dc01-overview of digital communications
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Ho Chi Minh City University of Technology
Faculty of Electrical and Electronics Engineering
Department of Telecommunications
Lectured by Ha Hoang Kha, Ph.D.
Ho Chi Minh City University of Technology
Email: [email protected]
Overview of
Digital Communications
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Content
Introduction on building blocks of digital
communication systems
Signal classification
Signal spectrum
Linear filters
Bandwidth
Danh gia: 30% BT+TN, 20% KT, 50% Thi
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References
Slides are adapted from the following resources:
Bernard Sklar, Digital Communications: Fundamentals
and Applications, Prentice Hall, 2nd ed, 2001.
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Communications
Main purpose of communication is to transfer information
from a source to a recipient via a channel or medium.
Basic block diagram of a communication system:
Source Transmitter Receiver
Recipient
Channel
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Brief Description
Source: analog or digital
Transmitter: transducer, amplifier, modulator,
oscillator, power amp., antenna
Channel: e.g. cable, optical fibre, free space
Receiver: antenna, amplifier, demodulator, oscillator,
power amplifier, transducer
Recipient: e.g. person, (loud) speaker, computer
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Types of information
Voice, data, video, music, email etc.
Types of communication systems
Public Switched Telephone Network
(voice,fax,modem)
Satellite systems
Radio,TV broadcasting
Cellular phones
Computer networks (LANs, WANs, WLANs)
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Information Representation
Communication system converts information into electrical electromagnetic/optical signals appropriate for the transmission medium.
Analog systems convert analog message into signals that can propagate through the channel.
Digital systems convert bits (digits, symbols) into signals
• Computers naturally generate information as characters/bits
• Most information can be converted into bits
• Analog signals converted to bits by sampling and quantizing (A/D conversion)
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Why digital?
Digital techniques need to distinguish between discrete symbols allowing regeneration versus amplification
Good processing techniques are available for digital signals, such as medium.
Data compression (or source coding)
Error Correction (or channel coding)(A/D conversion)
Equalization
Security
Easy to mix signals and data using digital techniques
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• Formatting/Source Coding
• Transforms source info into digital symbols (digitization)
• Selects compatible waveforms (matching function)
• Introduces redundancy which facilitates accurate decoding despite errors
It is essential for reliable communication • Modulation/Demodulation
• Modulation is the process of modifying the info signal to facilitate transmission
• Demodulation reverses the process of modulation. It involves the detection and retrieval of the info signal
- Types
- Coherent: Requires a reference info for detection
- Noncoherent: Does not require reference phase information
Basic Digital Communication
Transformations
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Basic Digital Communication
Transformations
• Coding/Decoding
Translating info bits to transmitter data symbols
Techniques used to enhance info signal so that they are less vulnerable to channel impairment (e.g. noise, fading, jamming, interference)
- Two Categories – Waveform Coding
- Produces new waveforms with better performance – Structured Sequences
Involves the use of redundant bits to determine the occurrence of error (and sometimes correct it) • Multiplexing/Multiple Access Is synonymous with
resource sharing with other users
• Frequency Division Multiplexing/Multiple Access (FDM/FDMA
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Performance Metrics
Analog Communication Systems
• Metric is fidelity: want
• SNR typically used as performance metric
Digital Communication Systems
• Metrics are data rate (R bps) and probability of bit
error
• Symbols already known at the receiver
• Without noise/distortion/sync. problem, we will never
make bit errors
ˆ ( ) ( )m t m t
ˆ( )bP p b b
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Main Points
Transmitters modulate analog messages or bits in case
of a DCS for transmission over a channel.
Receivers recreate signals or bits from received signal
(mitigate channel effects)
Performance metric for analog systems is fidelity, for
digital it is the bit rate and error probability.
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Why Digital Communications?
• Easy to regenerate the distorted signal
• Regenerative repeaters along the transmission path can detect a digital signal and retransmit a new, clean (noise free) signal
• These repeaters prevent accumulation of noise along the path
This is not possible with analog communication systems • Two-state signal representation
The input to a digital system is in the form of a sequence of bits (binary or M_ary) • Immunity to distortion and interference
• Digital communication is rugged in the sense that it is more immune to channel noise and distortion
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Why Digital Communications?
• Hardware is more flexible
• Digital hardware implementation is flexible and permits the use of microprocessors, mini-processors, digital switching and VLSI
Shorter design and production cycle
• Low cost
The use of LSI and VLSI in the design of components and systems have resulted in lower cost
• Easier and more efficient to multiplex several digital signals
• Digital multiplexing techniques – Time & Code Division Multiple Access - are easier to implement than analog techniques such as Frequency Division Multiple Access
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Why Digital Communications?
