Super-Resolution in Phase Space

0.17 MB·English

by  Ayush Bhandari

1
Super–Resolution in Phase Space
Ayush Bhandari , Yonina Eldar and Ramesh Raskar
† ‡ †
Media Laboratory, Massachusetts Institute of Technology
†
Cambridge, MA 02139–4307 USA.
Dept. of Electrical Engineering
‡
Technion–Israel Institute of Technology, Haifa 32000, Israel.
Email: [email protected] [email protected] [email protected]
• •
Abstract: This work considers the problem of super–resolution. The goal is to resolve a Dirac distribution from
knowledgeofitsdiscrete,low–pass,Fouriermeasurements.Classically,suchproblemshavebeendealtwithparameter
5 estimationmethods.Recently,ithasbeenshownthatconvex–optimizationbasedformulationsfacilitateacontinuous
1 timesolutiontothesuper–resolutionproblem.Herewetreatsuper–resolutionfromlow–passmeasurementsinPhase
0 Space. The Phase Space transformation parametrically generalizes a number of well known unitary mappings such
2 astheFractionalFourier,Fresnel,LaplaceandFouriertransforms.Consequently,ourworkprovidesageneralsuper–
resolution strategy which is backward compatible with the usual Fourier domain result. We consider low–pass
n
measurementsofDiracdistributionsinPhaseSpaceandshowthatthesuper–resolutionproblemcanbecastasTotal
a
Variationminimization.Remarkably,eventhougharesettingisquitegeneral,theboundsontheminimumseparation
J
distance of Dirac distributions is comparable to existing methods.
0
3
Index Terms

]
Fractional Fourier, low–pass, phase space, spike train, super–resolution, total variation.
T
I
.
s
c CONTENTS
[

I Introduction 1
1
v
II Phase Space Representation of Signals 3
2
6
6 III Sparse Signals in Phase Space 4
7
0 IV Super–Resolution in Phase Space 5
.
1 IV-A Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
0 IV-B Super–Resolution Via Convex Programming . . . . . . . . . . . . . . . . . . . . . . . . . 6
5
IV-C Remarks and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1
: IV-D An Application of Super–Resolution in Phase Space . . . . . . . . . . . . . . . . . . . . 8
v
i
X V Conclusion 9
r
a References 9
I. INTRODUCTION
ErnstAbbe’sfoundationalwork[1]in1873reportedanobservationregardinglackofresolvabilityofoptical
features beyond the diffraction limit. This problem is central to several areas of science and engineering such
as optics [2], [3], imaging [4], geophysics [5], depth sensing [6] and astronomy [7].
In its abstract form, the problem can be stated as follows: How can we recover a K–sparse signal (spike
train/Diracimpulses)withunknownlocationsandamplitudesfromtheknowledgeofitslow–passmeasurements
intheFourierdomain?Theproblemischallengingbecauseitasksforrecoveryofanon–bandlimitedsignalfrom
its projectiononto the subspace of bandlimitedfunctions.Two predominantapproachesexist in the literature to
super–resolution: optimization and parameter estimation based solutions.
2
∆∆∆: Minimum separation f : Cut–off frequency
c
Donoho [14] Kahane [15] Cande`s–G. [11] Moitra [16]
∆∆∆> 1 ∆∆∆> 5 log 1 ∆∆∆> 2 ∆∆∆> 1
fc fcr 2fc fc fc−1
(cid:16) (cid:17)
TABLE I: Exact Recovery Condition for Super–Resolution.
(cid:4) Optimization Based Super–Resolution: Early roots of ℓ norm based optimization can be traced back to
1
[8]. This approach has been revitalized due to the advent of compressed sensing [9]. A very recent idea in this
contextisthatofcontinuoussparsemodelingwherethesignalattributesareestimatedonacontinuouslydefined
grid [10], [11] (instead of its discrete counterpart, as is the case for the usual compressive sensing setup).
Bredies and Pikkarainen [12] present an optimization method in the space of signed measures. In their
setting, they consider discrete measurements while the solution space is infinite dimensional. In parallel, de
Castro and Gamboa [13] and Cande`s and Fernandez-Granda [11] consider a Total Variation (TV) formulation
for super–resolution.
