Operator Identification on the Circle
Gökhan Civan
Department of Mathematics
University of Maryland
College Park, MD 20742
Email: gcivan@math.umd.edu
Abstract—We study the identification problem for linear timevariant communication channels on the circle, extending earlier
work on the real line. We therefore resolve a suitably formulated
periodic version of the conjectures of Kailath and Bello regarding
channel identification.
I. I NTRODUCTION
A linear time-variant communication channel (e.g. mobile
communications, radar) can be modeled formally as an integral
operator [1] of the form
Z
f → κ(·, t)f (t) dt,
where κ is the kernel function. The problem of identifying
such a channel only from its action on a single properly chosen
input signal was addressed by Kailath [2]. He considered
a family of channels with time delay and Doppler spread
confined to a common support set, and conjectured that such a
family of channels is identifiable if and only if their common
support set has area at most 1. Kailath considered only
rectangular support sets; his conjecture was later extended by
Bello [3] to include arbitrary support sets. More recently, the
channel identification problem was addressed using modern
time-frequency analysis techniques. In [4], Kozek and Pfander
resolved a suitably formulated version of Kailath’s conjecture.
In [5], Pfander and Walnut resolved a suitably formulated
version of the more general conjecture of Bello. The theory
of ”operator sampling” or ”operator identification” which
emerged from these efforts has since been further developed
and elaborated on; see [6], [7], and the references therein.
We are interested in the extension of the theory to more
general groups. In this paper, we study the operator identification problem on the circle. Specifically, we consider operators
of the form
Z
XZ
g→
κ(·, t)g(t) dt =
η(z, ξ)Mξ Tz g dz,
T
ξ∈Z
T
where g is a function on T, κ is a function on T×T, and η is a
function on T × Z with compact support; the operators Tz and
Mξ are translation and modulation. We shall make precise
to which spaces these functions belong presently. Our main
results are Theorems 1 and 2, corresponding to the main results
in [5], which provide sufficient and necessary conditions for
identification, respectively. We follow the formulation and
c 2015 IEEE
978-1-4673-7353-1/15/$31.00 proof method in [5], and partly borrow from [4], introducing
modifications as needed for our new setting. Therefore, we
give credit to the original authors for some of the ideas and
constructions that we utilize.
II. F OUNDATIONAL T HEORY
Our development uses the language and techniques of timefrequency analysis. We suggest [8] as a basic reference. The
standard treatment of time-frequency analysis focuses on the
b = Z are
real line. Here, T = {z ∈ C : |z| = 1} and T
the groups that are of interest to us; we take the unit Haar
measure on T and the counting measure on Z. The standard
theory goes through for these (and more general) groups. We
shall therefore be content with mostly stating the basic facts
necessary for our discussion.
A. The Short-Time Fourier Transform
Let G be a group of the form TM × ZN . We have the
Schwartz space S(G) of rapidly decreasing functions, and
its dual S 0 (G), the space of tempered distributions. Clearly,
S(TM ) = C ∞ (TM ). One can show that S 0 (ZM ) is precisely
the set of all functions of polynomial growth. Since the
Fourier transform is an isomorphism between S 0 (ZM ) and
S 0 (TM ), the latter space consists precisely of Fourier series
with coefficients of polynomial growth.
For f ∈ S 0 (G) and g ∈ S(G) \ {0}, the short-time Fourier
b defined by
transform (STFT) is a function of (a, â) ∈ G × G
Z
Vg f (a, â) = hf, Mâ Ta gi =
f (t)g(t − a)(−t, â) dt. (1)
G
Here, the integral is formal unless f ∈ S(G). The operators
Ta and Mâ are translation and modulation, respectively. Note
that (z, ξ) = z ξ = Eξ (z) for z ∈ T and ξ ∈ Z. The STFT is
a continuous function and a tempered distribution in its own
right.
Let 1 ≤ p ≤ ∞. The modulation space norm of f ∈ S 0 (G)
is kf kM p = kVg f kLp . Different window functions g define
equivalent norms. The modulation space M p (G) is the set of
all f ∈ S 0 (G) with kf kM p < ∞. The space M p (G) is a
Banach space invariant under translations, modulations, and
(partial) Fourier transforms. (A partial Fourier transform is a
Fourier transform with respect to a single variable while all
the other variables are fixed.) Of course, here and elsewhere,
we shift to the appropriate dual group whenever a (partial)
Fourier transform is applied.
