Introduction to Communication Systems 2

EEE 330
Introduction to
Communication Systems
Lecture # 2
Signal and Systems
Signal Comparison
Review of the Fourier Series
Review of the Fourier Transform

Overview
The Objectives of Today’s Lecture
Signal and Systems
Signal Comparison
Overview/Motivation for Fourier Theory
Review of the Fourier Series
Review of the Fourier Transform
Reading
B.P. Lathi, Modern Digital and Analog Communication
Systems, 3rd Ed., Oxford University Press, 1998.
Chapter 2
Chapter 3

To Study Communication Systems
you must understand…
Signals and Systems
Fourier Analysis
Modulation Theory
We will study this in detail
Detection Theory
Given that this signal is corrupt at the receiver, how
do we determine the original signal?
Probability Theory
Since the transmit signal and noise are both unknown
to the receiver, we can use probability theory to
study communications systems

Signals and Systems
In this class we will rely on mathematical
representations of signals and systems to describe
communications
Relies on background obtained from EEE301
A system is characterized by inputs and outputs which
are mathematically modeled as signals
We will also mathematically represent the signals at
various points within a communications system
Mathematical representations of the various
components of the system can be viewed as
subsystems with input-output relationships defined by
Impulse response in the time domain
Transfer function in the frequency domain

System Representation
A system is any process that results in the
transformation of signals.
H is typically used to represent the system
x(t) is typically used to represent the excitation or
input to the system
y(t) is typically used to represent the response or
output of the system
Systems can have multiple inputs and/or mulitple
outputs
Example of a Single-Input Single Output system:

System Properties
There are several properties of systems
that are important to understand
Many properties allow us to make
simplifications in our analysis
Specific properties
Time Invariance
Linearity
Stability
Causality
Memory
Invertibility

Time-invariance
A system is time-invariant if a time-shift in
the input causes a time shift in the output
Ex: y(t) = sin(x(t))
y(t-t0) = sin(x(t-t0))
If a system is not time-invariant, then it is
time-varying.
Ex: y(t) = t x(t)
y(t-t0) = t x(t-t0) (t-t0) x(t-t0)≠

Linearity
A linear system is any system that
obeys the properties of scaling
(homogeneity) and superposition
(additivity)
If y(t) = H(x(t)) then
α y(t) = H(α x(t))
and
H(α x1(t)+ β x2(t))= α H(x1(t))+ β H(x2(t))

Stability
A stable system is one where the
output does not diverge as long as
the input does not diverge.
If the input is bounded then the output is
also bounded (BIBO system)
However, this is not always true.
x[n] = u[n] (unit step function – Bounded)
y[n] = H(x[n] ) =(n+1)u[n] (not bounded)

Causality
A causal system is one that is
nonanticipative; that is, the output
may depend on current and past
inputs, but not future inputs.
Ex: y[n] = H(x[n]) = x[n] – x[n-1]

Memory
A system is memoryless if its output
for each value of independent
variable is dependent only on the
input at the same time.
Ex: y[n] = H(x[n]) = [x[n] ]2

Invertibility
A system is called invertible, if
distinct inputs lead to distinct
outputs.
By observing output, you can determine
its input
Ex: y(t) = 2x(t) x(t) = 0.5 y(t)

Signals
A signal is a function representing a
physical quantity.
Signals are represented mathematically as
functions of one or more independent
variables.
Speech signal is represented by acoustic
pressure as a function of time.
Picture is represented by brightness function of
tqo spatial variables.
Although functions can operate on any type
of variable, we will be most concerned with
functions of time

Physically realizable functions
Have finite time duration (finite
energy!)
Occupy finite frequency spectrum
Are continuous
Have finite peak value
Are real-valued

Classification of Signals
Signals (or more specifically their
mathematical representations) can be
categorized according to a few major
features
Continuous Time vs. Discrete Time
Analog vs. Digital
Deterministic vs. Propabilistic (Random)
Power vs. Energy
Periodic vs. Aperiodic
Even vs. Odd

Contiuous Time vs. Discrete Time
This classification is determined by whether or not the time axis-
independent variable) is discrete (countable) or continuous.
A continuous-time signal are defined for a continuum of values of time
A discrete-time signal is only defined at discrete times.

