TLT-1406 Introduction to Communications Engineering
Spectrum, waveforms, and data transmission
1. General view to communication systems
2. The idea of modulation
3. Spectrum concepts
4. Modulation, bandpass transmission and complex signals
5. Digital transmission system model
6. Channel capacity
Core content:
• Understanding signal spectrum
• Understanding the idea and purpose of modulation
• Main elements of a digital transmission link
The material is a selected subset of the slides of the TLT-5206
Communication theory.
Recommended textbook for further information:
A.B. Carlson, Communication Systems, Fourth Edition, McGraw-Hill,
2001.
TLT-5206 / 1
Introduction
Information and messages
Telecommunications is transferring information from one place to
another in an electric or electromagnetic form.
In the following we refer to an information bearing message, which can
be a waveform, a bit sequence etc. The main task of a communications
system is to make a copy (as good as possible) of the transmitted
message in the receiver.
INFORMATION
SOURCE
TRANSMISSION
SYSTEM
INFORMATION
WELL
Original message can be analog (e.g. speech) or digital (e.g. text).
Regardless of the source format, the message can be transferred either
in analog or digital form.
· Division between analog and digital transmission is mostly based
on how source information is presented from transmission
system’s point of view.
For example, speech transmission (i.e., original message is analog):
· Analog transmission system: target is to transfer the original
message as little interfered and distorted as possible, e.g., using
appropriate carrier modulation.
· Digital transmission system: representing the original message by
a bit sequence (source coding, speech encoder), and transferring
the bits effectively using some specific communications waveforms
In analog transmission there are certain quality criterions (Signal-tonoise ratio, distortion, etc.). In digital transmission, instead, the objective
is to minimize the bit error probability.
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Information and messages (continuing)
All (at least most) physical transmission channels, and therefore, also
used transmission signals (i.e. waveforms), are analog and continuous
· E.g. voltage or current signal in a cable, electromagnetic wave in
radio path, etc.
INFORMATION
SOURCE
TRANSMITTER
TRANSMISSION
SYSTEM
INFORMATION
WELL
TRANSMISSION
CHANNEL
RECEIVER
Thus, telecommunications is fundamentally:
Producing effective message bearing waveform that performs as well
as possible in the used communications channel (transmission media).
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Elements of a communications system
Transmitter adapts the signal from the source to suit the transmission
channel. The most essential signal processing operations are, e.g.,
· Modulation (almost all communications systems)
· Coding (digital source and/or digital communications system)
Two fundamental resources are
· Transmission power and energy [mW,W,…]
· Bandwidth [kHz,MHz]
Note! Physical and/or official
regulations given by authorities
Transmission channel generates different undesired changes in the
message signal. In practice these changes appear also in the receiver,
but in the model they are all associated with the channel. For example:
· Attenuation
· Noise: additive random nature interference signal
· Interference: additive interference signal from some recognizable
source (e.g., adjacent channel transmission)
· Distortion: distortion affected by the non-ideal response of the
communications system (linear filtering, non-idealities, etc.)
· Transmission can be “one-way” or “two-way”:
o Simplex (one-way)
o Full-duplex (transmission and reception can be done “at the
same time”)
o Half-duplex (two-way transmission but not at the same time)
Receiver restores the message signal into the original form so that the
non-idealities created by the channel are minimized (amplification,
filtering, demodulation, decoding).
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Transmission channels
Electric wire
· Pair of wires (e.g., regular telephone connections)
· Coaxial cable
Radio path
· broadcasting (transmissions of broadcasting companies)
· microwave links
· satellite transmission
· cell networks
Optical fiber
Magnetic tape, etc.
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Physical limitations
The most essential physics law originated limitations in communications
systems are power/energy and bandwidth.
In nature there is always, e.g., thermal noise which may become as a
quality limiting factor (such as when increasing the transmission
distance → transmission loss increases).
On the other hand, all physical systems have a limited bandwidth.
Bandwidth will eventually become as a limiting factor, since it directly
affects the achievable information transmission rate. Especially systems
utilizing the radio path are often limited by the bandwidth.
For a channel whose bandwidth B and signal-to-noise power ratio
(S / N ) is known, we are able to calculate the largest possible capacity
C < B log2 (1 ∗ S / N )
This is known as the Hartley-Shannon law and it is one of the most
famous results in information theory related to communications
systems. The law will be more carefully studied in the end of the course
(information theory part).
The above mentioned fundamental resources (power/energy and
bandwidth) are anyhow those physical keystones, whose around the
contents of the course, e.g., regarding to wave form studies, are based
on (basically a physical layer of any communications system).
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Modulation
In modulation two separate waveforms are connected: a modulating
message signal and a properly chosen carrier. In below there are
examples of amplitude modulation using sine wave and square wave as
a carrier. In both cases the message signal is found in the envelope of
the modulated signal. In the receiver the envelope can be easily
distinguished from the received signal (demodulation).
Usually the carrier frequency is much larger than the highest existing
frequency in the message signal. Moreover, the frequency band of the
modulated signal is commonly located around the carrier frequency. As
a result, there is a frequency shift in modulation.
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Examples of analog modulation methods
Unmodulated carrier:
Low frequency signal (information signal):
Amplitude modulated carrier (information lies in amplitude variations):
Frequency modulated carrier (information lies in frequency variations):
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Advantages of the modulation
Transmission channel requirements
For instance, in radio transmissions the required antenna size is
proportional to the used wave length (at least 1/10 of the wave
length). E.g., with 100 Hz frequency a 300 km long antenna would
be necessary whereas with 100 MHz frequency a practical
antenna size would be 1 m.
Separation of different transmission stations
By setting an individual carrier frequency for each transmission, it
is possible to transmit several messages without interfering with
each other. Radio frequencies are divided into separate radio
bands by international agreements (different communications
services on each band).
