Nonlinear and Nonideal Sampling: Theory and Methods

Abstract

We study a sampling setup where a continuous-time signal is mapped by a memoryless, invertible and nonlinear transformation, and then sampled in a nonideal manner. Such scenarios appear, for example, in acquisition systems where a sensor introduces static nonlinearity, before the signal is sampled by a practical analog-to-digital converter. We develop the theory and a concrete algorithm to perfectly recover a signal within a subspace, from its nonlinear and nonideal samples. Three alternative formulations of the algorithm are described that provide different insights into the structure of the solution: A series of oblique projections, approximated projections onto convex sets, and quasi-Newton iterations. Using classical analysis techniques of descent-based methods, and recent results on frame perturbation theory, we prove convergence of our algorithm to the true input signal. We demonstrate our method by simulations, and explain the applicability of our theory to Wiener-Hammerstein analog-to-digital hybrid systems.

Full text

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5874IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 12, DECEMBER 2008
Nonlinear and Nonideal Sampling:
Theory and Methods
Tsvi G. Dvorkind, Yonina C. Eldar, Senior Member, IEEE, and Ewa Matusiak
Abstract—We study a sampling setup where a continuous-time
signal is mapped by a memoryless, invertible and nonlinear
transformation, and then sampled in a nonideal manner. Such
scenarios appear, for example, in acquisition systems where a
sensor introduces static nonlinearity, before the signal is sampled
by a practical analog-to-digital converter. We develop the theory
and a concrete algorithm to perfectly recover a signal within a
subspace, from its nonlinear and nonideal samples. Three alter-
native formulations of the algorithm are described that provide
different insights into the structure of the solution: A series of
oblique projections, approximated projections onto convex sets,
and quasi-Newton iterations. Using classical analysis techniques
of descent-based methods, and recent results on frame perturba-
tion theory, we prove convergence of our algorithm to the true
input signal. We demonstrate our method by simulations, and
explain the applicability of our theory to Wiener–Hammerstein
analog-to-digital hybrid systems.
Index Terms—Generalized sampling, interpolation, nonlinear
sampling, Wiener–Hammerstein.
I. INTRODUCTION
D
sets of numbers, which are related to continuous-time signals
through an acquisition process. One major goal, which is at the
heart of digital signal processing, is the ability to reconstruct
continuous-time functions, by properly processing their avail-
able samples.
In this paper, we consider the problem of reconstructing a
function from its nonideal samples, which are obtained after the
signalwasdistortedbyamemoryless(i.e.,static),nonlinear,and
invertible mapping.
The main interest in this setup stems from scenarios where
an acquisition device introduces a nonlinear distortion of
amplitudes to its input signal, before sampling by a practical
IGITAL signal processing applications are often con-
cerned with the ability to store and process discrete
ManuscriptreceivedFebruary26,2008;revisedJuly05,2008.Firstpublished
August 29, 2008; current version published November 19, 2008. The associate
editor coordinating the review of this manuscript and approving it for publica-
tion was Dr. Chong-Meng Samson See. This work was supported in part by the
Israel Science Foundation under Grant 1081/07 and by the European Commis-
sion in the framework of the FP7 Network of Excellence in Wireless COMmu-
nications NEWCOM++ (Contract 216715).
T. G. Dvorkind is with the Rafael Company, Haifa 2250, Israel (e-mail:
dvorkind@gmail.com).
Y. C. Eldar is with the Department of Electrical Engineering, Technion—Is-
rael Institute of Technology, Haifa 32000, Israel (e-mail: yonina@ee.tech-
nion.ac.il).
E. Matusiak is with the Faculty of Mathematics, University of Vienna, 1090
Wien, Austria (e-mail: ewa.matusiak@univie.ac.at).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2008.929872
Fig. 1. (a) Sampling setup. (b) Illustration of the memoryless nonlinear map-
ping.
analog-to-digital converter (ADC) [see Fig. 1(a)]. Nonlinear
distortions appear in a variety of setups and applications of
digital signal processing. For example, charge-coupled device
(CCD) image sensors introduce nonlinear distortions when
excessive light intensity causes saturation [1], [2]. Memoryless
nonlinear distortions also appear in the areas of power elec-
tronics [3] and radiometric photography [4], [5]. In some cases,
nonlinearity is introduced deliberately in order to increase the
possible dynamic range of the signal while avoiding amplitude
clipping, or damage to the ADC [6]. The goal is then to process
the samples in order to recover the original continuous-time
function.
Theusualassumptioninsuchproblemsisthatthesamplesare
ideal, i.e., they are pointwise evaluations of the nonlinearly dis-
torted, continuous-time signal. Even then, the problem may ap-
pear to be hard. For example, nonlinearly distorting a band-lim-
itedsignal,usuallyincreasesitsbandwidth.Thus,itmightnotbe
obvious how to adjust the sampling rate after the nonlinearity.
In [7], for instance, the author seeks sampling rates to recon-
struct a band-pass signal, which is transformed by a nonlinear
distortionoforderatmostthree.However,asnoticedbyZhu[8],
oversampling in such circumstances is unnecessary. Assuming
the band-limited setting and ideal sampling, Zhu showed that
perfect reconstruction of the input signal can be obtained, even
if the distorted function is sampled at the Nyquist rate of the
input. The key idea is to apply the inverse of the memoryless
nonlinearity to the given ideal samples, resulting in ideal sam-
ples of the band-limited input signal. Recovery is then straight-
forward by applying Shannon’s interpolation. Unfortunately, in
practice, ideal sampling is impossible to implement. A more ac-
curate model considers generalized sampling [9]–[12]. Instead
of pointwise evaluations of the continuous-time function, the
samplesaremodeledbyasetofinnerproductsbetweenthecon-
tinuous-timesignal and thesamplingfunctions. Thesesampling
functionsare relatedto thelinear partof theacquisition process,
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DVORKIND et al.: NONLINEAR AND NONIDEAL SAMPLING: THEORY AND METHODS5875
forexampletheycandescribetheantialiasingfilteringeffectsof
an ADC [9].