• Can combine different signal types – data, voice, text, etc.
• Data communication in computers is digital in nature whereas voice communication between people is analog in nature
• The two types of communication are difficult to combine over the same medium in the analog domain.
Using digital techniques, it is possible to combine both format for transmission through a common medium
Encryption and privacy techniques are easier to implement • Better overall performance
• Digital communication is inherently more efficient than analog in realizing the exchange of SNR for bandwidth
• Digital signals can be coded to yield extremely low rates and high fidelity as well as privacy
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Why Digital Communications?
Disadvantages
Requires reliable ―synchronization‖
Requires A/D conversions at high rate
Requires larger bandwidth
Nongraceful degradation
Performance Criteria
Probability of error or Bit Error Rate
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Goals in Communication System Design
To maximize transmission rate, R
To maximize system utilization, U
To minimize bit error rate, Pe
To minimize required systems bandwidth, W
To minimize system complexity, Cx
To minimize required power, Eb/No
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Digital Signal Nomenclature
Baud Rate
• Refers to the rate at which the signaling elements are
transmitted, i.e. number of signaling elements per
second.
Bit Error Rate
• The probability that one of the bits is in error or simply
the probability of error
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Classification Of Signals
Deterministic and Random Signals
A signal is deterministic means that there is no uncertainty
with respect to its value at any time.
Deterministic waveforms are modeled by explicit
mathematical expressions, example:
A signal is random means that there is some degree of
uncertainty before the signal actually occurs.
Random waveforms/ Random processes when examined
over a long period may exhibit certain regularities that can
be described in terms of probabilities and statistical
averages.
x(t) = 5cos(10t)
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Classification Of Signals
Periodic and Non-periodic Signals
A signal x(t) is called periodic in time if there exists a
constant T0 > 0 such that
t denotes time
T0 is the period of x(t).
0x(t) = x(t + T ) for - < t <
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Classification Of Signals
Analog and Discrete Signals
An analog signal x(t) is a continuous function of time; that is, x(t) is
uniquely defined for all t
A discrete signal x(kT) is one that exists only at discrete times; it is
characterized by a sequence of numbers defined for each time, kT,
where
k is an integer
T is a fixed time interval.
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Classification Of Signals
Energy and Power Signals
The performance of a communication system depends on the received signal energy; higher energy signals are detected more reliably (with fewer errors) than are lower energy signals
x(t) is classified as an energy signal if, and only if, it has nonzero but finite energy (0 < Ex < ∞) for all time, where:
An energy signal has finite energy but zero average power.
Signals that are both deterministic and non-periodic are classified as energy signals
T/2
2 2
xT / 2
E = x (t) dt = x (t) dtlimT
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Power is the rate at which energy is delivered.
A signal is defined as a power signal if, and only if,
it has finite but nonzero power (0 < Px < ∞) for all
time, where
Power signal has finite average power but infinite
energy.
As a general rule, periodic signals and random
signals are classified as power signals
Classification Of Signals
Energy and Power Signals
T/2
2
xT / 2
1P = x (t) dt
Tlim
T
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Dirac delta function δ(t) or impulse function is an
abstraction—an infinitely large amplitude pulse, with zero
pulse width, and unity weight (area under the pulse),
concentrated at the point where its argument is zero.
Sifting or Sampling Property
The Unit Impulse Function
(t) dt = 1
(t) = 0 for t 0
(t) is bounded at t 0
0 0( ) (t-t )dt = x(t ) x t
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Fourier Series
Let denote a periodic signal, where the subscript
T0 denotes the duration of periodicity.
The fundamental frequency is
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Fourier Series
Let denote a periodic signal, where the subscript
T0 denotes the duration of periodicity.
The cn are called the complex Fourier coefficients.
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Fourier Transform
Fourier transform
Inverse Fourier transform
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Fourier Transform
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Fourier Transform
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Fourier Transform
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Fourier Transform
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Fourier Transform
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Spectral Density
The spectral density of a signal characterizes the
distribution of the signal’s energy or power in the
frequency domain.
This concept is particularly important when considering
filtering in communication systems while evaluating the
signal and noise at the filter output.
The energy spectral density (ESD) or the power
spectral density (PSD) is used in the evaluation.
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Energy Spectral Density (ESD)
Energy spectral density describes the signal energy per unit
bandwidth measured in joules/hertz.
