A robust detector of known signal in non-Gaussian noise ...

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Signal Processing

Signal Processing 92 (2012) 2676–2688

0165-16

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n Corr

fax: þ1

E-m

mmand

URL

journal homepage: www.elsevier.com/locate/sigpro

A robust detector of known signal in non-Gaussian noise usingthreshold systems

Gencheng Guo, Mrinal Mandal n, Yindi Jing

Department of Electrical and Computer Engineering, 2nd Floor, ECERF Building, University of Alberta, Edmonton, AB, Canada T6G 2V4

a r t i c l e i n f o

Article history:

Received 10 November 2011

Received in revised form

17 February 2012

Accepted 23 April 2012Available online 2 May 2012

Keywords:

Signal detection in non-Gaussian noise

Threshold system

Robust detection

84/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.sigpro.2012.04.014

esponding author. Tel.: þ1 780 492 0294;

780 492 1811.

ail addresses: [email protected] (G. Guo)

[email protected] (M. Mandal), [email protected]

: http://www.ece.ualberta.ca/~mandal/ (M. M

a b s t r a c t

In this paper, we propose a threshold-system-based detector (TD) for detecting a known

deterministic signal in independent non-Gaussian noise whose probability density

function (pdf) is unknown but is symmetric and unimodal. The optimality of the

proposed TD is proved under the assumptions of white noise, small signal, and a large

number of samples. While previous TD designs need accurate information of the noise

pdf, the proposed TD is independent of the noise pdf, and thus is robust to the noise pdf.

The detection probability and the receiver operating characteristic (ROC) of the

proposed TD are analyzed both theoretically and numerically. It is shown that even

without knowing the noise pdf, the proposed TD has close performance to the optimal

detector designed with the noise pdf information. It also performs significantly better

than the matched filter (MF) when the noise pdf has heavy tails. The practical

implementation, robustness to both the noise pdf and the signal, and region of validity

of the proposed TD are also investigated.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

The detection of a known deterministic signal inunknown non-Gaussian noise is a problem of great interestin many fields, such as communications and image proces-sing [1,2]. For example, in watermark detection in discretecosine transform (DCT) domain, the signal is the watermark(or a signature), which is usually known, while the DCTcoefficients of an image is the noise, whose probabilitydensity function (pdf) is non-Gaussian and unknown ingeneral [3–5]. Other applications include the feature extrac-tion in images [6] and underwater communications, wherewe can have precise signal information or obtain a reliableestimation of the signal, while the noise pdf is non-Gaussianand is difficult to estimate [7–9].

ll rights reserved.

,

a (Y. Jing).

andal).

This detection problem has the following generaldiscrete-time model. Consider the binary hypothesisdetection problem described as

H0 : x½n� ¼w½n�, n¼ 0;1, . . . ,N�1,

H1 : x½n� ¼ s½n�þw½n�, n¼ 0;1, . . . ,N�1,

(ð1Þ

where w½n�’s are independent and identically distributed(i.i.d.) noises, whose distribution is non-Gaussian, ands½n�’s form a known deterministic signal sequence. Adecision H0 or H1 is made based on the observationsx½n�’s. If the noise pdf, referred to as fW(w), is known, theoptimal detector, which is the likelihood-ratio test (LRT),and the optimal test statistic ToðxÞ can be derived. Theoptimal detector decides on H1 if ToðxÞ4Z and H0 other-wise. The threshold Z can be calculated based no Bayesiancriteria (if the cost function is available) or Neyman–Pearson criteria from the desired level of false alarmprobability, denoted as PFA [1,2].

In general, ToðxÞ is nonlinear in s½n�, which complicatesthe design and implementation of the detector. To sim-plify the detector, a common method is to calculate the

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G. Guo et al. / Signal Processing 92 (2012) 2676–2688 2677

first order Taylor expansion of ToðxÞ about the signal s½n�,which leads to the locally optimal (LO) test statistic [1]

TLOðxÞ ¼XN�1

n ¼ 0

gðx½n�Þs½n�, ð2Þ

where

gðx½n�Þ ¼ �1

f W ðx½n�Þ

df W ðx½n�Þ

dx½n�: ð3Þ

The schematic of the LO detector is illustrated in Fig. 1.When the signal is weak compared with the noise level,the LO detector performs close to optimal. The LO teststatistic is linear in the signal, which leads to simplercorrelator structure; but it is in general nonlinear in theobservation x½n� and can still be highly complex inimplementation.

To further simplify g(x), suboptimal structures for g(x)have been proposed [10,11], one of which is the thresholdsystem (TS). The TS-based detector [9,12–21], referred toas threshold detector (TD), is especially efficient forproblems whose noise pdf has heavy tails. Its low com-plexity also leads to high detection speed. In [12,13], amulti-level TS is used and the parameters of the TS areoptimized to minimize the mean-square error betweenthe TS output and the output of the g(x) in (3). In [14],several other types of TS’s including the hard-limiter andthe amplifier-limiter are used in signal detection. Sahaet al. in [9,21] propose to use a three-level TS to detectsinusoidal signals with unknown amplitude and phase,and analyze the optimal TS design.

The aforementioned TD and the LO detector need fullknowledge of the noise pdf. In many practical applications,however, the noise pdf is unavailable. One can conductnoise pdf estimation [3,4]. Nevertheless, pdf estimation issubject to error; and in many cases, the pdf changes withtime, which makes the estimation challenging and unreli-able. When pdf estimation is not an option, the followingthree strategies are conventionally used.