In the context of generalized sampling, considerable effort
has been devoted to the study of purely linear setups, where
nonlinear distortions are absent from the description of the ac-
quisition model. The usual scenario is to reconstruct a function
by processing its generalized samples, obtained by a linear and
bounded acquisition device. Assuming a shift-invariant setup,
the authors of [9] introduce the concept of consistent recon-
struction in which a signal is recovered within the reconstruc-
tion space, such that when re-injected into the linear acquisition
device, the original sample sequence is reproduced. The idea
of consistency was then extended to arbitrary Hilbert spaces in
[10],[13],and[14].Insomesetups,theconsistencyrequirement
leads to large error of approximation. Instead, robust approx-
imations were developed in [12], in which, the reconstructed
function is optimized to minimize the so called minimax regret
criterion, related directly to the squared-norm of the reconstruc-
tion error. This approach guarantees a bounded approximation
error, however, there as well, the acquisition model is linear.
In this paper, we consider a nonlinear sampling setup, com-
bined with generalized sampling. The continuous-time signal is
first nonlinearly distorted and then sampled in a nonideal (gen-
eralized) manner. We assume that the nonlinear and nonideal
acquisition deviceisknowninadvance,and thatthesamplesare
noise free. In this general context, we develop the theory to en-
sure perfect reconstruction, and an iterative algorithm, which is
proved to recover the input signal from its nonlinear and gener-
alizedsamples.Thetheorywedevelopleadstosimplesufficient
conditions on the nonlinear distortion and the spaces involved,
that ensure perfect recovery of the input signal. If the signal is
not constrained to a subspace, then the problem of perfect re-
covery becomes ill-posed (see Section III) as there are infin-
itely many functions which can explain the samples. Therefore,
ourmaineffortconcernsthepracticalproblemofreconstructing
a function within some predefined subspace. For example, the
problem may be to reconstruct a band-limited function, though,
in this work we are not restricted to the band-limited setup.
Threealternativeformulationsofthealgorithmaredeveloped
thatprovidedifferentinsightintothestructureofthesolution:A
seriesofobliqueprojections[15],[16],anapproximatedprojec-
tions onto convex sets (POCS) method [17], and quasi-Newton
iterations [18], [19]. Under some conditions, we show that all
three viewpoints are equivalent, and from each formulation we
extract interesting insights into the problem.
Our approach relies on linearization, where at each iteration
we solve a linear approximation of the original nonlinear sam-
pling problem. To prove convergence of our algorithm we ex-
tend some recent results concerning frame perturbation theory
[20].Wealsoapplyclassicalanalysistechniqueswhichareused
to prove convergence of descent-based methods.
After stating the notations and the mathematical prelim-
inary assumptions in Section II, we formulate our problem
in Section III. In Section IV, we prove that under proper
conditions, perfect reconstruction of the input signal from its
nonlinear and generalized samples is possible. In Section V,
we suggest a specific iterative algorithm. The recovery method
relies on linearization of the underlying nonlinear problem,
and takes on the form of a series of oblique projections. In
Section VI, we develop a reconstruction based on the POCS
method and show it to be equivalent to the iterativeoblique-pro-
jections algorithm. In Section VII, we view the linearization
approach within the framework of frame perturbation theory.
This viewpoint leads to conditions on the nonlinear mapping
and the spaces involved, which ensure perfect recovery of
the input. In Section VIII, we formulate our algorithm as
quasi-Newton iterations, proving convergence of our method.
Some practical aspects are discussed in Section IX. Specifi-
cally, we explain how the algorithm should be altered, if some
of the mathematical preliminary assumptions do not hold in
practice. We also show how to apply our results to acquisition
devices that are modeled by a Wiener–Hammerstein system
[21]. Simulation results are provided in Section X. Finally,
in Section XI, we conclude and suggest future directions of
research. Some of the mathematical derivations are provided
within the appendixes.
II. NOTATIONS AND MATHEMATICAL PRELIMINARIES
We denote continuous-time signals by bold lowercase letters,
omittingthetimedependence,whenpossible.Theelementsofa
sequence
will be written with square brackets, e.g.,
Operators are denoted by upper case letters. The operator
represents the orthogonal projection onto a closed subspace
and
is the orthogonal complement of
an oblique projection operator [15], [16], with range space
and null space. The identity mapping is denoted by . The
range and null spaces are denoted by
tively. Inner products and norms are denoted by
, withbeing the Hilbert space involved. The norm
of a linear operatoris its spectral norm.
The Moore–Penrose pseudoinverse [22] and the adjoint of a
bounded transformation
are written as
tively. If
is a linear bounded operator with
closed range space, then the Moore–Penrose pseudoinverse
exists [22, pp. 321]. If in addition,
then.Therefore,foralinearandboundedbi-
jection
, and linear, bounded
exists.
An easy way to describe linear combinations and inner prod-
ucts is by utilizing set transformations. A set transformation
correspondingtoframe[23]vectors
by
for all
adjoint, if
, then
between two closed subspaces
is the sum set
the property
. For an operator
, we denote bythe set obtained by applying
vectors in
.
We denote by
less (i.e., static) mapping, which maps an input function
an output signal
. Being static, there is a functional
describingtheinput–outputrelationateachtimein-
stance , such that
The derivative of
is denoted by
.
,
. stands for
and, respec-
and
and, respec-
is a linear bijection on
with closed range,
isdefined
. From the definition of the
. A direct sum
andof a Hilbert space
with
and a subspace
to all
a nonlinear memory-
to
[see Fig. 1(b)].
. We will also use
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5876IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 12, DECEMBER 2008
the Fréchet derivative [24] of
an operator in
Definition 1: Anoperator
tiable at
, if there is a linear operator
that in a neighborhood of
to describein terms of
.
is Fréchet differen-
, such
(1)
We refer to
tive) of
operator is the operator itself.
In our setup,
is completely determined in terms of the composition
, i.e.,
setting,
, the Fréchet derivative of
, for any function , and for all
time instances . For example, the Fréchet derivative of the
memoryless operator
termined by the functional
, and
as the Fréchet derivative (or simply the deriva-
. Note that the Fréchet derivative of a linearat
is memoryless, so that the image
. In such a
satisfies at
evaluated at is de-
.
A. Mathematical Safeguards
OurmaintreatmentconcernstheHilbertspaceofreal-valued,
finite-energy functions. Throughout the paper we assume that
the sampling functions
form a frame [23] for the closure
of their span, which we denote by the sampling space
Thus, there are constants
.
such that
(2)
for all
, whereis the set transform corresponding to
.
To assure that the inner products
fined, we assume that the distorted function
for all. The latter requirement is satisfied if, for ex-
ample,
and is Lipschitzcontinuous.Indeed,in this
case
are well de-
is in
(3)
where
is a Lipschitz bound of
has finite energy, is when
and the input function
Lipschitz continuity of the nonlinear mapping can be guar-
anteed by requiring that
Throughout our derivations we also assume that
ible. In particular, this holds if the input–output distortion
curve
is a strictly ascending function,1i.e.,
. Another case in which
is Lipschitz continuous
has finite support.