Multiplexing
When messages are modulated using separated carrier
frequencies, multiple messages can be transmitted simultaneously
in the same channel.
Mitigation of noise and interference
When using appropriate modulation methods and increasing the
bandwidth the effects of noise can be reduced in the demodulated
signal (by not changing the transmission power, of course).
Implementation originated limitations
From communications electronics point of view, the relative
bandwidth (transmission bandwidth/carrier frequency) should be
about 1-10% → modulation method has to be selected depending
on the carrier frequency and transmission bandwidth.
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Frequency allocation of electromagnetic spectrum
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Coding
Coding is basically digital transmission related message signal
modification. Decoding means the inverse operation performed in the
receiver.
Examples:
1) ASCII-code: coding alphanumeric signs into binary data
2) When binary data is transmitted in a good quality channel,
transmission capacity can be increased by using 2M-level
symbols (each correspond to M binary data bits).
3) In error control coding, redundancy is added in the original
information message. This enables correction of some errors
occurred in the channel. Error correction capability depends on
the used code structure.
In real life digital transmission all these can be used along with some
other coding schemes.
NB! Introduction to digital transmission is presented in the last part of
the course.
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SIGNALS AND SPECTRA
Electric/electromagnetic signal: electric or electromagnetic time
dependable quantity. For example, voltage between the ends of the
wire pair as a function of time, or a electromagnetic wave in the radio
path.
In the following we concentrate on representing signals in time domain
and frequency domain and study how these two are connected.
In the frequency representation, a signal is divided into sinusoidal
components with specific frequencies.
The frequency representation (i.e. spectrum) is mathematically defined
by the Fourier transform (note that the plural of spectrum is spectra).
In case of periodic signals, spectrum can also be found by using Fourier
series.
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Sinusoidal signals
Sinusoidal signal can always be presented as follows
v(t ) < A cos(w0t ∗ f) < A cos(2p f0t ∗ f)
§ A is amplitude, w0 < 2p f0 is angular frequency ( f0 is
frequency) and f is phase angle (or just phase).
This is a periodic signal whose period is
T0 < 2p / w0 < 1/ f0
One of the signal maximums is found from
t < ,f / w0
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Spectrum of a sinusoidal signal
One-sided spectrum of a sinusoidal signal is depicted in the following
figure:
Amplitude
spectrum
Phase
spectrum
Amplitude spectrum and phase spectrum include a peak at frequency f0 .
From spectrum it is possible to observe the essential parameters of a
sinusoidal signal: frequency, amplitude and phase.
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Linear combination of sinusoidal signals
w(t ) < 7 , 10 cos(40pt , 60↓) ∗ 4 sin 120pt
This can be modified as:
w(t ) < 7 cos 2p 0t ∗ 10 cos(2p20t ∗ 120↓) ∗ 4 cos(2p60t , 90↓)
from which one-sided line spectrum can be drawn:
Amplitude
spectrum
Phase
spectrum
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Complex representation of sinusoidal signal
Real life signals are always real valued. However, a concept of complex
signals is often a useful and necessary tool in communications
engineering. Although in some of the studied cases one would manage
using only real valued signals, it is common in spectrum analysis to use
formalism that enables also utilization of complex signals.
Complex signals are closely related to so-called bandpass signals
(more carefully studied later in the course). For now, without a specific
physical background, a complex signal is defined as a signal or wave
form whose value (at any moment of time) is a complex number.
· In practice: two parallel real valued signals (Re and Im parts)
In this context Euler’s equations are frequently needed:
e ° j q < cos q ° j sin q
The way to another direction can be found by
e j q ∗ e,j q
cos q <
< Re éë e j q ùû
2
e jq , e,jq
sin q <
< Im éë e j q ùû
2j
Notations in the future:
1) In spectrum representations the free variable is f (unit Hz).
Angular frequency is defined as w < 2p f . Certain fixed
frequencies are denoted as f0, f1, fi , etc.
2) Phase angle is measured with respect to a cosine wave.
3) Amplitude is always positive: ,A cos wt < A cos(wt ° 180↓)
4) The unit for the phase angle is degree (° ), although it is usually
defined as radians.
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Two-sided spectrum
For real and periodic signals one can use a one-sided line spectrum.
Instead in future, we concentrate on two-sided spectrum which enables
also studying of complex signals.
In two-sided spectrum representation the base functions are complex
exponential functions (instead of real oscillators).
In case of real valued signal, two-sided spectrum is obtained by
substitution (Euler)
A cos(w0t ∗ f) <
A j f j w0t A , jf ,j w0t
e e
∗ e e
2
2
Considering the earlier example, the two-sided spectrum is now given
as:
w(t ) < 7 cos 2p 0t ∗ 10 cos(2p20t ∗ 120↓) ∗ 4 cos(2p60t , 90↓)
< ...
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Fourier-series
Objective: Finding out the frequency content (spectrum) of an arbitrary
periodic signal.
A periodic signal can be represented as an exponential Fourier series
development:
v(t ) <
⁄
å
n <,⁄
cne j 2pnf0t
where T0 is the period (cycle time), f0 <
frequency, and
cn < c(nf0 ) <
1
T0
ò v(t )e
T0
, j 2 pnf0t
1
is the fundamental
T0
dt
Coefficients cn are complex quantities which are usually represented in
the polar form:
cn < cn e j arg cn
Exponential Fourier series defines the two-sided (line) spectrum for a
periodic signal. It consists of multiples of f0 (can you see why?).
Number cn indicates the value of the amplitude spectrum at frequency
nf0 whereas arg(cn ) indicates the value of the corresponding phase
spectrum.
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Sinc-function
In future, we repeatedly use the sinc-function given as:
ìï sin pl
ï
sinc l < ïí pl
ïï 1
ïî
kun l ÷ 0
kun l < 0
Where does this come from?
In spectrum analysis following type of integrals come up frequently:
1
T
T /2
ò
,T / 2
e j 2p ftdt < ...
