Introduction to Digital Communications by Erol Seke For the course - - PowerPoint PPT Presentation
Introduction to Digital Communications by Erol Seke For the course Digital Communications OSMANGAZI UNIVERSITY The Goal Transfer information from source point to destination correctly (and in shortest possible time, in most cases)
by Erol Seke For the course “Digital Communications”
OSMANGAZI UNIVERSITY
Transfer information from source point to destination correctly (and in shortest possible time, in most cases)
Information Generator Information User
Source point Destination point
Noise Sources Information Channel
Information
The probability of an event occurring Low High High self-information Low self-information
Example : Consider the statement “We will have an exam tomorrow” The statement has highly valuable, since it is the beginning of a new semester and an exam at this time is very unusual (low probability)
Self-info of the event Probability of the event
)) ( log( ) ( 1 log ) ( E P E P E I
Noise Sources
(quantization noise and granular noise are evaluated in different contexts)
Channel input Channel output Noise is usually assumed to be (this assumption is not baseless)
N kS S
i Additive : White : |F(N )|=c (has the same power at all frequencies) Gaussian : Probability distribution function is Gaussian So it is AWGN
Little More Detailed Information System
Source coder / decoder : Codes for efficient transmission, removes redundancy (compression) Channel coder / decoder : Codes for reliable transmission, adds redundancy Some examples of (electronic) source, channel and destination Microphone – Twisted pair of wire - Amplifier Modem - Twisted pair of telephone line - Modem Radio transmitter – Air – Radio receiver Computer – Ethernet cable - Computer Computer’s data storage medium – Fiber optic network – Another computer’s storage Digital data generator – Magnetic disk – Digital data user Fax scanner – Telephone system – Fax printer Digital TV data from satellite – Atmosphere – Digital TV receiver TV Remote controller – Air – IR sensor/receiver on TV
Little More Detailed Communication System
The Problem
distortion/bandlimiting additive noise The noise is the most important obstacle in front of the fast and reliable data transmission
Q : What are other problems/factors that limit the data transmission rate and reliability?
terrestrial microwave AM radio
1 kHz 10 GHz 1 GHz 100 MHz 10 MHz 1 MHz 100 kHz 10 kHz 100 GHz 100 THz 1 PHz 100 km 100 m 1 km 10 km 10 m 1 m 100 mm 10 mm 1 mm 1 µm 10 nm 10 PHz 10 THz 1 THz 10 µm 100 µm
ELF VLF VF LF MF HF VHF UHF SHF EHF FIR MIR NIR
NUV
EUV SX Twisted pair coax FM TV satellite fiber radar
Din Transmitter
clk
Receiver
might be a twinax cable with common ground 1 2 3
bits
propagation delay
1 2 3
Simplest Design
transmitter receiver binary stream. noise binary stream.
channel
Let us assume that data and clock line lengths differ by 10 cm
(speed of e.m. wave on copper is about 2x108 m/s) Problem is : for a 1GHz clock, 0.5 ps is about half a clock cycle.
what you need/want what you get
Solution to clock lead/lag problem Solution is to generate clock from data at the receiver.
The data signal should necessarily be designed to allow such an operation
Transmitter Din
clk
sync. Receiver bit synchronizer received data signal generated clock
A Phase Locked Loop (PLL) can be used if there are enough transitions in the signal
Same approach is used in wireless comm. systems
Simplest System (what to send) mapping waveform generation baseband modulation receiver detector binary stream. noise binary stream.
channel
…010011010010…
A B Tb
… …
There are several waveform choices to represent 0/1 or 00/01/10/11 or … There are also voltage/power range and bit-period issues
is mapped to what?
