To appear in the IEEE Transactions on Signal ProcessingCurrent draft: 29 November 2010

Noise Benefits in Quantizer-Array Correlation Detection and Watermark Decoding

Ashok Patel, Member IEEE, and Bart Kosko, Fellow IEEE,

Abstract—Quantizer noise can improve statistical signaldetection in array-based nonlinear correlators in Neyman-Pearson and maximum-likelihood detection. This holds even forinfinite-variance symmetric alpha-stable channel noise and forgeneralized-Gaussian channel noise. Noise-enhanced correlationdetection leads to noise-enhanced watermark extraction based onsuch nonlinear detection at the pixel or bit level. This yields anoise-based algorithm for digital watermark decoding using twonew noise-benefit theorems. The first theorem gives a necessaryand sufficient condition for quantizer noise to increase thedetection probability of a constant signal for a fixed false-alarmprobability if the channel noise is symmetric and if the samplesize is large. The second theorem shows that the array mustcontain more than one quantizer for such a stochastic-resonancenoise benefit if the symmetric channel noise is unimodal. It alsoshows that the noise-benefit rate improves in the small-quantizernoise limit as the number of array quantizers increases. Thesecond theorem further shows that symmetric uniform quantizernoise gives the optimal rate for an initial noise benefit amongall finite-variance symmetric scale-family noise. Two corollariesgive similar results for stochastic-resonance noise benefits inmaximum-likelihood detection of a signal sequence with knownshape but unknown amplitude.

Index Terms—Maximum-likelihood detection, nonlinear cor-relation detectors, noise benefits, Neyman-Pearson detection,quantizer arrays, quantizer noise, scale-family noise, stochasticresonance.

I. NOISE BENEFITS IN NONLINEAR SIGNAL DETECTION

Noise can sometimes improve nonlinear signal processing[1]–[10]. This noise-benefit stochastic resonance (SR) effect insignal detection occurs when small amounts of noise improvesdetection performance while too much noise degrades it [11]–[15]. Such SR noise benefits arise in many physical and bio-logical signal systems from carbon nanotubes to neurons [16]–[23]. We focus here on the special case of SR for quantizer-array-based nonlinear correlators in Neyman-Pearson (NP) andmaximum-likelihood (ML) signal detection in non-Gaussianchannel noise. This channel noise can include symmetricα-stable (SαS) noise or generalized-Gaussian noise. Suchdetection problems occur in sonar, radar, and watermark de-tection [24]–[26]. The two theorems below prove that injectingsmall amounts of quantizer noise in these types of nonlineardetectors can improve their detection performance. These newSR theorems extend the growing list of formal proofs of SRnoise benefits [1]–[3], [8]–[14], [27]–[31].

Copyright c© 2010 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

Ashok Patel and Bart Kosko are with the Signal and Image Processing Insti-tute, Department of Electrical Engineering, University of Southern California,Los Angeles, CA 90089-2564, USA. Email: kosko@usc.edu

Figure 1 shows an SR noise benefit in the maximum-likelihood watermark extraction of the ‘yin-yang’ image em-bedded in the discrete-cosine transform (DCT-2) coefficientsof the ‘Lena’ image [32]. The ‘yin-yang’ image of Figure1(a) is the 64×64 binary watermark message embedded inthe mid-frequency DCT-2 coefficients of the 512×512 gray-scale ‘Lena’ image using direct-sequence spread spectrum[33]. Figure 1(b) shows the result when the ‘yin-yang’ figurewatermarks the ‘Lena’ image. Figure 1(c)-(g) shows that smallamounts of additive uniform quantizer noise improve thewatermark-extraction performance of the noisy quantizer-arrayML detector while too much noise degrades the performance.Uniform quantizer noise with standard deviation σ = 1 reducesmore than 33% of the pixel-detection errors in the extractedwatermark image. Section VI gives the details of such noise-enhanced watermark decoding.

The quantizer-array detector consists of two parts. It consistsof a nonlinear preprocessor that precedes a correlator and alikelihood-ratio test of the correlator’s output. This nonlineardetector takes K samples of a noise-corrupted signal and thensends each sample to the nonlinear preprocessor array of Qnoisy quantizers connected in parallel. Each quantizer in thearray adds its independent quantizer noise to the noisy inputsample and then quantizes this doubly noisy data sample intoa binary value. The quantizer array output for each sampleis just the sum of all Q quantizer outputs. The correlatorthen correlates these preprocessed K samples with the signal.The detector’s final stage applies either the Neyman-Pearsonlikelihood-ratio test in Section II or the maximum likelihood-ratio test in Section V.

Section III presents two SR noise-benefit theorems thatapply to broad classes of channel and quantizer noises forthe quantizer-array NP and ML detectors. Theorem 1 givesa necessary and sufficient condition for an SR noise benefitin Neyman-Pearson detection of a constant (“dc”) signal.The condition characterizes when the detection probability PD

will have a positive noise-based derivative—when dPDdσN

> 0.Theorem 1 applies to all symmetric channel noise and toall symmetric quantizer noise so long as the number Kof data samples is large. Corollary 1 in Section V gives asimilar condition for the ML detection of a known sequenceof unknown amplitude. It gives a simple method to find anear-optimal quantizer noise intensity for the ML detectionand does not need the error probability. Section VI uses thismethod for watermark decoding.

Theorem 2 of Section III contains three SR results forquantizer-array NP detectors when the quantizer noise comesfrom a symmetric scale-family probability density function

2

64×64 yin-yangwatermark image

Enlarged yin−yang Image

512 × 512 Lena Image Watermarked With Yin−Yang Image

Peak Signal−to−Noise Ratio PSNR = 46.6413dB

0 0.5 1 1.5 220

30

40

50

60

70

80

90

Standard Deviation σ of Uniform Quantizer Noise

Pix

el E

rrors

in W

ate

rmark

Extr

action

Watermark Extraction Errors Using Noisy Quatizer−Array−Based Detector

Q = 30Q → ∞

(a) (b) (c)

672 pixel errors

Linear Correlation Detection

75 pixel errors

σ = 0

48 pixel errors

σ = 1

79 pixel errors

σ = 3

(d) (e) (f) (g)

Fig. 1. Noise-enhanced digital watermark extraction using a noise-based algorithm: SR noise benefits in quantizer-array maximum-likelihood (ML) watermarkdecoding. (a) Binary 64×64 watermark ‘yin-yang’ image. (b) Watermarked 512×512 ‘Lena’ image. Direct-sequence spread spectrum embeds each messagebit of the ‘yin-yang’ image in a set of mid-frequency discrete cosine transform (DCT-2) coefficients of the gray-scale ‘Lena’ image. (c) Nonmonotonicquantizer-noise-enhanced watermark-detection performance plot of the array-based ML detectors. The noisy array detector had Q = 30 quantizers. Uniformquantizer noise decreased the pixel-detection error by more than 33 %. The solid U-shaped line shows the average pixel-detection errors of 200 simulationtrials. The dashed vertical lines show the total min-max deviations of pixel-detection errors in these simulation trials. The dashed U-shaped line shows theaverage pixel-detection errors of the limiting-array (Q → ∞) correlation detector. This dashed U-shaped line gives the lower bound on the pixel-detectionerror for any finite (Q <∞) quantizer-array detector with symmetric uniform quantizer noise. (d) Retrieved ’yin-yang’ image using the ML linear correlationdetector. (e) Retrieved ’yin-yang’ image using the ML noiseless quantizer-array detector. This nonlinear detector outperforms the linear correlation detector.(f) Retrieved ’yin-yang’ image using the ML noisy quantizer-array detector. The additive uniform quantizer noise improves the detection of the quantizer-arraydetector by more than 33% as the uniform quantizer noise standard deviation σ increases from σ = 0 to σ = 1. (g) Too much quantizer noise degrades thewatermark detection. The SR effect is robust against the quantizer noise intensity since the pixel-detection error in (g) is still less than the pixel-detectionerrors in (d).

(pdf) with finite variance. The first result shows that Q >1 is necessary for an initial SR effect if the symmetricchannel noise is unimodal. The second result is that the rateof the initial SR effect in the small quantizer noise limit(limσN→0

dPDdσN

) improves if the number Q of quantizers in thearray increases. This result implies that we should replace thenoisy quantizer-array nonlinearity with its deterministic limit(Q→∞) to achieve the upper-bound detection performance ifthe respective quantizer-noise cumulative distribution functionhas a simple closed form. The third result is that symmetricuniform quantizer noise gives the best initial SR effect rateamong all symmetric scale-family noise types. Corollary 2 inSection V extends Theorem 2 to the ML detection of a knownsequence of unknown amplitude. All these results hold forany symmetric unimodal channel noise even though we focuson SαS noise and symmetric generalized-Gaussian channelnoise. The scope of these new theorems extends well beyondwatermark decoding and detection. They show how quantizer

noise can enhance a wide range of array-based Neyman-Pearson and maximum-likelihood detection problems in non-Gaussian channel noise. Applications include radar, sonar,and telecommunications [24]–[26], [34]–[36] when optimaldetectors do not have a closed form or when we cannot easilyestimate channel noise parameters.

Array-based noise benefits have only a recent history. Stocks[10] first showed that adding quantizer noise in an array ofparallel-connected quantizers improves the mutual informa-tion between the array’s input and output. This produced atype of suprathreshold SR effect (or SSR as Stocks calls it[37]) because it did not require subthreshold signals [38].Then Rousseau and Chapeau-Blondeau [9], [13] used such aquantizer array for signal detection. They first showed the SReffect for Neyman-Pearson detection of time-varying signalsand for Bayesian detection of both constant and time-varyingsignals in different types of non-Gaussian but finite-variancechannel noise. We proved in [12] that noise in parallel arrays of

3

threshold neurons can improve the ML detection of a constantsignal in symmetric channel noise. Theorem 5 in [12] showedthat collective noise benefits can occur in a large numberof parallel arrays of threshold units even when an individualthreshold unit does not itself produce a noise benefit.

II. NEYMAN-PEARSON BINARY SIGNAL DETECTION INα-STABLE NOISE

This section develops the Neyman-Pearson hypothesis-testing framework for the two noise-benefit theorems thatfollow. The problem is to detect a known deterministic signalsk with amplitude A in additive white symmetric α-stable(SαS) channel noise Vk given K random samples X1, ..., XK :

H0 : Xk = Vk

H1 : Xk = Ask + Vk.(1)

such that the signal detection probability PD =P (Decide H1|H1 is true) is maximal while the false-alarm probability PFA = P (Decide H1|H0 is true) stays ata preset level τ . The Vk are independent and identicallydistributed (i.i.d.) zero-location SαS random variables. Weconsider only constant (dc) signals so that sk = 1 for all k. Sothe null hypothesis H0 states that the signal sk is not presentin the noisy sample Xk while the alternative hypothesis H1

states that sk is present.The characteristic function ϕ of the SαS noise random

variable Vk has the exponential form [39], [40]

ϕ(ω) = exp(jδω − γ|ω|α) (2)

where real δ is the location parameter, α ∈ (0, 2] is the char-acteristic exponent that controls the density’s tail thickness, γ= σα > 0 is the dispersion that controls the width of the bellcurve, and σ is the scale parameter. The bell curve’s tails getthicker as α falls from 2 to near zero. So energetic impulsesbecome more frequent for smaller values of α.SαS pdfs can model heavy-tailed or impulsive noise in ap-

plications that include underwater acoustic signals, telephonenoise, clutter returns in radar, internet traffic, financial data,and transform domain image or audio signals [24], [26], [40]–[44]. The only known closed-form SαS pdfs are the thick-tailed Cauchy with α = 1 and the thin-tailed Gaussian with α= 2. The Gaussian pdf alone among SαS pdfs has a finitevariance and finite higher-order moments. The mth lower-order moments of an α-stable pdf with α < 2 exist if andonly if m < α. The location parameter δ serves as a proxyfor the mean if 1 < α ≤ 2 and as a proxy for the median if0 < α ≤ 1.

The uniformly most powerful detector for the hypotheses in(1) is a Neyman-Pearson log-likelihood ratio test [45], [46]:

ΛNP(X) =K∑k=1

log(fα(Xk − sk))− log(fα(Xk))

H1><H0

λτ

(3)

because the random K samples X = X1, ..., XK are i.i.d.We choose λτ so that it has a preset false-alarm probability PFA

of PFA = τ . This SαS Neyman-Pearson detector (3) is hard to

implement because again the SαS pdf fα has no closed formexcept when α = 1 or α = 2. The NP detector (3) does reduceto the simpler test

ΛLin(X) =K∑k=1

skXk

H1><H0

λτ (4)

if the additive channel noise Vk is Gaussian (α = 2) [45].But this linear correlation detector is suboptimal when thechannel noise is non-Gaussian. Its detection performancedegrades severely as the channel noise pdf departs further fromGaussianity [45], [47] and thus when α < 2 holds.