(cid:4) Parameter Estimation Based Super–Resolution: The super–resolution problem has been studied in the
signal processing context with the goal of resolving overlappingechoes/time–delay estimation [17], multi–path
characterization [6] and deconvolution. Li and Speed [18] considered super–resolution in form of parametric
deconvolution of spike trains. Vetterli, Blu and co–workers [19], [20] developed the idea of super–resolution
as a sparse sampling problem. Eldar and co–workers developed super–resolution methods in the context of
sub–NyquistsamplingandtheXamplingframework [21].Finally,sincethesuper–resolutionproblemisclosely
linked with the spectral estimationproblem[18], MUSIC, ESPIRIT and matrixpencil[22], [23] based methods
have also been used.
Incontrasttoparameterestimationtechniqueswheretherecoveryconditionisbasedonthenumberofspikes,
non–parametricmethodsprovideaboundin theformofa minimumseparationcondition.Moreprecisely,letK
bethenumberofspikestobesuper–resolved,f bethecut–offfrequencyinFourierdomainandlet∆∆∆denotethe
c
minimumspacingbetween any two spikes. While parameterestimation methodsrequiref 2K+1 assuming
c
≥
that K is known a priori, non–parametric methods super–resolve the spikes provided that f >f (∆∆∆) where
c SR
f is some function.The worksof Donoho[14], Kahane [15], Cande`s and Fernandez-Granda[11] and Moitra
SR
[16]provideatheoreticalguaranteeforthesuper–resolutionproblemintermsoff .Wesummarizetherecovery
SR
guarantees in terms of f in Table I.
SR
The Fourier transform is well suited for examining signals which are linear combinations of sinusoids.
In many practical applications such as radar, sonar, holography, wave–physics and quantum optics, the basic
building blocks of the signals are not sinusoidal. Often polynomialphase, Fourier–like transformationsof form
eφ(t) are well suited for analysis of such signals. For this purpose, tools such the Fresnel transform [24],
Fractional Fourier transform [25] and the Chirp transform [26], [27] have been developed in the literature.
Our goalhereis to extendexistingsuper–resolutionresultsto integraltransformationsotherthan theFourier
transform.To this end, we cast the super–resolutionproblem in phase domain which parametricallygeneralizes
3
some of the well–known unitary transformations. Some examples are listed in Table II. We show that exact
recovery of spike trains from low–pass measurements in phase space is possible by minimizing the signal’s
TV–norm. This is accomplished using a convex program. Remarkably, even with the general construct of the
problem, our theoretical guarantee for exact recovery is the same as in [11].
Throughoutthepaper,R,CandZdenotesetsofreal,complexandintegers.Weusez todenotethecomplex
∗
conjugate of z C, z = 1(z+z ) is the real part of z and =√ 1. Discrete sequences are represented by
∈ ℜ 2 ∗ −
s[m],m Zwhiletheircontinuouscounterpartsarerepresentedbys(t),t R.TheL innerproductbetween
2
∈ ∈
functions f and g is denoted by f,g = fg dt. Function composition is denoted by (f g)(x)=f(g(x)).
∗
h i ◦
Convolution between functions f and g iRs defined as (f g)(t) = f(z)g(t z)dz. We use s to represent
∗ −
the row–vector of a discrete sequence s[m],m [0,...,M 1]. WRe use bold capitals to represent matrices,
∈ −
for exampleD and D is the pseudo–inverseof D. Calligraphic letters such as are used to denote operators.
†
L
Sets are denoted by capitalized roman font, S. The estimate of function/sequence µ is represented by µ.
e
II. PHASESPACE REPRESENTATIONOF SIGNALS
The term Phase Spacenaturallyarises in areasof optics [28] andmathematicalphysics[29]. The Fractional
Fourier transform (FrFT) [25], [30]–[32] and the Linear Canonical Transform (LCT) [28], [33], [34] are
instantiations of phase space transformations. These transformations have found a number of applications in
signalprocessing[25]andcommunicationproblemssuchasshift–invariantsampling/approximationtheory[30],
[34], operator theory, multi–carrier communications [32], array processing [31] and optical signal processing
[28]. In this paper, we will work with the Linear Canonical Transform (LCT) [29].