Analog vs. Digital
Analog signal can take any value for all t
Digital signal can take only finite number of distinct
values

Deterministic vs. Random
A deterministic signal is a signal in which each value of the signal is fixed
and can be determined by a mathematical expression, rule, or table.
The future values of the signal can be calculated from past values with complete
confidence.
If a signal is known only in terms of probabilistic description such as mean
value, mean squared value, and so on, it is a random signal.
The future values of a random signal cannot be accurately predicted and can
usually only be guessed based on the averages of sets of signals.

Power vs. Energy
Energy signals have finite energy
Every signal in real life is an energy signal
Power signal have finite and nonzero
power.
Power signal is of infinite duration

Periodic vs. Aperiodic
Periodic signals repeat with some period T.
A signal is called aperiodic if it is not
periodic.

Even vs. Odd
An even signal is any
signal f such that f(t)
=f(−t) .
Even signals can be easily
spotted as they are
symmetric around the
vertical axis.
An odd signal is a signal f
such that f(t) =−f(−t)

Signal Comparison (Orthogonality)
Orthogonality: Two complex signals are said to
be orthogonal over an interval t1≤ t ≤ t2, if
or
Significance:
Sum of weighted orthogonal signals are used to
represent any signal with minimum error
We can transmit signals over orthogonal signals
We can reject undesired signals to select just one that
we want, by filtering at the demodulator
Orthogonal signals are used in CDMA
2
1
*
1 2( ) ( ) 0
t
t
x t x t dt =∫
2
1
*
1 2( ) ( ) 0
t
t
x t x t dt =∫

Orthogonality
(Approximation of functions)
xn are N mutually orthogonal signals
cn are coefficients

Orthogonality (Transmitter)
Any two sinusoids that
are harmonically
related are orthogonal
over the whole cycle
All sinusoids are
orthogonal over the
interval -∞ to ∞.
This means we can
modulate information
over separate carriers
and “tune in” the
channel we want.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-1
0
1
sin(πt)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-1
0
1
sin(2πt)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-1
0
1
sin(4πt)
t

Orthogonality
(Demodulator)
0 1 2 3 4 5 6 7 8 9 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (kHz)
abs(ModulatedSignal)
f=3khz
Local
oscillator
0
∞
∫Performs operation
Signals orthogonal to
cos (2∗π∗3*10^3*t) will
be cancelled

Orthogonality (CDMA)
In CDMA the
spectrum is used
by N users
Each user is
assigned to a
unique code
The N codes are
orthogonal to each
other
3G systems
0 0.5 1 1.5 2
-1
-0.5
0
0.5
1
code1
0 0.5 1 1.5 2
-1
-0.5
0
0.5
1
time
code2

Signal Comparison (Correlation)
Correlation is related with the information
that how much two signals are similar
cn is the correlation coefficient and
normalizes the levels of g(t) and
x(t), which are complex signals.
1
( ) ( )n
g x
c g t x t dt
E E
∞
∗
−∞
= ∫
1 1nc− ≤ ≤
1
g xE E

Correlation
cn = 1 Two signals
are similar
Two best friends
cn = 0 Two signals
are orthogonal
Unrelated
Complete strangers
cn =-1 Two signals
are dissimilar
Worst enemies
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-1
0
1
g(t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-1
0
1
x1
(t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-1
0
1
x2
(t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-1
0
1
time
x3
(t)

Correlation (Contd.)
x2(t) is a shifted
version of x1(t),
hence they are
“IDENTICAL”
However cn=0
Use cross-
correlation instead
of correlation
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
x1
(t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
time
x2
(t)

Cross-Correlation
function
g(t)
z(t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
time
z(t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
g(t)

Cross-Correlation
The normalization factor is dropped
It can take any value
It plots the similarity index for all
relative time shifts between two
signals
For complex signals

Autocorrelation Function
The correlation of signal with itself is
called autocorrelation
The autocorrelation function for a real
signal g(t) is

Autocorrelation
Autocorrelation of a random noiseor
signal indicates the “periodicity”
structure of the signal
Truely random (unpredictable) noise
has Ψg(τ)=δ(τ).

