Table Of ContentOn Detecting Harmonic Oscillations
Anatoli Juditsky∗ Arkadi Nemirovski†
January 24, 2013
3
1
0
2
Abstract
n
a
J In this paper, we focus on the following testing problem: assume that we are
2 given observations of a real-valued signal along the grid 0,1,...,N−1, corrupted by
2 the standard Gaussian noise. We want to distinguish between two hypotheses: (a)
the signal is a nuisance – a linear combination of d harmonic oscillations of given
] n
T frequencies, and (b): signal is the sum of a nuisance and a linear combination of a
S
given number d of harmonic oscillations with unknown frequencies, and such that
. s
h the distance (measured in the uniform norm on the grid) between the signal and
t
a the set of nuisances is at least ρ > 0. We propose a computationally efficient test
m
for distinguishing between (a) and (b) and show that its “resolution” (the smallest
[ value of ρ for which (a) and (b) are distinguished with a given confidence 1−α)
1 is O(cid:16)(cid:112)ln(N/α)/N(cid:17), with the hidden factor depending solely on d and d and
n s
v
independent of the frequencies in question. We show that this resolution, up to a
8
2 factorwhichispolynomialind , d andlogarithmicinN, isthebestpossibleunder
n s
3 circumstances. We further extend the outlined results to the case of nuisances and
5
signalsclosetolinearcombinationsofharmonicoscillations, andprovideillustrative
.
1
numerical results.
0
3
1
:
v
1 Introduction
i
X
r
a In this paper, we address the detection problem as follows. A signal (two-sided sequence
of reals) x is observed on time horizon 0,1,...,N −1 according to
y = xN−1 +ξ,
0
where ξ ∼ N(0,I ) is the white Gaussian noise and zN−1 = [z ;...;z ]. Given y we
N 0 0 N−1
want to distinguish between two hypotheses:
• Nuisance hypothesis: x ∈ H , where H is comprised of all nuisances – linear
0 0
combinations of d harmonic oscillations of given frequencies;
n
∗LJK, Universit´e J. Fourier, B.P. 53, 38041 Grenoble Cedex 9, France, [email protected]
†Georgia Institute of Technology, Atlanta, Georgia 30332, USA, [email protected]
Research of the second author was supported by the ONR grant N000140811104.
1
• Signal hypothesis: x ∈ H (ρ), where H (ρ) is the set of all sequences x representable
1 1
as s+u with the “nuisance component” u belonging to H and the “signal compo-
0
nent” s being a sum of at most d harmonic oscillations (of whatever frequencies)
s
such that the uniform distance, on the time horizon in question, from x to all nui-
sance signals is at least ρ:
min(cid:107)xN−1 −zN−1(cid:107) ≥ ρ.
0 0 ∞
z∈H0
We are interested in a test which allows to distinguish, with a given confidence 1 − α,
between the above two hypotheses for as small “resolution” ρ as possible.
An approach to this problem which is generally advocated in the signal processing
literature is based on frequency estimation. The spectrum of the signal is first estimated
using noise subspace methods, such as multiple signal classification (MUSIC) [10, 11],
then the nuisance spectrum is removed from this estimation and the decision is taken
whether the remaining “spectral content” indicates the presence of a signal or is a noise
artifact (for detailed presentation of these techniques see [12, 5]). To the best of our
knowledge, no theoretical bounds for the resolution of such tests are available. A different
test for the case when no nuisance is present, based on the normalized periodogram,
has been proposed in [3]. The properties of this test and of its various modifications
were extensively studied in the statistical literature (see, e.g., [13, 4, 1, 5]). However,
theoretical results on the power of this test are limited exclusively to the case of sequence
x being a linear combination of Fourier harmonics e2πıkt/N, k = 0,1,...,N−1 under signal
hypothesis.
In this paper we show that a good solution to the outlined problem is offered by an
extremely simple test as follows.
(cid:110) (cid:111)N−1
Let F u = √1 (cid:80)N−1u exp{2πıkt/N} : CN → CN be the Discrete
N N t=0 t k=0
Fourier Transform. Given the observation y, we solve the convex optimization
problem
(cid:110) (cid:111)
Opt(y) = min (cid:107)F (y −zN−1)(cid:107) : z ∈ H
N 0 ∞ 0
z
and compare the optimal value with a threshold q (α) which is a valid upper
N
bound on the 1−α-quantile of (cid:107)F ξ(cid:107) , α ∈ (0,1) being a given tolerance:
N ∞
Prob {(cid:107)F ξ(cid:107) > q (α)} ≤ α.