The result of the integral can be conveniently represented by the sincfunction (see the next example).
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Example: Square wave or pulse train
The coefficients for the Fourier series can now be given as:
1
cn <
T0
<
T0 / 2
ò
,T0 / 2
1
v(t )e , j 2pnf0tdt <
T0
t /2
ò
,t / 2
Ae , j 2pnf0tdt
A
A sin pnf0t
At
sinc nf0t
<
e , j pnf0t , e j pnf0 t ( <
∋
,j 2pnf0T0
T0 pnf0
T0
In the following figure the amplitude and phase spectra are shown in
case of pulse ratio (or duty cycle) t /T0 < 1/ 4 . The sinc shaped
function sinc f t can be found from the envelope of the amplitude
spectrum. The amplitude of the DC-component is c0 < At /T0 , which
can easily be derived also from the time domain function (how?).
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Fourier transform and continuous spectrum
Objective: Defining the frequency content (spectrum) for an arbitrary
non-periodic signal.
Let’s study signal whose energy (compare to the power)
E <
⁄
ò
,⁄
v(t ) 2 dt
is finite. Practically this means that the signal is restricted in some
relatively small time span.
For this type of a signal the Fourier transform is defined as:
V (f ) < F Ζ v(t ) ∴ <
⁄
ò
,⁄
v(t )e ,j 2p ftdt
where V ( f ) is the spectrum of v(t ) .
With non-periodic signals the spectrum is continuous (meaning what?).
The spectrum has the following properties:
1) V (f ) is a complex function. V ( f ) is the amplitude spectrum
and argV ( f ) is the phase spectrum.
2) V (0) <
⁄
ò
,⁄
v(t )dt
3) If v ( t ) is real, then V (,f ) < V )( f ) , i.e.,
V (,f ) < V ( f )
and argV (,f ) < , argV ( f )
If needed, time function v(t ) can be obtained from V ( f ) with the inverse
Fourier transform:
v(t ) < F ,1 Ζ V ( f ) ∴ <
⁄
ò V (f )e j 2pftdf
,⁄
(cf. Fourier series)
TLT-5206 / 27
Example: Rectangular pulse
In future, a rectangular pulse is denoted as Ο(t / t ):
ì
ïï 1
Ο(t / t ) < í
ïï 0
î
t ; t /2
t = t /2
(draw by yourself)
Let’s now study signal v(t ) < AΟ(t / t ) . Its Fourier transform is given as
V (f ) <
t /2
ò
,t / 2
Ae ,j 2p ftdt <
At
sin p f t < At sinc f t
pf t
It can be seen that the spectrum of a rectangular pulse correspond with
the envelope shown in the previous example with the line spectrum of a
square wave. In addition, it is shown that most of the spectrum energy
is located between the frequencies f ; 1/ t . This means that the
spectrum of short (or narrow) pulses is wide.
· Reciprocal spreading phenomenon (universally valid)
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Calculating Fourier transforms in practice
Straightforward integration is basically possible only in few cases (text
book examples, etc.)
Other ways:
1) Tables & transform theorems (duality, frequency and time
translation, etc.)
· The most common elementary operations and their mapping
into the transform domain
2) Approximation. If function z%(t ) approximates function z (t ) , then
⁄
ò
,⁄
Z (f ) , Z%(f ) df <
2
⁄
ò
,⁄
z (t ) , z%(t ) 2 dt
It follows that the approximation in frequency domain can
become as accurate as desired by improving the approximation
accuracy in time domain.
Proof: try Rayleigh’s energy theorem …
3) Numeric methods: Discrete Fourier transform (DFT)
· Can be calculated, e.g., with Matlab software (see Matlab
exercises!).
· The differences between the continuous Fourier transform
and the DFT must be recognized!
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About the DFT
Background: Using the same methods as with continuous signals, it is
possible to define the Fourier transform and spectrum also for a
discrete signal (sequence of numbers) x (k ) .
Fourier transform of an N -point sequence x (k ) is defined as
X (e
j 2 p fTS
)<
N ,1
å x(k )e,j 2p fkT
S
k <0
(cf. continuous signal transform)
Here TS refers to the distance between two adjacent numbers in time
domain and notation X (e j 2p fTS ) tries to emphasize (i) the difference
compared to the continuous transform, (ii) and second, the periodicity of
the sequence transforms
· Above function X (e j 2p fTS ) repeats itself at integer 1/Ts multiples
(Why? Verify yourself…)
It follows that if the sequence x (k ) is a sample sequence taken from a
continuous signal x (t ), i.e.,
x (k ) < x (kTs ),
k < 0, K, N , 1 ,
and if X ( f ) does not include larger frequency components than 1/ 2Ts
(so there is no aliasing), then
X (e j 2p fTS ) < TS X ( f )
;
, 1/(2TS ) ′ f ′ 1/(2TS )
(A lot more about this will be discussed later…)
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About the DFT (continuing)
Now the actual discrete Fourier transform (DFT) of an N -point
sequence x (k ) is defined as:
X (n ) < DFT Ζ x (k ) ∴ <
N ,1
å x(k )e,j 2pkn / N
k <0
;
n < 0,1, K, N , 1
When compared to the previous one, it is shown that DFT consists of
samples from Fourier transform of a sequence at frequency points nf0
where f0 < 1/(NTs ) (this is DFT’s resolution):
X (n ) < X (e j 2p fkTS ) f <nf
0
f0 <1/(NTS )
;
n < 0,1, K, N , 1
Note! FFT (Fast Fourier transform) is, instead, just a set of methods to
calculate the DFT in an effective manner (see Matlab: “help fft”).
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Calculating the spectrum with the DFT
So, with DFT it is possible to define the spectrum of a signal by using a
sampled sequence of it. According to previous pages, the frequency
resolution of the DFT is f0 < 1/(NTs ) and it includes the frequency
band ,1/(2Ts ) ; f ; 1/(2Ts ) . Negative frequencies can also be
incorporated, since Xˆ(n ) can be understood periodic so that
X (N , n ) < X (,n ) < Ts X (,nf0 ) .
