# The finite harmonic oscillator sequences

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SETA'08

The finite harmonic oscillator
sequences

Shamgar Gurevich
Department of mathematics, UC Berkeley

Joint work with

Ronny Hadani (Chicago) & Nir Sochen (Tel Aviv)

SETA 08, Sep. 16, 2008
Goal
To construct a dictionary of sequences with
good correlation properties in time and
frequency

Fact: random sequences satisfy this property.
Problem: find a deterministic construction.

SETA 08, Sep. 16, 2008   The Oscillator Dictionary
Basic Set-Up

• Sequences are p-dimensional complex vectors.
• The Hilbert space of sequences is

H               (    p   ).
• Inner product
,                (t ) (t ).
t    p

SETA 08, Sep. 16, 2008         The Oscillator Dictionary
Basic operations

• Time translation:

L :H          H           (PxP matrix)

L (t )         (t        )               t   p
mod p
• Phase translation:
M :H                 H           (PxP matrix)
2 i t
p
M    (t )          e               (t )           p
SETA 08, Sep. 16, 2008        The Oscillator Dictionary
Phase translation matrix

1
i2       1
p
e                                   0
i2       2
M                                 e        p

0
i2   ( p 1)
p
e

SETA 08, Sep. 16, 2008                    The Oscillator Dictionary
The construction
• The dictionary of sequences is a disjoint
union of bases.

• First, we give a detail account of the
construction of one special basis.

• Then we indicate how to generalize it.
SETA 08, Sep. 16, 2008       The Oscillator Dictionary
First basis
Diagonalization of the DFT
4
( D FT )                       Id
Large multiplicities of eigenvalues
Goal: Find canonical basis of eigenfunctions for
the DFT.
Approach: Look for its symmetries.

SETA 08, Sep. 16, 2008      The Oscillator Dictionary
Intertwining time and phase
The DFT intertwines time and phase translations:

DFT L           M          DFT

DFT M              L           DFT

SETA 08, Sep. 16, 2008      The Oscillator Dictionary
Combinning time and phase translation
• The combined operation:

( , ):H            H                (PxP matrix)
i
p
( , ) e           M L

• Rewriting the two relations:
DFT              ( , )      (       , ) DFT              ( , )      p   p

SETA 08, Sep. 16, 2008           The Oscillator Dictionary
Characterization of the DFT
The following linear system for DFT:

DFT
: DFT          ( , )                (       , ) DFT

( , )   p       p   comprises p 2 constraints.

Theorem (Stone-von Neumann):
dim (sol              DFT
) 1.
DFT is characterized (up to a scalar) by the system.
SETA 08, Sep. 16, 2008       The Oscillator Dictionary
Another point of view
Rewriting the linear system for DFT as:

0         1
W
: DFT                                       DFT
1        0
W

• Note:                W   SL2 (    p   )     - the group of 2x2 matrices
with entries in            p
and det =1.

SETA 08, Sep. 16, 2008             The Oscillator Dictionary
Generalization
Consider the linear system for arbitrary g                        SL2 (   p   ):

a b
g
: (g)                                                  (g)        ( , )     p   p
c d
g

Theorem (Stone-von Neumann):
dim (sol             g
)     1.
The system characterizes a matrix                      ( g ) (up to a scalar).
SETA 08, Sep. 16, 2008      The Oscillator Dictionary
The Weil representation
Theorem (Schur): There exists a unique choice of matrices                               (g )

such that                   ( gh)   ( g ) ( h)                   g , h SL2 (   p   ).
The homomorphism
: SL ( 2     p   )        U (H )
is called the Weil representation.

Note that in this language:              (W ) C DFT .
SETA 08, Sep. 16, 2008            The Oscillator Dictionary
Solution: diagonalization of the DFT
• The Weil representation provides a homomorphism

: SL2 (      p   )         U (H )
:W           DFT
This homomorphism enables to change problems:
Finding the symmetry group of DFT – DIFICULT
Finding the symmetry group of W in SL2 (                      p   ) - EASY
SETA 08, Sep. 16, 2008             The Oscillator Dictionary
Symmetries of W in SL(2)
• Let
TW      g         SL2 (         p   ) | gW            Wg
• T can be explicitly described (finite rotations)

TW        SO2 (     p)           A SL2 (            p ) | AAt   Id

Main property: TW is commutative, i.e.,
gh       hg            g , h TW .
SETA 08, Sep. 16, 2008                The Oscillator Dictionary
Canonical symmetries of DFT
• Let
G          ( g ) U ( H ) | g TW

• Note                     W     TW               (W )        DFT    G

• Main property: G is commutative!
Proof.                ( g ) ( h)         ( gh)             (hg )   (h) ( g ).
SETA 08, Sep. 16, 2008            The Oscillator Dictionary
Facts from linear algebra
Every             A U (H )     is diagonalizable.

If A, B U ( H ) with AB BA then they can
be diagonalized simultaneously.

If Ai U ( H ); i I with Ai Aj Aj Ai i, j , then
they can be diagonalized simultaneously.

SETA 08, Sep. 16, 2008              The Oscillator Dictionary
In our situation
Apply to G                     ( g ) | g TW

The collection                     (g)        g TW    can be diagonalized
together.

The Hilbert spaces of sequences decomposes
H               H ( i ),
i

i     H ( i ) iff       (g)      i          i   (g)   i   g TW .
SETA 08, Sep. 16, 2008             The Oscillator Dictionary
The canonical basis for the DFT
Theorem: We have            dim( H ( i )) 1.
We resolved the degeneracies!
• Choose a representative from each invariant
subspace and get an orthonormal basis
BTW             i      H( i) .
• We call it “the canonical basis of
eigenfunctions for the DFT”.
SETA 08, Sep. 16, 2008    The Oscillator Dictionary
Pictorially

U (H )

get
SL(2,           p   )                         |TW         BTW
U (H )

TW
SETA 08, Sep. 16, 2008                The Oscillator Dictionary
Generalization

U (H )

get
|Ti         BTi
SL(2,            p   )
U (H )

Tp 2

T1   TW
SETA 08, Sep. 16, 2008             The Oscillator Dictionary
The oscillator dictionary

A deterministic collection of sequences

D                   BT , | D | order of p 3 .
T SL (2,   p)

SETA 08, Sep. 16, 2008                  The Oscillator Dictionary
Properties
Theorem.
Small auto-correlation (in time and frequency!)

1             ( , ) (0,0)
|            M L       | ! 2
i,         i
"               ( , ) \$ (0,0)
#     p

SETA 08, Sep. 16, 2008           The Oscillator Dictionary
Theorem (Cont.)

Small and stable cross-correlation

4
|    i,   M L      j    |"              ( , )
p

SETA 08, Sep. 16, 2008               The Oscillator Dictionary
Publications: ( Hompages: Gurevich, Hadani, Sochen)
• PNAS, “The finite harmonic oscillator and its associated
sequences”. July 2008.
• IEEE Trans. IT, “ The finite harmonic oscillator and its
applications to sequences, communications, and radar”,
Sep. 2008.
• JFAA, Special Issue on Sparsity, “On some deterministic
dictionaries that support sparsity”. To appear, 2008.

Thank you
SETA 08, Sep. 16, 2008     The Oscillator Dictionary

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