Represented as ψx(f), the squared magnitude spectrum
According to Parseval’s theorem, the energy of x(t):
Therefore:
The Energy spectral density is symmetrical in frequency about
origin and total energy of the signal x(t) can be expressed as:
2( ) ( )x f X f
2 2
x
- -
E = x (t) dt = |X(f)| df
x
-
E = (f) dfx
x
0
E = 2 (f) dfx
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Power Spectral Density (PSD)
The power spectral density (PSD) function Gx(f ) of the periodic
signal x(t) is a real, even, and nonnegative function of frequency
that gives the distribution of the power of x(t) in the frequency
domain.
PSD is represented as:
Whereas the average power of a periodic signal x(t) is
represented as:
Using PSD, the average normalized power of a real-valued
signal is represented as:
2
x n 0
n=-
G (f ) = |C | ( ) f nf
0
0
/2
2 2
x n
n=-0 / 2
1P x (t)dt |C |
T
TT
x x x
0
P G (f)df 2 G (f)df
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Autocorrelation
Autocorrelation of an Energy Signal
Correlation is a matching process; autocorrelation refers to
the matching of a signal with a delayed version of itself.
Autocorrelation function of a real-valued energy signal x(t)
is defined as:
The autocorrelation function Rx(τ) provides a measure of
how closely the signal matches a copy of itself as the copy
is shifted
τ units in time.
Rx(τ) is not a function of time; it is only a function of the time
difference τ between the waveform and its shifted copy.
xR ( ) = x(t) x (t + ) dt for - < <
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Autocorrelation of an Energy Signal
The autocorrelation function of a real-valued energy signal
has the following properties:
x xR ( ) =R (- )
x xR ( ) R (0) for all
x xR ( ) (f)
2
xR (0) x (t)dt
symmetrical in about zero
maximum value occurs at
the origin
autocorrelation and ESD form a
Fourier transform pair, as designated
by the double-headed arrows
value at the origin is equal to
the energy of the signal
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Autocorrelation of a Power Signal
Autocorrelation function of a real-valued power signal
x(t) is defined as:
When the power signal x(t) is periodic with period T0,
the autocorrelation function can be expressed as
/ 2
xT / 2
1R ( ) x(t) x (t + ) dt for - < < lim
T
TT
0
0
/ 2
x
0 / 2
1R ( ) x(t) x (t + ) dt for - < <
T
TT
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Autocorrelation of a Power Signal
The autocorrelation function of a real-valued periodic signal has
the following properties similar to those of an energy signal:
symmetrical in about zero
maximum value occurs at the origin
autocorrelation and PSD form a
Fourier transform pair
value at the origin is equal to the
average power of the signal
x xR ( ) =R (- )
x xR ( ) R (0) for all
x xR ( ) (f)G
0
0
T / 2
2
x
0 T / 2
1R (0) x (t)dt
T
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Random Signals
Random Variables
All useful message signals appear random; that is, the receiver does not know, a priori, which of the possible waveform have been sent.
Let a random variable X(A) represent the functional relationship between a random event A and a real number.
The (cumulative) distribution function FX(x) of the random variable X is given by
Another useful function relating to the random variable X is the probability density function (pdf)
( ) ( )XF x P X x
( )( ) X
X
dF xP x
dx
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Ensemble Averages
The first moment of a probability distribution of a random variable X is called mean value mX, or expected value of a random variable X
The second moment of a probability distribution is the mean-square value of X
Central moments are the moments of the difference between X and mX and the second central moment is the variance of X
Variance is equal to the difference between the mean-square value and the square of the mean
{ } ( )X Xm E X x p x dx
2 2{ } ( )XE X x p x dx
2
2
var( ) {( ) }
( ) ( )
X
X X
X E X m
x m p x dx
2 2var( ) { } { }X E X E X
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Random Processes
A random process X(A, t) can be viewed as a function of
two variables: an event A and time.
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Statistical Averages of a Random
Process
A random process whose distribution functions are
continuous can be described statistically with a probability
density function (pdf).
A partial description consisting of the mean and
autocorrelation function are often adequate for the needs of
communication systems.
Mean of the random process X(t) :
Autocorrelation function of the random process X(t)
{ ( )} ( ) ( )kk X X kE X t xp x dx m t
1 2 1 2( , ) { ( ) ( )}XR t t E X t X t
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Noise in Communication Systems
The term noise refers to unwanted electrical signals that are always present in electrical systems; e.g spark-plug ignition noise, switching transients, and other radiating electromagnetic signals.
Can describe thermal noise as a zero-mean Gaussian random process.
A Gaussian process n(t) is a random function whose amplitude at any arbitrary time t is statistically characterized by the Gaussian probability density function
21 1
( ) exp22
np n
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Noise in Communication Systems
The normalized or standardized Gaussian density function of a
zero-mean process is obtained by assuming unit variance.