In Strategy I, regardless of the noise pdf, the matchedfilter (MF), which is optimal for Gaussian noise, is used fordetection. This design is independent to the noise pdf, but itsperformance is generally poor for non-Gaussian noise [1,22].

In Strategy II, the noise is modeled by a specific pdfform, and the parameters of the pdf are estimated fromthe observations. The detection becomes a compositehypothesis testing, and generalized LRT is used [1,2].Detectors based on this strategy are not robust withrespect to errors in the noise pdf. Even with accuratenoise pdf information, the complexity of the detectors canbe a concern in practical implementation.

Fig. 1. Structure of the LO detector.

In Strategy III, g(x) is specified as a certain nonlinearfunction (e.g., TS) [19,20]. Compared with Strategy I,which is a detector optimal to Gaussian noise only,Strategy III can achieve better performance when thenoise is non-Gaussian. Compared with Strategy II, thisstrategy is expected to be more robust to errors in thenoise pdf. Naturally, when precise noise pdf informationis available, it is expected to perform worse. Hence, it ismore desirable for systems with unknown or constantlychanging noise pdf. Another advantage of it over StrategyII is its complexity. With Strategy III, we can control thecomplexity via the design of the nonlinear function g(x)and achieve the desired balance between complexity andperformance. Chapeau-Blondeau [19] proposes a maxi-mum a-posteriori probability detector, which uses abinary TS to detect DC signals in non-Gaussian noise.The detector is shown to have better performance thanthe MF. However, the optimal design of the binarizationthreshold of the TS is not addressed. We fill this gap in ourprevious work [20] by deriving the optimal thresholdanalytically. But the detector design is limited to thedetection of a DC signal in non-Gaussian noise whosepdf is known perfectly.

To the best of our knowledge, there is no previous workon TD for detecting an arbitrary signal in non-Gaussiannoise with unknown pdf, which is the focus of this paper.Our goal is to find a robust and low complexity detector thatalso enjoys near-optimal performance. We consider detec-tion problems where the noise pdf is unimodal and sym-metric. This covers a wide range of noise pdf’s, for example,the Gaussian mixture (GM) and the generalized Gaussian(GG). We propose a TD composed of a binary TS array and alinear correlator that is independent of the noise pdf. Theoptimality of the design is analyzed for the case of whitenoise, small signal, and a large number of samples. Simula-tion results show that the proposed TD performs very closeto the LO detector and is much better than the MF for noiseswith heavy pdf tails. Properties of the proposed detectorsuch as robustness, complexity, and region of validity arealso investigated.

The remainder of the paper is organized as follows. InSection 2, we present the detection problem and theproposed TD structure. The optimality of the TD designincluding the binary TS and the linear correlator is alsoproved. In Section 3, we derive the detection probabilityof the proposed TD and present simulation results. InSection 4, we discuss the robustness, implementationcomplexity, and region of validity of the proposed TD.Finally, we draw conclusions in Section 5. Involved proofsare included in the appendices.

2. Problem statement and proposed detector structure

In this section, we explain the detection problem andpresent the proposed detector.

2.1. Problem statement

Consider the detection problem described in (1). Thesignal s½n�’s are assumed to be known. The noise isassumed unknown but subject to the following

G. Guo et al. / Signal Processing 92 (2012) 2676–26882678

constraints: (1) fW(w) is symmetric about w¼0, i.e.,f W ð�wÞ ¼ f W ðwÞ; (2) fW(w) is unimodal; and (3) fW(w) iscontinuous. Thus, fW(w) has a unique maximum at w¼0;and is non-decreasing when wo0, non-increasing whenw40. The above assumptions are not overly restrictivefor practical applications [23]. For example, for under-water communication and DCT-domain watermarkingmentioned in Section 1, the noise pdf’s, althoughunknown, are shown to satisfy these assumptions[3,4,7,8]. Furthermore, we also assume that comparedwith the noise standard deviation s, the signal is weak[1], i.e., 9s½n�95s for n¼ 0;1, . . . ,N�1.

Because fW(w) is unknown and sometimes ever-chan-ging, the optimal and LO detectors, which require thenoise pdf information, cannot be realized. We aim atdesigning a detector whose parameters are independentof the noise pdf, thus robust to the noise pdf; but at thesame time, good detection probability is desired.

2.2. Proposed TD structure

The proposed TD is shown in Fig. 2, in which each datapoint x½n� is separately quantized at the threshold s½n�=2(by a TS with binarization threshold s½n�=2) to yield abinary y½n�. Then linear correlation is performed betweenthe sequence y½n� and the absolute value of the signal tobe detected 9s½n�9. A decision is made based on thecorrelation result with the threshold Z.

Here are more detailed explanations on different partsof the proposed TD. The multiway switch directs theobservation x½n� to its corresponding TS, denoted asTS½n�, and outputs y½n�. For simplicity in implementation,we use binary TS with the following structure. Whens½n�Z0, we employ

TS1ðtÞ : y½n� ¼1, x½n�Zt,

0, x½n�ot;

(ð4Þ

and when s½n�o0, we employ

TS2ðtÞ : y½n� ¼0, x½n�Zt,

1, x½n�ot:

(ð5Þ

The binarization threshold is set to be s½n�=2 for bothcases, i.e., t¼ s½n�=2. The linear correlator produces

Z ¼ TTDðyÞ ¼

1

N

XN�1

n ¼ 0

y½n�9s½n�9: ð6Þ

Fig. 2. Proposed TD structure.