Definition 1 (LCT): Let Λ= a b , with ad bc=1. The LCT of a function f(t),t R is a parametric,
c d − ∈
unitary, integral mapping, Λ :f(cid:2) f(cid:3)with respect to the time–frequencykernel kΛ,
L →
b
f,kΛ(,ω) b=0,
f(ω)= Λ[f](ω)d=ef h · i 6 (1)
L LCT √de−21cdω2f(dω) b=0,
b
where the transformation kernel is de|fined{zby } 
1 1
kΛ(t,ω)= exp  at2+dω2 2ωt . (2)
√ 2πb − 2b −
− (cid:18) (cid:19)
(cid:0) (cid:1)
Now since b=0 amounts to dilating the function √de−21cdω2f(dω) (see (1)), we will develop our results for
the case when b=0.
6
The LCT satisfies a useful operator composition property: LΛ1 ◦LΛ2 = LΛ3 with Λ3 = Λ2Λ1. This can
be used to show that,
f(t)= Λ−1[ f ](t)d=ef  f,kΛ−1(t,·) b6=0 (3)
LInverse–LCT D√bae−12cat2f(Eat) b=0.
b
TheLCTparametrizesanumbero|fwel{l–zknow}n,unitarytransformations,someofwhicharelistedinTableII.
For Λ= 0101 =ΛFT, the LCT amounts to the Fourier transform (upto a constant). (FT). Similarly, with the
−
2 2 rota(cid:2)tion m(cid:3)atrix Λθ (see Table II), we obtain the FractionalFourier transform(FrFT). Matrix factorization
×
4
TABLE II: Parametric Representation of Unitary Transformations
ParameterMatrix (Λ) CorrespondingTransform
cosθ sinθ =Λ Fractional Fourier Transform
sinθcosθ θ
−
0 1 =Λ Fourier Transform (FT)
(cid:2) 10 (cid:3)FT
−
0  =Λ Laplace Transform (LT)
(cid:2) 0 (cid:3) LT
cosθ sinθ Fractional Laplace Transform
(cid:2)sin(cid:3)θ cosθ
−
1b Fresnel Transform
(cid:2)01 (cid:3)
1b Bilateral Laplace Transform
(cid:2) 1(cid:3)
1 b , b 0 Gauss–Weierstrass Transform
(cid:2)0−1(cid:3) ≥
(cid:2)√12 e(cid:3)−0π/2 e−1π/2 Bargmann Transform
−
(cid:2) (cid:3)
shows that the LCT is related to the FT and the FrFT. The Fourier matrix factorization is simple:
a b b 0 1 0
 = ΛFT ⇔Λ=M1ΛFTM2,
c d d b−1 a/b 1
     
and leads to the implementation Λ = M Λ M . The Fractional Fourier matrix factorization is more
L L 1◦L FT◦L 2
involved. Relying on the Iwasawa Decomposition [35],
a b Γ 0 1 u
 =Λθ  ⇔Λ=ΛθDU,
c d 0 Γ−1 0 1
    
whereD is a diagonalmatrixwith Γ=√a2+c2 andU is an upper–triangularmatrixwith u=(ab+cd)/Γ2.
A useful operation that is linked with phase space is the convolution/filtering operator. For the Fourier
domain, we have the convolution–multiplication property: (f g) = Λ−1[ Λ[f] Λ[g]], with Λ = ΛFT.
∗ L L L
Unfortunately, this property is not preserved in phase space. To circumvent this problem, we use a version of
the FrFT convolution operator [30].
Definition 2 (LCT Convolution/Filtering): Let Λ denote the convolution/filteringoperation in LCT domain
∗
and be the usual Fourier domain convolution operator. Convolution of functions f and g in the LCT domain
∗
is defined by
(f Λg)(t)= e−a2tb2 f(t)e+a2tb2 g(t)e+a2tb2 . (4)
∗ √2πb ∗
(cid:16) (cid:17)
ConvolutionofModulatedFunctions
By following the steps in [30], it is easy to verify th|at the operat{izon in (4) adm}its a convolution–multiplication
property:
Λ[f Λg](ω)=e−d2ωb2f(ω)g(ω). (5)
L ∗
b b
III. SPARSE SIGNALS IN PHASE SPACE
Consider a K–sparse object/spike train modeled by,
K 1
s(t)= − c δ(t t ), t R (6)
k k
k=0 − ∈
X
where δ denotes the Dirac mass with weights c C that is activated on locations t [0,τ),k =
k k k
{ } ∈ { } ∈
0,...,K 1. Since we are dealing with a finite length signal s that lives on the interval [0,τ), we investigate
−
5
its representation in phase space using the Fractional Fourier series [36] analog of the LCT.
