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Design of detectors based on stochastic resonance
Signal Processing 83 (2003) 1193 – 1212 www.elsevier.com/locate/sigpro
Design of detectors based on stochastic resonance Aditya A. Saha∗ , G.V. Anand Acoustics Laboratory, Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560 012, India Received 24 January 2002; received in revised form 28 August 2002
Abstract This paper presents a study of the phenomenon of stochastic resonance in quantizers, and discusses the use of this phenomenon for the detection of weak sinusoidal signals in noise. Stochastic resonance in 2-level, symmetric 3-level, and symmetric multilevel quantizers is investigated. Expressions are derived for the signal-to-noise ratio (SNR) gain of the quantizers driven by a small amplitude sinsuoidal signal and i.i.d. noise. The gain depends on the probability density function (PDF) of the input noise, and for a given noise PDF, the gain can be maximized by a proper choice of the quantizer thresholds. The maximum gain GSR is less than unity if the input noise is Gaussian, but several non-Gaussian noise PDFs yield values of GSR exceeding unity. Thus, the quantizers provide an e5ective enhancement in the SNR, which can be utilized to design a nonlinear signal detector whose performance is better than that of the matched 6lter. The nonlinear detector in consideration consists of a stochastically resonating (SR) quantizer followed by a correlator. An asymptotic expression for the probability of detection of the SR detector is derived. It is shown that the detection performance of the SR detector is better than that of the matched 6lter for a large class of noise distributions belonging to the generalized Gaussian and the mixture-of-Gaussian families. ? 2003 Elsevier Science B.V. All rights reserved. Keywords: Stochastic resonance; Threshold nonlinearity; Quantizer; Suboptimal detector; Passive detection; Non-Gaussian noise; Marine noise
1. Introduction Detection of sinusoidal signals at low signal-tonoise ratios (SNRs) is a problem of great interest in the context of passive detection of targets in the ocean. The acoustic radiation from most targets contain distinct line components in the spectral domain, and are weak compared to the ambient noise. It is known [11] that a quadrature or incoherent matched 6lter is the optimal detector if a signal with unknown amplitude/phase is buried in Gaussian noise. Here, ∗ Corresponding author. Department of Physics, University of Alberta, Edmonton, AB, Canada T6G 2J1. E-mail address: [email protected] (A.A. Saha).
optimality refers to maximization of the probability of detection constrained by a 6xed probability of false alarm. The matched 6lter though easy to implement and analyze is not optimal under conditions of non-Gaussian noise prevalent in marine environments. Optimal detectors in non-Gaussian noise are nonlinear and are not easy to implement. Hence, suboptimal nonlinear detectors which are easier to implement are often employed for the detection of signals in non-Gaussian noise. The aim of the present study is to design easily implementable nonlinear detectors based on stochastic resonance, and to analyze the performance of these detectors. Stochastic resonance (SR) is the phenomenon of enhancement of signal transmission by certain
0165-1684/03/$ - see front matter ? 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0165-1684(03)00039-2
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Nomenclature 1 x[n] y[n] s[n] w[n] A1
Y1 y2 [n] y2 G Rin; out p a; b
noise standard deviation threshold normalized threshold input sequence output sequence incoming sinusoidal signal white noise sequence at the input amplitude of incoming sinusoid phase of incoming sinusoid 6rst Fourier coeFcient of the sequence E(y[n]) variance of y[n] average of the sequence E(y[n]) SNR gain of the system considered input, output SNR order of generalized Gaussian constants associated with generalized Gaussians mixing parameter of mixture-ofGaussians ratio of standard deviations of mixtureof-Gaussians
nonlinear systems resulting from the addition of noise to the system (see [7,13] for recent reviews). Quantizers and other static nonlinear systems exhibit stochastic resonance [1–3,5,6,8,9,16,17]. For an SR system driven by a sinusoidal signal and stationary white noise, the output SNR increases as the input noise intensity 2 is increased over a certain range of values of 2 . The probability of detection of a signal is maximized at the peak of the output SNR [10]. If the SNR gain is greater than unity, a combination of the SR system and a matched 6lter will yield, a detection performance that is better than that of the matched 6lter alone [4]. While conventional studies of stochastic resonance search for the optimal noise level, keeping the system 6xed [8,10], in detection schemes the problem is reversed; the system parameters are optimized for a given noise level and type. This has been demonstrated by Chapeau–Blondeau for the detection of pulse trains using quantizers with optimizable thresholds [2]. In this paper also, we follow a similar approach. The organization of the paper is as follows. In Section 2,
c R(:) P(:) f(:) F(:) C(:) ˜ C(:) (:) (:; :) H (:; :) T (:) 22 (:) Q 2 ( ) (:) 2
constant associated with mixture-ofGaussians ratio of component Gaussians for mixture-of-Gaussians probability of an event PDF of input noise probability distribution function of input noise the characteristic function for detector the normalized characteristic function for detector the Gamma function the incomplete Gamma function the continued fraction associated with the incomplete Gamma function test statistic for detector detector threshold noncentral chi-squared distribution noncentrality parameter of the chisquared distribution the right-tail probability of the noncentral chi-squared PDF
properties of stochastic resonance demonstrated by quantizers and their utility in signal detection are discussed qualitatively. It is shown that the maximum SNR gain is less than unity if the input noise is Gaussian, but the SNR gain can exceed unity if the noise is non-Gaussian. In Section 3, an algorithm for optimizing the SNR gain of the symmetric 3-level quantizer is developed, and is applied to generalized Gaussian and mixture-of-Gaussian noise probability density functions (PDFs). Section 4 presents a comparison between the optimal SNR gains of the symmetric 3-level, asymmetric 2-level and uniform symmetric multilevel quantizers. In Section 5, the monotonic dependence of the receiver operating characteristics on the SNR gain is established. It is also observed that if the noise PDF belongs to a subclass of the generalized Gaussian family, or a subclass of the Gaussian mixture family, the asymptotic performance of the quantizer-detector is signi6cantly better than that of the matched 6lter but not as good as that of the optimal nonlinear detector. Section 6 contains the conclusions.
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2. Stochastic resonance in quantizers We consider a symmetric 3-level quantizer with thresholds −1 ; 1 and quantization levels −1; 0; 1. If the quantizer is driven by a sequence x[n], the output sequence y[n] is given by the relation −1 for x[n] 6 − 1 ; for − 1 ¡ x[n] 6 1 ; y[n] = 0 (1) 1 for x[n] ¿ 1 : Let the input x[n] be the sum of an N -periodic sinusoidal signal s[n] with a known frequency and a zero-mean noise sequence w[n] consisting of independent and identically distributed (i.i.d) random variables with variance 2 : x[n] = s[n] + w[n]:
(2)
The incoming signal s[n] can be represented as s[n] = A1 cos(2$n=N − );
(3)
and is subthreshold such that A1 6 1 . In the absence of input noise, the output y[n] remains unchanged at the initial state. When noise is also present at the input, the output exhibits random transitions from one state to another. For a sinusoidal signal we need to consider the output SNR at the 6rst harmonic alone, which, as discussed in [6], is given by SNR out =
|Y1 |2 y2
;
(4)
where Y1 is the 6rst Fourier coeFcient of the sequence E(y[n]) and y2 is the average of the sequence y2 [n], and are given by Y1 =
N −1 1 E(y[n]) exp(j2$n=N ); N n=0
y2 =
1 N
N −1
(5)
n=0
Similarly, the input SNR is de6ned as SNR in =
A21 : 42
the other hand, the input SNR depends only on the input signal amplitude and the input noise variance. The SNR gain is de6ned as SNR out : (7) G= SNR in It follows G=
42 |Y1 |2 A21 y2
:
(6)
It may be noted that the output SNR depends on the input signal waveform and the input noise PDF. On
(8)
Plots of SNR gain vs. are shown in Figs. 1 and 2 for di5erent i.i.d. noise distributions belonging to the generalized Gaussian and the Gaussian mixture families. The probability density function of a generalized Gaussian random variable with mean 0 and variance 2 is given by f(&) = a= exp(−b|&=|p ); where a=
p 1=2 (3=p) ; 2 3=2 (1=p)
p ¿ 0;
b=
(3=p) (1=p)
(9) p=2 ;
(10)
and (:) is the gamma function. This family of probability distributions is fairly representative, as they span heavy-tailed PDFs (the Laplacian), less heavy-tailed PDFs (the Gaussian), and PDFs of 6nite support (the uniform distribution), and are generally used in the analysis of stochastic resonators [4,5,16]. The Laplacian, Gaussian and uniform PDFs belong to the family de6ned by (9) corresponding to p = 1; 2 and ∞, respectively. A Gaussian mixture PDF with mean 0 and variance 2 has the form 2 2 c & c √ exp − 2 f(&) = 2 2$ 1− c 2 &2 + ; (11) exp − 2 2 2 where c = [ + (1 − )2 ]1=2 ;
y2 [n]:
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0 ¡ ¡ 1; ¿ 0:
(12)
This family of probability distributions is a subclass of Middleton’s class A distributions and is widely used to model ocean acoustic noise [12,14]. Figs. 1 and 2 illustrate the trend of increasing output SNR gain with increasing input noise intensity, for a generalized Gaussian PDF and a Gaussian mixture PDF. This implies that over a certain range, increasing noise
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6
5
SNR gain G
4 p=0.5 3
p=10
2
p=1
1
p=2
0 0
1
2
3
4
5
6
7
8
9
10
Standard deviation of input noise Fig. 1. Variation of SNR gain of symmetric 3-level quantizer as a function of the standard deviation of input noise, with threshold 1 = 1. Input noise PDF is generalized Gaussian, with p = 0:5; 1; 2; 10.