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White Noise
The primary spectral characteristic of thermal noise is that its power spectral density is the same for all frequencies of interest in most communication systems
Power spectral density Gn(f )
Autocorrelation function of white noise is
The average power Pn of white noise is infinite
0( ) /2
n
NG f watts hertz
1 0( ) { ( )} ( )2
n n
NR G f
0( )2
Np n df
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The effect on the detection process of a channel with
additive white Gaussian noise (AWGN) is that the noise
affects each transmitted symbol independently.
Such a channel is called a memoryless channel.
The term ―additive‖ means that the noise is simply
superimposed or added to the signal
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Signal Transmission
through Linear Systems
A system can be characterized equally well in the time
domain or the frequency domain, techniques will be
developed in both domains
The system is assumed to be linear and time invariant.
It is also assumed that there is no stored energy in the
system at the time the input is applied
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Impulse Response
The linear time invariant system or network is characterized in the time domain by an impulse response h (t ),to an input unit impulse (t)
The response of the network to an arbitrary input signal x (t )is found by the convolution of x (t )with h (t )
The system is assumed to be causal,which means that there can be no output prior to the time, t =0,when the input is applied.
The convolution integral can be expressed as:
( ) ( ) ( ) ( )y t h t when x t t
( ) ( ) ( ) ( ) ( )y t x t h t x h t d
0
( ) ( ) ( )y t x h t d
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Frequency Transfer Function
The frequency-domain output signal Y (f )is obtained by taking the Fourier transform
Frequency transfer function or the frequency response is defined as:
The phase response is defined as:
( ) ( ) ( )Y f X f H f
( )
( )( )
( )
( ) ( ) j f
Y fH f
X f
H f H f e
1 Im{ ( )}( ) tan
Re{ ( )}
H ff
H f
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Random Processes and Linear Systems
If a random process forms the input to a time-
invariant linear system,the output will also be a
random process.
The input power spectral density GX (f )and the
output power spectral density GY (f )are related as:
2( ) ( ) ( )Y XG f G f H f
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Distortionless Transmission
The output signal from an ideal transmission line may have some time delay and different amplitude than the input
It must have no distortion—it must have the same shape as the input.
For ideal distortionless transmission:
Output signal in time domain
Output signal in frequency domain
System Transfer Function
0( ) ( )y t Kx t t
02( ) ( )
j ftY f KX f e
02( )
j ftH f Ke
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Ideal Transmission
The overall system response must have a constant magnitude response
The phase shift must be linear with frequency
All of the signal’s frequency components must also arrive with identical time delay in order to add up correctly
Time delay t0 is related to the phase shift and the radian frequency = 2f by:
t0 (seconds) = (radians) / 2f (radians/seconds ) (1.57a)
Another characteristic often used to measure delay distortion of a signal is called envelope delay or group delay:
(1.57b)1 ( )
( )2
d ff
df
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Ideal Filters
For the ideal low-pass filter transfer function with bandwidth Wf =
fu hertz can be written as:
Figure1.11 (b) Ideal low-pass filter
(1.58)
Where
(1.59)
(1.60)
( )( ) ( ) j fH f H f e
1 | |( )
0 | |
u
u
for f fH f
for f f
02( ) j ftj fe e
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Ideal Filters
The impulse response of the ideal low-pass filter:
0
0
1
2
2 2
2 ( )
0
0
0
( ) { ( )}
( )
sin 2 ( )2
2 ( )
2 sin 2 ( )
u
u
u
u
j ft
f
j ft j ft
f
f
j f t t
f
uu
u
u u
h t H f
H f e df
e e df
e df
f t tf
f t t
f nc f t t
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Ideal Filters
For the ideal band-pass filter
transfer function
For the ideal high-pass filter
transfer function
Figure1.11 (a) Ideal band-pass filter Figure1.11 (c) Ideal high-pass filter
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Theorems of
communication and
information theory are
based on the
assumption of strictly
bandlimited channels
The mathematical
description of a real
signal does not permit
the signal to be strictly
duration limited and
strictly bandlimited.
Bandwidth Dilemma
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Bandwidth Dilemma
All bandwidth criteria have in common the attempt to
specify a measure of the width, W, of a nonnegative
real-valued spectral density defined for all
frequencies f < ∞
The single-sided power spectral density for a single
heterodyned pulse xc(t) takes the analytical form:
2
sin ( )( )
( )
cx
c
f f TG f T
f f T
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Different Bandwidth Criteria
(a) Half-power bandwidth.
(b) Equivalent
rectangular or noise
equivalent bandwidth.
(c) Null-to-null bandwidth.
(d) Fractional power
containment
bandwidth.
(e) Bounded power
spectral density.
(f) Absolute bandwidth.
Fundamental Concepts 68 H. H. Kha