.2 – 5.1Churches 7.1 – 9.9Cinemas 3.1Rooms for speech 2.8

B. Sabine’s formula (for actual reverberation timewith average absorption less than or equal to 0.2)

aV

t 161.0

60 = secondswhere: V = room volume (m3)

a = total absorption units (m2 – metricSabine) (for a room: the sum of allabsorption of the ceiling, walls, floor,furnishings and occupants).

aV

t 049.0

60 = secondswhere: V = room volume (ft3)

a = total absorption units (ft2 – customarySabine)

Coefficient of absorption is the ratio of theabsorbed sound intensity to the incident soundintensity.

)(unitless I I

i

a=α

Note: " = 1 for perfect absorbent material

r ia I I I −= where: Ir = reflected sound intensity

Average absorption coefficient )(α

nnααααα ++++= ...321

Total absorption (a)

Aa α= (m2 or ft

2)where: A = surface area of the absorbent structure

(m2 or ft2)

C. Norris-Eyring’s formula (for actualreverberation time with average absorptioncoefficient greater than 0.2)

)1ln(161.0

60 α−−=

S V

t seconds

where: S = total surface area (m2)

α = average absorption coefficient of thereflecting surface

)1ln(049.0

60 α−−= S V

t seconds

where: S = total surface area (ft2)α = average absorption coefficient of thereflecting surface

A further correction may need to be added for higfrequency to allow for air absorption.

xV S V

t +−−= )1ln(

161.060 α

seconds

For values of " less than about 0.2 but frequenciabove 1000 Hz then a modified form of Sabinformula is considered.

V a

V t

+= 161.0

60 seconds

where: x = sound absorption/volume of air (m2/m3)

x per m 3 at a temperature of 20°CFreq(Hz)

30%RH×10 –3

40%RH×10 –3

50%RH×10 –3

60%RH×10 –3

70%RH×10 –3

80%RH×10 –3

1000 3.28 3.28 3.28 3.28 3.28 3.22000 11.48 8.2 8.2 6.56 6.56 6.54000 39.36 29.52 22.96 19.68 16.4 16

RH = Relative Humidity

Methods of measuring absorption coefficientA. Reverberation Chamber Method Note: The lowest frequency should not be lower ththe computed frequency from the formula belowensure a diffuse sound field where v is the volumethe room.

3180

v f lowest = Hz

Principle of reverberation chamber method

“A measurement of reverberation time is made firswithout, and then with the absorbent material in thchamber.”

Without the absorbent material,

aV

t 161.0

1 =




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With the absorbent material,

aaV

t δ+=

161.02

Therefore:

−=

12

11161.0t t

V aδ

In practice some slight correction needs to be madefor the behavior of sound in the chamber which canmake a difference of nearly 5%.

−

=

12

113.55t t v

V aδ

Absorption coefficient

S aδα =

where: V = volume of reverberation chambert1 = reverberation of the chamber withoutabsorbent materialt2 = reverberation of the chamber withabsorbent materiala = absorption of the chamber withoutabsorbent material4a = extra absorption due to the materialv = velocity of sound in airS = surface area under measurement, whichshould be a single area between 10 and 12 m2

B. Impedance Tube MethodAbsorption coefficient

221

21

)(4

A A A A

+=α

where: A1 and A2 are the maximum and minimumamplitudes of the resultant standing wave pattern reverberation of the chamber withoutabsorbent material

Note: " of impedance tube method is less than" ofreverberation chamber method.

Types of absorbentsA. Membrane or Panel absorbersThe absorption is highly dependent upon frequencyand is normally in the range of 50 to 500 Hz. Theyare often used in recording.

md f

60=

where: f = approximate resonant frequency

m = mass of the panel in kg/m2 d = depth of the air space in m

B. Helmholtz or Cavity or Volume ResonatorsResonant frequency (f) for a narrow-neck resonais approximately

V r l vr

f )2(

22

π

π

π +=

If there is no neck, l = 0

V r v

f 2

2π=

where: v = velocity of sound in airr = radius of the neckl = length of the neckV = volume of cavity

SATELLITE COMMUNICATIONS

• Communications SatelliteOrbit Location (Satellite Elevation category)(a) Low Earth Orbit (LEO) SatelliteOrbital height : 100 – 300 miOrbital velocity : 17,500 mphOrbital time (period) : 1.5 hoursSatellite Availability : 15 min per orbitTypical operating frequency : 1.0 GHz – 2.5 GHz

(b) Medium Earth Orbit (MEO) SatelliteOrbital height : 6,000 – 12,000 miOrbital velocity : 9,580 mphOrbital time (period) : 5 – 12 hoursSatellite Availability : 2 – 4 hours per orbiTypical operating frequency : 1.2 GHz – 1.66 GH

