Information Technology Reference
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with respect to the restriction of the Weil representation to a maximal non-split
torus in
SL
2
. The only difference appears to be the field of definition, which for
the harmonic oscillator is the reals and for the oscillator functions is the finite
field.
4
Applications
Two applications of the oscillator system will be described. The first application
is to the theory of discrete radar. The second application is to CDMA systems.
We will give a brief explanation of these problems, while emphasizing the relation
to the Heisenberg representation.
4.1
Discrete Radar
The theory of discrete radar is closely related [2] to the finite Heisenberg group
H.
A radar sends a signal
ϕ
(
t
)andobtainsanecho
e
(
t
). The goal [9] is to
reconstruct, in maximal accuracy, the target range and velocity. The signal
ϕ
(
t
)
and the echo
e
(
t
) are, principally, related by the transformation
e
(
t
)=
e
2
πiwt
ϕ
(
t
+
τ
)=
M
w
L
τ
ϕ
(
t
)
,
where the time shift
τ
encodes the distance of the target from the radar and
the phase shift encodes the velocity of the target. Equivalently, the transmitted
signal
ϕ
and the received echo
e
are related by an action of an element
h
0
∈
H
,
i.e.,
e
=
π
(
h
0
)
ϕ.
The problem of discrete radar can be described as follows.
Given a signal
ϕ
and an echo
e
=
π
(
h
0
)
ϕ
extract the value of
h
0
.
It is easy to show that
and it obtains its maximum
at
h
−
0
. This suggests that a desired signal
ϕ
for discrete radar should admit an
ambiguity function
A
ϕ
which is highly concentrated around 0
|
m
ϕ,e
(
h
)
|
=
|
A
ϕ
(
h
·
h
0
)
|
H
,whichisa
property satisfied by signals in the oscillator system (Property 2).
∈
Remark 2.
It should be noted that the system
S
O
is “large” consisting of order
of
p
3
signals. This property becomes important in a
jamming
scenario.
4.2
Code Division Multiple Access (CDMA)
We are considering the following setting.
-
There exists a collection of users
i
∈
I
, each holding a
bit
of information
(usually
b
i
is taken to be an
N
'th root of unity).
-
Each user transmits his bit of information, say, to a central antenna. In order
to do that, he multiplies his bit
b
i
by a private signal
ϕ
i
∈H
b
i
∈
C
and forms a
message
u
i
=
b
i
ϕ
i
.
