AustralieFrV1, Pmc, Corpus, bibRecord, 002A07

Fisher Information as a Metric of Locally Optimal Processing and Stochastic Resonance

Identifieur interne : 002A07 ( Pmc/Corpus ); précédent : 002A06; suivant : 002A08

Fisher Information as a Metric of Locally Optimal Processing and Stochastic Resonance

Auteurs : Fabing Duan ; François Chapeau-Blondeau ; Derek Abbott

Source :

PLoS ONE [ 1932-6203 ] ; 2012.

RBID : PMC:3320899

Abstract

The origins of Fisher information are in its use as a performance measure for parametric estimation. We augment this and show that the Fisher information can characterize the performance in several other significant signal processing operations. For processing of a weak signal in additive white noise, we demonstrate that the Fisher information determines (i) the maximum output signal-to-noise ratio for a periodic signal; (ii) the optimum asymptotic efficacy for signal detection; (iii) the best cross-correlation coefficient for signal transmission; and (iv) the minimum mean square error of an unbiased estimator. This unifying picture, via inequalities on the Fisher information, is used to establish conditions where improvement by noise through stochastic resonance is feasible or not.

Url:

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3320899

DOI: 10.1371/journal.pone.0034282
PubMed: 22493686
PubMed Central: 3320899

Links to Exploration step

PMC:3320899

Le document en format XML

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<author><name sortKey="Duan, Fabing" sort="Duan, Fabing" uniqKey="Duan F" first="Fabing" last="Duan">Fabing Duan</name>
<affiliation><nlm:aff id="aff1"><addr-line>College of Automation Engineering, Qingdao University, Qingdao, People's Republic of China</addr-line>
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<author><name sortKey="Chapeau Blondeau, Francois" sort="Chapeau Blondeau, Francois" uniqKey="Chapeau Blondeau F" first="François" last="Chapeau-Blondeau">François Chapeau-Blondeau</name>
<affiliation><nlm:aff id="aff2"><addr-line>Laboratoire d'Ingénierie des Systèmes Automatisés (LISA), Université d'Angers, Angers, France</addr-line>
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<author><name sortKey="Abbott, Derek" sort="Abbott, Derek" uniqKey="Abbott D" first="Derek" last="Abbott">Derek Abbott</name>
<affiliation><nlm:aff id="aff3"><addr-line>Centre for Biomedical Engineering (CBME) and School of Electrical & Electronic Engineering, The University of Adelaide, Adelaide, Southern Australia, Australia</addr-line>
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<front><div type="abstract" xml:lang="en"><p>The origins of Fisher information are in its use as a performance measure for parametric estimation. We augment this and show that the Fisher information can characterize the performance in several other significant signal processing operations. For processing of a weak signal in additive white noise, we demonstrate that the Fisher information determines (i) the maximum output signal-to-noise ratio for a periodic signal; (ii) the optimum asymptotic efficacy for signal detection; (iii) the best cross-correlation coefficient for signal transmission; and (iv) the minimum mean square error of an unbiased estimator. This unifying picture, via inequalities on the Fisher information, is used to establish conditions where improvement by noise through stochastic resonance is feasible or not.</p>
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<pmc article-type="research-article"><pmc-dir>properties open_access</pmc-dir>
  <front><journal-meta><journal-id journal-id-type="nlm-ta">PLoS One</journal-id>
<journal-id journal-id-type="iso-abbrev">PLoS ONE</journal-id>
<journal-id journal-id-type="publisher-id">plos</journal-id>
<journal-id journal-id-type="pmc">plosone</journal-id>
<journal-title-group><journal-title>PLoS ONE</journal-title>
</journal-title-group>
<issn pub-type="epub">1932-6203</issn>
<publisher><publisher-name>Public Library of Science</publisher-name>
<publisher-loc>San Francisco, USA</publisher-loc>
</publisher>
</journal-meta>
<article-meta><article-id pub-id-type="pmid">22493686</article-id>
<article-id pub-id-type="pmc">3320899</article-id>
<article-id pub-id-type="publisher-id">PONE-D-12-03221</article-id>
<article-id pub-id-type="doi">10.1371/journal.pone.0034282</article-id>
<article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject>
</subj-group>
<subj-group subj-group-type="Discipline-v2"><subject>Engineering</subject>
<subj-group><subject>Signal Processing</subject>
</subj-group>
</subj-group>
<subj-group subj-group-type="Discipline-v2"><subject>Mathematics</subject>
<subj-group><subject>Probability Theory</subject>
</subj-group>
</subj-group>
<subj-group subj-group-type="Discipline-v2"><subject>Physics</subject>
<subj-group><subject>Interdisciplinary Physics</subject>
</subj-group>
</subj-group>
</article-categories>
<title-group><article-title>Fisher Information as a Metric of Locally Optimal Processing and Stochastic Resonance</article-title>
<alt-title alt-title-type="running-head">Fisher Information for Stochastic Resonance</alt-title>
</title-group>
<contrib-group><contrib contrib-type="author"><name><surname>Duan</surname>
<given-names>Fabing</given-names>
</name>
<xref ref-type="aff" rid="aff1"><sup>1</sup>
</xref>
<xref ref-type="corresp" rid="cor1"><sup>*</sup>
</xref>
</contrib>
<contrib contrib-type="author"><name><surname>Chapeau-Blondeau</surname>
<given-names>François</given-names>
</name>
<xref ref-type="aff" rid="aff2"><sup>2</sup>
</xref>
</contrib>
<contrib contrib-type="author"><name><surname>Abbott</surname>
<given-names>Derek</given-names>
</name>
<xref ref-type="aff" rid="aff3"><sup>3</sup>
</xref>
</contrib>
</contrib-group>
<aff id="aff1"><label>1</label>
<addr-line>College of Automation Engineering, Qingdao University, Qingdao, People's Republic of China</addr-line>
</aff>
<aff id="aff2"><label>2</label>
<addr-line>Laboratoire d'Ingénierie des Systèmes Automatisés (LISA), Université d'Angers, Angers, France</addr-line>
</aff>
<aff id="aff3"><label>3</label>
<addr-line>Centre for Biomedical Engineering (CBME) and School of Electrical & Electronic Engineering, The University of Adelaide, Adelaide, Southern Australia, Australia</addr-line>
</aff>
<contrib-group><contrib contrib-type="editor"><name><surname>Perc</surname>
<given-names>Matjaz</given-names>
</name>
<role>Editor</role>
<xref ref-type="aff" rid="edit1"></xref>
</contrib>
</contrib-group>
<aff id="edit1">University of Maribor, Slovenia</aff>
<author-notes><corresp id="cor1">* E-mail: <email>fabing.duan@gmail.com</email>
</corresp>
<fn fn-type="con"><p>Conceived and designed the experiments: FD. Performed the experiments: FD FCB DA. Analyzed the data: FD FCB DA. Contributed reagents/materials/analysis tools: FCB DA. Wrote the paper: FD. Proofreading: FCB DA.</p>
</fn>
</author-notes>
<pub-date pub-type="collection"><year>2012</year>
</pub-date>
<pub-date pub-type="epub"><day>6</day>
<month>4</month>
<year>2012</year>
</pub-date>
<volume>7</volume>
<issue>4</issue>
<elocation-id>e34282</elocation-id>
<history><date date-type="received"><day>1</day>
<month>2</month>
<year>2012</year>
</date>
<date date-type="accepted"><day>25</day>
<month>2</month>
<year>2012</year>
</date>
</history>
