COMMUNICATIONS Formulas and Concepts V3.pdf

AMPLITUDE MODULATION

mt

2

=

m1

+ m22 + m32 ......

where: mt = total modulation modu lation index m1, m2, m3 = modulation modu lation index of signal V m V m having index 1, 2, 3 respectively V (t ) = V c sin ω c t + cos(ω c − ω m )t − cos(ω c + ω m )t 2 2 where: Vc = maximum voltage of the carrier signal Power Savings Vm = maximum voltage vo ltage of the original a. Single Sideband (SSB) modulating signal P + P C PS = LSB / USB c = 2!f c = frequency of o f the carrier signal P t m = 2!f m = frequency of the modulating signal

•

AM Wave

m=

V m V c

=

b. Single Sideband full carrier (SSBFC) P PS = LSB / USB P t

V max−V min V max+V min

where: m = modulation index Vmax = maximum peak-to-peak voltage swing of AM wave Vmin = minimum peak-to-peak voltage swing of AM wave AM wave equation in terms of modulation index V (t ) = V c sin ω c t +

•

mV c

2

cos(ωc − ω m )t −

mV c

•

Tuned Radio-Frequency (TRF)AM Receiver TRF Design formulas

cos(ω c + ω m )t

2

f r

BW = 2 f m

AM Power and Current 2 2 2 V carr V LSB V USB + + P t = R R R 2 2 2 m V c V m P = LSB = P USB = 8 R 8 R

P t

=1+

m2

1 2π LC

2

P c

= P c

f r BW

where: Q = quality factor f r r = = frequency BW = Bandwidth

m

=

V c

2 R

2

4

•

Superheterodyne Receiver f si = f s + 2 f i

where: f sisi = image frequency f s = signal frequency f i = intermediate frequency

2

 I t  m2   1 = +  2  I c 

2 where: Pt = total transmitted power Pc = unmodulated carrier power It = total transmitted current Ic = unmodulated carrier current m = modulation index P c

=

Q=

AM Bandwidth

where: BW = bandwidth f m = modulating signal frequency

•

c. Two independent Sidebands P = C PS = P t

α ρ

=

f si f s

−

= 1+ Q2ρ 2 f s f si

=

f image f RF

−

f RF f image

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

Note: The voltage should be in rms Amplitude Modulation with Multiple Signals

 mt 2   P t = P c 1 +   2   PDF created with pdfFactory trial version www.pdffactory.com

TELEVISION

P AV

Details of Horizontal Blanking Period Time, sec

Total line (H) H blanking H sync pulse Front porch Back porch Visible line time

63.5 0.15H-0.18H or 9.5-11.5 0.08H, or 4.75 ± 0.5 0.02H, or 1.27 0.06H, 3.81 52-54

Details of Vertical Blanking Period Time

Total field (V) V blanking Each V sync pulse Total of 6 V sync pulse Each equalizing pulse Each serration Visible field time

1/60s 0.0167s 0.05V-0.08V or 9.5-11.5 27.35 #s 3H = 190.5 #s 0.04H = 2.54 #s 0.07H = 4.4 #s 0.92V-0.95V, 0.015-0.016s

= P PEAK × DR

P PEAK

2

Minimum displayed range

=c

Rmin

Directional Gain

G dir =

4π θφ

Radar Range

R

P T A P SAO

=4 A P

(4π ) 2 P R min 4π AO

=

θ

PRT

2

R

=4

P T AO S

4πλ2 P R min where: R = Radar Range PT = Transmitted Power AP = antenna gain S = cross-sectional area of the target targ et A0 = captured area of an antenna PRmin = detected signal level in W

=λ

DR =

F D

L

RADAR Pulse (Waveform) PRT = PW + RT

1

λ2

Doppler Effect

where: $ = horizontal beam-width (radians) = the wavelength of the radar % = L = the dimension of o f the antenna in the direction of interest (i.e. width or height) (rad ians) φ = vertical beam-width (radians)

PRF =

PW

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

Q = 0.21 R − 0.52G + 0.31 B

NAVIGATIONAL AIDS

PW

Maximum unambiguous range PRT Runamb = c

I = 0.60 R − 0.28G − 0.32 B

Tip of sync = 100% Blanking level = 75% Black setup = 67.5% Maximum white = 10 to 15% or 12.5% (typical)

P AV PRT

where: PRT = Pulse Repetition Time PW = Pulse Width (#s) RT = Rest Time (#s) PRF = Pulse Repetition Frequency DR = Duty Cycle or o r Duty Ratio PAV = Average Power PPEAK = = Peak Power

Picture Information Encoding Y = 0.30 R + 0.59G + 0.11 B

Relative amplitude for the AM RF picture signal

=

=

2v cos θ λ

where: FD = frequency change between transmitter and reflected signal v = relative velocity between RADAR R ADAR and target % = = wavelength of o f the transmitted wave $ = angle between target direction and RADAR system

PW PRT

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TRANSMISSION LINES

Resistance, R

R

•

Electrical Characteristics Characteristic Impedance

Z

=

Z O

where: Z = R +jL Y = G +jC Z O Z O

=

R

=

L

R

&

Y &/m

at low frequency, &

G

L =

at high frequency, &

C =

=

Z SC Z OC

2 D

L

m

= 1.016 Z O ε r × 10 −3 × 10 −3

Z O

Z O

Open Line:

Z O

Teflon = 2.1 Polyethelene = 2.25 Polysterene = 2.7 Mylar = 3.1 Porcelain = 6

d

Z O

#H/ft

× 10 −3

#F/ft

=

C

276 ε r

log

= 60 ln D

Z O

=

d

ε r

Note: 40& ' Z0 ' 150&

#F/ft

Resistance, R

d

ε r

Z O

138

= 8.34 × 10 −8

ε r

log

D d

 1 + 1  Ω  d  m

f   D

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

L

2 D d

Note: 150& ' Z0 ' 600& Er: air = 1

r

#H/ft

= 120 ln 2 D =

m

Characteristic Impedance, Z 0

R

Characteristic Impedance, Z 0

Z O

D

d

Alternate formulas:

r

µ D H ln 2π d m 2πε F

= 1.016 Z O ε r × 10 −3

C = 1.016

where: L = Inductance C = Capacitance D = Separation between center to center d = diameter of the wire

C = 1.016

5d

Alternate formulas:

µ 2 D H L = ln d m π F πε

L

m

where: D = diameter of the outer conductor d = diameter of the inner conductor

For Parallel-wire line:

ln

a

Ω 100 − ft

f

ln

&

where: ZSC = short circuit impedance ZOC = open circuit impedance

C =

Ω

For coaxial line:

Also,

Z O

=

f

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

S/m

C

= 8.34 × 10 −8

R

= 0.1

Ω  1 + 1   d  100 − ft

f   D

where: D = diameter of the outer conductor (inches) d = diameter of the inner conductor (inches) f = frequency (MHz)

Mica = 6 Paper = 7 Nylon = 8 Silicon = 11.68 Water = 80 @20 ‘C


V ( d ) = V + e jγ d + V − e − jγ d

Complex Propagation constant, !