Definition 3 (Linear Canonical Series): Letf beacompactlysupportedfunctionsuchthatf =0, t [0,τ)
6 ∀ ∈
andzeroelsewhereandletκ = 2πb/τ,b=0.TheLinearCanonicalSeries(LCS)expansionofthefunction
b,τ
6
f is given by p
n=+
∞ 2π
f(t)=κb,τ f[n]kΛ(t,nω0b), ω0 = . (7)
τ
n=
X−∞
b
The LCS coefficients f are evaluated at ω =nω b,n Z,
0
∈
b
f[n]=κb,τ Λ[f](nω0b) κb,τ f,kΛ(,nω0b) . (8)
L ≡ h i
Note that by appropriately pabrameterizing Λ, one can easily design the basis functions for any of the
transformations listed in Table II. For example, with Λ = Λ in (8), we obtain the Fourier Series expansion
FT
of f.
We now compute the series coefficients for the sparse signal s(t) in (6). We use (8), to compute the LCS
coefficients
(8)
s[n]=κb,τ Λ[f](nω0b) = κb,τ s(t)kΛ∗ (t,nω0b)dt
L
Zτ
K 1
b =κb,τ − ckkΛ∗ (tk,nω0b). (9)
k=0
X
Plugging the coefficients into the LCS representation (7), we obtain the phase space representation of s(t)
m=+
(7) ∞ 2π
s(t) = κb,τ s[m]kΛ(t,mω0b), ω0 =
τ
m=
X−∞
= e−a2tb2 m=+∞b K−1cke2abt2ke−mω0tk emω0t (10)
τ
!
m= k=0
X−∞ X
yb[m]
= e−a2tb2 m=+∞|y[m]emω0t{.z } (11)
τ
m=
X−∞
b
From [11], we see that y[m] are precisely the Fourier series coefficients of τs(t)ea2tb2. Thus, even though we
are dealing with the phase space, the linear frequency modulated, sparse signal s is completely characterized
b
by its Fourier series coefficients y.
bIV. SUPER–RESOLUTION IN PHASE SPACE
A. Problem Formulation
Let sinc(t)= sin(πt). Consider the frequency modulated function,
πt
φLP(t)=(Ω/b)e−a2tb2 sinc((Ω/b)t). (12)
Notethatthisfunctionis(Ωπ)–bandlimitedbecauseinphasespacethefunctionφ (t)iscompactlysupported,
LP
or,
e+d2ωb2 ω
φLP(ω)= Λ[φLP](ω)= Π
L √2πb 2πΩ
(cid:16) (cid:17)
b
6
where Π(ω)=1, ω 61/2 and zero, elsewhere.
| |
With s defined in (11), its low–pass version is,
h(t)= (s ΛφLP)(t) (=4) e−a2tb2 1 y[m]eω0mt, (13)
∗ √2πbτ
Low–PassFiltering |m|6X⌊Ωτ/2b⌋
b
where is the floor operation. | {z }
⌊·⌋
Suppose we observe N discrete, low–pass measurements sampled with sampling rate T =b/Ω,
h[n]= h(t) , T =b/Ω, n=0,...,N 1. (14)
|t=nT −
By modulating h, we obtain measurements of the form,
y[n]= 2πbτe+a(n2Tb)2h[n]= y[m]eω0m(nT) (15)
pModulatedMeasurements |mX|6fc
b
wheref = Ωτ/2b .Invector–mat|rixnotatio{nz,wehave}low–passmeasurements,y=V y, whereV
c IDFT IDFT
⌊ ⌋ ∈
CN×(2fc−1) is the usual inverse–DFT matrix with elements [VIDFT]n,m = emω0(nT) and we assume that
b
N >2f 1 so that V is a full–rank marix.
