Fundamental of Transmissions

Dr. FarahmandUpdated: 2/9/2009

What is telecommunications? Conveying information between two points

or between one and multi-points Transmitting information wirelessly is

achieved via electromagnetic signals (E) Electric current flowing through a wire creates

magnetic field around the wire An alternating electric current flowing through a

wire creates electromagnetic waves Electromagnetic radiation is waves of energy These waves collectively called electromagnetic

spectrum

RXTX Medium

Signal Characteristics Analog (continuous) or digital

(discrete) Periodic or aperiodic Components of a periodic

electromagnet wave signal Amplitude (maximum signal

strength) – e.g., in V Frequency (rate at which the

a periodic signal repeats itself) – expressed in Hz

Phase (measure of relative position in time within a single period) – in deg or radian (2 = 360 = 1 period)

S t A ft

phase

A am plitude

f frequncy

T period f

( ) s in ( )

/

2

1

Period ic

S t S t T

:

( ) ( )

Sine Waves

Sound Wave Examples

Each signal is represented by x(t) = sin (2f.t)

A dual tone signal with f1 and f2 is represented by x(t) = sin (2f1.t) + sin (2f2.t)

f = 5Kz

f = 1Kz

Periodic Signal Characteristics The simplest signal is a

sinusoidal wave A sine wave can be

expressed in time or space (wavelength) Wavelength is the

distance the signal travels over a single cycle

Wavelength is a function of speed and depends on the medium (signal velocity)

vf

T f

v x m

1

3 1 0 8

/

/ sec

Exact speed light through vacuum is 299,792,458 m/s

f=v

Periodic Signal Characteristics A signal can be made of many frequencies

All frequencies are multiple integer of the fundamental frequency

Spectrum of a signal identifies the range of frequencies the signal contains

Absolute bandwidth is defined as: Highest_Freq – Lowest_Freq

Bandwidth in general is defined as the frequency ranges where a signal has its most of energies

Signal data rate Information carrying capacity of a signal Expressed in bits per second (bps) Typically, the larger frequency larger data rate

Example

Periodic Signal Characteristics

Consider the following signal Consists of two freq. component (f) and

(3f) with BW = 2f

S t ft f t

F undam en ta l freq f

M ax freq f

Abs BW f f f

( ) ( / ) s in ( ) ( / ) s in ( ( ) )

_

_

_

4 2 4 3 2 3

3

3 2

f 3f

BW

What is the Max amplitude of this

component?

http://www.jhu.edu/~signals/listen-new/listen-newindex.htm

Second harmonic

Periodic Signal CharacteristicsS(t)=sin(2ft)

S(t)=1/3[sin(2f)t)]

S(t)= 4/{sin(2pft) +1/3[sin(2f)t)]}

Frequency Domain Representation

S(t)= 4/{sin(2ft) +1/3[sin(2f)t)]}

frequency domain function for a single square pulse that has the value 1{s(t)=1} between –X/2 and X/2, and is 0 {s(t)=1} elsewhere

Refer to NOTES!

Data Rate & Frequency Example:

What is f1? What is f2? Which case has larger

data rate? (sending more bits per unit of time)

0 1 0 0

1 0

Case 1: f1

1 msec

Case 2: f2

f1 = 2(1/10^-3)=2KHz Case I data rate=one bit per (0.25msec) 4 Kbps f2 = 1 KHz data rate=2Kbps

Case 1 has higher data rate (bps)

Bandwidth and Data Rate Case 1:

Assume a signal has the following components: f, 3f, 5f ; f=10^6 cycles/sec What is the BW? What is the period? How often can we send a bit? What is the data rate? Express the signal equation in time domain

Case 2: Assume a signal has the following components: f, 3f, 5f; f=2x10^6 cycles/sec What is the BW? What is the period? How often can we send a bit? What is the data rate?

Case 3: Assume a signal has the following components: f, 3f ; f=2x10^6 cycles/sec What is the BW? What is the period? How often can we send a bit? What is the data rate? Express the signal equation in time domain

BW=4MHzT=1usec

1 bit every 0.5usecData rate=2*f=2bit/usec=2MHz

BW=8MHz (5 x 2 - 2=8)T= ½ usec

1 bit every 0.25usecData rate=2*f=2bit/0.5 usec=4MHz

BW=4MHzT=0.5 usec

1 bit every 0.25usecData rate=2*f=4bit/usec=4MHz

Remember: Greater BW larger cost but Lower BW more distortion;

Nyquist Formula and Bandwidth Assuming noise free system and assuming that only one

bit is provided to represent the signal: Nyquist’s formula states the limitation of the data rate

due to the bandwidth: If the signal transmission rate is 2B, then a

signal with frequency of less or equal B is required to carry this signal: TR(f)=2BfB

If bandwidth is B (Hz) the highest signal rate that can be carried is 2B (bps): f=BTR(f)B

Example: if the highest frequency is 4KHz (bandwidth) a sampling rate of 8 Kbps is required to carry the signal

Remember: Channel Capacity = (number of bit) x (signal bandwidth)

Channel CapacityNyquist’s formulation when multilevel signaling is present

channel capacity (C) is the tightest upper bound on the amount of information that can be reliably transmitted over a communications channel (max. allowable data rate)

What if the number of signal levels are more than 2 (we use more than a single bit to represent the sate of the signal)?