<permissions><copyright-statement>Duan et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.</copyright-statement>
<copyright-year>2012</copyright-year>
</permissions>
<abstract><p>The origins of Fisher information are in its use as a performance measure for parametric estimation. We augment this and show that the Fisher information can characterize the performance in several other significant signal processing operations. For processing of a weak signal in additive white noise, we demonstrate that the Fisher information determines (i) the maximum output signal-to-noise ratio for a periodic signal; (ii) the optimum asymptotic efficacy for signal detection; (iii) the best cross-correlation coefficient for signal transmission; and (iv) the minimum mean square error of an unbiased estimator. This unifying picture, via inequalities on the Fisher information, is used to establish conditions where improvement by noise through stochastic resonance is feasible or not.</p>
</abstract>
<counts><page-count count="6"></page-count>
</counts>
</article-meta>
</front>
<body><sec id="s1"><title>Introduction</title>
<p>Fisher information is foremost a measure of the minimum error in estimating an unknown parameter of a probability distribution, and its importance is related to the Cramér-Rao inequality for unbiased estimators <xref ref-type="bibr" rid="pone.0034282-Cover1">[1]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
. By introducing a location parameter, the de Bruijn's identity indicates that the fundamental quantity of Fisher information is affiliated with the differential entropy of the minimum descriptive complexity of a random variable <xref ref-type="bibr" rid="pone.0034282-Cover1">[1]</xref>
. Furthermore, in known weak signal detection, a locally optimal detector, acting as the small-signal limited Neyman-Pearson detector, has favorable properties for small signal-to-noise ratios <xref ref-type="bibr" rid="pone.0034282-Capon1">[3]</xref>
. With sufficiently large observed data and using the central limit theorem, it is demonstrated that the locally optimal detector is asymptotically optimum and the Fisher information of the noise distribution is the upper bound of the asymptotic efficacy <xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
–<xref ref-type="bibr" rid="pone.0034282-Song1">[7]</xref>
. For weak random signal detection, the second order Fisher information is also associated with the maximum asymptotic efficacy of the generalized energy detector <xref ref-type="bibr" rid="pone.0034282-Poor1">[4]</xref>
–<xref ref-type="bibr" rid="pone.0034282-Song1">[7]</xref>
.</p>
<p>However, the fundamental nature of Fisher information is not adequately recognized for processing weak signals. To extend the heuristic studies of <xref ref-type="bibr" rid="pone.0034282-Cover1">[1]</xref>
–<xref ref-type="bibr" rid="pone.0034282-Song1">[7]</xref>
, in this paper, we will theoretically demonstrate that, for a weak signal buried in additive white noise, the performance for locally optimal processing can be generally measured by the Fisher information of the noise distribution. We show this for the following signal processing case studies: (i) the maximum output signal-to-noise ratio for a periodic signal; (ii) the optimum asymptotic efficacy for signal detection; (iii) the best cross-correlation coefficient for signal transmission; and (iv) the minimum mean square error of an unbiased estimator. The physical significance of Fisher information is that it provides a unified bound for characterizing the performance for locally optimal processing. Furthermore, we establish the Fisher information condition for stochastic resonance (SR) that has been studied for improving system performance over several decades <xref ref-type="bibr" rid="pone.0034282-Benzi1">[8]</xref>
–<xref ref-type="bibr" rid="pone.0034282-Hnggi1">[32]</xref>
. In our recent work <xref ref-type="bibr" rid="pone.0034282-Duan1">[28]</xref>
, it is established that improvement by adding noise is impossible for detecting a weak known signal. Here, based on Fisher information inequalities, we further prove that SR is not applicable for improving the performance of locally optimal processing in the considered cases (i)–(iv). This result generalizes a proof that existed previously only for a weak periodic signal in additive Gaussian noise <xref ref-type="bibr" rid="pone.0034282-Dykman1">[12]</xref>
, <xref ref-type="bibr" rid="pone.0034282-DeWeese1">[33]</xref>
. However, beyond these restrictive conditions, the observed noise-enhanced effects <xref ref-type="bibr" rid="pone.0034282-Zozor1">[9]</xref>
–<xref ref-type="bibr" rid="pone.0034282-ChapeauBlondeau2">[11]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kay2">[26]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Duan1">[28]</xref>
–<xref ref-type="bibr" rid="pone.0034282-Greenwood1">[30]</xref>
 show that SR can provide a signal processing enhancement using the constructive role of noise. The applications of SR to nonlinear signal processing are of practical interest.</p>
</sec>
<sec id="s2"><title>Results</title>
<p>In many situations we are interested in processing signals that are very weak compared to the noise level <xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Capon1">[3]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
. It would be desirable in these situations to determine an optimal memoryless nonlinearity in the following study cases.</p>
<sec id="s2a"><title>Output signal-to-noise ratio for a periodic signal</title>
<p>First, consider a static nonlinearity with its output<disp-formula><graphic xlink:href="pone.0034282.e001"></graphic>
<label>(1)</label>
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where the function <inline-formula><inline-graphic xlink:href="pone.0034282.e002.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is a memoryless nonlinearity and the input is a signal-plus-noise mixture <inline-formula><inline-graphic xlink:href="pone.0034282.e003.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The component <inline-formula><inline-graphic xlink:href="pone.0034282.e004.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is a known weak periodic signal with a maximal amplitude <inline-formula><inline-graphic xlink:href="pone.0034282.e005.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (<inline-formula><inline-graphic xlink:href="pone.0034282.e006.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) and period <inline-formula><inline-graphic xlink:href="pone.0034282.e007.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Zero-mean white noise <inline-formula><inline-graphic xlink:href="pone.0034282.e008.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, independent of <inline-formula><inline-graphic xlink:href="pone.0034282.e009.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, has probability density function (PDF) <inline-formula><inline-graphic xlink:href="pone.0034282.e010.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and a root-mean-square (RMS) amplitude <inline-formula><inline-graphic xlink:href="pone.0034282.e011.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. It is assumed that <inline-formula><inline-graphic xlink:href="pone.0034282.e012.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 has zero mean under <inline-formula><inline-graphic xlink:href="pone.0034282.e013.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, i.e. <inline-formula><inline-graphic xlink:href="pone.0034282.e014.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, which is not restrictive since any arbitrary <inline-formula><inline-graphic xlink:href="pone.0034282.e015.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 can always include a constant bias to cancel this average <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