.
is invert-
.
1The results of this work can be extended for the strictly descending case as
well; see Section IX.
In summary, our hypothesis is that the slope of the nonlinear
distortion satisfies
(4)
for some
To reconstruct functions within some closed subspace
, let to be a Riesz basis [23] of
sponding set transformation
, , and all.
of
. Then the corre-
satisfies
(5)
for some fixed
, and all.
III. PROBLEM FORMULATION
Our problem is to reconstruct a continuous-time signal
from samples of, which is obtained by a nonlinear, memo-
ryless and invertible mapping
of, i.e.,
ThisnonlineardistortionofamplitudesisillustratedinFig.1(b),
where a functional
describes the input–output relation of the
nonlinearity.Ourmeasurementsaremodelledasthegeneralized
samples of
, with the th sample given by
(6)
Here,
be the sampling space, which is the closure of
to be the set transformation corresponding to
notation, the generalized samples (6) can be written as
is theth sampling function. We defineto
and
. With this
(7)
By the Riesz representation theorem, the sampling model (7)
can describe any linear and bounded sampling scheme.
Animportantspecialcaseofsampling,iswhen
variant (SI) subspace, obtained by equidistant shifts of a gener-
ator
isashift-in-
AnexampleisanADCwhichperformsprefilteringpriortosam-
pling, as shown in Fig. 2. In such a setting, the sampling vectors
are shifted and mirrored versions of the
prefilter impulse response [9]. Furthermore, the sampling func-
tions form a frame for the closure of their span [23], [25] if and
only if
for some. Here we denote,
(8)
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DVORKIND et al.: NONLINEAR AND NONIDEAL SAMPLING: THEORY AND METHODS5877
Fig. 2. Filtering with impulse response ????? followed by ideal sampling. The
sampling vectors are ???? ? ????.
where
generator
is the continuous-time Fourier transform of the
, andis the set of frequencies
.
If there are no constraints on
constructing this input from the samples
Specifically, there are infinitely many functions of
can yield the known samples. Indeed, any signal of the form
is a possible candidate, where
vector in
and
for which
, then the problem of re-
becomes ill-posed.
which
is an arbitrary
(9)
istheorthogonalprojectionof
is uniquely determined by the samples . However, in many
practical problems we assume some subspace structure on the
input signal. The assumption of a signal being band-limited is
probably the most common scenario, though, in this work, we
are not limited to the band-limited setup. Our formulation treats
the problem of reconstructing
bitrary closed subspace
we consider is illustrated in Fig. 1.
To recover
from its samples, we need to determine a bi-
jection between this function and the sample sequence. Unfor-
tunately, though we restrict the solution to a closed linear sub-
space
of , it is still possible to have infinitely many func-
tionsin
whichcan explainthesamples. Forexample,evenfor
a simplified setup of our problem where
identity mapping, there are infinitely many consistent solutions
if
but
we have
the direct sum condition
ontothesamplingspace,which
which is known to lie in an ar-
. The overall sampling scheme
is replaced by the
.Indeed,forany
. If, however,andsatisfy
(10)
thenit is wellknown[10],[14] thatfor
tent solution exists. In that case, the consistent solution is also
the perfect reconstruction of the true input
and (10) holds, then the problem is trivial.
In our setup, however,
a linear operator. Instead of ignoring the effects of the nonlin-
earity, yet another simplification of the problem might be to
assume that the samples are ideal, i.e.,
case, we can resort to Zhu’s sampling theorem [8], by applying
to the samples . Presuming that indeed
by this approach we then obtain the ideal samples
The problem then reduces to that of recovering a signal
subspace
from its ideal samples, which has been treated ex-
tensively in the sampling literature (see, for example, [26]).
As mentioned, however, in our setup the signal is distorted
by a nonlinear mapping
, and generalized (rather than ideal)
sampling takes place. Hence, approaches which ignore the non-
linearity or the nonideal nature of the sampling scheme are sub-
, a uniqueconsis-
. Thus, if
. Furthermore,is not even
. In that
,
of .
in a
optimal, and in general, will not lead to perfect recovery of the
input .InSectionX,wedemonstratethatbyapplying
rectly to the samples and then recovering
reconstruction performance. Nonetheless, we will show that if
(10) is satisfied, and under proper assumptions on the nonlinear
distortion
, then there is a unique function
canexplainthemeasuredsamples.Buildingonthisresultwede-
velop the theory and a concrete iterative method for obtaining
perfect reconstruction of a signal in a subspace, despite the fact
that it is measured through a nonlinear and nonideal acquisition
device.
Beforetreatingthisgeneralcase,wenotethattherearespecial
setups for which it is possible to reconstruct the function
a closed form. An example of such a setup is presented in the
following theorem.
Theorem 1: Let
be a periodic function with period
that satisfies the Dirichlet conditions. Let
schitz continuous, memoryless and invertible mapping. If
is sampled with the sampling functions
di-
leads to suboptimal
within, which
in
be a Lip-
(11)
and the generator
satisfies
(12)
for all
eralized samples (6). The reconstruction is given by
, thencan be reconstructed from the gen-
(13)
where
istheconvolutionalinverseof
is the continuous-time Fourier transform of .
Proof: See Appendix I.
In Theorem 1, a special choice of sampling functions is em-
ployed in order to reconstruct a periodic function. In the gen-
eral case, however, the sampling functions do not satisfy (11).
Hence,therestofthispaperwilltreatamuchbroadersetting,al-
lowing the use of arbitrary sampling and reconstruction spaces.
In particular, the more standard setup of SI spaces is included in
our framework.
,and
IV. UNIQUENESS
In this section, we prove that under proper conditions on
and thespaces involved,theproblemofperfectlyreconstructing
the input signal from its nonlinear and generalized samples
indeed has a unique solution. Specifically, we show that if
the subspace
(i.e., the space obtained by applying the
Fréchet derivative
to each vector in
sum
for all
determined by its samples.
Theorem 2: Assume
Then there is a unique
such that
Proof: Assumethat there are two functions
which both satisfy
) satisfies the direct
, thenis uniquely
for all.
.
