The Finite Balian-Low Conjecture
Mark Lammers and Simon Stampe
Index Terms—Finite Gabor Systems, Zak Transform, BalianLow Theorem

0 1
0 0

0 0

T =


0 0
1 0
Abstract—We present a conjecture for a finite version of the
celebrated Balian-Low Theorem for Gabor systems in L2 (R). We
proceed to prove a special case of the conjecture for C9 .
0
1
0
0
0
1
..
.
···
···
···
..
.
···
···
0
0
0
0

0
0

0

,


1
0
I. I NTRODUCTION
The Balian-Low Theorem for L2 (R) states that if g(x) ∈
L (R) and the critically sampled Gabor system {g(x −
n)e2πimx }n,m∈Z produces an orthonormal basis, then either
d
d
k dx
g(x)k2 = ∞ or k dw
ĝ(w)k2 = ∞ [1], [2], [3]. We will
generalize this to the finite case but first we point out some
immediate hurdles. We will consider finite Gabor systems for
CN so the window function of the finite Gabor system will
be a vector g in CN . It is natural to use a finite difference
operator, D, to play the role of the derivative and the discrete
Fourier transform, F, to play the role of the continuous Fourier
transform. However, neither kDgk2 or kDFgk2 will ever
be infinite because we are in a finite vector space. To get
around this we will instead consider time-frequency measure
kDgk2 + kDFgk2 and consider it’s growth as the dimension
of the vector space grows. This is motivated by the GaussHermite differential equation, i.e., the time-independent
d2 f
Schrödinger equation with potential x2 , G(f ) = − 2 +
dx
x2 f = λf. Numerous authors have used variations on the
discrete versions of the Gauss-Hermite differential equation
[4], [5], [6], [7], [8], [9] to produce discrete versions of
the Hermite functions for the DFT, i.e., eigenvectors for a
difference operator corresponding to a discrete version of the
differential operator G. In this proposition for a finite version
of the Balian-Low Theorem we will conjecture what the
minimizer of the measure kDgk2 +kDFgk2 is for orthonormal
Gabor systems, and that the minimizer of this finite timefrequency measure goes to infinity as the dimension N of
CN goes to infinity for orthonormal Gabor systems.
2
II. P RELIMINARIES
First we present notation to generalize Gabor systems to
CN . We use matrix versions of translation and modulation,
Mark Lammers is with the Department of Mathematics, University of North Carolina at Wilmington, Wilmington, NC 28403, USA.
email:lammersm@uncw.edu
Simon Stampe is a medical student at SUNY Upstate Medical University.
email:sstampe@gmail.com
c
978-1-4673-7353-1/15/$31.00 2015
IEEE

1
0

0

M =


0
0
0
e2πi/N
0
0
0
0
0
e
4πi/N
···
···
0
0
0
..
.
···
···
···
..
.
0
0
0
0
0
e2πi(N −2)/N
0
0
e2πi(N −1)/N
Next we define the circulant difference operator to be
D = I − T , where I is the identity matrix, and we let F be
2πi
the usual discrete Fourier transform matrix, Fn,m = (e N )mn
m, n = 0 · · · N − 1.
For simplicity of this note we will only be considering
finite Gabor systems of vector spaces CN where N = d2
for some integer d. This allows us to take the same number
of translations as modulations in producing the orthonormal
Gabor systems. For clarity we note that Fd , Td , Md Id are
the DFT transform, translation, modulation, and the identity
operators for Cd . Similarly letting N = d2 , we define FN ,
TN , MN IN as the DFT transform, translation, modulation,
2
and the identity operators for CN . Letting g ∈ Cd for
2
some integer d, we define a finite Gabor system for Cd as
d−1
md
nd
{TN MN g}m,n=0 .
Because of the way we eventually define a finite Zak
transform we now introduce a tensor product notation. We
let ⊗ represent the Kronecker tensor product between two
matrices. That is, if A is an m × n matrix and B is p × q
matrix A ⊗ B is the mp × nq block matrix given by


a11 B . . . a1n B

.. 
..
A ⊗ B =  ...
.
. 
an1 B
...
ann B
Let us note a few important properties of the Kronecker
product:
1) (A ⊗ B)(C ⊗ D) = AC ⊗ BD.
2) (A ⊗ B)−1 = A−1 ⊗ B −1 .
3) (A ⊗ B)∗ = A∗ ⊗ B ∗ .
4) A ⊗ (B + C) = A ⊗ B + A ⊗ C.
Using this notation we can write our finite Gabor systems
d−1
md
as {TNnd MN
g}m,n=0
= {Tdn ⊗ Mdm g}d−1
m,n=0 but the real