. The input signal-to-noise ratio for <inline-formula><inline-graphic xlink:href="pone.0034282.e016.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 can be defined as the power contained in the spectral line <inline-formula><inline-graphic xlink:href="pone.0034282.e017.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 divided by the power contained in the noise background in a small frequency bin <inline-formula><inline-graphic xlink:href="pone.0034282.e018.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 around <inline-formula><inline-graphic xlink:href="pone.0034282.e019.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-ChapeauBlondeau1">[10]</xref>
, this is<disp-formula><graphic xlink:href="pone.0034282.e020"></graphic>
<label>(2)</label>
</disp-formula>
with <inline-formula><inline-graphic xlink:href="pone.0034282.e021.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 indicating the time resolution or the sampling time in a discrete-time implementation and the temporal average defined as <inline-formula><inline-graphic xlink:href="pone.0034282.e022.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-ChapeauBlondeau1">[10]</xref>
. Here, we assume the sampling time <inline-formula><inline-graphic xlink:href="pone.0034282.e023.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and observe the output <inline-formula><inline-graphic xlink:href="pone.0034282.e024.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 for a sufficiently large time interval of <inline-formula><inline-graphic xlink:href="pone.0034282.e025.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (<inline-formula><inline-graphic xlink:href="pone.0034282.e026.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) <xref ref-type="bibr" rid="pone.0034282-ChapeauBlondeau1">[10]</xref>
. Since <inline-formula><inline-graphic xlink:href="pone.0034282.e027.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is periodic, <inline-formula><inline-graphic xlink:href="pone.0034282.e028.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is in general a cyclostationary random signal with period <inline-formula><inline-graphic xlink:href="pone.0034282.e029.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-ChapeauBlondeau1">[10]</xref>
. Similarly, the output signal-to-noise ratio for <inline-formula><inline-graphic xlink:href="pone.0034282.e030.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is given by<disp-formula><graphic xlink:href="pone.0034282.e031"></graphic>
<label>(3)</label>
</disp-formula>
with nonstationary expectation <inline-formula><inline-graphic xlink:href="pone.0034282.e032.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and nonstationary variance <inline-formula><inline-graphic xlink:href="pone.0034282.e033.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-ChapeauBlondeau1">[10]</xref>
.</p>
<p>In the case of <inline-formula><inline-graphic xlink:href="pone.0034282.e034.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, we have a Taylor expansion of the expectation at a fixed time <inline-formula><inline-graphic xlink:href="pone.0034282.e035.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 as<disp-formula><graphic xlink:href="pone.0034282.e036"></graphic>
<label>(4)</label>
</disp-formula>
where we assume the derivatives <inline-formula><inline-graphic xlink:href="pone.0034282.e037.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and <inline-formula><inline-graphic xlink:href="pone.0034282.e038.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 exist for almost all <inline-formula><inline-graphic xlink:href="pone.0034282.e039.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (similarly hereinafter) <xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
. Thus, we have<disp-formula><graphic xlink:href="pone.0034282.e040"></graphic>
<label>(5)</label>
</disp-formula>
where <inline-formula><inline-graphic xlink:href="pone.0034282.e041.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and <inline-formula><inline-graphic xlink:href="pone.0034282.e042.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, compared with <inline-formula><inline-graphic xlink:href="pone.0034282.e043.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, can be neglected as <inline-formula><inline-graphic xlink:href="pone.0034282.e044.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (<inline-formula><inline-graphic xlink:href="pone.0034282.e045.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) <xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
. The above derivations of Eqs. (4) and (5) are exact in the asymptotic limit for weak signals, and have been generally adopted in <xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
.</p>
<p>Substituting Eqs. (4) and (5) into Eq. (3), we have<disp-formula><graphic xlink:href="pone.0034282.e046"></graphic>
<label>(6)</label>
</disp-formula>
where the expectation <inline-formula><inline-graphic xlink:href="pone.0034282.e047.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is simply the Fisher information <inline-formula><inline-graphic xlink:href="pone.0034282.e048.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 of the noise PDF <inline-formula><inline-graphic xlink:href="pone.0034282.e049.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
, and the equality occurs as<disp-formula><graphic xlink:href="pone.0034282.e050"></graphic>
<label>(7)</label>
</disp-formula>
by the Cauchy-Schwarz inequality for a constant <inline-formula><inline-graphic xlink:href="pone.0034282.e051.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
.</p>
<p>Noting Eqs. (2) and (6), the output-input signal-to-noise ratio gain <inline-formula><inline-graphic xlink:href="pone.0034282.e052.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is bounded by<disp-formula><graphic xlink:href="pone.0034282.e053"></graphic>
<label>(8)</label>
</disp-formula>
with equality achieved when <inline-formula><inline-graphic xlink:href="pone.0034282.e054.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 takes the locally optimal nonlinearity <inline-formula><inline-graphic xlink:href="pone.0034282.e055.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 of Eq. (7). Here, for a standardized PDF <inline-formula><inline-graphic xlink:href="pone.0034282.e056.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 with zero mean and unity variance <inline-formula><inline-graphic xlink:href="pone.0034282.e057.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the scaled noise <inline-formula><inline-graphic xlink:href="pone.0034282.e058.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 has its PDF <inline-formula><inline-graphic xlink:href="pone.0034282.e059.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and the Fisher information satisfies <inline-formula><inline-graphic xlink:href="pone.0034282.e060.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Cover1">[1]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Brown1">[34]</xref>