,
. Then
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5878IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 12, DECEMBER 2008
implying that
For each time instance , we have some interval of ampli-
tudes
, where we have assumed, without loss of
generality, that
. Since by (4) the nonlinear dis-
tortion curve
is continuous and differentiable, by the mean
value theorem, there is a scalar
value
.
and an intermediate
such that
(14)
Defining a function
using operator notations
, we may rewrite (14) for all
(15)
The resulting function
and by the right-hand side of (15) it is also a function
in
. By the direct sum assumption
must have
bijection),
.
In the sequel we will state simple conditions on
spaces
and, which assure that the direct sum assumptions
of Theorem 2 are met in practice. In particular, we will show
that if
is smooth enough, then (10) is sufficient to ensure
uniqueness.
lies within the subspace
, we
is a, or equivalently (since
and the
V. RESTORATION VIA LINEARIZATION
UndertheconditionsofTheorem2,thereisa uniquefunction
which is consistent with the measured samples . There-
fore, all we need is to find
A natural approach to retrieve
nonlinear equations. As this method relies on linearization, we
first need to explain in detail, how to recover the input
it is related to
through a linear mapping.
satisfying
is to iteratively linearize these
.
if
A. Perfect Reconstruction for Linear Schemes
In this section we adopt a general formulation within some
Hilbert space
(which is not necessarily
where
is related tothrough a linear, bounded and bijective
mapping
, i.e.,
). Assume a model
The case
[10], [13], [14]. Assuming that the direct sum condition (10) is
satisfied, it is known that
from thesamples of , byoblique projecting
onto the reconstruction space
was previously addressed in the literature [9],
can be perfectly reconstructed
of (9) along
:
(16)
The extension of this result to any linear continuous and con-
tinuously invertible
(not necessarily
note that since the solution
lies within
constrained to the subspace
more, in our context
will play the role of the operator
) is simple. First
, the function
. Further-
is
of
Fig. 3. Perfect reconstruction with linear invertible mapping.
Theorem 2, and we will derive conditions to assure that the di-
rect sum
holds. Then, perfect reconstruction
of any
is given by
(17)
Also note that due to the direct sum assumption
, the oblique projection operator
(e.g., [25]). Finally, to obtain
This simple idea is illustrated in Fig. 3 and summarized in the
following theorem.
Theorem 3: Let
a linear, continuous and bijective mapping satisfying
. Then we can reconstruct
by
is well defined
itself, we apply to (17).
and letwith
from the samples
(18)
where
for
In (18) we have used
special case
solution [9], [13], [14] of (16).
is a set transformation corresponding to a Riesz basis
.
. For the
we obtain the expected oblique projection
B. Iterating Oblique Projections
We now use the results on the linear case to develop an iter-
ative recovery algorithm in the presence of nonlinearities. The
true input
is a function consistent with the measured samples
, i.e., it satisfies
(19)
To recover
initial guess
linear operator
we may first linearize (19) by starting with some
and approximating the memoryless non-
using its Fréchet derivative at
(20)
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DVORKIND et al.: NONLINEAR AND NONIDEAL SAMPLING: THEORY AND METHODS5879
Fig. 4. Schematic of the iterative algorithm.
whereforbrevitywedenoted
(20) yields
.Rewriting(19)using
(21)
The left-hand side of (21) describes a vector
withinthesubspace
ator
. The right hand is the resulting sample sequence. Since
by (4) the linear derivative operator
on
, we may apply the result of Theorem 3 as long as the di-
rect sum condition
identifying
withinTheorem3,theuniquesolutionof(21)
is
which lies
andissampledbytheanalysisoper-
is a bounded bijection
is satisfied. Specifically,
(22)
where
repeated, using
Assuming that for each iteration
, we may summarize the basic form of the algorithm by
such that. The process can now be
as the new approximation point.
we have
(23)
where we used
equality. This leads to the following interpretation of the algo-
rithm:
• Using the current approximation
, the error within the sampling space.
• Solve a linear problem of finding a function within the re-
construction space
, consistent with
The solution is
• Update the current estimate
tion term.
This idea is described schematically in Fig. 4.
Finally, note that in practice, we need only to update the rep-
resentation coefficients of
within
, where the set transformation
Riesz basis for
, andare the coefficients. This results
in a discrete version of the algorithm (23):
in the last
, calculate
when.
.
using the resulting correc-
. Thus, we may write
corresponds to a
(24)
where we used
, and
. Note that the pseudoinverse is well defined due to
the direct sum assumption
[25].
VI. THE POCS POINT OF VIEW
Adifferentapproachfortacklingourproblemcanbeobtained
by the POCS algorithm [17]. In this section we will show the
equivalencebetweenthePOCSmethodandtheiterativeoblique
projections (23).
Firstnotethattheunknowninputsignalliesintheintersection
of two sets: The subspace
and the set
(25)
of all functions which can yield the known samples. For non-
linear
, the set of (25) is in general nonconvex. POCS
methods are successfully used even in problems where we it-
erate projections between a convex set and a nonconvex one,
assuming that it is known how to compute the projections onto
the sets involved (e.g., [27] and [28]). Unfortunately, in our
problem, it is hard to compute the orthogonal projection onto
. However, we can approximate
convex) subset. Replacing the operator
around some estimate
, i.e.,
allows us to locally approximate
using an affine (and hence
with its linearization
,
by the set
(26)
where we define
(27)
Note that when
contains a residual term due to approximating
derivative.
We point out that the set
, but since by (4)
. In addition, since
rose pseudoinverse of
Therefore, we may rewrite
is linear, . For nonlinear,also
by its Fréchet
is never empty; indeed,
is bijective, then
is also bounded, the Moore-Pen-
is well defined (see Section II).
as the affine subset
(28)
Given a vector , its projection onto
is given by
(29)
Using (29), we now apply the POCS algorithm to this approx-
imated problem of finding an element within the intersection
.
Startingwithsomeinitialization
the form
,thePOCSiterationstake
(30)
where
the intersection
the iterations (30) are known to converge [17] to some element
within
.Onceconvergencehasbeenobtained,theprocess
is held fixed, andis the iteration index. As long as
is non empty, for any initialization,
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5880IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 12, DECEMBER 2008
canberepeatedbysettinganewlinearizationpoint
and defining a new approximation set,
Combining (30) with (29) we have
. Continuing this expansion, while expressing
the result in terms of the initial guess
, similarly to (28).
, leads to
Substituting, in the limit we have
(31)
Interestingly, the infinite sum (31), can be significantly sim-
plified if again we assume that the direct sum conditions
(32)
are satisfied for all . In fact, the POCS method becomes equiv-
alent to the approach presented in the previous section.
Theorem 4: Assume
Then iterations (31) are equivalent to (23).