Review of the Fourier Series and the
Fourier Transform

Motivation
If a system is linear, the response due to a sum of
signals is the sum of the responses to each individual
signal
System analysis can be simplified by decomposing an
input signal into a sum of simpler signals
The system output can then be found as the sum of
the system responses to these simpler signals
A physically meaningful way of decomposing signals is
to represent them as a sum (or integral) of sinusoids
Periodic signals – Fourier Series
Aperiodic signals – Fourier Transform
Periodic signals can also be represented using the
Fourier Transform
This gives rise to the idea of the frequency domain

Fourier Theory
Two basic types of signals
Power signals
Energy signals
We can represent a signal in time or in frequency
Fourier Representations
Fourier Series
Representation valid for all time if signal is periodic
(i.e., power signals)
Representation is valid only over a certain interval for
aperiodic signals
Fourier Transform
Applies directly to energy signals
Requires introduction of the impulse for application to
power signals

Fourier Theory (cont.)
Fourier Theory tells us that signals can be represented
as weighted sums (or integrals) of sinusoids.
The “amount” of each sinusoid is equivalent to the
“frequency domain” information of a particular signal
If the signal is periodic, the signal can be represented
as an infinite sum of sinusoids whose frequencies are
integer multiples of the fundamental frequency, fo.
If a signal is aperiodic we can take the limit of the
Fourier Series as the period goes to infinity. The result
is the Fourier Transform
The Fourier Transform doesn’t technically apply to
periodic signals.
However we can create a FT through the use of the
delta function

Trigonometric Fourier Series
Trigonometric Fourier Series
where To is the period, and
CompacttrigonometricFourierSeries

Exponential Fourier Series
We can represent a periodic signal g(t) with period T0
exactly by the sum of complex sinusoids
where
The above integral must converge
This is termed the Exponential Fourier Series
We can represent the relationship between g(t) and Dn
as
( ) FS
ng t D←→

Exponential Fourier Series (Cont’d)
Dn are complex numbers in general.
For any real signal, | Dn | is even
function and the phase is always
an odd function
It still represents a signal on one
period.

Negative frequencies ?
Negative frequencies arise as a necessary
implication of the exponential phasor view
of the signals, required to be able to
represnt purely real signal.
Frequency is not negative
cos( )
2
j t j t
e e
t
ω ω
ω
−
+
=
sin( )
2
j t j t
e e
t
j
ω ω
ω
−
−
=

Energy Signals
The Fourier Series applies to periodic signals
which are also power signals.
They all have a “line spectrum”
However, we would like to analyze both
power signals and energy signals.
Energy signals have a “continuous spectrum”
There is some energy at every frequency in the
signal spectrum
Thus, we need a more powerful analysis
tool.
The Fourier Transform is the answer

Fourier Transform
It is the Fourier series in the limit
or
G(ω)
has the same amplitude symmetry and phase
anti-symmetry properties of exp. FS
For a single pulse g(t), it gives the envelope of
exp. FS that is obtaines if g(t) is repeated
periodically.

FT and Exp. FS
Continuous
Spectrum
Line
Spectrum

G(ω)
In general G(ω) is a complex number
G(ω) = | G(ω) |*exp(jθG(ω))
For real signals g(t)
G(ω)= G*(-ω) (Conjugate symmetry
property)
Hence
|G(-ω)|= |G(ω)| (even function of ω)
θG(-ω)= -θG(ω) (odd function of ω)

Inverse Fourier Transform
Inverse Fourier Transform reconstructs the
signal from its spectrum
g(t) and G(ω) forms a FT pair

The Frequency Domain
The original signal g(t) is said to be in the time domain
since its argument represents time
The Fourier Transform G(ω) representation is said to be in
the frequency domain since its argument ω represents
frequency
Notes:
Frequency is the reciprocal of time
The Fourier Transform is referred to as an analysis of the
signal g(t) since it extracts the frequency components of
g(t) at each value of ω
The Inverse Fourier Transform is referred to as synthesis
since it recombines the components G(ω) to obtain the
original signal g(t)
The physical meaning of G(ω) depends on the meaning of
g(t). If g(t) has units of volts, G(ω) has units volts/Hz.
Thus it represents how much of the voltage signal is present
at each frequency.

