Uncertainty in time–frequency representations on finite Abelian groups and applications

20 Pages·2007·0.37 MB·English

Uncertainty in time–frequency representations on
finite Abelian groups and applications
Felix Krahmer † G¨otz E. Pfander ‡ Peter Rashkov ‡
Keywords: Uncertainty principles, short–time Fourier transforms, Gabor frames, sparsity, signal
recovery.
ABSTRACT
Classical and recent results on uncertainty principles for functions on finite Abelian groups relate the
cardinality of the support of a function to the cardinality of the support of its Fourier transforms.
We obtain corresponding results relating the support sizes of functions and their short–time Fourier
transforms. We use our findings to construct a class of equal norm tight Gabor frames that are max-
imally robust to erasures. Also, we discuss consequences of our findings to the theory of recovering
and storing signals with sparse time–frequency representations.
1. INTRODUCTION
Uncertainty principles establish restrictions on how well localized the Fourier transform of a well
localized function can be and vice versa [DS89, Gr¨o03, FS97]. In the case of a function defined on
a finite Abelian group, localization is generally expressed through the cardinality of the support of
the function. Due to its relevance for compressed sensing and, in particular, for the recovery of lossy
signals under the assumption of restricted spectral content [CRT06], the uncertainty principle for
functions on finite Abelian groups has recently drawn renewed interest.
In this realm, a classical result on uncertainty states that the product of the number of nonzero
entries in a vector representing a nontrivial function on an Abelian group and the number of nonzero
¨
entries in its Fourier transform is not smaller than the order of the group [DS89, MOP04]. This
result can be improved for any nontrivial Abelian group [Mes06]. For example, for groups of prime
order, the sum of the number of nonzero entries in a vector and the number of nonzero entries in its
Fourier transform exceeds the order of the group [Tao05].
The objective of this paper is to establish corresponding results for joint time–frequency repre-
sentations, that is, to obtain restrictions on the minimal cardinality of the support of joint time–
frequency representations of functions defined on finite Abelian groups. For example, let us consider
the simplest time–frequency representation of a function, namely the one that is given by the tensor
product of a function and its Fourier transform. In this case, the classical uncertainty principle for
nontrivial functions on finite Abelian groups states that the cardinality of the support of this tensor
is at least the order of the group.
In this paper though our focus lies on time–frequency representations given by short–time Fourier
transforms. It is easy to see that, again, the cardinality of the support of any short–time Fourier
‡School of Engineering and Science, Jacobs University, 28759 Bremen, Germany.
†Courant Institute of Mathematical Science, New York University, New York USA
1
transform of a nontrivial function defined on a finite Abelian group is bounded below by the order of
the group. As seen below, we can improve this bound by using the subgroup structure of the groups
and/or by allowing only well-chosen window functions. For example, we establish in Theorem 4.4
that for any group of prime order and for almost every window function on the group, the sum
of the cardinality of the support of the analyzed function and the cardinality of the support of its
short–time Fourier transform exceeds the square of the order of the group.
In addition to the above, we give applications of our results to the theory of so-called Gabor
frames and the theory of sparse signal recovery. For instance, the results on the cardinality of the
support of short–time Fourier transforms can be translated into criteria for the recovery of encoded
signals from a channel with erasures.
The paper is organized as follows. In Section 2 we give a brief but self-contained account of the
Fourier transformation and of the short–time Fourier transformation for functions defined on finite
Abelian groups. Section 3 reviews uncertainty principles that relate the cardinality of the support
of functions with the cardinality of the support of their Fourier transforms. In addition, we provide
numerical evidence on the achieved support set pairs for the Fourier transformation on groups of
order less than or equal to 16.
Section 4 is devoted to uncertainty inherent in the short–time Fourier transformation. There,
a discussion of general results is followed by results for functions defined on cyclic groups of prime
order. Results on other finite Abelian groups are given. These are based on the subgroup structure
of the underlying group as were recent improvements to the classical uncertainty result for Fourier
transforms obtained in [Mes06]. We conclude our discussion of the cardinality of the support set
of short–time Fourier transforms with a question on the possible cardinalities of the support of
short–time Fourier transforms with respect to a optimally chosen window functions. Our results are
complemented by numerical experiments.