γ = α

+ jβ =

ZY

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

=ω

 R  = 4.343   Z  O  LC =

ω V P

= 2π

radians/length

λ

1

V P =

dB/length

LC

m/s

where: V p = propagation velocity

•

Loading Conditions

Note: The zero reference is at the load not on the generator.

I (d ) =

= I S e V R = V S e −γ L −2γ L P R = P S e

= I R e V S = V R e γ L 2γ L P S = P R e

where: IR , IS, VR , VS = receiving and sending end current and voltages respectively PR , PS = power at the receiving and sending end ) = complex propagation constant L = length of the transmission lone Zin = input impedance ZL = load impedance 2. ZL " Z0 (Mismatch) Z O2 Z in = Z L

Z in

for % /4 line

I (d ) = Z ( d ) =

1

I (d ) = Z ( d ) =

Z O

V ( d ) I ( d )

= jZ 0 tan(β d )

2 jV + sin( β d ) Z 0 V (d ) I ( d )

= − jZ 0 cot(β d )

Γ R = 1 3. ZL = Z0 (matched load) V ( d ) = V + e jβ d

I (d ) =

V + e jβd Z 0

Z ( d ) = Z 0

Γ R = 0 4. ZL = jX (pure reactance)

(V + e β − V − e − β ) j d

Z 0

2. ZL # $ (open circuit) V ( d ) = 2V + cos( βd )

Load boundary characteristics V (d ) = V + e jβd + V − e − jβd

I ( d ) =

2V + cos(βd )

Γ R = −1

 Z + Z tanh γ L   for L > % /4 = Z O  L O  Z Z tanh L + γ  O L 

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

j d

Four Cases (loss-less transmission line) 1. ZL # 0 (short circuit) V ( d ) = 2 jV + sin( β d )

γ L

I S

j d

where: V(d) = line voltage at point d I(d) = line current at point d Z0 = characteristic impedance of the line V+ = incident voltage V – = reflected voltage ) = complex propagation constant for lossyline ( = complex propagation constant for lossless line d = distance from the load

with ZL = Z0 then Zin = Z0 I R

Z O

(V + e γ − V − e − γ )

Loss-less transmission line

1. ZL = Z0 (match load) − γ L

1

-

Reactive impedance can be realized with transmission lines terminated by a short or by an open circuit.

j d

Loss-less transmission line


Z in

= jZ 0 tan(βL)

-

Reflection coefficient has a unitary magnitude, as in the case of short and open circuit load.

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

0 < L < L

λ

4 L

λ

2

=

Im( Z in ) > 0

λ

Resonance Inductance

4

λ

Z in

→∞

4

< L < λ

Im( Z in ) < 0

Parallel Resonance Capacitance

2

=

λ

Z in

=0

2

< L < 3λ

Im( Z in ) > 0

Series Resonance Inductance

4

3λ L = 4 3λ < L < λ 4

Z in

→∞

Im( Z in ) < 0


Shorted Transmission Line – Fixed Frequency Parallel L = 0 Z in → ∞

0 < L < L

λ

4 L

λ

2 L

Im( Z in ) < 0

λ

4

=λ

Z in

=0

4

< L <

Resonance Capacitance

λ

Im( Z in ) > 0


2

→∞

=λ

Z in

3 < L < λ

Im( Z in ) < 0

2


4

=

3λ 4

3λ < L < λ 4

Z in

=0

Im( Z in ) > 0


B. Voltage Standing Wave Ratio (SWR) V inc + V ref V VSWR = max = V min V inc − V ref

where: V+ = Vinc = incident (forward) voltage V – = Vref = reflected (reverse) voltage C. Current Standing Wave Ratio (SWR) I inc + I ref I ISWR = max = I min I inc − I ref

where: Iinc = incident (forward) current Iref = reflected (reverse) current Note: SWR = VSWR = ISWR In dB: SWRdB = 20 log SWR Coefficient of reflection, % V ref I ref Z L − Z 0

Γ =

Z L + Z 0

=

SWR − 1 SWR + 1

- for purely resistive '

Z 0

=

Z 0 Z L

&

where: Z0’ = Characteristic impedance of the quarter-wave matching transformer 2. Stub

Procedure of using stubs: a. Calculate the load admittance b. Calculate the stub susceptance c. Connect the stub to the load, the resulting admittance being the load conductance G. d. Transform conductance to resistance, and calculate Z0’ of the quarter-wave transformer. ANTENNA Antenna Characteristics

•

1 + Γ 1 − Γ Note: The greater the SWR, the greater the mismatch

I inc

=

Solutions to mismatch condition: 1. Quarter-wave transformer matching

•

Degree of Mismatch A. Standing Wave Ratio (SWR) Z R SWR = 0 = L (whichever is larger) R L Z 0

V inc

=

G = 10 log

P 2 P 1

where: G(dB) = antenna gain in decibels P1 = power of unidirectional antenna P2 = power of reference antenna

SWR =


ERP = P in G

ERP = P rad D

+F = field strength in the most optimum

direction +B = field strength in the opposite direction

where: G = power gain (unitless) Pin = power delivered to the feedpoint For an isotropic antenna: P T = P rad But for a unidirectional antenna: P T = ERP Rrad

=

P rad I 2

where: R rad = radiation resistance Prad = power radiated by the antenna I = current at the feedpoint Radiation resistance for l not in excess of & /8

ξ

l  = 790    λ 

P d

where: R rad = radiation resistance Prad = power radiated by the antenna

=

Rrad

Gain over isotropic = 0 dB Beamwidth = 360º Types of Antenna A. Dipole Antenna a. Half-wave dipole

Gain over isotropic = 2.14 dB Beamwidth = 55º b. Folded half-wave dipole

Gain over isotropic = 5.64 dB Beamwidth = 45º

= ηP in G = η D

P rad

where: * = antenna efficiency (1 for lossless ant.) R rad = antenna radiation resistance R T = antenna radiation resistance = R rad and R d (ohmic resistance) D = directivity (maximum directive gain)

φ

B. Beam Antenna a. Yagi-Uda Antenna

Gain over isotropic = 7.14 dB Beamwidth = 25º b. Rhombic Antenna

Gain over isotropic = 5.14 dB

f r Q

C. Loop Antenna

Gain over isotropic = 3.14 dB Beamwidth = 200º

= 70 λ

D

where: BW = bandwidth f r = antenna resonant frequency Q = antenna quality factor φ = beamwidth A FB