ξ∼N(0,IN) N ∞ N
If Opt(y) ≤ q (α), we accept the nuisance hypothesis, otherwise we claim
N
that a signal is present.
It isimmediately seenthat theoutlined testrejects thenuisance hypothesiswhen itis true
with probability at most α 1. Our main result (Theorems 3.1) states that the probability
to reject the signal hypothesis when it is true is ≤ α, provided that the resolution ρ is not
too small, specifically,
(cid:113)
ρ ≥ C(d +d ) ln(N/α)/N (!)
n s
1This fact is completely independent of what the nuisance hypothesis is – it remains true when H is
0
an arbitrary set in the space of signals.
2
with an appropriately chosen universal function C(·).
Some comments are in order.
• We show that in our detection problem the power of our test is nearly as good as
it can be: precisely, for every pair d , d and properly selected H , no test can
n s 0
(cid:113)
distinguish (1 − α)-reliably between H and H (ρ) when ρ < O(1)d ln(1/α)/N
0 1 s
from now on, O(1)’s are appropriately chosen positive absolute constants.
• We are measuring the resolution in the “weakest” of all natural scales, namely, via
the uniform distance from the signal to the set of nuisances. When passing from the
√
uniform norm to the normalized Euclidean norm |xN−1| = (cid:107)xN−1(cid:107) / N ≤ (cid:107)xN(cid:107) ,
0 2 0 2 0 ∞
an immediate lower bound on the resolution which allows for reliable detection
(cid:113)
becomes O(1) ln(1/α)/N. In the case when, as in our setting, signals obeying H
0
and H (ρ) admit parametric description involving K parameters, this lower bound,
1
up to a factor logarithmic in N and linear in K, is also an upper resolution bound,
and the associated test is based on estimating the Euclidean distance from the
signal underlying the observations to the nuisance set. Note that, in general, the
√
|·| -norm can be smaller than (cid:107)·(cid:107) by a factor as large as N, and the fact that
2 ∞
“energy-based” detection allows to distinguish well between parametric hypotheses
(cid:113)
“separated” by O( ln(N/α)/N) in | · | norm does not automatically imply the
2
(cid:113)
possibility to distinguish between hypotheses separated by O( ln(N/α)/N) in the
uniform norm2. The latter possibility exists in the situation we are interested in
duetotheparticularstructureofourspecificnuisanceandnon-nuisancehypotheses;
this structure allows also for a dedicated non-energy-based test.
• For the sake of definiteness, throughout the paper we assume that the observation
noise is the standard white Gaussian one. This assumption is by no means critical:
whatever is the observation noise, with q (α) defined as (an upper bound on) the
N
(1−α)-quantile of (cid:107)F ξ(cid:107) , the above test (1−α)-reliably distinguishes between the
N ∞ √
hypotheses H and H (ρ), provided that ρ ≥ C(d +d )q (α)/ N. For example,
0 1 n s N
the results of Theorems 3.1 and 3.2 remain valid when the observation noise is of
the form ξ = {ξ = (cid:80)∞ γ η }N−1 with deterministic γ , (cid:80) |γ | ≤ 1, and
t τ=−∞ τ t−τ t=0 τ τ τ
independent η ∼ N(0,1).
t
• The main observation underlying the results on the resolution of the above test
is as follows: when x is the sum of at most d harmonic oscillations, (cid:107)F x(cid:107) ≥
√ N ∞
C(d) N(cid:107)xN−1(cid:107) with some universal positive function C(d). This observation
0 ∞
originates from [8] and, along with its modifications and extensions, was utilized,
for the time being in the denoising setting, in [9, 2, 6, 7]. It is worth to mention that
it also allows to extend, albeit with degraded constants, the results of Theorems 3.1
and 3.2 to multi-dimensional setting.