An important special case is the NP detector for Cauchy (α= 1) channel noise. The zero-location Cauchy random variableVk (α = 1) has the closed-form pdf

fVk(vk) =1π

γ

γ2 + v2k

(5)

for real vk and positive dispersion γ. The NP detector isnonlinear for such Cauchy channel noise and has the form

ΛCNP(X) =

K∑k=1

log(

γ2 + (Xk)2

γ2 + (Xk −Ask)2

)H1><H0

λτ .

(6)

So the NP Cauchy detector (6) does not have a simplecorrelation structure. It is also more computationally complexthan the NP linear correlation detector (4). But the NP Cauchydetector performs well for highly impulsive SαS noise cases[26], [40]. Section III shows that three other simple nonlinearcorrelation detectors can perform as well or even better thanthe Cauchy detector does when the SαS noise is mildlyimpulsive (when α ≥ 1.6).

The locally optimal detector has the familiar correlationstructure [40], [48]

ΛLO(X) =K∑k=1

sk−f ′α(Xk)fα(Xk)

=K∑k=1

sk gLO(Xk)H1><H0

λτ

(7)

and coincides with the linear correlator (4) for Gaussian (α =2) channel noise Vk. The score function gLO is nonlinear forα < 2. The locally optimal detector (7) performs well whenthe signal amplitude A is small. But this test is not practicalwhen fα does not have a closed form because gLO requires bothfα and f ′α. So researchers have suggested other suboptimaldetectors that preserve the correlation structure but that replacegLO with different zero-memory nonlinear functions g [46],[49]–[52]. These nonlinearities range from simple ad-hoc soft-limiters

gSL(Xk) =

Xk if |Xk| ≤ c1 if Xk > c−1 if Xk < −c

(8)

and hole-puncher functions

gHP(Xk) =Xk if |Xk| ≤ c0 else (9)

4

to more complex nonlinearities that better approximate gLO.The latter may use a scale-mixture approach [51] or a simpli-fied Cauchy-Gaussian mixture model [52].

The next section presents the two main SR theorems for thenonlinear correlation detectors that replace the deterministicnonlinearity gLO with a noisy quantizer-array-based randomnonlinearity gNQ or with its deterministic limit gN∞. We showthat these detectors enjoy SR noise benefits. We then comparetheir detection performances with the Cauchy detector (6) andwith the nonlinear correlation detectors based on the simplesoft-limiter and hole-puncher nonlinearities (8)-(9).

III. QUANTIZER NOISE BENEFITS INNONLINEAR-CORRELATION-DETECTOR-BASED

NEYMAN-PEARSON DETECTION

This section presents the two main SR noise-benefit theo-rems for Neyman-Pearson detectors. The Appendix gives theproof of Theorem 2. We start with the nonlinear correlationdetector:

ΛNQ(X) =K∑k=1

sk gNQ(Xk)H1><H0

λ (10)

where

gNQ(Xk) =1Q

Q∑q=1

sign(Xk +Nq − θ). (11)

Here λ is the detection threshold, θ is the quantization thresh-old, and sign(Xk +Nq − θ) = ±1 for q = 1, ..., Q. We choseθ = A

2 because both the channel noise Vk and the quantizernoise Nq are symmetric.

We assume that the additive quantizer noise Nq has asymmetric scale-family [53] noise pdf fN(σN, n) = 1

σNfN( nσN

).Here σN is the noise standard deviation and fN is the standardpdf for the whole family [53]. Then the noise cumulativedistribution function (CDF) is FN(σN, n) = FN( nσN

) whereFN is the standard CDF for the whole family. Scale-familydensities include many common densities such as the Gaussianand uniform but not the Poisson. We assume that the quantizernoise random variables Nq have finite variance and are inde-pendent and come from a symmetric scale family noise. Thequantizer noise can arise from electronic noise such as thermalnoise or from avalanche noise in analog circuits [54], [55].The noisy quantizer-array detector (10)-(11) is easy to useand requires only one bit to represent each quantizer’s output.This favors sensor networks and distributed systems that havelimited energy or that allow only limited data handling andstorage [56], [57].

Define next µi(σN) and σ2i (σN) as the respective population

mean and population variance of ΛNQ under the hypothesisHi (i = 0 or i = 1) when σN is the quantizer noise intensity:µi(σN) = E(ΛNQ|Hi) and σ2

i (σN) = V ar(ΛNQ|Hi). Then µ0(σN)= −µ1(σN) and σ2

0(σN) = σ21(σN) for all σN because both

the additive channel noise V and the quantizer noise N aresymmetric. The mean µi and variance σ2

i of the test statisticΛNQ depend on both V and N . So µi and σ2

i depend onthe noise intensities σV and σN . We write these two termsas µi(σN) and σ2

i (σN) because we control only the quantizer

noise intensity σN and not the channel noise intensity σV . TheAppendix derives the complete form of µi(σN) and σ2

i (σN) inthe respective equations (71) and (91) as part of the proof ofTheorem 2.

The additive structure of ΛNQ in (10) gives rise to a keysimplification. The pdf of ΛNQ is approximately Gaussianfor both hypotheses because the central limit theorem [53]applies to (10) if the sample size K is large since therandom variables X1, ..., XK have finite variance and areindependent and identically distributed (i.i.d.). Then Theorem1 gives a necessary and sufficient inequality condition for theSR effect in the quantizer-array detector (10)-(11). This SRcondition depends only on µ1(σN) and σ2

1(σN) and on theirfirst derivatives. It is equivalent to d

dσN(lnµ1(σN)

σ1(σN) ) > 0 sinceµ1 > 0 in (72).

Theorem 1: Suppose that the detection statistic ΛNQ in (10)has sufficiently large sample size K so that it is approx-imately conditionally normal: ΛNQ|H0 ≈ N(µ0(σN), σ2

0(σN))and ΛNQ|H1 ≈ N(µ1(σN), σ2

1(σN)) where µ0(σN) = −µ1(σN)and σ2

0(σN) = σ21(σN). Then the inequality

σ1(σN) µ′1(σN) > µ1(σN) σ′1(σN) (12)

is necessary and sufficient for the SR noise benefit dPDdσN

> 0in Neyman-Pearson signal detection based on the nonlineartest statistic ΛNQ.

Proof: We first derive an approximate linear form for the de-tection threshold λτ : λτ ≈ zτσ0(σN) + µ0(σN). The Neyman-Pearson detection rule based on ΛNQ rejects H0 if ΛNQ >λτ because we choose the detection threshold λτ such thatP (ΛNQ > λτ |H0) = τ . We also need to define the constant zτso that 1−Φ(zτ ) = τ where Φ(z) =

∫ z−∞(1/2

√π)e−x

2/2dx.Then standardizing ΛNQ under the assumption that the nullhypothesis H0 is true gives

P (ΛNQ > λτ |H0)

= P

(ΛNQ − µ0(σN)

σ0(σN)>λτ − µ0(σN)σ0(σN)

)(13)

≈ P

(Z >

λτ − µ0(σN)σ0(σN)

)(14)

for Z ∼ N(0,1) by the central limit theorem

= 1− Φ(λτ − µ0(σN)σ0(σN)

). (15)

So 1−Φ(zτ ) ≈ 1−Φ(λτ−µ0(σN)σ0(σN)

)and thus zτ ≈ λτ−µ0(σN)

σ0(σN) .So the detection threshold λτ has the approximate linear formλτ ≈ zτσ0(σN) + µ0(σN).

Standardizing ΛNQ under the assumption that the alternativehypothesis H1 is true likewise gives the detection probabilityPD as

PD(σN) = P (ΛNQ > λτ |H1) from (10) (16)

= P

(ΛNQ − µ1(σN)

σ1(σN)>λτ − µ1(σN)σ1(σN)

)(17)

≈ P

(Z >

λτ − µ1(σN)σ1(σN)

)(18)

for Z ∼ N(0,1) by the central limit theorem

5

0 0.5 1 1.5 2 2.5 3 3.5 4

−0.2

0

0.2

0.4

0.6

0.8

1

Standard deviation σN of the additive uniform quantizer noise N

σ1(σ

η)μ

1’(

ση)

− μ

1(σ

η)σ

1’(

ση)

SR Condition: SR occurs if and only if σ

1(ση)μ

1’(ση) − μ

1(ση)σ

1’(ση) > 0

No SRRegion

SRRegion

Optimal quantizernoise intensity σ

N opt

0 0.5 1 1.5 2 2.5 3 3.5 40.8

0.82

0.84

0.86

0.88

0.9

0.92

Standard deviation σN of the additive uniform quantizer noise N

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Gaussian ApproximationActual

Optimal quantizernoise intensity

σN opt

No SR RegionSR Region

Vk ~ SαS(α = 1.85, γ =2.67, δ = 0)

Number of samples K = 50False−alarm probability P

FA = 0.1

Signal strength A = 1

Quantizer noise Nq ~ U(0,σ

N)

Number of Quantizers Q = 16

(a) (b)Fig. 2. SR noise benefits based on inequality (12) for constant (dc) signal detection in α-stable channel noise. (a) The plot of σ1(σN)µ′1(σN) - µ1(σN)σ′1(σN)versus the standard deviation σN of zero-mean additive uniform quantizer noise. The zero crossing occurs at the quantizer noise standard deviation σNopt . (b)The solid line and circle markers show the respective plots of the detection probabilities PD with and without the Gaussian approximation of the ΛNQ pdf.Adding small amounts of quantizer noise N improved the detection probability PD by more than 7.5%. This SR effect occurred until inequality (12) held.So σNopt maximized the detection probability.

= 1− P(Z ≤ λτ − µ1(σN)

σ1(σN)

)(19)

= 1− Φ(λτ − µ1(σN)σ1(σN)

)(20)

≈ 1− Φ(zτσ0(σN) + µ0(σN)− µ1(σN)

σ1(σN)

)because λτ ≈ zτσ0(σN) + µ0(σN) (21)

= 1− Φ(zτσ1(σN)− µ1(σN)− µ1(σN)

σ1(σN)

)(22)

because µ0(σN) = −µ1(σN) and σ20(σN) = σ2

1(σN)

= 1− Φ(zτ −

2µ1(σN)σ1(σN)

). (23)

Then the normal pdf φ(z) = dΦ(z)dz and the chain rule of

differential calculus give

dPDdσN

≈ 2φ(zτ −

2µ1(σN)σ1(σN)

)× σ1(σN)µ′1(σN)− µ1(σN)σ1

′(σN)σ2

1(σN)(24)

because zτ is a constant. So σ1(σN)µ′1(σN) > µ1(σN)σ′1(σN) isnecessary and sufficient for the SR effect (dPD

dσN> 0) because

φ is a pdf and thus φ ≥ 0.

Figure 2 shows a simulation instance of the SR inequalitycondition in Theorem 1 for constant (dc) signal detectionin impulsive infinite-variance channel noise. The signal hasmagnitude A = 0.5 and we set the false-alarm probability PFA

to PFA = 0.1. The channel noise is SαS with parameters α =1.85, γ = 1.71.85 = 2.67, and δ = 0. The detector preprocesseseach of the K = 50 noisy samples Xk with Q = 16 quantizersin the array. Each quantizer has quantization threshold θ =A/2 and adds the independent uniform quantizer noise N tothe noisy sample Xk before quantization. Figure 3(a) plots thesmoothed difference σ1(σN)µ′1(σN) - µ1(σN)σ′1(σN) versus the

standard deviation σN of the additive uniform quantizer noise.We used 105 simulation trials to estimate µ1(σN) and σ1(σN)and then used the difference quotients

µ1(σNj )−µ1(σNj−1 )

σNj−σNj−1and

σ1(σNj )−σ1(σNj−1 )

σNj−σNj−1to estimate their first derivatives. Figure

3(b) shows that adding small amounts of quantizer noise Nimproved the detection probability PD by more than 7.5%. ThisSR effect occurs until the inequality (12) holds in Figure 3(a).Figure 3(b) also shows the accuracy of the Gaussian (centrallimit theorem) approximation of the detection statistic ΛNQ’spdf. Circle marks show the detection probabilities computedfrom the 105 Monte-Carlo simulations. The solid line plotsthe detection probability PD in (23).

Theorem 2 states that it takes more than one quantizer toproduce the initial SR effect and that the rate of initial SReffect increases as the number Q of quantizers increases. Itfurther states that uniform quantizer noise gives the maximalinitial SR effect among all possible finite-variance symmetricscale-family quantizer noise. Theorem 2 and Corollary 2involve an initial SR effect that either increases the detectionprobability PD or decreases the error probability Pe for smallamounts of noise. We define the SR effect as an initial SReffect if there exists some b > 0 such that PD(σN) > PD(0) orthat Pe(σN) < Pe(0) for all σN ∈ (0, b). Theorem 2 followsfrom Theorem 1 if we substitute the expressions that wederive in the Appendix for µ1(σN), µ′1(σN), σ1(σN), andσ′1(σN) and then pass to the limit σN → 0. The complete proofis in the Appendix because it is lengthy and uses real analysis.