C B M

M

meber

M M

n

2

2

2

2

2

lo g ( )

R e :

lo g ( ) ln ( ) / ln ( ) C = Maximum theoretical

Channel Capacity in bps M = number of discrete signals

(symbols) or voltage levels n = number of bits per symbol

Remember: More bits per symbol more complexity!

Example: Log2(8)=ln(8)/ln(2)=3

Channel Capacity Example: Voice has a BW of 3100 Hz. calculate the

channel capacity Assuming we use 2 signal levels Assuming we use 8 signal levels

channel capacity required to pass a voice signal:

Channel capacity (or Nyquist capacity) is 2 x 3100 cycles/sec = 6.1Kbps – note in this case one bit is being used to represent two distinct signal levels.

If we use 8 signal levels: channel capacity: 2x3100x3=18600 bps

S/N Ratio

The signal and noise powers S and N are measured in watts or volts^2, so the signal-to-noise ratio here is expressed as a power ratio, not in decibels (dB)

SNRSigna lPow er w att Vo lt

N o isePow er w att Vo lt

m eber

y y x

Pow er dB Pout P in

Pow er dBm P mW mW

dB

x

1 0

1 0

1 0

1 0 1

1 0

2

2

1 0

1 0

1 0

lo g( / )

( / )

R e :

lo g

( ) lo g ( / )

( ) lo g ( ( ) / )

Example: Assume signal strength is 2 dBm and noise strength is 5 mW. Calculate the SNR in dB.

2dBm 1.59 mWSNR = 10log(1.59/5)=-5dB

Signal ImpairmentsAttenuation

Strength of a signal falls off with distance over transmission medium

Attenuation factors for guided media: Received signal must have

sufficient strength so that circuitry in the receiver can interpret the signal

Signal must maintain a level sufficiently higher than noise to be received without error

Typically signal strength is reduced exponentially

Expressed in dB

Attenuation is greater at higher frequencies, causing distortion

A ttenua tion dBd

A ttenua tion dBd

Where

w aveleng th d d is ce

( ) lo g ( )

( ) lo g ( )

:

; tan

1 04

2 04

1 02

1 0

Signal ImpairmentsAttenuation Impacts

Lowers signal strength Requires higher SNR Can change as a

function of frequency More of a problem in

analog signal (less in digital)

Higher frequencies attenuate faster

Using equalization can improve – higher frequencies have stronger strength

Signal ImpairmentsDelay Distortion

In bandlimited signals propagation velocity is different for different frequencies Highest near the center

frequency Hence, bits arrive out of

sequence resulting in intersymbol

interference limiting the maximum bit

rate!

Categories of Noise

Thermal Noise Intermodulation noise Crosstalk Impulse Noise

Thermal Noise Thermal noise due to agitation of

electrons Present in all electronic devices and

transmission media Cannot be eliminated Function of temperature Particularly significant for satellite

communication When the signal is received it is very weak

Thermal Noise Amount of thermal noise to be found in a

bandwidth of 1Hz in any device or conductor is:

N0 = noise power density in watts per 1 Hz of bandwidth

k = Boltzmann's constant = 1.3803 10-23 J/K T = temperature, in Kelvins (absolute

temperature) – zero deg. C is 273.15 Expressed in dBW 10log(No/1W)

W/Hz k0 TN

Thermal Noise Noise is assumed to be independent of frequency Thermal noise present in a bandwidth of B Hertz (in

watts):

or, in decibel-watts

TBN k

BTN log10 log 10k log10

W/Hz k0 TN

Thermal Noise (MATLAB Example)%MATLAB CODE:T= 10:1:1000; k= 1.3803*10^-23;B=10^6;No=k*T;N=k*T*B;N_in_dB=10*log10(N);semilogy(T,N_in_dB)title(‘Impact of temperature in

generating thermal noise in dB’)xlabel(‘Temperature in Kelvin’)ylabel(‘Thermal Noise in dB’)

0 100 200 300 400 500 600 700 800 900 1000

-102.15

-102.16

-102.17

-102.18

-102.19

-102.2

Impact of temperature in generating thermal noise in dB

Temperature in Kelvin

The

rmal

Noi

se in

dB

Other Types of Noise Intermodulation noise – occurs if signals with different

frequencies share the same medium Interference caused by a signal produced at a frequency

that is the sum or difference of original frequencies Crosstalk – unwanted coupling between signal paths Impulse noise – irregular pulses or noise spikes

Short duration and of relatively high amplitude Caused by external electromagnetic disturbances, or

faults and flaws in the communications system

Question: Assume the impulse noise is 10 msec. How many bits of DATA are corrupted if we are using a

Modem operating at 64 Kbps with 1 Stop bit?