The Frequency Domain
We can think of the Fourier Transform and the
Inverse Fourier Transform as means for moving
between the time and frequency domains
Note that no information is lost in the transformation
and both are equivalent representations of a signal
This is sometimes
termed the
“Analysis equation”
This is sometimes
termed the
“Synthesis equation”

Existence of FT
Not all the signals are Fourier
transformable
The existence of FT is assured for any
g(t) satisfying the Dirichlet’s
conditions, i.e.

FT of Rectangular pulse
-τ/2 τ/2

Time vs. Frequency
τ =0.1
τ =0.01

Time vs. Frequency
Time and frequency are reciprocal
If a function speeds up in time, it slows down in
frequency
If a signal changes rapidly it requires more high
frequency components
Signals which change rapidly in time are said to have
a large bandwidth (a measure of the frequency
content)
If a function slows down in time, it speeds up in
frequency
If a signal changes slowly in time it requires less high
frequency components and more low-frequency
components
Signals which change slowly in time are said to have
a small bandwidth

Definitions of Bandwidth for
Baseband Signals
Bandwidth is a term used to describe a positive
frequency range over which the signal has significant
content. There are various definitions for bandwidth
including:
Absolute Bandwidth (Babs)
Defined as B where G(ω)=0 ω>B
3-dB Bandwidth (half-power bandwidth - (B3dB))
Defined as B where
X-dB Bandwidth
Defined as B where
First Null Bandwidth (Bfirst null)
For baseband systems this is equal to the frequency of
the first null in the spectrum
( ) ( )10 10 max
20log ( ) 20log ( ) -X >G G Bω ω ω<
2
2 max
( )
( ) >
2
G
G B
ω
ω ω<

Properties of Fourier Transform
Time-Frequency Duality
Symmetry
Linearity
Scaling
Time-shifting
Frequency-shifting
Convolution and multiplication
Time-differentiation and Time-Integration
Refer to Table 3.2 on pg 101 for the properties

Time-Frequency Duality
Due to the similar nature of the Fourier Transform
and the Inverse Fourier Transform, there is the
duality property.
Whenever we derive any result, we can be sure
that it has a dual

Symmetry (part of duality)
If
then
Example:

Linearity
If g(t)=α g1(t)+β g2(t)
then
G(ω)=α G1(ω)+β G2(ω)
α g1(t)+β g2(t) α G1(ω)+β G2(ω)

Scaling
If
then for a real constant a

Scaling - Interpretation
Scaling property states that the time compression
of signal results in the spectral expansion, and time
expansion of signal results in the spectral
compression.
Time Compression: α > 1.
Scaling a signal in time by α speeds the signal up in time.
The resulting transform is scaled by 1/α which slows the
transform down in frequency – this means that more of the
larger frequency values are present to accomplish faster
changes.
Time Expansion: α < 1.
Scaling a signal in time by 1/α slows the signal down in
time.
The resulting transform is scaled by α which speeds it up in
frequency – this means that more low frequency values are
present to account for slower changes.

Time and Frequency-shifting
Time-Shifting
If
then
Frequency-shifting
If
then

Frequency-Shifting and Modulation
Since
then
Amplitude
modulation
Carrier

Example
g(t) = rect(t) |G(ω)| = |sinc(ω/2)|

Example– cont’d.
z(t)=rect(t)cos(200πt), ωo=200π
Z(ω)=0.5*(sinc((ω−200π)/2)+ sinc((ω+200π)/2))

Bandpass signals
Low pass
Bandwidth: 2πB
Band pass
Bandwidth: 4πB
If a linear combination
of these two band pass
signals will be a band
pass signal

Convolution and multiplication
If
then
and
Thus, convolution in the time domain results in
multiplication in the frequency domain while
multiplication in the time domain results in
convolution in the frequency domain.
This can greatly simplify some system analysis
BW= B1 BW= B2
BW= B1+B2

Time-Differentiation
and Time-Integration
If then
time-differentiation
and time-integration

Summary
In this lecture we have discussed
Signals and systems
Fourier series
Fourier Transform.
The Fourier Transform is useful for providing
a frequency domain representation of
periodic and aperiodic signals that is valid for
all time.
Understanding the relationship between time
and frequency is perhaps one of the most
important concepts in this course.

Introduction to Communication Systems 2

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Introduction to Communication Systems 2