Filter Recovery in Infinite Spatially Invariant
Evolutionary System Via Spatiotemporal Trade Off
Sui Tang
Department of Mathematics
Vanderbilt University
Nashville, TN
Email: sui.tang@vanderbilt.edu
Abstract—We consider the problem of spatiotemporal sampling in an evolutionary process x(n) = An x where an unknown operator A driving an unknown initial state x is to
be recovered from a combined set of coarse spatial samples
{x|Ω0 , x(1) |Ω1 , · · ·, x(N ) |ΩN }. In this paper, we will study the
case of infinite dimensional spatially invariant evolutionary
process, where the unknown initial signals x are modeled as
`2 (Z) and A is an unknown spatial convolution operator given
by a filter a ∈ `1 (Z) so that Ax = a ∗ x. We show that
{x|Ωm , x(1) |Ωm , · · ·, x(N ) |Ωm : N ≥ 2m − 1, Ωm = mZ} contains
enough information to recover the Fourier spectrum of a typical
low pass filter a, if x is from a dense subset of `2 (Z). The idea is
based on a nonlinear, generalized Prony method similar to [2].
We provide an algorithm for the case when a and x are both
compactly supported. Finally, We perform the accuracy analysis
based on the spectral properties of the operator A and the initial
state x and verify them in several numerical experiments.
I. I NTRODUCTION
A. The dynamical sampling problem
In situations of practical interest, physical systems evolve
in time under the action of well-behaved operators such as
diffusion processes. Sampling of such an evolving system
is done by sensors or measurement devices that are placed
at various locations and can be activated at different times.
For practical reasons, we aim to reconstruct any states in the
evolutionary process using as few sensors as possible, but
allow one to take samples at different time levels. This setting
has not been studied within the classical approach in sampling
theory, where the samples are taken simultaneously at only one
time level. Dynamical sampling is a newly proposed sampling
framework. It involves studying the time-space patterns formed
by the locations of the measurement devices and the times
of their activation. Mathematically speaking, suppose x is
an initial distribution that is evolving in time satisfying the
evolution rule:
x t = At x
where {At }t∈[0,∞) is a family of evolution operators satisfying the condition A0 = I. Dynamical sampling asks the
question: when do coarse samplings taken at varying times
{x|Ω0 , (At1 x)|Ω1 , . . . , (AtN x)|ΩN } contain the same information as a finer sampling taken at the earliest time? One goal of
dynamical sampling is to find all spatiotemporal sampling sets
(χ, τ ) = {Ωt , t ∈ τ } such that certain classes of signals x can
be recovered from the spatiotemporal samples xt (Ωt ), t ∈ τ .
In the above cases, the evolution operators are assumed to
be known. It has been well-studied in the context of various
evolutionary systems in a very general setting, see [3], [12],
[8], [9], [10].
Another important problem arises when the evolution operators are themselves unknown or partially known. In this case,
we are interested in finding all spatiotemporal sampling sets
and certain classes of evolution operators so that the family
{At }t∈[0,∞) or their spectrum can be identified. We call such
a problem the unsupervised system identification problem in
dynamical sampling.
B. Problem Statement
We are going to introduce the notion of infinite dimensional spatially invariant evolutionary system and uniform
spatiotemporal sampling problem. Let x ∈ `2 (Z) be an
unknown initial spatial signal and the evolution operator A
be given by an unknown convolution filter a ∈ `1 (Z) such
that Ax = a ∗ x. At time t = n ∈ N, the signal x has evolved
to be xn = An x = an ∗ x, where an = a ∗ a · · · ∗ a. We
call this evolutionary system spatially invariant. In this paper,
we are interested in the recovery of unknown filter a that
drives the evolutionary process. Without loss of generality,
let m be a positive odd integer (m > 1) and denote by
Sm : `2 (Z) → `2 (Z) the sampling operator on Ωm = mZ,
i.e., (Sm x)(k) = x(mk). At time level t = l, we have partial
observations
yl = Sm (al ∗ x)
(1)
The special case we are going to consider can be stated as
follows:
Under what conditions on a, m, N, x can a be recovered
from the spatiotemporal samples {yl : l = 0, · · · , N − 1}, or
equivalently, from {x|Ωm , (a ∗ x)|Ωm , · · · , (aN −1 ∗ x)|Ωm }?