· Frequency band can be increased by reducing the sampling
interval.
· Frequency resolution can be improved by increasing the sampling
interval or by adding the number of samples (e.g. by adding 0samples in the end of the sequence)
· If x (t ) ÷ 0 when t ; 0 or t ″ NTs signal must be cut in the
sampling. This induces distortion in the signal. The effects of
distortion can be mitigated by using appropriate window functions.
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A general bandpass signal
Let’s study a bandpass signal vbp (t ) whose spectrum Vbp ( f ) is
centralized around a specific centre frequency fc :
Vbp ( f ) < 0
when
f ; fc , W ja
f = fc ∗ W
(In principal the centre frequency can be arbitrary chosen within this
frequency band. In practical cases the selection is somewhat obvious.)
A sinusoid of frequency fc can be spotted in the waveform. The
narrower is the band, the more the signal resembles the pure sinusoid.
Small variation in the envelope A(t ) and/or in the phase f(t ) can be
seen in the signal. Mathematically this can be represented as (intuition)
vbp (t ) < A(t ) cos ∋ wct ∗ f(t ) (
It is worth of noticing that the envelope is generally considered as nonnegative, A(t ) ″ 0 . Consequently, the possible changes in the sign
correspond to ±180° phase rotation.
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Representation of a bandbass signal (continuing)
According to the previous, there are two different representation forms
for the bandpass signal. Both of them are defined by two separate time
functions:
5) A(t ), f(t )
Envelope and Phase
6) vi (t ), vq (t )
In-phase and Quadrature (I/Q) components
Both of the representation forms are used a lot in future. Besides, both
of the forms have their own unique strengths (e.g., waveform behavior,
spectral analysis).
The connection between these representation forms is found as follows:
vi (t ) < A(t ) cos f(t )
vq (t ) < A(t ) sin f(t )
...and the same in reverse direction:
A(t ) <
vi2 (t ) ∗ vq2 (t )
f(t ) < arctan
vq (t )
vi (t )
Instead of using terms in-phase and quadrature component, they are
often referred as I and Q components.
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Lowpass equivalent signal
A spectrum of a bandpass signal can be represented using spectra of
in-phase and quadrature components:
vbp (t ) < vi (t ) cos(wct ) , vq (t ) sin(wct )
↔
1
j
Vbp ( f ) < ∋Vi ( f , fc ) ∗ Vi ( f ∗ fc ) ( ∗ ∋Vq ( f , fc ) , Vq ( f ∗ fc ) (
2
2
To assure that the bandpass signal is bounded within frequencies
fc , W ′ f ′ fc ∗ W , in-phase and quadrature components must be
lowpass signals (why?):
Vq (f ) < Vi (f ) < 0
when
f =W
Now, it is possible to formulate a lowpass equivalent signal:
vlp (t ) < 12 ∋ vi (t ) ∗ jvq (t ) (
< 12 A(t )e j f(t )
Generally, this is a complex signal, whose interpretation directly
corresponds with the earlier phasor representation. Its spectrum is
Vlp ( f ) < 21 ∋Vi ( f ) ∗ jVq ( f ) (
< Vbp ( f ∗ fc )u( f ∗ fc )
Thus, this is the positive part of the bandpass signal’s spectrum
relocated around the zero frequency.
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Spectrum of a lowpass equivalent signal
A passband signal/system can be effectively modelled using the
lowpass equivalent model, both in analytical studies and simulations.
* Complex signal models are a necessary tools for doing this!
Modulation: baseband => passband
Demodulation: passband => baseband
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About complex signals
Based on the previous, a physical real valued bandpass signal
vbp (t ) < vi (t ) cos(wct ) , vq (t ) sin(wct )
< A(t ) cos(wct ∗ f(t ))
< Re Ζ vlp (t )e j wct ∴
can be described using a complex valued lowpass equivalent
vlp (t ) < vi (t ) ∗ jvq (t )
< A(t )e j f(t )
whose
· Real and imaginary parts at any time instant express the I and Q
components of a physical bandpass signal
· Amplitude and phase at any time instant express the envelope and
phase of a physical bandpass signal
In general, complex signals do not involve any strange “mystique”
properties
· Simply two parallel real valued signals representing the real and
imaginary parts of a complex signal
· Processing and modification performed by using complex
arithmetic
NB! This is one of the topics that will be discussed more detailed in the
7cr course extension
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INTRODUCTION TO DIGITAL TRANSMISSION
Digital transmission has become more and more popular in every fields
of communications during the last 0 years. GSM introduced around 1992
All new communications systems under development are based on
digital transmission.
Contents:
1) The elements of digital transmission system
2) The advantages of digital transmission
3) Introduction to baseband digital transmission
· Digital PAM system
4) Introduction to digital carrier modulation
· I/Q modulated PAM/PSK/QAM; digital frequency modulation
5) Introduction to the information theory
In course TLT-5406 Digital transmission this topic is studied more
extensively
· will be lectured in periods 3 and 4
· http://www.cs.tut.fi/kurssit/TLT-5406/
Furthermore, more detailed issues regarding specific techniques, such
as CDMA and OFDM(A), are discussed in courses
· TLT-5606 Spread spectrum techniques
· TLT-5706 Multicarrier techniques
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The elements of digital transmission system
The system includes possibly a conversion of an analog message
signal (e.g. speech) into the digital form (sampling and quantization)
and vice versa.
The transmitting end of the actual transmission system converts the
digital signal into an analog waveform which is then transmitted into the
channel. The receiver converts the analog waveform again into the
digital form.
The transmission chain includes:
Source coding/decoding: To reduce the bit rate of the digital
message signal by removing some existing redundancy
(“compressing”)
One of the most essential results of the information theory is
that the source coding and the channel coding can be
performed independently between each other .
