Recapitulation

Focus topics
 Signals
• Fourier series; for what it is used
• Spectrum of a signal
• Sampling theorem; conditions for
reproduction of a signal from samples
• The concept of digital signal
• Fixed and variable rate digital data; typical
sources of these type of data
Representation of a signal in the frequency domain –
periodic signals

Any periodic signal can be

developed in a Fourier series
 Generalization for nonperiodic signals

The Fourier transform (linear transformation)

 
X(f) =  x(t)e-j2ftdt  x(t) =  X(f) ej2ftdf
- -

angular frequency =2f generalized frequency f(-, )

 Nyquist-Shannon sampling theorem

A perfectly lowpass signal ….

x(t)  X(f)
X(f)
f |X(f)|=0 when |f|> B
0
-B B

… can be ideally retrieved from samples (lowpass filtering) ….

{ xn }  x(t) p
= 2
… if they are being taken at minimum with the Nyquist frequency:

fp = 1  2 B
Tp
 Representation of a digital signal

B(k-1)T bkT …………………………..

T time

State of a given bit signal in a message code

(eg. b0) ia a given signalling time slot

Bit rate = number of bits per second

Total bit rate = bit rate  number of bits in a message
General forms of data
 Signals representing typical messages
• Masking effects in sound audibility
• Audibility threshold; logarithmic measure of
loudness
• The way of representing still images;
description of pixel features
• The way of representing movies; image
scanning in TV; bandwidth of TV signal (on
what it depends?)
 Audio masking features

masking tone
Acoustic pressure [dB]

masked tone

masking threshold

threshold in silence

frequency [Hz]
 Sounds - dynamics

Source of sound Pressure relative

[Pascal] [dB]
Pain threshold 100 134
Jet engine 6 – 200 110 – 140
Disco 2 100
Road traffic 0.2 – 0.6 80 – 90
Car 0.02 – 0.2 60 – 80
Talk 0.002 – 0.02 40 – 60
Very silent room 0.0002 – 0.0006 20 – 30
Leaves noise 0.00006 10
Audibility threshold 0.00002 0
 Structure of a still image (representation)

Image representation:
• 2-D matrix of pixels
• every pixel has following
features:
• colour intensity (intensity
of colour components)

Intensities of features
expressed as corresponding
signals

Lena – a test image

Signal representing a still image

The signal representing

a single line

Scanning time [s]

Still image signal spectrum (BW)

f [Hz]
fmax
fsyncl

Line synchro
 Video

Numerous images per second  fast scanning required (200 ns per

line)  high-frequency components

Parameter 525/60 625/50 unit

Lines per frame 525 625 lines
Lines per half-frame 262.5 312.5 lines
Frames per second 29.97 25 1/s
Half-frames per sec. 59.94 50 1/s
Line duration ~52.9 64 s
Video signal 4.28 6.5 MHz
bandwidth
half-frames  interleaved
scanning
 Video

Example: bandwidths for black-white video

luminance
576i720 – 5.15MHz
720i 1280 – 11.44MHz
1080i 1920 – 25.45MHz

576p720 – 10.30MHz
720p 1280 – 22.88MHz
1080p 1920 – 50.90MHz

For colour vodeo (without coding)  3 signals of identical bandwidth

 Physical means of transportation
• Properties of twisted pairs; reason for twisting
• Properties of optical fiber; single and multi-
mode fiber
• Physical phenomena of signal power
attenuation in wires, optical fiber, radio
• Rain fading in radio; physical cause of rain
fading
 Transfer characteristics

(f) 0 1km 1km

2km 2km
3km 3km
-50
4km 4km
5km

-100
0 0.2 0.6 0.8 1.0 1.2 MHz 0 10 20 s

• Wire gauge (diameter) influences shape of transfer function

attenuation: several tens to

several hundreds of dB/km
 Crosstalk

Far-end crostalk

Near-end crosstalk

• Crosstalk from other wires adds noise to transmitted signals

• Reflections at points where wave impedance changes
 Types of optical fibers
Core  Cladding  Coating 