Theorem 2: Suppose that the channel noise pdf is uniformlybounded and continuously differentiable at −A/2. Then(a) Q > 1 is necessary for the initial SR effect in theNeyman-Pearson detection of a constant signal in symmetricunimodal channel noise V if the test statistic is the nonlineartest statistic ΛNQ.

6

0 0.5 1 1.5 2 2.5 3 3.5 40.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Standard deviation σN of the additive uniform quantizer noise N

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Channel noise Vk ~ SαS(α = 1.85, γ =2.67, δ = 0)

Optimal SαS detector

Quantizer noise Nq ~ U(0,σ

N)

Number of samples K = 50False−alarm probability P

FA = 0.1

Signal strength A = 1Q = 1

Q = 4

Q = 8

Q = 16

Q = 32

Q → ∞

0 0.5 1 1.5 2 2.5 30.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

Standard deviation σN of the additive quantizer noise

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Signal strength A = 1Number of samples K = 50Number of quantizers Q = 16False−alarm probability P

FA = 0.1

Symmetric α−stable Channel noiseV

k ~ SαS(α = 1.85, γ =2.67, δ = 0)

Uniform

Laplacian

Gaussian

Discrete bipolar

(a) (b)

Fig. 3. Initial SR effects in quantizer-array correlation detectors for constant (dc) signal detection in impulsive symmetric α-stable (SαS) channel noise (α= 1.85). (a) Initial SR effects for zero-mean uniform quantizer noise. The solid lines show that the detection probability PD improves at first as the quantizernoise intensity σN increases. The dashed line shows that the SR effect does not occur if Q = 1 as Theorem 2(a) predicts. The solid lines also show that therate of the initial SR effect increases as the number Q of quantizers increases as Theorem 2(b) predicts. The thick horizontal dash-dot line shows the detectionprobability of the optimal SαS NP detector (3). The limiting-array (Q → ∞) detector gave almost the same detection performance as the optimal SαSNP detector gave. (b) Comparison of initial SR effects in the quantizer-array correlation detector for different types of symmetric quantizer noise. Symmetricuniform noise gave the maximal rate of the initial SR effect as Theorem 2(c) predicts. Symmetric discrete bipolar noise gave the smallest SR effect and wasthe least robust.

(b) Suppose that the initial SR effect occurs in the quantizer-array detector (10)-(11) with Q1 quantizers and with somesymmetric quantizer noise. Then the rate of the initial SReffect in the quantizer-array detector (10)-(11) with Q2

quantizers is larger than the rate of the initial SR effect withQ1 quantizers if Q2 > Q1.(c) Zero-mean uniform noise is the optimal finite-variancesymmetric scale-family quantizer noise in that it gives themaximal rate of the initial SR effect among all possiblefinite-variance quantizer noise in the Neyman-Pearsonquantizer-array detector (10)-(11).

Figure 3 shows simulation instances of Theorem 2. Thethin dashed line in Figure 3(a) shows that the SR effect doesnot occur if Q = 1 as Theorem 2(a) predicts. The solid linesshow that the initial SR effect increases as the number Qof quantizers increases as Theorem 2(b) predicts. Q = 32quantizers gave a 0.925 maximal detection probability andthus gave an 8% improvement over the noiseless detectionprobability of 0.856. The thick dashed line in Figure 4 showsthe upper bound on the detection probability PD that any noisyquantizer-array detector (10)-(11) with symmetric uniformquantizer noise can achieve if it increases the number Q ofquantizers in its array. The thick horizontal dash-dot line showsthe detection probability of the optimal SαS NP detector (3).This line does not depend on the quantizer-noise standarddeviation σN and so is flat because the optimal SαS NPdetector (3) does not use the quantizer noise N . The limiting-array (Q → ∞) detector gave almost the same detectionperformance as the optimal SαS NP detector gave.

Theorem 2(b) implies that the limit limQ→∞ PD(gNQ) gives

the upper bound on the detection probability of the noisyquantizer-array detector (10)-(11). The right side of (11) isthe conditional sample mean of the bipolar random variableYk,q = sign(Xk + Nq − θ) given Xk because the quantizernoise random variables Nq are i.i.d. Then the conditionalexpected value has the form

E[Yk,q|Xk = xk] = 1− 2FN(σN, θ − xk) (25)

= 1− 2FN(θ −Xk

σN

) (26)

where θ = A/2. The CDF FN(σN, ·) is the CDF of thesymmetric scale-family quantizer noise Nq that has standarddeviation σN and that has standard CDF FN for the entirefamily. So the strong law of large numbers [53] implies thatthe sample mean gNQ(Xk) in (11) converges with probabilityone to its population mean in (26):

limQ→∞

gNQ(Xk) = 1− 2FN(θ −Xk

σN

) (27)

= 2FN(Xk − θσN

)− 1 (28)

Equality (28) follows because the quantizer noise issymmetric. So Theorem 2(b) implies that the detectionprobability PD(gN∞) of the limiting nonlinear correlationdetector

ΛN∞(X) =K∑k=1

skgN∞(Xk)H1><H0

λ (29)

with gN∞(Xk) = 2FN(Xk − θσN

)− 1 (30)

gives the upper bound on the detection performance of thequantizer-array detector (10)-(11) for θ = A/2 when the

7

quantizer noise N has standard deviation σN and when thescale-family CDF is FN .

We can use this limiting non-noisy nonlinear correlationdetector (29)-(30) if the quantizer noise CDF FN(xk−A/2σN

)has a closed form. Simulations show that the detection per-formance of the noisy quantizer-array detector quickly ap-proaches the detection performance of the limiting quantizer-array (Q → ∞) detectors. So we can often use the noisyquantizer-array detector (10)-(11) with Q near 100 to get adetection performance close to that of the limiting quantizer-array (Q→∞) detector.

The limiting nonlinearity gN∞(Xk) is easy to use for sym-metric uniform quantizer noise because it is a shifted soft-limiter with shift θ = A/2:

gN∞(Xk) =

Xk −A/2 if |Xk −A/2| ≤ c1 if (Xk −A/2) > c−1 if (Xk −A/2) < −c

(31)

where c =√

3σN . Figure 3(a) shows that the limiting nonlinearcorrelation detector (29)-(30) with the shifted soft-limiternonlinearity (31) gives almost the same detection performanceas the optimal SαS detector (3). We used the numericalmethod of [58] to compute the SαS pdf fα for α = 1.85.Figure 4 shows that the limiting (Q → ∞) quantizer-arraydetector performed better than the Cauchy detector, the soft-limiter detector, and the hole-puncher detector for medium-to-low impulsive SαS noise cases (1.6 ≤ α ≤ 1.9) and for smallfalse-alarm probabilities (PFA ≤ 0.1).

The limiting-array nonlinearity (30) is monotone non-decreasing while the asymptotic behavior of the locally op-timal nonlinearity in (7) is gLO(Xk) ≈ (α + 1)/Xk. Sosmall signal strength A implies that quantizer-array detectorscannot perform better than nonlinear correlation detectors withnonmonotonic nonlinearities such as [49], [50]

gLSO-P(Xk) =

(α+1)Xk

c2 if |Xk| ≤ c(α+1)Xk

else(32)

or such as [46]

g(Xk) =aAskXk

1 + bAskX2k

. (33)

Figure 3(b) shows simulation instances of Theorem 2(c).It compares the initial SR noise benefits for different typesof simple zero-mean symmetric quantizer noises such asLaplacian, Gaussian, uniform, and discrete bipolar noise whenthere are Q = 16 quantizers in the array. Symmetric uniformnoise gave the maximal rate of the initial SR effect as Theorem2(c) predicts. It also gave the maximal SR effect (maximalincrease in the detection probability) compared to Laplacian,Gaussian, and discrete bipolar noise. Theorem 2(c) guaranteesonly a maximal rate for the initial SR effect. It does notguarantee a maximal SR effect for symmetric uniform noise.So some other type of symmetric quantizer noise may givethe maximal SR effect in other detection problems. Figure3(b) also shows that symmetric discrete bipolar noise gavethe smallest SR effect and was the least robust. The SR effectwas most robust against Laplacian quantizer noise.

IV. MAXIMUM-LIKELIHOOD BINARY SIGNAL DETECTIONIN SYMMETRIC α-STABLE CHANNEL NOISE OR

GENERALIZED-GAUSSIAN CHANNEL NOISE

Consider next the maximum-likelihood (ML) detection of adeterministic signal sequence of known shape sk but unknownamplitude A in either additive i.i.d. SαS channel noise Vk orgeneralized-Gaussian channel noise Vk. We assume that thenoise pdf has unknown parameters. The ML detection uses Krandom samples X1, ..., XK to decide between the equallylikely null hypothesis H0 and alternative hypothesis H1:

H0 : Xk = −Ask + Vk

H1 : Xk = Ask + Vk.(34)

The ML decision rule minimizes the average decision-error probability Pe = p0P (Decide H1|H0 is true) +p1P (Decide H0|H1 is true) [59]. The prior probabilities p0

and p1 are equal: p0 = P (H0) = P (H1) = p1 = 1/2. The MLdetector for (34) is a log-likelihood ratio test [45], [46]:

ΛML(X) =K∑k=1

log(f(Xk-Ask))− log(f(Xk+Ask))

H1><H0

0

(35)

The optimal ML detector (35) again does not have a closedform when the channel noise is SαS except when α = 1 andα = 2. Nor does the optimal ML detector have a correlationstructure if α < 2 and thus if the SαS channel noise Vk isnot Gaussian.

The optimal ML detector for the hypothesis test (34) hasthe different form

ΛGGML(X) =

K∑k=1

[|Xk +Ask|r − |Xk −Ask|r]

H1><H0

0

(36)

if the symmetric channel noise variables Vk are i.i.d. general-ized Gaussian random variables with pdf

fVk(vk) = De−B|vk|r

. (37)

Here r is a positive shape parameter, B is an intensityparameter, and D is a normalizing constant. The generalizedGaussian family [60] is a two-parameter family of symmetriccontinuous pdfs. Its scale-family pdf f(σ, v) has the form

f(σ, v) =1σfgg

( vσ

)=

1σ

r

2[Γ(3/r)]1/2

[Γ(1/r)]3/2e−B|

vσ |r (38)

where fgg is the standard pdf of the family, σ is the standarddeviation, and Γ is the gamma function. This family of pdfsincludes all normal (r = 2) and Laplace (r = 1) pdfs. It includesin the limit (r →∞) all continuous uniform pdfs on boundedreal intervals. It can also model symmetric platykurtic densitieswhose tails are heavier than normal (r < 2) or symmetricleptokurtic densities whose tails are lighter than normal (r >2). Applications include noise modeling in image, speech, andmultimedia processing [61]–[64].

8

10−3

10−2

10−1

1

0.9

0.8

0.7

0.6

False−alarm Probability PFA

Pro

ba

bili

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f D

ete

ctio

n P

D

Quantizer−array−based (Q → ∞) nonlinearityCauchy detectorSoftlimiter nonlinearityHolepuncher nonlinearity

SαS channel noise (α = 1.6, γ = 1, δ = 0)Signal Strength A = 1.5Number of Samples K = 50

1.6 1.65 1.7 1.75 1.8 1.85 1.90.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

Characteristic exponent α of SαS channel noise

De

tectio

n P

rob

ab

ility

P

D

Limiting (Q → ∞) ArrayHole PuncherSoftlimiterCauchy

Vk ~ SαS( γ =2, δ = 0)

Number of samples K = 50False−alarm probability P

FA = 0.005

Signal strength A = 1.1

(a) (b)

Fig. 4. Comparison of Neyman-Pearson detection performance of four nonlinear detectors for different values of (a) false-alarm probabilities PFA and (b)characteristic exponent or tail-thickness parameter α of SαS channel noise dispersion γ when the signal strength A was A = 1.1 and the number K ofsamples was K = 50. The limiting (Q→∞) quantizer-array detector performed better than the Cauchy detector, the soft-limiter detector, and the hole-puncherdetector for medium-to-low impulsive SαS noise cases (1.6 ≤ α ≤ 1.9) and for small false-alarm probabilities (PFA ≤ 0.1).

Generalized Gaussian noise can apply to watermark detec-tion or extraction [33], [65]–[67]. The generalized-GaussianML detector (36) does not use the scale parameter σ butwe still need joint estimates of σ and r. The generalized-Gaussian ML detector (36) applies to watermark extraction inimages when generalized Gaussian random variables Vk modelmid-frequency discrete cosine transform (DCT-2) coefficients[33], [68] or subband discrete wavelet transform coefficients[65], [69]. But the mid-frequency DCT-coefficients of manyimages may have thicker tails than generalized Gaussian pdfshave. And using the generalized-Gaussian ML detector (36)may be difficult for non-Gaussian (r 6= 2) noise because(36) requires joint estimation of the signal amplitude A andnoise parameters and because (36) also requires exponentiationwith floating point numbers when r 6= 2. So Briassouli andTsakalides [26] have proposed using instead the Cauchy pdfto model the DCT coefficients.