Channel coding/decoding: To reduce the effects of errors
produced in the transmission channel (error control coding)
Almost in any reasonable channel, an arbitrary small bit error
probability can be achieved by adding redundancy in the
transmitted signal.
Modulation/demodulation: Converting a digital signal into an
analog waveform and vice versa.
Channel: Here harmful noise, interference and distortion are
encountered.
While designing the system, one target could be minimizing the
required bandwidth or/and the transmission power/energy.
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Block diagram of digital transmission system
In the following, when referring to the digital transmission system we
mean the part of the chain, in which the interfaces are the input of the
channel coding and the output of the channel decoding.
Therefore, source coding and decoding are not included here
(compression of information, speech/audio/video coding, etc.)
Based on this definition we are able to design and analyze these types
of systems independent of the nature of the transmitted information
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Fundamental parameters of a digital transmission system
The external operation of the previously defined system can be
described with the following parameters:
· Transmission rate (bits/s)
· Error probability
· Propagation delay and the delay caused by the signal processing
and other processing
From system’s external operation point of view only these parameters
have significance
· The fact that how, e.g., a certain transmission rate is achieved, is
not important for the end user
=> more degrees of freedom / flexibility when compared with the
pure analog transmission
Thus, the internal operation of the system (coding, waveforms,…) can
be optimized with respect to to the properties of the used transmission
medium (bandwidth, transmission power,..) so that the requirements for
the external operation are fulfilled!
Notice that also the requirements for the external operation parameters
are strongly application specific
· E.g., intelligible speech vs. file transfer
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BASEBAND DIGITAL TRANSMISSION
Bits, Symbols, and waveforms
The starting point in the baseband digital transmission is the utilization
of pulse amplitude modulation (PAM) for transmission of binary bit
sequences or generally multilevel symbol sequences.
A multilevel symbol is produced when, e.g., 4 bits are combined into
one symbol. In this case, 24 < 16 different symbol levels are required
to represent all possible bit combinations. Generally, B bits can be
represented using 2B levels.
The number of used symbol levels (i.e., the number of bits / symbol) is
selected based on the requirements of the application and transmission
channel so that different symbol levels can be reliably distinguished in
the receiver.
When several bits are combined into one symbol, the used symbol rate
(baud rate) can be decreased. This affects directly the required
bandwidth as we will see later on.
Simple example:
bit sequence :
01442443
1 0 0 1144244
1 0 03 K
symbol sequence :
-3A
9A
K
bits/s
symb/s
binary signal:
16-level signal:
NB! Bit rate, symbol rate and symbol alphabet size (M) are related as
fbit < log2 (M ) ≥ fsym
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Pulse shapes
A digital PAM signal is composed of pulses scaled (weighted) with the
transmitted symbol values (from here the designation “PAM”).
Therefore, the waveform transmitted to the channel is given as
x (t ) <
å ak p(t , kT )
k
< ... ∗ a 0 p(t ) ∗ a1p(t , T ) ∗ ...
Here T is the symbol interval (the symbol rate is fsym < 1/T ) and
p(t ) is the fundamental pulse shape whose amplitude is scaled
according to the transmitted symbol value ak .
It is important that pulses representing consecutive symbols do not
interfere with each other in the reception. In ideal case the following
condition is fulfilled:
1 when t < 0
So called Nyquist criterion to
ì
ï
ï
p(t ) < í
avoid inter-symbol
ï
t
T
T
0
when
<
°
,
°
2
,...
ï
interference.
î
This can be practically implemented in two different ways:
) Using short pulses that are not superimposed in time
o Simple implementation, however, the bandwidth is not the
smallest possible
o Example: a rectangular pulse with the symbol interval duration
) Using pulses that are superimposed in time, but we take care that
the above condition is still fulfilled
o The used bandwidth can be minimized, however, the
implementation is more complex (e.g., symbol synchronization
in the receiver
o This is called as Nyquist pulse shaping and it is an essential
element in developing modern communications systems
TLT-5206 / 347
Pulse shapes (continuing)
Example: Illustration of PAM signal composed of individual pulses using
the rectangular pulse and a little bit smoother (and longer) pulse.
· Short rectangular pulse:
· Smoother and longer pulse:
TLT-5206 / 349
Spectral contents of a digital PAM signal
Spectral contents of a digital transmission signal depend on the symbol
sequence properties and the used pulse shape (quite obviously).
Now, let’s assume that
· Symbol sequence ak is a discrete random signal with power
spectral density of Ga (e j 2p fT )
· The Fourier transform of the used pulse shape is P ( f )
In this case, the power spectral density of a digital PAM signal is
1
Gx ( f ) <
P (f ) 2 Ga (e j 2p fT )
T
The latter term is commonly
constant => pulse shape
determines the spectrum.
Continuous and discrete time signals appear here, and therefore, the
mathematical derivation of the above result is not trivial.
On the other hand, the result is quite reasonable from the filter
interpretation point of view that will be discussed later on
· PAM signal generation using a transmitter filter
Here the general starting point is obviously that the spectrum of the
transmitted signal must be fitted into the properties of the channel
· E.g., in cables the attenuation is not constant in the used
frequency band (it is increased in high frequencies). That is why
the signal power is desired to be centered in the low frequencies
where the attenuation is the lowest. This will also decrease the
crosstalk issues in cable systems and radio frequency
interference.
· On the other hand, in AC coupled systems the spectrum is desired
to be zero at the zero frequency.
· Thus, the above result offers tools to process the PAM signal
spectrum (i.e., functions Ga (e j 2p fT ) and P ( f ) ).
TLT-5206 / 364
Pulse design
Time domain
Requirement: inter-symbol interference (ISI) is zero
p(t )
p(0) < 1
p(mT ) < 0 , m < °1, °2, K
Pulse form is otherwise free.