= 10 log

P F P B

= 20 log

r

Isotropic Antenna

RT

BW =

λr

30 P T

sin θ =

•

= P in − P rad

η

60π Le I

where: +F = magnitude of field strength r = distance Le = antenna length I = current amplitude $ = the angle of the axis of the wire and the point of maximum radiation

2

Rrad

=

ξ F ξ B

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

V = k ( 2π f ) BAN

where: V = voltage induced in a loop antenna k = physical proportional factor B = field strength flux, V/m A = loop area, m2 N = number of turns D. Antenna with parabolic reflector


G

=

Aeff Aiso

=

kA s Aiso

2

D  = π k     λ  2

Aiso

=

λ2 4π

A s

=

π D 2

4 where: Aeff = effective aperture or antenna capture area Aiso = isotropic area k = illumination factor D = diameter of parabolic reflector

where: l 1 =

λ1

= the length of the longest element 2 d1 = the distance between the longest element and the second element r = design factor which is between 0.7 and 0.98

Antenna Height

For a straight vertical antenna with h ' % /4

2

 D  G = 6   λ 

with k = 0.65

he

λ π sin

5λ Parabolic dipole: D = 2

2πh

Elevation Pattern: –3dB beamwidth

Azimuth Pattern: –3dB beamwidth

=

56λ

=

70λ

λ

Note: he the antenna effective height is ½ to - of the actual height. FIBER OPTICS

•

w

D  = 7.5    λ 

Nature of Light

E P = hf

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

E. Helical Antenna 2

 D  NS G = 15   λ  λ 52 φ=

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

•

π D NS λ λ

Snell’s Law

n1 sin θ1

= n2 sin θ 2

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

where: G = Power gain φ = beamwidth D = helix diameter N = number of turns S = pitch between turns % = wavelength L = center-line axis length , NS

Note: 1 Å = 10 –10 m 1 micron = 10 –6 m nair = 1.0003 , 1

Note: If pitch is not given S = % /4 n =

f H

=

c

λn

πh λ

h

2

F. Log-Periodic Antenna Design factor formulas: l l l r = 2 = 3 = 4 l 1 l 2 l 3

sin 2

where: he = effective height h = actual height

Horn Antenna (Pyramidal)

G

=

r =

d 2 d 1 f L

=

d 3 d 2

=

=

d 4 d 3

c v

where: n = refractive index c = speed of light v = velocity of light at material with refractive index of n

c

λ1

Note: Angle of incidence and refraction are measured from normal


sin θ C =

"c = connector attenuation "s = total splice losses

n2 n1

f m = fiber margin L = distance between repeaters

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

sin θ in (max) = n1 cosθ C where: $in(max) = acceptance angle = acceptance cone half angle NA =

2 n1

−

1 5

Z = B∆t

where: Z = system length B = maximum bit rate 2t = total fiber dispersion RADIO WAVE PROPAGATION

•

The Electromagnetic Wave Velocity of propagation

2 n2

Where: NA = Numerical Aperture

•

V p

Mode of Propagation

µ

1 N = V 2 2 V = π

d

λO

n12

∆=

− n 22 = π n1

d

λO

1

=

µε

m/s

= µ r µ 0

= ε r ε 0

ε

where: # = permeability of the medium (H/m) 3 = permittivity of the medium (F/m) NA

− n2

The Power Density ERP P T GT

℘=

n1

where: N = number of modes V = V number d = diameter % = wavelength NA = numerical Aperture n1 = refractive index of core n2 = refractive index of cladding 2 = fractional index difference

•

Optical Fiber System Design Mathematical Analysis

The power budget is the basis of the design of an optical fiber link.

=

A

W/m2

2

4πr

The Electric Field Intensity or Strength

ξ

= α H =

30 P T GT r

V/m

where: " = characteristic impedance of free space, & H = rms value of magnetic field intensity or strength (A/m) The characteristic impedance of a medium

α =

µ ε

&

Characteristic impedance in free space

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 power Pr = receiver sensitivity (minimum received power) "f = fiber attenuation


µo

= 4π × 10 −7 = 1.26 × 10 −6 H/m

10 −9 = 8.854 × 10 −12 F/m εo = 36π 4π × 10 −7 = 120πΩ = 377Ω α= 10 −9 36π

The Attenuation of Power Density and Electric Field Intensity r ℘ A℘ ( dB) = 10 log 1 = 20 log 2 r 1 ℘2

Aξ (dB) = 10 log

•

ξ1 ξ2

= 20 log

r 2 r 1

The effects of environment to propagation of radio waves Refractive indices of different materials

H2O Glass Quartz Crystal Glycerin Diamond

1.33 1.50 1.54 1.47 2.42

Snell’s Law

sin θ1 V 2 k = = 1 n2 sin θ 2 V 1 k 2 where: $2 = angle of refraction $1 = angle of incidence V2 = refracted wave velocity in medium 2 V1 = incident wave velocity in medium 1 k 1 = dielectric constant of medium 1 k 2 = dielectric constant of medium 2 n1 = refractive index of medium 1 n2 = refractive index of medium 2 n1

=

n=

c V p

k

where: n = refractive index c = velocity of light in free space V p = velocity of light in a given medium Resultant field strength between waves traveling in different (direct and reflected paths)

δ

=

The Propagation Modes The Radio Frequency Spectrum Band Name Frequency (MHz) Propagation

VLF LF MF HF VHF UHF SHF EHF

0.01 – 0.03 0.03 – 0.3 0.3 – 3.0 3.0 – 30 30 – 300 300 – 3,000 3,000 – 30,000 30,000 – 300,000

Ground Wave Ground Wave Ground Wave Sky Wave Space Wave Space Wave Space Wave Space Wave

A. The Ground (Surface) Wave Method The field strength at a distance ( ') αht I

ξ

=

λ r

The signal receive at that distance if a receiving antenna is in place V = ξhr where: " = characteristic impedance of free space

ht and hr = effective height of the transmitting and receiving antennas I = antenna current r = distance from transmitting antenna B. The Ionosphere The refractive index of the ionosphere

sin θ i 81 N = 1− 2 sin θ r f where: N = number of free electrons per m3 f = frequency of radio wave (Hz) n=

=

ξ r = 2ξ d sin 2π

•

δ V/m 2λ

2hat har d

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

The Ionospheric Layers D Layer – average height 70 km, with an average

thickness of 10 km. E Layer – existing at a height about 100 km, with a thickness of 25 km. F1 Layer – exists at a height 180 km, daytime thickness is about 20 km. F2 Layer – height ranges from 250 – 400 km in daytime and at night it falls to a height of 300 km where it combines with F1 layer, approximate thickness at about 200 km. The height of the ionospheric layer d h =