Other Types of Noise - Example

Intermodulation noise

Crosstalk

Impulse noise

Channel Capacity with Noise and Error An application of the channel capacity concept to an

additive white Gaussian noise channel with B Hz bandwidth and signal-to-noise ratio S/N is the Shannon–Hartley theorem:

Establishing a relation between error rate, noise, signal strength, and BW

If the signal strength or BW increases, in the presence of noise, we can increase the channel capacity

Establishes the upper bound on achievable data rate (theoretical) Does not take into account impulse and attenuation

Note: S/N is not in dB and it

is log base 2!

Noise Impact on Channel Capacity

Presence of noise can corrupt the signal Unwanted noise can cause more damage to

signals at higher rate For a given noise level, greater signal strength

improves the ability to send signal Higher signal strength increases system nonlinearity

more intermodulation noise Also wider BW more thermal noise into the system

increasing B can result in lower SNR

Example of Nyquist Formula and Shannon–Hartley Theorem What is the wavelength associated

with the highest energy level? Calculate the BW of this signal. Assuming the SNR = 24 dB,

Calculate the maximum channel capacity.

Using the value of the channel capacity, calculate how many signal levels are required to generate this signal?

How many bits are required to send each signal level?

Express the mathematical expression of this signal in time domain.

What type of signal, more likely, is this? (TV, Visible light, AM, Microwave) – Next slide

3MHz 4MHz

=108 m B=4-3=1 MHz

SNRdB(24)log-1(24/10)102.4= 251

C=Blog2(1+S/N)=8MbpsC=2Blog2MM=16

2n=Mn=4Signal Type: AM

/4

/3x4

http://www.adec.edu/tag/spectrum.html

Radio-frequency spectrum: commercially exploited bands

http://www.britannica.com/EBchecked/topic-art/585825/3697/Commercially-exploited-bands-of-the-radio-frequency-spectrum

Expression Eb/N0

Ratio of signal energy per bit to noise power density per Hertz R = 1/Tb; R = bit rate; Tb = time required to send one bit; S = Signal Power (1W = 1J/sec) Eb=S.Tb No = Thermal noise (W/Hz)

The bit error rate for digital data is a function of Eb/N0

Given a value for Eb/N0 to achieve a desired error rate, parameters of this formula can be selected

As bit rate R increases, transmitted signal power must increase to maintain required Eb/N0

TR

S

N

RS

N

Eb

k

/

00

Note that as R increases power must increase as well to maintain signal quality

SNR & Expression Eb/N0

Using Thermal noise within the bandwidth of B Hertz (in watts): N=NoxB

Using Shannon’s Theorem – Channel Capacity in the presence of noise

The relation between SNR and Eb/No will be (R=C=Data rate)

C/B expressed in bps/Hz and called Spectral Density

12/

/12

)1(log/

)1(log

/

/

2

2

BC

BC

NS

NS

SNRBC

SNRBC

R

B

N

S

RBN

SB

RN

S

N

Eb )( 000

)12( /

0

BCb

C

B

N

E

Q: What will be Eb/No if the spectral density is 6 bps/Hz???

Probability of Error

Question: Assume we require Eb/No = 8.4 dB

to achieve bit error rate of 10^-4.

Assume temperature is 17oC and data rate is

set to 2.4 Kbps. Calculate the required level of the received

signal in W and dBW. 8.4 dB

10^-4

Probability of Error

Question: Assume we require Eb/No = 8.4 dB

to achieve bit error rate of 10^-4.

Assume temperature is 17oC and data rate is

set to 2.4 Kbps. Calculate the required level of the received

signal in W and dBW. 8.4 dB

10^-48.4 dB 6.91

17oC 290oKelvinR=2400 bps

K=1.38*10^-23Eb/No=S/(KTR)S=-161 dBW

Review: Power in Telecommunication Systems

Remember:

Example 1: if P2=2mW and P1 = 1mW 10log10(P2/P1)=3.01 dB

Example 2: if P2=1KW and P1=10W 20dB What if dB is given and you must find P2/P1?

P2/P1 = Antilog(dB/10) = 10 dB/10 . Example 3: if dB is +10 what is P2/P1?

P2/P1 = Antilog(+10/10) = 10 +10/10 = 10

yxyy Hencexthenx loglog)10log(10

Colors and Wavelengths

Color RedOrange YellowGreenBlueViolet

Wavelength (nm)780 - 622 622 - 597597 - 577577 - 492492 - 455455 - 390

Frequency (THz)384 - 482482 - 503503 - 520520 - 610610 - 659659 - 769

1 terahertz (THz) = 10^3 GHz = 10^6 MHz = 10^12 Hz,  1 nm = 10^-3 um = 10^-6 mm = 10^-9 m.

The white  light is a mixture of the colors of the visible spectra.

Wavelengths is a common way of describing light waves.  Wavelength = Speed of light in vacuum / Frequency.

f=v

Colors and Wavelengths

Colors and Wavelengths

References Online calculator:

http://www.std.com/~reinhold/BigNumCalc.html

Wavelengths and lights http://www.usbyte.com/common/approximate_wavelength.htm

& http://eosweb.larc.nasa.gov/EDDOCS/Wavelengths_for_Colors.html

Learn about decibel http://www.phys.unsw.edu.au/jw/dB.html