Proof: See Appendix II.
holds for all.
VII. LINEARIZATION AS FRAME PERTURBATION THEORY
We have seen that under the direct sum conditions (32)
both the iterative oblique projections and the approximated
POCS method take on the form (23). Furthermore, as stated in
Theorem 2, such direct sum conditions are also vital to prove
uniqueness of the solution. Ensuring that for any linearization
point
,thedirectsum
section we derive sufficient (and simple to verify) conditions
on the nonlinear distortion
, which assure this condition.
The key idea is to view the linearization process, which led to
the modified subspace
, as a perturbation of the original
space
. If the perturbation is ‘small enough’, and
, then we can prove (32). Before proceeding with the mathe-
maticalderivations,itisbeneficialtogeometricallyinterpretthis
ideafor
,asillustratedinFig.5.Aslongas
holds (here, the line defined by the subspace
dicular to
) and is sufficiently close to
guarantee that
and the perturbed subspace
As shown in Fig. 5, the concept of an angle between spaces
is simple for
. The natural extension of this idea in
arbitrary Hilbert spaces is given by the definition of the cosine
and sine between two subspaces
[9], [16]:
holdsis nottrivial.Inthis
is not perpen-
, we can also
, since the angle between
is smaller than 90 .
,of some Hilbert space
(33)
Fig. 5. Subspace ? and its perturbation. As long as the sum of maximal angles
between ? and ?, and ? with its perturbation ? ??? (i.e., ? and ? , respec-
tively) is smaller than 90 , the direct sum (32) is satisfied.
Throughout this section we will also use the relations [16]
(34)
We start by showing that
(in terms of Fig. 5, the angle
stating a sufficient condition on the nonlinear distortion which
guarantees that
is ‘sufficiently close’ to
is smaller than 90 degrees), by
(35)
along the iterations. We then use (35), to derive sufficient con-
ditions for the direct sum
of Fig. 5, the latter means that
degrees.
To show (35),we rely onrecent results concerning frame per-
turbations on a subspace, and extend these to our setup.
Proposition1: If
.
Proof: The direct sum condition
equivalent to
[16]. The requirement
by relying on the following lemma.
Lemma 1 [20, Theorem 2.1]: Let
, with frame bounds. For a sequence
, and assume that there exists a constant
such that
to hold. In terms
is also smaller than 90
,then
is
and
can be guaranteed
be a frame for
in, let
(36)
for all finite scalar sequences
a)
is a Bessel sequence2with bound
;
b) if , then
. Then the following holds:
2?? ? is a Bessel sequence if the right-hand side inequality of (2) is satisfied.
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DVORKIND et al.: NONLINEAR AND NONIDEAL SAMPLING: THEORY AND METHODS5881
If, in addition to b),
c)
, then
is a frame forwith bounds
;
,
d)
Substituting
where we take
transformation
notations as
is isomorphic to.
for,for and,
to be an orthonormal basis of
, we can rewrite condition (36) using operator
with set
(37)
Now
where we used the orthonormality of
Thus, (37) holds with
this means that as long as
in the last equality.
. By Lemma 1, part b),
(38)
then
(39)
To avoid expressions which require the computation of
, we can further lower bound (39) by
(40)
To also establish
Lemma 1, by exploiting the special relation between
(i.e., connected by an invertible, bounded and self
adjoint mapping).
Lemma2: Assume
invertible and self adjoint mapping. Then
Proof: Let
for some
. Taketo be a set transformation of a Riesz basis
of
, with Riesz bounds
we now extend
and
withabounded,
.
, such that
as in (5). Since
it is suffi-
cient to show that
(41)
Exploiting the fact that
write
is bijective and self adjoint, we may
Since andtheoperator
have
, and some
isbounded,wemust
in
that
for every unit norm
strictly positive . Thus, the left-hand side of (41) can be lower
bounded by
where we used
equality.
Having established that
in the last in-
and
, it follows from [16] that
, completing the proof.
We now show that if
are nonlinear distortions for which (32) holds.
Theorem 5: Assume
is ‘sufficiently close’ tothen there
. If
(42)
then
.
Before stating the proof, note that (42) also implies
, which by Proposition 1 ensures
. We also note that since the direct
sum condition
guarantees that
the norm bound is meaningful as the right-hand side of (42) is
positive. In the special case
(42) becomes
requirement of Proposition 1.
Proof: See Appendix III.
We have seen that the initial direct sum condition
and the curvature bound (42) are sufficient to ensure that the
direct sum condition
by Theorem 2, there is also a unique solution to our nonlinear
sampling problem.
As a final remark, note that by relating the curvature bounds
(4) with (42) yields
,
we have
which is the less restrictive
and
is satisfied. Consequently,
(43)
and
(44)
This means that a sufficient condition for our theory to hold,
is to have an ascending nonlinear distortion, with a slope no
larger than two. In Section IX, we suggest some extensions of
our algorithm to the case in which these conditions are violated.
VIII. CONVERGENCE: THE NEWTON APPROACH
InSectionsVandVI,wesawthatunderthedirectsumcondi-
tions
the iterative oblique projections and
the approximated POCS method take the form (23). We now
establish that with a small modification, algorithm (23) is guar-
anteed to converge. Furthermore, it will converge to the input
signal
.
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5882IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 12, DECEMBER 2008
To this end, we interpret the sequence version of our algo-
rithm (24), as a quasi-Newton method [18], [19], aimed to min-
imize the consistency cost function
(45)
where
(46)
is the error in the samples with a given choice of
that by expressing the function
tation coefficients, we obtained an unconstrained optimization
problem.
Thetrueinput
hasrepresentationcoefficients ,forwhich
attains theglobalminimumofzero.SincebyTheorem2thereis
only one such function,
of (45) has a unique global minimum.
Unfortunately, since
is nonlinear,
and nonconvex. Obviously, without some knowledge about the
global structure of the merit function
mization methods cannot guarantee to trap the global minimum
of
. They can, however, find a stationary point of
vector
where the gradient
We now establish a key result, showing that if the direct sum
conditions are satisfied, then a stationary point of
be the global minimum.
Theorem 6: Assume
Then a stationary point of
is also its global minimum.
Proof:
Assume that
point of
. Then, the gradient of
. Note
in terms of its represen-
is in general nonlinear
(e.g., convexity), opti-
, i.e., a
is zero.
must also
for all.
is a stationary
at
. Denoting
, we can also rewrite
. Assume to the con-
is not the zero vector.
is
and using definition (46) of
trary that
Then, since
, also
. Byis not the zero function within
the direct sum
, we must have that
is not the zero vector, contradicting the
fact that a stationary point has been reached.