The ML Cauchy detector has the form

ΛCML(X) =

K∑k=1

log(γ2 + (Xk +Ask)2

γ2 + (Xk −Ask)2

)H1><H0

0.

(39)

It does not use exponentiation with floating point numbers.But the nonlinear detectors (36) and (39) require that wejointly estimate the signal amplitude and the parameters ofthe channel-noise pdf. This joint estimation is not easy in theML case.

We next analyze a noisy quantizer-array correlation statisticgNQ and its limit gN∞. Neither uses the value of A for the MLdetection of (34). These nonlinearities are versions of (11) and(30) with θ = 0. We show next that the results of Theorems 1and 2 also hold for the ML correlation detectors based on gNQ

and gN∞.

V. QUANTIZER NOISE BENEFITS INMAXIMUM-LIKELIHOOD DETECTION

The next two corollaries apply Theorems 1 and 2 to the MLdetection problem in (34). We first restate the noisy quantizer-array correlation statistic gNQ (11) and its limit gN∞ (30) withθ = 0 for the ML detection of the signal sk in (34):

ΛNQ(X) =K∑k=1

skgNQ(Xk)H1><H0

0 (40)

where gNQ(Xk) =1Q

Q∑q=1

sign(Xk +Nq) (41)

and

ΛN∞(X) =K∑k=1

skgN∞(Xk)H1><H0

0 (42)

where gN∞(Xk) = 2FN(Xk

σN

)− 1. (43)

We use θ = 0 because the pdfs of the random samples Xk aresymmetric about −Ask and Ask given the hypotheses H0 andH1 of (34) and because both hypotheses are equally likely.The two ML detectors (40)-(43) require that we know thequantizer noise Nq and its intensity σN . But they do not requirethat we know the signal amplitude A or the channel noise pdfparameters r or γ. The generalized-Gaussian ML detector (36)and the Cauchy ML detector (39) do require such knowledge.

Corollary 1 below requires that the mean and varianceof the detection statistics ΛNQ and ΛN∞ in (40) and (42)obey µ0(σN) = −µ1(σN) and σ2

0(σN) = σ21(σN) for all σN .

These equalities hold because (40)-(43) imply that ΛNQ|H0

= −ΛNQ|H1 and ΛN∞|H0 = −ΛN∞|H1. The pdf of ΛNQ isapproximately Gaussian for both hypotheses because thecentral limit theorem applies to (40) and (42) if the sample

9

0.2 0.4 0.6 0.8 1 1.2 1.4

0.006

0.01

0.02

0.03

0.04

0.05

0.06

Standard deviation σN of the Gaussian quantizer noise

Pro

ba

bili

ty o

f E

rro

r P

e

Q = ∞Optimal Generalized−Gaussian Detector

Q = 32

Q = 16

Q = 6

Q = 3

Q = 1

Channel noise Vk ~ GG(r = 1.2, σ = 2)

Quantizer noise Nq ~ N(0,σ

N)

Number of samples K = 75Signal strength A = 0.5

0.2 0.4 0.6 0.8 1 1.2 1.4

0.01

0.009

0.008

0.007

Standard deviation σN of quantizer noise

Pro

ba

bili

ty o

f E

rro

r P

e

Discrete bipolar

Uniform

Laplacian

Gaussian

Channel noise Vk ~ GG(r = 1.2, σ = 2)

Signal strength A = 0.5Number of samples K = 75Number of quantizers Q = 16

(a) (b)

Fig. 5. Initial SR effects in maximum-likelihood quantizer-array detection. The signal was a bipolar sequence with amplitude A = 0.5. The channel noisewas generalized-Gaussian with parameters r = 1.2 and σ = 2. (a) Initial SR effects for zero-mean Gaussian quantizer noise. The initial SR effect does notoccur if Q = 1 as Corollary 2(a) predicts. The rate of the initial SR effect increased as the number Q of quantizers increased as Corollary 2(b) predicts. Thethick dashed line shows the error probability Pe of the respective limiting-array (Q→∞) detector. This detection performance was nearly optimal comparedto the optimal generalized-Gaussian detector (36) (thick horizontal dashed-dot line). (b) Comparison of initial SR noise benefits in the maximum-likelihoodquantizer-array detector (40)-(41) for four different types of quantizer noise. Symmetric uniform noise gave the maximal rate of the initial SR effect asCorollary 2(c) predicts. But Gaussian noise gave the best peak SR effect because it had the largest decrease in error probability. Laplacian quantizer noisegave the most robust SR effect and had almost the same peak SR effect as Gaussian noise had. Symmetric discrete bipolar noise gave the smallest SR effectand was least robust.

size K is large since the random variables skgNQ(Xk) andskgN∞(Xk) are independent even though they are notidentically distributed. This holds for uniformly boundedrandom variables that satisfy the Lindeberg condition [70]:limK→∞ V ar(ΛNQ) = ∞ and limK→∞ V ar(ΛN∞) = ∞. Thevariables skgNQ(Xk) and skgN∞(Xk) are uniformly boundedso long as the sequence sk is bounded. The Lindebergcondition then holds because the noise pdfs have infinitesupport since the noise is generalized Gaussian or SαS.Then the SR noise-benefit conditions of Theorems 1 and 2also hold for the quantizer-array ML detector (40)-(41) andfor its limiting-array (Q → ∞) ML detector (42)-(43). Theproof of Corollary 1 mirrors that of Theorem 1 but uses theerror probability as the performance measure. We state it forcompleteness because it is brief.

Corollary 1: Suppose that the detection statistic ΛNQ in (40)has sufficiently large sample size K so that it is approx-imately conditionally normal: ΛNQ|H0 ≈ N(µ0(σN), σ2

0(σN))and ΛNQ|H1 ≈ N(µ1(σN), σ2

1(σN)) where µ0(σN) = −µ1(σN)and σ2

0(σN) = σ21(σN). Then the inequality

σ1(σN)µ′1(σN) > µ1(σN)σ′1(σN) (44)

is necessary and sufficient for the SR effect (dPe(σN)dσN

< 0) inthe ML detection of (34) using the quantizer-array detector(40)-(41).

Proof: The ML detection rule for ΛNQ rejects H0 if ΛNQ > 0.The null hypothesis H0 and alternative hypothesis H1 partitionthe sample space. So the probability Pe of average decision

error [59] is

Pe(σN)= p0P (ΛNQ > 0|H0) + p0P (ΛNQ < 0|H1) (45)

from (40)

=12P

(ΛNQ − µ0(σN)

σ0(σN)>−µ0(σN)σ0(σN)

)+

12P

(ΛNQ − µ1(σN)

σ1(σN)<−µ1(σN)σ1(σN)

)(46)

since p0 = p1

≈ 12P

(Z>−µ0(σN)σ0(σN)

)+

12P

(Z<−µ1(σN)σ1(σN)

)(47)

for Z ∼ N(0, 1)

=12

(1− Φ

(−µ0(σN)σ0(σN)

))+

12

Φ(−µ1(σN)σ1(σN)

)(48)

=12

Φ(−µ1(σN)σ1(σN)

)+

12

Φ(−µ1(σN)σ1(σN)

)(49)

since µ0(σN) = −µ1(σN) and σ20(σN) = σ2

1(σN)

= Φ(−µ1(σN)σ1(σN)

). (50)

Then the normal pdf φ(z) = dΦ(z)dz and the chain rule of

differential calculus give

dPe(σN)dσN

≈ − φ(−µ1(σN)σ1(σN)

)× σ1(σN)µ′1(σN)− µ1(σN)σ1

′(σN)σ2

1(σN). (51)

So σ1(σN)µ′1(σN) > µ1(σN)σ′1(σN) is necessary and sufficient

10

for the SR effect (dPedσN

< 0) because φ is a pdf and thus φ ≥0.

The SR condition (44) in Corollary 1 allows us to find anear-optimal quantizer-noise standard deviation σNopt for theML detector if we have enough samples of the receivedsignal vector X under both hypotheses H0 and H1 in (34).Then the pdf of ΛNQ is a mixture of two equally likelyGaussian pdfs: 1

2N(−µ1(σN), σ21(σN)) + 1

2N(µ1(σN), σ21(σN)).

So Λ2NQ/σ

21(σN) is a non-central chi-square random variable

[53] with noncentrality parameter µ21(σN) and 1 degree of

freedom. Then

V ar(ΛNQ) = σ21(σN) + µ2

1(σN). (52)V ar(Λ2

NQ) = 2σ21(σN) + 4σ2

1(σN)µ21(σN) (53)

Putting µ21(σN) from (52) in (53) gives a quadratic equation in

σ21(σN) with solution

σ21(σN) = V ar(ΛNQ)

− [V ar2(ΛNQ)− V ar(Λ2NQ)/2]1/2. (54)

So the real part of (54) gives the consistent estimator σ21(σN)

of σ21(σN) for large sample size if we replace the popula-

tion variances of ΛNQ and Λ2NQ with their sample variances

because then the right side of (54) is a continuous functionof consistent estimators. Then σ2

1(σN) can replace σ21(σN) in

(52) to give the consistent estimator µ21(σN) of µ2

1(σN). Thesame received vector X allows us to compute σ2

1(σN) andµ2

1(σN) for all values of σN by (40)-(43). Then a zero-crossingof ln

µ1(σNj )

σ1(σNj ) − lnµ1(σNj−1 )

σ1(σNj−1 ) estimates the optimal quantizer-noise standard deviation σNopt for a small step-size of ∆σN

= σNj -σNj−1because the SR condition (44) is equivalent to

ddσN

(lnµ1(σN)σ1(σN) ) > 0 since µ1 > 0. Section VI uses this zero-

crossing method to find a near-optimal standard deviation ofuniform quantizer noise for the limiting array detector (42)-(43) in watermark decoding.

The lengthy proof of Corollary 2 below is nearly the sameas the proof of Theorem 2 in the Appendix. It replaces thedetection probability PD with the error probability Pe anduses a zero threshold. So we omit it for reasons of space.

Corollary 2: Suppose that the channel noise pdf is uniformlybounded and continuously differentiable. Then(a) Q > 1 is necessary for the initial SR effect in thequantizer-array detector (40)-(41) for the ML detection of(34) in any symmetric unimodal channel noise.(b) Suppose that the initial SR effect occurs with Q1

quantizers and with some symmetric quantizer noise in thequantizer-array detector (40)-(41). Then the rate of the initialSR effect in the quantizer-array detector (40)-(41) with Q2

quantizers is larger than the rate of the initial SR effect withQ1 quantizers if Q2 > Q1.(c) Zero-mean uniform noise is the optimal finite-variancesymmetric scale-family quantizer noise in that it gives themaximal rate of the initial SR effect among all possiblefinite-variance quantizer noise in the ML quantizer-arraydetector (40)-(41).

The simulation results in Figures 5 and 6 show the re-spective predicted SR effects in the ML detection (34) ofsignal sk in generalized-Gaussian channel noise and in SαSchannel noise for quantizer-array detectors. The signal wasa bipolar sequence with amplitude A = 0.5. The respectivesample sizes were K = 75 and K = 50 for these detectors.The channel noise was generalized-Gaussian with parametersr = 1.2 and σ = 2 in Figure 5. It was SαS with α = 1.7and γ = 0.51.7 in Figure 5. The thin dashed lines in Figure5(a) and 6(a) shows that the SR effect does not occur if Q= 1 as Theorem 2(a) predicts. Figures 5(a) and 6(a) alsoshow that the rate of the initial SR effect increases as thenumber Q of quantizers increases as Corollary 2(b) predicts.The thick dashed line in Figure 5(a) shows the error probabilityof the limiting-array (Q → ∞) ML correlation detector(42) with limiting-array Gaussian-quantizer-noise nonlinearitygN∞(Xk) = 2Φ(XkσN

)− 1 where we have replaced FN in (43)with the standard normal CDF Φ. The thick horizontal dash-dot line shows the error probability of the optimal generalized-Gaussian ML detector (36). The limiting-array (Q → ∞)detector does not require that we know the signal amplitudeA. It still gave almost the same detection performance as theoptimal generalized-Gaussian detector gave.

The simulation results in Figures 5(b) and 6(b) show the ini-tial SR-rate optimality of symmetric uniform quantizer noise.The symmetric uniform quantizer noise gave the maximal rateof the initial SR effect as Corollary 2(c) predicts. Gaussiannoise gave the best peak SR effect in Figure 5(b) in thesense that it had the maximal decrease in error probability.Symmetric uniform noise gave the highest peak SR effectin Figure 6(b) when compared with symmetric Laplacian,Gaussian, and discrete bipolar noise even though Corollary2(c) does not guarantee such optimality for the peak SR effect.Symmetric discrete bipolar noise gave the smallest SR effectand was least robust.