2Indeed,letH statethatthesignalis0,andH (ρ)statethatthesignalis≥ρatt=0andiszerofor
0 1
all other t’s. These two hypotheses cannot be reliably distinguished unless ρ ≥ O(1), that is, the (cid:107)·(cid:107)
∞
(cid:112)
resolution in this case is much larger than O( ln(N/α)/N).
3
Therestofthispaperisorganizedasfollows. Insection2wegiveadetaileddescription
of the detection problems (P ), (P ), (N ) and (N ), we are interested in (where (P ) is
1 2 1 2 2
the problem we have discussed so far). Our test is presented in section 3 where we also
provide associated resolution bounds for these problems. Next in section 4 we present
lower bounds on “good” (allowing for (1 − α)-reliable hypotheses testing) resolutions,
whileinsection5wepresentsomenumericalillustrations. Theproofsofresultsofsections
3 and 4 are put into section 6.
2 Problem description
Let S stand for the space of all two-sided real sequences z = {z ∈ R}∞ . Assume that
t t=−∞
a discrete time signal x ∈ S is observed on time horizon 0 ≤ t < N according to
y = xN−1 +ξ, ξ ∼ N(0,I ), (1)
0 N
where (and from now on) for z ∈ S and integers p ≤ q, zq stands for the vector
p
[z ;z ;...;z ].
p p+1 q
In the sequel, we are interested in the case when the signal is a linear combination of
a given number of harmonic oscillations. Specifically, let ∆ stand for the shift operator
on S:
(∆z) = z , x ∈ S.
t t−1
Let Ω be the set of all unordered collections w = {ω ,...,ω } of d reals which are
d 1 d
“symmetric mod 2π,” meaning that for every a, the number of ω(cid:48)s equal, modulus 2π, to
i
a is exactly the same as the number of ω ’s equal, modulus 2π, to −a. We associate with
i
w ∈ Ω the real algebraic polynomial
d
d
(cid:89)
p (ζ) = (1−exp{ıω }ζ)
w (cid:96)
(cid:96)=1
and the subspace S[w] of S, comprised of x ∈ S satisfying the homogeneous finite-
difference equation
p (∆)x ≡ 0. (2)
w
In other words, S[{ω ,...,ω }] is comprised of all real two-sided sequences of the form
1 d
d
(cid:88)
x = [p (t)cos(ω t)+q (t)sin(ω t)]
t (cid:96) (cid:96) (cid:96) (cid:96)
(cid:96)=1
with real algebraic polynomials p (·), q (·) of degree < m , where m is the multiplicity,
(cid:96) (cid:96) (cid:96) (cid:96)
mod2π, of ω in w. We set
(cid:96)
(cid:91)
S = S[w].
d
w∈Ω
d
Remark 2.1 In what follows, we refer to the reals ω constituting w ∈ Ω as to fre-
i d
quencies of a signal from S[w]. A reader would keep in mind that the number of “actual
frequencies” in such a signal can be less than d: frequencies in w different from 0mod2π
and πmod2π go in “symmetric pairs” (ω,ω(cid:48) = −ωmod2π), and such a pair gives rise to
a single “actual frequency.”
4
Given a positive integer N, real (cid:15) ≥ 0, and w ∈ Ω , we set
d
SN,(cid:15)[w] = {x ∈ S : (cid:107)[p (∆)x]N−1(cid:107) ≤ (cid:15)}.
w 0 ∞
Finally, we denote SN,(cid:15) the set
d
SN,(cid:15) = (cid:91) SN,(cid:15)[w].
d
w∈Ω
d
When N is clear from the context, we shorten the notations SN,(cid:15)[w], SN,(cid:15) to S(cid:15)[w] and
d
S(cid:15), respectively.
d
In the definitions above, it was tacitly assumed that d is a positive integer. It makes
sense to allow also for the case of d = 0. By definition, Ω is comprised of the empty
0
collection w = ∅ and p (ζ) ≡ 1. With this convention, SN,(cid:15)[∅] = {x ∈ S : (cid:107)xN−1(cid:107) ≤ (cid:15)}.
∅ 0
Observe that the family SN,(cid:15), d ≥ 2, is quite rich. For instance, it contains “smoothly
d
varyingsignals”(caseofw = 0mod2π),alongwith“fastvarying”–amplitude-modulated
i
and frequency-modulated signals (see [2, 6] for more examples).