2 tan θ

The critical frequency (f c)

f c

NOISE

= MUF cos θ = 9

N max

The Maximum Usable Frequency (MUF) f c MUF = = f c sec θ

cos θ

The Optimum Working Frequency (OWF) or Frequency of Optimum Transmission (FOT) OWF = FOT = 0.85 MUF

•

Noise Calculation N = kTB

where: N = noise power k = Boltzmann’s constant T = resistor temperature B = bandwidth of the system Note: 17 °C/290 K is the typical noise temperature V n

C. The Space Wave Propagation The Radio Horizon Distance d d EC = 1 2 k Re = kR0

= 4kTBR

in #V

where: Vn = noise voltage R = resistance generating the noise Series Resistors

2 2 2 V n = V n1 + V n2 + V n3 + ... where: EC = Earth’s Curvature R e = effective earth’s radius Parallel Resistors R 0 = earth’s radius , 3960 mi , 6371 km 2 2 2 k = correction factor for relatively flat earth I n = I n1 + I n2 + I n3 + ... k = 4/3 1 _________________ Ns = surface refractivity k = (1- (0.0466)e^0.005577Ns T

T

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

d = d 1

+ d 2 = 4

ht

+4

hr

where: ht and hr = in meters d, d1 and d2 = in kilometers d =

2ht + 2hr where: ht and hr = in feet d = in miles

For a diode, the rms noise current

= 2eI D B

typically in #A where: e = charge of an electron (1.6×10-19 C) ID = direct diode current B = bandwidth of the system I n

I n

where: I0 = negligible reverse saturated current I. Addition of noise due to several sources

V nT = 4kTBRT

The correction factor (k) k = [1 − 0.04665e 0.005577 N s ]−1

where: Ns = surface refractivity D. Tropospheric Scatter Wave (Troposcatter) Propagation

Operates at the UHF band (between n 350 MHz to 10 GHz (and used to link multi-channel telephone links). The common frequencies are 0.9 GHz, 2 GHz and 5 GHz.

= 2e( I D + 2 I o ) B

II. Addition of noise due to several amplifiers in cascade Req = R1 + R2 '+ R3 '+... + Rn '

Req

= R1 +

R2

( A1 )2

+

R3

( A1 )2 ( A2 )2

+ ... +

III. Signal-to-Noise Ratio S S (dB) = 10 log R


Rn

( A1 )2 ...( An −1 )2

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

NF =

N i S o

Total information sent H = Ct Power required

P n

N o F (dB) = 10 log NF

For a noiseless receiver, NF = 1;

F = 0 dB

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

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

P 2

where: Pn = power required in the n-level code P2 = power level required in the binary code n = number of levels in a code

•

Noise Measurements Units dBrn (dB above reference noise) N dBrn = 10 log −12

1 × 10 W dBrn = dBm + 90

For a pure tone: dBa = 10 log

For attenuator elements

VI. Overall Noise Factor (Friis’ Formula) NF n − 1 NF 2 − 1 NF 3 − 1 + + ... + NF = NF 1 + G1 G1G 2 G1G 2 G3 ...G n −1 Overall Noise Temperature T e T e 3 T e n T e = T e 1 + 2 + + ... + G1 G1G2 G1G2G3 ...Gn −1

For F1A weighted:

dBa = dBm + 82

dBrnC (dB above reference noise, C-message weighted) dBrnC = dBm + 90 pWp (picowatts, psophometrically weighted) ( psop hom etricV 2 ) × 10 −12 pWp =

600Ω

•

Information Theory Hartley Law C = 2 B log 2 n

dBmp = 10 log

bps

where: C = channel capacity B = channel bandwidth (Hz) n = number of coding levels (2 for binary, 8 for octal, 10 for decimal etc.) Shannon-Hartley Law C = B log 2 (1 + S / N ) C = 3.32 B log(1 + S / N )

N

1 × 10 −11.5 dBa = dBm + 85

= T p ( L − 1)

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

VII.

= (n − 1) 2

dBa (dB above adjusted noise)

For a noiseless receiver, T e = 0 K T e

bits

pWp

10 −3

Transmission level point

TLP ( dB) = 10 log TLP dB S dBm 0

S S 0TLP

= S dBm − S dBm 0 = S dBm − TLP dB

bps bps where: S/N = signal-to-noise ratio (absolute value)

dBa0 (dBa at 0 dBm level point) dBa0 = dBa − TLP dB

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

dBrnC0 (dBrnC at 0 dBm level point) dBrnC 0 = dBrnC − TLP dB


ANGLE MODULATION

•

Angle Modulation Characteristics Phase Deviation/ Modulation Index PM waveform: m = ∆θ = K 1V m

where: m = 2$ = modulation index or peak phase deviation (radians) K 1 = deviation sensitivity of the PM modulator (rad/V) Vm = peak modulating signal amplitude (V)

where: Pt = total transmitted power in an anglemodulated waveform (modulation or no modulation) VC = peak amplitude of the carrier signal R = load resistor Bandwidth Requirements for Angle-Modulated Waves Low-index modulation (narrowband FM) B ≈ 2 f m High-index modulation B ≈ 2∆ f

FM waveform:

m=

K 2V m f m

where: K 2 = deviation sensitivity of the FM modulator (rad/V-s) f m = modulating signal frequency (Hz) Frequency Deviation PM waveform: ∆ f = K 1V m f m

where: 2f = peak frequency deviation of PM waveform (Hz) FM waveform:

∆ f = K 2V m where: 2f = peak frequency deviation of FM waveform (Hz) Percent Modulation (FM or PM) ∆ f % mod ulation = actual × 100 ∆ f max

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

where: n = number of significant sidebands Using Carson’s Rule (approximate bandwidth) B = 2( ∆ f + f m ) Noise and Angle Modulation

Maximum phase deviation due to an interfering single-frequency sinusoid:

∆θ ≈

V n V c

radians

where: 2$ = peak phase deviation due to interfering signal Vn = peak amplitude of noise voltage Vc = peak amplitude of carrier voltage Maximum frequency deviation due to an interfering single-frequency sinusoid:

 V  f n Hertz ∆ f ≈  n   V  c 

where: 2f actual = actual frequency deviation of carrier in hertz 2f max = maximum frequency deviation allowed for communication system

where: 2f = peak frequency deviation due to interfering signal f n = noise modulating frequency

Deviation Ratio

FM Noise Analysis

D. R. =

∆ f max ∆ f m (max)

= Φf m N  Φ = sin −1     S  δ N

where: D.R. = deviation ratio of an FM waveform 2f m(max) = maximum modulating frequency Power Relations in an Angle-Modulated Wave 2 V c P t = 2 R