In [2], Aldroubi and Kristal consider the recovery of unknown d × d matrix B and unknown initial state x ∈ `2 (Zd )
from coarse spatial samples of its successive states {B k x, k =
0, 1, · · · }. Given an initial sampling set Ω ⊂ Zd = {1, · · · , d},
they employ a new method related to Krylov subspace methods
to show how large li should be to recover all the eigenvalues of
B that can possibly be recovered from spatiotemporal samples
{B k x(i) : i ∈ Ω, k = 0, 1, · · · , li − 1}. Our setup is very
similar to the special case of regular invariant dynamical
sampling problem in [2]. In this special case, they employ
a generalization of the well-known Prony method that uses
these regular undersampled spatiotemporal data first for the
recovery of the Fourier spectrum of the correlating filter.
Since the filter is a typical low pass filter with symmetry
and monotonicity condition, it is completely determined by
its Fourier spectrum. By using techniques developed in [10],
one can recover the initial state. In this paper, we will address
the infinite dimensional analog of this special case.
The remainder of the paper is organized as follows: In
section II, we discuss the noise free case, and propose a
generalized Prony method to show that we can reconstruct
a via spatiotemporal samples {yl }N
l=1 , provided N ≥ 2m − 1.
In section III, we provide accuracy analysis of the generalized
Prony method and the estimation results are formulated in
the rigid `∞ norm. In section IV, we do several numerical
stimulations to verify some estimation results. Finally, we
summarize the work in section V.
C. Notations
Let us introduce some relevant notations. Let M = (mij )
be a n × n matrix, the infinity norm of M, is defined by
n
X
||M ||∞ = max ( |mij |)
1≤i≤n
j=1
For a vector z = (zi ) ∈ Cn , we define the infinity norm
||zz ||∞ = max |zi |. It is easy to see that
i=1,··· ,n
||M ||∞ =
max
||Mzz ||∞
z ||∞ =1
z ∈Cn ,||z
We use z T and M T to denote their nonconjugate transpose.
II. N OISE - FREE RECOVERY
We consider the recovery of a frequently encountered case
in applications when the filter a ∈ `1 (Z) is a typical low pass
filter and â(ξ) is real, symmetric and strictly decreasing on
[0, 12 ]. The symmetry reflects the fact that there is often no
preferential direction for physical kernels and monotonicity is
a reflection of energy dissipation. Without loss of generality,
we also assume a is a normalized filter, i.e., |â(ξ)| ≤ 1, â(0) =
1. Let µ denote the Lebesgue measure on T, and X be a
subclass of `2 (Z) defined by
X = {x ∈ `2 (Z) : µ({ξ ∈ T : x̂(ξ) = 0}) = 0}
Clearly, X is a dense class of `2 (Z) under the norm topology.
In noise free scenario, our first result shows, provided our
initial state x ∈ X, that we can completely determine a via a
series of consecutive uniform undersampled states.
Theorem 1. Let x ∈ X be the initial signal and the evolution
operator A be a convolution operator given by a ∈ `1 (Z)
so that â(ξ) is real, symmetric, and strictly decreasing on
[0, 21 ]. Then a can be exactly recovered from measurements
y0 , · · · , y2m−1 defined in (1).
2m−1
Proof. We show that the regular subsampled data {yl }l=0
contains enough information to recover the Fourier spectrum
of a on T up to a measure zero set. It is easy to show there
exists a measurable set E with µ(E) = 1 such that for each
ξ
), · · · , x̂( ξ+m−1
)} is non-vanishing. For each
ξ ∈ E, {x̂( m
m
1
ξ ∈ E − {0, 2 }, define the Hankel matrix


ŷ0 (ξ) ŷ1 (ξ) ... ŷm−1 (ξ)
 ŷ1 (ξ) ŷ2 (ξ) ... ŷm (ξ) 

Hm (ξ) = 
(2)
..