(c) Geostationary or Geosynchronous (GEO)Satellite

Orbital height : 22,300 miOrbital velocity : 6,879 mphOrbital time (period) : 24 hoursSatellite Availability : 24 hours per orbitTypical operating frequency : 2 GHz – 18 GHz

THE GEOSYNCHRONOUS SATELLITEAltitude : 19,360 nmi

: 22,284 smi: 35,855 km

Period : 23 hr, 56 min, 4.091 s (onesidereal day)

Orbit inclination : 0°




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Velocity : 6879 mphCoverage : 42.5% of earth’s surface (0°

elevation) Number of satellites : Three for global coverage

with some areas of overlap(120° apart)

Areas of no coverage : Above 81° north and southlatitude

Advantages : Simpler ground stationtracking: No handover problem: Nearly constant range: Very small Doppler shift

Disadvantages : Transmission delay: Range loss (free space loss)

Spatial separation : 3° – 6° [Typically 4°(equivalent to at least 1833miles of separation distance)or more]

Satellite classification according to sizeSize Mass Cost

Large Satellite > 1,000 kg > $ 100 MSmall Satellite 500 – 1,000 kg $ 50 – 100 MMini-Satellite 100 – 500 kg $ 5 – 20 MMicro-Satellite 10 – 100 kg $ 2 – 3 M Nano-Satellite < 10 kg < $ 1 M

• Satellite Orbital Dynamics

3

2

AP =α where:" = semi-major axis (km)A = constant (unitless)A = 42241.0979 for earthP = mean solar earth days [ratio of the timeof one sidereal day (23 hours and 56minutes) to the time of one revolution ofearth (24 hours)]P = 0.9972

For a satellite to stay in orbit, the centrifugal force

caused by its rotation around earth should be equalto the earth’s gravitational pull. g c F F =

2)( h Rmm

G F e s g +=

)(

2

h Rvm

F sc +=

where: Fc = centrifugal forceFg = gravitational forceG = gravitational constant (6.670×10-11)ms = mass of satellite

me = mass of earth (5.98×1024 kg)v = velocityR = earth’s radius (, 3960 mi , 6371 km)h = satellite height

Satellite velocity in orbit

)(

104 11

kmkm h R

v

+

×= m/s

Satellite height

R R gT

h −= 3 222

4π km

where: T = satellite periodg = gravitational acceleration (9.81×10-3 km/s2)

The escape velocity of earth is 25,000 mph or fromthe formula:

Escape velocity gR2=

The minimum acceptable angle of elevation is 5°.

Satellite Range (distance from an earth station)

ββ sincos)( 222 R Rh Rd −−+= where:( = angle of elevation

Note: ( = 0°, d is maximum, satellite is farthest( = 90°, d = h, satellite is nearest

• Frequency AllocationThe most common carrier frequencies used fSATCOM are the 6/4 and 14/12 GHz bands.

Frequency bands used in satellitecommunications

Frequency Band225 – 390 MHz P350 – 530 MHz J

1530 – 2700 MHz L2500 – 2700 MHz S3400 – 6425 MHz C7250 – 8400 MHz X10.95 – 14.5 GHz Ku17.7 – 21.2 GHz Ka27.5 – 31 GHz K36 – 46 GHz Q46 – 56 GHz V56 – 100 GHz W




19/24

Microwave frequency bandsBand designation Frequency range (GHz)

L 1 – 2S 2 – 4C 4 – 8X 8 – 12Ku 12 – 18K 18 – 27Ka 27 – 40

Millimeter 40 – 300Submillimeter >300

Earth coverage is approximately one-third of theearth’s surface with approximate antenna beamwidthof 17°.

• The Satellite System Parameters1. Transmit Power and Bit Energy

b

t bt b f P

T P E == where: E b = energy of a single bit (Joules/bit)

Pt = total carrier power (watts)T b = time of a singe bit (seconds)f b = bit rate (bps)

2. Effective Isotropic Radiated Power (EIRP)t r G P EIRP =

where: Pr = total power radiated from an antennaGt = transmit antenna power gain

3. Equivalent Noise Temperature (T e))1( −= NF T T oe

where: To = temperature of the environment (K) NF = noise factor (absolute value)

4. Noise Density (N o)

eo kT BW N

N ==

5.

