Geometric representation of signals can provide a compact
characterization of signals and can simplify analysis of their
performance as modulation signals.
Orthonormal bases are essential in geometry. Let
s
1
t
s
2
t…
s
M
t
s
1
t
s
2
t
…
s
M
t
be a set of signals.
Define
ψ
1
t=
s
1
t
E
1
ψ
1
t
s
1
t
E
1
where
E
1
=∫0T
s
1
2td
t
E
1
t
0
T
s
1
t
2
.
Define
s
21
=〈
s
2
,
ψ
1
〉=∫0T
s
2
t
ψ
1
t¯d
t
s
21
s
2
ψ
1
t
0
T
s
2
t
ψ
1
t
and
ψ
2
t=1
E
2
^
(
s
2
t−
s
21
ψ
1
)
ψ
2
t
1
E
2
^
s
2
t
s
21
ψ
1
where
E
2
^
=∫0T
s
2
t−
s
21
ψ
1
t2d
t
E
2
^
t
0
T
s
2
t
s
21
ψ
1
t
2
In general
ψ
k
t=1
E
k
^
(
s
k
t−∑
j
=1k−1
s
kj
ψ
j
t)
ψ
k
t
1
E
k
^
s
k
t
j
1
k
1
s
kj
ψ
j
t
(1)
where
E
k
^
=∫0T
s
k
t−∑
j
=1k−1
s
kj
ψ
j
t2d
t
E
k
^
t
0
T
s
k
t
j
1
k
1
s
kj
ψ
j
t
2
.
The process continues until all of the M
M signals are exhausted. The results are
N N orthogonal signals with unit
energy,
ψ
1
t
ψ
2
t…
ψ
N
t
ψ
1
t
ψ
2
t
…
ψ
N
t
where
N≤M
N
M
.
If the signals
s
1
t…
s
M
t
s
1
t
…
s
M
t
are linearly independent, then
N=M
N
M
.
The M M signals can be represented
as
s
m
t=∑
n
=1N
s
mn
ψ
n
t
s
m
t
n
1
N
s
mn
ψ
n
t
(2)
with
m∈12…M
m
1
2
…
M
where
s
mn
=〈
s
m
,
ψ
n
〉
s
mn
s
m
ψ
n
and
E
m
=∑
n
=1N
s
mn
2
E
m
n
1
N
s
mn
2
.
The signals can be represented by
s
m
=
s
m1
s
m2
⋮
s
mN
s
m
s
m1
s
m2
⋮
s
mN
ψ
1
t=
s
1
tA2T
ψ
1
t
s
1
t
A
2
T
(3)
s
21
=−(AT)
s
21
A
T
(5)
ψ
2
t=(
s
2
t−
s
21
ψ
1
t)1
E
2
^
=(−A+ATT)1
E
2
^
=0
ψ
2
t
s
2
t
s
21
ψ
1
t
1
E
2
^
A
A
T
T
1
E
2
^
0
(6)
Dimension of the signal set is 1 with
E
1
=
s
11
2
E
1
s
11
2
and
E
2
=
s
21
2
E
2
s
21
2
.
ψ
m
t=
s
m
t
E
s
ψ
m
t
s
m
t
E
s
where
E
s
=∫0T
s
m
2td
t
=A2T4
E
s
t
0
T
s
m
t
2
A
2
T
4
s
1
=
E
s
000
s
1
E
s
0
0
0
,
s
2
=0
E
s
00
s
2
0
E
s
0
0
,
s
3
=00
E
s
0
s
3
0
0
E
s
0
, and
s
4
=000
E
s
s
4
0
0
0
E
s
∀
m
n
:
d
mn
=|
s
m
−
s
n
|=∑
j
=1N
s
mj
−
s
nj
2=2
E
s
m
n
d
mn
s
m
s
n
j
1
N
s
mj
s
nj
2
2
E
s
(7)
is the Euclidean distance between signals.
Set of 4 equal energy biorthogonal signals.
s
1
t=st
s
1
t
s
t
,
s
2
t=
s
⊥
t
s
2
t
s
⊥
t
,
s
3
t=−st
s
3
t
s
t
,
s
4
t=−
s
⊥
t
s
4
t
s
⊥
t
.
The orthonormal basis
ψ
1
t=st
E
s
ψ
1
t
s
t
E
s
,
ψ
2
t=
s
⊥
t
E
s
ψ
2
t
s
⊥
t
E
s
where
E
s
=∫0T
s
m
2td
t
E
s
t
0
T
s
m
t
2
s
1
=
E
s
0
s
1
E
s
0
,
s
2
=0
E
s
s
2
0
E
s
,
s
3
=−
E
s
0
s
3
E
s
0
,
s
4
=0−
E
s
s
4
0
E
s
. The four signals can be geometrically represented using the
4-vector of projection coefficients
s
1
s
1
,
s
2
s
2
,
s
3
s
3
, and
s
4
s
4
as a set of constellation points.
d
21
=|
s
2
−
s
1
|=2
E
s
d
21
s
2
s
1
2
E
s
(8)
d
12
=
d
23
=
d
34
=
d
14
d
12
d
23
d
34
d
14
(9)
d
13
=|
s
1
−
s
3
|=2
E
s
d
13
s
1
s
3
2
E
s
(10)
d
13
=
d
24
d
13
d
24
(11)
Minimum distance
d
min
=2
E
s
d
min
2
E
s