In Section 5 we give applications of the results of Section 4 to communications engineering. In
Section 5.1 we discuss the identification/measurement problem for time–varying operators/channels.
Also, we consider channel coding for the transmission of information through channels with erasures.
In addition, we show the existence of equal norm tight frames of Gabor type. In Section 5.2 we
briefly discuss connections of our work to the recovery of signals which have a sparse representation
in a given dictionary.
2. BACKGROUND AND NOTATION
For any finite set A we set CA = {f : A −→ C}. For |A| = |B| = n, CA ∼= CB ∼= Cn as vector
spaces, where |A| denotes the cardinality of the set A. For M ∈ Cm×n and A ⊆ {0,1,...,n−1}
and B ⊆ {0,1,...,m−1}, we let M denote the |B|×|A|–submatrix of M with columns and
A,B
rows enumerated from the index sets B and A. For f ∈ CA, we use the now customary notation
kfk = |suppf| where suppf = {a ∈ A : f(a) 6= 0}. Clearly, k·k is not a norm.
0 0
Throughout this paper, G denotes a finite Abelian group. The dual group of characters Gb of G is
the set of homomorphisms ξ ∈ CG which map G into the multiplicative group S1 = {z ∈ C : |z| = 1}
[Ben97, Kat76, Ter99]. The set Gb is an Abelian group under pointwise multiplication and, as is
∼
customary, we shall write this commutative group operation additively. Note that G = Gb as groups
2
andPontryagindualityimpliesthatGbbcanbecanonicallyidentifiedwithG,afactwhichisemphasized
by writing hξ,xi for ξ(x).
The Fourier transform Ff = fb∈ CGb of f ∈ CG is given by fb(ξ) = P f(x)hξ,xi, ξ ∈ Gb. The
x∈G
inversion formula for the Fourier transformation is f(x) = 1 P fb(ξ)hξ,xi, x ∈ G. It implies that
|G| ξ∈Gb
kfk22 = |G1| Pξ∈Gb|fb(ξ)|2 = |G1|kfbk22, where kfk2 := (Px∈G|f(x)|2)21. Further, this fact together with
kξk2 = |G|12 for all ξ ∈ Gb implies that the normalized characters |G|−12ξ, ξ ∈ Gb form an orthonormal
basis for CG, and P hξ,xi = 0 if ξ 6= 0 and P hξ,xi = 0 if x 6= 0.
x ξ
For n ∈ N and ω = e2πi/n, the discrete Fourier matrix W of the cyclic group Z is de-
Zn n
fined by WZn = (ωrs)nr,−s=10. Identifying CZn with Cn, we have fb = WZnf. An arbitrary finite
Abelian group G can be represented as a direct product of cyclic groups Z ×Z ×...×Z where
d1 d2 dm
d ,...,d can be chosen to be powers of prime numbers. A character in the dual group Gb is
1 m
then given byh(ξ ,ξ ,...,ξ ),(x ,x ,...,x )i = hξ ,x ihξ ,x i...hξ ,x i,where (ξ ,ξ ,...,ξ ) ∈
1 2 m 1 2 m 1 1 2 2 m m 1 2 m
Zb ×Zb ×...×Zb ∼= Gb. The discrete Fourier matrix W for G = Z ×Z ×...×Z is the Kro-
d1 d2 dm G d1 d2 dm
necker product of the Fourier matrices for the groups Z ,Z ,...,Z , that is, W = W ⊗W ⊗
d1 d2 dm G d1 d2
...⊗W . For example, we have
dm
1 1 1 1 ! 1 1 1 1 !
1 i −1 −i 1 −1 1 −1
W = and W = .
Z4 1 −1 1 −1 Z2×Z2 1 1 −1 −1
1 −i −1 i 1 −1 −1 1
Note that for appropriately chosen bijections S : {0,1,...,|G|−1} −→ G and S : {0,1,...,|G|−1}
1 2
−→ Gb we have fb◦S = W (f ◦S ) for f ∈ CG.