S N

=

δ S δ N

where: 4 N = frequency deviation of the noise 5 = phase shift (radians) 4S = frequency deviation of the carrier


DIGITAL COMMUNICATION

•

Frequency Shift-Keying ∆ F MI = F a

•

Quadrature Amplitude Modulation 8QAM Binary 8QAM Input Output

000 001 010 011 100 101 110 111

where: MI = modulation index 2F = frequency deviation Fa = modulating frequency For worst-case condition (alternating 1’s and 0’s) F − F s MI = m F b

where: Fm = mark frequency Fs = space frequency F b = input bit rate

0.795V 1.848V 0.795V 1.848V 0.795V 1.848V 0.795V 1.848V B. E . =

–135° –135° –45° –45° +135° +135° +45° +45°

F b BW

Condition for synchronization nF b F m = F s =

where: B.E. = Bandwidth Efficiency F b = transmission rate BW = bandwidth

•

Digital Modulation Summary Modulation No. of BW Baud Bit(s)

2 where: n = any odd whole integer Phase Shift-Keying M-ary encoding N = log 2 M

PSK BPSK QPSK 8 PSK 8 QAM 16 PSK 16 QAM

M = 2

N

where: N = number of bits M = number of output conditions possible with n bits 1. Quaternary or Quadrature Phase Shift Keying (QPSK) Binary QPSK Input Output Phase

00 01 10 11

–135° –45° +135° +45°

2. Eight PSK (8PSK) Binary Input

000 001 010 011 100 101 110 111

8PSK Output

–112.5° –157.5° –67.5° –22.5° +112.5° +157.5° +67.5° +22.5°

1 1 2 3 3 4 4

f b f b f b/2 f b/3 f b/3 f b/4 f b/4

f b f b f b/2 f b/3 f b/3 f b/4 f b/4

B.E.

'1

1 2 3 3 4 4

•

Sampling Nyquist sampling theorem states that the minimum

sampling rate that can be used for a given PCM code is twice the highest audio input frequency f s ≥ 2 f a where: f s = minimum Nyquist sampling rate f a = highest frequency to be sampled

•

PCM code

qemax DR

=

= resolution 2

V max V min

=

V max resolution

In dB: DRdb

= 20 log

V max V min

where: qemax = quantization error DR = dynamic range


Vmin = equal to the resolution Vmax = maximum voltage that can be decoded by the DAC

The apparent loudness and loudness levels

0 – 15 dB 15 – 30 dB 30 – 60 dB 60 – 80 dB 80 – 130 dB 130 dB

To determine the number of bits required for a PCM code 2 n − 1 ≥ DR

where: n = number of PCM bits (excluding sign bit) Coding Efficiency

Coding _ efficiency =

Minimum _ no. _ of _ bits Actual _ no. _ of _ bits

× 100

Analog Companding a. -law companding – used in U.S. and Japan

Notes: 0 dB – threshold of hearing 60 dB – average conversation 120 dB – threshold of pain 150 dB – permanent damage to hearing Sound Pressure Levels of common sound sources Source SPL (dB)

Faintest audible sound Whisper Quiet residence Soft stereo in residence Speech range Cafeteria Pneumatic jack hammer Loud crowd noise Accelerating motorcycle Rock concert Jet engine (75 feet away)

 V  V max ln1 + µ in  V max    V out = ln(1 + µ ) where: Vmax = maximum uncompressed analog input amplitude Vmin = amplitude of the input signal at a particular distant of time # = parameter used to define amount of compression Vout = compressed output amplitude b. A-law companding – used in Europe A V out = V max

1 + ln A

0≤

V in V max

≤

1

where: A = parameter used to define the amount of compression

•

The Sound Generation Octave f n = f a 2 n −1

where: f n = frequency of the nth octave f a = fundamental frequency n = 1, 2, 3 … Phon

Phon = 40 + 10 log 2 (sone )

Basic Formulas Sound Velocity

v = f λ

A

 V  1 + ln A in  V max    1 ≤ V in ≤ A V out = V max 1 + ln A A V max

ACOUSTICS

0 20 30 40 50 – 70 80 90 100 100 120 140

•

V in V max

very faint faint moderate loud very loud deafening

Sound Velocity in Gases

v

=

γ P O ρO

where: ) = ratio of the specific heat at constant volume Po = the steady pressure of the gas (N/m2) 6o = the steady or average density of the gas (kg/m3) In dry air (experimental) v = 331.45 ± 0.05 v = 1087.42 ± 0.16

m/s ft/s

Velocity of sound in air for a range of about 20° Celsius change on temperature v = 331.45 ± 0.607T C m/s

= 1052.03 ± 1.016T F

ft/s where: TC = temperature in degrees Celsius


v

TF = temperature in degrees Fahrenheit

The Sound Intensity Level

I L

For TC > 20°C,

T K

v = 331.45

m/s

273 where: TK = temperature in Kelvin

P O

where: W = sound power in watts Wo = reference sound power = 10-12 W 2

 P  I 10 log = 10 log  =  I O  P O 

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 P 2 P 2 I = = ρv 410 where: 6 = density of air

W/m2

v = velocity of sound in air 6v = characteristic impedance of air to sound = 410 rayls in air The total intensity, IT I T = I 1 + I 2 + I 3

+ ... + I n

The total pressure, PT

P T

=

P 12 + P 22 + P 32

+ ... + P n2

Sound Intensity coming from (a) a point source (isotropic) in free space W I = 4πr 2 (b) a source at ground level W I = 2πr 2

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

PWL = 10 log W + 120

Sound Pressure Level

P

I O

2

 P  = 10 log    P O 

The Sound Power Level (PWL) W PWL = 10 log W O

Recall: T K = T C + 273 T R = T F + 460 9 T F = T C + 32 5 5 T C = (T F − 32 ) 9

SPL = 20 log

= 10 log

I

The Relation of SPL and PWL (a) for a sound produced in free space by an isotropic source SPL = PWL − 20 log r − 11 (b) for a sound produced at ground level SPL = PWL − 20 log r − 8

•

Room Acoustics Optimum reverberation (at 500 to 1000 Hz) Room Reverberation Function time (s)

Recording and broadcast studios Elementary classrooms Playhouses, intimate drama production Lecture and conference rooms Cinema Small Theaters High school auditoriums General purpose auditoriums Churches

0.45 – 0.55 0.6 – 0.8 0.9 – 1.1 0.9 – 1.1 0.8 – 1.2 1.2 – 1.4 1.5 – 1.6 1.5 – 1.6 1.4 – 3.4

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

reverberation time computation) t 60 = r (0.0123 V + 0.1070) seconds where: V = room volume (m3) r = 4 for speech = 5 for orchestra = 6 for choir