…
bandpass modulation Spectrum and Media Sharing Issues transmitter binary stream. receiver binary stream. problem ! media (air) does not belong to anyone
f
transmitter-1 tr-2 tr-3 tr-4 tr-n available bandwidth is shared among transmitters (must be managed by a central authority) How? : bandpass modulation
f
baseband signal spectrum
f
transmitted signal spectrum tr-5
fc
Modulation means : controlling a quantity/feature/property by another quantity in communications : controlling amplitude/phase/frequency/shape of a signal by another signal baseband modulation : doing modulation and still keeping the resulting signal in baseband baseband : a frequency band that centers at the zero frequency (lower frequencies implied) bandpass modulation : doing modulation in order to move the center frequency of the
Why? 1) adapt the characteristics of the signal to the channel's and improve efficiency 2) share the transmission media by other transmitters 3) in case of RF, improve antenna efficiency 4) other reasons; spread spectrum, security, reliability, necessity etc.
fiber optics
Basic System
baseband modulator receiver detector binary stream. noise binary stream.
channel
bandpass modulator transmitter bandpass demodulator receiver Since the spectrum of the signal is very important property and changed several times on the communication path, we urgently need to recall some junior- sophomore knowledge on frequency/spectrum issues. carrier synchronization bit/symbol sync
1
) sin( ) cos( ) (
n
n
t n c t n b t y
Any periodic waveform (with some limits, of course) can be constructed by an infinite sum of sin and cos signals where
is called the nth harmonic of the fundamental frequency
2 T fo 1
and
t y(t) T
Example periodic waveform
Fourier Series
The coefficients bn and cn can be calculated using
2 / 2 /
) ( 1
T T
t y T b
2 / 2 /
) cos( ) ( 2
T T
dt t n t y T b
2 / 2 /
) sin( ) ( 2
T T
dt t n t y T c , 3 , 2 , 1 n
1
,..., 1 , n
) 1 ( 1
2 / 2 /
T T T
dt T b
T T
T
t n t n Tn dt t n dt t n T b
2 / 2 / 2 / 2 /
) sin( ) sin( 2 ) cos( ) cos( 2
) sin( ) 2 sin( ) sin( ) sin( 1 ) / 2 sin( ) / 2 sin( 1
2 / 2 /
n T T T
b n n n n T t n T t n n
(we see this from the figure, thus no need for integration. It is an odd function) (mean value is zero, just as seen in the figure)
2 / 2 / 2 / 2 /
) / 2 cos( ) / 2 cos( 1 ) sin( ) sin( 2
T T T
T t n T t n n dt t n dt t n T c
) cos( 1 2 1 ) cos( ) cos( 1 1 n n n n n cn
... 3 , 1
) sin( 4 ) (
n
t n t y
Therefore
is n n even is n n
n
, 4 , ) 1 ( 1 2
interpretation: infinite sum of odd harmonics of fundamental frequency. The magnitude of the sin- waves decreases inversely with the harmonic number
cn n
2 / 2 /
) cos( ) ( 2
T T
dt t n t y T b
2 / 2 /
) sin( ) ( 2
T T
dt t n t y T c
) sin( ) cos( ) (
n
n
t n c t n b t y 2 ) cos(
t jn t jn
e t n
2 ) sin( j e e t n
t jn t jn
Euler eq.’s
n t jn n
a t y
) (
2 / 2 /
) ( 1
T T t jn n
dt e t y T a
exponential Fourier series coefficients of exponential Fourier series
2
: Fundamental Angular Frequency
: Harmonics
: Zero frequency component or DC value (or mean) Any periodic signal which satisfies Dirichlet conditions can be represented by a weighted sum of (possibly infinite number of) sinusoids with different magnitude and delay (phase)
Example : Periodic Pulse Signal
T sinc
(envelope)
2
(zero crossing (doesn’t have to coincide with a component))
Larger To means tighter components (not periodic) the number of components per cycle
if
(continuous) We call it the Fourier Transform
Fourier Transform
2 / 2 /
) ( 1
T T t jn n
dt e t y T a
Making the period T infinity in order to handle arbitrary (not periodic) waveforms
dt e t y Y
t j
) ( ) (
2
T
As
T
and the spectrum covers everywhere (continuous) We no longer have coefficients for linear sum but continuous function for linear integral, so
n t jn n
a t y
) (
d
e Y t y
t j
) ( 2 1 ) (
) ( ) ( t x X
F
) ( ) (
1
X t x
F ) ( ) ( X t x
The notation is Forward transform Inverse transform Transform pair
What do we find using Fourier Transform?