VI. WATERMARK DECODING USING THE NOISYNONLINEAR-CORRELATION DETECTOR

The above noisy nonlinear detectors and their limiting-array(Q → ∞) detectors can benefit digital watermark extraction.We will demonstrate this for blind watermark decoding basedon direct-sequence spread spectrum in the DCT domain [33],[71], [72]. Digital watermarking helps protect copyrightedmultimedia data because it hides or embeds a mark or signalin the data without changing the data too much [73], [74].Transform-domain watermarking techniques tend to be morerobust and tamper-proof than direct spatial watermarking [74],[75]. So we use the popular direct-sequence spread-spectrumapproach to watermarking in the DCT domain because suchspreading gives a robust but invisible watermark and becauseit allows various types of detectors for blind watermarkextraction [33], [71], [72].

DCT watermarking adds or embeds the watermark signalsW [k] of an M1 ×M2 binary watermark image B(n) in theDCT-2 coefficients V [k] of an L1×L2 host image H(n). Herek = (k1, k2) and n = (n1, n2) are the respective 2-D indices inthe transform domain and in the spatial domain. We apply an

11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

10−6

10−5

10−4

10−3

Standard deviation σN of additive Laplacian quantizer noise

Err

or

Pro

babili

ty P

e

Q = 31

Q = 1

Q = 7

Q = 3

Q = 15

Channel noise Vk ~ SαS(α = 1.7, γ = 0.51.7, δ = 0)

Signal amplitude A = 0.5Number of samples K = 50Number of quantizers Q = 1, 3, 7, 15, 31

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

10−6

10−5

10−4

Err

or

Pro

babili

ty P

e

Standard deviation σN of additive quantizer noise

Discrete bipolar

Uniform

Gaussian

Laplacian

Signal amplitude A = 0.5Number of samples K = 50Number of quantizers Q = 15

Channel noise Vk ~ SαS(α = 1.7, γ = 0.51.7, δ = 0)

(a) (b)

Fig. 6. Initial SR effects in the quantizer-array maximum-likelihood detection of a known bipolar sequence sk of unknown amplitude A in infinite-varianceSαS channel noise with α = 1.7 and γ = 0.51.7. (a) Initial SR effects for Laplacian quantizer noise. The initial SR effect did not occur if Q = 1 as Corollary2(a) predicts. The rate of initial SR effect increased as the number Q of quantizers increased as Corollary 2(b) predicts. (b) Comparison of initial SR effectsin the quantizer-array maximum-likelihood detection of a deterministic bipolar sequence sk of unknown amplitude A in infinite-variance SαS channel noisewith different types of quantizer noise. The symmetric uniform noise gave the maximal rate of the initial SR effect as Corollary 2(c) predicts. Symmetricdiscrete bipolar noise gave the smallest SR effect and was least robust. Symmetric uniform noise had the peak SR effect in the sense that it had the largestdecrease in error probability. Gaussian noise gave almost the same peak SR effect as uniform noise did. The SR effect was most robust against Laplacianquantizer noise.

8 × 8 block-wise DCT-2 transform [72]. Using a block-wiseDCT involves less computation than using a full-image DCT.

The watermark image B is an M1 ×M2 black-and-whiteimage such that about half its pixels are black and half arewhite. This watermark image B gives a watermark messageb = (b1, ..., bM) with M (= M1 ×M2) bipolar (±1) bits suchthat bi = 1 if B(n1, n2) = 255 (white) and such that bi = -1if B(n1, n2) = 0 (black) where i = n1 + (n2 − 1)×M1.

A secret key picks a pseudorandom assignment of disjointsubsets Di of size or cardinality K from the set D of mid-frequency DCT-coefficients for each message bit bi. DenoteIi as the set of 2-D indices of the DCT-coefficient set Di:Ii = k : Vi[k] ∈ Di. Then |Di| = |Ii| = K. The secretkey also gives the initial seed to a pseudorandom sequencegenerator that produces a bipolar spreading sequence si[k] oflength K (k ∈ Ii) for each message bit bi [33], [59]. Eachsi is a pseudorandom sequence of K i.i.d. random variablesin 1,−1. The cross-correlation among these M spreadingsequences siMi=1 equals zero while their autocorrelationequals the Kronecker delta function.

We embed the information of bit bi in each DCT-coefficientVi[k] of the set Di using the corresponding spreading sequencesi[k] of length K. Each bipolar message bit bi multiplies itsbiploar pseudorandom spreading sequence si to “spread” thespectrum of the original message signal over many frequen-cies. We use the psychovisual DCT-domain perceptual maska[k] = AT [k] of [26] to obtain the watermark embeddingstrength that reduces the visibility of the watermark in thewatermarked image [76]. Here T [k] is the known shape of theperceptual mask and 0 < A < 1 is the known or unknownscaling factor. This perceptual mask a[k] also multiplies the

pseudorandom sequence si[k] to give the watermark signalWi[k] = bia[k]si[k] for each k-pixel in the DCT domain.We then add Wi[k] to the host-image DCT-2-coefficient Vi[k]∈ Di. This gives the watermarked DCT-2 coefficient Xi[k]= Vi[k] + bia[k]si[k]. Then the inverse block-wise DCT-2transform gives a watermarked image HW [n].

Retrieving the hidden message b requires that we know thepseudorandom assignment of DCT-coefficients Vi[k] | k ∈Di to each message bit bi in b and that we also know thepseudorandom sequence si for each bi. Then an attacker can-not extract the watermark without the secret key. So supposethat we do know the secret key. Then watermark decoding justtests M binary hypotheses:

H0 (bi = −1) : Xi[k] = −Asi[k] + Vi[k]H1 (bi = +1) : Xi[k] = Asi[k] + Vi[k] (55)

for all k ∈ Ii and for i = 1, ...,M . Here si = si[k]T [k] is theknown signal sequence and A is the known or unknown scalingfactor of the perceptual mask. Define Xi = Xi[k]|k ∈ Ii.The optimal decision rule to decode the message bit bi is theML rule

ΛW (Xi) =∑k∈Ii

log(f(Xi[k] | H1)f(Xi[k] | H0)

)H1><H0

0 (56)

because we assume that the DCT-2 coefficients Vi[k] are i.i.d.random variables and that the message bits bi are equally likelyto be -1 or 1. So the ML detection rule (56) becomes a simplelinear correlator

ΛLin(Xi) =∑k∈Ii

si[k]Xi[k]H1><H0

0 (57)

12

if the DCT-2 coefficients are Gaussian random variables. TheML detection rule (56) becomes the generalized-Gaussian(GG) detector or decoder

ΛGGML(Xi) =

∑k∈Ii

[|Xi[k]+Asi[k]|r − |Xi[k]-Asi[k]|r]

H1><H0

0.(58)

or the Cauchy detector

ΛCML(Xi) =

∑k∈Ii

log(γ2 + (Xi[k] +Asi[k])2

γ2 + (Xi[k]−Asi[k])2

)H1><H0

0.

(59)

if the respective DCT-2 coefficients have a generalized Gaus-sian or Cauchy pdf. But these optimal ML detectors requirethat we know the the scaling factor A of the perceptual maskand that we know the pdf parameters r and γ. Both thesuboptimal quantizer-array detector (40)-(41)

ΛNQ(Xi) =∑k∈Ii

si[k]gNQ(Xi[k])H1><H0

0 (60)

where gNQ(Xi[k]) =1Q

Q∑q=1

sign(Xi[k] +Nq) (61)

and its limiting-array nonlinear correlation detector (42)-(43)

ΛN∞(Xi) =∑k∈Ii

si[k]gN∞(Xi[k])H1><H0

0

(62)

where gN∞(Xi[k]) = 2FN(Xi[k]σN

)− 1 (63)

require that we know the quantizer noise Nq and its intensityσN . They do not require that we know the scaling factor Aof the perceptual mask or the channel noise pdf parametersr or γ. We use the zero-crossing method of section V tofind a near-optimal quantizer-noise standard deviation for thesearray-based detectors. The noise-based algorithm below sum-marizes the processes of watermark embedding and watermarkdecoding.

Figure 1 shows a pronounced SR noise benefit in thewatermark decoding of the quantizer-array detector (60)-(61)and its limiting-array nonlinear correlation detector (62)-(63)when the quantizer noise is symmetric uniform noise. Thesimulation software was Matlab. Figure 1(a) shows the ‘yin-yang’ image. We used this binary (black and white) 64×64image as a hidden message to watermark the 512×512 gray-scale ‘Lena’ image. Figure 1(b) shows the ‘Lena’ imagewatermarked with the ‘yin-yang’ image such that its peaksignal-to-noise ratio (PSNR) was 46.6413 dB. We define thePSNR as the ratio of the maximum power of the original hostimage’s pixels H[n] to the average power of the differencebetween the original image H and the watermarked host imageHW :

PSNR = 10log10

[ maxn∈1,...,L1×1,...,L2 H2[n]

|H[n]−HW [n]|2/(L1 × L2)

]. (64)

Watermark Embedding1. Compute the 8 × 8 block-wise DCT-2 transform V [k] of

the L1×L2 host image H[n].

2. Let D be the set of mid-frequency DCT-2 coefficients of all

8 × 8 DCT blocks of V [k].

3. Convert the M1×M2 binary (black-and-white) watermark

image B(n) into an M1 ×M2 = M-bit bipolar watermark

message b = (b1, ..., bM ) such that bi = 1 if

B(n1, n2) = 255 (white) or bi = -1 if B(n1, n2) = 0 (black)

where i = n1 + (n2 − 1)×M1.

4. Use a secret key to generate M pseudorandom disjoint

subsets Di of size K from the set D.

5. Let Ii be the set of two-dimensional indexes of the

DCT-coefficient set Di.

6. Use the secret key to generate M pseudorandom bipolar

spreading sequences si[k] of length K where si[k] = ±1

for all k ∈ Ii and i = 1, ..., M .

7. For each message bit bi: compute the watermark signals

Wi[k] = bia[k]si[k] for all k ∈ Ii where a[k] is the

perceptual mask in [26] with the scale factor A.

8. For each message bit bi: compute the watermarked DCT-2

coefficients Xi[k] = Vi[k] + bia[k]si[k] for all k ∈ Ii.

9. Compute the inverse block-wise DCT-2 transform using

the watermarked DCT-coefficients X[k] to get the

watermarked image HW [n].

Watermark Decoding1. Compute the 8×8 block-wise DCT-2 transform coefficients

X[k] of the L1×L2 watermarked host image HW [n].

2. Use the secret key to obtain the index sets Ii for i = 1, ..., M .

3. Obtain M sets of watermarked DCT-coefficients

Xi = Xi[k] | k ∈ Ii for i = 1, ..., M .

4. Use the secret key to reproduce

the pseudo spreading sequences siMi=1.

5. Find the decoded message bits biMi=1 using one of the

following ML decoders for M binary hypothesis tests in (55):

Without Noise Injection

If the user knows the scaling factor A of the perceptual mask

∗ Use the Cauchy detector (59) (estimate the dispersion γ) OR

∗ Use the GG detector (59) (estimate the shape parameter r)

Else

∗ Use the Cauchy detector (59) (estimate both A and γ) OR

∗ Use the limiting-array correlation detector (62)-(63) for

uniform quantizer noise and use the zero-crossing method of

Section V to find a near-optimal qunatizer-noise intensity OR

With Noise Injection

∗ Use the noisy quantizer-array detector (60)-(61) with quantizer

number Q = 100 and use the zero-crossing method of

Section V to find a near-optimal qunatizer-noise intensity.

End

Each DCT-coefficient set Di of the ‘Lena’ image in Figure

13

1(a) hides one message bit of the ‘yin-yang’ image using aMatlab-generated pseudorandom bipolar spreading sequencesi. The watermark had a constant amplitude (a[k] = A) and sodid not involve psychovisual properties. The solid U-shapedline in Figure 1(c) shows the average pixel-detection errorsof the ML noisy quantizer-array detector (60)-(61) for 200randomly generated secret keys. The dashed vertical linesshow the total min-max deviation of the pixel-detection errorsin these simulation trials. The dashed U-shaped line shows theaverage pixel-detection errors of the limiting-array (Q→∞)correlation detector (62)-(63) where gN∞ is the soft-limiternonlinearity gSL of (8) because the quantizer noise is symmetricuniform noise. So the thick dashed line gives the lowerbound on the pixel-detection error that any quantizer-arraydetector with symmetric uniform quantizer noise can achieveby increasing the number Q of quantizers in its array.

Figures 1(d)-(g) show the extracted ’yin-yang’ image usingthe ML linear correlation detector (57) and the ML noisyquantizer-array detector (60)-(61) according to the noise-based algorithm. The noisy quantizer-array nonlinear detectoroutperforms the linear correlation detector. Figures 1(e)-(f)show that adding uniform quantizer noise plainly improvedthe watermark decoding. The pixel-detection errors decreasedby more than 33% as the uniform quatizer noise standarddeviation σ increased from σ = 0 to σ = 1. Figure 1(g)shows that too much quantizer noise degraded the watermarkdetection. But the SR effect was robust against the quantizernoise intensity because the pixel-detection error in (g) was stillless than the pixel-detection errors in (d) and (e).