2 G 1
The translation operator T , x ∈ G, is the unitary operator on CG given by T f(y) = f(y−x),
x x
y ∈ G. Similarly, the modulation operator M , ξ ∈ Gb, is the unitary operator defined by M f = f·ξ,
ξ ξ
where here and in the followingf·g denotes the pointwise product of f,g ∈ CG. We have Mdf = T fb.
ξ ξ
We refer to the unitary operators π(λ) = M ◦ T for λ = (x,ξ) ∈ G×Gb as time–frequency shift
ξ x
operators.
Theshort–timeFouriertransformationV : CG −→ CG×Gb withrespecttothewindowg ∈ CG\{0}
g
is given by [FK98, FKL07, Gr¨o01, Gr¨o03]
X
V f(x,ξ) = hf,π(x,ξ)gi = f(y)g(y−x)hξ,yi,f ∈ CG,(x,ξ) ∈ G×Gb.
g
y∈G
The inversion formula for the short–time Fourier transform is
X
f(y) = 1 V f(x,ξ)g(y−x)hξ,yi, y ∈ G, (1)
|G|kgk2 g
2
(x,ξ)∈G×Gb
that is, f can be composed of time–frequency shifted copies of any given g ∈ CG\{0}. Further,
p
kV fk = |G|kfk kgk . The so-called Gabor system {π(x,ξ)g} is clearly not an or-
g 2 2 2 (x,ξ)∈G×Gb
thonormal basis if |G| > 1 since it consists of |G|2 vectors in a |G| dimensional space.
For g ∈ CG and x ∈ G, we define the |G|×|G|–diagonal matrix
 g(S (0)+x) 0 
1
g(S (1)+x)
1
Dx,g =  ... .
0 g(S (|G|−1)+x)
1
3
Then, the |G|×|G|2–full Gabor system matrix with respect to g is given by
A = (D W |D W | ··· |D W )∗, (2)
G,g S1(0),g G S1(1),g G S1(|G|−1),g G
where M∗ denotes the adjoint of the matrix M. For example, for G = Z , and g = (1,2,3,4)T,
4
1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 !∗
2 2i −2 −2i 3 3i −3 −3i 4 4i −4 −4i 1 i −1 −i
A := .
Z4,(1,2,3,4)T 3 −3 3 −3 4 −4 4 −4 1 −1 1 −1 2 −2 2 −2
4 −4i −4 4i 1 −i −1 i 2 −2i −2 2i 3 −3i −3 3i
Similarly, for the group G = Z2 we have
2
1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 !∗
2 −2 2 −2 1 −1 1 −1 4 −4 4 −4 3 −3 3 −3
A := . (3)
Z2,(1,2,3,4)T 3 3 −3 −3 4 4 −4 −4 1 1 −1 −1 2 2 −2 −2
2
4 −4 −4 4 3 −3 −3 3 2 −2 −2 2 1 −1 −1 1
Using the enumeration S : {0,1,...,|G|2−1} −→ G×Gb that is given by the lexicographic order
induced by S and S described above on G×Gb, we have V f ◦S = A f. Therefore, we shall refer
1 2 g G,g
to A as short–time Fourier transform matrix with respect to the window g. Clearly, the rows of
G,g
A represent the vectors in the Gabor system {π(λ)g} , and formula (1) implies that A∗ A
G,g λ∈G×Gb G,g G,g
is a multiple of the identity matrix.
3. UNCERTAINTY PRINCIPLES FOR THE FOURIER TRANSFORM ON FINITE
ABELIAN GROUPS
The following uncertainty theorem for functions defined on finite Abelian groups is the natural
starting point for our discussion [DS89].
Theorem 3.1. Let f ∈ CG\{0}, then kfk ·kfbk ≥ |G|.