We have the continuous inclusions S(G) ⊆ M 1 (G) ⊆
L (G) ∩ C0 (G), M 2 (G) = L2 (G), and M p1 (G) ⊆ M p2 (G)
for p1 ≤ p2 . The space M 1 (G) is Feichtinger’s algebra (also
denoted by S0 (G)), and M ∞ (G) is its dual. The duality is
defined via the pairing
Z
0
hf, f i =
(2)
Vg f Vg f 0 ,
p
b
G×G
where f ∈ M (G) and f 0 ∈ M ∞ (G). This pairing is
independent of the chosen window function g as long as
kgk2 = 1. Moreover, if f 0 ∈ Lp (G), then f 0 ∈ M ∞ (G),
and
Z
(3)
hf, f 0 i =
f f 0.
1
G
The duality pairing is invariant under (partial) Fourier transforms.
Since T is compact, the modulation spaces in this case are
much simpler than what the definition might suggest. One can
show that the Fourier transform is an isomorphism between
M p (TM ) and `p (ZM ) (see [9] for a proof in a different
context). In particular, M 1 (TM ) is Wiener’s algebra A(TM ).
Note that δ ∈ M ∞ (TM ).
On G × G, we have the Fourier transform in the second
variable, denoted by F2 , and the coordinate transform (a, t) →
(t, t − a), denoted by TG .
Finally, we indicate that the theory of the STFT can be
extended to include window functions in M 1 (G).
B. The Spaces M 1 (T × T) and M 1 (T × Z)
Let f ∈ M 1 (T × T). Since M 1 (T × T) ∼
= `1 (Z × Z),
X
f=
cξ1 ,ξ2 Eξ1 ⊗ Eξ2
ξ1 ,ξ2 ∈Z
with convergence in M 1 (T × T). Here, {cξ1 ,ξ2 } ∈ `1 (Z × Z).
It is almost immediate that, for a ∈ T, f (a, ·) ∈ M 1 (T) ∼
=
`1 (Z), and


X X

f (a, ·) =
cξ1 ,ξ2 aξ1  Eξ2
ξ2 ∈Z
ξ1 ∈Z
1
with convergence in M (T). We compute
X
F2 f =
cξ1 ,ξ2 F2 (Eξ1 ⊗ Eξ2 )
ξ1 ,ξ2 ∈Z
=
X
bξ
cξ1 ,ξ2 Eξ1 ⊗ E
2
=
cξ1 ,ξ2 Eξ1 ⊗ δξ2
ξ1 ,ξ2 ∈Z
with convergence in M 1 (T × Z). It is almost immediate that
F2 f (a, ·) = f (a, ·)b .
hg, δi = g(1)
(4)
The ”almost immediate” conclusions that we have drawn here
really follow from the continuity of the inclusion M 1 (G) ⊆
C0 (G) for various choices of G. The last equality may seem
like a tautology, but this is the case only when f ∈ S(T × T).
(g ∈ M 1 (T)).
(5)
C. Gabor Analysis in Finite Dimensions
Let ωK = e2πi/K for some positive integer K. For c ∈ CK ,
let [5]
A(c) = (A0 |A1 | · · · |AK−1 ),
pq K−1
where Ak is the matrix (cp+k ωK
)p,q=0 . Here and elsewhere,
indices over Z/KZ are modulo K. The columns of A(c)
coincide with the vectors Mâ Ta c for a, â ∈ Z/KZ. In other
words, A(c) is the Gabor system generated by c. It is shown in
[10] (and earlier in [11] for K prime) that there exists c ∈ CK
such that every K × K minor of A(c) is invertible; in fact,
the set of all such c is a dense open set of full measure. (The
result in [11] for K prime is stronger in that the invertibility
of every minor is asserted.)
III. K ERNELS AND O PERATORS
2
defines an integral operator f →
R Every κ ∈ L (T × T)
2
κ(·,
t)f
(t)
dt
from
L
(T)
to L2 (T) with operator norm
T
kκkL2 . These are precisely the so-called Hilbert-Schmidt operators. Alternatively, these operators are characterized as those
defined by (infinite) matrices with finite Frobenius norms;
the matrices can be defined with respect to the standard
orthonormal basis {Eξ }ξ∈Z .