intensity enhances signal transmission and the quantizer can be characterized as a stochastic resonant (SR) system. From the point of view of signal detection, the most signi6cant outcome of these computations is that the peak value of SNR gain exceeds unity in the examples involving non-Gaussian noise. For these distributions, a detector incorporating the quantizer discussed above may be expected to perform better than the linear detector since no linear system can provide an SNR gain exceeding unity.
where w[n] are zero-mean, unit-variance i.i.d. random variables with PDF f(&) and probability distribution function F(&). It follows that the mean and variance of y[n] are: E[y[n]] = P(y[n] = 1) − P(y[n] = −1) = 1 − F( − Acn ) − F(− − Acn ); y2 [n] = P(y[n]
= 1) + P(y[n] = −1) − {E[y[n]]}2
= 1 − F( − Acn ) + F(− − Acn ) − {E[y[n]]}2 ;
(14)
where 3. Detector optimization
A = A1 =;
For a symmetric 3-level quantizer whose input– output relation is given by (1), the input sequence can be rewritten as x[n] = A1 cos(2$n=N − ) + w[n];
(13)
= 1 =;
cn = cos(2$n=N − ): (15)
The parameter , which we call the normalized threshold, is the ratio of the quantizer threshold 1 to the noise standard deviation . For the nondegenerate and degenerate forms of (1), 1 is positive and zero,
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3.5 α = 0.7 α = 0.75
3
α = 0.8
SNR gain G
2.5
2
1.5
1
0.5
0 0
1
2
3
4
5
6
7
8
9
10
Standard deviation of input noise Fig. 2. Variation of SNR gain of symmetric 3-level quantizer as a function of the standard deviation of input noise, with threshold 1 = 1. Input noise PDF is a Gaussian mixture, with = 0:7; 0:75; 0:8 and = 5.
respectively. Therefore it follows that is always nonnegative. Before proceeding further, we make the following assumptions: (i) The signal amplitude A1 is small compared to the noise standard deviation , i.e. A = A1 =1. This assumption stems from the fact that our primary concern is the detection of ‘weak’ signals in noise. (ii) The PDF f(&) is symmetric. This assumption is valid for most marine noise models. The assumptions made above, together with Taylor expansions about , simplify the expressions for the mean and variance as follows: E[y(n)] = F( + Acn ) − F( − Acn ) = 2AF()cn + O(A3 ); y2 [n] = 2 − F( + Acn ) − F( − Acn ) − {E[y[n]]}2 = 2[1 − F()] + O(A2 ):
Therefore the 6rst Fourier coeFcient Y1 of the sequence E(y[n]), and y2 the average variance observed over one period de6ned in (5), can be expressed as |Y1 |2 = A2 f2 () + O(A4 );
(16)
y2 = 2[1 − F()] + O(A2 ):
(17)
Under the small signal approximation A1, ignoring all but the lowest powers of A in the expansions, we get the following expression for SNR gain: G=
4|Y1 |2 A2 y2
=
2f2 () : 1 − F()
(18)
It is apparent that the gain depends on: (a) the input noise distribution and (b) the normalized threshold . For a given marine acoustic noise distribution, therefore, the SNR gain and thereby the detection performance can be maximized by determining the value of for which G is maximum. The steps of the optimization algorithm required to do so are as
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follows: (i) Since G is not necessarily unimodal, the set of all local maxima can be found from the roots of the equation dG=d = 0. Let the set of roots and the corresponding local maxima be denoted by S and G(S), respectively:
dG d2 G S = : = 0; 60 : d d2 We know dG 2f() [2(1 − F())f () + f2 ()]: = d (1 − F())2 For every such that dG=d = 0, assuming that f(:) is twice di5erentiable, d2 G 2f() f (): = 2 d 1 − F()
(19)
The expression for S, the set of all local maxima, thus simpli6es to S = {: C() = 0; f () 6 0};
(20)
where C() is the characteristic function, de6ned by C() = 2[1 − F()]f () + f2 ():
(21)
(ii) This optimal value of is therefore the member of S corresponding to the maximum of G(S). = Argmax{G(S)}:
(22)
(iii) The optimal value of the quantizer threshold is given by 1 = . It is now apparent that the optimal value of the threshold 1 is proportional to , the constant of proportionality being dependent on the PDF of the input noise. Therefore, when operating the detector, the threshold has to be varied linearly with the noise standard deviation. A similar observation has been made for systems with AR(1) nonlinearity [16]. For noise PDFs belonging to the families of generalized Gaussians and mixtures of Gaussians, a means of improving the computational accuracy and eFciency of the algorithm is discussed in Appendix A. The quantizer with optimal threshold 1 , together with the necessary detection statistic (shown in
Section 5), shall henceforth be called the SR detector. We note that the SNR gain of a system is a good measure of its usefulness for a detector. This is shown explicitly in Section 5, where other relevant details of the SR, linear, and optimal nonlinear detectors are also mentioned. In the next section we analyze the relative performance of these detectors based on the values of their SNR gains. 3.1. Results: generalized Gaussians For noise distributions belonging to the family of generalized Gaussians, if p 6 1, the characteristic function C() de6ned in (21) only has a root at 0. For p 6 1, the optimal value of is zero, i.e. the optimal 3-level quantizer has the degenerate form −1 for x[n] 6 0; (23) y[n] = 1 for x[n] ¿ 0: For p ¿ 1 however, C() has a positive root. In Fig. 3 and 4 the maximum value of G and the optimal value of , denoted by GSR and opt respectively, are plotted as functions of the parameter p of the generalized Gaussian family. For the Gaussian distribution (p = 2), GSR 0:81, which is very close to the lowest value of GSR . For the Laplacian distribution (p = 1), GSR = 2. For 0 ¡ p ¡ 1, GSR increases very sharply as p is reduced, and GSR → ∞ as p → 0+ . The explanation of the divergence of GSR is as follows. ∀p 6 1, the optimal normalized threshold = 0, and therefore from (9), f() = a. The dominant behavior of a and therefore of f( = 0) can be found by applying Stirling’s formula (limx→∞ (x) ∼ 1=2 ) to (10). Thus we get f( = 0) ∼ xx exp(−x)(2$=x) √ 1=2 1=2 3=2p ((2$ 3) =2)p 3 , which diverges as p → 0+ . Consequently GSR also diverges. For p ¿ 2, GSR increases slowly as p is increased and GSR → ∞ as p → ∞. The most signi6cant aspect of these graphs is that we can identify ranges of p; p 6 1:55 and p ¿ 4:1 for which GSR ¿ 1 and that for the corresponding PDFs therefore the SR detector shall outperform the linear detector. 3.2. Results: mixture of Gaussians It is apparent from (11) that PDFs belonging to the Gaussian mixture family can be uniquely
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1199
35
30
25
Gain G in dB
20
15
10
optimal detector SR detector
5 matched filter 0
5 0
1
2
3
4
5
6
7
8
9
10
Index of generalized Gaussian p Fig. 3. Variation of the gain of the optimal detector, SR detector and matched 6lter in generalized Gaussian noise as a function of the index p of generalized Gaussian PDF.
characterized by two parameters: and . Therefore, GSR and opt are plotted as functions of both and in Figs. 5 and 6. It is observed that as → 0 or → 1, the PDF tends to a Gaussian, and the gain GSR falls below unity. But for intermediate values of ; GSR → ∞ as → ∞. This divergence of GSR can also be explained by asymptotic analysis. As → ∞, the optimal normalized threshold → 0, √ c= → (1 − )1=2 and c= 2$ exp(−(c)2 =2) → (() where ((:) denotes the Dirac delta function. Thus we √ have lim→∞ f()=(()+(1−)3=2 = 2$ exp(−(1− )2 =2), which is a divergent (generalized) function. Consequently GSR diverges. The values of (; ) for which GSR ¿ 1 are identi6able from the region enclosed by the contour GSR = 1 in Fig. 7. For these values of (; ) and for the corresponding PDFs, therefore, the performance of the SR detector shall be superior to that of the linear detector. For a more explicit comparison the SNR gains of the SR, linear and optimal nonlinear detectors are plotted
for a 6xed value of with varying in Fig. 8, and then for a 6xed value of with varying in Fig. 9. 4. Other quantizers In this section we compare the performance of the 3-level symmetric quantizer to two other types of quantizer widely studied in connection to stochastic resonance. They are the asymmetric 2-level quantizer [5,6] and the (2M + 1)-level symmetric uniform quantizer family [8]. 4.1. Two-level asymmetric quantizer The input–output relation of a 2-level quantizer is given by −1 for x[n] 6 1 ; (24) y[n] = 1 for x[n] ¿ 1 :
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1.8 1.6
Normalized threshold γopt
1.4 1.2 1
0.8 0.6 0.4 0.2 0 0
1
2
3
4
5
6
7
8
9
10
Index of generalized Gaussian p Fig. 4. Variation of normalized threshold of SR detector in generalized Gaussian noise as a function of the index p of generalized Gaussian PDF.
Making the same assumptions as for the 3-level quantizer we obtain E[y[n]] = 1 − 2F() + 2Af()cn ; y2 [n] = 4F()[1 − F()]:
4.2. (2M + 1)-level uniform symmetric quantizer
The SNR gain G2 for the 2-level quantizer therefore is given by G2 () = =
f2 () F()[1 − F()] G3 () ; 2F()
(25)
where G3 () is the SNR gain of the 3-level quantizer in (18). For nonnegative values of ; 2F() ¿ 1. Therefore we get the following inequality relating the optimal gains of the 2- and 3-level quantizer: max G2 () 6 max G3 ():
Therefore we can conclude that the detection performance of the 2-level quantizer can never exceed that of the 3-level quantizer.