The threshold Z of the test statistic is calculated from thedesired false alarm level using

R1Z f Zðz;H0Þ dz¼ PFA.

2.3. Optimality of the proposed TD

In this subsection, we prove the optimality of thebinarization threshold of the TS and the correlator designin (6).

Our TS design problem can be stated as follows: for theone-dimensional binary detection problem where x½n� isthe observation and y½n� (output of the TS½n�) is thedetection result, find the optimal binarization threshold.For the optimality measure, Neyman–Pearson criteria iswidely used, i.e., to find the maximum detection prob-ability PD for a given PFA. However, with the Neyman–Pearson criteria, the threshold optimization requires thenoise pdf, which is unknown in our model. Furthermore,for different values of PFA, the optimal threshold will bedifferent, which complicates the implementation of theTD. Therefore, in this paper, we use the area under thereceiver operating characteristic (ROC) curve (AUC) as theoptimality measure [24]. It fully characterizes the detect-ability of a detector and is shown to be a good detect-ability measure according to Area Theorem [25]. Mostimportantly, it results in an optimal binarization thresh-old independent of PFA.

We first prove the optimality of the optimal binariza-tion threshold s½n�=2.

Theorem 1. For each observation x½n�, consider the binary

detection problem with the binary TS output y½n� as the test

statistic. The binarization threshold that leads to the max-

imum AUC is t¼ s½n�=2.

Proof. See Appendix A.

It is noteworthy that the proposed binarization thresh-old is optimal only for the single observation x½n� and maynot be globally optimal for the overall detection problem.As will be shown later, however, this design leads to arobust TD and close-to-optimal performance.

Next, we prove the optimality of the test statistic in(6).

Theorem 2. Consider the detection problem with observa-

tion y, the outputs of the TS array in the proposed TD. When

9s½n�95s and Nb1, the test statistic TTDðyÞ in (6) is the

optimal LRT.

Proof. See Appendix B.

The proposed TD has simple structure, which is easy inimplementation. Theorems 1 and 2 in addition show itsadvantage in performance.

3. Investigation on detection probability

In this section, the detection probability of the pro-posed TD is analyzed and simulation results are pre-sented. The performance of previously proposeddetectors is also investigated for comparison.


3.1. Detection probability of the proposed TD

To obtain the PD of the proposed TD for a given PFA, weneed to derive the pdf’s of TTD

ðyÞ under H1 and H0. TTDðyÞ

is the summation of N independent random variablesy½n�, n¼ 0;1, . . . ,N�1 with a weight of 9s½n�9. Because y½n�’sare independent, according to the central limit theorem,when N is large, the distribution of TTD

ðyÞ can be approxi-mated as Gaussian. Therefore, we only need to calculatethe mean and the variance of TTD

ðyÞ.Denote the probabilities of y½n� ¼ 1 under Hypothesis

H0 and H1, respectively, as

p0½n�9Pðy½n� ¼ 1;H0Þ, p1½n�9Pðy½n� ¼ 1;H1Þ: ð7Þ

It can be derived that (see Appendix B for the calculations)

p0½n� �1

2�9s½n�9

2f W ð0Þ, p1½n� �

1

2þ

9s½n�92

f W ð0Þ: ð8Þ

The probability of y½n� ¼ 0 under Hypothesis Hi is 1�pi½n�

for i¼0,1. The mean and the variance of y½n� underHypothesis Hi are

Efy½n�;Hig ¼ 1 � pi½n�þ0 � ð1�pi½n�Þ ¼ pi½n�, ð9Þ

Varfy½n�;Hig ¼ 12� pi½n�þ02

� ð1�pi½n�Þ�E2fy½n�;Hig ¼ pi½n�ð1�pi½n�Þ, ð10Þ

where E stands for the expectation. With (9) and (10), andthe independence of y½n�’s, the mean and variance ofTTDðyÞ are

mi ¼ EfTTDðyÞ;Hig ¼ E

1

N

XN�1

n ¼ 0

y½n�9s½n�9

( )¼

1

N

XN�1

n ¼ 0

pi½n�9s½n�9,

ð11Þ

s2i ¼VarfTTD

ðyÞ;Hig ¼ Var1

N

XN�1

n ¼ 0

y½n�9s½n�9

( )

¼1

N2

XN�1

n ¼ 0

s2½n�pi½n�ð1�pi½n�Þ: ð12Þ

Thus,

f Zðz;H0Þ ¼N ðm0,s20Þ, f Zðz;H1Þ ¼N ðm1,s2

1Þ, ð13Þ

where N ðm,s2Þ is the Gaussian distribution with mean mand variance s2. The pdf’s in Eq. (13) are the same asthose for a standard Gaussian binary detection problem. Itcan be derived that for a given PFA, the PD of the proposedTD can be expressed as

PTDD ¼ Q

s0Q�1ðPFAÞþm0�m1

s1

!, ð14Þ

where Q ð�Þ is the complementary cumulative distributionfunction (cdf) of the standard normal pdf and Q�1

ð�Þ is theinverse of Q ð�Þ.

For a DC signal, i.e., s½n� ¼ A, n¼ 0;1, . . . ,N�1, pi½n�’s areidentical, denoted as pi. We have mi ¼ A � pi ands2

i ¼ A2� pi � ð1�piÞ=N. The result in (14) reduces to the

one in [20].