We also watermarked ‘Lena’ and six other known images(‘Elaine’, ‘Goldhill’, ‘Pirate’, ‘Peppers’, ‘Bird’, and ‘Tiffany’)with the ‘yin-yang’ image using a perceptual mask based onthe psychovisual properties in [26]. Figure 7 shows these sixwatermarked images. We used the scaling factor A of thepsychovisual perceptual mask such that the PSNRs of allthese watermarked images remained between 43 dB and 47dB. We used over 200 simulation trials for each of theseseven images. A set of 200 randomly generated secret keysallocated the DCT-2-coefficients and produced the spreadingsequences for the watermarking process of each host image.We then applied the ML Cauchy detector, the limiting-array(Q→∞) detector for the symmetric uniform quantizer noise,and the generalized-Gaussian detector for various values oftheir respective pdf parameters γ, σN , and r to each of theseven watermarked images to decode the watermark usingeach secret key.

Table 1 shows the minimal values of the average pixel-detection errors for each host image and for each detector.The numbers in parentheses show the pixel decoding errorsfor the Cauchy detector and the generalized-Gaussian detectorwhen the detectors used the scale factor A of the pyschovisualperceptual mask. The linear detector and the limiting arraydetector did not need the watermark scale factor A. Thelimiting array detector for the symmetric uniform quantizernoise performed substantially better than did the other twononlinear detectors when the detectors did not use the scalefactor A. The Cauchy detector gave the best performanceotherwise. The computational complexity for all the detectors

(a) Elaine (b) Goldhill (c) Pirate

(d) Bird (e) Peppers (f) Tiffany

Fig. 7. Six different 512×512 host images watermarked with the ‘yin-yang’image in Figure 1(a) using a perceptual mask [26] based on the psychovisualproperties: (a) Elaine, (b) Goldhill, (c) Pirate, (d) Bird, (e) Peppers, and(f) Tiffany. The peak signal-to-noise ratio of each watermarked image wasbetween 43 dB to 47 dB.

ImagesMinimal Average Pixel Errors in Watermark Extractionlimiting array Cauchy GG Linear

Detector Detector Detector DetectorElaine 59 131 (61) 144 (63) 156

Goldhill 112 350 (109) 390 (120) 560Pirate 105 357 (90) 392 (98) 645Bird 178 317 (173) 360 (190) 561

Peppers 26 120 (24) 153 (29) 376Lena 7 79 (5) 100 (8) 242

Tiffany 0.3 27 (0.2) 39 (0.22) 151

Table 1: Minimal values of average watermark-decoding performance for theCauchy detector, the limiting-array (Q → ∞) detector, and the generalized-Gaussian (GG) detector. The watermarking used a perceptual mask [26] basedon psychovisual properties.

was only of order K for all arithmetic and logical operationsif the length of the signal was K. The limiting array detectoralso had the least complexity among the nonlinear detectorswhile the generalized-Gaussian detector had the most.

Psychovisual watermarking greatly reduced decoding errorscompared with constant-amplitude watermark. Consider the‘Lena’ image: Table 1 and Figure 1(c)-(d) show that thelimiting-array detector had 7 versus 43 pixel errors and thelinear detector had 242 versus 672 pixel errors.

Figure 8 shows the estimation accuracy of the zero-crossingmethod in Section V for the limiting-array detector (62)-(63) inthe watermark decoding of the ‘Bird’ image. The method givesthe near-optimal quantizer-noise standard deviation of 1.3 inFigure 8(a). Figure 8(b) shows the true optimal quantizer-noisestandard deviation of 1.38 for the pixel detection error.

Noise can also benefit DCT-domain watermark decoding forsome other types of nonlinear detectors. Researchers [77]–[79]have found noise benefits in DCT-domain watermark decodingbased on parameter-induced stochastic resonance [80]. Theirapproaches differed from ours in two main ways. They useda pulse-amplitude-modulated antipodal watermark signal butdid not use pseudorandom bipolar spreading sequences toembed this watermark signal in the DCT coefficients. Theyfurther used nonlinear but dynamical detectors to processthe watermarked DCT coefficients. Sun et al. [77] used a

14

0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75−0.06

−0.03

0

0.03

0.06

0.09

0.12

Standard Deviation σN of Uniform Quantizer Noise

d ln

(μ1(σ

N)/

σ1(σ

N))

/ d

σN

estimated σ*

opt = 1.3

(a)

0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75170

180

190

200

210

220

Standard Deviation σN of Uniform Quantizer Noise

Pix

el E

rrors

Watermark Decoding Errors in The Bird Image For The Limiting Quatizer−Array−Based Detector

σN opt

= 1.38

(b)

Fig. 8. Finding a near-optimal quantizer-noise standard deviation for thelimiting-array detector (62)-(63) in the watermark decoding of the ‘Bird’image.

monostable system with selected parameters for a given wa-termark. Wu and Qiu [81] and Sun and Lei [78] used a singlebistable detector while Duan et al. [79] used an array ofbistable saturating detectors. These bistable detectors requiretunning two system parameters besides finding the optimaladditive noise type and its variance. These detectors are alsosubthreshold systems. Their dynamical nature made the errorprobability in the decoding of one watermark bit depend onthe value of the previous watermark bit. Our noisy quantizer-array detectors produced suprathreshold SR noise benefits [5].The detection error probabilities for the watermark bits werealso independent.

VII. CONCLUSION

Noise-enhanced quantizer-array correlation detectionleads naturally to noise-enhanced watermark decodingbecause we can view digital watermarking systems as digitalcommunication systems [82] with statistical hypothesis testingat the pixel level. Such noise benefits will occur only ifthe symmetric unimodal channel noise is not Gaussian andonly if the array contains more than one quantizer. Thisnoise-enhancement technique should apply to other problems

of signal processing and communications that involve sometype of nonlinear statistical hypothesis testing. The paperalso showed that uniform noise gives the optimal initial rateof the SR noise benefit among all finite-variance symmetricscale-family quantizer noise for both Neyman-Pearson andmaximum-likelihood detection. Finding practical algorithmsfor the overall optimal quantizer noise remains an openresearch question.

APPENDIX: Proof of Theorem 2

Proof: Part (a) The definition of the initial SR effect andTheorem 1 imply that the initial SR effect occurs if and onlyif there exists some b > 0 such that inequality (12) holds forall quantizer-noise intensities σN ∈ (0, b). Then multiply bothsides of (12) by 2σ1(σN) to get

µ1(σN)σ21′(σN) < 2σ2

1(σN)µ′1(σN) (65)

for all σN ∈ (0, b) as a necessary condition for the initialSR effect. We will prove that inequality (65) does not holdfor all quantizer-noise intensities σN in some positive intervalif Q = 1. We first derive the equations for µ1(σN), µ′1(σN),σ2

1(σN), and σ21′(σN). We then show that the right side of (65)

is negative in some noise-intensity interval (0, h) while theleft side of (65) is positive in the same interval.

Define first the random variables Yk = gNQ(Xk) for k =1, ..., K. Then ΛNQ =

∑Kk=1 Yk and the population mean of

ΛNQ|H1 is

µ1(σN) = E(ΛNQ|H1) =K∑k=1

E(Yk|H1). (66)

The Yk|H1 are i.i.d. random variables with population mean

E(Yk|H1) =∫ ∞−∞

E(Yk|Xk = x;H1)fXk|H1(x)dx (67)

=∫ ∞−∞

E(Yk|Vk = v;H1)fV (v −A)dv. (68)

Here A is the signal amplitude and E(Yk|Xk = x;H1) isthe conditional mean with received signal Xk = x when thealternative hypothesis H1 is true. Then (68) follows becauseXk|H1 = Vk + A and because the Vk are i.i.d. channel-noiserandom variables with common pdf fV . Write E(Yk|Vk =v;H1) = E(Yk|v;H1) for brevity.

Define next Yk,q = sign(Xk +Nq − θ) where θ = A/2 andthe Nq are finite-variance i.i.d. scale-family quantizer-noiserandom variables with variance σ2

N and CDF FN . Then

E(Yk|v;H1) = E(Yk,q|v,H1) = 1− 2FN (A

2− v) (69)

= 1− 2FN(A2 − vσN

) (70)

where FN is the standard CDF for the scale family of thequantizer noise. So (66), (68) and (70) imply that

µ1(σN) =∫ ∞−∞

K

[1− 2FN(

A2 − vσN

)

]fV (v −A)dv. (71)

> 0 (72)

15

because 1 − 2FN(A2 −vσN

) is a non-decreasing odd functionaround A/2 and because fV (v−A) is a symmetric unimodaldensity with mode A.

We next derive an expression for µ′1(σN) and then find thelimit limσN→0

µ′1(σN)σN

. Note first that

µ′1(σN) =∂

∂σN

∫ ∞−∞

K

[1− 2FN(

A2 − vσN

)

]fV (v −A)dv

(73)

=∫ ∞−∞

(A2 − vσN

)K

σN

fN(A2 − vσN

)fV (v −A)dv (74)

because of the distributional derivatives [83] of µ1(σN) and[1− 2FN(

A2 −vσN

)]

with respect to σN . This allows us to in-terchange the order of integration and differentiation [83] in(73)-(74). Next substitute

A2 −vσN

= n in (74) to get

µ′1(σN)σN

=∫ ∞−∞

nKfV (−A2 − nσN)

σN

fN(n)dn (75)

= −∫ ∞

0

nKfV (−A2 +nσN)− fV (−A2 − nσN)

σN

fN(n)dn

(76)

because fN is a symmetric pdf. The mean-value theorem [84]implies that for any ε > 0∣∣∣∣∣fV (−A2 + nσN)− fV (−A2 − nσN)

σN

∣∣∣∣∣ ≤ 2n sup|−A2 −u|≤ε

f ′V (u)

(77)

for all |nσN| ≤ ε. The supremum in (77) is finite for somesmall ε because the pdf derivative f ′V is continuous at −A2 .Then Lebesgue’s dominated convergence theorem [84] allowsus to commute the limit and integral in (76) because the rightside of (77) bounds or dominates the left side:

limσN→0

µ′1(σN)σN

=∫ ∞

0

−nK limσN→0

fV (-A2 +nσN )− fV (-A2 -nσN )σN

fN(n)dn

by dominated convergence (78)

= −K∫ ∞

0

2n2f ′V (−A2

)fN(n)dn (79)

by L’Hospital’s rule [85]

= −Kf ′V (−A2

)∫ ∞

0

2n2fN(n)dn (80)

= − 2Kf ′V (−A2

) (81)

because fN is the symmetric pdf of the zero-mean and unit-variance quantizer noise N . The unimodality of the symmetricchannel noise V implies that (81) is negative. Then µ′1(σN) isnegative for all noise intensities σN in some interval (0, h).Then (81) also implies that

limσN→0

µ′1(σN) = 0 (82)

because σN → 0 in µ′1(σN)σN

.

We now derive expressions for the population varianceσ2

1(σN) of ΛNQ|H1 and its distributional derivative σ21′(σN). The

Yk|H1 are i.i.d. random variables. So

σ21(σN) = V ar (ΛNQ|H1) (83)

= K V ar(Yk|H1) (84)= K

[E(Y 2

k |H1)− E2(Yk|H1)]

(85)

where E(Yk|H1) =µ1(σN)K

(86)

and E(Y 2k |H1) =

∫ ∞−∞

E(Y 2k |v,H1)fV (v −A)dv.

(87)

Expand the integrand term E(Y 2k |v,H1) as follows:

E(Y 2k |v;H1)

=1QE(Y 2

k,q|v,H1) +Q− 1Q

E2(Yk,q|v,H1) (88)

=1Q

+Q− 1Q

E2(Yk,q|v,H1) (89)

because E(Y 2k,q|v,H1) = 1 by the definition of Yk,q

=1Q

+Q− 1Q

[1− 2FN(

A2 − vσN

)

]2

(90)

because of (70). Put (90) in (87) and then put (86) and (87)in (85). This gives

σ21(σN) =

∫ ∞−∞

K

QfV (v −A)dv − µ2

1(σN)K

+∫ ∞−∞

K(Q− 1)Q

[1− 2FN(

A2 − vσN

)

]2

fV (v −A)dv.

(91)

Then the distributional derivative of σ21(σN) with respect to σN

is

σ21′(σN) =

[∫ ∞−∞

K(Q− 1)Q

2[1− 2FN(A2 − vσN

)]

(A2 − vσN

)1σN

fN(A2 − vσN

)fV (v −A)dv

]− 2µ1(σN)µ′1(σN)/K. (92)

Equation (92) implies that σ21′(σN) is positive for all

σN ∈ (0, h) if Q = 1 because µ′1(σN) is negative in thequantizer-noise-intensity interval (0, h) and because of (72).So the left side of (65) is positive and the right side of (65)is negative for all σN ∈ (0, h) if Q = 1. Hence Q > 1 isnecessary for the initial SR effect in the Neyman-Pearsondetection of a dc signal A in symmetric unimodal channelnoise V using the nonlinear test statistic ΛNQ.