Multimode fiber
(step index)

Multimode fiber
(step index)

Singlemode
fiber (SI)
 Attenuation in optical fibers

 [dB/km]

wavelength nm
 Transfer function

Dispersion:

• intermodal (MM) MM SI <5MHz, <1km

• chromatic compensators MM GI <50MHz, <40km

• polarization SM 20...160 GHz, 200...1500km

 Propagation of radiowaves

Free-space propagation

At d distance from antenna the

field intensity decreased ~ d2

+
PROPAGATION DISTURBANCES
Propagation of radiowaves - disturbances

altering direction
Deflection
Refraction
Diffraction wave weakenning

Attenuation
Interference signal filtering
 Temporary disturbances

Radiowave attenuation in over 10 GHz range strongly depends on

rainfalls and polarization

Per cent of time during which

rainfall exceeds the indicated value
 Analog modulation and multiplexing
• Properties of AM-DSB and AM-SSB
modulations
• Cohenrent and envelope demodulation of AM
signals
• Properties of FM modulation
• Concept and properties of FDM and TDM
multiplexing
 Amplitude modulation (AM)

g(t)=A·[ 1+m·x(t) ]·cos(2f0t)

 Amplitude modulation (AM)

Spectra of AM

• Modulating data appears in the form

of signal components at frequencies
DSB slightly higher and lower than that of
the carrier
• The components are called
SC DSB sidebands
• The lower sideband (LSB) appears
at frequencies below the carrier
frequency
• The upper sideband (USB) appears
at frequencies above the carrier
frequency
 AM demodulation (2)

 Coherent detector (AM – DSB SC )

x(t)Acos(2f0t+0)+z(t) xd(t) = ½AAgcos(0 - g)·x(t)

Ring Lowpass
modulator filter

Agcos(2fgt+g)

Stablilized
sine wave Sm(f)
generator

-2f0 0 2f0
 AM demodulation (1)

 Envelope detector (AM - DSB)

AM LF signal
signal
(HF)

x(t)Acos(2f0 t)+z(t)

1º A[1+mx(t)] >> z(t) practically usefull

2º A[1+mx(t)] << z(t) only noise at the demodulator output

 Frequency modulation (FM)

FM in time domain

𝑡
• 𝑔 𝑡 = 𝐴𝑐 cos[2𝜋𝑓𝑐 t +2𝜋∆𝑓 0
𝑥 ()d]

Frequency modulation (FM) f(t)= fc + ∆fx(t)

 FM signal spectrum

2f

f – deviation (amplitude of
frequency alteration e.g. 75kHz) BT = 2(f + B) –
conventional bandwidth
BT >> B
 Properties of analog modulations

gain Bandwidth (BT)

AM-DSB ½ 2Bm
AM-
DSB-SC 1 2Bm
AM-
Wide 1 Bm
SSB-SC
bandwidth

FM* large *
2(f+Bm)
2
 BT 
  

 Bm 
 Frequency division multiplexing (FDM, analog)

narrowband signals

x1(t)
mod
x2(t)
mod
x3(t)
mod
f0 f0 +f f0 +2f f

f0 f0 +f f0 +2f N–
channels

M1  cos(2f0)
Applications:
M2  cos(2f0+f)
M3  cos(2f0+2f) radio and television systems
 Time division multiplexing (TDM, analog)

N – channels

Wideband spectrum of
2 the TDM

M1   (nT0)
M2   (nT0 +)
M3   (nT0 +2)

At present, analog applications superceded by digital ones

 Digital modulations
• Constellations of binary and multilevel PSK,
QAM
• Consequencies of the use of multilevel
signalling
• Spectra of PAM, PSK, QAM signals;
bandwidths
• PAM, PSK, QAM, FSK signals in the time
domain
 Techniques for narrowband bandpass, M-QAM
• M-QAM – the most general narrowband bandpass
digital modulation
• Simultaneous modulation of amplitude and phase