α = average absorption coefficient of the

Optimum volume/person for various types of hall

Types of halls Concert halls Italian-type opera houses Churches Cinemas Rooms for speech

Optimum volume/person (m3) 7.1 4.2 – 5.1 7.1 – 9.9 3.1 2.8

B. Sabine’s formula (for actual reverberation time with average absorption less than or equal to 0.2) 0.161V t 60 = seconds a

where: V = room volume (m3) a = total absorption units (m2 – metric Sabine) (for a room: the sum of all absorption of the ceiling, walls, floor, furnishings and occupants). 0.049V t 60 = seconds a

3

where: V = room volume (ft ) a = total absorption units (ft2 – customary Sabine) Coefficient of absorption is the ratio of the

absorbed sound intensity to the incident sound intensity. α

=

I a I i

= I i − I r

Average absorption coefficient (α )

=

α1 + α 2

Total absorption (a) a = α A

=

A further correction may need to be added for higher frequency to allow for air absorption. 0.161V t 60 = seconds − S ln(1 − α ) + xV For values of " less than about 0.2 but frequencies above 1000 Hz then a modified form of Sabine’s formula is considered. 0.161V t 60 = seconds a + V

where: x = sound absorption/volume of air (m2/m3) 3

x per m at a temperature of 20°C Freq (Hz)

1000 2000 4000

30%RH ×10 –3

40%RH ×10 –3

50%RH ×10 –3

60%RH ×10 –3

70%RH ×10 –3

80%RH ×10 –3

3.28 3.28 3.28 3.28 11.48 8.2 8.2 6.56 39.36 29.52 22.96 19.68

3.28 6.56 16.4

3.28 6.56 16.4

RH = Relative Humidity

where: Ir = reflected sound intensity

α

0.049V seconds − S ln(1 − α ) where: S = total surface area (ft2) α = average absorption coefficient of the reflecting surface t 60

(unitless)

Note: " = 1 for perfect absorbent material I a

reflecting surface

+ α 3 + ... + α n

Methods of measuring absorption coefficient A. Reverberation Chamber Method

Note: The lowest frequency should not be lower than the computed frequency from the formula below to ensure a diffuse sound field where v is the volume of the room. 180 f lowest = 3 Hz v

n 2

2

(m or ft ) where: A = surface area of the absorbent structure (m2 or ft 2) C. Norris-Eyring’s formula (for actual reverberation time with average absorption coefficient greater than 0.2) 0.161V t 60 = seconds − S ln(1 − α ) where: S = total surface area (m2)

Principle of reverberation chamber method

“A measurement of reverberation time is made first without, and then with the absorbent material in the chamber.” Without the absorbent material, 0.161V


t 1 =

a

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

With the absorbent material, 0.161V t 2

=

a + δa

B. Helmholtz or Cavity or Volume Resonators

Therefore:

 1 1  −   t  2 t 1 

δa = 0.161V 

In practice some slight correction needs to be made for the behavior of sound in the chamber which can make a difference of nearly 5%.  V   1 1  δa =  55.3   −  



v

  t 2

t 1 

Absorption coefficient

α =

δa

B. Impedance Tube Method Absorption coefficient 4 A1 A2

=

( A1 + A2 ) 2 where: A1 and A2 are the maximum and minimum amplitudes of the resultant standing wave pattern reverberation of the chamber without absorbent material Note: " of impedance tube method is less than " of reverberation chamber method. Types of absorbents A. Membrane or Panel absorbers

The absorption is highly dependent upon frequency and is normally in the range of 50 to 500 Hz. They are often used in recording. 60 f =

If there is no neck, l = 0 f =

md

where: f = approximate resonant frequency

v

2r

2π V where: v = velocity of sound in air r = radius of the neck l = length of the neck V = volume of cavity

S

where: V = volume of reverberation chamber t1 = reverberation of the chamber without absorbent material t2 = reverberation of the chamber with absorbent material a = absorption of the chamber without absorbent material 4a = extra absorption due to the material v = velocity of sound in air S = surface area under measurement, which should be a single area between 10 and 12 m2

α

Resonant frequency (f) for a narrow-neck resonator is approximately vr 2π f = 2π (2l + πr )V

SATELLITE COMMUNICATIONS

•

Communications Satellite Orbit Location (Satellite Elevation category) (a) Low Earth Orbit (LEO) Satellite

Orbital height Orbital velocity Orbital time (period) Satellite Availability Typical operating frequency

: 100 – 300 mi : 17,500 mph : 1.5 hours : 15 min per orbit : 1.0 GHz – 2.5 GHz

(b) Medium Earth Orbit (MEO) Satellite


: 6,000 – 12,000 mi : 9,580 mph : 5 – 12 hours : 2 – 4 hours per orbit : 1.2 GHz – 1.66 GHz

(c) Geostationary or Geosynchronous (GEO) Satellite


: 22,300 mi : 6,879 mph : 24 hours : 24 hours per orbit : 2 GHz – 18 GHz

THE GEOSYNCHRONOUS SATELLITE

Altitude Period Orbit inclination


: 19,360 nmi : 22,284 smi : 35,855 km : 23 hr, 56 min, 4.091 s (one sidereal day) : 0°

: 6879 mph : 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 south latitude Advantages : Simpler ground station tracking : 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 1833 miles of separation distance) or more]

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

Velocity Coverage

Satellite classification according to size Size Mass Cost

Large Satellite Small Satellite Mini-Satellite Micro-Satellite Nano-Satellite

•

> 1,000 kg 500 – 1,000 kg 100 – 500 kg 10 – 100 kg < 10 kg

> $ 100 M $ 50 – 100 M $ 5 – 20 M $2–3M <$1M

Satellite velocity in orbit

v

2 3

α = AP

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

F g

=G

h=

m/s

gT 2 R 2

km − R 4π 2 where: T = satellite period g = gravitational acceleration (9.81×10-3 km/s2) 3

The escape velocity of earth is 25,000 mph or from the formula: Escape velocity = 2 gR The minimum acceptable angle of elevation is 5°. Satellite Range (distance from an earth station)

d = ( R + h)

2

− R 2 cos 2 β − R sin β

where: ( = angle of elevation

•

Frequency Allocation

The most common carrier frequencies used for SATCOM are the 6/4 and 14/12 GHz bands.

