Introduction to Communication Systems: Part II

&
Review of Signals & Systems: Part I

Satyajit Thakor
IIT Mandi
Wireless channel not EM waves
sound pressure waves

I Examples: Underwater acoustic, radar, sonar, cellular/mobile,

infrared, etc.
EM waves are
not
Why acoustic waves signals and
underwater communication
used for
e.g for submarines

EM waves cannot travel long distance

Ans
underwater due to heavy attenuation

waves are modulated to send information

Acoustic

in the air EM waves

Radar to see

Sonar to see under water acoustic waves

Wireless frequency range
o

limitedource

Telecom service
providers buy
spectrumfrom
gov
free
Receiver

I The function of receiver is to recover (almost perfectly /

approximately) the transmitted information from the signal
received via the channel.
I The design of receiver depends on the properties of the channel
and how the transmitter processes the baseband signal.

frequency band wireline

Channel properties e.g
or wireless

transmitted AM signal
AM receiver for
transmitted FM signal
FM receiver for
Mathematical models for communication channels

We will consider the following basic mathematical channel models.

These models reflect the most important characteristics of the
transmission medium.
I Additive noise channel
I Linear time-invariant filter channel
I Linear time-variant filter channel
Signals

I Signals are used to transmit information over a communication

channel.
I What is a signal?
Typically a function of time

I What is a function?
f X Y X domain Y co domain
I Real discrete time signal x[n], x : Z ! R. Example:

1 n2 + 1
x[n] = n + , x : Z ! R, n 7!
n n
I Complex continuous time signal x(t), x : R ! C. Example:

x(t) = ej✓t , x : R ! C, t 7! ej✓t

Basic operations on x(t)

I Time shifting x(t t0 ) sett alt to

if delayed
É
to 0
t
it to co advanced o
sittaldelayed by to
I Time reversal x( t) art Kft

E I
a 0
I Time scaling x(at) sett recat

slat
if act expanded a 2

if asi compressed
FIs It
Types of signals

I Continuous-time and discrete-time

set attotto EIR
Ex CT A cos
A cos antont 0 ne I
DT n

Here N n is obtained by sampling selt at t n

I Real and complex
A cos Catfot A
Ex Real
Complex A ei attotto
I Periodic and nonperiodic
alt t
sit act To
Periodic if 7 To 0

such To is the fundamental period

Smallest
is fundamental frequency
to
I Deterministic and random
To random processes
Types of signals

I Causal and noncausal

20
net is causal if sett 0
I Even, odd and Hermitian symmetry
salt
sett is even if aC t
C t ret
is odd if K

For ans
retiia I It
rect MIRI rott DIE
Hermitfiphustle sect is called Hermitian if
and its imaginary part is odd
its real part is even
Energy-type and power-type signals

I Energy content of a signal x(t) is defined as

Z +1 Z +T /2
2
Ex = |x(t)| dt = lim |x(t)|2 dt.
1 T !1 T /2

I Power content is defined as

Z +T /2
1
Px = lim |x(t)|2 dt.
T !1 T T /2

I Signal x(t) is energy-type i↵ Ex < 1.

I Signal x(t) is power-type i↵ 0 < Px < 1.
Some important signals A sin 2Tfot A cos attot 12

I Real sinusoidal signal: x(t) = A cos(2⇡f0 t + ✓)

I Complex sinusoidal or exponential: x(t) = Aej(2⇡f0 t+✓) In
I It is a spiral in the complex plane: plot (homework)
I Euler’s formula/identity
Tcstt sint
to
I Cartesian and polar representation of a point/signal in the
complex plane
kilt Cartesian
ret serct
e
out Polar
net

cos Cret
ret
art
Int
I
nicts
1Mt 1 I
net arctom 2T
Some important signals

I Unit step signal (Heaviside step):

(
1 t 0
u 1 (t) =
0 t<0

I Rectangular pulse:
TA
1

1m
(
1 1
1 t
2 2
⇧(t) =
0 otherwise

Ct U Ct
Note that Ct U

at discontinuity point t
except
Some important signals
convolution
I Triangular signal:
8 Δ Ct TAFE TA
>
<t + 1 1t0
(t) =
>
t+1 0t1 IT 2 IT t 2 dz
:
0 otherwise

I Signum signal:
8 24 Ct 1
>
< 1 t < 0 Sgult
asgult
sgn(t) = 1 t > 0 Cercept at discontinuity
>
:
0 t=0 Point t o
st

5inch
I Sinc signal: (

W.hr
sin(⇡t)
⇡t t 6= 0
sinc(t) =
1 t=0 i
Some important signals

I Impulse signal/Dirac delta function is a “function” (t) such

that Z 1
(t) (t)dt = (0)
1

for test function (t) continuous at the origin.

I Sifting property: The impulse signal sifts/extracts/samples the
value of test function at the origin under the integral sign.
I Note that, (0) = 1 (undefined) and 0 otherwise.
I Caution: Impulse is not a signal or function. Why?
I As the limit of some signals:
✓ ◆
1 t 1 1 t2
lim ⇧ , lim p e 2 2
✏#0 ✏ ✏ !0 2⇡ 2
I Recall/refer the textbook for more properties of (t).
Homework

I Revise the basic concepts in real analysis, linear algebra, signals

and systems, etc.
I For example: di↵erentiation, integration, limit, sequences, linear
operators, trigonometric functions and identities, Cartesian,
polar coordinate systems
I Solve problems on signals and systems given in reference books.

I Reading:
Chapter 2 of J. G. Proakis and M. Salehi, Fundamentals of
Communication Systems