3.2. Detection probability of the MF and the LO detector

If the MF is used, which is optimal for Gaussian noise,the test statistic is given by

TMFðxÞ ¼

1

N

XN�1

n ¼ 0

s½n�x½n�:

Again, according to the central limit theorem, TMFðxÞ is

Gaussian. We have

EfTMFðxÞ;H0g ¼

1

NEXN�1

n ¼ 0

s½n�w½n�

( )¼ 0,

VarfTMFðxÞ;H0g ¼

1

N2Var

XN�1

n ¼ 0

s½n�w½n�

( )¼

s2

N2

XN�1

n ¼ 0

s2½n�,

EfTMFðxÞ;H1g ¼

1

NEXN�1

n ¼ 0

s½n�ðs½n�þw½n�Þ

( )¼

1

N

XN�1

n ¼ 0

s2½n�,

VarfTMFðxÞ;H1g ¼

1

N2Var

XN�1

n ¼ 0

s½n�ðs½n�þw½n�Þ

( )

¼s2

N2

XN�1

n ¼ 0

s2½n�:

Therefore, for a given PFA, the PD of the MF can beexpressed as

PMFD ¼ Q Q�1

ðPFAÞ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN�1n ¼ 0 s2½n�

s2

s0@

1A: ð15Þ

For comparison, we also present the detection prob-ability of the LO detector [1]

PLOD ¼Q Q�1

ðPFAÞ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIXN�1

n ¼ 0

s2½n�

vuut0@

1A, ð16Þ

where

I¼

Z 1�1

df W ðwÞ

dw

� �2

f W ðwÞdw: ð17Þ

3.3. Simulation results

In this section, we present and compare the PD’s andthe ROCs of the proposed TD, the MF, the LO detector, andthe detector in [9] via simulation. Note that the perfor-mance of the MF is utilized as the lower bound bench-mark and the one of the LO detector as the upper boundbenchmark.

We consider two signals:

(1)
sinusoidal signal
s½n� ¼ A sinð0:02pnþfÞ, ð18Þ

where A¼0.1 and f¼ 0 unless otherwise stated;
(2) a randomly generated signal
s½n� ¼ B � l½n�, ð19Þ

where l½n�’s are randomly generated by the uniformdistribution on ½0;1� and B¼0.1 unless otherwise

Fig.sign


stated. The sinusoidal signal is largely used in signaldetection. The random signal is used to model aknown signal with an arbitrary structure. We setN¼100.

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Theoretical ROC from (14)ROC from Monte Carlo simulation

Sinusoidal signal in GM

Sinusoidal signal in GG

Random signal in GM

Random signal in GG

Fig. 4. Comparison between theoretical and simulated ROCs for the

proposed TD. GM is with a¼ 0:3, b¼ 5, s¼ 1; GG is with b¼ 0:9, s¼ 1;

and N¼100.

As to the noise, GM noise and GG noise are consideredbecause they are widely used in practical applicationssuch as underwater noise [7–9] and DCT coefficients [3,4].The GM pdf has three parameters a, b, and s, and isdefined as

f W ðwÞ ¼c

sffiffiffiffiffiffi2pp a exp �

c2w2

2s2

� �þ

1�ab

exp �c2w2

2b2s2

!" #,

ð20Þ

where c¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaþð1�aÞb2

q, 0oao1, b40. The GG pdf has

two parameters b and s, and is defined as

f W ðwÞ ¼c1ðbÞs

exp �c2ðbÞw

s

�� 2=ð1þbÞ� �, ð21Þ

where c1ðbÞ ¼ ðG1=2ð32 ð1þbÞÞÞ=ðð1þbÞG

3=2ð12ð1þbÞÞÞ, c2ðbÞ ¼

½ðGð32 ð1þbÞÞÞ=ðGð12ð1þbÞÞÞ�

1=ð1þbÞ and Gð�Þ is the Gammafunction. We set ða,b,sÞ ¼ ð0:3,5;1Þ for GM noise andðb,sÞ ¼ ð0:9,1Þ for GG noise unless otherwise stated.

First we compare the theoretical results on PD for boththe sinusoid signal (18) and the random signal (19) in theGM noise (20). PFA is set to be 0.01. We show the detectionprobabilities of the proposed TD and other schemes fordifferent energy-to-noise ratios (ENRs) defined as10 log10

PN�1n ¼ 0 s2½n�=s2

� �dB. By having the magnitude A

of the sinusoidal signal range from 0.045 to 0.45, and themagnitude B of the random signal range from 0.055 to0.55, the ENR ranges from �10 dB to 10 dB. The PD’s ofthe proposed TD, the MF, and the LO detector arecalculated using (14)–(16), respectively. Note that forthe proposed TD, when calculating PTD

D using (14),mi,si, i¼ 0;1 are calculated by (11) and (12), in whichpi½n�, i¼ 0;1 are calculated according to (8). For the LOdetector, the value of I given in (17) is calculated numeri-cally. From Fig. 3 we can see that the proposed TD has

3. PD versus ENR of the proposed TD, the MF, and the LO detector in GM no

al. (b) Random signal.