.



reason we introduce it is to develop a notation for the finite
Zak transform. The Zak transform,
X
Z(f (x)) (t, v) =
f (t − n)e−2πinv ,
n∈Z
2
is a unitary map from L (R) to L2 ([0, 1]2 ). It is a tool that is
used prominently in studying Gabor systems and has numerous
nice properties with regard to them. In particular, for f ∈
L2 (R),
Z(f (x − n))(t, v) = e2πint Z(f (x)) (t, v)
N is divisible by four yields one repeated eigenvalue but
this can be addressed (see section B of [16] for example).
It follows that eigenvectors of G are also eigenvectors of F
since F commutes with G. We refer to the columns of H as
Hermite vectors and H itself as the Hermite Matrix. The
eigenvector for G corresponding to the smallest eigenvector
will be denoted g0 and will be referred to as the Gaussian
vector. As expected, the Gaussian vector is an eigenvector
of the DFT matrix F with corresponding eigenvalue 1, i.e.,
Fg0 = g0 .
and
Z(e2πimx f (x))(t, v) = e2πimv Z(f (x)) (t, v) .
Perhaps most importantly for this note is the fact that if a
critically sampled Gabor system {g(x − n)e2πimx }n,m∈Z
produces an orthonormal basis, then |Z(g(x))(t, v)| = 1.
A finite version of the Zak transform appears in [10], [11].
This representation maps a vector in Cnm to a Cn × Cm
matrix, a natural extension of the continuous version which
maps from L2 (R) to L2 ([0, 1]2 ). We will be using a version
introduced in [12] which is simply a unitary operator from
CN to itself, where N = d2 for some integer d. Define
ZN = Fd ⊗ Id , ZN : CN → CN to be the N-dimensional
−1
finite Zak transform. Note with this notation ZN
= Z ∗.
Additionally, with this notation the properties of original Zak
transform can be expressed as:
Theorem II.1. [10], [11], [12] Let the operators be defined
as above. Then ZN TNd = (Md ⊗ Id )ZN , ZN MN = (Td∗ ⊗
1
d
= (Id ⊗ Md )ZN .
Mdd )ZN and ZN MN
Corollary II.2. [10], [11], [12] The finite Gabor system {Tdn ⊗ Mdm g}d−1
m,n=0 forms an orthonormal basis iff
|(ZN g)(j)|2 = N1 .
Corollary II.3. [10], [11] The finite Gabor system {Tdn ⊗
Mdm g}d−1
m,n=0 is a basis iff |(ZN g)(j)| > 0.
III. F INITE H ERMITE F UNCTIONS
Letting ∆ = D∗ D, we now define the finite Gauss-Hermite
difference operator as G = F ∗ ∆F + ∆. It can be shown
that F ∗ ∆F is a diagonal matrix and therefore acts like a
multiplication operator and that the Fourier matrix commutes
with G , FGF ∗ = G see [5], [13], [14], [6], [8]. We should
note that the sign on our discrete Laplacian is positive instead
of negative as it is in the differential operator. This is referred
to the quantum repulsive operator in [15]. Both operators yield
a diagonalization in the finite case ([13] Sec 6.2) but defining
G as we have has several advantages from our prospective.
All the eigenvalues of G are positive, G is invertible and
it provides a nice connection with the finite time-frequency
measure hGf, f i = kDf k2 + kDFf k2 that we will be using
in our Finite Balian-Low Conjecture. The eigenvalues of G
are unique, at least when N is not divisible by 4, so the
eigenvectors of G are orthogonal and G has an orthogonal
diagonalization of the form G = HΛH ∗ . The case when
Fig. 1. Gaussian for N=100 from the matrix operations above.
IV. Z AK TRANSFORM OF THE G AUSSIAN IN THE FINITE
DOMAIN .
It is well known that in the L2 (R) setting the Zak transform
of the Gaussian function has a zero and it is precisely this that
allows one to eventually conclude that not only do critically
sampled Gabor systems not form an orthonormal basis, they
do not form a basis at all. Things in our finite setting are
a bit different. While it can be shown that if d is even the
finite Zak transform of the Gaussian vector does indeed have
a zero, one can compute odd cases where it does not. This
is already a big difference from the L2 (R) case but as the
dimension grows in the odd case the Jacobi theta type function,
i.e., the Zak transform of the Gaussian [17], gets closer to
having a 0 component. Therefore, the associated Gabor system
will become very poorly conditioned. In the next section we
will want to throw away the modulus of the components of
the vector and only keep the phase, so we visualize the Zak
transform of the Gaussians as a complex vector instead of the
usual surface plot that only represents the modulus. Plotting