Frequency domain
The ideal bandlimited sinc-pulse is not a practical solution. Practical
pulses include typically about 10 -100% excess bandwidth compared to
the sinc-pulse. Consequently, the overall bandwidth is
W < (1 ∗ a)
1
2T
where a < 0.1, ... , 1 is so called roll-off factor.
In practice, the pulse shape filter P ( f ) has a symmetric transition band
with respect to the minimum bandwidth 1/(2T ) .
The Nyquist criterion guarantees zero
ISI and leads to the specific shape of
the spectrum.
Raised-cosine filter is a common way
to generate such pulse shapes.
TLT-5206 / 368
Raised-cosine pulses
The idea: Controlling the length of an ideal sinc-pulse using a window
function. Here the pulse duration is decreased by increasing the excess
bandwidth a . In this case the oscillation of the pulse attenuates faster
but the bandwidth increases.
é cos(apt /T ) ù
é sin(pt /T ) ù é cos(apt /T ) ù
ú
ê
ú
p(t ) < ê
<
sinc(
t
/
T
)
úê
êë 1 , (2at /T )2 úû
êë pt /T úû êë 1 , (2at /T )2 úû
The Fourier transform is:
ìï
ïïT
ïï
ïT
é pT
P ( f ) < ïí
1 , cos ê
ïï 2
ë a
ïï
ï0
ïîï
ζ
∋
f ,
(|
1∗a ù
ú
2T û
; f ′
1,a
2T
1,a
1∗a
′ f ′
2T
2T
1∗a
; f ″
2T
;
p(t )
P (f )
,
1
2T
1
2T
f
TLT-5206 / 357
BASEBAND DIGITAL TRANSMISSION SYSTEM BASED ON
THE NYQUIST PULSE SHAPING
Channel
Typically modeled with a linear filter (describes the linear distortion) and
additive noise (usually normally distributed).
· In mobile systems the channel varies as a function of time, and
therefore, also the filter models have to be time-dependent.
Coder
Converts the incoming bit stream into a symbol sequence where the
symbols are taken from a specific alphabet. Some examples:
1) Binary alphabet, the symbol sequence is basically the same as
the bit sequence
2) Two consecutive bits are described using the alphabet
{,3, , 1, ∗ 1, ∗ 3} (or more generally using the complex
alphabet {,1, , j, ∗ 1, ∗ j } )
TLT-5206 / 358
About the transmitter elements
Transmitter filter
Converts the discrete symbol sequence into a continuous-time signal.
The impulse response of this filter g(t ) is now the transmitted pulse
shape.
Thus, the transmitted waveform is given as
S (t ) <
⁄
å
m <,⁄
Am g (t , mT )
where Am is the transmitted symbol at time instant mT , and 1/T is
the symbol rate.
Consequently, the waveform is composed of pulses scaled with the
symbol values. Depending on the used pulse shape, the pulses may
overlap in time.
Below there is a simple example, in which the pulse shape g(t ) is a
rectangular pulse of length T :
TLT-5206 / 359
Channel
The received waveform can be given as
R(t ) < b(t ) ) S (t ) ∗ N (t )
<
<
<
⁄
ò b(t )S (t , t )d t ∗ N (t )
,⁄
⁄
⁄
Am g(t , mT , t ) d t ∗ N (t )
ò b(t )må
<,⁄
,⁄
⁄
å
m <,⁄
Amh(t , mT ) ∗ N (t )
Here h(t ) is the received pulse shape
h(t ) <
⁄
ò
,⁄
b(t )g(t , t ) d t < b(t ) ) g (t )
Example:
If the channel is steeply bandlimited
1
ì
ï
ï
B( f ) < í
ï
0
ï
î
f ; BW
f ″ BW
then the rectangular pulse shape (in previous page) is not feasible,
since it would be strongly distorted in the channel
· Magnitude of the distortion depends on the ratio between the
bandwidth BW and the symbol duration T (why?)
In future optimizing the pulse shape within the given band limitations is
anyway one of the most important objectives.
TLT-5206 / 360
Elements of the receiver
Development and functioning of the receiver is generally more critical
· Detection of transmitted bits reliably from the received signal
(noisy and distorted)
Basic receiver functionality is depicted below.
Timing recovery
Defines the correct timing and sampling time for the received pulses.
There are often specific components in the transmitted signal that make
the synchronization easier. However, this is not always necessary (in
carrier modulated system also the carrier synchronization is important).
Receiver filter
1) Attenuates noise and interference outside the transmission band
2) Affects the pulse shape
3) If the transmission channel is known, the receiver filter can
compensate linear distortion caused by the channel (equalization
using the inverse transfer function)
· Channel’s transfer function is not usually known, and
therefore, adaptive techniques are important
· In practice part of the receiver filtering (e.g., channel
equalization) is performed using discrete time filters (after
sampling)
NB! The signal after the receiver filter f (t ) is given as
Q(t ) < f (t ) ) R(t ) < ... <
⁄
å
m <,⁄
Am p(t , mT ) ∗ N ϒ(t )
where p(t ) < g(t ) ) b(t ) ) f (t ) is the overall pulse shape and
N ϒ(t ) < f (t ) ) N (t ) describes the filtered (colored) noise.
TLT-5206 / 361
Elements of the receiver (continuing)
Sampling
In sampling the continuous-time signal is sampled to create the
corresponding discrete time signal. In ideal case the sample is taken at
the moment in which the sample corresponds best with the transmitted
symbol (effects of other symbols is minimized)
Detection (decision making)
µ k of the transmitted symbol sequence Ak is
In detection an estimate A
created using the received sample sequence Qk . Conventionally this is
based on some decision thresholds.
Example: Let’s consider a ternary alphabet {,1, 0, ∗ 1} . In this case the
decisions could be done according to the following figure:
Here we have applied so called minimum distance principle which will
maximize the probability of correct decisions with certain assumptions
(e.g., noise distribution type).
