DC Gram-Schmidt procedure and constellations

1. Gram-Schmidt
procedure
and
Constellations

2. INTRODUCTION
❑ The Gram-Schmidt orthogonalization procedure is a straightforward way by which
an appropriate set of orthonormal functions can be obtained from any given signal
set.
❑ PASSBAND SYSTEM
Passband transmission shifts the signal to be transmitted in frequency to a higher
frequency and then transmits it

3. Gram-Schmidt procedure
First orthonomal waveform:-
Second orthonomal waveform:-
Substiution:-

4. Gram-Schmidt procedure
with constant k:
Expressing as linear
combination:-
; N <= M

5. Gram-Schmidt procedure
By, sk
(t) expression
we can say that,
(or)
N-dimensional signal space with coordinates {sk
i, i=1,2,,…N}.
Any signal can be represented geometrically as a point in the signal space
spanned by the orthonormal functions {fn
(t)}.

6. EXAMPLE
signals {si(t), i=1,2,3,4}
and its corresponding
signals

7. EXAMPLE
We see that C12
=0
s2
(t) and f2
(t) are orthogonal
Consequently, s4
(t) is a linear combination of f1
(t) and
f3
(t) and and, hence, f4
(t) = 0 .

8. CONSTELLATIONS
A constellation diagram is a representation of a signal modulated by a digital
modulation scheme such as quadrature amplitude modulation or phase-shift keying.
The distance of a point from the origin represents a measure of the amplitude or power of
the signal.

9. PROPERTIES
MODULATION SCHEME:
► Bandwidth occupied by the modulation , dimension of the modulated signal
► Bandwidth occupied by the modulation , signal_points per dimension
► Probability of bit error is proportional to the distance between the closest points in the constellations

10. THANK YOU