(26)
The input–output relation of a (2M +1)-level quantizer is given by −M for x[n] 6 − (2M − 1)1 ; m for (2m − 1)1 ¡x[n]6(2m + 1)1 ; y[n] = m = 0; ±1; : : : ; ±(M − 1); M for x[n] ¿ (2M − 1)1 : (27) Once again making the same assumptions as for the 2- and 3-level quantizers and proceeding similarly we
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1201
Fig. 5. Three-dimensional plot of the gain of the SR detector in mixture-of-Gaussian noise as a function of the parameters and of the noise PDF.
Fig. 6. Three-dimensional plot of the normalized threshold of the SR detector in mixture-of-Gaussian noise as a function of the parameters and of the noise PDF.
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1 0.9 0.8
Mixing parameter α
0.7 0.6 GSR=8
0.5 0.4
GSR=4
0.3 GSR=2
0.2 GSR=1 0.1 0 3
4
5
6
7
8
9
10
Ratio of standard deviations β Fig. 7. Contour plot of the gain of the SR detector in mixture-of-Gaussian noise as a function of the parameters and of the noise PDF.
get after some tedious but straight forward algebra G = M
2[
m=1
M
m=1
f((2m − 1))]2
(2m − 1)[1 − F((2m − 1))]
:
(28)
The maximum value of G, denoted by GSR , is plotted vs. (2M + 1) in Fig. 10 for di5erent input noise distributions belonging to the generalized Gaussian family. It is observed that no signi6cant increase in SNR gain can be achieved by increasing the number of quantizer levels. Henceforth the discussion is restricted exclusively to symmetric 3-level quantizers.
reduced to the following test: Decide H1 is true if T (x) ¿ ; Decide H0 is true if T (x) ¡ : Here T (x) is the test statistic, a function of the data vector x[n], and is the detector threshold. If the amplitude and phase of the sinusoidal signal are unknown, the test statistic for the optimal detector and the more conventional quadrature detector share the following generic form: 2 −1 1 N T (x) = g(x[n]) exp(−j2$f0 n : (29) N n=0
5. Detection statistics In statistical detection theory it is customary to state the signal detection problem as that of testing two hypotheses: ‘signal present’ and ‘signal absent’, denoted by H1 and H0 respectively. The decision can be
The test statistic T (x) is therefore the periodogram of the transformed data vector g(x[n]) at the frequency f0 . It can be shown [11] that for this class of test statistic, under hypothesis H0 the random variable (2=2 )T has chi-squared distribution with two degrees of freedom, and that under hypothesis H1 , the random variable (2=2 )T has noncentral chi-squared distribution
A.A. Saha, G.V. Anand / Signal Processing 83 (2003) 1193 – 1212
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60
50 optimal detector
Gain G in dB
40
30
20
10 SR detector matched filter
0 3
4
5
6
7
8
9
10
Ratio of standard deviations β Fig. 8. Variation of the gain of the optimal detector, SR detector and matched 6lter in mixture-of-Gaussian noise as a function of the parameter of the noise PDF, with = 0:2.
with two degrees of freedom and noncentrality parameter de6ned by GEs GRin ; = = 22 2
(30)
where G is the SNR gain of the system, Es is the energy of the signal, 2 is the noise power and Rin is the input SNR. Hence, the probability of false alarm is given by PF = P(T ¿ ; H0 ) = exp(− =2 );
(31)
and the probability of detection is given by 2
PD = P(T ¿ ; H1 ) = Q 2 ( ) (2 = ); 2
by
1 : PD = Q 2 ( ) 2 ln 2 PF
For the quadrature detector or incoherent matched 6lter, gMF (:) is the identity transformation, which is the optimal transformation in Gaussian noise: gMF (x[n]) = x[n]:
where Q 2 ( ) (:) denotes the right-tail probability of the 2 noncentral chi-squared PDF with two degrees of freedom and noncentrality parameter . The corresponding receiver operating characteristic (ROC) is given
(34)
Since the SNR gain of the matched 6lter is unity, GMF = 1 and the noncentrality parameter is given by MF =
(32)
(33)
Es Rin = : 22 2
(35)
For non-Gaussian noise the optimal nonlinear transformation gopt can be derived from the Neyman Pearson criterion and is de6ned [11] as follows: gopt (&) = −2
f (&) ; f(&)
(36)
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18 optimal detector
16 14
Gain G in dB
12 10 8 6 4 2
SR detector matched filter
0 -2 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Mixing parameter α Fig. 9. Variation of the gain of the optimal detector, SR detector and matched 6lter in mixture-of-Gaussian noise as a function of the parameter of the noise PDF, with = 2:5.
where f (&) is the derivative of the PDF f(&). The SNR gain Gopt and the resulting noncentrality parameter opt of the resulting optimal nonlinear detector [11] are given by ∞ (f (&))2 Gopt = 2 d&; (37) f(&) −∞ opt =
Gopt Es Gopt Rin = : 2 2 2
(38)
As shown in [11] the optimal nonlinear transformation yields an e5ective SNR gain Gopt whose value is greater than unity if the noise is non-Gaussian. We can obtain a suboptimal detector of sinusoidal signals by replacing the optimal nonlinear transformation gopt (:) de6ned in (36) by a stochastic resonator (SR) operating at maximum gain. The SNR gain can be maximized by choosing the threshold 1 according to the optimization algorithm outlined in Section 3. The test
statistic of the SR detector is therefore given by N −1 2 1 gSR (x[n]) exp(−j2$f0 n) ; TSR (x) = N
(39)
n=0
where gSR (x[n]) = y[n] is the output of the stochastic resonator. Now the expressions for PF and PD have the form as in (31) and (32), but the parameter is rede6ned as SR =
GSR Es : 22
(40)
Since the ROC for the stochastic resonator is now given by 1 PD = Q 2 ( SR ) 2 ln 2 PF
(41)
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Fig. 10. Variation of maximum SNR gain GSR of multilevel symmetric quantizers as a function of the number of quantization levels 2M + 1. Input noise PDF is generalized Gaussian with p = 1; 1:5; 2; 4:5; 5.
1 0.9
optimal detector
0.8 SR detector
0.7 0.6
matched filter
0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 11. ROCs of linear, SR and optimal nonlinear detectors in generalized Gaussian noise with p = 0:55.
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and Q 2 ( ) (x) increases monotonically with for all 2 x, it follows that (i) By maximizing GSR we shall maximize PD for a given PF , thereby optimizing the detection performance of the SR detector. (ii) If GSR ¿ 1 , the SR detector performs better than the conventional matched 6lter. In Section 3, we had observed that for generalized Gaussians with p 6 1:55 and ¿ 4:1, the SR detector shall outperform the linear detector. For the values of p = 0.55 and 20, which are representative of these two regions, this is apparent from the ROCs of the matched 6lter, SR and optimal nonlinear detectors for the low input SNR of Rin = −10 dB, in Figs. 11 and 13, respectively. The degree of improvement in these regions increases markedly as p → 0+ and p → ∞, as shown in Figs. 12 and 14. Similarly, for Gaussian mixtures, for any ordered pair (; ) for which GSR ¿ 1, the SR detector shows an improvement over the matched 6lter. For
(; ) = (0:3; 10) this is borne out from the ROCs of the matched 6lter, SR and optimal nonlinear detectors for the low input SNR of Rin = −10 dB in Fig. 15. When the given value of is kept constant, the degree of improvement increases markedly as is increased, as shown in Fig. 16. This is because, as observed in Section 3, for a 6xed value of such that 0 ¡ ¡ 1, GSR diverges as → ∞. When the given value of is kept constant, the ROC shows improvement as is increased from 0, but then decreases as approaches 1. This is illustrated for = 20 in Figs. 17 and 18. The plots of the SNR gains and the ROCs also illustrate an interesting trend 6rst reported by Chapeau– Blondeau [4]: quantizer detectors show greater improvement over matched 6lter detectors as the noise PDF becomes more heavy-tailed or leptokurtic in nature, as when p → 0+ for generalized Gaussians, and → ∞ for a 6xed for Gaussian mixtures. A tentative explanation is as follows: quantizers, being threshold systems, always have bounded outputs. Thus, upon quantization, the expected output noise power is necessarily 6nite, regardless of the
1 p=0.4 0.9 p=0.45 0.8 p=0.5 0.7
p=0.55 p=0.65
0.6
p=0.9 0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fig. 12. Variation of ROCs of the SR detector in generalized Gaussian noise as p → 0.
1
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1207
1 0.9 0.8 0.7
optimal detector
PD
0.6
SR detector
0.5
matched filter
0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PFA Fig. 13. ROCs of linear, SR and optimal nonlinear detectors in generalized Gaussian noise with p = 20.
input. For increasingly leptokurtic PDFs however, the expected input noise power diverges. Therefore, for matched 6lters, the expected output noise power diverges. Thus, quantizers ‘damp’ the degenerative effects of noise more e5ectively than do matched 6lters, for more heavy-tailed or leptokurtic PDFs, hence the pronounced improvement for detectors with quantizer nonlinearity for such PDFs. 6. Conclusions We have investigated the phenomenon of stochastic resonance exhibited by quantizers. The case of weak sinusoidal signals is considered, and expressions for the output SNR gain of 2-level quantizer, symmetric 3-level quantizer and uniform symmetric (2M + 1)-level quantizer are derived in terms of the noise PDF, noise variance, and the quantizer threshold. The SNR gain depends only on the ratio of the quantizer threshold and the noise standard deviation, and not separately on the values of these parameters.