We detail now the hypothesis testing problems about the sequence x via observation
y given by (1). In what follows d and d are given positive integers, and ρ, (cid:15) , (cid:15) are
s n n s
given positive reals.
(P ) The “basic” hypothesis testing problem we consider is that of testing of a simple
1
nuisance hypothesis {x = 0} against the alternative that a signal x ∈ S “is
ds
present,” meaning that the uniform norm of the signal on the observation window
[0,...,N − 1] exceeds certain threshold ρ > 0. In other words, we consider the
following set of hypotheses:
H = {x = 0},
0
(cid:110) (cid:111)
H (ρ) = x ∈ S : (cid:107)xN−1(cid:107) ≥ ρ .
1 ds 0 ∞
(P ) We suppose that x ∈ S decomposes into “signal” and “nuisance”:
2
x = s+u,
where s is the signal of interest and a nuisance u belongs to a subspace S[w],
assumed to be known a priori. We consider a composite nuisance hypothesis that
x is a “pure nuisance”, and the alternative (signal hypothesis) that useful signal
s does not vanish, and the deviation, when measured in the uniform norm on the
observation window, of “signal+nuisance” from the nuisance subspace is at least
ρ > 0. Thus we arrive at the testing problem: given w ∈ Ω decide between the
dn
hypotheses
H = {x = u ∈ S[w]},
0
(cid:40) (cid:41)
x = u+s : u ∈ S[w], s ∈ S ,
H (ρ) = ds
1 such that min {(cid:107)[x−z]N−1(cid:107) : z ∈ S[w]} ≥ ρ.
z 0 ∞
Clearly, problem (P ) is a particular case of (P ) with d = 0 (and thus S = {0}
1 2 n dn
is a singleton).
5
(N ) Given w ∈ Ω , consider the nonparametric nuisance hypothesis that the nuisance
1 dn
u ∈ SN,(cid:15)n[w] with some known w. The signal hypothesis is that the useful signal
s ∈ S is present, and x = s+u deviates from the nuisance set on the observation
ds
window by at least ρ > 0 in the uniform norm:
H = {x = u ∈ SN,(cid:15)n[w]},
0
(cid:40) (cid:41)
x = u+s : u ∈ SN,(cid:15)n[w], s ∈ S ,
H (ρ) = ds
1 such that min {(cid:107)[x−z]N−1(cid:107) : z ∈ SN,(cid:15)n[w]} ≥ ρ.
z 0 ∞
(N ) The last decision problem is a natural extension of (N ): we consider the prob-
2 1
lem of testing a nonparametric nuisance hypothesis against a nonparametric signal
alternative that the useful signal s ∈ SN,(cid:15)s is present:
ds
H = {x = u ∈ SN,(cid:15)n[w]},
0
(cid:40) x = u+s : u ∈ SN,(cid:15)n[w], s ∈ SN,(cid:15)s, (cid:41)
H (ρ) = ds
1 and such that min {(cid:107)[x−z]N−1(cid:107) : z ∈ SN,(cid:15)n[w]} ≥ ρ.
z 0 ∞
Note that problem (N ) is a particular case of (N ) with (cid:15) = 0.
1 2 s
In the sequel, we refer to the sequences obeying H (resp., H = H (ρ)) as nuisance (resp.
0 1 1
signal) sequences.
Let ϕ(·) be a test, i.e. a Borel function on RN taking values in {0,1}, which receives
on input observation (1) (along with the data participating in the description of H and
0
H ). The event {ϕ(y) = 1} corresponds to rejecting the hypothesis H , while {ϕ(y) =
1 0
0} implies that H is rejected. The quality of the test is characterized by the error
1
probabilities – the probabilities of rejecting erroneously each of the hypotheses:
ε (ϕ;H ) = sup Prob {ϕ(y) = 1}, ε (ϕ;H (ρ)) = sup Prob {ϕ(y) = 0}.
0 0 x 1 1 x
x∈H0 x∈H1(ρ)
We define the risk of the test as
Risk(ϕ,ρ) = max{ε (ϕ;H ),ε (ϕ;H (ρ))}.