c IDFT
−
Havingobtainedy, the questionthen is:howmanysamplesofy are sufficientforcompletecharacterization
of y[m] (1=0) Kk=−01 cke+2abt2k e−mω0tk? Indeed, from spectral estimation theory [22], we know that we
b b
need at least 2PK+1(cid:16)values of y(cid:17)to solve for ck,tk k. Consequently, whenever
b { }
b f = Ωτ >K τ >K, (16)
c 2b ⇔ 2T
(cid:4) (cid:5) (cid:4) (cid:5)
the system of equations is complete in y meaning that we have at least 2K+1 values of y and we can solve
for c ,t . Thus (16) provides a bound on the minimum sampling density in phase space.
k k
{ } b b
B. Super–Resolution Via Convex Programming
Given N measurements y, we obtain y using y=VI†DFTy (15). From the phase space development of the
problem, we know that
b b
y[m](1=0) K−1 cke+2abt2k e−mω0tk (17)
k=0
Xτ (cid:16) (cid:17)
b = µ(t)e−mω0tdt, m 6fc = Ωτ/2b
| | ⌊ ⌋
Z0
where
µ(t)= K−1cke+2abt2kδ(t tk) (18)
k=0 −
X ρk
and ρk d=ef cke+2abt2k are the new weights for s(t). | {z }
We are now left to solve the standard super–resolution problem [11] where one has access to the low–pass
measurementsy[m] andthe signalto be super–resolvedis prescribedin (18). Hence,the problemof recovering
b
7
µ from y can now be solved by using,
Super–ResolutioninPhaseSpace(PrimalProblem)
b
mµeinkµkTV subject to (cid:26)y[m](1=7)Z0τµ(t)e−mω0tdt(cid:27)|m|6fc
e b e
(19)
(18)
where, for the model assumed in (18), the TV–norm amounts to, µ = ρ = c . We note that
k kTV k| k| k| k|
in principle µ = s , however,due to thenatureof phasespace measPurements,thePreal/complexweights
k kTV k kTV
ck k need to be demodulated using the linear frequency modulation term e+2abt2k which depends on Λ.
{ } e
The problem in (19) seeks to recover the infinite–dimensional variable µ from finitely many constraints
set up in (17). This continuous optimization problem has a tractable dual problem. As was shown in [11], a
b
semidefiniteprogram(SDP)canbeusedtorecoverµ bycomputing t firstandthenrecovering c using
k k k k
{ } { }
a least squares fit. The SDP equivalent [11] of the convex dual of (19) is,
e
SemidefiniteProgram(DualProblem)
max y,u subject to,
u,M ℜh i
M u
b
0, [M] =δ
 u∗ 1 ≻≻≻ j∈S2,m∈S1 m,m+j j
X
 
where S1 = [1,2fc+1 j], S2 = [0,2fc], M C(2fc+1)×(2fc+1) is some Hermitian matrix and u C2fc+1
− ∈ ∈
is a complex vector.
The SDP inputy results in a vector u. In order to recoverthe locations t , we constructthe polynomial
k k
{ }
of degree N =4f ,
0 c
b
p (z)=1 u zk, z C. (20)
N0 − |k|≤2fc k ∈
X
The roots of p4fc(z),z =eω0t lead to the locations {tk}k. Knowing y together with the estimates, {tk}k, we
use the constraints in (17) to set up a system of equations which leads to amplitude estimates ck =ρke−2abet2k.
b e
(see (18)). Finally, we recover our super–resolved signal, s(t) = Kk=−01ckδ t−tk . Stepwisee proecedure for
super–resolution in phase space is outlined in Algorithm 1. P (cid:0) (cid:1)
e e e
In view of [11], let us invoke the definition of minimum distance ∆∆∆ = inf t t . With f =
{tk}k:tk6=tl| k− l| c
Ωτ/2b , the exact recovery requirement for phase space is as follows:
⌊ ⌋
Theorem 1 (Exact Recovery in Phase Space): Let the support set of s(t) in (6) be S = t . If the
k k
minimum distance obeys the bound∆∆∆(S)fc >2, then s(t) is a unique solution to (19). (cid:8) (cid:9)
e
Theproofofthistheoremisastraight–forwardconsequenceof[11].Moreover,duetoinherentFourierstructure
of the phase space problem, our work may benefit from the ideas discussed in [13], [14], [16], [23].