Thus, if the noise PDF and noise variance are known, the quantizer threshold that maximizes the SNR gain can be readily determined. Two families of noise PDF relevant to the ocean acoustics scenario, viz. the generalized Gaussian family and the mixture of Gaussian family, are considered. It is shown that, for a large number of noise PDFs belonging to either family, the peak SNR gain of the symmetric 3-level quantizer exceeds unity. Two other important results are the following: (a) the peak SNR gain of the symmetric 3-level quantizer is greater than that of the 2-level quantizer for all PDFs and (b) if the quantization is uniform, improvement in SNR gain obtained by increasing the number of quantization levels is not very signi6cant. Hence, only 3-level quantizers are considered in the discussion of application of quantizers for sinusoidal signal detection. The application of stochastic resonance in quantizers for detection of sinusoidal signals in i.i.d. non-Gaussian noise is discussed, and the performance of the SR detector is compared with that of the optimal detector and the matched 6lter. Asymptotic
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1 0.9 0.8 0.7
p=90 p=60 p=40
PD
0.6
p=30 p=15 p=2
0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PFA
Fig. 14. Variation of ROCs of the SR detector in generalized Gaussian noise as p → ∞. 1 optimal detector 0.9 0.8 SR detector
0.7 0.6 PD
matched filter 0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5 PFA
0.6
0.7
0.8
0.9
1
Fig. 15. ROCs of linear, SR and optimal nonlinear detectors in mixture-of-Gaussian noise with (; ) = (0:3; 10).
A.A. Saha, G.V. Anand / Signal Processing 83 (2003) 1193 – 1212
1209
1 β = 50
0.9
β = 40
0.8
β = 30
0.7
β = 25 β = 20
0.6 PD
β = 16 0.5
β = 10
0.4
β=2
0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PFA Fig. 16. Variation of ROCs of the SR detector in mixture-of-Gaussian noise as → ∞, with = 0:3.
expressions for the probability of false alarm PF and probability of detection PD of the SR detector are derived for the case when the signal amplitude and phase are unknown but the signal frequency is known. This characterizes the acoustic signatures of most target vessels. An analysis of the degradation in performance of such an SR detector due to an error in the estimate of the signal frequency shall be presented in a future investigation. It is shown that the performance of the SR detector depends solely on its SNR gain GSR . If GSR is greater than unity, the SR detector performs better than the matched 6lter. If Gopt denotes the e5ective SNR gain of the optimal detector it is observed that for Laplacian noise Gopt = GSR = 2, but in general Gopt ¿ GSR . Hence, in general, the asymptotic performance of the optimal detector is better that of the SR detector. The advantage of the SR detector over the optimal detector is that the nonlinear transformation involved, that of quantization, is simple, easy to implement and remains invariant in a variety of marine environments.
Acknowledgements This project was sponsored by the Naval Physical and Oceanographic Laboratory, Kochi, India, under Project Number PC 4048. The authors are grateful to the referees for having suggested corrections and re6nements to the original manuscript.
Appendix A. Step (ii) of the algorithm to 6nd the optimal value of the quantizer threshold 1 as formulated in Section 3 involves calculating the zeros of the characteristic function C() de6ned in (20). There are two problems encountered here: (i) C() is a rapidly decreasing function. Calculating the zeros of such a function is prone to inaccuracy and is characterized by slow rate of convergence.
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1 0.9 0.8 α = 0.65 0.7 α = 0.45 0.6 PD
α = 0.25 0.5
α = 0.05
0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PFA Fig. 17. Variation of ROCs of the SR detector in mixture-of-Gaussian noise for = 0:05; 0:25; 0:45; 0:65, with = 20.
(ii) Each evaluation of C() requires numerical computation of an integral for the term [1 − F()], which contributes a considerable overhead.
The incomplete Gamma function is de6ned as ∞ e−t t −1+a dt; (A.2) (z; a) =
The accuracy and convergence for the root 6nding procedure for the families of generalized Gaussians and mixtures of Gaussians can be improved by:
which can also be expressed as
(i) Calculating the zeros of the normalized charac˜ teristic function C() de6ned as
where H (z; a) is a continued fraction with the representation
˜ C() = C()=f2 (); 2
(A.1)
where f () is a rapidly decreasing but posi˜ tive function. It follows that C() and C() have the same zero crossings. Therefore, normalization improves the numerical accuracy without a5ecting the location of the zeros or roots. (ii) Expressing the term [1 − F()] in terms of the complement of the incomplete Gamma function, which can be represented in terms of a standardized continued fraction [15], thus avoiding the numerical computation of an integral.
z
(z; a) = e−z z a H (z; a);
H (z; a) =
(A.3)
1 z+1−a−
1(1−a) 2(2−a) z+3−a− z+5−a− ···
:
(A.4)
For the family of generalized Gaussians ∞ p ae−b|t| dt [1 − F()] =
=
a (bp ; 1=p): b1=p p
(A.5)
Combining (21), (A.3) and (A.5) we arrive at the following expression for the characteristic
A.A. Saha, G.V. Anand / Signal Processing 83 (2003) 1193 – 1212
1211
1 0.9 0.8
α = 0.65 α = 0.88
0.7
α = 0.95
PD
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PFA Fig. 18. Variation of ROCs of the SR detector, in mixture-of-Gaussian noise for = 0:65; 0:88; 0:95, with = 20.
function C(): C() = [1 − 2bp H (bp ; 1=p)]f2 ():
(A.6)
˜ The normalized characteristic function C() is therefore ˜ C() = [1 − 2bp H (bp ; 1=p)]:
(A.7)
The expression for f () for the local extrema of G is given by f () = −pbp−2 [p − 1 − pbp ]f():
(A.8)
R() = e−(1−1=
For the family of mixtures of Gaussians [1 − F()] = √ ((c)2 =2; 1=2) 2 $ 1− + √ ((c)2 =22 ; 1=2): 2 $
˜ function C(): ˜ C() = 1 − (c)2 R()H ((c)2 =2) + (1 − )=H ((c)2 =22 ) × R() + (1 − )= R() + (1 − )=3 × : (A.10) R() + (1 − )= For convenience of notation H (:; 1=2) is denoted by H (:) and the quantity R() is introduced which can be de6ned as the ratio of the two component Gaussians by the formula 2
)(c)2 =2
:
(A.11)
(A.9)
Combining (21), (A.3) and (A.9) we arrive at the following expression for the normalized characteristic
Similarly the expression for f () at the local extrema of G is given by 2 2 c3 f () = − √ e−(c) =2 2$ ×[{1 − (c)2 }R() + (1 − )=3 ×{1 − (c)2 =2 }]:
(A.12)
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References [1] F. Chapeau-Blondeau, Stochastic resonance in the Heaviside nonlinearity with white noise and arbitrary periodic signal, Phys. Rev. E 53 (1996) 5469–5472. [2] F. Chapeau-Blondeau, Stochastic resonance and optimal detection of pulse trains by threshold devices, Digital Signal Process. 9 (1997) 162–177. [3] F. Chapeau-Blondeau, Input–Output gain for signal in noise in stochastic resonance, Phys. Lett. A 232 (1997) 41–48. [4] F. Chapeau-Blondeau, Nonlinear test statistic to improve signal detection in non-Gaussian noise, IEEE Signal Process. Lett. 7 (2000) 205–207. [5] F. Chapeau-Blondeau, X. Godivier, Noise-assisted signal transmission in a nonlinear electronic comparator: experiment and theory, Signal Processing 56 (1997) 293–303. [6] F. Chapeau-Blondeau, X. Godivier, Theory of Stochastic resonance in signal transmission by static nonlinear systems, Phys. Rev. E 55 (1997) 1478–1495. [7] L. Gammaitoni, P. Hanggi, P. Jung, F. Marchesoni, Stochastic resonance, Rev. Mod. Phys. 70 (1998) 223–287. [8] L. Gammaitoni, F. Marchesoni, S. Santucci, Stochastic resonance and the dithering e5ect in threshold physical systems, Phys. Rev. E 52 (5) (1995) 4691–4698. [9] X. Godivier, J. Rojas-Varela, F. Chapeau-Blondeau, Noiseassisted signal transmission via stochastic resonance in a diode nonlinearity, Electron. Lett. 33 (1997) 1666–1668.
[10] M.E. Inchiosa, A.R. Bulsara, Signal detection statistics of stochastic resonators, Phys. Rev. E 53 (1996) 2021–2024. [11] S.M. Kay, Fundamentals of Statistical Signal Processing, Vol. II: Detection Theory, Prentice-Hall, Englewood Cli5s, NJ, l998. [12] F.W. Machell, C.S. Penrod, G.E. Ellis, Statistical characteristics of ocean acoustic noise processes, in: E.J. Wegman, S.C. Schwartz, J.B. Thomas (Eds.), Topics in Non-Gaussian Signal Processing, Springer, New York, 1989, pp. 29–57 (Chapter 3). [13] S. Mitaim, B. Kosko, Adaptive stochastic resonance, Proc. IEEE 86 (1998) 2152–2183. [14] D.R. Powell, G.R. Wilson, Class A Modeling of ocean acoustic noise processes, in: E.J. Wegman, S.C. Schwartz, J.B. Thomas (Eds.), Topics in Non-Gaussian Signal Processing, Springer, New York, 1989, pp. 17–28 (Chapter 2). [15] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C: The Art of Scienti6c Computing, Cambridge University Press, Cambridge, MA, 1995. [16] S. Zozor, P.-O. Amblard, Stochastic resonance in discrete-time nonlinear AR(1) models, IEEE Trans. Signal Process. 47 (1) (1999) 108–122; S. Zozor, P.-O. Amblard, IEEE Trans. Signal Process. 49 (5) (2001) 1107–1109. [17] S. Zozor, P.-O. Amblard, On the use of stochastic resonance in sine detection, Signal Processing 82 (3) (2002) 353–367.