0 0 1 1
Let α ∈ (0,1/2) be given. In this paper we address the following question: for the testing
problems above, what is the smallest possible ρ such that one can distinguish (1 − α)-
reliably between the hypotheses H and H = H (ρ) via observation (1) (i.e., is such
0 1 1
that Risk(ϕ,ρ) ≤ α). In the sequel, we refer to such ρ as to α-resolution in the testing
problem in question, and our goal is to find reasonably tight upper and lower bounds on
this resolution along with the test underlying the upper bound.
3 Basic Test and Upper Resolution Bounds
In this section, we present a simple test which provides some upper bounds on the α-
resolutions in problems (P ) – (N ).
1 2
6
Preliminaries. Let Γ = {µ = exp{2πıτ/N} : 0 ≤ τ < N −1} be the set of roots of
N τ
1 of degree N, and let F : CN → C(Γ ) be the normalized Fourier transform:
N N
1 N−1
[F f](µ) = √ (cid:88) f µt, µ ∈ Γ . (3)
N t N
N
t=0
Note that (3) can also be seen as a mapping from S to C(Γ ).
N
Given a tolerance α ∈ (0,1/2), let q (α) be the (1−α)-quantile of (cid:107)F ξ(cid:107) , so that
N N ∞
Prob {(cid:107)F ξ(cid:107) ≥ q (α)} ≤ α.
ξ∼N(0,IN) N ∞ N
In the sequel we use the following immediate bound for q (·):
N
(cid:104) (cid:16) (cid:17) (cid:113) (cid:105)
inf max ErfInv sα , N−2 , for N even,
q (α) ≤ 0≤s≤1 4 2(1−s)α
N inf0≤s≤1max(cid:104)ErfInv(cid:16)s2α(cid:17), (cid:113)2(N1−−s1)α(cid:105), for N odd
(cid:113)
(cid:16) ln(N/α) (4)
Here a (cid:16) b means that the ratio a/b is in-between positive absolute constants, and
ErfInv(α) is the inverse error function: √1 (cid:82)∞ e−s2/2ds = α.
2π ErfInv(α)
The test we are about to consider (and which we refer to as basic test in the sequel) is
as follows:
1. given y, we solve the convex optimization problem
Opt (y) = min(cid:107)F (y −zN−1)(cid:107) , (5)
Z N 0 ∞
z∈Z
where the set Z is defined according to
{0}, for problem (P ),
1
Z = S[w], for problem (P ), (6)
2
S(cid:15)[w], for problems (N ) and (N ).
1 2
2. We compare Opt (y) to q (α), where α is a given tolerance: if Opt ≤ q (α), we
Z N N
accept H , otherwise we accept H .
0 1
We describe now the properties of the basic test as applied to problems (P ), (P ), (N )
1 2 1
and (N ).
2
Theorem 3.1 The risk of the basic test as applied to problems (P ), (P ) is bounded by
1 2
α, provided that d = d +d > 0 and
∗ n s
(cid:113)
ρ ≥ O(1)d3ln(2d )q (α)N−1/2 = O(1)d3ln(2d ) N−1ln(N/α) (7)
∗ ∗ N ∗ ∗
with properly chosen positive absolute constants O(1).
The result for the nonparametric problems (N ) and (N ) is similar:
1 2
7
Theorem 3.2 The risk of the basic test as applied to problems (N ), (N ) is bounded by
1 2
α, provided that d = d + d > 0, ρ satisfies (7) with properly selected O(1)’s and, in
∗ n s
addition, (cid:15) and (cid:15) are small enough, specifically,
n s
1 1
Ndn+2(cid:15) +Nds+2(cid:15) ≤ O(1)q (α) (8)
n s N
with properly selected positive absolute constant O(1).
The proofs of Theorems 3.1, 3.2 are relegated to section 6.
Theorems 3.1 and 3.2 provide us with upper resolution bounds independent of the
frequencies constituting w and w. When (cid:15) , (cid:15) are “small enough,” so that (8) holds
n s
true (we refer to the corresponding range of problems’ parameters as the parametric
zone), our upper bound on α-resolution in all testing problems of interest is essentially
(cid:113)
the same as in the case of (cid:15) = (cid:15) = 0 — it is C(d + d ) ln(N/α)/N with the factor
n s n s
C(d) = O(1)d3ln(2d) depending solely on d.
4 Lower Resolution Bounds
The lower resolution bounds of this section complement the upper bounds of section 3.