Note that the combination of Theorems 2 and 6 implies that
whenthedirectsumconditions
optimization methods which are able to trap stationary points of
, also retrieve the true input signal
we have obtained simple sufficient conditions for these direct
sums to hold. This leads to the following corollary.
Corollary1: Assumethattheslopeofthenonlineardistortion
satisfies
aresatisfied,
. Also, in Theorem 5
(47)
and that
a stationary point
. Then any algorithm which can trap
in (45) also recovers the true inputof
.
We now interpret (24) as a quasi-Newton method aimed to
minimize the cost function
of (45). For descent methods in
general, and quasi-Newton specifically, we iterate
(48)
with
tively. To minimize , the search direction is set to be a descent
direction:
andbeing the step size and search direction, respec-
(49)
where,
. The step size
[18] (also known as the Armijo and curvature conditions) by
using the following simple backtracking procedure:
for some positive-definite matrix3
is chosen to satisfy the Wolfe conditions
Set
Repeat until
by setting(50)
Itiseasytoseethatthesequenceversionofouralgorithm(24)
is in the form (48) with
This search direction is gradient related:
and .
(51)
Thelastequalityfollowsfromthefactthatwhen
is satisfied then [14]
.
Using this Newton point of view, we now suggest a slightly
modified version of our algorithm, which converges to coeffi-
cients
of the true input
Theorem 7: The algorithm of Table I will converge to coeffi-
cients
of the true input, if
1)
and the derivative
, and by (46),
.
satisfies the bound
;
2)
Before stating the proof, note that condition 1 implies
for all. The only difference with the
basic version (24) of the algorithm, is by introducing a step size
and stating requirement 2. The latter technical condition is
needed to assure convergence of descent based methods.
Proof: By Corollary 1 we only need to show that the algo-
rithmconverges to a stationary point of . We start byfollowing
known techniques for analyzing Newton-based iterations. The
is Lipschitz continuous.
3With quasi-Newton methods ?
tionally efficient approximation of the Hessian inverse. Though in our setup it is
possible to show that ? is related to ???? ? with a matrix approximating the
Hessian inverse, we will not claim for computational efficiency here. Nonethe-
less, we will use the term quasi-Newton when describing our method.
? ?? ???? ? where ? is a computa-
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DVORKIND et al.: NONLINEAR AND NONIDEAL SAMPLING: THEORY AND METHODS5883
TABLE I
THE PROPOSED ALGORITHM
standard analysis is to show that Zoutendijk condition [18] is
satisfied:
(52)
where
(53)
is the cosine of the angle between the gradient
search direction
. If
and the
(54)
that is, the search direction never becomes perpendicular to the
gradient, then Zoutendijk condition implies that
so that a stationary point is reached.
To guarantee (52) we rely on the following lemma.
Lemma 3 [18, Theorem 3.2]: Consider iterations of the form
(48), where
is a descent direction and
conditions. If
is bounded below, continuously differentiable
and
is Lipschitz continuous then (52) holds.
Inourproblem,
isadescentdirectionandthebacktracking
procedure guarantees that the step size
conditions [18], [19]. Also,
partial derivatives of
with respect to
by
. Thus, all that is left is to prove
Lipschitz continuity of
(which will also imply that
continuousmapping,andthus,
This is proven in Appendix IV.
We now establish (54). Using (51)
,
satisfies the Wolfe
satisfies the Wolfe
and the
exist and are given
is a
iscontinuouslydifferentiable).
(55)
where we used the notations
. Since
is upper, it is sufficient to show that
bounded. Now [29],
, where
(56)
Since
that
for all
for all . Therefore,
, there exists ansuch
so that
that the suggested quasi-Newton algorithm converges to a sta-
tionary point of , by Theorem 6, we also perfectly reconstruct
the coefficients of the input .
and (54) is satisfied. Having proved
IX. PRACTICAL CONSIDERATIONS
We have presented a Newton based method for perfect recon-
struction of the input signal. Though the suggested algorithm
hasaninterestinginterpretationintermsofiterativeobliquepro-
jections and approximated POCS, it is definitely not the only
choice of an algorithm one can use. In fact, any optimization
method, which is known to trap a stationary point of the con-
sistency merit function (45), will, by Theorem 6, also trap its
(unique) global minimum of zero. Thus, we presented here con-
ditions on the sampling space
the nonlinear distortion
, for which minimization of the merit
function (45) leads to perfect reconstruction of the input signal.
Wewillnowexplainhowsomeoftheconditionsonthespaces
involved and the nonlinear distortion can be relaxed.
For some applications, the bounds (43) and (44) might not be
satisfied everywhere but only for a region of input amplitudes.
For example, the mapping
tinuous only on a finite interval. Also, the derivative
for an input amplitude of zero. Thus, conditions (43) and (44)
are violated unless we restrict our attention to input functions
which are a priori known to have amplitudes within
some predefined, sufficiently small interval. Restricting our at-
tention to amplitude bounded signals can be obtained by min-
imizing (45) with constraints of the form
There are many methods of performing numerical optimization
with constraints. For example, one common approach is to use
Newtoniterationswhileprojectingthesolutionateachsteponto
thefeasibleset(e.g.,[30]).Anotherexampleistheuseofbarrier
methods [18].
Note, however, that the functional (45) is optimized with re-
spect to the representation coefficients
uous-time signal
itself. Thus, it is imperative to link ampli-
tude bounds on
to its representation coefficients , in cases
where amplitude constraints should be incorporated. It is pos-
sibletodothatifwealsoassumethatthereconstructionspace
is a reproducing kernel Hilbert space (RKHS) [31], [32], which
are quite common in practice. For example, any shift-invariant
frame of
corresponds to a RKHS [33]. In particular, the sub-
space of band-limited functions is a RKHS. Formally, if
a RKHS, then for any
,whereisthekernelfunctionofthespace[31],
[32]. Thus, we can bound the amplitude of
energy, which can be accomplished by controlling the norm of
its representation coefficients.
, the restoration space and
is Lipschitz con-
is zero
.
and not the contin-
is
,
by controlling its
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5884 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 12, DECEMBER 2008
Fig. 6. Extension of the memoryless nonlinear setup to Wiener–Hammerstein
sampling systems.
An additional observation concerns the special case of
. In such a setup, at theth approximation point, the gra-
dientof(45)is
is positive, we can use
eliminatingtheneedtocomputeobliqueprojectionsateachstep
of the algorithm.