· Basically, statistical decision making (detection theory) is a large
field of applied mathematics which will be more discussed in the
TLT-506 Digital transmission course.
Decoding
Describes the detected symbol sequence back into the bit stream
according to the used alphabet.
TLT-5206 / 366
Pulse shaping in baseband PAM system
Inter-symbol interference is essential in the sampling process in the
receiver. The observed pulse shape p(t ) < g(t ) ) b(t ) ) f (t ) depends on
the transmitter filter g(t ) , the receiver filter f (t ), and the channel b(t ) .
Now the corresponding transfer function is P ( f ) < G (f )B( f )F (f ) .
The objective is that the cascade of the three transfer function
P ( f ) < G ( f )B( f )F ( f )
fulfils the Nyquist criterion.
This is so called zero-forcing criterion, since it forces the inter-symbol
interference to zero. However, when the channel noise is included, it
does not necessarily offer the optimal solution (as seen later on)
The transfer function of the channel is usually fixed or it cannot be
affected.
Transmission and receiver filters are designed together. Here the
following solutions are available:
4) Pulse shaping in the transmitter, the receiver filter approximates
the ideal lowpass filter whose bandwidth is (1 ∗ a)/ 2T . Simple
from implementation point of view.
5) Matched filter pair is theoretically the optimal solution. In this
case the impulse responses are mirror images and the
amplitude responses are the same. If the channel does not
considerably affect the pulse shape, this gives the optimal
solution.
6) Transmitter filter is designed according to the item (1). The
receiver filter strives adaptively to minimize ISI with certain
criterions. Feasible in the sense that the channel’s transfer
function is rarely known (at least while designing the filter). On
the other hand, implementing an adaptive filter, instead of a
fixed one, is more complicated.
TLT-5206 / 369
Raised-cosine pulses (continuing)
Many of the Nyquist filters, also the raised-cosine filters, have also so
called square root Nyquist versions
· Here the idea is that an individual pulse does not fulfill the Nyquist
criterion but two consecutive filters (in cascade) do.
· I.e. one square root filter in the transmitter and one in the receiver.
Expressions:
p (t ) <
4a
t
t
t
cos([1 ∗ a ]p ) ∗ sin([1 , a ]p )
T
T
T
t
t
p [1 , (4a )2 ]
T
T
ì
ïïï T
ïï
ï T
é pT
P ( f ) < ïí
1 , cos ê
ïï 2
ë a
ïï
ïï 0
ï
î
ζ
∋
f ,
(|
1∗a ù
ú
2T û
; f ′
1,a
2T
1,a
1∗a
′ f ′
2T
2T
1∗a
; f ″
2T
;
Design and analysis in Matlab, e.g., using the “rcosine” function
In case of an ideal channel, square-root
raised-cosine filters in both transmitter and
receiver is the ideal solution providing
0 ISI.
TLT-5206 / 375
CARRIER MODULATION IN DIGITAL TRANSMISSION
In carrier modulation the baseband signal is relocated around a desired
carrier frequency (...in frequency domain).
...or…from digital transmission point of view:
In carrier modulated digital transmission, we develop effectively bit
carrying waveforms, whose power/energy is located in the desired
part of the spectrum.
Contents:
· Complex quadrature (I/Q) modulation and complex constellations:
QAM, PSK
TLT-5206 / 377
Complex (I/Q) modulation
Here
s(t ) <
x (t ) <
⁄
å
m <,⁄
am g(t , mT )
é jw t ⁄
ù
2 Re ê e c å am g(t , mT ) ú
êë
úû
m <,⁄
and wc < 2p fc where fc is the carrier frequency. (scaling factor 2 is
included to match the powers of the transmitted signal x (t ) and the
baseband signal s(t ) ).
Spectrum illustration is given below:
Modulating signal spectrum
Modulated signal spectrum
X (f )
WLP < (1 ∗ a)
1
2T
NB! Bandwidth of the modulated signal
WBP < (1 ∗ a)
1
T
WBP < (1 ∗ a)
1
T
TLT-5206 / 378
I/Q modulation
Practically g ( t ) (transmit filter) is always a real pulse:
⁄
2 cos(wct )
å
Re[am ]g(t , mT )
, 2 sin(wct )
å
Im[am ]g(t , mT )
x (t ) <
m <,⁄
⁄
m <,⁄
Hence, the real and imaginary parts of the baseband signal modulate
the cosine and sine components of the carrier.
Although this is the practical implementation principle, complex notation
is used in future (since it is much simpler).
NB: The term bandpass PAM is quite often used in case of the I/Q
modulation.
TLT-5206 / 379
Bandpass PAM receiver
Two equivalent structures:
1) Based on complex signals:
2) Based on real signals:
Spectra interpretation using complex signals:
TLT-5206 / 381
Constellations
In I/Q modulated PAM, the symbol values are complex numbers. If we
assume that the transmit filter g(t ) is real (practically always true) and
we denote the complex symbol using its amplitude and phase as
am < rme jfm , the modulated signal can be further written as
x (t ) <
<
<
é jw t ⁄
ù
2 Re ê e c å amg (t , mT ) ú
êë
úû
m <,⁄
é ⁄ j w t jf
ù
2 Re ê å e c rme m g (t , mT ) ú
êë m <,⁄
úû
2
⁄
å
m <,⁄
rm cos(wct ∗ fm )g(t , mT )
Consequently, it is possible to think the modulated waveform so that
· The amplitude and phase in a certain symbol period defines the
amplitude and phase of the modulated carrier
· Changes between consecutive symbols depend on pulse shaping
The utilized complex symbols (the symbol alphabet) can be illustratively
presented with a constellation figure.
Often used constellations are, e.g.,
· QAM: constellation points are distributed in uniformly spaced grid
· PSK: constellation points are distributed uniformly in a circle
Also many other constellations are available.