As we would like to be able to feed distributional elements
into our operators, the L2 framework is not suitable. We next
describe a refinement of the above class of operators in order
to suit our needs. Our kernel function κ shall be an element of
the smaller space M 1 (T × T). In this case, we get a bounded
linear operator H : M ∞ (T) → M 1 (T) as follows. We define
Hf (a) = hκ(a, ·), f i;
we conjugate f so that the resulting operator is linear instead
of conjugate-linear. Let us compute this pairing in terms of
the series expansions obtained previously. Recall that κ is
defined by a sequence {cξ1 ,ξ2 } ∈ `1 (Z × Z). It follows from
the characterization M ∞ (T) ∼
= `∞ (Z) that f is defined by a
bounded sequence {dξ }ξ∈Z . By (3) and the invariance of (2)
under the Fourier transform,


X X

Hf (a) =
cξ1 ,ξ2 aξ1  dξ2 .
ξ2 ∈Z
ξ1 ,ξ2 ∈Z
X
Here is another seemingly tautological equality which can be
proved using series expansions:
ξ1 ∈Z
Since the convergence is absolute, we can rearrange this series
to obtain


X X

Hf (a) =
cξ1 ,ξ2 dξ2  aξ1 .
ξ1 ∈Z
ξ2 ∈Z
It follows that Hf ∈ M 1 (T), and that H is defined by matrix
multiplication via the matrix (cξ1 ,ξ2 )ξ1 ,ξ2 ∈Z . Moreover, the
operator norm of H is bounded by CkκkM 1 , where C does
not depend on κ. However, we actually have the following
stronger form of continuity: If fj → 0 in the weak* topology
of M ∞ (T), then Hfj → 0 in the norm topology of M 1 (T).
(Since {fj } is convergent in the weak* topology of M ∞ (T), it
is bounded in the norm topology of M ∞ (T). This allows one
to apply the dominated convergence theorem.) Since M 1 (T) is
sequentially weak* dense in M ∞ (T), H is determined by its
values on M 1 (T). A simple calculation shows that H satisfies
operator identification problem. However, it is reasonable to
additionally require that eg be bounded and stable so that
errors on one side correspond in a controllable fashion to errors
on the other side. (An operator is stable if it is injective with
a bounded inverse [5].) Note that the continuity of eg is not
automatic, unless g ∈ L2 (T), since we put the L2 norm on
HW instead of the M 1 norm.
hHg, f i = hκ, f ⊗ gi (f, g ∈ M 1 (T)).
We shall use the following notation. For a positive integer
K, let IK be the image of [0, 1/K) under the exponential map
x → e2πix . The measure of a Borel set will be denoted by the
absolute value notation.
(6)
Here is a summary of our discussion so far. Let H be the set
of all linear maps from M ∞ (T) to M 1 (T) that are continuous
in the weak* sense described above. We have demonstrated an
injective map from M 1 (T×T) to H. It is not too hard to show
that this map is a bijection. In other words, we have a version
of the Schwartz kernel theorem.
Applying F2 and TT , we can recast (6) in the form
1
hHg, f i = hη, Vg f i (f, g ∈ M (T)).
ν∈Z
Since the convergence is absolute, the function (z, ξ, t) →
Mξ Tz g(t) on T × Z × T is continuous and bounded, which
guarantees the existence of the double integral in the definition
of the following function on T:
XZ
ϕ:t→
η(z, ξ)Mξ Tz g(t) dz.
ξ∈Z
IV. S UFFICIENT C ONDITION FOR O PERATOR
I DENTIFICATION
Let W be a subset of T × Z. Let
HW = {H ∈ H : supp ηH ⊆ W }
be equipped with the norm kHk = kηH kL2 . For g ∈ M ∞ (T),
let eg : HW → L2 (T) be defined by eg (H) = Hg. In
other words, eg is evaluation by g. If eg is injective for
some g ∈ M ∞ (T), then we have a positive answer to the
ck δk ,
k=0
k
where δk is the Dirac distribution on T at the point ωK
, and
ck ∈ C.