We start with the parametric setting (P ) and (P ). Through this section, c (d ,d ) are
1 2 i n s
properly selected positive and monotone functions of their arguments.
Proposition 4.1 Given integers d ≥ 0, d ≥ 1, and s real α ∈ (0,1/2), consider
n s
problems (P ) and (P ) with the data d (d = 0 in the case of problem (P )), d , α and
1 2 n n 1 s
dn
(cid:122) (cid:125)(cid:124) (cid:123)
w = {0,...,0}. Then for properly selected c (d ,d ) and all N ≥ c (d ,d ) the α-resolution
0 n s 0 n s
ρ in the problems (P ) and (P ) associated with the outlined data admits the lower bound
1 2
(cid:113)
O(1)d ln(1/α)/N.
s
We see that in the problem (P ) α-resolution grows with d at least linearly. Note
1 s
that by Theorem 3.1, this growth is at most cubic (more precisely, it is not faster than
O(1)d3ln(d )). Beside this, we see that the upper bounds on α-resolution for problems
s s
(P ) and (P ) stemming from Theorem 3.1 and associated with the basic test coincide,
1 2
within a factor depending solely on d ,d ,N and logarithmic in N, with lower bounds on
n s
α-resolution.
We have the following lower bound on the α-resolution in the problem (N ).
1
Proposition 4.2 Given integers d > 0, d ≥ 2 and reals α ∈ (0,1/2), (cid:15) ≥ 0, consider
n s n
dn
(cid:122) (cid:125)(cid:124) (cid:123)
problem (N ) with the data d , d , N, (cid:15) , α and w = {0,...,0}. Then for properly selected
1 n s n
c (d ,d ) > 0 depending solely on d ,d and for all N satisfying
i n s n s
N ≥ c (d ,d ), (9)
0 n s
the α-resolution ρ (α) in the problem (N ) associated with the outlined data satisfies:
∗ 1
8
(cid:113)
(i) in the range 0 ≤ (cid:15) ≤ c (d ,d )N−dn−1/2 ln(1/α),
n 1 n s
(cid:113)
ρ (α) ≥ c (d ,d ) ln(1/α)/N;
∗ 2 n s
(ii) in the range
(cid:113) (cid:113)
c (d ,d )N−dn−1/2 ln(1/α) ≤ (cid:15) ≤ c (d ,d )N ln(1/α), (10)
3 n s n 1 n s
(cid:104) (cid:105) 1 (cid:113)
ρ (α) ≥ c (d ,d ) (cid:15) Ndn+1/2[ln(1/α)]−1/2 2dn+3 N−1ln(1/α).
∗ 4 n s n
In the case of the problem (N ) we have a similar lower bound on α-resolution when
2
(cid:15) ≤ (cid:15) .
n s
Proposition 4.3 Given integers d > 0, d ≥ 2 and reals α ∈ (0,1/2), (cid:15) ≥ 0, consider
n s n
dn
(cid:122) (cid:125)(cid:124) (cid:123)
problem (N ) with the data d , d , N, (cid:15) , (cid:15) , α and w = {0,...,0}.
2 n s n s
Assume that 0 ≤ (cid:15) ≤ (cid:15) . Then for properly selected c (d ,d ) > 0 depending solely on
n s i n s
d ,d and all N ≥ c (d ,d ) the α-resolution ρ (α) in the problem (N ) associated with
n s 0 n s ∗ 2
the outlined data satisfies:
(cid:113)
(i) in the range 0 ≤ (cid:15) ≤ c (d ,d )N−ds−1/2 ln(1/α),
s 1 n s
(cid:113)
ρ (α) ≥ c (d ,d ) ln(1/α)/N;
∗ 2 n s
(ii) in the range
(cid:113) (cid:113)
c (d ,d )N−ds−1/2 ln(1/α) ≤ (cid:15) ≤ c (d ,d ) ln(1/α), (11)
3 n s s 1 n s
(cid:104) (cid:105) 1 (cid:113)
ρ (α) ≥ c (d ,d ) (cid:15) Nds+1/2[ln(1/α)]−1/2 2ds+1 N−1ln(1/α)
∗ 4 n s s (12)
1 ds
≥ c5(dn,ds)(cid:15)s2ds+1 (ln(1/α))2ds+1 .
