Digital Signal Processing Reference
In-Depth Information
FOURIER-series as orthogonal function systems :
Sawtooth (2 Hz) = Sinus (2 Hz) + Sinus (4 Hz) + Sinus (6 Hz) + . . .
Orthogonal basis vectors of a tridimensional space
z
Time domain
0, 5
0, 4
0, 3
0, 2
0, 1
0, 0
-0,1
-0,2
-0,3
-0,4
-0,5
0, 5
0, 4
0, 3
0, 2
0, 1
0, 0
-0,1
-0,2
-0,3
-0,4
-0,5
v
= 2,5
i
+ 3
j
+ 2
k
k
j
i
Sinus (12 Hz) lacks!
v
P
y
0
50
150
250
350
450
550
650
750
850
950
ms
Frequency domain
x
0,35
0,30
0,25
Sinus (12 Hz) lacks!
0,20
0,15
0,10
0,05
0,00
0,0
5,0
10,0
15,0
20,0
25,0
30,0
Hz
Illustration 271:
Orthogonality
How can a point P in space be reached? Quite easily by following these instructions: go 2.5 standard steps
(of length 1) in x-direction (2.5
i
) plus 3 standard steps in y-direction (3
j
) plus 2 standard steps in z-direc-
tion (2
k
). Thus the vector leading to P can be defined as v = 2.5
i
+ 3
j
+ 2
k
.
Vectors are underlined and in
bold print. They represent numeric values which indicate a direction.
i
,
j
, and
k
are linearly independent because they are in a vertical, i.e. orthogonal position to each other.
This means: Point P could never be reached without one of the three standard vectors
i
,
j
and
k
. The
equation
v
= 2.5
i
+ 3
j
+ 2 (
i
+
j
)
would be wrong.
i
,
j
and
k
are indispensable if you want to set up a three-
dimensional space, i.e. if you want to be able to reach any point in this space.
Orthogonal function systems
: a periodic sawtooth of 2 Hz contains, as we know, all integer multiples of
the frequencies of 2Hz, i.e. sinusoidal oscillations of 2,4,6, … Hz up to “infinitely high” frequencies.
This infinite number of discrete frequencies form - in the above mentioned sense - a“vector space with an
infinite number of dimensions”. The reason is: if only one of these frequencies is missing - in the Illustra-
tion at the top it is the sinusoidal oscillation of 12 Hz - the sawtooth oscillation cannot be reconstructed as
a sinusoidal oscillation. In that sense each of these innumerable oscillations is indispensable and their
total constitutes the minimum number of oscillations required to form the sawtooth. In mathematical terms
this means: all these sinusoidal oscillations have an "orthogona"l relationship towards each other.
Orthogonal Frequency division Multiplex (OFDM)
The new European digital broadcasting standard DAB operates using a special DMT
procedure: OFDM (Orthogonal Frequency Division Multiplex). "Orthogonal" means "to
be in a vertical position to…" and is a basic term in mathematics, in particular in vector
space mathematics. Illustration 271 shows the connection between orthogonality and
modulation technology.
The carrier frequencies used in OFDM are always integer
multiples of a basic frequency.