..
..


.
.
···
.
ŷm−1 (ξ) ŷm (ξ) ... ŷ2m−2 (ξ)
and the vector b m (ξ) = (ŷm (ξ), · · · , ŷ2m−1 (ξ))T
Cm , we can show that there is a unique q (ξ)
(q0 (ξ), · · · , qm−1 (ξ))T ∈ Cm such that
Hm (ξ)qq (ξ) = −bbm (ξ)
∈
=
(3)
Form the Prony polynomial pξ [z] = z m + qm−1 (ξ)z m−1 +
· · · + q0 (ξ) , it turns out pξ [z] has m distinct roots {â( ξ+i
m ) :
i = 0, · · · , m − 1}. Due to symmetry and monotonicity
condition of â, we can find the roots of pξ [z] and order them
appropriately to get {â( ξ+i
m ) : i = 0, · · · , m − 1}. Therefore,
we can recover â(ξ) except for a measure zero set. The
conclusion follows by applying inverse Fourier transformation
on â.
Theorem 1 addresses the infinite dimensional analog of
Theorem 4.1 in [2]. Once a is recovered, we can recover
x using techniques developed in [8]. In general, for any
a ∈ `1 (Z), one can show the recovery of range of â on
the Torus except for a measure zero set only with minor
modifications of the above proof.
Definition 1. Let a = (a(n))n∈Z , the support set of a is
defined by Supp(a) = {k ∈ Z : a(k) 6= 0}. If Supp(a) is
a finite set, a is said to be a finite impulse response filter.
The term finite impulse response arises because the filter
output is computed as a weighted, finite term sum, of past,
present, and perhaps future values of the filter input. If a
is a finite impulse response filter with support contained
in {−r, −r + 1, · · · , r}, we can get the following results
immediately.
Corollary 1. In addition to the assumptions of Theorem 1, if
a is a finite impulse response filter supported in {−r, −r +
1, · · · , r}, then it is enough to recover {â(ηi ) : i = 1, · · · , r}
at r distinct locations via equation (3).
The proof of Theorem 1 doesn’t give a practical method in
general, since it involves computing the Fourier transformation
of infinite seuqences and solving the roots of uncountablely
many polynomials. However, if we know in prior Supp(a)
and Supp(x) are contained in {−r, −r + 1, · · · , r} for some
r ∈ N+ , then {yl }2m−1
consists of sequences supported in
l=0
{−2mr, · · · , 2mr}. We are able to compute {ŷl (ξ)}2m−1
for
l=0
any ξ ∈ T. In this case, the proof of Theorem 1 essentially
provides an algorithm for the recovery of the Fourier spectrum
of a. We summarize it as follows:
Algorithm II.1. Input: Choose ξ ∈ T − {0, 12 }
2m−1
1) From measurements {yl }2m−1
l=0 , compute {ŷl (ξ)}l=0 .
Form the Hankel matrix Hm (ξ) and b m (ξ) as in (2). Test
the rank of Hm (ξ), if rank(Hm (ξ)) = m, solve q (ξ) via
the following equation.
Hm (ξ)qq (ξ) = −bbm (ξ)
2) For q (ξ), form the Prony polynomial pξ [z] =
m−1
P
zm +
ql (ξ)z l . Find roots of pξ [z] and order
its lower degree coefficients. Hence, for a small perturbation,
although the roots of the perturbed polynomial p̃ξ may not be
real, we can order them according to their modulus and have
a one to one correspondence with the roots of pξ .
Definition 2. Let ξ ∈ T − {0, 12 }, consider the set {â( ξ+i
m ) :
i = 0, · · · , m − 1} that consists of m distinct nodes.