Proof. We follow the proof method in [5]. We can assume
without loss of generality that W ⊆ T × {0, 1, 2, . . .}. Let K
be a positive integer such that W ⊆ T × [0, K − 1]. We also
pick K large enough so that W is a subset of a union of at
k
× {q} (0 ≤ k, q ≤ K − 1);
most K sets of the form IK ωK
this is possible because of (measure theoretic) outer regularity
and the hypothesis that W is compact with |W | < 1. Let g
be as in the statement of the theorem with ck chosen so that
every K × K minor of A(c) is invertible.
Let H ∈ HW . We define
T
Multiplying the Fourier series expansion of η with (8), we get a
series expansion for the integrand. One can integrate this series
term by term to discover that ϕ ∈ M 1 (T). Fubini’s theorem
now gives hϕ, f i = hη, Vg f i, so ϕ = Hg. This representation
of H shows that H can be interpreted as a weighted sum of
time-frequency shifts. It can be verified by direct calculation
that the measure of supp η quantifies the extent of the timefrequency ”spread” in the output of H.
The upshot of this section is that the space M 1 (T × Z) of
spreading functions, the space M 1 (T×T) of kernel functions,
and the set H are all identified with each other. See [12] for a
discussion of the Schwartz kernel theorem over more general
groups.
K−1
X
g=
(7)
Here, η = F2 TT κ is the so-called spreading function of H.
Note that η, Vg f ∈ M 1 (T × Z). Let us express Hg more
directly in terms of η. Recall that g is defined by a sequence
{cν } ∈ `1 (Z). Then, for t, z ∈ T and ξ ∈ Z,
X
X
Mξ Tz g(t) =
cν Mξ Tz Eν (t) =
cν tξ+ν z −ν . (8)
ν∈Z
Theorem 1. Suppose that W is compact with |W | < 1. There
exists g ∈ M ∞ (T) such that eg is bounded and stable. In fact,
g can be chosen to be of the form
−k
κH,k (t, z) = κH (t−1 ωK
z, z)1IK ×T (t, z)
(t, z ∈ T).
It is straightforward to check that
−k
κH,k (t−1 ωK
, zt−1 ) = κH (z, zt−1 )1I −1 ω−k ×T (t, z),
K
K
and hence we have the partitioning
κH (z, zt−1 ) =
K−1
X
−k
κH,k (t−1 ωK
, zt−1 ).
k=0
Let ηH,k = F2 κH,k . We can show using (4) that
−kξ
−k
ηH,k (t, ξ) = t−ξ ωK
ηH (t−1 ωK
, ξ)1IK ×Z (t, ξ)
(ξ ∈ Z),
or
−k
ηH,k (t−1 ωK
, ξ) = tξ ηH (t, ξ)1I −1 ω−k ×Z (t, ξ),
K
K
and hence we have the partitioning
−ξ
ηH (t, ξ) = t
K−1
X
−k
ηH,k (t−1 ωK
, ξ).
(9)
k=0
Fix t ∈ IK . For 0 ≤ p ≤ K − 1, we compute
p
Hg(t−1 ωK
)=
K−1
X
k=0
p
k
ck κH (t−1 ωK
, ωK
)
(10)
=
=
=
=
=
K−1
X
k=0
K−1
X
k=0
K−1
X
k=0
K−1
X
k=0
K−1
X
p+k
−k p+k
cp+k κH (t−1 ωK
ωK , ωK
)
p+k
cp+k κH,k (t, ωK
)
p+k
cp+k F2−1 ηH,k (t, ωK
)
cp+k
K−1
X
cp+k
(p+k)q
ηH,k (t, q)ωK
(11)
K−1
X
ηH,k (t, q)ωK
(p+k)q
(12)
pq kq
cp+k ωK
ωK ηH,k (t, q),
(13)
q=0
k=0
=
X
q∈Z
k=0
=
Theorem 2. Suppose that W is open with |W | > 1. There
exists no g ∈ M ∞ (T) for which eg is stable.
p
p+k
cp+k κH (t−1 ωK
, ωK
)
K−1
X K−1
X
T
=
k=0 q=0
where (10) follows from (5), (11) follows from (4), and (12)
follows from the fact that W ⊆ T × [0, K − 1]. Let
kq
ηH,k (t, q))0≤k,q≤K−1 .