close performance to the LO detector and is significantlybetter than the MF for both signals. At PD¼0.3, theproposed TD is about 4 dB better than the MF, and is only1 dB worse than the LO detector. The advantage of theproposed TD over the MF is even bigger at higher PD

levels.We now compare the theoretical ROCs calculated by

(14) with the ROCs obtained from Monte Carlo simula-tions for the proposed TD. To obtain the ROC from MonteCarlo simulations, observations are generated under H0

and H1 (20 000 times for each) and the test statistic iscalculated as Z ¼ TTD

ðyÞ for all observations. f Zðz;HiÞ, i¼

0;1 are approximated as the normalized histograms of the20 000 outputs for both hypothesis, from which the ROCis generated. We use the sinusoid signal (18) and therandom signal (19) in GM noise and GG noise, respec-tively. Fig. 4 shows that the ROCs calculated from (14) areconsistent with the ROCs obtained from simulations forall cases.

ise with a¼ 0:3, b¼ 5, and s¼ 1 for N¼100 and PFA ¼ 0:01. (a) Sinusoidal


Next, we compare the ROCs of the proposed TD, theMF, the LO detector, and the detector in [9] obtained fromMonte Carlo simulations in Fig. 5. The detector in [9] hasthe detection structure in Fig. 1, in which g(x) is designedas a three-level TS and a quadrature MF is used instead ofthe replica-correlator due to unknown parameters insignal. The optimal design of the three-level TS requiresnoise pdf information, thus this detector can only be usedwhen the noise pdf is available. Here, we change thequadrature MF to linear MF, which is a better design forthe detection of known signals, and to make it compar-able with the proposed TD. The optimal thresholds of thethree-level TS in [9] are numerically calculated to be 0.08for the GM noise and 0.01 for the GG noise. We show theROCs for the detection of the sinusoidal signal (18) in GMnoise in Fig. 5(a) and GG noise in Fig. 5(b), respectively. Itcan be seen that the ROCs of the proposed TD are close tothose of the LO detector and the detector in [9], especiallyfor the GG noise, although it requires no noise pdfinformation. The proposed TD has a significant improve-ment compared to the MF. To quantify the difference, wecalculate the average uniform loss (AUL) between twoROCs (i and k) as

AULik ¼

Z 1

010 log10

PiDPkD

� �dPFA: ð22Þ

For the sinusoidal detection in GM noise, compared withthe LO detector, the proposed TD has an AUL of 0.195 dB(4.59% degradation). Compared with the proposed TD, theMF has an AUL of 0.642 dB (15.93% degradation). Com-pared with the detector in [9], the proposed TD has anAUL of 0.0265 dB (0.61% degradation). For the sinusoidaldetection in GG noise, compared with the LO detector, theproposed TD has an AUL of 0.0493 dB (1.14% degradation)only. Compared with the proposed TD, the MF has an AULof 0.386 dB (9.29% degradation). Compared with thedetector in [9], the proposed TD has an AUL of0.0059 dB (0.14% degradation). It should be mentionedthat the closeness of the proposed TD to the optimal

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

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LO detectorProposed TDDetector in [9]MF

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Fig. 5. Comparison of the ROCs obtained from Monte Carlo simulations in dete

with a¼ 0:3, b¼ 5, s¼ 1. (b) GG noise with b¼ 0:9, s¼ 1.

detector varies with the noise pdf parameters. In general,the proposed TD performs closer to the optimal detectorwhen the noise pdf has a heavier tail.

In the next experiment, we compare the ROCs for thedetection of the random signal (19) in GM noise and GGnoise. The ROCs are shown in Fig. 6. Again we can see thatthe proposed TD has close performance to the LO detectorand the detector in [9], and is a lot better than the MF. Forthe detection in GM noise, compared with the LO detec-tor, the proposed TD has an AUL of 0.199 dB (4.69%degradation). Compared with the detector in [9], theproposed TD has an AUL of 0.04 dB (0.93% degradation).Compared with the proposed TD, the MF has an AUL of0.597 dB (14.74% degradation). For the detection in GGnoise, compared with the LO detector, the proposed TDhas an AUL of 0.016 dB (0.37% degradation) only. Com-pared with the detector in [9], the proposed TD has anAUL of �0.031 dB (0.72% increase). Compared with theproposed TD, the MF has an AUL of 0.41 dB (9.9%degradation).

4. Discussions

In this section, we discuss properties of the proposedTD, including the robustness, implemental complexity,and region of validity.

4.1. Robustness

The robustness in signal detection refers to the stabi-lity of the detection performance to changes in para-meters in the system. For example, we can observe from(16) and (17) that the detection probability of the LOdetector depends on PFA, s½n�, the form of fW(w) and itsparameters. Let rm, m¼ 1, . . . ,M be the set of parametersinvolved in the robustness evaluation. In this paper, weuse a quantitative measure proposed in [26] to evaluate

0 0.2 0.4 0.6 0.8 10

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cting the sinusoid signal (18) in GM and GG noise, N¼100. (a) GM noise

0 0.2 0.4 0.6 0.8 10

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etec

tion


Fig. 6. Comparison of the ROCs in detecting the random signal (19) in GM and GG noise, N¼100. (a) GM noise with a¼ 0:3, b¼ 5, s¼ 1. (b) GG noise

with b¼ 0:9, s¼ 1.


the robustness, which is defined as

F9 1þXM

m ¼ 1

@PD

@rm

� �2 !�1=2

: ð23Þ

It reflects how PD fluctuates with changes/inaccuracy inrm’s. It is normalized to be between 0 and 1. A lower Fmeans lower robustness.