= F g F c

( Rkm + hkm )

Satellite height

For a satellite to stay in orbit, the centrifugal force caused by its rotation around earth should be equal to the earth’s gravitational pull. m s me

=

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

Satellite Orbital Dynamics

F c

4 × 1011

=

ms v 2

( R + h) ( R + h) 2 where: Fc = centrifugal force Fg = gravitational force G = gravitational constant (6.670×10-11) ms = mass of satellite PDF created with pdfFactory trial version www.pdffactory.com

Frequency bands used in satellite communications Frequency Band

225 – 390 MHz 350 – 530 MHz 1530 – 2700 MHz 2500 – 2700 MHz 3400 – 6425 MHz 7250 – 8400 MHz 10.95 – 14.5 GHz 17.7 – 21.2 GHz 27.5 – 31 GHz 36 – 46 GHz 46 – 56 GHz 56 – 100 GHz

P J L S C X Ku Ka K Q V W

Microwave frequency bands Band designation Frequency range (GHz)

L S C X Ku K Ka Millimeter Submillimeter

1–2 2–4 4–8 8 – 12 12 – 18 18 – 27 27 – 40 40 – 300 >300

7. Gain-to-Equivalent Noise Temperature Ratio G Gr + G ( LNA)

T e

=

T e

The satellite system link equations Uplink Equations C At P r ( L p Lu ) Ar At P r ( L p Lu )

N o

=

kT e

=

k

×

G T e

Expressed in dB

 G  4π D  = 10 log At P r − 20 log   + 10 log   N o  λ   T e  − 10 log Lu − 10 log k  G  C (dBK −1 ) = EIRP (dBW ) − L p (dB) +    N o  T e  − Lu (dB) − k ( DBWK ) C

Earth coverage is approximately one-third of the earth’s surface with approximate antenna beamwidth of 17°.

•

The Satellite System Parameters 1. Transmit Power and Bit Energy P E b = P t T b = t f b

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) EIRP = P r Gt

where: Pr = total power radiated from an antenna Gt = transmit antenna power gain 3. Equivalent Noise Temperature (T e) T e = T o ( NF − 1)

where: To = temperature of the environment (K) NF = noise factor (absolute value) 4. Noise Density (N o)

N o

=

N BW

= kT e

5. Carrier-to-Noise Density Ratio C C

N o

=

kT e

6. Energy Bit-to-Noise Density Ratio C

E b N o

=

f b N BW

=

CBW Nf b

Uplink Equations C At P r ( L p Ld ) Ar

N o

=

kT e

=

At P r ( L p Ld ) k

×

G T e

Expressed in dB

 G  4π D  = 10 log At P r − 20 log   + 10 log   N o  λ   T e  − 10 log Ld − 10 log k  G  C (dBK −1 ) = EIRP (dBW ) − L p (dB) +    N o  T e  − Ld (dB) − k ( DBWK ) C

MULTIPLEXING

•

Frequency Division Multiplexing

Voice band frequency (VF): 0 – 4 kHz Basic voice band (VB) circuit is called 3002 Channel: 300 – 3000 Hz band Note: The basic 3002 channel can be subdivided into 24 narrower 3001 (telegraph) channels that have been frequency-division multiplexed to form a single 3002 channel. A. Basic Group f c = (112 − 4n)

kHz where: f c = channel carrier frequency n = channel number


17 18 D25 D26 D27 D28

Lower sideband f LSB = f c – (0 to 4 kHz) Upper sideband f USB = f c + (0 to 4 kHz) B. Basic Supergroup f c = (372 + 48n)

kHz

where: f c = group carrier frequency n = group number Lower sideband f LSB = f c – (60 to 108 kHz) Upper sideband f USB = f c + (60 to 108 kHz) C. Basic Mastergroup Two categories of mastergroups U600 – may be further multiplexed and used for

higher-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 Frequencies L600 Mastergroup Supergroup Carrier frequency (kHz)

1 2 3 4 5 6 7 8 9 10

612 Direct 1116 1364 1612 1860 2108 2356 2724 3100

U600 mastergroup Supergroup Carrier frequency (kHz)

13 14 15 16

1116 1364 1612 1860

2108 2356 2652 2900 3148 3396

Summary of AT&T’s FDM Hierarchy

Group Supergroup

= 12 VB channels = 5 Groups = 60 VB channels Mastergroup = 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 Hierarchy

Group Supergroup

= 12 VB channels = 5 Groups = 60 VB channels Mastergroup = 5 Supergroups = 25 Groups = 300 VB channels Supermastergroup = 3 Mastergroups = 15 Supergroups

•

Time Division Multiplexing Summary of Digital Multiplex Hierarchy (North American)

Line Type

Digital Signal

Bit rate (Mbps)

Channel Capacity

Services Offered

Medium

T1

DS – 1

1.544

24

Twisted pair

T1C

DS – 1C

3.152

48

T2

DS – 2

6.312

96

T3

DS – 3

44.736

672

T4M

DS – 4

274.176

4032

T5

DS – 5

560.160

8064

VB telephone VB telephone VB tel, picture phone VB tel, picture phone, TV Same as T3 except more capacity Same as T3 except more capacity


Twisted pair Twisted pair, #wave coax, #wave coax, optical fiber optical fiber

Summary of CEPT 30 + 2 PCM Multiplex Hierarchy (European) Level Data Rate (Mbps) Channel Capacity

1 2 3 4 5

Level

2.048 8.448 34.368 139.264 564.992

30 120 480 1920 7680

Speech Measurement

For typical single talker average power in dBm: P(dBm) = VU reading – 1.4 dB

Japanese Multiplex Hierarchy Data Rate (Mbps) Channel Capacity

1 2 3 4 5

1.544 6.312 32.064 97.728 565.148

3.) Maximum intelligibility for voice frequency: 1,000 and 3,000 Hz. 4.) Maximum voice energy is located between 250 and 500 Hz.

24 96 480 1440 7680

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

where: N = number of speakers

•

The telephone Set Pulse Dialing

To transmit a digit, it takes 0.1 second per pulse + 0.5 second inter-digital delay time.

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

DTMF Frequencies Frequencies 1209 Hz 1336 Hz 1 2 697 Hz 4 5 770 Hz 7 8 852 Hz * 0 941 Hz

With CCITT system, a 125-#s frame is divided into 32 equal time slots. E – 1 Carrier Framing and alarm channel TS 0

Voice Voice Common Voice Voice channel Channel Signaling Channel Channel 1 2 – 15 channel 16 – 29 30 TS 1 TS 2 – 16 TS 17 TS 18 – 30 TS 31

Minimum BW

Average DC

Clock Recovery

Error Detection

UPNRZ BPNRZ UPRZ BPRZ BPRZ-AMI

f b/2 f b/2 f b f b f b/2

+ V/2 0V + V/2 0V 0V

Poor Poor Good Best Good

No No No No Yes

TELEPHONY

•

Introduction

1.) 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.