C. Remarks and Discussion
(cid:4) Backward Compatibility With the choice of parameter matrix Λ=Λ (cf. Table II), our result coincides
FT
with the usual, Fourier domain case of super–resolution [11], [13]. Furthermore, Λ=Λ relates to the case of
θ
Fractional Fourier domain for which our result generalizes a previous known result [37].
8
Algorithm 1: Super–Resolution in Phase Space.
(13)
Input:Low-passsamplesh[n] = (s∗ΛφLP)(nT)
Modulate Samplesh[n]→ 2πb|τ|e+a(n2Tb)2h[n]=y[n]
Data:y=V† y,y∈C2fpc+1
IDFT
M u
Solve SbDP: mu,aMxbℜhy,ui subject to (cid:20) u∗ 1 (cid:21)≻0
Construct Polynomial: pbN0(z)=1− |k|62fcukzk
Support: pN0 eω0t =0→ tk k P
OWuetigphutts::s(mρte)kin=(cid:12)(cid:12)(cid:12)(cid:0)yb[mKk=]−−(cid:1)01PckKδk=−t01−(cid:8)ρeektke(cid:9)−,{ωc0ketk=(cid:12)(cid:12)(cid:12)2ρ→ke−{ρek2a}bket2k}k
P (cid:0) (cid:1)
e e e e e
(cid:4) Exact Recovery Condition Even though our super–resolution naturally extends to a number of well known
unitary transformations, the exact recovery condition remains unchanged. Hence re-formulating the super–
resolution problem in context of phase space comes at no extra cost in the sense of recovery requirement.
D. An Application of Super–Resolution in Phase Space
Bandlimted signals are compactly supported in the Fourier domain. When a bandlimited signal is corrupted
by additive impulsive noise or AIN, the holes/zeros in the spectrum are filled by the spectral components that
characterize the impulsive noise which is essentially non–bandlimited.Wolf [38] used the idea of curve–fitting
the out–of–bandcomponentsfor identification of impulsive noise components.Here, we formulate the problem
of denoising linear frequency modulated (LFM) signals that are corrupted by AIN. Since LFM signals are the
basis functionsof phase space transformations,it is clear that such signals are bandlimited in the LCT domain.
Consider a bandlimited LFM signal
rBL(t)=κb,τ rBL[m]kΛ(t,mω0b),
m M
| |≤
X
with r [m] = 0, m > M and let r(t) = r (t)+sb(t) be the signal corrupted by AIN. Clearly, r(t) is
BL BL
| |
non–bandlimited in phase space due to s(t). Suppose we observe low–pass filtered samples of r(t), that is,
b
(r ΛφLP)(nT)=e−a(n2Tb)2 (c1y1[m]+c2y2[m])eω0mnT
∗
h[n] |mX|6fc ybr[m]
b b
where c1,c2 are know|n cons{taznts, y}1[m] = rBL[m]e−d(|m2ωb0b)2 an{dzy2[m] =} y[m] (as in (17)) where fc =
Ωτ/2b τ/2T . Again, let us define y[n]=h[n]ea(n2Tb)2,n=0,...,N 1,N 2fc+1. Provided that
⌊ ⌋ ≡⌊ ⌋ b b b −b ≥
f M +2K+1, we have,
c
≥
c y [m]+c y [m] m M
1 1 2 2
y [m]= | |≤ ,
r 
c2yb2[m] b |m|>M
b
which leads to complete characterization of s(bt) since the 2K +1 values of y2[m] can be used with (19) to
solve for s(t). With y=y , m>M we can use Algorithm 1 for exact denoising of r(t).
r
b
b b
9
V. CONCLUSION
We develop a method for super–resolution in phase space. The phase space transformation generalizes a
numberofwellknowntransforms(see TableII).Moreprecisely,weareconcernedwithrecoveryofspiketrains
fromtheirlow–passsamples.Forthispurpose,wefilterthespiketrainwithakernelwhichisbandlimitedinphase
space. We show that even though we are dealing with a general class of parametric transformations, the low–
pass samples are completely characterized by chirp–modulatedFourier series. Having made this link, we show
that the recovery of spikes from their low–rate measurements can be cast as a total–variation minimization—a
problem that can be tackled by convex programming. In closing, our work extends the recent results of [11]
without altering the exact recovery condition. That said, the cut–off frequency is a function of the transform
being used for investigation. Our work warrants future research, specially for the case of additive noise.
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