. It is known that a standardized Gaussian PDF <inline-formula><inline-graphic xlink:href="pone.0034282.e061.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 has the minimal Fisher information <inline-formula><inline-graphic xlink:href="pone.0034282.e062.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and any standardized non-Gaussian PDF <inline-formula><inline-graphic xlink:href="pone.0034282.e063.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 has the Fisher information <inline-formula><inline-graphic xlink:href="pone.0034282.e064.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
. It can be seen that, the linear system <inline-formula><inline-graphic xlink:href="pone.0034282.e065.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 has its output signal-to-noise ratio <inline-formula><inline-graphic xlink:href="pone.0034282.e066.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 in Eq. (3). Thus, the output-input signal-to-noise ratio gain <inline-formula><inline-graphic xlink:href="pone.0034282.e067.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 in Eq. (8) also clearly represents the expected performance improvement of the nonlinearity <inline-formula><inline-graphic xlink:href="pone.0034282.e068.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 over the linear system <inline-formula><inline-graphic xlink:href="pone.0034282.e069.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
</sec>
<sec id="s2b"><title>Optimum asymptotic efficacy for signal detection</title>
<p>Secondly, we consider the observation vector <inline-formula><inline-graphic xlink:href="pone.0034282.e070.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 of real-valued components <inline-formula><inline-graphic xlink:href="pone.0034282.e071.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 by<disp-formula><graphic xlink:href="pone.0034282.e072"></graphic>
<label>(9)</label>
</disp-formula>
where the components <inline-formula><inline-graphic xlink:href="pone.0034282.e073.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 form a sequence of independent and identically distributed (i.i.d.) random variables with PDF <inline-formula><inline-graphic xlink:href="pone.0034282.e074.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and the known signal components <inline-formula><inline-graphic xlink:href="pone.0034282.e075.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 are with the signal strength <inline-formula><inline-graphic xlink:href="pone.0034282.e076.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
. For the known signal sequence <inline-formula><inline-graphic xlink:href="pone.0034282.e077.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, it is assumed that there exists a finite (non-zero) bound <inline-formula><inline-graphic xlink:href="pone.0034282.e078.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 such that <inline-formula><inline-graphic xlink:href="pone.0034282.e079.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and the asymptotic average signal power is finite and non-zero, i.e. <inline-formula><inline-graphic xlink:href="pone.0034282.e080.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
. Then, the detection problem can be formulated as a hypothesis-testing problem of deciding a null hypothesis <inline-formula><inline-graphic xlink:href="pone.0034282.e081.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (<inline-formula><inline-graphic xlink:href="pone.0034282.e082.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) and an alternative hypothesis <inline-formula><inline-graphic xlink:href="pone.0034282.e083.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (<inline-formula><inline-graphic xlink:href="pone.0034282.e084.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) describing the joint density function of <inline-formula><inline-graphic xlink:href="pone.0034282.e085.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 with<disp-formula><graphic xlink:href="pone.0034282.e086"></graphic>
<label>(10)</label>
</disp-formula>
Consider a generalized correlation detector<disp-formula><graphic xlink:href="pone.0034282.e087"></graphic>
<label>(11)</label>
</disp-formula>
where the memoryless nonlinearity <inline-formula><inline-graphic xlink:href="pone.0034282.e088.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 has zero mean under <inline-formula><inline-graphic xlink:href="pone.0034282.e089.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, i.e. <inline-formula><inline-graphic xlink:href="pone.0034282.e090.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
. In the asymptotic case of <inline-formula><inline-graphic xlink:href="pone.0034282.e091.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and <inline-formula><inline-graphic xlink:href="pone.0034282.e092.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the test statistic <inline-formula><inline-graphic xlink:href="pone.0034282.e093.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, according to the central limit theorem, converges to a Gaussian distribution with mean <inline-formula><inline-graphic xlink:href="pone.0034282.e094.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and variance <inline-formula><inline-graphic xlink:href="pone.0034282.e095.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 under the null hypotheses <inline-formula><inline-graphic xlink:href="pone.0034282.e096.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
. Using Eqs. (4) and (5), <inline-formula><inline-graphic xlink:href="pone.0034282.e097.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is asymptotically Gaussian with mean <inline-formula><inline-graphic xlink:href="pone.0034282.e098.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and variance <inline-formula><inline-graphic xlink:href="pone.0034282.e099.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 under the hypothesis <inline-formula><inline-graphic xlink:href="pone.0034282.e100.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
.</p>
<p>Given a false alarm probability <inline-formula><inline-graphic xlink:href="pone.0034282.e101.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the asymptotic detection probability <inline-formula><inline-graphic xlink:href="pone.0034282.e102.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 for the generalized correlation detector of Eq. (11) can be expressed as <xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
<disp-formula><graphic xlink:href="pone.0034282.e103"></graphic>
<label>(12)</label>
</disp-formula>
with <inline-formula><inline-graphic xlink:href="pone.0034282.e104.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and its inverse function <inline-formula><inline-graphic xlink:href="pone.0034282.e105.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
. Thus, for fixed <inline-formula><inline-graphic xlink:href="pone.0034282.e106.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and <inline-formula><inline-graphic xlink:href="pone.0034282.e107.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (since the signal is known), <inline-formula><inline-graphic xlink:href="pone.0034282.e108.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is a monotonically increasing function of the normalized asymptotic efficacy <inline-formula><inline-graphic xlink:href="pone.0034282.e109.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 given by <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