1) For 0 ≤ k ≤ m − 1, the “distance” between â( ξ+k
m ) with
other m − 1 nodes is measured by the quantity
l=0
δk (ξ) =
them to get {â( ξim+i ) : i = 0, · · · , m − 1}.
Q
j6=k
0≤j≤m−1
Output:{â( ξ+i
m )
: i = 0, · · · , m − 1}.
Remark 1. Note in this case, Hm (ξ) is not invertible only
at finitely many locations of T, Hm (ξ) is invertible with
probability 1.
Let {ξi : i = 1, · · · K} be a subset of T satisfying the
k
for k = 0, · · · , m − 1, and Km >
condition |ξi − ξj | =
6 m
ξ +i
r. Assume we have recovered {â( jm ) : i = 0, · · · , m −
1, j = 1, · · · , K} via Algorithm II.1, by Corollary 1, we can
completely determine a.
III. ACCURACY A NALYSIS
In previous sections, we show that if we are able to
compute the spectral data {ŷl (ξ)}2m−1
at ξ, we can recover the
l=0
Fourier spectrum {â( ξ+i
)
:
i
=
0,
·
·
·
,
m − 1} by Algorithm
m
II.1. However, a critical issue remains. We need to analyze
the accuracy of solution achieved by Algorithm II.1. The
motivation to study the accuracy comes from two aspects. On
one hand, for the case when one or both of a and x are not
compactly supported, although we only have access to a finite
section of each exact measurement yl ∈ `2 (Z) for practical
reasons , we may have a good approximation ỹl of yl , so that
||yb̃l (ξ) − ŷl (ξ)||∞ ≤ l 1. Consequently, we can employ
Algorithm II.1 to compute an approximation of the Fourier
spectrum of a. A natural question to ask is how large the
error will be between the approximate solutions and the actual
solutions. On the other hand, even for the case when both x
and a are compactly supported, what if we have noise in the
2m−1
process of computing {ŷl (ξ)}l=0
? We can summarize our
accuracy analysis problem in the following:
Assume the measurements are given by {ỹl }2m−1
compared
l=0
to (1) so that ||ŷl (ξ) − yb̃l (ξ)||∞ ≤ l for all ξ ∈ T. Given
an estimate = maxl |l |, how large can the error in the
reconstructed parameters in Step I and Step II of Algorithm
II.1 be in the worst case in terms of , and the true parameters.
Our accuracy analysis will consist of two steps. Suppose our
measurements are perturbed from {yl }2m−1
to {ỹl }2m−1
l=0
l=0 . For
any ξ, we first measure the perturbation of q (ξ) in terms of `∞
norm. This step is linear and standard. Then we measure the
perturbation of the roots. It is well known that the roots of a
polynomial are continuously dependent on the small change of
1
ξ+k
|â( ξ+j
m ) − â( m )|
2) For 0 ≤ k ≤ m, the kth elementary symmetric function
generated by the m nodes is denoted by

1
if k=0,
P
ξ+j2
ξ+jk
ξ+j1
σk (ξ) =
â( m )â( m ) · · · â( m ) otherwise.

0≤j1 <···<jk ≤m−1
(4)
For 0 ≤ k, i ≤ m − 1, the kth elementary symmetric
function generated by m − 1 nodes with â( ξ+i
m ) missing
(i)
is denoted by σk (ξ).
Proposition 1. Let the perturbed measurements {ỹl }2m−1
be
l=0
b̃
given with error satisfying ||yl (ξ) − yˆl (ξ)||∞ ≤ for all l. Let
H̃m (ξ) and b̃bm (ξ) be formed by {yb̃l (ξ)}2m−1
in the same way
l=0
as in (2). Assume Hm (ξ) is invertible and is sufficient small
so that H̃m (ξ) is also invertible. Denote by q̃q (ξ) the solution
of H̃m (ξ)q̃q (ξ) = −b̃bm (ξ). Form the Prony polynomial p̃ξ by
ˆ ξ+i ) : i = 0, · · · , m − 1} be its roots , then
q̃q (ξ) and let {ã(
m
we have the following estimates as → 0.