Part (b) Take the limit σN → 0 on both sides of (24) to findthe rate of the initial SR effect near a zero quantizer-noiseintensity:

limσN→0

dPDdσN

= 2φ(zτ −2µ1(0)σ1(0)

)

×σ1(0)µ′1(0)− µ1(0) lim

σN→0σ1′(σN)

σ21(0)

(93)

16

= − 2φ(zτ −2µ1(0)σ1(0)

)µ1(0) lim

σN→0σ1′(σN)

σ21(0)

. (94)

because of (82). We know that

limσN→0

σ21′(σN) = 2σ1(0) lim

σN→0σ1′(σN) (95)

because σ21′(σN) = 2σ1(σN)σ1

′(σN). Equations (94) and (95)imply that the rate of the initial SR effect increases iflimσN→0 σ

21′(σN) decreases. Further:

limσN→0

σ21′(σN) = lim

σN→0

[∫ ∞−∞

K(Q− 1)Q

2[1− 2FN(n)]

× nfN(n)fV (−A2− σNn)dn

](96)

if we substituteA2 −vσN

= n in (92). Then

limσN→0

σ21′(σN) =

2K(Q− 1)fV (−A2 )Q

×∫ ∞−∞

[1− 2FN(n)]nfN(n)dn. (97)

The integral of (97) is negative because [1 − 2FN(n)]nis non-positive. Then limσN→0 σ

21′(σN) decreases as the

number Q of quantizers increases because Q2−1Q2

> Q1−1Q1

if Q2 > Q1. So the initial SR effect with Q2 quantizers islarger than that of the detector with Q1 quantizers if Q2 > Q1.

Part (c) Fix the channel noise V and the number Q ofquantizers and choose the symmetric scale-family quantizernoise N . Equations (94) and (95) imply that the rate ofincrease in the detection probability with respect to σN ismaximal if limσN→0 σ

21′(σN) achieves its minimal value. We

want to find the symmetric standard (zero-mean and unit-variance) pdf fN of the zero-mean unit-variance scale-familyquantizer noise N that minimizes∫ ∞

−∞[1− 2FN(n)]nfN(n)dn. (98)

Suppose first that the quantizer noise N is a symmetricdiscrete random variable on the sample space ΩN = n` ∈ R :` ∈ L ⊆ Z where n0 = 0. Suppose also that n−` = −n` andn` > 0 for all ` ∈ N = 1, 2, 3, .... Let PN(n`) = P (N = n`)= p`

2 denote the probability density of this symmetric standarddiscrete quantizer noise N where PN(n−`) = p`

2 = PN(n`) forall ` ∈ N and PN(n0) = p0. So we replace the integral (98)with the appropriate convergent series:

∞∑`=−∞

[1− 2FN(n`)]n`PN(n`). (99)

Then we proceed to find the density PN that minimizes (99).We first expand (99) as

∞∑`=−∞

[1− 2FN(n`)]n`PN(n)

=∞∑

`=−∞

n`PN(n`)− 2∞∑

`=−∞

FN(n`) n` PN(n`) (100)

= − 2∞∑

`=−∞

FN(n`) n` PN(n`) (101)

= − 2∞∑`=1

[FN(n`)− FN(−n`)] n` PN(n`) (102)

≤ 0 . (103)

Equality (101) follows because N is zero-mean while (102)follows because the density PN(n) is symmetric ( n−` = −n`and PN(−n`) = PN(n`) ). Then we need to find the densityPN that maximizes 2

∑∞`=1[FN(n`) − FN(−n`)] n` PN(n`)

because FN(n`)− FN(−n`) is nonnegative. So write

2∞∑`=1

[FN(n`)− FN(−n`)] n` PN(n`)

= 2∞∑`=1

[FN(n`)− FN(−n`)]√PN(n`) n`

√PN(n`).

(104)

Then the Cauchy-Schwarz inequality [84] gives

2∞∑`=1

[FN(n`)− FN(−n`)] n` PN(n`)

≤ 2

[ ∞∑`=1

[FN(n`)− FN(−n`)]2PN(n`)

] 12

×

[ ∞∑`=1

n2`PN(n`)

] 12

(105)

= 2

[ ∞∑`=1

[FN(n`)− FN(−n`)]2PN(n`)

] 12 1√

2(106)

because N is symmetric zero-mean and

unit-variance noise and so∞∑`=1

n2`PN(n`) = 1/2

=√

2[

(p1 + 2p0)2

4p1

2+

(p2 + 2p1 + 2p0)2

4p2

2

+(p3 + 2p2 + 2p1 + 2p0)2

4p3

2+ ...

] 12

(107)

because N is symmetric and soPN(n`) = PN(−n`) = p`/2 for all ` ∈ N

≤√

2

[(p0 + p1 + p2 + ...)3 + (p0+p1+p2+...)3

3

8

] 12

(108)

by the multinomial theorem

=√

2[

1 + 13

8

] 12

because∑∞`=0 p` = 1 (109)

= 1/√

3. (110)

So (110) gives the upper bound on (99) for any discretesymmetric standard quantizer noise N .

Suppose now that the quantizer noise pdf fN is continuous.Then we want to find the symmetric standard pdf fN of thezero-mean unit-variance scale-family quantizer noise N that

17

maximizes ∫ ∞−∞|[1− 2FN(n)]nfN(n)| dn (111)

because the integrand of (97) is non-positive. So write∫ ∞−∞|[1− 2FN(n)]nfN(n)| dn

=∫ ∞−∞

∣∣∣[1− 2FN(n)]√fN(n)n

√fN(n)

∣∣∣ dn. (112)

Again the Cauchy-Schwarz inequality [84] gives∫ ∞−∞|[1− 2FN(n)]nfN(n)| dn

≤[∫ ∞−∞

[1− 2FN(n)]2fN(n)dn] 1

2[∫ ∞−∞

n2fN(n)dn] 1

2

(113)

=[∫ ∞−∞

[1− 2FN(n)]2fN(n)dn] 1

2

(114)

because N has zero mean and unit variance

=[E

[(1− 2FN(N)

)2]] 1

2

(115)

=[E[(1− 2U)2

]] 12 (116)

= 1/√

3. (117)

Equalities (116)-(117) follow because U = FN(N) is a uniformrandom variable in [0, 1] for any continuous quantizer noiseN [53]. Inequality (113) becomes an equality if and only ifthe FN satisfy

[1− 2FN(n)]2 = κn2 (118)

for some constant κ on the support of fN [84]. But (118)implies that

FN(n) =12

+√κn

2for all n ∈ [

−1√κ,

1√κ

] (119)

for κ = 1/3 because FN is the CDF of a standard quantizernoise. The same CDF in (119) implies that N is uniformlydistributed in [−

√3,√

3]. So symmetric uniform quantizernoise achieves the upper bound 1/

√3 of (117) and (110). So

zero-mean uniform noise gives the maximal rate of the initialSR effect among all finite-variance continuous scale-familyquantizer noise.

REFERENCES

[1] H. Chen, P. K. Varshney, S. M. Kay, and J. H. Michels, “Theory ofstochastic resonance effects in signal detection: Part I - fixed detectors,”IEEE Transactions on Signal Processing, vol. 55, no. 7, pp. 3172–3184,July 2007.

[2] M. Guerriero, P. Willett, S. Marano, and V. Matta, “Speedier sequentialtests via stochastic resonance,” in Acoustics, Speech and Signal Process-ing, 2008. ICASSP 2008. IEEE International Conference on, 31 2008-April 4 2008, pp. 3901–3904.

[3] S. Kay, “Can detectability be improved by adding noise?” IEEE SignalProcessing Letters, vol. 7, no. 1, pp. 8–10, Jan 2000.

[4] B. Kosko, Noise. Viking/Penguin, 2006.

[5] M. D. McDonnell and D. Abbott, “What is stochastic resonance?definitions, misconceptions, debates, and its relevance to biology,” PLoSComputational Biology, vol. 5, no. 5, p. e1000348, May 2009.

[6] S. Mitaim and B. Kosko, “Adaptive stochastic resonance,” Proceedingsof the IEEE, vol. 86, no. 11, pp. 2152–2183, 1998.

[7] A. Patel and B. Kosko, “Stochastic resonance in noisy spiking retinaland sensory neuron models,” Neural Networks, vol. 18, no. 5-6, pp.467–478, 2005.

[8] ——, “Optimal noise benefits in Neyman-Pearson and inequality-constrained statistical signal detection.” IEEE Transactions on SignalProcessing, vol. 57, pp. 1655–1669, 2009.

[9] D. Rousseau and F. Chapeau-Blondeau, “Constructive role of noise insignal detection from parallel array of quantizers,” Signal Processing,vol. 85, no. 3, pp. 571–580, March 2005.

[10] N. G. Stocks, “Information transmission in parallel threshold arrays:Suprathreshold stochastic resonance,” Phys. Rev. E, vol. 63, no. 4, p.041114, Mar 2001.

[11] A. R. Bulsara and A. Zador, “Threshold detection of wideband signals:A noise-induced maximum in the mutual information,” Phys. Rev. E,vol. 54, no. 3, pp. R2185–R2188, Sep 1996.

[12] A. Patel and B. Kosko, “Error-probability noise benefits in thresholdneural signal detection.” Neural Networks, vol. 22, no. 5-6, pp. 697–706, July/August 2009.

[13] D. Rousseau, G. V. Anand, and F. Chapeau-Blondeau, “Noise-enhancednonlinear detector to improve signal detection in non-Gaussian noise,”Signal Processing, vol. 86, no. 11, pp. 3456–3465, November 2006.

[14] A. A. Saha and G. V. Anand, “Design of detectors based on stochasticresonance,” Signal Process., vol. 83, no. 6, pp. 1193–1212, 2003.

[15] P.-O. Amblard, S. Zozor, M. D. McDonnell, and N. G. Stocks, “Poolingnetworks for a discrimination task: noise-enhanced detection,” Proceed-ings of SPIE, vol. 6602, no. 1, pp. 66 020S–1–12, 2007.

[16] L. Gammaitoni, “Stochastic resonance and the dithering effect in thresh-old physical systems,” Physical Review E, vol. 52, no. 5, pp. 4691–4698,November 1995.

[17] P. Hanggi, “Stochastic resonance in biology,” ChemPhysChem, vol. 3,pp. 285–290, 2002.

[18] I. Y. Lee, X. Liu, C. Zhou, and B. Kosko, “Noise-enhanced detectionof subthreshold signals with carbon nanotubes,” IEEE Transactions onNanotechnology, vol. 5, no. 6, pp. 613–627, November 2006.

[19] A. Patel and B. Kosko, “Stochastic resonance in continuous and spikingneuron models with Levy noise,” IEEE Transactions on Neural Net-works, vol. 19, no. 12, pp. 1993–2008, Dec. 2008.

[20] D. Applebaum, “Extending stochastic resonance for neuron models togeneral levy noise,” IEEE Transactions on Neural Networks, vol. 20,no. 12, pp. 1993 – 1995, December 2009.

[21] W. C. Stacey and D. M. Durand, “Stochastic resonance improves signaldetection in hippocampal CA1 neurons,” Journal Neurophysiology,vol. 83, pp. 1394–1402, 2000.

[22] N. Stocks, D. Appligham, and R. Morse, “The application ofsuprathreshold stochastic resonance to cochlear implant coding.” Fluc-tuation and Noise Letters, vol. 2, pp. L169–L181, 2002.

[23] K. Wiesenfeld and F. Moss, “Stochastic resonance and the benefits ofnoise: from ice ages to crayfish and squid,” Nature (London), vol. 373,p. 33, 1995.

[24] A. Benerjee, P. Burlina, and R. Chellappa, “Adaptive target detectionin foliage-penetrating SAR images using alpha -stable models,” IEEETransactions on Image Processing, vol. 8, no. 12, pp. 1823–1831,December 1999.

[25] M. Bouvet and S. C. Schwartz, “Comparison of adaptive and robustreceivers for signal detection in ambient underwater noise,” IEEETransactions on Acoustics, Speech, and Signal Processing, vol. ASSP-37, no. 4, p. 621, 1989.

[26] A. Briassouli, P. Tsakalides, and A. Stouraitis, “Hidden messages inheavy-tails: DCT-domain watermark detection using alpha-stable mod-els.” IEEE Transactions on Multimedia, vol. 7, no. 4, pp. 700–715, 2005.

[27] S. Kay, J. Michels, H. Chen, and P. Varshney, “Reducing probabilityof decision error using stochastic resonance,” Signal Processing Letters,IEEE, vol. 13, no. 11, pp. 695–698, Nov. 2006.

