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The characterization and construction of the oscillator system is representa-
tion theoretic and we devote the rest of the survey to an intuitive explanation
of the main underlying ideas. As a suggestive model example we explain first
the construction of the well known system of chirp (Heisenberg) signals, de-
liberately taking a representation theoretic point of view (see [2,5] for a more
comprehensive treatment).
2
Model Example (Heisenberg System)
F
p
→
C
×
the character
ψ
(
t
)=
e
2
π
p
t
. We consider the
pair of orthonormal bases
Δ
=
Let us denote by
ψ
:
and
Δ
∨
=
{
δ
a
:
a
∈
F
p
}
{
ψ
a
:
a
∈
F
p
}
,where
1
ψ
a
(
t
)=
√
p
ψ
(
at
).
2.1 Characterization of the Bases
Δ
and
Δ
∨
Let
L
:
be the time shift operator
L
ϕ
(
t
)=
ϕ
(
t
+ 1). This operator is
unitary and it induces a homomorphism of groups
L
:
H→H
F
p
→
U
(
H
)givenby
L
τ
ϕ
(
t
)=
ϕ
(
t
+
τ
) for any
τ
∈
F
p
.
Elements of the basis
Δ
∨
are character vectors with respect to the action
L
,
i.e.,
L
τ
ψ
a
=
ψ
(
aτ
)
ψ
a
for any
τ
∈
F
p
. In the same fashion, the basis
Δ
consists of
character vectors with respect to the homomorphism
M
:
F
p
→
U
(
H
) generated
by the phase shift operator
M
ϕ
(
t
)=
ψ
(
t
)
ϕ
(
t
).
2.2 The Heisenberg Representation
The homomorphisms
L
and
M
can be combined into a single map
π
:
F
p
×
F
p
→
2
τw
M
w
◦
L
τ
. The plane
F
p
×
F
p
is called the
time-frequency plane
and will be denoted by
V
.Themap
π
is not an homomorphism since, in general, the operators
L
τ
and
M
w
do not commute. This deficiency can be corrected if we consider the group
H
=
V
π
(
τ, ω
)=
ψ
1
U
(
H
) which sends a pair (
τ, w
) to the unitary operator
−
×
F
p
with multiplication given by
(
τ
,w
,z
)=(
τ
+
τ
,w
+
w
,z
+
z
+
1
2
(
τw
−
τ
w
))
.
(
τ, w, z
)
·
The map
π
extends to a homomorphism
π
:
H
→
U
(
H
)givenby
π
(
τ, w, z
)=
ψ
2
τw
+
z
M
w
◦
1
−
L
τ
.
The group
H
is called the
Heisenberg
group and the homomorphism
π
is called
the
Heisenberg representation.
2.3 Maximal Commutative Subgroups
The Heisenberg group is no longer commutative, however, it contains various
commutative subgroups which can be easily described. To every line
L
V
,
which pass through the origin, one can associate a maximal commutative sub-
group
A
L
=
⊂
{
(
l,
0)
∈
V
×
F
p
:
l
∈
L
}
. It will be convenient to identify the sub-
group
A
L
with the line
L
.