We have also assumed that the nonlinearity is defined by a
strictly ascending functional
then we can always invert the sign of our samples, i.e., process
thesequence
insteadof ,tomimicthecasewherewesample
, instead of . Once convergence of the al-
gorithm is obtained, the sign of the resulting representation co-
efficients should be altered. Thus, our algorithm and the theory
behind it, also apply to strictly descending nonlinearities.
Finally, we point out that the developed theory imposed the
nonlinearity
to have a slope
upper bound
. This is, however, merely a sufficient con-
dition. In practice, we have also simulated nonlinearities with a
largerslope,and thealgorithmstillconverged (seetheexamples
within Section X).
.Sincetheoperator
as a descent direction,
. Ifis strictly descending
which is no larger than the
A. Extension to Wiener–Hammerstein Systems
Throughout the paper we have assumed that the nonlinear
distortion caused by the sensor is memoryless. Such a model
is a special case of Wiener, Hammerstein and Wiener–Ham-
merstein systems [21]. A Wiener system is a composition of a
linear mapping followed by a memoryless nonlinear distortion,
while in a Hammerstein model these blocks are connected in re-
verse order. Wiener–Hammerstein systems combine the above
two models, by trapping the static nonlinearity, with dynamic
and linear models from each side. We can address such systems
by noting that we can absorb the first linear mapping into the
structural constraints
, and use the last linear operator to
define a modified set of generalized sampling functions. Thus,
it is possible to extend our derivations to Wiener–Hammerstein
acquisition devices as well. This concept is illustrated in Fig. 6.
X. SIMULATIONS
In this section, we simulate different setups of reconstructing
a signal in a subspace, from samples obtained by a nonlinear
and nonideal acquisition process.
A. Band-limited Example
We start by simulating an optical sampling system described
in [34]. There, the authors implement an acquisition device
Fig. 7. Optical sampling system. For high-gain signals, the optical modulator
introduces nonlinear amplitude distortions.
receiving a high frequency, narrowband electrical signal. The
signal is then converted to its baseband using an optical mod-
ulator. In [34] a small-gain electrical signal is used, such that
the transfer function of the optical modulator is approximately
linear. Here, however, we are interested in the nonlinear distor-
tion effects. Thus, we simulate an example of a high-gain input
signal, such that the optical modulator exhibits memoryless
nonlinear distortion effects when transforming the electrical
input
to an optical output. The sampling system is shown in
Fig. 7.
We note that the ability to introduce high-gain signals is very
important in practice as it allows to increase the dynamic range
of a system, improving the signal to interference ratio. We now
show that by applying the algorithm of Theorem 7, we are able
to eliminate the nonlinear effects caused by the optical modu-
lator.
The input–output distortion curve of the optical modu-
lator is known in advance, and can be described by a sine
wave [35]. Here, we simulate a nonlinear distortion which is
given by
. For input signals
the optical modulator introduces a strictly as-
cendingdistortioncurve.Notice,however,thattotestapractical
scenario, we apply our method to a nonlinear distortion having
a maximal slope of five, which is larger than the bound (44).
In this simulation the input signal
frequency, narrowband components; a contribution at a carrier
frequency of
550 MHz and at
width of each component is set to 8 MHz. Hence, the input
lies in a subspace spanned by
and
The support of the input signal (in the Fourier domain) is
depicted in Fig. 8(a).
Astheinputsignalisacquiredbythenonlinearopticalsensor,
the input–output sine distortion curve introduces odd order har-
monics at the output. Already the third order harmonics con-
tribute energy in the region of 1.5–2 GHz. The resulting signal
is sampled by an (approximately linear) optical detector (photo-
diode)andanelectricalADC.Weapproximateboththesestages
by an antialiasing low-pass filter followed by an ideal sampler,
similar to the description of Fig. 2. The antialiasing filter is
chosen to have a transition band in the range 620 MHz–1GHz,
andthesamplingrateis2GHz.Thus,incompliancewithFig.2,
the sampling functions
sions of the impulse response of this antialiasing filter, with
2 GHz. The original input, the resulting output
and the frequency response of the antialiasing filter are
showninFig.8(b).Noticetheharmonicsin ,introducedbythe
optical modulator. Also note that the sampling rate of 2 GHz, is
below the Nyquist rate of
.
Since we have prior knowledge of
(here, a subspace composed of two narrowband regions around
which satisfy
is composed of two high
600 MHz. The band-
, where16 MHz.
are shifted and mirrored ver-
being an element of
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DVORKIND et al.: NONLINEAR AND NONIDEAL SAMPLING: THEORY AND METHODS5885
Fig. 8. (a) Input signal ? is composed of two high-frequency components,
modulated at 550 and 600 MHz. (b) The input signal ?, the distorted output
? ? ????, and the frequency response of the antialiasing filter serving as the
generalized sampling function.
550 and 600 MHz) we are able to apply the algorithm of The-
orem 7. In Fig. 9(a), we show the true input
imation
obtained by a single iteration of the algorithm. In
Fig. 9(b) we show the result after the third iteration. The con-
sistency error in the samples appears in the title of each figure.
At the seventh iteration the algorithm has converged (to the true
input signal), within the numerical precision of the machine.
If we disregard the nonlinearity (i.e., by assuming that
), then the solution will be to perform an oblique projection
onto the reconstruction space:
tialize our algorithm with
, as shown in Fig. 9(a). Evidently, ac-
counting for the nonlinearity improves this result.
Another possibility is to assume that the samples are ideal,
i.e., toassumethat are pointwise evaluationsofthefunction ,
andapply
priortointerpolation.Unfortunately,sincein
practice the samples are nonideal, many of them receive values
and its approx-
. Since we ini-
, it is simple to show that
Fig. 9. (a) True input signal ?, and its approximation ? obtained by a single
iteration of the algorithm. (b) True input signal ?, and its approximation ?
obtained at the third iteration. The consistency error is stated in the title of each
plot.
outside the range
cannot even be performed, since the domain of the arcsin func-
tion is restricted to the
add-hoc approach by normalizing the samples which have ex-
cessive values to
. As evident from the time and fre-
quency-domain plots of Fig. 10, this approach results in poor
approximation of the input signal, giving in this example a spu-
rious-free dynamic rage of only 16 dB. Our proposed method,
in contrast, perfectly reconstructs the input signal, theoretically
giving an infinite spurious-free dynamic range.