cos( ) sin( )
j t
e t j t
dt e t y Y
t j
) ( ) (
d
e Y t y
t j
) ( 2 1 ) (
n t jn n
a t y
) (
How do we calculate FT or FS in real applications? We can not! We can only mimic FT and calculate DFT using finite number of samples from y(t) Since this is a correlation/convolution/inner-product of two signals, we are actually measuring their similarity. Or, how much of a sinusoid with frequency is in the signal y(t) Therefore Y() are the magnitude&phase of the sinusoid with frequency So that y(t) is a sum of those sinusoids as If y(t) is periodic, then
Discrete Fourier Transform
1 / 2
] [ ] [
N n N kn j
e n x k X
1 / 2 1
] [ ] [
N k N kn j N
e k X n x
Having N samples that assumably span a complete period of a periodic waveform (that is, the samples repeat every N) the Discrete Fourier Transform (DFT) pair is defined as
t n
N samples We get N complex numbers representing magnitude&phase of the sinusoids whose sum make up the continuous periodic signal (hopefully! remember Nyquist)
DFT
Why FT? : Some properties of the signal may be seen easier in frequency domain than in time domain
Short Time Fourier Transform
window width In 1946 Dennis Gabor calculated the Fourier Transform for short time periods on a long time signal in order to expose timing information of local features in the signal
t f |Xs(f)|
An example STFT in discrete domain
Mimic Continuous Transform
It is not possible to find a single transform coefficient set for time variant signals but for periodic signals it may be possible to mimic continuous FT by heavily oversampling the signal. Higher the number of samples per period, higher the resemblance to continuous FT
128 bit binary data N=128 FFT (magnitude) N=12800 FFT (magnitude) FFT of 100 times oversamples data zoom-in exposes sinc structure
interpolation of Fourier coefficients sampling
sampling
reconstruction
FT IFT DFT IDFT
spatial interpolation
( ) x t ( ) Y f [ ] x n [ ] Y k
t f
n k
Discrete Time Relation
Some Properties of Fourier Transform
df f Y f X dt t y t x ) ( ) ( ) ( ) (
df f X dt t x
2 2
) ( ) (
Parseval’s relation Rayleigh’s property
) ( ) ( ) ( ) ( ) ( ) ( f Y f X t y t x t y t x F F F
Convolution
) ( ) ( t X f x F
) ( ) ( t X f x F
Duality
( ) ( ) ( ) ( ) Ax t By t AX f BY f F
Linearity
Power and Energy of a Signal
The energy supplied from the source x(t) and spent on the 1 ohm resistor is defined as
dt t x Ex
2
) (
2 2
2 1
) ( lim
T T
dt t x P
T T x
T x
dt t x P
2 1
) (
x(t)
1 Ω
x(t) Volts
If the signal x(t) does not diminish over time then the energy may not be finite. In such cases we define power of the signal as For periodic signals with period To if
x
E
it is an energy type signal It is common to define power as the energy delivered per unit time.
) 8 cos( ) ( t t y
Find the power and energy of the waveform Solution
T a a
dt t x T P
2
) ( 1
16 / 1 16 / 1 2 8 1
) 8 ( cos 1 dt t P 2 1 32 ) 16 sin( 2 8
16 / 1 16 / 1
t t P
Since
2 1
it is a power signal. Therefore it is not an energy signal. So,
E
. Example
Power and Energy Spectral Densities
df f X dt t x
2 2
) ( ) (
According to Rayleigh’s property
dt t x Ex
2
) (
and the definition of energy And for a nonperiodic signal
df f G P
x x
) (
2
) ( ) ( f X f
x
Energy Energy Spectral Density Similarly, the Power Spectral Density for a periodic signal is defined by the equation
df f X Ex
2
) (
Power Power Spectral Density
2
) ( 1 lim ) ( f X T f G
T T x
Linear System
) (t h ) (t x
d t h x t y ) ( ) ( ) ( ) ( f X ) ( f H ) ( ) ( ) ( f H f X f Y ) ( ) ( ) ( t h t x t y Fourier pairs ) (t h
?
n(t) s(t) r(t)
The usual problem
?
) ( f Gx
2
) ( ) ( ) ( f H f G f G
x Y
) ( f H