0 0
A complementary result characterizes those f for which the bound in Theorem 3.1 is sharp
[DS89, Smi90, MO¨P04]. Namely, if k divides |G|, then there exists f ∈ CG with kfk = k and
0
kfbk = |G|. Further, if kfk kfbk = |G| and suppf contains the identity element, then suppf is a
0 k 0 0
subgroup of G. A generalization of Theorem 3.1 to non Abelian groups is given in [Mes92] and those
f achieving the respective lower bounds are described in [Kan07].
p
Theorem 3.1 implies the weaker inequality kfk + kfbk ≥ 2 |G| for f ∈ CG \ {0}. If G is a
0 0
cyclic group of prime order, then this inequality and also Theorem 3.1 can be improved significantly
[Fre04, Tao05].
Theorem 3.2. Let G = Z with p prime. Then kfk +kfbk ≥ |G|+1 holds for all f ∈ CG\{0}.
p 0 0
As illustrated in [Tao05], Theorem 3.2 follows from combining Chebotarev’s theorem on roots of
unity which states that every minor of the Fourier transform matrix W , p prime, is nonzero [EI76,
Z
p
SL96, Tao05, Fre04], with
Proposition 3.3. Let M ∈ Cm×n. Then kfk +kMfk ≥ m+1 for all f ∈ Cn if and only if every
0 0
minor of M is nonzero. Moreover, if every minor of M ∈ Cm×n is nonzero and k,l are given with
k +l ≥ m+1, then there exists f ∈ Cn with kfk = k and kMfk = l.
0 0
Proposition 3.3 in turn can be obtained using the following observation that will also be used in
numerical experiments below.
4
Lemma 3.4. For M ∈ Cm×n and 1 ≤ k ≤ m, 1 ≤ l ≤ n, there exists f ∈ Cn with kfk = k and
0
kMfk = l if and only if there exist sets A ⊆ {0,...,n−1} and B ⊆ {0,...,m−1} with |A| = k,
0
|B| = m−l, and for all a ∈ A and y ∈ Bc, we have
rankM = rankM = rankM −1 < |A|.
A\{a},B A,B A,B∪{y}
We refer for the proofs of Proposition 3.3 and Lemma 3.4 to [KPR07].
Theorem 3.2 improves on Theorem 3.1 but it applies only to cyclic groups of prime order since
¨
any other finite Abelian group G has proper subgroups leading to zero minors in W [MOP04]. See
G
[CR06] for estimates on the probability that for randomly chosen sets T ⊆ G and Ω ⊆ Gb with
|T|+|Ω| ≤ G there exists f ∈ CG with suppf = T and suppfb= Ω.
Meshulam improved the bound in the classical uncertainty relation given in Theorem 3.1 for non-
trivial finite Abelian groups of non-prime order [Mes06]. He defined for 0 < k ≤ |G| the function
θ(G,k) = min(cid:8)kfbk : f ∈ CG and 0 < kfk ≤ k(cid:9). (4)
0 0
Using this notation, Theorem 3.2 implies that θ(Z ,k) = p+1−k. The main result in [Mes06] is
p
Theorem 3.5. For k ≤ |G|, let d be the largest divisor of |G| which is less than or equal to k and
1
let d be the smallest divisor of |G| which is larger than or equal to k. Then
2
|G|
θ(G,k) ≥ (d +d −k). (5)
1 2
d d
1 2
A streamlined version of the Meshulam’s proof by induction of Theorem 3.5 can be found
in [KPR07] and [LM05] contains a non-inductive proof thereof.
Tao realized that Theorem 3.5 simply states that all possible support pairs (kfk ,kfbk ) lie in
0 0
the convex hull of the points (|H|,|G/H|), where H ranges over all subgroups of G [Mes06]. The
¨
results from [MOP04] mentioned below Theorem 3.1 imply that the vertex points (|H|,|G|/|H|) are
attained, but little more is known about the set {(kfk ,kfbk ), f ∈ CG}.