18

[28] H. Chen, P. K. Varshney, S. M. Kay, and J. H. Michels, “Noise enhancednonparametric detection.” IEEE Transactions on Information Theory,vol. 55, no. 2, pp. 499–506, 2009.

[29] M. D. McDonnell, N. G. Stocks, and D. Abbott, “Optimal stimulusand noise distributions for information transmission via suprathresholdstochastic resonance,” Physical Review E, vol. 75, no. 6, pp. 061 105–+,2007.

[30] S. Zozor and P.-O. Amblard, “Stochastic resonance in locally optimaldetectors,” Signal Processing, IEEE Transactions on, vol. 51, no. 12,pp. 3177 – 3181, dec. 2003.

[31] F. Chapeau-Blondeau and D. Rousseau, “Noise-enhanced performancefor an optimal bayesian estimator,” IEEE Transactions on Signal Pro-cessing, vol. 52, no. 5, pp. 1327–1334, May 2004.

[32] D. C. Munson, “A note on Lena,” IEEE Transactions on Image Process-ing, vol. 5, no. 1, p. 3, 1996.

[33] J. R. Hernandez, A. Member, F. Perez-Gonzalez, M. Amado, andO. Prez-gonzlez, “DCT-domain watermarking techniques for still im-ages: Detector performance analysis and a new structure,” IEEE Trans.on Image Processing, vol. 9, pp. 55–68, 2000.

[34] M. Thompson, D. Halverson, and G. Wise, “Robust detection in nom-inally Laplace noise,” IEEE Transactions on Communications, vol. 42,no. 234, pp. 1651 –1660, 1994.

[35] D. Middleton, “Non-Gaussian noise models in signal processing fortelecommunications: New methods and results for class A and classB noise models,” IEEE Transactions on Information Theory, vol. 45,no. 4, pp. 1129–1149, may 1999.

[36] S. M. Saberali, H. Amindavar, and J. Ritcey, “Blind detection insymmetric non-Gaussian noise with unknown pdf using maximumentropy method with moment generating function constraints,” SignalProcessing, vol. 90, no. 3, pp. 891 – 899, 2010.

[37] N. G. Stocks, “Suprathreshold stochastic resonance in multilevel thresh-old systems,” Phys. Rev. Lett., vol. 84, no. 11, pp. 2310–2313, March2000.

[38] M. D. McDonnell, N. G. Stocks, C. E. M. Pearce, and D. Abbott,Stochastic Resonance: From Suprathreshold Stochastic Resonance toStochastic Signal Quantization. Cambridge University Press, 2008.

[39] M. Grigoriu, Applied Non-Gaussian Processes. Prentice Hall, 1995.[40] C. L. Nikias and M. Shao, Signal processing with alpha-stable distri-

butions and applications. Wiley-Interscience, 1995.[41] R. J. Adler, R. E. Feldman, and M. S. Taqqu, Eds., A practical guide to

heavy tails: statistical techniques and applications. Cambridge, MA,USA: Birkhauser Boston Inc., 1998.

[42] A. Achim, S. Member, A. Bezerianos, and P. Tsakalides, “NovelBayesian multiscale method for speckle removal in medical ultrasoundimages,” IEEE Trans. Medical Imaging, vol. 20, pp. 772–783, 2001.

[43] P. G. Georgiou, S. Member, P. Tsakalides, and C. Kyriakakis, “Alpha-stable modeling of noise and robust time-delay estimation in the pres-ence of impulsive noise,” IEEE Transactions on Multimedia, vol. 1, pp.291–301, 1999.

[44] W. Willinger, M. S. Taqqu, R. Sherman, and D. V. Wilson, “Self-similarity through high-variability: Statistical analysis of Ethernet LANtraffic at the source level,” IEEE/ACM Transactions on Networking,vol. 5, no. 1, pp. 71–86, 1997.

[45] G. Tsihrintzis and C. Nikias, “Performance of optimum and suboptimumreceivers in the presence of impulsive noise modeled as an alpha-stableprocess,” IEEE Transactions on Communications, vol. 43, no. 234, pp.904–914, March 1995.

[46] S. Zozor, J.-M. Brossier, and P.-O. Amblard, “A parametric approach tosuboptimal signal detection in α -stable noise,” IEEE Transactions onSignal Processing, vol. 54, no. 12, pp. 4497–4509, Dec. 2006.

[47] S. Ambike, J. Ilow, and D. Hatzinakos, “Detection for binary transmis-sion in a mixture of gaussian noise and impulsive noise modeled as analpha-stable process,” Signal Processing Letters, IEEE, vol. 1, no. 3, pp.55–57, Mar 1994.

[48] E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses, 3rd ed.Springer, 2008.

[49] C. Brown and A. Zoubir, “A nonparametric approach to signal detectionin impulsive interference,” Signal Processing, IEEE Transactions on,vol. 48, no. 9, pp. 2665–2669, Sep 2000.

[50] C. Brown, “Score functions for locally suboptimum and locally subop-timum rank detection in alpha-stable interference,” Proceedings of the11th IEEE Signal Processing Workshop on Statistical Signal Processing,SSP2001, pp. 58–61, 2001.

[51] E. Kuruoglu, W. Fitzgerald, and P. Rayner, “Near optimal detectionof signals in impulsive noise modeled with a symmetric α-stabledistribution,” Communications Letters, IEEE, vol. 2, no. 10, pp. 282–284, Oct 1998.

[52] X. Li, L. Jin, and S. Wang, “A simplified non-Gaussian mixture modelfor signal lo detection in α-stable interference,” in Proceedings of the2008 Congress on Image and Signal Processing, vol. 5. IEEE ComputerSociety, 2008, pp. 403–407.

[53] G. Casella and R. Berger, Statistical Inference. Duxbury ResourceCenter, June 2001.

[54] B. Hayes, “Randomness as a resource,” American Scientist, vol. 89,no. 4, p. 300304, 2001.

[55] M. Epstein, L. Hars, R. Krasinski, M. Rosner, and H. Zheng, “Designand implementation of a true random number generator based on digitalcircuit artifacts,” Lecture Notes in Computer Science, vol. 2779, pp.152–165, 2003.

[56] N. Kimura and S. Latifi, “A survey on data compression in wirelesssensor networks,” in Proceedings of the International Conference onInformation Technology: Coding and Computing (ITCC’05) - VolumeII. IEEE Computer Society, 2005, pp. 8–13.

[57] H. C. Papadopoulos, G. W. Wornell, and A. V. Oppenheim, “Low-complexity digital encoding strategies for wireless sensor networks,”in Proc. ICASSP98, 1998, pp. 3273–3276.

[58] J. Nolan, “Numerical calculation of stable densities and distributionfunctions.” Communications in Statistics, vol. 13, pp. 759–774, 1997.

[59] J. Proakis and M. Salehi, Digital Communications, 5th ed. McGrawHill, 2008.

[60] S. Nadarajah, “A generalized normal distribution,” Journal of AppliedStatistics, vol. 32, no. 7, pp. 685–694, September 2005.

[61] Y. Bazi, L. Bruzzone, and F. Melgani, “Image thresholding based onthe EM algorithm and the generalized Gaussian distribution.” PatternRecognition, vol. 40, no. 2, pp. 619–634, 2007.

[62] M. Do and M. Vetterli, “Wavelet-based texture retrieval using general-ized Gaussian density and Kullback-Leibler distance,” Image Processing,IEEE Transactions on, vol. 11, no. 2, pp. 146–158, Feb 2002.

[63] S. Gazor and W. Zhang, “Speech probability distribution,” SignalProcessing Letters, IEEE, vol. 10, no. 7, pp. 204–207, July 2003.

[64] R. Krupinski and J. Purczynski, “Approximated fast estimator for theshape parameter of generalized Gaussian distribution,” Signal Process.,vol. 86, no. 2, pp. 205–211, 2006.

[65] Q. Cheng and T. Huang, “An additive approach to transform-domaininformation hiding and optimum detection structure,” Multimedia, IEEETransactions on, vol. 3, no. 3, pp. 273–284, Sep 2001.

[66] A. Nikolaidis and I. Pitas, “Asymptotically optimal detection for additivewatermarking in the DCT and DWT domains,” Image Processing, IEEETransactions on, vol. 12, no. 5, pp. 563–571, May 2003.

[67] J. Wang, G. Liu, Y. Dai, J. Sun, Z. Wang, and S. Lian, “Locally optimumdetection for Barni’s multiplicative watermarking in DWT domain,”Signal Process., vol. 88, no. 1, pp. 117–130, 2008.

[68] A. C. Bovik, The Essential Guide to Image Processing. AcademicPress, June 2009.

[69] G. V. D. Wouwer, P. Scheunders, and D. V. Dyck, “Statistical texturecharacterization from discrete wavelet representations,” IEEE Transac-tions on Image Processing, vol. 8, pp. 592–598, 1999.

[70] P. Billingsley, Probability and Measure, 3rd ed. New York: Wiley-Interscience, 1995.

[71] M. Barni, F. Bartolini, V. Cappellini, and A. Piva, “A DCT-domainsystem for robust image watermarking,” Signal Processing, vol. 66,no. 3, pp. 357 – 372, 1998.

[72] I. J. Cox, M. L. Miller, J. A. Bloom, J. Fridrich, and T. Kalker, DigitalWatermarking and Steganography, 2nd ed. Morgan Kaufmann, 2007.

[73] M. Arnold, S. D. Wolthusen, and M. Schmucker, Techniques andApplications of Digital Watermarking and Content Protection. ArtechHouse Publishers, July 2003.

[74] B. Furht and D. Kirovski, Multimedia Security Handbook. CRC, 2005.

19

[75] G. Langelaar, I. Setyawan, and R. Lagendijk, “Watermarking digitalimage and video data. A state-of-the-art overview,” Signal ProcessingMagazine, IEEE, vol. 17, no. 5, pp. 20–46, Sep 2000.

[76] A. J. Ahumada and H. A. Peterson, “Luminance-model-based DCTquantization for color image compression,” in Society of Photo-OpticalInstrumentation Engineers (SPIE) Conference Series, B. E. Rogowitz,Ed., vol. 1666, Aug. 1992, pp. 365–374.

[77] S. Sun and P. Qui, “Algorithm for digital watermarking based onparameter-induced stochastic resonance,” Journal of Communications,vol. 88, no. 8, pp. 48 – 55, 2005.

[78] S. Sun and B. Lei, “On an aperiodic stochastic resonance signal pro-cessor and its application in digital watermarking,” Signal Processing,vol. 88, no. 8, pp. 2085 – 2094, 2008.

[79] F. Duan, D. Abbott, and F. m. c. Chapeau-Blondeau, “The applicationof saturating detectors to a DCT-domain watermarking scheme,” Fluc-tuation and Noise Letters, vol. 8, no. 1, pp. L65–L79, 2008.

[80] F. Duan and B. Xu, “Parameter-induced stochastic resonance and base-band binary PAM signals transmission over an AWGN channel,” I. J.Bifurcation and Chaos, vol. 13, no. 2, pp. 411–425, 2003.

[81] G. Wu and Z. Qiu, “A novel watermarking scheme based on stochasticresonance,” in 8th International Conference on Signal Processing, vol. 2,2006.

[82] S. Baudry, J. Delaigle, B. Sankur, B. Macq, and H. Maitre, “Analysisof error correction strategies for typical communication channels inwatermarking,” Signal Processing, vol. 81, pp. 1239–1250, 2001.

[83] D. Jones, The Theory of Generalized Functions, 2nd ed. CambridgeUniversity Press, 1982.

[84] G. B. Folland, Real Analysis: Modern Techniques and Their Applica-tions, 2nd ed. Wiley-Interscience, 1999.

[85] W. Rudin, Principles of Mathematical Analysis, 3rd ed. McGraw-Hill,1976.

Dr. Ashok Patel is a postdoctoral researcher and alecturer in the Department of Electrical Engineeringand in the Department of Industrial & SystemsEngineering at the University of Southern California(USC). He received his Ph.D. degree in ElectricalEngineering, M.A. degree in applied mathematics,and M.S. degree in electrical engineering from USC.He received his B.E. and M.E. degrees in electricalengineering from Gujarat University, Ahmedabad,India.

Dr. Bart Kosko is a professor of electrical engineer-ing and law at the University of Southern California(USC), a past director of USC’s Signal and ImageProcessing Institute, a licensed attorney, and haschaired or cochaired several international confer-ences on neural and fuzzy systems. He holds degreesin philosophy, economics, mathematics, electricalengineering, and law. He has published the textbooksNeural Networks and Fuzzy Systems and Fuzzy Engi-neering, the trade books Fuzzy Thinking and Heaven

in a Chip, the novel Nanotime, edited the volume Neural Networks for SignalProcessing, and co-edited the volume Intelligent Signal Processing. His mostrecent book is Noise.