Design of detectors based on stochastic resonance Aditya A. Saha∗ , G.V. Anand Acoustics Laboratory, Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore 560 012, India Received 24 January 2002; received in revised form 28 August 2002
Abstract This paper presents a study of the phenomenon of stochastic resonance in quantizers, and discusses the use of this phenomenon for the detection of weak sinusoidal signals in noise. Stochastic resonance in 2-level, symmetric 3-level, and symmetric multilevel quantizers is investigated. Expressions are derived for the signal-to-noise ratio (SNR) gain of the quantizers driven by a small amplitude sinsuoidal signal and i.i.d. noise. The gain depends on the probability density function (PDF) of the input noise, and for a given noise PDF, the gain can be maximized by a proper choice of the quantizer thresholds. The maximum gain GSR is less than unity if the input noise is Gaussian, but several non-Gaussian noise PDFs yield values of GSR exceeding unity. Thus, the quantizers provide an e5ective enhancement in the SNR, which can be utilized to design a nonlinear signal detector whose performance is better than that of the matched 6lter. The nonlinear detector in consideration consists of a stochastically resonating (SR) quantizer followed by a correlator. An asymptotic expression for the probability of detection of the SR detector is derived. It is shown that the detection performance of the SR detector is better than that of the matched 6lter for a large class of noise distributions belonging to the generalized Gaussian and the mixture-of-Gaussian families. ? 2003 Elsevier Science B.V. All rights reserved. Keywords: Stochastic resonance; Threshold nonlinearity; Quantizer; Suboptimal detector; Passive detection; Non-Gaussian noise; Marine noise
1. Introduction Detection of sinusoidal signals at low signal-tonoise ratios (SNRs) is a problem of great interest in the context of passive detection of targets in the ocean. The acoustic radiation from most targets contain distinct line components in the spectral domain, and are weak compared to the ambient noise. It is known [11] that a quadrature or incoherent matched 6lter is the optimal detector if a signal with unknown amplitude/phase is buried in Gaussian noise. Here, ∗ Corresponding author. Department of Physics, University of Alberta, Edmonton, AB, Canada T6G 2J1. E-mail address: [email protected] (A.A. Saha).
optimality refers to maximization of the probability of detection constrained by a 6xed probability of false alarm. The matched 6lter though easy to implement and analyze is not optimal under conditions of non-Gaussian noise prevalent in marine environments. Optimal detectors in non-Gaussian noise are nonlinear and are not easy to implement. Hence, suboptimal nonlinear detectors which are easier to implement are often employed for the detection of signals in non-Gaussian noise. The aim of the present study is to design easily implementable nonlinear detectors based on stochastic resonance, and to analyze the performance of these detectors. Stochastic resonance (SR) is the phenomenon of enhancement of signal transmission by certain
0165-1684/03/$ - see front matter ? 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0165-1684(03)00039-2
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Nomenclature 1 x[n] y[n] s[n] w[n] A1
Y1 y2 [n] y2 G Rin; out p a; b
noise standard deviation threshold normalized threshold input sequence output sequence incoming sinusoidal signal white noise sequence at the input amplitude of incoming sinusoid phase of incoming sinusoid 6rst Fourier coeFcient of the sequence E(y[n]) variance of y[n] average of the sequence E(y[n]) SNR gain of the system considered input, output SNR order of generalized Gaussian constants associated with generalized Gaussians mixing parameter of mixture-ofGaussians ratio of standard deviations of mixtureof-Gaussians
nonlinear systems resulting from the addition of noise to the system (see [7,13] for recent reviews). Quantizers and other static nonlinear systems exhibit stochastic resonance [1–3,5,6,8,9,16,17]. For an SR system driven by a sinusoidal signal and stationary white noise, the output SNR increases as the input noise intensity 2 is increased over a certain range of values of 2 . The probability of detection of a signal is maximized at the peak of the output SNR [10]. If the SNR gain is greater than unity, a combination of the SR system and a matched 6lter will yield, a detection performance that is better than that of the matched 6lter alone [4]. While conventional studies of stochastic resonance search for the optimal noise level, keeping the system 6xed [8,10], in detection schemes the problem is reversed; the system parameters are optimized for a given noise level and type. This has been demonstrated by Chapeau–Blondeau for the detection of pulse trains using quantizers with optimizable thresholds [2]. In this paper also, we follow a similar approach. The organization of the paper is as follows. In Section 2,
c R(:) P(:) f(:) F(:) C(:) ˜ C(:) (:) (:; :) H (:; :) T (:) 22 (:) Q 2 ( ) (:) 2
constant associated with mixture-ofGaussians ratio of component Gaussians for mixture-of-Gaussians probability of an event PDF of input noise probability distribution function of input noise the characteristic function for detector the normalized characteristic function for detector the Gamma function the incomplete Gamma function the continued fraction associated with the incomplete Gamma function test statistic for detector detector threshold noncentral chi-squared distribution noncentrality parameter of the chisquared distribution the right-tail probability of the noncentral chi-squared PDF
properties of stochastic resonance demonstrated by quantizers and their utility in signal detection are discussed qualitatively. It is shown that the maximum SNR gain is less than unity if the input noise is Gaussian, but the SNR gain can exceed unity if the noise is non-Gaussian. In Section 3, an algorithm for optimizing the SNR gain of the symmetric 3-level quantizer is developed, and is applied to generalized Gaussian and mixture-of-Gaussian noise probability density functions (PDFs). Section 4 presents a comparison between the optimal SNR gains of the symmetric 3-level, asymmetric 2-level and uniform symmetric multilevel quantizers. In Section 5, the monotonic dependence of the receiver operating characteristics on the SNR gain is established. It is also observed that if the noise PDF belongs to a subclass of the generalized Gaussian family, or a subclass of the Gaussian mixture family, the asymptotic performance of the quantizer-detector is signi6cantly better than that of the matched 6lter but not as good as that of the optimal nonlinear detector. Section 6 contains the conclusions.
A.A. Saha, G.V. Anand / Signal Processing 83 (2003) 1193 – 1212
2. Stochastic resonance in quantizers We consider a symmetric 3-level quantizer with thresholds −1 ; 1 and quantization levels −1; 0; 1. If the quantizer is driven by a sequence x[n], the output sequence y[n] is given by the relation −1 for x[n] 6 − 1 ; for − 1 ¡ x[n] 6 1 ; y[n] = 0 (1) 1 for x[n] ¿ 1 : Let the input x[n] be the sum of an N -periodic sinusoidal signal s[n] with a known frequency and a zero-mean noise sequence w[n] consisting of independent and identically distributed (i.i.d) random variables with variance 2 : x[n] = s[n] + w[n]:
(2)
The incoming signal s[n] can be represented as s[n] = A1 cos(2$n=N − );
(3)
and is subthreshold such that A1 6 1 . In the absence of input noise, the output y[n] remains unchanged at the initial state. When noise is also present at the input, the output exhibits random transitions from one state to another. For a sinusoidal signal we need to consider the output SNR at the 6rst harmonic alone, which, as discussed in [6], is given by SNR out =
|Y1 |2 y2
;
(4)
where Y1 is the 6rst Fourier coeFcient of the sequence E(y[n]) and y2 is the average of the sequence y2 [n], and are given by Y1 =
N −1 1 E(y[n]) exp(j2$n=N ); N n=0
y2 =
1 N
N −1
(5)
n=0
Similarly, the input SNR is de6ned as SNR in =
A21 : 42
the other hand, the input SNR depends only on the input signal amplitude and the input noise variance. The SNR gain is de6ned as SNR out : (7) G= SNR in It follows G=
42 |Y1 |2 A21 y2
:
(6)
It may be noted that the output SNR depends on the input signal waveform and the input noise PDF. On
(8)
Plots of SNR gain vs. are shown in Figs. 1 and 2 for di5erent i.i.d. noise distributions belonging to the generalized Gaussian and the Gaussian mixture families. The probability density function of a generalized Gaussian random variable with mean 0 and variance 2 is given by f(&) = a= exp(−b|&=|p ); where a=
p 1=2 (3=p) ; 2 3=2 (1=p)
p ¿ 0;
b=
(3=p) (1=p)
(9) p=2 ;
(10)
and (:) is the gamma function. This family of probability distributions is fairly representative, as they span heavy-tailed PDFs (the Laplacian), less heavy-tailed PDFs (the Gaussian), and PDFs of 6nite support (the uniform distribution), and are generally used in the analysis of stochastic resonators [4,5,16]. The Laplacian, Gaussian and uniform PDFs belong to the family de6ned by (9) corresponding to p = 1; 2 and ∞, respectively. A Gaussian mixture PDF with mean 0 and variance 2 has the form 2 2 c & c √ exp − 2 f(&) = 2 2$ 1− c 2 &2 + ; (11) exp − 2 2 2 where c = [ + (1 − )2 ]1=2 ;
y2 [n]:
1195
0 ¡ ¡ 1; ¿ 0:
(12)
This family of probability distributions is a subclass of Middleton’s class A distributions and is widely used to model ocean acoustic noise [12,14]. Figs. 1 and 2 illustrate the trend of increasing output SNR gain with increasing input noise intensity, for a generalized Gaussian PDF and a Gaussian mixture PDF. This implies that over a certain range, increasing noise
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6
5
SNR gain G
4 p=0.5 3
p=10
2
p=1
1
p=2
0 0
1
2
3
4
5
6
7
8
9
10
Standard deviation of input noise Fig. 1. Variation of SNR gain of symmetric 3-level quantizer as a function of the standard deviation of input noise, with threshold 1 = 1. Input noise PDF is generalized Gaussian, with p = 0:5; 1; 2; 10.
intensity enhances signal transmission and the quantizer can be characterized as a stochastic resonant (SR) system. From the point of view of signal detection, the most signi6cant outcome of these computations is that the peak value of SNR gain exceeds unity in the examples involving non-Gaussian noise. For these distributions, a detector incorporating the quantizer discussed above may be expected to perform better than the linear detector since no linear system can provide an SNR gain exceeding unity.