Zak transforms of the Gaussian vectors as 3 dimensional
−
parameterized curves →
r (n) = hn, real(f (n)), imag(f (n))i,
in Fig.2. we see the even case intersects the black line
representing real and imaginary parts of zero while the odd
case does not.
V. T HE C ONJECTURE .
2
Definition V.1. Let f ∈ Cd be a vector with no zero
entries. Define the pointwize normalization of f
as
f (k)
.
J
is
highly
non
linear
but
it
preserves
J (f )(k) = d|f
(k)|
the phase of each entry of the vector and kJ (f )k = 1.
From Corollary II.2 is is clear that the inverse Zak
transforms of pointwise normalized functions will produce
We are now ready for our Finite Balian-Low Conjecture.
d−1
Conjecture V.2. Let {Tdn ⊗ Mdm f }m,n=0
be a Gabor system
for CN , N = d2 that forms an orthonormal basis.
1) As d → ∞, kDf k2 + kDFf k2 → ∞ .
∗
2) If d is odd, (ZN g0 )(j) 6= 0 and ZN
J (ZN g0 ) is a
2
2
minimizer of kDf k + kDFf k .
Fig. 2. Zak transform of Gaussian for N=100 and N=81 case.
Fig. 3. Zak transform of Gaussian for N = 289 and it’s pointwise
normalization.
∗
finite Gabor systems {Tdn ⊗ Mdm (ZN
J (f ))}d−1
m,n=0 that are
orthonormal bases. In fact, this is just the matrix version of
−1/2
(g) in
creating a window for a Parseval frame given by Sg
2
the L (R) case where Sg is the frame operator for the Gabor
system {g(x − na)e2πimbx }n,m∈Z .
∗
J (f ). We have already
Let’s consider a special case of ZN
observed that when d is odd ZN (g0 ) does not have to have
∗
a zero so below we consider the graph of ZN
J (Zn g0 ) for
N = 289 along with it’s Fourier transform.
VI. S PECIAL C ASE OF BALIAN -L OW C ONJECTURE FOR
EIGENVECTORS OF F WITH N = 9.
In this section we will prove the conjecture is true for N = 9
and the window function of the orthonormal Gabor system
{Tdn ⊗Mdm g}d−1
m,n=0 is an eigenvector of the Fourier transform.
Assuming that the window function is an eigenvector of the
Fourier transform, it is natural condition to consider given that
the Gaussian vector is the minimizer of the time-frequency
measure over all norm one vectors. Additionally, it is natural
to assume that the minimum occurs when kDf k = kDFf k
and any eigenvector of F satisfies this condition. For the sake
of simplicity we let Z9 = Z and F9 = F for the rest of this
section.
Theorem VI.1. Let N = d2 = 9 and let {Tdn ⊗ Mdm g}d−1
m,n=0
be a Gabor system that forms an orthonormal basis. If g is
a an eigenvector of F, then Z ∗ J (Zg0 ) is a minimizer of
kDf k2 + kDFf k2 .
Proof. First we compute J (Zg0 ) to get
(J (Zg0 ))T =
i
1h
1 1 1 1 eπi/9 e5πi/9 1 e−πi/9 e−5πi/9 .
3
Now we let g be an eigenvector of F and Θ = dZg. If the
Gabor system with window g generates an orthonormal basis,
the modulus of each component of Θ is one. To complete our
proof it is enough to show that there is a minimizer of the
time-frequency measure g satisfying the conditions above so
that Zg = J (Zg0 ). We denote the components of Θ by
ΘT = eiθ0 eiθ1 eiθ2 eiθ3 eiθ4 eiθ5 eiθ6 eiθ7 eiθ8 .
Because g is an eigenvector of F and all eigenvalues are
modulus 1, the time frequency measure becomes
kDgk2 + kDF gk2 = 2kDgk2
2
= 4 − hZ(T + T ∗ )Z ∗ Θ, Θi.
9
−1
Fig. 4. ZN
J (ZN g0 ) and it’s Fourier transform for N = 289.
It should be noted that in the example above
only the real parts of these vectors were plotted
because the imaginary part was of magnitude less than
10−15 . It should also be noted that in this example
−1
−1
max(FN ZN
J (ZN g0 ) − ZN
J (ZN g0 )) < 1.8 · 10−14 giving
strong indication that this is also an eigenvector of the Fourier
transform corresponding to eigenvalue 1.
A straightforward computation results in
hZ(T + T ∗ )Z ∗ Θ, Θi
= ei(θ1 −θ0 ) + ei(θ2 −θ0 ) + ei(θ0 −θ1 ) + ei(θ2 −θ1 ) + ei(θ0 −θ2 )
+ ei(θ1 −θ2 ) + ei(θ4 −θ3 ) + ei(θ0 −θ3 −2π/3) + ei(θ3 −θ4 )
+ ei(θ5 −θ4 ) + ei(θ3 −θ5 +2π/3) + ei(θ4 −θ5 ) + ei(θ7 −θ6 )
+ ei(θ8 −θ6 +2π/3) + ei(θ6 −θ7 ) + ei(θ8 −θ7 )
+ ei(θ6 −2π/3−θ8 ) + ei(θ7 −θ8 ) .
Now combining terms above we get:
hZ(T + T ∗ )Z ∗ Θ, Θi
= 2(cos(θ1 − θ0 ) + cos(θ0 − θ2 ) + cos(θ2 − θ1 )
+ cos(θ4 − θ3 ) + cos(θ5 − θ3 − 2π/3)
+ cos(θ5 − θ4 ) + cos(θ7 − θ6 )
+ cos(θ6 − θ8 − 2π/3) + cos(θ8 − θ7 )).
(1)
Let P = ZF Z ∗ then Θ = dZg is an eigenvector of P
with the same eigenvalue since ZF g = P Zg implying that
λZg = P Zg.
Applying P to Θ we get

