2.7.1 Orthogonal Vector Space
The
goal is to represent a signal as a sum of an orthogonal set
of signals
For that, lets review vectors and then extend the results to
signals
Consider a three dimensional Cartesian vector space,
described by orthogonal

2.8 Trigonometric Fourier Series
Consider
class of periodic signals g(t) with period T0
A well-known complete orthogonal basis formed by the
trigonometric set
A sinusoid of angular frequency is called the nth harmonic of the
sinusoid of the angular fre

Example
We would like to verify the following by an example
(extension of example 2.7):
Periodic (Fourier series)
Aperiodic (Fourier Transform)
T0
(t ) exp( t / 2) for 0 t
0.504
Dn
1 j 4n
Example (continue)
g (t ) exp( at ) for 0 t , a 0
G ( f ) g (t

The Exponential Fourier Series
As we discussed before, the orthogonal basis is NOT unique
We have studies Trigonometric Fourier Series
An equivalent but simpler form: Exponential Fourier Series
Complete Orthogonal basis
In other words:
1, , , , ,
The Expo

Review of Last Lecture
We can consider Continuous-time signals as vectors
g(t)
x(t)
Inner product of two signals is given by
Norm of a signal is given by
Review of Last Lecture (2)
We can project a signal g(t) over a signal x(t), call it cx(t).
We sa

2.5.4 Energy of sum of orthogonal signals
=
Find the Energy of the sum of them:
z(t)=x(t)+y(t)
0
0
(x and y are
Orthogonal)
Energy of sum of orthogonal signals (2)
=0
x
z
y
This is the famous Pythagorean theorem!
Example
Two
signals x and y each have u

EEL 3552: Analog and Digital
Communication Fundamentals
Instructor: Dr. Nazanin Rahnavard
University of Central Florida
Spring 2016
Instructor Information
Instructor: Dr. Nazanin Rahnavard (you can call me Dr. Naz or
Naz)
Got my PhD from Georgia Institu

EEL 3552: Analog and Digital
Communication Fundamentals
Instructor: Dr. Nazanin Rahnavard
University of Central Florida
Spring 2016
Why is Modulation Needed?
1. Ease of Radiation/Transmission
Radiating antenna should be on the order of a fraction of
the

Periodic vs Aperiodic signal
A signal g(t) is said to be periodic if for a T0>0 we have
for all t
The smallest value that satisfies above equation is called the
period of g(t)
Periodic signal of period T0.
A periodic signal starts from and extends to

Amplitude Modulation
g (t ) cos( 2f 0t )
1
G f f0 G f f 0
2
g (t ) : modulating signal
cos( 2f 0t ) : carreir
g (t ) cos( 2f 0t ) : modulated signal
g (t ) : when cos(2f 0t ) 1
g (t ) cos(2f 0t )
g (t ) : when cos(2f 0t ) 1
Shifting the phase spectru

Unit Impulse Function (2)
The unit impulse function is defined as
, t 0
(t )
0, t 0
Shows the
area
(t )
1
t
(t )dt 1
1
Multiplication by an impulse
(t )
(t )
(t ) (t )
1
(0)
Similarly:
(0)
Sampling property of unit impulse
Sampling or sifting pr

Complex Numbers
Supplementary for EEL3552
Two equivalent representation of a complex number Z
1. Z x jy :
x Recfw_Z , y Imcfw_Z
2. Z re :
j
r | Z |: magnitude of Z
Z : phase of Z
Imaginary
Z
y
r
Real
x
-y
Z*
Conjugate of Z is represented by Z* and is de