where w[n] are zero-mean, unit-variance i.i.d. random variables with PDF f(&) and probability distribution function F(&). It follows that the mean and variance of y[n] are: E[y[n]] = P(y[n] = 1) − P(y[n] = −1) = 1 − F( − Acn ) − F(− − Acn ); y2 [n] = P(y[n]
= 1) + P(y[n] = −1) − {E[y[n]]}2
= 1 − F( − Acn ) + F(− − Acn ) − {E[y[n]]}2 ;
(14)
where 3. Detector optimization
A = A1 =;
For a symmetric 3-level quantizer whose input– output relation is given by (1), the input sequence can be rewritten as x[n] = A1 cos(2$n=N − ) + w[n];
(13)
= 1 =;
cn = cos(2$n=N − ): (15)
The parameter , which we call the normalized threshold, is the ratio of the quantizer threshold 1 to the noise standard deviation . For the nondegenerate and degenerate forms of (1), 1 is positive and zero,
A.A. Saha, G.V. Anand / Signal Processing 83 (2003) 1193 – 1212
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3.5 α = 0.7 α = 0.75
3
α = 0.8
SNR gain G
2.5
2
1.5
1
0.5
0 0
1
2
3
4
5
6
7
8
9
10
Standard deviation of input noise Fig. 2. Variation of SNR gain of symmetric 3-level quantizer as a function of the standard deviation of input noise, with threshold 1 = 1. Input noise PDF is a Gaussian mixture, with = 0:7; 0:75; 0:8 and = 5.
respectively. Therefore it follows that is always nonnegative. Before proceeding further, we make the following assumptions: (i) The signal amplitude A1 is small compared to the noise standard deviation , i.e. A = A1 =1. This assumption stems from the fact that our primary concern is the detection of ‘weak’ signals in noise. (ii) The PDF f(&) is symmetric. This assumption is valid for most marine noise models. The assumptions made above, together with Taylor expansions about , simplify the expressions for the mean and variance as follows: E[y(n)] = F( + Acn ) − F( − Acn ) = 2AF()cn + O(A3 ); y2 [n] = 2 − F( + Acn ) − F( − Acn ) − {E[y[n]]}2 = 2[1 − F()] + O(A2 ):
Therefore the 6rst Fourier coeFcient Y1 of the sequence E(y[n]), and y2 the average variance observed over one period de6ned in (5), can be expressed as |Y1 |2 = A2 f2 () + O(A4 );
(16)
y2 = 2[1 − F()] + O(A2 ):
(17)
Under the small signal approximation A1, ignoring all but the lowest powers of A in the expansions, we get the following expression for SNR gain: G=
4|Y1 |2 A2 y2
=
2f2 () : 1 − F()
(18)
It is apparent that the gain depends on: (a) the input noise distribution and (b) the normalized threshold . For a given marine acoustic noise distribution, therefore, the SNR gain and thereby the detection performance can be maximized by determining the value of for which G is maximum. The steps of the optimization algorithm required to do so are as
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A.A. Saha, G.V. Anand / Signal Processing 83 (2003) 1193 – 1212
follows: (i) Since G is not necessarily unimodal, the set of all local maxima can be found from the roots of the equation dG=d = 0. Let the set of roots and the corresponding local maxima be denoted by S and G(S), respectively:
dG d2 G S = : = 0; 60 : d d2 We know dG 2f() [2(1 − F())f () + f2 ()]: = d (1 − F())2 For every such that dG=d = 0, assuming that f(:) is twice di5erentiable, d2 G 2f() f (): = 2 d 1 − F()
(19)
The expression for S, the set of all local maxima, thus simpli6es to S = {: C() = 0; f () 6 0};
(20)
where C() is the characteristic function, de6ned by C() = 2[1 − F()]f () + f2 ():
(21)
(ii) This optimal value of is therefore the member of S corresponding to the maximum of G(S). = Argmax{G(S)}:
(22)
(iii) The optimal value of the quantizer threshold is given by 1 = . It is now apparent that the optimal value of the threshold 1 is proportional to , the constant of proportionality being dependent on the PDF of the input noise. Therefore, when operating the detector, the threshold has to be varied linearly with the noise standard deviation. A similar observation has been made for systems with AR(1) nonlinearity [16]. For noise PDFs belonging to the families of generalized Gaussians and mixtures of Gaussians, a means of improving the computational accuracy and eFciency of the algorithm is discussed in Appendix A. The quantizer with optimal threshold 1 , together with the necessary detection statistic (shown in
Section 5), shall henceforth be called the SR detector. We note that the SNR gain of a system is a good measure of its usefulness for a detector. This is shown explicitly in Section 5, where other relevant details of the SR, linear, and optimal nonlinear detectors are also mentioned. In the next section we analyze the relative performance of these detectors based on the values of their SNR gains. 3.1. Results: generalized Gaussians For noise distributions belonging to the family of generalized Gaussians, if p 6 1, the characteristic function C() de6ned in (21) only has a root at 0. For p 6 1, the optimal value of is zero, i.e. the optimal 3-level quantizer has the degenerate form −1 for x[n] 6 0; (23) y[n] = 1 for x[n] ¿ 0: For p ¿ 1 however, C() has a positive root. In Fig. 3 and 4 the maximum value of G and the optimal value of , denoted by GSR and opt respectively, are plotted as functions of the parameter p of the generalized Gaussian family. For the Gaussian distribution (p = 2), GSR 0:81, which is very close to the lowest value of GSR . For the Laplacian distribution (p = 1), GSR = 2. For 0 ¡ p ¡ 1, GSR increases very sharply as p is reduced, and GSR → ∞ as p → 0+ . The explanation of the divergence of GSR is as follows. ∀p 6 1, the optimal normalized threshold = 0, and therefore from (9), f() = a. The dominant behavior of a and therefore of f( = 0) can be found by applying Stirling’s formula (limx→∞ (x) ∼ 1=2 ) to (10). Thus we get f( = 0) ∼ xx exp(−x)(2$=x) √ 1=2 1=2 3=2p ((2$ 3) =2)p 3 , which diverges as p → 0+ . Consequently GSR also diverges. For p ¿ 2, GSR increases slowly as p is increased and GSR → ∞ as p → ∞. The most signi6cant aspect of these graphs is that we can identify ranges of p; p 6 1:55 and p ¿ 4:1 for which GSR ¿ 1 and that for the corresponding PDFs therefore the SR detector shall outperform the linear detector. 3.2. Results: mixture of Gaussians It is apparent from (11) that PDFs belonging to the Gaussian mixture family can be uniquely
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1199
35
30
25
Gain G in dB
20
15
10
optimal detector SR detector
5 matched filter 0
5 0
1
2
3
4
5
6
7
8
9
10
Index of generalized Gaussian p Fig. 3. Variation of the gain of the optimal detector, SR detector and matched 6lter in generalized Gaussian noise as a function of the index p of generalized Gaussian PDF.
characterized by two parameters: and . Therefore, GSR and opt are plotted as functions of both and in Figs. 5 and 6. It is observed that as → 0 or → 1, the PDF tends to a Gaussian, and the gain GSR falls below unity. But for intermediate values of ; GSR → ∞ as → ∞. This divergence of GSR can also be explained by asymptotic analysis. As → ∞, the optimal normalized threshold → 0, √ c= → (1 − )1=2 and c= 2$ exp(−(c)2 =2) → (() where ((:) denotes the Dirac delta function. Thus we √ have lim→∞ f()=(()+(1−)3=2 = 2$ exp(−(1− )2 =2), which is a divergent (generalized) function. Consequently GSR diverges. The values of (; ) for which GSR ¿ 1 are identi6able from the region enclosed by the contour GSR = 1 in Fig. 7. For these values of (; ) and for the corresponding PDFs, therefore, the performance of the SR detector shall be superior to that of the linear detector. For a more explicit comparison the SNR gains of the SR, linear and optimal nonlinear detectors are plotted
for a 6xed value of with varying in Fig. 8, and then for a 6xed value of with varying in Fig. 9. 4. Other quantizers In this section we compare the performance of the 3-level symmetric quantizer to two other types of quantizer widely studied in connection to stochastic resonance. They are the asymmetric 2-level quantizer [5,6] and the (2M + 1)-level symmetric uniform quantizer family [8]. 4.1. Two-level asymmetric quantizer The input–output relation of a 2-level quantizer is given by −1 for x[n] 6 1 ; (24) y[n] = 1 for x[n] ¿ 1 :
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1.8 1.6
Normalized threshold γopt
1.4 1.2 1
0.8 0.6 0.4 0.2 0 0
1
2
3
4
5
6
7
8
9
10
Index of generalized Gaussian p Fig. 4. Variation of normalized threshold of SR detector in generalized Gaussian noise as a function of the index p of generalized Gaussian PDF.
Making the same assumptions as for the 3-level quantizer we obtain E[y[n]] = 1 − 2F() + 2Af()cn ; y2 [n] = 4F()[1 − F()]:
4.2. (2M + 1)-level uniform symmetric quantizer
The SNR gain G2 for the 2-level quantizer therefore is given by G2 () = =
f2 () F()[1 − F()] G3 () ; 2F()
(25)
where G3 () is the SNR gain of the 3-level quantizer in (18). For nonnegative values of ; 2F() ¿ 1. Therefore we get the following inequality relating the optimal gains of the 2- and 3-level quantizer: max G2 () 6 max G3 ():
Therefore we can conclude that the detection performance of the 2-level quantizer can never exceed that of the 3-level quantizer.