eiθ0
eiθ3
eiθ6
eiθ2





 i(θ −2π/9)
5
PΘ = 
 ei(θ −4π/9)
 e 8


eiθ1

 ei(θ4 −π/9)
ei(θ7 −2π/9)







 = λΘ






First we notice that the operator P fixes the first component.
So eiθ0 = λeiθ0 , which implies the eigenvalue λ must be
1 since eiθ0 6= 0. Now matching up the terms we get the
following relations:
θ0
θ1
θ4
θ5
θ7
θ8
= θ0
= θ2 = θ3 = θ6
= θ4
= θ4 + 4π/9
= θ4 − 2π/9
= θ4 − 6π/9.
Fig. 5. Plot of 4 − 49 [3 + 2 cos(u) + 2 cos(u + 2π/9) + 2 cos(4π/9)].
We observe that this is the same vector we obtained calculating
J (Zg0 ) .
Remark VI.2. We point out that the argument used in the
proof above to show that the λ = 1, generalizes to all
dimensions because in [11] they show the matrix P is the
product of a permutation matrix and a unitary diagonal
matrix that always keeps the first component fixed. That is,
d−1
if {Tdn ⊗ Mdm g}m,n=0
is a Gabor system that forms an
orthonormal basis and g is a an eigenvector of F, then the
eigenvalue must be 1.
VII. N UMERICAL R ESULTS
Using MATLAB we calculate the time-frequency measure
∗
kDN gk2 + kDN FN gk2 for g = ZN
J (Zg0 ) for odd dimensions of d. Using a logarithmic regression we fit these points to
the curve 6.08 ln(d) + 0.63 with a coefficient of determination
r2 = 0.9996 in Fig. 6 below.
Substituting the equations above into equation 1 we get
hZ(T + T ∗ )Z ∗ Θ, Θi
= 2[2cos(θ1 − θ0 ) + 2cos(θ4 − θ1 ) + cos(θ4 − θ1 − 2π/9)
+ 2cos(4π/9) + 1],
and so the measure becomes
kDgk2 + kDF gk2
4
= 4 − [2 cos(θ1 − θ0 ) + 2 cos(θ4 − θ1 )
9
+ 2 cos(θ4 − θ1 − 2π/9) + 2 cos(4π/9) + 1].
Fig. 6. The time-frequency measure kDN gk2 + kDN FN gk2 of g =
Z ∗ J (Zg0 ) as a function of d (black squares). Graph of 6.08 ln(d) + 0.63
(blue dashes).
We observe that θ0 only appears once in the expression so
the relation θ0 = θ1 must occur in any minimization. Letting
u = θ4 − θ1 we get the single variable expression:
R EFERENCES
kDgk2 + kDF gk2
4
= 4 − [3 + 2 cos(u) + 2 cos(u − 2π/9) + 2 cos(4π/9)]
9
which has critical points when u = π9 + nπ and a minimum
when u = π9 . Choosing θ1 = 0 gives us that θ4 = 0 and this
gives us the minimizer we were seeking:
Zg T =
i
1h
1 1 1 1 eπi/9 e5πi/9 1 e−πi/9 e−5πi/9 .
3
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