The results of items (i) in Propositions 4.2 and 4.3 say that when d ,d are fixed, N
n s
is large and (cid:15) , (cid:15) are small enough so that the problem parameters are in the parametric
n s
zone (i.e., (8) holds), Theorem 3.2 describes “nearly correctly” (i.e., up to factors de-
pending solely on d ,d ,N and logarithmic in N) the α-resolution in problems (N ) and
n s 1
(N ): within such a factor, the α-resolution for problems (N ), (N ), same as for problems
2 1 2
(cid:113)
(P ), (P ), is ln(1/α)/N. Besides, items (ii) in Propositions 4.2 and 4.3 show that when
1 2
((cid:15) ,(cid:15) ) goes “far beyond” the range (8), the α-resolution in problems (N ), (N ) becomes
n s 1 2
(cid:113)
“much worse” than ln(1/α)/N.
5 Numerical Results
Below we report on preliminary numerical experiments with the basic test.
9
5.1 Problem (N )
1
The goal of the first series of experiments was to quantify “practical performance” of the
basic test as applied to problem (N ).
1
Organization of experiments. We consider problem (N ) on time horizon 0 ≤ t < N
1
for N ∈ {128,512,1024} with reliability threshold α = 0.01. In these experiments d = 4,
n
the frequencies in w are selected at random, (cid:15) = 0.01, and d = 4 (note that N and (cid:15) are
n s n
deliberately chosen not to satisfy (8)). As explained in section 3, the above setup data
specify the basic test for problem (N ), and our goal is to find the “empirical resolution”
1
of this test. To this end, we ran 10 experiments as follows. In a particular experiment,
• We draw at random w ∈ S , a shift s¯∈ S[w] and basic nuisance u ∈ S(cid:15)n[w].
4
• We generate a “true signal” x according to x = λs¯+ u, where λ > 0 is (nearly)
λ
as small as possible under the restriction that with x = x , the basic test “rejects
λ
reliably” the hypothesis H , namely, rejects it in every one of 15 trials with x = x
0 λ
and different realizations of the observation noise ξ, see (1)3.
• For the resulting λ, we compute ρ = min (cid:107)x − u(cid:107) , which is the output of
λ ∞
u∈S(cid:15)n[w]
the experiment. We believe that the collection of 10 outputs of this type gives a
good impression on the “true resolution” of the basic test. As a byproduct of an
experiment, we √get also the (cid:107) · (cid:107)∞-closest to xλ point ux ∈ S(cid:15)n[w]; the quantity
r = (cid:107)x −u (cid:107) / N can be thought of as a natural in our context “signal-to-noise
λ x 2
ratio.”
The results are presented in table 1. We would qualify them as quite compatible with
the theory we have developed: both empirical resolution and empirical signal-to-noise
ratio decreases with N as N−1/2. The “empirically observed” resolution ρ for which the
basic test (1−α)-reliably, α = 0.01, distinguishes between the hypotheses H and H (ρ)
0 1
(cid:113)
associated with problem (N ) is ≈ 6 ln(N/α)/N.
1
Comparison with MUSIC. An evident alternative to the basic test is (a) to apply the
standard MUSIC algorithm [11] in order to recover the spectrum of the observed signal,
(b) to delete from this spectrum the “nuisance frequencies”, and (c) to decide from the
remaining data if the signal of interest is present. Our related numerical results are, to the
best of our understanding, strongly in favor of the basic test. Let us look at figure 1 where
we present four MUSIC pseudospectra (we use pmusic function from MATLAB Signal
Processing Toolbox) of the observations associated with signals x obeying the hypothesis
H (ρ) (magenta) and of the observations coming from the (cid:107)·(cid:107) -closest to x nuisance (i.e.,
1 ∞
obeying the hypotheses H ) u (blue). ρ was chosen large enough for the basic test to
0 x
accept reliably the hypothesis H (ρ) when it is true. We see that while sometimes MUSIC
1
3Since a run of the test requires solving a nontrivial convex program, it would be too time-consuming
to replace 15 trials with few hundreds of them required to check reliably that the probability to reject
H , the signal being x , is at least the desired 1−α=0.99.
0 λ
10