(26)
The input–output relation of a (2M +1)-level quantizer is given by −M for x[n] 6 − (2M − 1)1 ; m for (2m − 1)1 ¡x[n]6(2m + 1)1 ; y[n] = m = 0; ±1; : : : ; ±(M − 1); M for x[n] ¿ (2M − 1)1 : (27) Once again making the same assumptions as for the 2- and 3-level quantizers and proceeding similarly we
A.A. Saha, G.V. Anand / Signal Processing 83 (2003) 1193 – 1212
1201
Fig. 5. Three-dimensional plot of the gain of the SR detector in mixture-of-Gaussian noise as a function of the parameters and of the noise PDF.
Fig. 6. Three-dimensional plot of the normalized threshold of the SR detector in mixture-of-Gaussian noise as a function of the parameters and of the noise PDF.
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1 0.9 0.8
Mixing parameter α
0.7 0.6 GSR=8
0.5 0.4
GSR=4
0.3 GSR=2
0.2 GSR=1 0.1 0 3
4
5
6
7
8
9
10
Ratio of standard deviations β Fig. 7. Contour plot of the gain of the SR detector in mixture-of-Gaussian noise as a function of the parameters and of the noise PDF.
get after some tedious but straight forward algebra G = M
2[
m=1
M
m=1
f((2m − 1))]2
(2m − 1)[1 − F((2m − 1))]
:
(28)
The maximum value of G, denoted by GSR , is plotted vs. (2M + 1) in Fig. 10 for di5erent input noise distributions belonging to the generalized Gaussian family. It is observed that no signi6cant increase in SNR gain can be achieved by increasing the number of quantizer levels. Henceforth the discussion is restricted exclusively to symmetric 3-level quantizers.
reduced to the following test: Decide H1 is true if T (x) ¿ ; Decide H0 is true if T (x) ¡ : Here T (x) is the test statistic, a function of the data vector x[n], and is the detector threshold. If the amplitude and phase of the sinusoidal signal are unknown, the test statistic for the optimal detector and the more conventional quadrature detector share the following generic form: 2 −1 1 N T (x) = g(x[n]) exp(−j2$f0 n : (29) N n=0
5. Detection statistics In statistical detection theory it is customary to state the signal detection problem as that of testing two hypotheses: ‘signal present’ and ‘signal absent’, denoted by H1 and H0 respectively. The decision can be
The test statistic T (x) is therefore the periodogram of the transformed data vector g(x[n]) at the frequency f0 . It can be shown [11] that for this class of test statistic, under hypothesis H0 the random variable (2=2 )T has chi-squared distribution with two degrees of freedom, and that under hypothesis H1 , the random variable (2=2 )T has noncentral chi-squared distribution
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1203
60
50 optimal detector
Gain G in dB
40
30
20
10 SR detector matched filter
0 3
4
5
6
7
8
9
10
Ratio of standard deviations β Fig. 8. Variation of the gain of the optimal detector, SR detector and matched 6lter in mixture-of-Gaussian noise as a function of the parameter of the noise PDF, with = 0:2.
with two degrees of freedom and noncentrality parameter de6ned by GEs GRin ; = = 22 2
(30)
where G is the SNR gain of the system, Es is the energy of the signal, 2 is the noise power and Rin is the input SNR. Hence, the probability of false alarm is given by PF = P(T ¿ ; H0 ) = exp(− =2 );
(31)
and the probability of detection is given by 2
PD = P(T ¿ ; H1 ) = Q 2 ( ) (2 = ); 2
by
1 : PD = Q 2 ( ) 2 ln 2 PF
For the quadrature detector or incoherent matched 6lter, gMF (:) is the identity transformation, which is the optimal transformation in Gaussian noise: gMF (x[n]) = x[n]:
where Q 2 ( ) (:) denotes the right-tail probability of the 2 noncentral chi-squared PDF with two degrees of freedom and noncentrality parameter . The corresponding receiver operating characteristic (ROC) is given
(34)
Since the SNR gain of the matched 6lter is unity, GMF = 1 and the noncentrality parameter is given by MF =
(32)
(33)
Es Rin = : 22 2
(35)
For non-Gaussian noise the optimal nonlinear transformation gopt can be derived from the Neyman Pearson criterion and is de6ned [11] as follows: gopt (&) = −2
f (&) ; f(&)
(36)
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18 optimal detector
16 14
Gain G in dB
12 10 8 6 4 2
SR detector matched filter
0 -2 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Mixing parameter α Fig. 9. Variation of the gain of the optimal detector, SR detector and matched 6lter in mixture-of-Gaussian noise as a function of the parameter of the noise PDF, with = 2:5.
where f (&) is the derivative of the PDF f(&). The SNR gain Gopt and the resulting noncentrality parameter opt of the resulting optimal nonlinear detector [11] are given by ∞ (f (&))2 Gopt = 2 d&; (37) f(&) −∞ opt =
Gopt Es Gopt Rin = : 2 2 2
(38)
As shown in [11] the optimal nonlinear transformation yields an e5ective SNR gain Gopt whose value is greater than unity if the noise is non-Gaussian. We can obtain a suboptimal detector of sinusoidal signals by replacing the optimal nonlinear transformation gopt (:) de6ned in (36) by a stochastic resonator (SR) operating at maximum gain. The SNR gain can be maximized by choosing the threshold 1 according to the optimization algorithm outlined in Section 3. The test
statistic of the SR detector is therefore given by N −1 2 1 gSR (x[n]) exp(−j2$f0 n) ; TSR (x) = N
(39)
n=0
where gSR (x[n]) = y[n] is the output of the stochastic resonator. Now the expressions for PF and PD have the form as in (31) and (32), but the parameter is rede6ned as SR =
GSR Es : 22
(40)
Since the ROC for the stochastic resonator is now given by 1 PD = Q 2 ( SR ) 2 ln 2 PF
(41)
A.A. Saha, G.V. Anand / Signal Processing 83 (2003) 1193 – 1212
1205
Fig. 10. Variation of maximum SNR gain GSR of multilevel symmetric quantizers as a function of the number of quantization levels 2M + 1. Input noise PDF is generalized Gaussian with p = 1; 1:5; 2; 4:5; 5.
1 0.9
optimal detector
0.8 SR detector
0.7 0.6
matched filter
0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 11. ROCs of linear, SR and optimal nonlinear detectors in generalized Gaussian noise with p = 0:55.
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and Q 2 ( ) (x) increases monotonically with for all 2 x, it follows that (i) By maximizing GSR we shall maximize PD for a given PF , thereby optimizing the detection performance of the SR detector. (ii) If GSR ¿ 1 , the SR detector performs better than the conventional matched 6lter. In Section 3, we had observed that for generalized Gaussians with p 6 1:55 and ¿ 4:1, the SR detector shall outperform the linear detector. For the values of p = 0.55 and 20, which are representative of these two regions, this is apparent from the ROCs of the matched 6lter, SR and optimal nonlinear detectors for the low input SNR of Rin = −10 dB, in Figs. 11 and 13, respectively. The degree of improvement in these regions increases markedly as p → 0+ and p → ∞, as shown in Figs. 12 and 14. Similarly, for Gaussian mixtures, for any ordered pair (; ) for which GSR ¿ 1, the SR detector shows an improvement over the matched 6lter. For
(; ) = (0:3; 10) this is borne out from the ROCs of the matched 6lter, SR and optimal nonlinear detectors for the low input SNR of Rin = −10 dB in Fig. 15. When the given value of is kept constant, the degree of improvement increases markedly as is increased, as shown in Fig. 16. This is because, as observed in Section 3, for a 6xed value of such that 0 ¡ ¡ 1, GSR diverges as → ∞. When the given value of is kept constant, the ROC shows improvement as is increased from 0, but then decreases as approaches 1. This is illustrated for = 20 in Figs. 17 and 18. The plots of the SNR gains and the ROCs also illustrate an interesting trend 6rst reported by Chapeau– Blondeau [4]: quantizer detectors show greater improvement over matched 6lter detectors as the noise PDF becomes more heavy-tailed or leptokurtic in nature, as when p → 0+ for generalized Gaussians, and → ∞ for a 6xed for Gaussian mixtures. A tentative explanation is as follows: quantizers, being threshold systems, always have bounded outputs. Thus, upon quantization, the expected output noise power is necessarily 6nite, regardless of the
1 p=0.4 0.9 p=0.45 0.8 p=0.5 0.7
p=0.55 p=0.65
0.6
p=0.9 0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fig. 12. Variation of ROCs of the SR detector in generalized Gaussian noise as p → 0.
1
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1207
1 0.9 0.8 0.7
optimal detector
PD
0.6
SR detector
0.5
matched filter
0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PFA Fig. 13. ROCs of linear, SR and optimal nonlinear detectors in generalized Gaussian noise with p = 20.
input. For increasingly leptokurtic PDFs however, the expected input noise power diverges. Therefore, for matched 6lters, the expected output noise power diverges. Thus, quantizers ‘damp’ the degenerative effects of noise more e5ectively than do matched 6lters, for more heavy-tailed or leptokurtic PDFs, hence the pronounced improvement for detectors with quantizer nonlinearity for such PDFs. 6. Conclusions We have investigated the phenomenon of stochastic resonance exhibited by quantizers. The case of weak sinusoidal signals is considered, and expressions for the output SNR gain of 2-level quantizer, symmetric 3-level quantizer and uniform symmetric (2M + 1)-level quantizer are derived in terms of the noise PDF, noise variance, and the quantizer threshold. The SNR gain depends only on the ratio of the quantizer threshold and the noise standard deviation, and not separately on the values of these parameters.
