An optimal Gabor frame set-up
Optimizing frame bounds for the Gaussian
window over separable lattices
Markus Faulhuber (University of Vienna)
joint work with Stefan Steinerberger (Yale University)
Gabor frames and lattices
Theorem (F, Steinerberger)
A Gabor frame (or Weyl-Heisenberg frame) for L2 (Rd ) is generated by a (fixed, non-zero) window function
g ∈ L2 (Rd ) and an index set Λ ⊂ R2d and is denoted by G(g, Λ). It consists of time-frequency shifted versions
of g. We say λ = (x, ω) ∈ Rd × Rd is a point in the time-frequency plane and use the following notation for
a time-frequency shift by λ
For the standard Gaussian window g0 (t) = 21/4 e−πt and (α, β) ∈ F(g0 ) with 1 < (αβ)−1 = n ∈ N fixed, the
following holds.
1 √1
√
The lower frame bound is uniquely maximized by the parameter pair
, n . Furthermore, the upper
n
frame bound is uniquely minimized by the same pair of lattice parameters. Hence, for any integer redundancy
greater than 1, the frame condition number B/A is uniquely minimized by the square lattice.
π(λ)g(t) = Mω Tx g(t) = e2πiω·t g(t − x),
x, ω, t ∈ Rd
G(g, Λ) = {π(λ)g | λ ∈ Λ}.
In order to be a frame, G(g, Λ) has to fulfil the frame property
X
Akf k22 ≤
|hf, π(λ)gi|2 ≤ Bkf k22 ,
(1)
(2)
2
6
6
5
5
∀f ∈ L2 (Rd )
(3)
4
4
λ∈Λ
3
3
for some positive constants 0 < A ≤ B < ∞ called frame bounds. A lattice Λ ⊂ R2d is generated by an
invertible (non-unique) 2d × 2d matrix S, in the sense that Λ = SZ2d . The volume of the lattice is defined as
vol(Λ) = | det(S)|
(4)
2
2
1
1
1
0.0
and its redundancy or density is given by the inverse of the volume
ρ(Λ) =
0.5
1.0
1.5
2.0
2.5
3.0
6
1
6
6
1
1
6
6
Figure 3: The lower frame bound for redundancy 6. Linear scaling (left) and logarithmic scaling (right).
1
.
vol(Λ)
(5)
15
15
A lattice is called separable if the generating matrix can be chosen as
αI 0
S=
, α, β ∈ R+
0 βI
(6)
10
10
hence, we write Λ = αZd × βZd . For a window function g ∈ L2 (Rd ) we denote the full frame set by
Ff ull (g) = {Λ ⊂ R2d lattice | G(g, Λ) is a frame}
5
5
(7)
and the reduced frame set by
1
0.0
F(g) = {(α, β) ∈ R+ × R+ | G(g, αZd × βZd ) is a frame}.
(8)
Explicit formulas
0.5
1.0
1.5
2.0
6
1
6
6
1
1
6
6
Figure 4: The upper frame bound for redundancy 6. Linear scaling (left) and logarithmic scaling (right).
Outline of the proof
−1
1/4 −πt2
For the standard Gaussian window g0 (t) = 2 e
and 1 < (αβ) = n ∈ N according to the work of
Janssen in 1996 [5] the lower and the upper frame bound can be expressed as
2 2 2 2 1
(n β )
1
1
n e
e3 (n β )
θ4
θ
−
((1
−
(−1)
))
θ
θ
,
(9)
A = A(β) =
4
3
n
2β 2
2
2β 2
2
2 2 2 2 1
1
(n β )
1
(n β )
n e
B = B(β) =
θ3
θ3
− ((1 − (−1) ))θ3
θe3
.
(10)
2
2
n
2β
2
2β
2
The functions involved are special cases of Jacobi’s theta functions. In particular
X
2
θ3 (s) =
e−πk s
(11)
k∈Z
θe3 (s) =
X
2s
e−π(2k+1)
(12)
X
2
(−1)k e−πk s
(13)
The proof considers 4 different cases. We distinguish between even and odd redundancy and between the
lower and upper frame bound. All cases have in common that we can set n = 1 in formulas (9) and (10) by
the following property.
Let Fn (s) = f (ns)f (n/s), n > 0 fixed. If the expression
s
For the upper frame bound the proof requires some standard estimates using geometric series to show that
s
Some frame bound conjectures
2
(14)
k≥1
k∈Z
Conjecture. For the standard Gaussian window g0 (t) = 21/4 e−πt among all Λ ∈ Ff ull (g0 ) with ρ(Λ) = 2 the
frame condition number B/A is minimized by a hexagonal lattice.
s>0
is strictly decreasing/increasing, then Fn (s) takes its global maximum/minimum for s = 1.
Even redundancy. For the lower bound the proof is straight forward by using the triple product representation
of the theta function
Y
2
(15)
θ4 (s) =
1 − e−2kπs 1 − e−(2k−1)πs .
k∈Z
θ4 (s) =
f 0 (s)
,
f (s)
θ30 (s)
θ3 (s)
and the identity
s
is strictly increasing for s ≥ 1
θ30 (s)
θ0 (1/s)
1
+ 1/s 3
=−
θ3 (s)
θ3 (1/s)
2
(16)
(17)
which was proved by the authors.
Odd redundancy. The difference to the previous case is the appearance of the θe3 (s)θe3 (1/s) term. First, we
check that
θe0 (s)
is strictly decreasing for s > 0
(18)
s 3
θe3 (s)
1.5
1
0.5
which implies that θe3 (s)θe3 (1/s) assumes its global maximum for s = 1. Therefore, the statement for the
upper frame bound follows immediately since we subtract a maximum from a minimum. For the lower frame
bound the conclusion follows by some standard estimates on the partial sums of the appearing terms and
comparing term-wise since all series are absolutely convergent.
0
−0.5
−1
−1.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
References
2
Figure 1: A hexagonal lattice structure of density 2.
This problem goes back to the work by Strohmer & Beaver in 2003 [6]. Regarding the work by Floch, Alard
& Berrou in 1995 [2] we have a similar conjecture for separable lattices.
2
Conjecture. For the standard Gaussian window g0 (t) = 21/4e−πt among
all (α, β) ∈ F(g0 ) with (αβ)−1 = 2
the frame condition number B/A is minimized by the pair √12 , √12 , i.e. the square lattice.
[1] Markus Faulhuber and Stefan Steinerberger. Optimal Gabor frame bounds for separable lattices & estimates for
Jacobi theta functions. arXiv preprint, 2016.
[2] Bernard Le Floch, Michel Alard, and Claude Berrou. Coded orthogonal frequency division multiplex [TV broadcasting]. Proceedings of the IEEE, 83(6):982–996, 1995.
[3] Karlheinz Gröchenig. Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis.
Birkhäuser, Boston, MA, 2001.
[4] Karlheinz Gröchenig. The Mystery of Gabor Frames. Journal of Fourier Analysis and Applications, 20(4):865–
895, 2014.
1.5
1
[5] A.J.E.M. Janssen. Some Weyl-Heisenberg frame bound calculations. Indagationes Mathematicae, 7(2):165–183,
1996.
0.5
[6] Thomas Strohmer and Scott Beaver. Optimal OFDM design for time-frequency dispersive channels. Communications, IEEE Transactions, 51(7):1111–1122, July 2003.
0
−0.5
Acknowledgment: The author was support by the Austrian Science Fund (FWF): [P26273-N25]. The author
wishes to thank Karlheinz Gröchenig for many fruitful discussions on the topic.
−1
−1.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Figure 2: A square lattice structure of density 2.
Contact: [email protected]