<disp-formula><graphic xlink:href="pone.0034282.e110"></graphic>
<label>(13)</label>
</disp-formula>
with equality being achieved when <inline-formula><inline-graphic xlink:href="pone.0034282.e111.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 in Eq. (7). This result also indicates that the asymptotic optimal detector is just the locally optimal detector established by the Taylor expansion of the likelihood ratio test statistic <inline-formula><inline-graphic xlink:href="pone.0034282.e112.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (<inline-formula><inline-graphic xlink:href="pone.0034282.e113.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) in terms of the generalized Neyman-Pearson lemma <xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
.</p>
<p>Interestingly, with <inline-formula><inline-graphic xlink:href="pone.0034282.e114.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 achieved by a linear correlation detector (<inline-formula><inline-graphic xlink:href="pone.0034282.e115.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 in Eq. (11)) as a benchmark <xref ref-type="bibr" rid="pone.0034282-Lu1">[5]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
, the asymptotic relative efficiency<disp-formula><graphic xlink:href="pone.0034282.e116"></graphic>
<label>(14)</label>
</disp-formula>
provides an asymptotic performance improvement of a generalized correlation detector over the linear correlation detector when both detectors operate in the same noise environment <xref ref-type="bibr" rid="pone.0034282-Lu1">[5]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
.</p>
<p>Next, consider the weak random signal components <inline-formula><inline-graphic xlink:href="pone.0034282.e117.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 has PDF <inline-formula><inline-graphic xlink:href="pone.0034282.e118.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 with zero mean <inline-formula><inline-graphic xlink:href="pone.0034282.e119.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and variance <inline-formula><inline-graphic xlink:href="pone.0034282.e120.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 in the observation model of Eq. (9) <xref ref-type="bibr" rid="pone.0034282-Lu1">[5]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
. Here, the signal components <inline-formula><inline-graphic xlink:href="pone.0034282.e121.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 are i.i.d. Then, this random signal hypothesis test becomes <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
<disp-formula><graphic xlink:href="pone.0034282.e122"></graphic>
</disp-formula>
for determining whether the random signal is present or not. Consider a generalized energy detector<disp-formula><graphic xlink:href="pone.0034282.e123"></graphic>
<label>(15)</label>
</disp-formula>
where we also assume <inline-formula><inline-graphic xlink:href="pone.0034282.e124.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and then <inline-formula><inline-graphic xlink:href="pone.0034282.e125.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Furthermore, in the asymptotic case of <inline-formula><inline-graphic xlink:href="pone.0034282.e126.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the expectation <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
<disp-formula><graphic xlink:href="pone.0034282.e127"></graphic>
<label>(16)</label>
</disp-formula>
Thus, the efficacy of a generalized energy detector is defined as <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
<disp-formula><graphic xlink:href="pone.0034282.e128"></graphic>
<label>(17)</label>
</disp-formula>
where <inline-formula><inline-graphic xlink:href="pone.0034282.e129.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is treated as the signal strength parameter and <inline-formula><inline-graphic xlink:href="pone.0034282.e130.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is the second order Fisher information <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Song1">[7]</xref>
. It is noted that the equality of Eq. (17) is achieved as <inline-formula><inline-graphic xlink:href="pone.0034282.e131.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 for a constant <inline-formula><inline-graphic xlink:href="pone.0034282.e132.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
. Given a false alarm probability <inline-formula><inline-graphic xlink:href="pone.0034282.e133.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the asymptotic detection probability <inline-formula><inline-graphic xlink:href="pone.0034282.e134.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 for the generalized energy detector of Eq. (15) is a monotonically increasing function of the efficacy <inline-formula><inline-graphic xlink:href="pone.0034282.e135.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Lu1">[5]</xref>
–<xref ref-type="bibr" rid="pone.0034282-Song1">[7]</xref>
.</p>
</sec>
<sec id="s2c"><title>Cross-correlation coefficient for signal transmission</title>
<p>Thirdly, we transmit a weak aperiodic signal <inline-formula><inline-graphic xlink:href="pone.0034282.e136.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 through the nonlinearity <inline-formula><inline-graphic xlink:href="pone.0034282.e137.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 of Eq. (1) <xref ref-type="bibr" rid="pone.0034282-Collins1">[13]</xref>
. Here, the signal <inline-formula><inline-graphic xlink:href="pone.0034282.e138.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is with the average signal variance <inline-formula><inline-graphic xlink:href="pone.0034282.e139.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the zero mean and the upper bound A (<inline-formula><inline-graphic xlink:href="pone.0034282.e140.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). For example, <inline-formula><inline-graphic xlink:href="pone.0034282.e141.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 can be a sample according to a uniformly distributed random signal equally taking values from a bounded interval. The input cross-correlation coefficient of <inline-formula><inline-graphic xlink:href="pone.0034282.e142.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and <inline-formula><inline-graphic xlink:href="pone.0034282.e143.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is defined as <xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Collins1">[13]</xref>
<disp-formula><graphic xlink:href="pone.0034282.e144"></graphic>
<label>(18)</label>
</disp-formula>
Using Eqs. (4) and (5), the output cross-correlation coefficient of <inline-formula><inline-graphic xlink:href="pone.0034282.e145.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and <inline-formula><inline-graphic xlink:href="pone.0034282.e146.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is given by<disp-formula><graphic xlink:href="pone.0034282.e147"></graphic>
<label>(19)</label>
</disp-formula>
