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In-Depth Information
Remark 1.
Note the main differences between the Heisenberg and the oscillator
systems. The oscillator system consists of order of
p
3
signals, while the Heisen-
berg system consists of order of
p
2
signals. Signals in the oscillator system admit
an ambiguity function concentrated at 0
V
(thumbtack pattern) while signals
in the Heisenberg system admit ambiguity function concentrated on a line (see
Figures 1, 3).
∈
3
The Oscillator System
Reflecting back on the Heisenberg system we see that each vector
ϕ
∈
S
H
is
characterized in terms of action of the additive group
G
a
=
F
p
. Roughly, in
comparison, each vector in the oscillator system is characterized in terms of
action of the multiplicative group
G
m
=
F
p
. Our next goal is to explain the last
assertion. We begin by giving a model example.
Given a multiplicative character
2
χ
:
G
m
→
C
×
, we define a vector
χ
∈H
by
χ
(
t
)=
1
√
p
1
χ
(
t
)
,
t
=0
,
−
0
,
t
=0
.
B
std
=
χ
:
χ
=1
,where
G
m
is the dual
G
m
,χ
We consider the system
∈
group of characters.
3.1
Characterizing the System
B
std
For each element
a
be the unitary operator acting
by scaling
ρ
a
ϕ
(
t
)=
ϕ
(
at
). This collection of operators form a homomorphism
ρ
:
G
m
→
∈
G
m
let
ρ
a
:
H→H
U
(
H
).
B
std
are character vectors with respect to
ρ
, i.e., the vector
χ
Elements of
satisfies
ρ
a
χ
=
χ
(
a
)
χ
for every
a
G
m
. In more conceptual terms, the action
ρ
yields a decomposition of the Hilbert space
∈
=
H
χ
,
H
into character spaces
H
where
χ
runs in the group
G
m
. The system
B
std
consists of a representative unit
vector for each space
H
χ
,
χ
=1.
3.2 The Weil Representation
We would like to generalize the system
B
std
in a similar fashion to the way we
generalized the bases
Δ
and
Δ
∨
in the Heisenberg setting. In order to do this
we need to introduce several auxiliary operators.
Let
ρ
a
:
∈
F
p
,
be the operators acting by
ρ
a
ϕ
(
t
)=
σ
(
a
)
ϕ
(
a
−
1
t
)
(scaling), where
σ
is the unique quadratic character of
H→H
,
a
F
p
,let
ρ
T
:
be
the operator acting by
ρ
T
ϕ
(
t
)=
ψ
(
t
2
)
ϕ
(
t
) (quadratic modulation), and finally
let
ρ
S
:
H→H
H→H
be the operator of Fourier transform
ν
√
p
ρ
S
ϕ
(
t
)=
ψ
(
ts
)
ϕ
(
s
)
,
s
∈
F
p
2
A multiplicative character is a function
χ
:
G
m
→
C
×
which satisfies
χ
(
xy
)=
χ
(
x
)
χ
(
y
) for every
x, y ∈ G
m
.