3 6 9 #

Network Call Progress Tones Tone Frequency (Hz)

Dial Tone Ringback Busy Signal

Line-Encoding Summary Encoding Format

1477 Hz

•

350 + 440 440 + 480 480 + 620

Switching and Signaling n( n − 1) N =

2 where: N = number of connections n = number of subscribers

•

Traffic Engineering Measurement of Telephone Traffic A = C × T

where: A = traffic intensity in Erlangs C = designates the number of calls originated during a period of 1 hr (calls/hr or calls/min) T = the average holding time, usually given in hours (hr/call or min/call)

Nature of Speech

1.) Sound pressure wave of speech contains frequencies: 100 Hz to 10 kHz. 2.) Speech power range: 10 to 1,000 #W.

A =

S t

where: S = sum of all the holding time (min) t = observation period (1 hr or 60 min)


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

General Formula for Mobile radio Propagation Path Loss: P r = P t – 134.4 – 38.4logr 1 + 20logh1 +20logh2 + Gt + Gm

Grade _ of _ service =

Number _ of _ lost _ calls Total _ no. _ of _ offered _ calls

•

GSM Network Radio-Path Propagation Loss

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 D q = R

 d  ∆ P = 40 log 1    d 2  (40 dB/decade path loss)

Frequency Reuse factor, K

 h '  ∆G = 20 log 1    h1 

q

(A base station antenna height gain of 6dB/octave) where: 2P = the difference in two receive signal strengths based on two different path lengths d1 and d2 2G = the difference in two receive signal strengths based on two different antenna heights h1 and h1’ Receive signal in decibels For non-obstructive path

P r = P ro

 r   he '    − γ log  + 20 log  + α  r  h1   o 

For obstructive path

K =

3

Radio Capacity A. Analog, FDMA and TDMA cellular system Bt m= 2  C  Bc   3  I 

where: Bt = total allocated spectrum Bc = channel bandwidth (C/I) = required carrier-to-interference ratio in linear values B. CDMA cellular system M m = K

where: M = total number of voice channels K = frequency reuse factor

 r  P r = P ro − γ log   + L + α  r o  where: r = distance between the base and the mobile unit in mi or km he’ = effective antenna height L = shadow loss Pro = 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) Base-station antenna height (h 1) Base-station power at the antenna Base-station antenna gain (G t) Mobile-unit antenna height ro Mobile-unit antenna gain (G m)

= 3 K

q2

Antenna Separation Requirement A. At the Base Station h

d

= 11

where: h = antenna height d = spacing between two antennas B. At the Mobile Unit

900 MHz 30.46 m 10 watts 6 dBd 3m 1.6 km 0 dBd

A separation of a half-wavelength between two mobile antennas is required at 850 MHz. Therefore, the separation between two antennas needs to be only 0.18 m (about 6 inches) at the cellular frequency of 850 MHz.


Cell Splitting Formulas

S =

A

N =

2

3.464r where: N = number of cells A = total area to be covered r = radius inscribed in the hexagon

3λ Re L

where: S = separation R e = effective earth’s radius L = path length

• MICROWAVE COMMUNICATIONS

Path Characteristics Free-space attenuation or path Loss, L p 2

 4πd  L P =    λ  L P = 32.4 + 20 log d km + 20 log f MHz L P = 92.4 + 20 log d km + 20 log f GHz L P = 36.6 + 20 log d mi + 20 log f MHz L P = 96.6 + 20 log d mi + 20 log f GHz

•

Microwave Passive Repeater Gain of a passive repeater 4π A cos α G P = 20 log A P 2

sin θ 2 λ where: A = actual surface area of the repeater (ft2) Acos" = AP or Aeff = projected or effective area of the passive repeater (ft2)

= A

Antenna Gain, G

Also, G P

2

 D  G = 6   λ  G = −52.6 + 20 log D ft + 20 log f MHz G = 7.5 + 20 log D ft + 20 log f GHz G = −42.3 + 20 log Dm + 20 log f MHz G = 17.7 + 20 log Dm + 20 log f GHz

= 22.1 + 20 log A P + 40 log f GHz

Beamwidth of a fully illuminated passive repeater

θ

=

58.7λ L

where: L = effective linear dimension of the repeater in the direction in which the beamwidth is to be measured Net Path Loss (NPL) NPL( dB) = GT − L P 1

+ G P − L P 2 + G R

where: GT = transmit antenna gain LP1 = path loss on path 1 GP = passive repeater gain LP2 = path loss on path 2 GR = receive antenna gain Near field and far field conditions 1 πλd '

k

If

1 k

1 k

=

4 A

− L fTB

RSLdBm = (minimum RF input)dBm + FM dB

where: Po = transmitter output LfTA = total fixed losses at the transmitter side which includes feeder loss, connector loss, branching loss, waveguide loss etc. LfRB = total fixed losses at the receiver side GT = transmitter antenna gain GR = receiver antenna gain min. RF input = practical receiver threshold FM = fade margin

< 2.5 , near-field condition exists

Fade Margin – an attenuation allowance so that

> 2.5 , far-field condition exists

anticipated fading will still keep the signal above specified minimum RF input.

where: d’ = length of the path in question (i.e., the shorter distance)

•

Receive Signal Level (RSL) RSLdBm = P odBm − L fTA + GT − L P + G R

Protection Switching

The antenna separation required for optimum operation of space diversity system may be calculated using the formula:

FM = 30 log d km

+ 10 log 6abf GHz − 10 log(1 − R) − 70

where: a = roughness factor = 4 for smooth terrain, including over water = 1 for average terrain, with some roughness = 0.25 for mountainous, very rough b = 0.5 for hot, humid coastal areas


= 0.25 for normal interior temperature or subarctic areas = 0.125 for mountainous or very dry but nonreflective areas System Gain, G S GS = P OdBm – (minimum RF input)dBm System Reliability Estimates b) Based on Propagation R = (1 − U ndp ) × 100%

where: Undp = non-diversity outage probability for a given path U ndp

1.5

= abf

−6

3

d (1.25 × 10 )10

− FM 10

where: f = frequency in GHz d = distance in miles c) Based on equipment R = (1 − U ) × 100%

U =

MTTR MTBF + MTTR

where: U = unavailability or probability of outage MTTR = Mean Time To Repair MTBF = Mean Time Between Failures or Mean Time Before Failures Also, Outage

U =

8760 _ hours Note: Downtime or Outage time (in hours per year) A =

MTBF MTBF + MTTR

where: A = Availability

Fresnel Zone Radius/Clearance/Height

R ft

= 72.1

nd 1 d 2 f GHz d mi

where: d1 and d2 are distances in miles n = number of Fresnel Zone (n = 1 for 1st FZ; n = 2 for 2nd FZ, etc.) Rm

= 17.3

Fn = F1(n) F2 = 0.6(F1) F3 = 0.6(F2)

nd 1d 2 f GHz d km

where: d1 and d2 are distances in kilometers PDF created with pdfFactory trial version www.pdffactory.com

COMMUNICATIONS Formulas and Concepts V3.pdf

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