which has its maximal value as <inline-formula><inline-graphic xlink:href="pone.0034282.e148.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 of Eq. (7). Then, the cross-correlation gain <inline-formula><inline-graphic xlink:href="pone.0034282.e149.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is bounded by<disp-formula><graphic xlink:href="pone.0034282.e150"></graphic>
<label>(20)</label>
</disp-formula>
</p>
</sec>
<sec id="s2d"><title>Mean square error of an unbiased estimator</title>
<p>Finally, for the <inline-formula><inline-graphic xlink:href="pone.0034282.e151.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 observation components <inline-formula><inline-graphic xlink:href="pone.0034282.e152.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, we assume the signal <inline-formula><inline-graphic xlink:href="pone.0034282.e153.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 are with an unknown parameter <inline-formula><inline-graphic xlink:href="pone.0034282.e154.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. As the upper bound <inline-formula><inline-graphic xlink:href="pone.0034282.e155.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (<inline-formula><inline-graphic xlink:href="pone.0034282.e156.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), the Cramér-Rao inequality indicates that the mean squared error of any unbiased estimator of the parameter <inline-formula><inline-graphic xlink:href="pone.0034282.e157.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is lower bounded by the reciprocal of the Fisher information <xref ref-type="bibr" rid="pone.0034282-Cover1">[1]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
 given by<disp-formula><graphic xlink:href="pone.0034282.e158"></graphic>
<label>(21)</label>
</disp-formula>
which indicates that the minimum mean square error of any unbiased estimator is also determined by the Fisher information <inline-formula><inline-graphic xlink:href="pone.0034282.e159.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 of a distribution, as <inline-formula><inline-graphic xlink:href="pone.0034282.e160.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is given.</p>
<p>Therefore, just as the Fisher information represents the lower bound of the mean squared error of any unbiased estimator in signal estimation <xref ref-type="bibr" rid="pone.0034282-Cover1">[1]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
, the physical significance of the Fisher information <inline-formula><inline-graphic xlink:href="pone.0034282.e161.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(<inline-formula><inline-graphic xlink:href="pone.0034282.e162.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) is that it provides a unified upper bound of the performance for locally optimal processing in the considered signal processing cases.</p>
<p>Aiming to explain the upper bound of the performance for locally optimal processing as Fisher information, we here show an illustrative example in <xref ref-type="fig" rid="pone-0034282-g001">Fig. 1</xref>
. Consider the generalized Gaussian noise with PDF<disp-formula><graphic xlink:href="pone.0034282.e163"></graphic>
<label>(22)</label>
</disp-formula>
where <inline-formula><inline-graphic xlink:href="pone.0034282.e164.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, <inline-formula><inline-graphic xlink:href="pone.0034282.e165.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 for a rate of exponential decay parameter <inline-formula><inline-graphic xlink:href="pone.0034282.e166.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Kay1">[2]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
. The corresponding locally optimal nonlinearity is <inline-formula><inline-graphic xlink:href="pone.0034282.e167.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and the output-input signal-to-noise ratio gain in Eq. (8) is <inline-formula><inline-graphic xlink:href="pone.0034282.e168.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (solid line), as shown in <xref ref-type="fig" rid="pone-0034282-g001">Fig. 1</xref>
. For comparison, we also operate the sign nonlinearity <inline-formula><inline-graphic xlink:href="pone.0034282.e169.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and the linear system <inline-formula><inline-graphic xlink:href="pone.0034282.e170.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 in the generalized Gaussian noise. The output-input signal-to-noise ratio gain in Eq. (8) of <inline-formula><inline-graphic xlink:href="pone.0034282.e171.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is <inline-formula><inline-graphic xlink:href="pone.0034282.e172.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (dashed line), as shown in <xref ref-type="fig" rid="pone-0034282-g001">Fig. 1</xref>
. For the linear system <inline-formula><inline-graphic xlink:href="pone.0034282.e173.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, Eq. (8) indicates that <inline-formula><inline-graphic xlink:href="pone.0034282.e174.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (dotted line) for <inline-formula><inline-graphic xlink:href="pone.0034282.e175.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, as plotted in <xref ref-type="fig" rid="pone-0034282-g001">Fig. 1</xref>
. It is seen in <xref ref-type="fig" rid="pone-0034282-g001">Fig. 1</xref>
 that, only for <inline-formula><inline-graphic xlink:href="pone.0034282.e176.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the performance of <inline-formula><inline-graphic xlink:href="pone.0034282.e177.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 attains that of the locally optimal nonlinearity of <inline-formula><inline-graphic xlink:href="pone.0034282.e178.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. This is because, the nonlinearity <inline-formula><inline-graphic xlink:href="pone.0034282.e179.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is just the locally optimal nonlinearity for Laplacian noise (<inline-formula><inline-graphic xlink:href="pone.0034282.e180.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), and the Fisher information limit <inline-formula><inline-graphic xlink:href="pone.0034282.e181.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is achieved. Likewise, for Gaussian noise (<inline-formula><inline-graphic xlink:href="pone.0034282.e182.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), the linear system <inline-formula><inline-graphic xlink:href="pone.0034282.e183.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is optimal and the output-input SNR gain <inline-formula><inline-graphic xlink:href="pone.0034282.e184.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. It is noted that the above analyses are also valid for the asymptotic relative efficiency of Eq. (14) and the cross-correlation gain of Eq. (20).</p>
<fig id="pone-0034282-g001" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0034282.g001</object-id>
<label>Figure 1</label>
<caption><title>The output-input signal-to-noise ratio gain</title>
<p><inline-formula><inline-graphic xlink:href="pone.0034282.e185.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<bold>.</bold>