0 0
In the following, we address the question whether for some given Abelian group G and (k,l)
chosen with l ≥ θ(G,k) ≥ |G| (d +d −k) there exists f ∈ CG with kfk = k and kfbk = l. This
d1d2 1 2 0 0
(cid:8) (cid:9)
question has been considered earlier in [FKLM05] where the set (kfk ,kfbk ), f ∈ G has been
0 0
described for G = Z and G = Z .
6 8
First, we state an affirmative positive result for cyclic groups. It follows from Example 5.6 in
[Smi90] and the proof of Proposition 4.5 in [Kut03].
Proposition 3.6. Let G = Z , n ∈ N. If 0 < k,l ≤ |G| satisfy l+k ≥ |G|+1, then there exists a
n
function f ∈ CG with kfk = k and kfbk = l.
0 0
For product groups G = H ×H , positive results on possible support pairs (kfk ,kfbk ) can
1 2 0 0
be obtained on the basis of the support pairs for H and H . These follow from the fact that for
1 2
f ⊗f ,f ∈ CH1,f ∈ CH2 we have kf ⊗f k = kf k ·kf k and kf\⊗f k = kfb⊗fbk = kfbk ·kfbk
1 2 1 2 1 2 0 1 0 2 0 1 2 0 1 2 0 1 0 2 0
[Gr¨o06].
5
Y−pr Y−nu Y−co ? N−co N−nu N−pr
Figure 1. Color coding used in Figures 2–5 to describe membership to subsets of N2 or N3. Y-pr
indicates a proof is known that the corresponding value is in the set considered. Y-nu implies that
there is numerical evidence that the value is in the set and Y-co indicates that we conjecture that
the value is in the set. N-pr indicates a proof is known that the corresponding value is not in the
set, and N-nu and N-co are defined accordingly. The color adjacent to ? implies that no judgement
is made here.
A negative result for the groups G = Z for, p ≥ 5 prime, stating that there exists no f ∈ CZ2p
2p
with kfk = 3 and kfbk = p−1 is included in [KPR07].
0 0
The numerical results collected in Figure 2 are based on Lemma 3.4 and they show that the
set of all possible pairs (kfk ,kfbk ) is not easily described in general. The computations needed to
0 0
obtain Figure 2 are quite involved. For example, the computations showing that there is no vector
on Z with five nonzero entries and whose Fourier transform has nine nonzero entries include the
16
(cid:16) (cid:17)(cid:16) (cid:17)
16 16
calculation of the singular values of = 49969920 five by seven matrices.
5 7
4. UNCERTAINTY PRINCIPLES FOR SHORT–TIME FOURIER TRANSFORMS ON
FINITE ABELIAN GROUPS
We now turn to discuss minimum support conditions on time–frequency representations of elements
in CG, in particular, for the short–time Fourier transform V f ∈ CG×Gb of a function f ∈ CG with
g
respect to a window g ∈ CG. For background on uncertainty principles in joint time–frequency
representations see [Gr¨o03, HL05]
But first, we consider the simplest joint time–frequency representation of f which is given by
the tensor product f⊗fb. Similarly, in electrical engineering the so-called Rihaczek distribution
R : G×Gb −→ C given by Rf(x,ω) = f(x)fb(ω)hω,xi is considered. Theorem 3.1 implies that
kRfk = kf⊗fbk = kfk kfbk ≥ |G|. Figure 3 lists all possible pairs (kfk ,kRfk ) for f ∈ CZ4 and
0 0 0 0 0 0
f ∈ CZ22.
The following result resembles Theorem 3.1. It is given for functions on the real line as so-called
weak uncertainty principle in [Gr¨o03].