In what follows, we derive the expressions of F for theproposed TD, the MF, and the LO detector in the case ofthe sinusoidal signal (18) in GM noise. We consider therobustness with respect to inaccuracy in the noise para-meters (a,b,s) and the signal parameters (A,f), respec-tively. Analysis on the robustness with respect to otherfactors, such as the noise pdf form, is more involved andleft for future work. We define

Fn9 1þX

rn ¼ fa,b,sg

@PD

@rn

� �20@

1A�1=2

, ð24Þ

which is the robustness to the noise parametersrn ¼ fa,b,sg; and

Fs9 1þX

rs ¼ fA,fg

@PD

@rs

� �20@

1A�1=2

, ð25Þ

which is the robustness to the signal parametersrs ¼ fA,fg.

For the proposed TD, PTDD depends on pi½n� via mi,s2

i fori¼0,1 as shown in (11) and (12). From (8), we can see thatpi½n� only depends on the signal s½n� and the value of thenoise pdf at 0. Using (8), (11) and (12), we obtain

m0 �1

2N

XN�1

n ¼ 0

9s½n�9½1�9s½n�9f W ð0Þ�,

s20 �

1

4N2

XN�1

n ¼ 0

s2½n�½1�s2½n�f 2W ð0Þ�,

m1 �1

2N

XN�1

n ¼ 0

9s½n�9½1þ9s½n�9f W ð0Þ�,

s21 �

1

4N2

XN�1

n ¼ 0

s2½n�½1�s2½n�f 2W ð0Þ� � s

20:

Using these approximations in (14), we have

PTDD � Q Q�1

ðPFAÞ�2vf W ð0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv�uf 2

W ð0Þq

0B@

1CA, ð26Þ

where we have defined

v9XN�1

n ¼ 0

s2½n�, u9XN�1

n ¼ 0

s4½n�: ð27Þ

For any parameter rn 2 fa,b,sg, we have

@PTDD

@rn

¼

ffiffiffiffi2

p

rv�uf 2

W ð0Þh i�3=2

v2 exp �1

2Q�1ðPFAÞ�

2vf W ð0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv�uf 2

W ð0Þq

0B@

1CA

2264

375 @f W ð0Þ

@rn

: ð28Þ

From (20), we obtain f W ð0Þ ¼ ðc=sffiffiffiffiffiffi2ppÞ½aþðð1�aÞ=bÞ�,

where c¼


q. Thus

@f W ð0Þ

@a ¼c

sffiffiffiffiffiffi2pp 1�

1

b

� �þ

1�b2

2ðaþð1�aÞb2Þ

!f W ð0Þ, ð29Þ

@f W ð0Þ

@b¼

c

sffiffiffiffiffiffi2pp �

1�ab2

!þ

ð1�aÞbaþð1�aÞb2

!f W ð0Þ, ð30Þ

@f W ð0Þ

@s ¼�1

s f W ð0Þ: ð31Þ

Using (28)–(31) in (24), we can obtain FTDn .


For the robustness to the signal, for rs ¼ fA,fg, we have

@PTDD

@rs

¼�f W ð0Þffiffiffiffiffiffi

2pp exp �

1

2Q�1ðPFAÞ�

2vf W ð0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv�uf 2

W ð0Þq

0B@

1CA

2264

375

½v�2uf 2W ð0Þ�

@v

@rs

þvf 2W ð0Þ

@u

@rs

ðv�uf 2W ð0ÞÞ

3=2, ð32Þ

where

@v

@A¼ 2A

XN�1

n ¼ 0

sin2ð0:02pnþfÞ,

@u

@A¼ 4A3

XN�1

n ¼ 0

sin4ð0:02pnþfÞ, ð33Þ

@v

@f¼ A2

XN�1

n ¼ 0

sinð0:04pnþ2fÞ,

@u

@f¼ 4A4

XN�1

n ¼ 0

sin3ð0:02pnþfÞ cosð0:02pnþfÞ: ð34Þ

Using (33) and (34) in (32) and using (32) in (25), we canobtain FTD

s .For the MF, the detection probability is shown in (15),

which depends on the noise parameter s2 and the signalparameters rs ¼ fA,fg. It can be shown that

@PMFD

@s ¼1ffiffiffiffiffiffi2pp exp �

1

2Q�1ðPFAÞ�

ffiffiffivp

s

� �2 !

�

ffiffiffivp

s2

� �: ð35Þ

@PMFD

@rs

¼1ffiffiffiffiffiffi2pp exp �

1

2Q�1ðPFAÞ�

ffiffiffivp

s

� �2 !

�1

2sffiffiffivp

� �@v

@rs

� �:

ð36Þ

FMFn can be calculated by using (35) in (24). FMF

s can becalculated by using (33) and (34) in (36) and using (36) in(25).

For the LO detector, PD relies on s and I defined in (17),and I is influenced by all parameters a,b,s in (20). Forrn ¼ fa,b,sg, we have

@PLOD

@rn

¼�1

2

ffiffiffiffiffiffiffiffiv

2pI

rexp �

1

2Q�1ðPFAÞ�

ffiffiffiffiffivIp� �2

� �@I

@rn

� �,

ð37Þ

where

@I

@rn

¼

Z 1�1

2f 0W ðwÞ@f 0W ðwÞ

@rm

f W ðwÞ�ðf0W ðwÞÞ

2@f W ðwÞ

@rm

f 2W ðwÞ

0BB@

1CCA dw:

ð38Þ

Note that we resort to numerical calculation because theclose form result of (38) is unavailable. For the signalparameters rs ¼ fA,fg, we have

@PLOD

@rs

¼1ffiffiffiffiffiffi2pp exp �

1

2Q�1ðPFAÞ�

ffiffiffiffiffivIp� �2

� ��

ffiffiIp

2ffiffiffivp

!@v

@rs

� �:

ð39Þ

By using (37) in (24) and (39) in (25), FLOn and FLO

s can becalculated.