. In that case, the operation
interval. One can then use an
B. Non-Band-Limited Example
As another example, which departs from the band-limited
setup, assume that the input signal
lies in a shift-invariant
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5886 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 12, DECEMBER 2008
Fig. 10. True input signal ?, and its approximation, obtained by (falsely) as-
suming ideal samples, restricted to the ?????? interval. (a) Time-domain plot.
(b) Frequency-domain plot.
subspace, spanned by integer shifts of the generator
, whereis the unit step function. For example,
can be the impulse response of an RC circuit, with the delay
constant RC set to one. We choose the nonlinear distortion as
the inverse tangent function, i.e.,
sampling scheme is given by local averages of the form
. Accordingly, the sampling space is also shift in-
variant, with the generator
we show the original input signal
for approximating it from the nonlinear and nonideal samples.
First, one can neglect the nonlinear effects of the acquisition de-
vice, assuming that
is the identity mapping. As with the pre-
vious example, in this case the approximation takes the form of
an oblique projection onto the reconstruction space along
which also results from the first iteration of our algorithm. As
another option, one can assume an ideal sampling scheme. In
that case,
are assumed to be the ideal samples of
from which the signal is reconstructed. As can be seen from
, and the
. In Fig. 11(a)
and two naive forms
,
,
Fig. 11. (a) Original input ???? (solid), and two forms of approximation: Ig-
noring the nonlinearity (dotted) and ignoring the nonideality of the sampling
scheme (dashed). (b) True input signal ?, and its approximation ? obtained at
the sixth iteration.
the figure, this also leads to inexact restoration of the input. On
the other hand, our algorithm takes into account the nonlinear
distortion and the nonideality of the sampling scheme, yielding
perfect reconstruction of the signal [Fig. 11(b)].
Though out of the scope of the developed theory, it is inter-
esting to investigate the influence of quantization noise. For this
purpose,werepeatthelastsimulation,whenquantizingthesam-
ples with a quantization step of 0.1. Naturally, in such a setup
there is no reason to expect perfect reconstruction of the input
signal.Instead,wewillbesatisfiedifanapproximation
found, which can explain the nonlinear, nonideal and quantized
sample sequence. For that purpose, our algorithm was slightly
rectified by quantizing at each iteration the samples error vector
accordingtothequantizationstep.Theresultsof
this simulation are presented in Fig. 12. As can be seen from the
is
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DVORKIND et al.: NONLINEAR AND NONIDEAL SAMPLING: THEORY AND METHODS5887
Fig. 12. Nonlinear, generalized and quantized sampling case. (a) True input
signal ?, and its approximation ? obtained at the sixth iteration (dashed). (b)
True quantized samples, and the quantized sample sequence of the approxima-
tion ? .
figure, the approximation is not identical to the input function
, yet, they both produce the same quantized sample sequence.
Weemphasize,however,thatthedevelopedtheorydoesnottake
noisy samples into account, and we do not claim that this modi-
fied algorithm will produce consistent approximations under all
circumstances.
XI. CONCLUSION
In this paper, we addressed the problem of reconstructing a
signal in a subspace from its generalized samples, which are
obtained after the signal is distorted by a memoryless nonlinear
mapping. Assuming that the nonlinearity is known, we derived
sufficient conditions on the nonlinear distortion curve and the
spaces involved, which ensure perfect reconstruction of the
input signal from these nonlinear and nonideal samples. The
developed theory shows that for such setups, the problem of
perfect reconstruction can be resolved by retrieving stationary
points of a consistency cost function. We then developed an
algorithm for this problem, which was also demonstrated
by simulations. Finally, we explained how to extend these
derivations to nonlinear and nonideal acquisition devices which
can be modeled by a Wiener–Hammerstein system. Future
extensions of this research may consider the important problem
of estimating the nonlinearity of the system from the known
samples, and approximating the signal from noisy samples.
APPENDIX I
PROOF OF THEOREM 1
Under the conditions of the theorem,
mits a Fourier series expansion, and can be written as
for some Fourier coefficients
(6), we then have
ad-
. By
Thus, the Fourier coefficients
izedsamples throughdiscrete-timeconvolution
.
To stably reconstruct
the discrete-time Fourier transform (DTFT) of the sequence
to be invertible and bounded. Since the continuous-time Fourier
transform of
at is
then , being the uniform samples of
, has the DTFT
ofare related to the general-
with
from the sample sequence we need
,
with interval
(57)
Replacing
below and above if and only if (12) holds.
The coefficients
with. Finally, the restoration of
, we obtain that (57) is bounded from
can now be determined, by convolving
takes the form (13).
APPENDIX II
PROOF OF THEOREM 4
Beforestatingtheproof,weneedthefollowinglemma,which
shows that the direct sum condition is invariant under linear
continuous and invertible mappings. Throughout this section,
we use the more general notation of inner products and norms
within some Hilbert space
(not necessarily
Lemma 4: Let
be a linear, continuous and continuously
invertible mapping. Then
.
Proof: Assume
, then
assumption
. To show
some
. Then there is a decomposition
some
. Thus,
).
if and only if
. If there is some
contradicting the
, take
for
and
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5888IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 12, DECEMBER 2008
constitute a decomposition of . The proof in the other direction
is similar.
By Lemma 4,
. Furthermore,
implies [12], [16]
if and only if
Noting that
mapping
,
we conclude that the
istruncating.Thus,foranyinitialization
(58)
and
(59)
Combining (58) and (59) with (31) gives
(60)
It is still left to show that the right-hand side of (60) reduces
to the right-hand side of (23). To simplify the notations, we will
denote
.Since
andwe rewrite (60) as
(61)
Lemma5: Thetransformation
satisfies
.
Proof: It is simple to see that for any
. Furthermore, due to the direct sum condition this
mapping is onto, that is
can replace
Therefore
,
. Also note that we
.
where we used
the last line.
CombiningLemma 5with(61) and (27)yields thatthePOCS
iterations reduce to
in the transition to
(62)
Now
where we used
transition to the last line. Thus, we can rewrite (62) as
for bijectivein the
where we usedin the last equality.
APPENDIX III
PROOF OF THEOREM 5
Our proof will follow the geometric insight described by
Fig. 5. For that matter we first define
(63)
to be the maximal angles between the spaces, restricted to
the interval
. We now show that
are both lower bounded by
we will derive a condition on the nonlinear distortion, assuring
that
.
Since,
bound
and
. Then,
we can lower
(64)
We then have
where we used
equality. The condition
in the last
also implies [16]
. Similarly,implies
and
. Thus, we end up with
(65)
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