−1
||qq (ξ) − q̃q (ξ)||∞ < ||Hm
(ξ)||∞ (1 + mβ1 (ξ)) + O(2 ) (5)
where β1 (ξ) =
max |σk (ξ)|. As a result, we achieve the
k=1,··· ,m
following first order estimation
ξ+i
ξ+i
−1
)−â(
)| < Ci (ξ)(1+mβ1 (ξ))||Hm
(ξ)||∞ +O(2 )
m
m
(6)
m−1
P k ξ+i
where Ci (ξ) = δi (ξ) · (
|â ( m )|).
ˆ
|ã(
k=0
Therefore it is important to understand the relation between
−1
the behavior of ||Hm
(ξ)||∞ and our system parameters, i.e,
−1
m, a and x. Next, we are going to estimate ||Hm
(ξ)||∞ and
reveal their connection with the spectral properties of a and
x.
Theorem 2. Assume Hm (ξ) is invertible, then we have the
lower bound estimation
−1
||Hm
(ξ)||∞ ≥ m2
where β2 (i, ξ) =
estimation
max
max
i=0,··· ,m−1
k=0,··· ,m−1
β2 (i, ξ)δi (ξ)
|x̂( ξ+i
m )|
(7)
|σki (ξ)|, and the upper bound
Ac c ur ac y of s olut ion,log ǫ = −14
20
10
Q
(δi (ξ)
−1
||Hm
(ξ)||∞
2
<m
max
(1 +
j6=i
i=0,··· ,m−1
2
|â( ξ+j
m )|))
|x̂( ξ+i
m )|
15
10
10
10
(8)
Corollary 2. If |x̂(ξ)| ≤ M for every ξ ∈ T, then
−1
−1
||Hm
(ξ)||∞ → ∞ as m → ∞. In fact, ||Hm
(ξ)||∞ ≥
m−1
O(2
).
5
10
0
10
−5
10
−10
|∆ 0 ( ξ ) |
10
IV. NUMERICAL EXPERIMENT
|δ 0 ( ξ ) δ 2 ( ξ ) |
−15
10
4
In this section, we provide some simple numerical simulations to verify some theoretical accuracy estimations in section
III.
10
6
8
10
10
10
12
10
14
10
10
16
18
10
10
20
10
|δ 0 ( ξ ) δ 2 ( ξ ) |
(a) Dependence on the distances between nodes
Accuracy of solution
−4
10
A. Experiment Setup
−6
Suppose our filter a is supported in {−2, · · · , 2}. For
example, â(ξ) = 0.1 + 0.8cos(2πξ) + 0.1cos(4πξ), x is dirac
at the center so that x̂(ξ) = 1, and m = 3.
1) Choose ξ1 , ξ2 , · · · , ξd , and calculate ŷl (ξi ) and the perturbed yb̃l (ξi ) = ŷl (ξi ) + l for l = 0, · · · , 5, where yl is
defined as in (1) and l 1.
2) Use Algorithm II.1 to calculate the roots of pξi and
the perturbed roots of p̃ξ respectively, then compute
ˆ ξi +k ) − â( ξi +k )|.
|∆k (ξi )| = |ã(
m
m
10
−8
10
−10
10
−12
10
|∆ 1 ( ξ ) |
|∆ 2 ( ξ ) |
|∆ 0 ( ξ ) |
ǫ
1
ǫ2
−14
10
−16
10
−14
−13
10
−12
10
−11
10
−10
10
−9
10
10
|ǫ|
(b) Depdendence on the measurements error
B. Experiment Results
1) Sharpness of estimation (6) and (8). Fix m and x, our
estimation (6) and (8) suggest that the error |∆k (ξ)| =
ˆ ξ+k ) − â( ξ+k )| in the worst possible case could be
|ã(
m
m
proportional to δk (ξ)δ(ξ)2 . In this experiment, we choose
six points ξ1 < · · · < ξ6 < 12 such that δ0 (ξi )δ(ξi ) grows
geometrically at rate 103 , and set the noise level ∼
10−14 . In Figure 1(a), we plot the value of ∆0 (ξi ) =
ˆ ξi ) − â( ξi )| for i = 1, · · · , 6. We can see that ∆0 (ξi )
|ã(
3
3
grows propotionally to the gowth of δ0 (ξ)δ 2 (ξ), which
verified the sharpness of estimation(6) and (8).