Proposition 4.1. kV fk ≥ |G| for f,g ∈ CG\{0} with equality for f = g = δ.
g 0
Proof. Clearly kV δk = |G|. For f,g ∈ CG\{0}, the result follows from
δ 0
|G| kfk2 kgk2 = kV fk2 ≤ kV fk kV fk2 ≤ kV fk kfk2 kgk2. (cid:3)
2 2 g 2 g 0 g ∞ g 0 2 2
Next, we derive lower bounds on kV fk which depend on both, kfk and kgk .
g 0 0 0
Proposition 4.2. For f,g ∈ CG\{0}, we have for θ defined in (4)
kV fk ≥ max{ θ(G,kgk )θ(G,kfbk ), θ(G,kfk )θ(G,kgk ) }. (6)
g 0 0 0 0 b 0
6
4 4 5 6
4
3
2 2
2
2 4 2 4 5 2 3 4 6
7 8 8 8
6 6 6
4 4 4
2 2 2
7 2 4 6 8 2 4 6 8 2 4 6 8
9 9 10 12
10
8
9
6 6 8
6
5 6
4
3 3 4
3
2 2
3 6 9 3 6 9 2 4 5 6 8 10 2 3 4 6 8 9 10 12
12 14 15 16
10 12 12 14
9 10 12
8 8 190 10
6 7 8
6 6 6
4 5
3 4 4
3
2 2 2
2 3 4 6 8 9 10 12 2 4 6 7 8 10 12 14 3 5 6 910 12 15 2 4 6 8 10 12 14 16
16 16 16 16
14 14 14 14
12 12 12 12
10 10 10 10
8 8 8 8
6 6 6 6
4 4 4 4
2 2 2 2
2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16
Figure 2. The set (cid:8)(kfk ,kfbk ), f ∈ CG\{0}(cid:9) for all Abelian groups of order less than or equal to
0 0
16 with exception of the groups of prime order Z and Z . The groups (row wise from left to right)
11 13
are Z , Z2, Z , Z ; Z , Z , Z ×Z , Z3; Z , Z2, Z , Z ; Z ×Z , Z , Z , Z ; Z ×Z , Z2, Z2×Z ,
4 2 5 6 7 8 2 4 2 9 3 10 12 2 6 14 15 16 2 8 4 2 4
Z4. The color code used is given in Figure 1, and justified by the results in Section 3.
2
7
4 4
3 3
2 2
1 1
2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16
Figure 3. For groups G = Z and Z2 all possible pairs (kfk ,kRfk ) are colored red, those pairs
4 2 0 0
that are not achieved by some f ∈ CG are colored blue in accordance with the color code given in
Figure 1.
Proof. We shall prove kV fk ≥ θ(G,kfk )θ(Gb,kgk ). Then (6) follows from kV fk = kV fbk
g 0 0 b 0 g 0 g 0
b
and θ(G,k) = θ(Gb,k) for any k, or, alternatively from kV fk = kV gk . To see (6), observe first that
g 0 f 0
theso-calledsymplecticFouriertransformationF = R◦F−1◦F ,thatis,thecompositionofaFourier
s G
Gb
transformation F on G, an inverse Fourier transformation F−1 on Gb, and the axis transformation
G
(cid:16) (cid:17) Gb
0 1
R : F 7→ F ◦ obeys the same uncertainty principle as the Fourier transformation on the
1 0
group G×Gb. For f,g ∈ CG, we calculate
XX XXX
F V f(r,ρ) = V f(x,ξ)hρ,xihξ,ri = f(t)g(t−x)hξ,ti hρ,xihξ,ri
s g g
x∈Gξ∈Gb x∈Gξ∈Gb t∈G
XX X X
= f(t)g(t−x)hρ,xi hξ,r−ti = |G| f(r)g(r−x)hρ,xi
x∈G t∈G ξ∈Gb x∈G
= |G|hρ,rif(r)g(ρ)
b
and note that suppF V f = suppf×suppg. A simple tensor argument implies that kV fk =
s g b g 0
kF−1(cid:0)F V f(cid:1)k ≥ θ(G,kfk )θ(Gb,kgk ). (See Proposition 3.9 in [KPR07] for details.) (cid:3)
s s g 0 0 b 0
For f,g ∈ CZp\{0}, p prime, Proposition 4.2 gives the lower bound
kV fk ≥ max{ (p+1−kgk )(p+1−kfbk ), (p+1−kfk )(p+1−kgk ) }
g 0 0 0 0 b 0
which is improved below.
