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2.4 Bases Associated with Lines
Restricting the Heisenberg representation
π
to a subgroup
L
yields a decom-
position of the Hilbert space
H
into a direct sum of one-dimensional subspaces
=
χ
H
χ
,
where
χ
runs in the set
L
∨
of (complex valued) characters of the
group
L
. The subspace
H
χ
consists of vectors
ϕ ∈H
such that
π
(
l
)
ϕ
=
χ
(
l
)
ϕ
.
In other words, the space
H
χ
consists of common eigenvectors with respect to
the commutative system of unitary operators
H
{
π
(
l
)
}
l∈L
such that the operator
π
(
l
) has eigenvalue
χ
(
l
).
Choosing a unit vector
ϕ
χ
L
∨
∈H
χ
for every
χ
∈
we obtain an orthonor-
.Inparticular,
Δ
∨
and
Δ
are recovered as the
bases associated with the lines
T
=
L
∨
}
mal basis
B
L
=
{
ϕ
χ
:
χ
∈
{
(
τ,
0) :
τ
∈
F
p
}
and
W
=
{
(0
,w
):
w
∈
F
p
}
respectively. For a general
L
the signals in
B
L
are certain kinds of chirps. Con-
cluding, we associated with every line
L
B
L
,
and
overall we constructed a system of signals consisting of a union of orthonormal
bases
⊂
V
an orthonormal basis
S
H
=
{
ϕ
∈B
L
:
L
⊂
V
}
.
For obvious reasons, the system
S
H
will be called the
Heisenberg
system
.
2.5
Properties of the Heisenberg System
It will be convenient to introduce the following general notion. Given two sig-
nals
φ, ϕ
∈H
, their matrix coecient is the function
m
φ,ϕ
:
H
→
C
given by
m
φ,ϕ
(
h
)=
φ, π
(
h
)
ϕ
. In coordinates, if we write
h
=(
τ, w, z
)then
m
φ,ϕ
(
h
)=
ψ
−
2
τw
+
z
1
.When
φ
=
ϕ
the function
m
ϕ,ϕ
is called the
am-
biguity
function of the vector
ϕ
and is denoted by
A
ϕ
=
m
ϕ,ϕ
.
The system
φ,
M
w
◦
L
τ
ϕ
S
H
consists of
p
+ 1 orthonormal bases
1
, altogether
p
(
p
+1)
signals and it satisfies the following properties [2,5]
1.
Auto-correlation
.
For every signal
ϕ
∈B
L
the function
|
A
ϕ
|
is the charac-
teristic function of the line
L
, i.e.,
=
0
,v/
∈
L,
|
A
ϕ
(
v
)
|
1
,v
∈
L.
2.
Cross-correlation
. For every
φ
∈B
L
and
ϕ
∈B
M
where
L
=
M
we have
1
√
p
,
|
m
ϕ,φ
(
v
)
|≤
for every
v ∈ V
.If
L
=
M
then
|m
ϕ,φ
|
is the characteristic function of some
translation of the line
L
.
3.
Supremum
.
Asignal
ϕ
1
√
p
∈
S
H
is a unimodular function, i.e.,
|
ϕ
(
t
)
|
=
for
every
t
∈
F
p
,inparticularwehave
1
√
p
max
{|
ϕ
(
t
)
|
:
t
∈
F
p
}
=
1.
1
Note that
p
+ 1 is the number of lines in
V
.