For the detector in [9], since its close form PD is notavailable, its robustness measure is not calculated here. Butsimulation results are provided for comparison in Fig. 8.

In Fig. 7, we plot FTDn , FMF

n , FLOn and FTD

s , FMFs , FLO

s asfunctions of s2, a, b, A, and f. We set PFA ¼ 0:1, N¼100. Ineach subfigure of Fig. 7, the robustness measures areshown as functions of one of the parameters while theother parameters are fixed. The fixed parameter valuesare set to be s2 ¼ 1, a¼ 0:3, b¼ 5, A¼0.1, f¼ 0. Forexample, in Fig. 7(a), the values of the robustness measureare shown as s2 ranges from 1 to 9 while a¼ 0:3, b¼ 5,A¼0.1, f¼ 0. It can be observed that among the threedetectors, the MF always has the highest robustness tothe noise parameters. The robustness of the proposed TDis superior to that of the LO detector, and is close to FMF

n

for some parameter values. In terms of the signal, notsurprisingly, due to its dependency on the signal, therobustness of the proposed TD is a lot worse than the MF.It is also worse than the LO detector for all phase valuesand small amplitude values, but the difference is small.For large amplitude, the proposed TD is more robust tosignal than the LO detector.

In Fig. 8, we show the ROCs obtained via Monte Carlosimulation under inaccurate parameters of the noise pdfand the signal for the sinusoidal signal (18) withA¼ 0:1, f¼ 0 in the GM noise with a¼ 0:3, b¼ 5, s¼ 1.In Fig. 8(a), the ROCs based on accurate signal informationbut an inaccurate estimation of the noise pdf, wherea¼ 0:4, b¼ 2:5, s¼ 1, are shown. We can see that theperformance of the LO detector and the performance ofthe detector in [9] are worse than the proposed TD due toinaccurate noise estimation. Compared with the ROCsshown in Fig. 5, the AUL of the LO detector is 0.392 dB(9.45% degradation) and the AUL of the detector in [9] is0.4 dB (9.65% degradation). The AUL of the proposed TDand the MF is 0, which shows that they are immune toinaccuracy in noise pdf. In Fig. 8(b), the ROCs based onaccurate noise information but an inaccurate estimationof the signal amplitude and phase, where A¼ 0:12,f¼ 0:1p, are shown. We can observe that the proposedTD still performs slightly worse than the LO detector,comparable to the detector in [9], and significantly betterthan the MF. Compared with the ROCs shown in Fig. 5, theAULs of the LO detector, the proposed TD, the detector in[9], and the MF are 0.022 dB (0.5% degradation), 0.034 dB(0.7% degradation), 0.022 dB (0.5% degradation), and0.018 dB (0.4% degradation), respectively. This reveals thatthe proposed TD, although is based on perfect signal infor-mation, can bear inaccuracy in signal to some extent. Inparticular, we compare the robustness of the proposed TDwith the detector in [9], which depends on both the noiseand the signal. The above simulation shows that the pro-posed TD is significantly more robust to the noise pdf. As tothe robustness to the signal, the proposed TD is inferior, asexpected, but still comparable to the detector in [9].

4.2. Implementation complexity

In this part, the implementation complexity of theproposed TD is discussed. The multiway switch and thebinary TS array can both be easily implemented in circuit

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


Pro

babi

lity

of d

etec

tion

LO detectorProposed TDDetector in [9]Matched filter

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


Pro

babi

lity

of d

etec

tion

LO detectorProposed TDDetector in [9]Matched filter

Fig. 8. ROCs under inaccurate estimations of the noise pdf and the signal. (a) ROCs under inaccurate noise pdf. (b) ROCs under inaccurate signal.

1 2 3 4 5 6 7 8 90.75

0.8

0.85

0.9

0.95

1

σ 2

Rob

ustn

ess

mea

sure

Φn

ΦnLO using(24),(37),(38)

ΦnTD using(24),(28)−(31)

ΦnMF using(24),(35)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

α

Rob

ustn

ess

mea

sure

Φn

ΦnLO using(24),(37),(38)

ΦnTD using(24),(28)−(31)


1 2 3 4 5 6 7 8

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

β

Rob

ustn

ess

mea

sure

Φn

ΦnLO using(24),(37),(38)

ΦnTD using(24),(28)−(31)


0.05 0.1 0.15 0.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

A

Rob

ustn

ess

mea

sure

Φs

ΦnLO using(25),(33),(34),(39)

ΦnTD using(25),(32)−(34)

ΦnMF using(25),(33),(34),(36)

−0.5 0 0.5 1 1.50.2

0.25

0.3

0.35

0.4

φ

Rob

ustn

ess

mea

sure

Φs

ΦnLO using(25),(33),(34),(39)

ΦnTD using(25),(32)−(34)

ΦnMF using(25),(33),(34),(36)

Fig. 7. Comparison of robustness measure. (a) Fn versus s2, (b) Fn versus a, (c) Fn versus b, (d) Fs versus A, (e) Fs versus f.


design. The correlator is also easy in implementation,since it is linear in both the absolute value of the signal9s½n�9 and the TS output. In addition, the structure of theproposed TD can be further simplified. In the proposedTD, an array of TSs instead of one TS is used because s½n�

can take different values, which requires different binar-ization thresholds. Thus, for different samples in thesignal sequence with the same value, the same TS canbe used. If the signal is DC, only one TS is needed and themultiway switch is not necessary. In this case, theproposed TD structure reduces to the one we proposedin [20]. If the signal is periodic and the sampling time is

appropriate, we only need to consider the samples in oneperiod and the required number of TSs can be reduced. Ingeneral, based on the given s½n�, we can divide the range ofs½n� values into groups. For example, the s½n�’s whose valueis between C�E and CþE can be put into one group andthe TS with binarization threshold C=2 is applied, where C

is a constant and E is a positive number. This reduces theimplementation complexity with some penalty on theperformance. One can balance performance and complex-ity by adjusting the number of groups.

