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