Proposition 4.3. Let G = Z , p prime. For f,g ∈ CG\{0},
p
(cid:26)
|G|(|G|+1)−kfk kgk if kfk +kgk > |G|;
kV fk ≥ 0 0 0 0
g 0 |G|(|G|+1)−(|G|+1−kfk )(|G|+1−kgk ) if kfk +kgk ≤ |G|.
0 0 0 0
Proof. Note that for all x ∈ G, V f(x,·) = hf,π(x,·)gi represents the Fourier transform of a
g
vector of the form f ·T g¯, that is,
x
X \
V f(x,ξ) = hf,π(x,ξ)gi = f(y)g(y −x)hξ,xi = f ·T g¯(ξ), x ∈ G,ξ ∈ Gb.
g x
y∈G
8
\
As long as f ·T g¯ 6= 0, Theorem 3.2 applies and so kf ·T g¯k +kf ·T g¯k ≥ |G|+1. For K := {x :
x x 0 x 0
f ·T g¯ 6= 0} we get
x
X \ X
kV fk = kf ·T g¯k ≥ |K|(|G|+1)− kf ·T g¯k = |K|(|G|+1)−kfk kgk ,
g 0 x 0 x 0 0 0
x∈K x∈G
X
where kf ·T g¯k = kfk kgk follows from a simple counting argument.
x 0 0 0
x∈G
We shall now estimate |K| using the Cauchy-Davenport inequality, which states that for non-
emptysubsetsAandB ofG = Z ,pprime,|A+B| ≥ min(|A|+|B|−1,|G|),whereA+B = {a+b : a ∈
p
A,b ∈ B} [K´ar05]. Now K = {x : f·T g¯ 6= 0} = {x : {(suppg¯)+x}∩suppf 6= ∅} = suppf−suppg¯.
x
We set A = suppf,B = suppg¯, and obtain |K| = |suppf −suppg¯| ≥ min(kfk +kgk −1,|G|).
0 0
If kfk + kgk ≥ |G| + 1, then |K| = |G| and, hence, kV fk ≥ |G|(|G| + 1) − kfk kgk . If
0 0 g 0 0 0
kfk +kgk ≤ |G|, then |K| ≥ kfk +kgk −1 and so
0 0 0 0
kV fk ≥ (kfk +kgk −1)(|G|+1)−kfk kgk = |G|(|G|+1)−(|G|+1−kfk )(|G|+1−kgk ).
g 0 0 0 0 0 0 0
(cid:3)
To establish support size constraints for short–time Fourier transformations for a given group G
analytically is quite tedious since it requires to check all combinations of kfk and kgk . For the
0 0
case G = Z , however, we have assembled all possible and impossible combinations in Figure 4. A
3
derivation of the entries can be found in Appendix 6.4 in [KPR07].
In the following, we shall fix the window g and vary only the analyzed function f. First we
provide a short–time Fourier transform version of Theorem 3.2.
Theorem 4.4. Let G = Z , p prime. For almost every g ∈ CG, we have
p
kfk +kV fk ≥ |G|2 +1 (7)
0 g 0
for all f ∈ CG \{0}. Moreover, for 1 ≤ k ≤ |G| and 1 ≤ l ≤ |G|2 with k +l ≥ |G|2 +1 there exists
f with kfk = k and kV fk = l.
0 g 0
3 5
kgk 4
0
3
2
3 2 5
2
1 1 1 1
1 2 3 4 5 10 15 20 25
1 2 3 4 5 6 7 8 9
kfk
0
kV fk kV fk
g 0 g 0
Figure 4. The set (cid:8)(kfk ,kgk ,kV fk ), f,g ∈ CG\{0}(cid:9) for G = Z ,Z . The color coding is given
0 0 g 0 3 5
in Figure 1 and justified by Proposition 4.3 and Theorem 4.4.
We picture this result for G = Z and G = Z in Figure 5. Note that Theorem 4.4 follows from
5 7
Proposition 3.3 together with Theorem 4 from [LPW05] which we state as
9
Theorem 4.5. For almost every g ∈ CZp, p prime, we have that every minor of AZp,g is nonzero.
