Lecture 2

Description of telecommunication
signals

Types of signals
Signals’ features in time and frequency domains
Representation of bandpass signals
Representation of discrete time signals
Representation of digital signals
 The concept of a signal

Signal represents evolution in time (or other independent

variable, e.g. displacement) of a physical quantity. May
be a vector function.

The role of signals in telecom:

- physical representation of a
message
- message carrier in a process
of transfering over distances

Generated in coders and modulators

 Description of signals

• Any signal can be described as a function of an

independent variable(s), typically time (time domain
representation).
• Equivalently, the functions can be transformed to an
another domain, typically frequency. Linear
transformations are 1 to 1 mappings (bijective), so
preserve all features of an original representation.
• Compact descriptions: use some features that
characterize the signal well (signal model) plus some
parameters
Description of signals in the time domain
Description of signals in the time domain – signal features

Qualitative description:
• periodic / non-periodic
• impulse
• harmonic

….. and more cathegories to come …

 Types of signals (1)

signals

deterministic random

periodic/non-periodic nonstationary stationary

impulse
normally distrib normally distrib
harmonic

finite/infinite energy

finite/infinite power
 Types of signals (2)

Continuous time Discrete time

Discrete value Continuous value

practically all real life

signals made artifically
continuous functions sampling operation

made artifically made artifically

quantization operation discontinuous functions
 Types of signals (2 cont)

Continuous time Discrete time

Continuous value

x R x R
tD

tR

x D x D
Discrete value

t R tR
 Description of signals in the time domain

Quantitative description:
• duration (also conventional)
• peak value
• mean value
• energy
• power (mean square)
• root mean square value
 Signal features in time domain

signal duration
A
Unit Unit Equivalent
u(  )
x(t)

second (s) 1s herc (Hz)


1 Hz
T authentic
millisecond (ms) 10–3 s kiloherc (KHz) 103 Hz

microsecond (s) 10–6 s megaherc (MHz) 106 Hz

x( t )

nanosecond (ns) 10–9 s gigaherc (GHz)

o( t )
109 Hz
0

picosecond (ps) 10–12 s teraherc (THz) T 1012conventional

Hz
t
 Signal features in time domain

• signal mean (DC value):

• energy:

• mean power (mean square):

• root mean square:

Signal features in time domain -values expressed in dB

dB is a way to represent a relative value

Voltage: dBmV, dBV

current: dBmA, dBA

power: dBmW, dBW

Signal features in time domain - power values expressed in
dB

Power ratio [dB]

0.01 -20
0.1 -10
1 0
10 10
100 20
1000 30
104 40
Signal features in time domain - examples of signals

Finite duration & finite energy

square pulse

Td=1
xp=1
W=1
Signal features in time domain - examples of signals

Infinite duration & finite energy

Sa function

Td=
xp=1
xave=0
W=/0
=2πf
 Signal features in time domain - examples of signals

Infinite duration & infinite energy (finite power)

harmonic function

Td=
xpp=2X0
xave=0
P=0.5(X0)2
W= 
 Signal features in time domain - examples of signals

Unitary step function 1(t)

Infinite duration &

infinite energy
(finite power)

Td= 
xp=1
W= 
 Examples of signals

Dirac delta distribution

Model of a physically non-

realizable signal of a unity power
and zero duration

Td=0
xp=
P=
W=1
Representation of a signal in the frequency domain
Representation of a signal in the frequency domain –
periodic signals

An example periodic signal

Representation of a signal in the frequency domain –
periodic signals
Representation of a signal in the frequency domain –
periodic signals

Any periodic signal can be

developed in a Fourier series
Physical interpretation of signal representation in
frequency domain
 Representation of a signal in the frequency domain –
periodic signals

2 harmonics

9 harmonics

all harmonics
 Generalization for nonperiodic signals

The Fourier transform (linear transformation)

 
X(f) =  x(t)e  x(t) =  X(f) e df
-j2ft j2ft
dt
- -
angular frequency =2f generalized frequency f(-, )
 Physical interpretation of signal representation in
frequency domain

Properties of a physical object (e.g. electrical filter)

described in the frequency domain:

x(t) y(t)

Y(j) =T(j)X(j)
Example signals and their spectra

Spectrum of a unity pulse

Example signals and their spectra

Spectrum of a high frequency pulse

 Example signals and their spectra

Spectrum of a sine wave (example of a periodic signal)

 Energy spectrum, power spectrum

Signal energy, Parseval theorem

  
E   x t dt  X   d  X   d
2 1 2 1 2

2   0



S    X  
2 Energy spectrum (spectral
distribution)
 Energy spectrum, power spectrum

X T  
2
1 
P
2 
  T 
lim
   T 
d
Signal power
S  

X T  
2

S    lim
Power spectrum (spectral
distribution)
T  T
 Energy spectrum, power spectrum – random signals

 
 