Carrier-to-Noise Density Ratio

eo kT C

N C =

6. Energy Bit-to-Noise Density Ratio

b

b

o

b

Nf CBW

BW N f C

N E ==

7. Gain-to-Equivalent Noise Temperature Ratio

ee T LNAGGr

T G )(+=

The satellite system link equationsUplink Equations

e

u pr t

e

r u pr t

o T

G

k

L L P A

kT

A L L P A

N

C ×==)()(

Expressed in dB

k L

T G D

P A N C

u

er t

o

log10log10

log104log20log10

−−

+

−=λπ

)()(

)()()( 1

DBWK k dB L

dBK T G

dB LdBW EIRP N C

u

e p

o

−−

+−= −

Uplink Equations

e

d pr t

e

r d pr t

o T G

k

L L P A

kT

A L L P A

N C ×==

)()(

Expressed in dB

k L

T G D

P A N C

d

er t

o

log10log10

log104log20log10

−−

+

−=λπ

)()(

)()()( 1

DBWK k dB L

dBK T GdB LdBW EIRP

N C

d

e p

o

−−

+−= −

MULTIPLEXING

• Frequency Division MultiplexingVoice band frequency (VF): 0 – 4 kHzBasic voice band (VB) circuit is called 3002Channel: 300 – 3000 Hz band

Note: The basic 3002 channel can be subdivided i24 narrower 3001 (telegraph) channels that ha been frequency-division multiplexed to form a sin3002 channel.

A. Basic Group)4112( n f c −= kHz

where: f c = channel carrier frequencyn = channel number




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Lower sideband f LSB = f c – (0 to 4 kHz)

Upper sideband f USB = f c + (0 to 4 kHz)

B. Basic Supergroup)48372( n f c += kHz

where: f c = group carrier frequencyn = group number

Lower sideband f LSB = f c – (60 to 108 kHz)

Upper sideband f USB = f c + (60 to 108 kHz)

C. Basic MastergroupTwo categories of mastergroupsU600 – may be further multiplexed and used forhigher-capacity microwave radio.L600 – used for low-capacity microwave systems.

Basic Mastergroup bandwidth:L600 (60 – 2788 kHz)

BW = 2728 kHz

U600 (564 – 3084 kHz) BW = 2520 kHz

The Supergroup Carrier FrequenciesL600 Mastergroup

Supergroup Carrier frequency (kHz)1 6122 Direct3 11164 13645 16126 18607 21088 23569 272410 3100

U600 mastergroupSupergroup Carrier frequency (kHz)

13 111614 136415 161216 1860

17 210818 2356

D25 2652D26 2900D27 3148D28 3396

Summary of AT&T’s FDM HierarchyGroup = 12 VB channelsSupergroup = 5 Groups

= 60 VB channelsMastergroup = 10 Supergroups

= 50 Groups= 600 VB channels

Jumbogroup = 6 Mastergroups= 60 Supergroups= 300 Groups= 3600 VB channels

Superjumbogroup = 3 Jumbogroups= 18 Mastergroups= 180 Supergroups= 900 Groups= 10800 VB channels

Summary of CCITT’s FDM HierarchyGroup = 12 VB channelsSupergroup = 5 Groups

= 60 VB channelsMastergroup = 5 Supergroups

= 25 Groups= 300 VB channels

Supermastergroup = 3 Mastergroups= 15 Supergroups

• Time Division MultiplexingSummary of Digital Multiplex Hierarchy (North

American)LineType

DigitalSignal

Bit rate(Mbps)

ChannelCapacity

ServicesOffered

Medium

T1 DS – 1 1.544 24 VBtelephone

Twisted pair

T1C DS – 1C 3.152 48 VBtelephone

Twisted pair

T2 DS – 2 6.312 96 VB tel, picture- phone

Twisted pair#wave

T3 DS – 3 44.736 672 VB tel, picture-

phone, TV

coax, #wave

T4M DS – 4 274.176 4032 Same asT3 except

morecapacity

coax, opticafiber

T5 DS – 5 560.160 8064 Same asT3 except

morecapacity

optical fiber




21/24

Summary of CEPT 30 + 2 PCM MultiplexHierarchy (European)

Level Data Rate (Mbps) Channel Capacity1 2.048 302 8.448 1203 34.368 4804 139.264 19205 564.992 7680

Japanese Multiplex HierarchyLevel Data Rate (Mbps) Channel Capacity

1 1.544 242 6.312 963 32.064 4804 97.728 14405 565.148 7680

CCITT Time-Division Multiplexed CarrierSystem (European Standard PCM-TDM System)

With CCITT system, a 125-#s frame is divided into32 equal time slots.