2) Dependence of ∆k (ξ) on the measurement error .
In this experiment, we fix some 0 < ξ < 21 , we only
change the magnitude of the error = max l such
l=0,··· ,5
√
that grows geometrically at rate 10 and keep other
parameters unchanged. We plot the value of ∆k (ξ) =
ˆ ξ+k ) − â( ξ+k )| of different noise levels. The results
|ã(
3
3
are presented in Figure 1(b), we show that |∆k (ξ)| ∼
O() for k = 0, 1, 2.
−1
3) The infinity norm of Hm
(ξ). In this experiment, we
choose m = 2, 3, · · · , 6 and ξ = 0.3. We compute and
−1
plot the value of ||Hm
(ξ)||∞ for different m. The results
−1
are presented in Figure 1(c). It is shown that ||Hm
(ξ)||∞
grows geometrically.
V. C ONCLUSION
In this paper, we investigate the conditions under which we
can recover a typical low pass convolution filter a ∈ `1 (Z) and
a vector x ∈ `2 (Z) from the combined regular subsampled
version of the vector x, · · · , AN −1 x defined in (1), where
4
10
3
10
2
10
1
10
−1
||H m
( ξ ) || ∞
0
10
2
2.5
3
3.5
4
4.5
5
5.5
6
m
(c) The infinity norm of
−1
Hm
(ξ)
Fig. 1. Experiment Results
Ax = a ∗ x. A generalized Prony method is proposed to show
that {x|Ωm , x(1) |Ωm , · · ·, x(N ) |Ωm : N ≥ 2m − 1, Ωm = mZ}
contains enough information to recover a almost surely. Our
accuracy estimates are formulated in very simple geometric
terms involving Fourier spectral function of a, x and m,
shedding some light on the structure of the problem. Our
results suggest that when the generalized Prony method is
used, the parameters of the problem are coupled to each
other, in the sense that the accuracy of recovering the nodes
{â( ξ+i
m ) : i = 0, · · · , m − 1} depends on the values of all
the parameters at once. This unfavorable behavior is reflected
by our numerical experiments in section IV. For example, the
accuracy of recovering the node â( ξ+k
m ) not only depends on
its separation with other nodes δk (ξ)(see Defintion 2), but also
depends on the minimum separation δ(ξ) =
max
k=0,··· ,m−1
δk (ξ)
among the nodes. The classical Prony method performs poorly
when noisy sampled data are given. In our case, we have similar issues, since our Hankel matrix Hm (ξ) is ill conditioned.
In practice, we can employ denoising techniques to process
sampled data such as Cadzow denoising algorithm to make the
method more robust to noise. However, we believe that a full
answer to our somewhat rigid `∞ formulation of the accuracy
problem may contribute to the understanding of limitations of
using Prony type methods in spatiotemporal sampling.
VI. FUTURE WORK
Finding stable solution of Prony-type systems is generally
considered to be a difficult problem, and in recent years many
algorithms have been devised for this task, such as nonlinear
least squares, ESPRIT. We believe these algorithms can be
adapted to our case and make our solution become more robust
to noise, which we posit as our future work.
ACKNOWLEDGMENT
The author would like to thank Akram Aldroubi for his
endless encouragements and insightful comments in the process of creating this work. The author also thanks Llya Kristal
and Yang Wang for their helpful discussions related to this
work. The author is indebted to anonymous referees for their
useful comments to improve the quality of the presentation of
this paper. This research was supported in part by NSF Grant
DMS-1322099.
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