η H (t) = (ωK
Then (13) is equivalent to
(14)
Since W is a subset of a union of at most K sets of the form
k
IK ωK
× {q} (0 ≤ k, q ≤ K − 1), it follows from (9) that at
most K entries of η H (t) are nonzero. In fact, we can choose
K 2 −K entries that are necessarily zero independent of t (and
e H (t) be η H (t) with such K 2 − K entries
certainly H). Let η
e
removed. Let A(c) be A(c) with the corresponding columns
removed. Then (14) becomes
e η H (t).
xH (t) = A(c)e
(15)
e
Since A(c)
is invertible,
e −1 xH (t).
e H (t) = A(c)
η
a
≤
kxH (t)k22
2
≤b
ξ∈Z
Z Z
=
η(z)δ(ξ)g(tz −1 )tξ−v dt dz
T
η(z)g(t)t−v z −v dt dz
T
It follows that |Pcg(ν)| ≤ kĝk`∞ |η̂(ν)|. We now lift the
restriction that g ∈ M 1 (T). Let {gj } be a sequence in M 1 (T)
such that gj → g in the weak* topology of M ∞ (T). Then
Pcg j → Pcg uniformly, and ĝj → ĝ in the weak* topology
of `∞ (Z). In particular, ĝj is bounded in the norm topology
of `∞ (Z). Let C 0 be a bound. Passing to the limit, we get
|Pcg(ν)| ≤ C 0 |η̂(ν)|. Note that the same bound C 0 works for
any time-frequency shift of g.
We next define a synthesis operator using P as an atom.
Observe that Mq+pB P TωK
k M−pB ∈ H for 0 ≤ k ≤ K − 1
and p, q ∈ Z. We can show using (7) that
ηMq+pB P Tωk
ke
η H (t)k22 .
Integrating these inequalities on IK gives
a2 kHk2 ≤ keg (H)k2L2 ≤ b2 kHk2 ,
which is the desired conclusion.
= M(pB,1) T(ωK
k ,q) ηP .
We define the synthesis operator U : `c (Z × Z × Z/KZ) → H
by
X K−1
X
U (σ) =
σ(p, q, k)Mq+pB P TωK
k M−pB .
p,q∈Z k=0
Here, `c (Z × Z × Z/KZ) is endowed with the `2 norm, and
H is endowed with the norm kHk = kηH kL2 . We compute
kηU (σ) k2L2 = k
X K−1
X
2
σ(p, q, k)M(pB,1) T(ωK
k ,q) ηP k 2
L
p,q∈Z k=0
X K−1
X X
2
=
k
σ(p, q, k)M(pB,1) T(ωK
k ,q) ηP k 2
L
q∈Z k=0
V. N ECESSARY C ONDITION FOR O PERATOR
I DENTIFICATION
We next prove the following counterpart to Theorem 1.
M−pB
K
(16)
e −1 k−1 and b = kA(c)k
e
Let a = kA(c)
2 . Taking Frobenius
2
norms in (15) and (16) gives
ke
η H (t)k22
T
= η̂(ν)ĝ(ν).
and
xH (t) = A(c)η H (t).
T
ξ∈Z
XZ Z
T
p
xH (t) = (Hg(t−1 ωK
))0≤p≤K−1
2
Proof. We follow the ideas in [4] and [5]. Let K be a positive
integer such that W contains J = K + 2 sets of the form
k
IK ωK
× {q} (0 ≤ k ≤ K − 1, q ∈ Z); this is possible because
of the hypothesis that W is open with |W | > 1. Choose J such
sets contained in W . Let B = K + 1 and θ = eπi(1/K−1/B) .
Note that the center of IB θ coincides with the center of IK .