 The output-input signal-to-noise ratio gain <inline-formula><inline-graphic xlink:href="pone.0034282.e186.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 versus the exponential decay parameter <inline-formula><inline-graphic xlink:href="pone.0034282.e187.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 of the generalized Gaussian noise for the locally optimal nonlinearity <inline-formula><inline-graphic xlink:href="pone.0034282.e188.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (solid line), the sign nonlinearity <inline-formula><inline-graphic xlink:href="pone.0034282.e189.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (red line) and the linear system <inline-formula><inline-graphic xlink:href="pone.0034282.e190.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (dotted line), respectively.</p>
</caption>
<graphic xlink:href="pone.0034282.g001"></graphic>
</fig>
</sec>
<sec id="s2e"><title>Fisher information condition for stochastic resonance</title>
<p>Stochastic resonance (SR), being contrary to conventional approaches of suppressing noise, adds an appropriate amount of noise to a nonlinear system to improve its performance <xref ref-type="bibr" rid="pone.0034282-Benzi1">[8]</xref>
–<xref ref-type="bibr" rid="pone.0034282-Hnggi1">[32]</xref>
. SR emerged from the field of climate dynamics <xref ref-type="bibr" rid="pone.0034282-Benzi1">[8]</xref>
, and the topic has flourished in physics <xref ref-type="bibr" rid="pone.0034282-Gammaitoni1">[15]</xref>
–<xref ref-type="bibr" rid="pone.0034282-Gosaka1">[19]</xref>
 and neuroscience <xref ref-type="bibr" rid="pone.0034282-Collins1">[13]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Bezrukov1">[14]</xref>
, <xref ref-type="bibr" rid="pone.0034282-McDonnell1">[20]</xref>
. The notion of SR has been widened to include a number of different mechanisms <xref ref-type="bibr" rid="pone.0034282-Gammaitoni1">[15]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Zaikin1">[17]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Stocks1">[25]</xref>
, and SR effects have also been demonstrated in various extended systems <xref ref-type="bibr" rid="pone.0034282-Zozor1">[9]</xref>
–<xref ref-type="bibr" rid="pone.0034282-McDonnell1">[20]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Stocks1">[25]</xref>
 and complex networks <xref ref-type="bibr" rid="pone.0034282-Perc2">[21]</xref>
–<xref ref-type="bibr" rid="pone.0034282-Gan1">[24]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Ward1">[27]</xref>
.</p>
<p>An open question concerning SR is that, under the asymptotic cases of weak signal and large sample size, can SR play a role in locally optimal processing? Here, based on the Fisher information inequalities, we will demonstrate that SR is inapplicable to performance improvement for locally optimal processing.</p>
<p>For a given observation <inline-formula><inline-graphic xlink:href="pone.0034282.e191.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, we add the extra noise <inline-formula><inline-graphic xlink:href="pone.0034282.e192.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, independent of the initial noise <inline-formula><inline-graphic xlink:href="pone.0034282.e193.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and the signal <inline-formula><inline-graphic xlink:href="pone.0034282.e194.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, to <inline-formula><inline-graphic xlink:href="pone.0034282.e195.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Then, the updated data <inline-formula><inline-graphic xlink:href="pone.0034282.e196.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Here, the composite noise <inline-formula><inline-graphic xlink:href="pone.0034282.e197.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 has a convolved PDF<disp-formula><graphic xlink:href="pone.0034282.e198"></graphic>
<label>(23)</label>
</disp-formula>
where <inline-formula><inline-graphic xlink:href="pone.0034282.e199.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is the PDF of noise <inline-formula><inline-graphic xlink:href="pone.0034282.e200.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Currently, the weak signal <inline-formula><inline-graphic xlink:href="pone.0034282.e201.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is corrupted by the composite noise <inline-formula><inline-graphic xlink:href="pone.0034282.e202.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and then the performance measures of locally optimal processing in Eqs. (6), (13), (17), (19) and (21) should be replaced with <inline-formula><inline-graphic xlink:href="pone.0034282.e203.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (<inline-formula><inline-graphic xlink:href="pone.0034282.e204.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). It can be shown by the Cauchy-Schwarz inequality that <xref ref-type="bibr" rid="pone.0034282-Brown1">[34]</xref>
<disp-formula><graphic xlink:href="pone.0034282.e205"></graphic>
<label>(24)</label>
</disp-formula>
<disp-formula><graphic xlink:href="pone.0034282.e206"></graphic>
<label>(25)</label>
</disp-formula>
This is because that, if <inline-formula><inline-graphic xlink:href="pone.0034282.e207.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, then using <inline-formula><inline-graphic xlink:href="pone.0034282.e208.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 and the Cauchy-Schwarz inequality <xref ref-type="bibr" rid="pone.0034282-Brown1">[34]</xref>
<disp-formula><graphic xlink:href="pone.0034282.e209"></graphic>
<label>(26)</label>
</disp-formula>
Similarly, substituting <inline-formula><inline-graphic xlink:href="pone.0034282.e210.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 into Eq. (26), we also obtain <inline-formula><inline-graphic xlink:href="pone.0034282.e211.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 of Eq. (25).</p>
<p>Therefore, in asymptotic cases of weak signal and large sample size, Eqs. (24) and (25) show that SR cannot improve the performance of the above four locally optimal processing cases by adding more noise. However, the asymptotic limits of weak signal and large sample size are well delimited, and may not be met in practice. It is interesting to note that, under less restrictive conditions, noise-enhanced effects have been observed in fixed locally optimal detectors <xref ref-type="bibr" rid="pone.0034282-Zozor1">[9]</xref>
, suboptimal detectors <xref ref-type="bibr" rid="pone.0034282-Kay2">[26]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Chen1">[29]</xref>
, the optimal detector with finite sample sizes <xref ref-type="bibr" rid="pone.0034282-ChapeauBlondeau2">[11]</xref>
 or non-weak signals <xref ref-type="bibr" rid="pone.0034282-ChapeauBlondeau2">[11]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Stocks1">[25]</xref>
, soft-threshold systems <xref ref-type="bibr" rid="pone.0034282-Greenwood1">[30]</xref>
 and the dead-zone limiter detector <xref ref-type="bibr" rid="pone.0034282-Duan1">[28]</xref>
 by utilizing the constructive role of noise.</p>
<p>We here present an illustrative example of SR that occurs outsides restrictive conditions, where a suboptimal detector is adopted for Gaussian noise. Consider a generalized correlation detector of Eq. (11) based on the dead-zone limiter nonlinearity<disp-formula><graphic xlink:href="pone.0034282.e212"></graphic>
<label>(27)</label>
</disp-formula>
with response thresholds at <inline-formula><inline-graphic xlink:href="pone.0034282.e213.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Kassam1">[6]</xref>