2
 1  X T  
P  E  Tlim d  Average power of a random process
 2  T 
 

 2
 X    
S    E  lim Average power spectrum (spectral
T
 distribution) of a random process
T  T 
 
Signal features in the frequency domain
Signal features in the frequency domain
 Signal features in the frequency domain

signal bandwidth:

Width of the spectral interval where energy (power)

of the signal is located
 Properties of signal spectrum

Scaling property

f
x t   X  
1
  

„Compressing” signal in time results in greater

dispersion in the frequency domain and vice versa
 Properties of signal spectrum

Scaling property

1
f1( t )
x(t)f2( t ) |X(f)|
0

1
6 4 2 0 2 4 6
t

Short signal duration  wide spectrum

long signal duration  narrow spectrum
 Properties of signal spectrum

Modulation property

xt  cos(2f ot )  X  f  f o   X  f  f o 
1
2
X(f)

X(f + fo)/2 X(f - fo)/2

-fo +fo
Representation of bandpass signals in the time domain
Time domain representation of bandpass signals

Definition of a bandpass signal:

x(t)  X(f) i |X(f)|=0 gdy |f - f0 |> 2B

|X(f)|

f
- f0
f0
2B 2B
 Time domain representation of bandpass signals

Decomposition to modulated in-phase and quadrature harmonic

components ……
in-phase component quadrature component

g(t) = x(t) cos(2f0t) - y(t) sin(2f0t)

gI (t) gQ (t)

where: x(t) & y(t) – components of a bandpass signal which can be get
from Hilbert transforms of the signal
 Time domain representation of bandpass signals

…… or equivalently

g(t) = a(t) cos[2f0t +(t)]

A harmonic signal jointly modulated in amplitude and phase.

Conclusions:
any bandpass signal is a combination of two amplitude
modulated quadrature harmonic signals
or equivalently …
is an amplitude and phase modulated harmonic signal
Time domain representation of bandpass signals

y(t)
ac

as
x(t)

In-phase component amplitude

Quadrature component amplitude

 Time domain representation of bandpass signals

alternative description……
a(t)
ac

as
(t)
 Time domain representation of bandpass signals

in-phase component quadrature component

…. there is how to connect the two representations

Conversion to discrete time signals
 Conversion to discrete-time signals

Sampling reads values of a signal at discrete time instances

x(t) x(nTp)
sampling

x(t) x(nTp)

t t


x(t)  { xn }= x(nTp) (t - n Tp)
p = -2
 Nyquist-Shannon sampling theorem

A perfectly lowpass signal ….

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

Conclusion: any lowpass signal can be perfectly decribed by

its (infinite number of) samples
 Representation of a lowpass signal

r(t)
 
x(t) = x(nTp) Sa[2B(t - n Tp)] = Sa(2Bt) * x(nTp) (t - n Tp)
- - { cn }

 Tp (f/B)  G(f) =
1
Tp
 X(f - n fp)
TpG(f)

X(f+ 2f ) X(f- f ) X(f) X(f- f ) X(f- 2f )

p p p p
f

- 2fp - fp 0 fp 2fp
 Conversion to discrete values

x(nTp)  L gdzie xR, L D

x(nTp)

t t

Tp .. 3Tp .. 5Tp Tp .. 3Tp .. 5Tp

Usually discrete time signals are an intermediate form

which then undergoes conversion to discrete-time &
discrete-value signals (digital)
 Conversion to discrete values (ADC)

L 
x(t)  { cn }=  cn (t - n Tp)
3
-
2Zmax

2
Xq=L where cnCD
1
0,
0
{ cn }= 1,
7
....

In an effect of analog-to-digital conversion:

Ln=[Un/]
 Reconversion to analog signal

Quantization
error (noise)

After digital-to analog

conversion:

Un=Ln
Representation of a digital signal
 Representation of a digital signal

A digital signal is (typically) binary coded …

Binary coding format is common in digital systems

ci  bk-1,bk, ….., b0 i=0,..,I-1

bl ={0,1} l=0,…,k-1
k - the length of a sequence (code)

Digitized analog signals:

In the simplest case: the code is a binary coded number of the
quantization interval within which the signal sample has fallen
 Representation of a digital signal
message A (111 code)
message B (000 code)
s0
Here is an example of how 3-
bit data can be represented by
s1 a signal
 s0 t  b0 
 s t    b 
 1   1
s2  s 2 t  b2 

Bit Rate = number of bits in

a message / signalling time
Tms
 Representation of a digital signal

b0 ………………………….. bk-2 bk-1

Tb time

Tms
State of a given bit signal in a message code Tb 
(eg. b0) ia a given signalling time slot k
Bit rate = number of bits per second
 Binary signal in the frequency domain

X(f)
A AT
u(  )
x(t)
f
A(t/T) 
w( t )

T 2/T
A·Sa(T)
t

Spectrum of a pulse signal

 Binary signal in the frequency domain

power spectrum
Obligatory readings:
•S. Haykin, „Communication systems” (4-th ed
available on internet)

Next lecture: „Characteristic

features of typical messages