• General formula for QAM signal

g i (t )  Ai cos(2f 0t  i )  ai cos(2f 0t )  bi sin(2f 0t )

e.g. 8QAM = 2 amplitudes, 4 i  0... M  1
phase values
 Techniques for narrowband bandpass, M-QAM

• Example: 16-QAM 64-QAM

constellations
Techniques for narrowband bandpass, other modulations

M-level phase shift keying (PSK)

2 2
g (t )  cos[ (m  1)] cos[2f 0t ]  sin[ (m  1)]sin[2f 0t ]
M M
011

010 001

110 000

111 100

101

8-PSK
 Multilevel signalling

• Value of a modulated parameter can take an

arbitrary number of levels
– 2-level (binary), the simplest and reference for
other types
– M-level (usually M=2k)
• Multilevel signalling reduces bandwidth
requirements by the factor:
k=log2(M)

• However, nonlinearily increases bit error rate

Techniques for narrowband lowpass, B-PAM
Two level digital modulation:

Umax

0 1 0

Umin
=1/Tb

Typical PAM spectrum

 Spectra of multilevel signals

….this example is for a modulation with pulse carrier

 Spectra of M-QAM, M-PSK, M-ASK signals

16-QAM spectrum

G(f) B-ASK spectrum

f0
¼B

Generally:
B

B=2/(Tblog2M)=2Rblog2M
 Techniques for bandpass, other modulations

M-level frequency shift keyng (FSK)

Example: B-FSK (used in GSM)

 Digital modulations
• Concept of the multicarrier modulation;
properties; applications
• The concept of symbol (binary sequence)
detection in PAM, PSK, QAM
Techniques for wideband vs narrowband channels

Channel

In a narrowband channel – a single carrier modulation sufficient

In a wideband channel:
• multicarrier

• spread spectrum
 Use of multiple carriers - ADSL
głos ULUL DL
 Detection of the transmitted symbols/messages

s1 t   n(t )
s2 t   n(t ) m1
Physical Decision m2
sM t   n(t )
means of
circuit
transportation

mM
 Detection of the transmitted symbols/messages

|s0(t)-r0(t)|<  symbol=M0

ac 1
Umax

as
0 1 0

Umin
0
Symbol detection in M-QAM (example: 4-QAM)

noiseless nonnegligible noise

Q Q
01 00 01 00

I I
11 10 11 10

Each point corresponds to a given binary sequence.

signalled within signalling time interval of lenght Ti
 Techniques for digital channel
• Amount of information in a message
• The concept of entropy of a data source; when
entropy maximizes?
• The concept of prediction based compression;
what conditions achievability of compression
 Measure of amount of information

Emission of any message is random

(probability).
The less probable message is … the more I=-log2(pk)
information it provides and vice versa!!!

N
H=- pk·log2(pk) Average amount of information
in a discrete source
1

If 2 messages in a source (ON/OFF) i p0=p1=1/2

Conventionally as a
H=1 (shannon, bit) measure of amount of
information
 Example: a binary source
Average amount of information in a single binary symbol

Probability of taking either 0 or 1 value

Average amount of information in a single bit maximizes when both values equally
probable (p0=p1=1/2)
 Lossless compression
Average codeword length generated by encoder per a single message:

K 1
L   pk  l k Number of bits of a k-th message
k 0 codeword

Shannon rule: average codeword in any lossless coding has the lower bound:

Representation can be
L  H ( ) adjusted so number of bits
is minimum!!
 Compression basics

• Transform symbols (signals), so their probability

distribution is known (possibly exponential)

– Coding that respects probability (frequency) :

frequent symbols assigned shorter
codewords
more rare symbols can be assigned
longer codewords(VLC)
 Predictive source coding

• Conditional probability for a symbol is used (condition = a given

sequence of symbols had emerged) :

…………….

• Signal sample is expressed as a function of preceding samples

(regression); equivalent to prediction

…………….