Thus, if the noise PDF and noise variance are known, the quantizer threshold that maximizes the SNR gain can be readily determined. Two families of noise PDF relevant to the ocean acoustics scenario, viz. the generalized Gaussian family and the mixture of Gaussian family, are considered. It is shown that, for a large number of noise PDFs belonging to either family, the peak SNR gain of the symmetric 3-level quantizer exceeds unity. Two other important results are the following: (a) the peak SNR gain of the symmetric 3-level quantizer is greater than that of the 2-level quantizer for all PDFs and (b) if the quantization is uniform, improvement in SNR gain obtained by increasing the number of quantization levels is not very signi6cant. Hence, only 3-level quantizers are considered in the discussion of application of quantizers for sinusoidal signal detection. The application of stochastic resonance in quantizers for detection of sinusoidal signals in i.i.d. non-Gaussian noise is discussed, and the performance of the SR detector is compared with that of the optimal detector and the matched 6lter. Asymptotic
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1 0.9 0.8 0.7
p=90 p=60 p=40
PD
0.6
p=30 p=15 p=2
0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PFA
Fig. 14. Variation of ROCs of the SR detector in generalized Gaussian noise as p → ∞. 1 optimal detector 0.9 0.8 SR detector
0.7 0.6 PD
matched filter 0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5 PFA
0.6
0.7
0.8
0.9
1
Fig. 15. ROCs of linear, SR and optimal nonlinear detectors in mixture-of-Gaussian noise with (; ) = (0:3; 10).
A.A. Saha, G.V. Anand / Signal Processing 83 (2003) 1193 – 1212
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1 β = 50
0.9
β = 40
0.8
β = 30
0.7
β = 25 β = 20
0.6 PD
β = 16 0.5
β = 10
0.4
β=2
0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PFA Fig. 16. Variation of ROCs of the SR detector in mixture-of-Gaussian noise as → ∞, with = 0:3.
expressions for the probability of false alarm PF and probability of detection PD of the SR detector are derived for the case when the signal amplitude and phase are unknown but the signal frequency is known. This characterizes the acoustic signatures of most target vessels. An analysis of the degradation in performance of such an SR detector due to an error in the estimate of the signal frequency shall be presented in a future investigation. It is shown that the performance of the SR detector depends solely on its SNR gain GSR . If GSR is greater than unity, the SR detector performs better than the matched 6lter. If Gopt denotes the e5ective SNR gain of the optimal detector it is observed that for Laplacian noise Gopt = GSR = 2, but in general Gopt ¿ GSR . Hence, in general, the asymptotic performance of the optimal detector is better that of the SR detector. The advantage of the SR detector over the optimal detector is that the nonlinear transformation involved, that of quantization, is simple, easy to implement and remains invariant in a variety of marine environments.
Acknowledgements This project was sponsored by the Naval Physical and Oceanographic Laboratory, Kochi, India, under Project Number PC 4048. The authors are grateful to the referees for having suggested corrections and re6nements to the original manuscript.
Appendix A. Step (ii) of the algorithm to 6nd the optimal value of the quantizer threshold 1 as formulated in Section 3 involves calculating the zeros of the characteristic function C() de6ned in (20). There are two problems encountered here: (i) C() is a rapidly decreasing function. Calculating the zeros of such a function is prone to inaccuracy and is characterized by slow rate of convergence.
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1 0.9 0.8 α = 0.65 0.7 α = 0.45 0.6 PD
α = 0.25 0.5
α = 0.05
0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PFA Fig. 17. Variation of ROCs of the SR detector in mixture-of-Gaussian noise for = 0:05; 0:25; 0:45; 0:65, with = 20.
(ii) Each evaluation of C() requires numerical computation of an integral for the term [1 − F()], which contributes a considerable overhead.
The incomplete Gamma function is de6ned as ∞ e−t t −1+a dt; (A.2) (z; a) =
The accuracy and convergence for the root 6nding procedure for the families of generalized Gaussians and mixtures of Gaussians can be improved by:
which can also be expressed as
(i) Calculating the zeros of the normalized charac˜ teristic function C() de6ned as
where H (z; a) is a continued fraction with the representation
˜ C() = C()=f2 (); 2
(A.1)
where f () is a rapidly decreasing but posi˜ tive function. It follows that C() and C() have the same zero crossings. Therefore, normalization improves the numerical accuracy without a5ecting the location of the zeros or roots. (ii) Expressing the term [1 − F()] in terms of the complement of the incomplete Gamma function, which can be represented in terms of a standardized continued fraction [15], thus avoiding the numerical computation of an integral.
z
(z; a) = e−z z a H (z; a);
H (z; a) =
(A.3)
1 z+1−a−
1(1−a) 2(2−a) z+3−a− z+5−a− ···
:
(A.4)
For the family of generalized Gaussians ∞ p ae−b|t| dt [1 − F()] =
=
a (bp ; 1=p): b1=p p
(A.5)
Combining (21), (A.3) and (A.5) we arrive at the following expression for the characteristic
A.A. Saha, G.V. Anand / Signal Processing 83 (2003) 1193 – 1212
1211
1 0.9 0.8
α = 0.65 α = 0.88
0.7
α = 0.95
PD
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PFA Fig. 18. Variation of ROCs of the SR detector, in mixture-of-Gaussian noise for = 0:65; 0:88; 0:95, with = 20.
function C(): C() = [1 − 2bp H (bp ; 1=p)]f2 ():
(A.6)
˜ The normalized characteristic function C() is therefore ˜ C() = [1 − 2bp H (bp ; 1=p)]:
(A.7)
The expression for f () for the local extrema of G is given by f () = −pbp−2 [p − 1 − pbp ]f():
(A.8)
R() = e−(1−1=
For the family of mixtures of Gaussians [1 − F()] = √ ((c)2 =2; 1=2) 2 $ 1− + √ ((c)2 =22 ; 1=2): 2 $
˜ function C(): ˜ C() = 1 − (c)2 R()H ((c)2 =2) + (1 − )=H ((c)2 =22 ) × R() + (1 − )= R() + (1 − )=3 × : (A.10) R() + (1 − )= For convenience of notation H (:; 1=2) is denoted by H (:) and the quantity R() is introduced which can be de6ned as the ratio of the two component Gaussians by the formula 2
)(c)2 =2
:
(A.11)
(A.9)
Combining (21), (A.3) and (A.9) we arrive at the following expression for the normalized characteristic
Similarly the expression for f () at the local extrema of G is given by 2 2 c3 f () = − √ e−(c) =2 2$ ×[{1 − (c)2 }R() + (1 − )=3 ×{1 − (c)2 =2 }]:
(A.12)
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References [1] F. Chapeau-Blondeau, Stochastic resonance in the Heaviside nonlinearity with white noise and arbitrary periodic signal, Phys. Rev. E 53 (1996) 5469–5472. [2] F. Chapeau-Blondeau, Stochastic resonance and optimal detection of pulse trains by threshold devices, Digital Signal Process. 9 (1997) 162–177. [3] F. Chapeau-Blondeau, Input–Output gain for signal in noise in stochastic resonance, Phys. Lett. A 232 (1997) 41–48. [4] F. Chapeau-Blondeau, Nonlinear test statistic to improve signal detection in non-Gaussian noise, IEEE Signal Process. Lett. 7 (2000) 205–207. [5] F. Chapeau-Blondeau, X. Godivier, Noise-assisted signal transmission in a nonlinear electronic comparator: experiment and theory, Signal Processing 56 (1997) 293–303. [6] F. Chapeau-Blondeau, X. Godivier, Theory of Stochastic resonance in signal transmission by static nonlinear systems, Phys. Rev. E 55 (1997) 1478–1495. [7] L. Gammaitoni, P. Hanggi, P. Jung, F. Marchesoni, Stochastic resonance, Rev. Mod. Phys. 70 (1998) 223–287. [8] L. Gammaitoni, F. Marchesoni, S. Santucci, Stochastic resonance and the dithering e5ect in threshold physical systems, Phys. Rev. E 52 (5) (1995) 4691–4698. [9] X. Godivier, J. Rojas-Varela, F. Chapeau-Blondeau, Noiseassisted signal transmission via stochastic resonance in a diode nonlinearity, Electron. Lett. 33 (1997) 1666–1668.
[10] M.E. Inchiosa, A.R. Bulsara, Signal detection statistics of stochastic resonators, Phys. Rev. E 53 (1996) 2021–2024. [11] S.M. Kay, Fundamentals of Statistical Signal Processing, Vol. II: Detection Theory, Prentice-Hall, Englewood Cli5s, NJ, l998. [12] F.W. Machell, C.S. Penrod, G.E. Ellis, Statistical characteristics of ocean acoustic noise processes, in: E.J. Wegman, S.C. Schwartz, J.B. Thomas (Eds.), Topics in Non-Gaussian Signal Processing, Springer, New York, 1989, pp. 29–57 (Chapter 3). [13] S. Mitaim, B. Kosko, Adaptive stochastic resonance, Proc. IEEE 86 (1998) 2152–2183. [14] D.R. Powell, G.R. Wilson, Class A Modeling of ocean acoustic noise processes, in: E.J. Wegman, S.C. Schwartz, J.B. Thomas (Eds.), Topics in Non-Gaussian Signal Processing, Springer, New York, 1989, pp. 17–28 (Chapter 2). [15] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C: The Art of Scienti6c Computing, Cambridge University Press, Cambridge, MA, 1995. [16] S. Zozor, P.-O. Amblard, Stochastic resonance in discrete-time nonlinear AR(1) models, IEEE Trans. Signal Process. 47 (1) (1999) 108–122; S. Zozor, P.-O. Amblard, IEEE Trans. Signal Process. 49 (5) (2001) 1107–1109. [17] S. Zozor, P.-O. Amblard, On the use of stochastic resonance in sine detection, Signal Processing 82 (3) (2002) 353–367.