Outline of a proof of Theorem 4.5. It suffices to show that each square submatrix (A ) has
Zp,g A,B
determinant nonzero for almost every g.
To this end, choose A ⊆ Zp and B ⊆ Zp×Zcp with |A| = |B| and set PA,B(z) = det(AZp,z)A,B,
z = (z ,z ,...,z ). To show that P 6= 0, we shall locate a term in the polynomial in standard
0 1 p−1 A,B
form which has a nonzero coefficient. To construct this term, we determine first the maximal possible
exponent of z in one of the terms of P that are not trivially zero. Next, we determine the maximal
0
exponent that z can have in a monomial where the maximal exponent of z is attained and so on.
1 0
Using generalized Vandermonde determinants, it can then be shown that the coefficient of this
“maximal” term within P can be expressed as a product of different minors of the discrete Fourier
A,B
matrix W . For p prime, all these minors are nonzero, so the polynomial P has a nonzero coefficient
Z
p
for this “maximal term”, hence is not identically 0, and nonzero almost everywhere. We have
Q
P = P 6≡ 0, which implies that for g ∈/ Z = {z : P(z) = 0}, every minor of A is
A,B P Zp,g
A,B:|B|=|A|
nonzero. Since P 6≡ 0, Z has Lebesgue measure 0. (cid:3)
P
Clearly, this proof of Theorem 4.5 is also based on Chebotarev’s theorem on roots of unity. Also,
Chebotarev’s theorem on roots of unity and, therefore, Theorem 3.2 can be obtained as a corollary
to Theorem 4.5 as shown in Appendix 6.5 of [KPR07].
It is easy to see that if g ∈ CZp satisfies (7) then kgk = kgk = p, that is, g(x) 6= 0 for all x ∈ Z
0 b 0 p
and g(ξ) 6= 0 for all ξ ∈ Zc [LPW05]. In addition, we have
b p
Proposition 4.6. There exists a unimodular g ∈ CZp, p prime, that is, a g with |g(x)| = 1 for all
x ∈ Z , satisfying the conclusions of Theorem 4.4.
p
Proof. Theorem 4.5 implies that all minors of A are nonzero polynomials in the polynomial
Zp,g
ring C[z ,...,z ]. Let P be the product of all these minor polynomials, which, by assumption, is
0 n−1
nonzero. We have to show that P(g) 6= 0 for some g ∈ CZp with |g(x)| = 1 for all x ∈ G.
This follows since the only polynomial P with P(g) = 0 whenever |g(x)| = 1 for all x ∈ G is
trivial, P ≡ 0, which we show below using induction over the number of variables n.
The case n = 1 follows since any nonzero polynomial in one variable has only finitely many zeros;
only P ≡ 0 vanishes for all z ∈ S1 = {z : |z| = 1}. Next, we consider a polynomial P of n variables
which we regard as a polynomial in z with coefficients in the polynomial ring C[z ,...,z ], that
n−1 0 n−2
is,
P(z ) = Q (z ,...,z )zm +Q (z ,...,z )zm−1 +···+Q (z ,...,z )
n−1 m 0 n−2 n−1 m−1 0 n−2 n−1 0 0 n−2
For any fixed (c ,...,c ) ∈ (S1)n−1 we have
0 n−2
Q (c ,...,c )zm +Q (c ,...,c )zm−1 +···+Q (c ,...,c ) = 0
m 0 n−2 n−1 m−1 0 n−2 n−1 0 0 n−2
for all z ∈ S1, hence, all its coefficients Q (c ,...,c ), k = 0,...,m vanish. In other words,
n−1 k 0 n−2
we have that Q ∈ C[z ,...,z ], k = 0,...,m vanish on (S1)n−1, which, by induction hypothesis,
k 0 n−2
implies that all Q ≡ 0 and therefore P ≡ 0. (cid:3)
k
Proposition 3.3 and Proposition 4.7 below combine to show that the condition “G = Z with p
p
prime” is necessary for the existence of g ∈ CG satisfying (7).
10