We define an auxiliary operator in H which we shall later
use to restrict the identification problem to a more manageable
subspace. Let η ∈ C ∞ (T) with 0 ≤ η ≤ 1, η = 1 on IB θ,
o
and η = 0 outside IK
. Let P be that operator in H with
ηP = η ⊗ δ. We first study the asymptotic behavior of P . Let
g ∈ M ∞ (T). Suppose first that g ∈ M 1 (T). We compute


Z
XZ

Pcg(ν) =
ηP (z, ξ)Mξ Tz g(t) dz  t−v dt
p∈Z
(17)
=
X K−1
X
q∈Z k=0
k
X
p∈Z
σ(p, q, k)M(pB,1) ηP k2L2
(18)
=
X K−1
X
X
kηP
q∈Z k=0
=
≥
=
|η(z)
T
X
σ(p, q, k)z pB |2 dz
(19)
p∈Z
X K−1
XZ
q∈Z k=0
≤ C 0 d(λν − (pJ + j)),
p∈Z
X K−1
XZ
q∈Z k=0
= C 0 |η̂(λ−1 (λν − λqj + j − (pJ + j))|
σ(p, q, k)M(pB,1) 1k2L2
X
|
σ(p, q, k)z pB |2 dz
IB θ p∈Z
X K−1
X XZ
|σ(p, q, k)|2 dz
(20)
IB θ
q∈Z k=0 p∈Z
VI. C ONCLUSION
1
= kσk2`2 ,
B
where (17) follows from the fact that the indicated translations
of ηP have disjoint supports, (18) follows from the translation invariance of the L2 norm, and (20) follows from the
Pythagorean theorem. We have shown that U is stable. We
now carry out the previous computation in a different direction
starting from (19). Note that
|η|2 ≤ 1IB + 1IB θ + 1IB θ2 .
Then
kηU (σ) k2L2
=
XZ
X K−1
q∈Z k=0
≤
|η(z)
T
X
σ(p, q, k)z pB |2 dz
p∈Z
2 Z
X K−1
XX
q∈Z k=0 j=0
|
IB θ j
X
σ(p, q, k)z pB |2 dz
p∈Z
3
= kσk2`2 .
B
We have shown that U is bounded. Therefore, U is bounded
and stable. Since `c (Z × Z × Z/KZ) is dense in `2 (Z × Z ×
Z/KZ), and both `2 (Z × Z × Z/KZ) and H are complete,
U extends uniquely to a bounded and stable operator U :
`2 (Z × Z × Z/KZ) → H.
Let {(qj , kj )}J−1
j=0 ⊆ Z × Z/KZ be the collection of those
indices corresponding to the J sets chosen at the beginning.
Note that the map (p, j) → pJ + j from Z × {(qj , kj )}J−1
j=0
to Z is a bijection. The inverse of this map induces a linear
isometry ι : `2 (Z) → `2 (Z × Z × Z/KZ). Let g ∈ M ∞ (T).
Let Ag be the composition of the following sequence of maps:
ι
U
eg
F
`2 (Z) → `2 (Z × Z × Z/KZ) → HW → L2 (T) → `2 (Z).
Note that the replacement of H with HW is valid because the
image of U ◦ ι is in HW . Since ι, U , and F are all stable, in
order to show that eg is not stable, it suffices to show that Ag
is not stable. The matrix of Ag is
(aν,pJ+j = (P T
k
ωKj
J−1
where d(x) = maxj=0
|η̂(λ−1 (x−λqj +j))|. Since η̂ ∈ S(Z),
−s
d(x) ≤ Cs (1 + |x|) for all s > 0. We have shown that the
entries of Ag decay rapidly away from the slanted diagonal of
slope λ > 1, where the slope is defined assuming that the row
indices represent the horizontal axis and the column indices
represent the vertical axis. As shown in [13], this property
implies that Ag is not stable.
M−pB g)b (ν − qj − pB))ν,pJ+j∈Z .
Let us investigate how the entries of Ag decay. Extend η̂ from
Z to R by defining η̂(x) = η̂(dxe). Let λ = J/B > 1. We
have
|aν,pJ+j | ≤ C 0 |η̂(ν − qj − pB)|
We would like to point out certain differences between
the operator identification results on the real line and the
circle. The identifying signal in Theorem 1 consists of a finite
sequence of Dirac impulses whereas an infinite sequence of
Dirac impulses is required on the real line. More notably, in
Theorem 2, we discretized the right hand side of eg via the
Plancherel theorem. On the real line, the discretization of the
right hand side is achieved with a Gabor frame generated by
a Gaussian window, using the deep result that a Gabor system
generated by a Gaussian window below the critical density is
necessarily a frame.
ACKNOWLEDGMENT
I would like to extend many thanks to John Benedetto
for helpful discussions, blackboard computations, and his
friendship. I would also like to thank Kasso Okoudjou for
being available to help me understand certain points.
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