TLT-5406/103
CONSTELLATIONS
Symbol values are complex in I/Q modulation. They
determine the amplitude and phase of the modulated carrier
at the sampling instants. The transitions between adjacent
symbols depend on the used pulse shaping.
Example: Square pulse shaping,
symbol sequence 1 -1 3j -5j 1
6
4
2
0
-2
-4
-6
0
0.5
1
1.5
2
2.5
3
3.5
Time in symbol intervals
4
4.5
5
The complex symbol values (alphabet) can be represented
by constellation:
• QAM: points are situated on a regular rectangular grid
• PSK: points are situated on a circle
• Also other constellations are available, but not in wide use.
TLT-5206 / 382
Example constellations
4-PSK (QPSK)
· Alphabet size 2B < 4 , each symbol represents B < 2 bits.
· Symbols: Am < be j fm ; fm Î {0, p 2 , p, 3p 2}
· Information is in the phase of the modulated waveform
16-QAM
· Alphabet size 2B < 16 , each symbol represents B < 4 bits.
· Symbols: Am < am,I ∗ jam,Q ; am,I , am ,Q Î {°c, °3c}
· Information is in the amplitude and phase of the modulated
waveform
More general:
· M-PSK: 2B constellation points uniformly distributed in a circle
· M-QAM: 2B square formed (2B / 2 ≥ 2B / 2 ) constellation (uniformly
spaced “grid”)
TLT-5206 / 383
Noisy constellation
Because of the noise, the received samples are not perfectly matching
the constellation points. Now, if we draw noisy samples from the
complex domain, the following types of results can be seen:
Symbol error rate as
Im{Qk }
a function of SNR is
an important performance metric.
Re{Qk }
It is calculated
based on the properties
of Gaussian distribution.
Im{Qk }
Re{Qk }
If the physical channel noise is Gaussian distributed, also the deviation
around the ideal constellation points is distributed in the same way
(why?). Subsequently, Gaussian “clouds” are seen around the points.
Generally, the objective in decision making is to select that specific
symbol (from the used alphabet) which the received complex sample
Qk would most probably be representing. Intuitively the most
µ k which minimizes the distance
reasonable
selection is the symbol A
2
µ k . It can be shown that with certain presumptions this is also
Qk , A
the optimal selection (that maximizes the probability of the correct
decision).
By this way a decision region can be formed around every constellation
point. This region includes all those points that are closer to that specific
symbol compared to any other symbols in the alphabet.
Here are examples of decision regions for the 4-PSK and 16-QAM:
Im{Qk }
Re{Qk }
Im{Qk }
Re{Qk }
TLT-5206 / 392
SHORT INTRODUCTION TO INFORMATION THEORY
In this section we touch the very basics of such concepts as
information, entropy, and channel capacity.
In the general electrical communication context, by using these ideas, it
is possible to determine the largest possible information transmission
rate through a given channel
· This is called the channel capacity
· Formulated by the so called Shannon-Hartley law
Even though it is usually not possible to achieve the channel capacity in
a practical system, it is an important reference point when evaluating
the performance of practical systems.
In fact, the Shannon-Hartley law is one of the most important
fundamental laws of nature in the field of communication theory, and it
is quite useful also in practical engineering work.
In general, the main idea here is to shortly introduce these concepts.
Much more detailed treatment will be given in the following course
TLT-5400/5406 Digital Transmission
Sources and references:
· E. A. Lee and D.G. Messerschmitt, “Digital Communication”,
Kluwer, 1988/1994/2003.
· S. Benedetto and E. Biglieri, “Principles of Digital Transmission”,
Kluwer, 1999.
· B. Sklar, “Digital Communications”, Prentice-Hall 2001.
TLT-5206 / 401
CHANNEL CAPACITY
Addressing the fundamental physical limits for the amount of
information (per time unit or channel use), which in theory can be
communicated error-free over a given channel.
SOURCE
X
CHANNEL
Y
SINK
The source is here modeled as a sequence of independent
observations of a source random variable X .
The observation consists of another random variable Y .
Based on the earlier discussions, the average amount of information at
source per unit time is H (X ).
The question is now: How much of this information can “pass” through
the channel ?
And in general: What’s the maximum amount of information transfer
rate that a given channel can support ?
These are addressed in the following.
TLT-5206 / 404
Example 1: Binary Symmetric Channel
1–p
x=0
p
y=0
Y
X
p
x=1
1–p
y=1
· channel bit flip probability p
· input probabilities PX (1) < q , PX (0) < 1 , q
· an example of discrete-input discrete-output system model
The maximum of mutual information and thus the capacity is obtained
for q < 1/ 2 which yields (…)
C s < 1 ∗ p log2 p ∗ (1 , p) log2 (1 , p)
g
p < 1/2
g
p=0 or p < 1
Þ Cs < 0
(input and output are independent of each other)
Þ C s < 1 (error-free binary channel)
TLT-5206 / 405
Example 2: Capacity of Continuous-Time AWGN Channels
Physical system bandwidth W [Hz] and the channel adds white
Gaussian noise (AWGN) to the transmitted waveform.
Then it turns out that the maximum mutual information and thus the
capacity is of the form (bits per second here)
C < W log2 (1 ∗
S
)
N
where S denotes the received signal power and N the additive noise
power within the signal band. This is generally known as ShannonHartley law.
When deriving this result (TLT-5400/5406), it is assumed that also the
information bearing signal is Gaussian distributed (this is indeed
needed to maximize the mutual information).
Obviously none of the practical digital transmission systems transmit
Gaussian signals but are typically based on discrete symbol alphabets
(binary, QPSK, 16QAM, etc.)
Then one crucial question is how much of the available maximum
capacity is lost when using the discrete alphabets.
The answer is: Not much, when designed and implemented properly.
This (among others) will be addressed in much more details in the
Digital Transmission course ... :o)