E – 1 CarrierFraming

and alarmchannel

Voicechannel

1

VoiceChannel2 – 15

CommonSignalingchannel

VoiceChannel16 – 29

VoiceChannel

30TS 0 TS 1 TS 2 – 16 TS 17 TS 18 – 30 TS 31

Line-Encoding SummaryEncodingFormat

MinimumBW

AverageDC

ClockRecovery

ErrorDetection

UPNRZ f b/2 + V/2 Poor NoBPNRZ f b/2 0 V Poor NoUPRZ f b + V/2 Good NoBPRZ f b 0 V Best No

BPRZ-AMI f b/2 0 V Good Yes

TELEPHONY

• Introduction1.) Typical sounds produced by humans: 100 to

1000 Hz.2.) Peak sensitivity of human hearing: 4 kHz.3.) Upper frequency limit for hearing: 18 to 20 kHz.4.) Lower frequency limit for hearing: 18 to 20 Hz.

Nature of Speech1.) Sound pressure wave of speech contains

frequencies: 100 Hz to 10 kHz.2.) Speech power range: 10 to 1,000#W.

3.) Maximum intelligibility for voice frequency:1,000 and 3,000 Hz.

4.) Maximum voice energy is located between 250and 500 Hz.

Speech MeasurementFor typical single talker average power in dBm:

P(dBm) = VU reading – 1.4 dB

For more than one speaker over the channel P(dBm) = VU reading – 1.4 + 10logN

where: N = number of speakers

• The telephone SetPulse DialingTo transmit a digit, it takes 0.1 second per pulse +0.5 second inter-digital delay time.

DTMF FrequenciesFrequencies 1209 Hz 1336 Hz 1477 Hz

697 Hz 1 2 3770 Hz 4 5 6852 Hz 7 8 9941 Hz * 0 #

Network Call Progress TonesTone Frequency (Hz)

Dial Tone 350 + 440Ringback 440 + 480

Busy Signal 480 + 620

• Switching and Signaling

2)1( −= nn N

where: N = number of connectionsn = number of subscribers

• Traffic EngineeringMeasurement of Telephone Traffic

T C A ×=

where: A = traffic intensity in ErlangsC = designates the number of calls originatduring a period of 1 hr (calls/hr or calls/minT = the average holding time, usually givenin hours (hr/call or min/call)

t S

A = where: S = sum of all the holding time (min)

t = observation period (1 hr or 60 min)




22/24

Note: 1 Erlang = 36 ccs (Century Call Seconds orHundred Call Seconds)

callsoffered of noTotal callslost of Number

serviceof Grade _ _ _ . _

_ _ _ _ _ =

• GSM NetworkRadio-Path Propagation Loss

=∆

2

1log40d d

P

(40 dB/decade path loss)

=∆

1

1 'log20hh

G

(A base station antenna height gain of 6dB/octave)where:2 P = the difference in two receive signal

strengths based on two different path lengths

d1 and d2 2 G = the difference in two receive signalstrengths based on two different antennaheights h1 and h1’

Receive signal in decibelsFor non-obstructive path

αγ +

+

−=

1

'log20log

hh

r r

P P eo

ror

For obstructive pathαγ ++

−= Lr r

P P o

ror log

where: r = distance between the base and the mobileunit in mi or kmhe’ = effective antenna heightL = shadow lossPro = received signal at a reference distance r o r o = usually equal to 1 mi (1.6 km)" = correction factor

Standard Condition:Frequency (f o) 900 MHzBase-station antenna height (h 1) 30.46 mBase-station power at the antenna 10 wattsBase-station antenna gain (G t) 6 dBdMobile-unit antenna height 3 mr o 1.6 kmMobile-unit antenna gain (G m) 0 dBd

General Formula for Mobile radio PropagationPath Loss: P r = P t – 134.4 – 38.4logr 1 + 20logh 1 +20logh 2 + G t + G m where: Pt and Pr are in decibels above 1mW, r 1 is in

kilometers, h1 and h2 are in meters, and Gt and Gm are in decibels

Cochannel Interference Reduction Factor

(CIRF), q

R D

q =

Frequency Reuse factor, K

K q 3= 3

2q K =

Radio CapacityA. Analog, FDMA and TDMA cellular system

=

I C

B Bm

c

t

32

where: Bt = total allocated spectrumBc = channel bandwidth(C/I) = required carrier-to-interference ratioin linear values

B. CDMA cellular system

K

M m =

where: M = total number of voice channelsK = frequency reuse factor

Antenna Separation RequirementA. At the Base Station

11=d h

where: h = antenna heightd = spacing between two antennas

B. At the Mobile UnitA separation of a half-wavelength between twmobile antennas is required at 850 MHz. Therefothe separation between two antennas needs to only 0.18 m (about 6 inches) at the cellufrequency of 850 MHz.




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