. For the generalized Gaussian noise of Eq. (22), the normalized asymptotic efficacy <inline-formula><inline-graphic xlink:href="pone.0034282.e214.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 in Eq. (13) of <inline-formula><inline-graphic xlink:href="pone.0034282.e215.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 can be rewritten as<disp-formula><graphic xlink:href="pone.0034282.e216"></graphic>
<label>(28)</label>
</disp-formula>
where <inline-formula><inline-graphic xlink:href="pone.0034282.e217.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is the cumulative distribution function of the standardized generalized Gaussian noise PDF <inline-formula><inline-graphic xlink:href="pone.0034282.e218.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Duan1">[28]</xref>
. For a fixed response threshold <inline-formula><inline-graphic xlink:href="pone.0034282.e219.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (<inline-formula><inline-graphic xlink:href="pone.0034282.e220.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 without loss of generality), we plot the the normalized asymptotic efficacy <inline-formula><inline-graphic xlink:href="pone.0034282.e221.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (solid line) of the dead-zone limiter nonlinearity <inline-formula><inline-graphic xlink:href="pone.0034282.e222.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 as a function of the RMS amplitude <inline-formula><inline-graphic xlink:href="pone.0034282.e223.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 of Gaussian noise (<inline-formula><inline-graphic xlink:href="pone.0034282.e224.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), as shown in <xref ref-type="fig" rid="pone-0034282-g002">Fig. 2</xref>
. It is clearly seen in <xref ref-type="fig" rid="pone-0034282-g002">Fig. 2</xref>
 that the SR effect appears, and <inline-formula><inline-graphic xlink:href="pone.0034282.e225.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 achieves its maximum <inline-formula><inline-graphic xlink:href="pone.0034282.e226.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 at a non-zero level of <inline-formula><inline-graphic xlink:href="pone.0034282.e227.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. If the original Gaussian noise RMS <inline-formula><inline-graphic xlink:href="pone.0034282.e228.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, we can add independent Gaussian noise <inline-formula><inline-graphic xlink:href="pone.0034282.e229.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 with its RMS amplitude <inline-formula><inline-graphic xlink:href="pone.0034282.e230.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 to increase <inline-formula><inline-graphic xlink:href="pone.0034282.e231.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 to the maximum <inline-formula><inline-graphic xlink:href="pone.0034282.e232.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0034282-Duan1">[28]</xref>
. However, <inline-formula><inline-graphic xlink:href="pone.0034282.e233.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 is a suboptimal nonlinearity for Gaussian noise, and the locally optimal detector is the linear correlation detector based on the linear system <inline-formula><inline-graphic xlink:href="pone.0034282.e234.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 in Eq. (11). It is seen in <xref ref-type="fig" rid="pone-0034282-g002">Fig. 2</xref>
 that <inline-formula><inline-graphic xlink:href="pone.0034282.e235.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 can not overperform <inline-formula><inline-graphic xlink:href="pone.0034282.e236.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (dashed line), even we can add the appropriate amount of noise to exploit constructive role of noise in <inline-formula><inline-graphic xlink:href="pone.0034282.e237.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<fig id="pone-0034282-g002" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0034282.g002</object-id>
<label>Figure 2</label>
<caption><title>The normalized asymptotic efficacy</title>
<p><inline-formula><inline-graphic xlink:href="pone.0034282.e238.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<bold>.</bold>
 The normalized asymptotic efficacy <inline-formula><inline-graphic xlink:href="pone.0034282.e239.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 of the dead-zone limiter nonlinearity <inline-formula><inline-graphic xlink:href="pone.0034282.e240.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (solid line) and the linear system <inline-formula><inline-graphic xlink:href="pone.0034282.e241.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 (red line) as a function of the RMS amplitude <inline-formula><inline-graphic xlink:href="pone.0034282.e242.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 of Gaussian noise (<inline-formula><inline-graphic xlink:href="pone.0034282.e243.jpg" mimetype="image"></inline-graphic>
</inline-formula>
).</p>
</caption>
<graphic xlink:href="pone.0034282.g002"></graphic>
</fig>
</sec>
</sec>
<sec id="s3"><title>Discussion</title>
<p>In this paper, for a weak signal in additive white noise, it is theoretically demonstrated that the optimum performance for locally optimal processing is upper bounded by the Fisher information of the noise distribution, and this is uniformly obtained in (i) the maximum output signal-to-noise ratio ratio for a periodic signal; (ii) the optimum asymptotic efficacy for signal detection; (iii) the best cross-correlation coefficient for signal transmission; and (iv) the minimum mean square error of an unbiased estimator. Based on the Fisher information inequalities, it is demonstrated that SR cannot improve locally optimal processing under the usual conditions. However, outside these restrictive conditions of weak signal and large sample size, improvement by addition of noise through SR can be achieved, and becomes an attractive option for nonlinear signal processing. The analysis in the paper has focused on the simplest case of additive white noise as an essential reference, and an interesting extension for future work is to examine the affect of considering different forms of colored noise <xref ref-type="bibr" rid="pone.0034282-Gammaitoni1">[15]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Zeng1">[31]</xref>
, <xref ref-type="bibr" rid="pone.0034282-Hnggi1">[32]</xref>
.</p>
</sec>
<sec sec-type="methods" id="s4"><title>Methods</title>
<p>Under the assumption of weak signals, the Taylor expansion of the noise PDF is utilized in Eqs. (4), (5), (16) and (21). The Cauchy-Schwarz inequality is extensively used in Eqs. (6), (13), (17), (19) and (26).</p>
</sec>
</body>
<back><fn-group><fn fn-type="conflict"><p><bold>Competing Interests: </bold>
The authors have declared that no competing interests exist.</p>
</fn>
<fn fn-type="financial-disclosure"><p><bold>Funding: </bold>
This work is sponsored by the NSF of Shandong Province, China (No. ZR2010FM006). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.</p>
</fn>
</fn-group>
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Fisher Information as a Metric of Locally Optimal Processing and Stochastic Resonance

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