For the LO detector, in general, the g(x) component in (3)is nonlinear in the observations, and its implementation can


be highly complicated. Further, since it requires perfectnoise pdf information, its complexity even increases if thenoise pdf estimation component is taken into account. Thedetector in [9] is simple in implementation, but it alsorequires noise pdf estimation and, in addition, a numericaloptimization of the three-level TS threshold [21]. Hence itmay be even more complicated in implementation than theLO detector.

4.3. Region of validity

The proposed TD is applicable for noises with zero-mean, unimodal, and symmetric pdf. However, it may notperform efficiently for all noises in this category. There-fore, it is helpful to address the validity of the proposedTD corresponding to the form and the parameters of thenoise pdf. Using the MF as the benchmark, we define thevalidity of the proposed TD based on whether it has ahigher detection probability than the MF. Accordingly, fora given type of noise pdf, the validity region is defined asthe set of the parameters of the noise pdf with which theproposed TD is superior to the MF.

In general, theoretical derivation of the validity regionis difficult because for a certain noise pdf, the close-formof the ROC is usually unavailable. We often need to resortto numerical method and conduct simulation for a largenumber of noise pdf parameters and signal forms. Never-theless, we show in the following theorem that when thesignal is weak, and the number of samples is large, thevalidity region reduces to a simple form, which is inde-pendent of the signal s½n�’s and, for some cases, even thenoise variance s2.

Theorem 3. If 9s½n�95s and Nb1, the validity region of the

proposed TD with respect to the MF is independent of the

signal sequence s½n�’s and PFA, and can be approximated as

the set of noise pdf parameters that satisfy f W ð0Þ41=2s.

Proof. To find the validity region, we need to solve

PTDD 4PMF

D 3Qs0Q�1

ðPFAÞþm0�m1

s1

!

4Q Q�1ðPFAÞ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN�1n ¼ 0 s2½n�

s2

s0@

1A: ð40Þ

Using (26), we can recast (40) as

Q Q�1ðPFAÞ�

2vf W ð0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv�uf 2

W ð0ÞÞÞq

0B@

1CA4Q Q�1

ðPFAÞ�

ffiffiffivp

s

� �, ð41Þ

where v,u are defined in (27). Since 9s½n�95s, ignoring thesecond order term uf 2

W ð0Þ in (41), we have

Q Q�1ðPFAÞ�2f W ð0Þ

ffiffiffivp� �

4Q Q�1ðPFAÞ�

ffiffiffivp

s

� �

3f W ð0Þ41

2s : &

Theorem 3 shows that the validity region only dependson f W ð0Þ and s, but is independent of PFA and s½n�. For theGM noise and the GG noise, we further investigate the

validity region in the following corollary and show that itis also independent of s.

Corollary 1. For the GM noise, the validity region of the

proposed TD is

ða,bÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaþð1�aÞb2

qaþ 1�a

b

� �� 4

ffiffiffiffip2

r� : ð42Þ

For the GG noise, the validity region of the proposed TD is

b9b 2 ½0:42,1� �

: ð43Þ

For both cases, the validity region of the proposed TD is

independent of the noise variance.

Proof. For the GM noise in (20), according to Theorem 3,we have

f W ð0Þ41

2s3f W ð0Þ ¼c

sffiffiffiffiffiffi2pp aþ 1�a

b

� �4

1

2s

3cffiffiffiffiffiffi2pp aþ 1�a

b

� �4

1

2

3


qaþ 1�a

b

� �4

ffiffiffiffip2

r:

Similarly, for the GG noise in (21), we have

f W ð0Þ41

2s3f W ð0Þ ¼c1ðbÞs 4

1

2s

3c1ðbÞ ¼G1=2 3

2 ð1þbÞ�

ð1þbÞG3=2 12 ð1þbÞ� 4 1

2

3fb9b 2 ½0:42,1�g: &

Corollary 1 provides a simple way to calculate thevalidity regions for the two noise pdf’s. In the remainingof this subsection, we show simulation results on theregion of validity.

First, for the GM noise, we justify the analytical resultsin Theorem 3 and Corollary 1. We set PFA ¼ 0:1 andN¼100, and simulate the validity regions for three detec-tion problems. Problem I: sinusoidal signal (18) in GMnoise with s2 ¼ 1; Problem II: sinusoidal signal (18) in GMnoise with s2 ¼ 4; and Problem III: random signal (19) inGM noise with s2 ¼ 1. For all problems, we obtain PTD

D ’sfrom simulation for different ða,bÞ’s, and determine thevalidity regions by PTD

D 4PMFD ¼ 0:2828. These validity

regions are compared with the analytic result, which iscalculated using (42) in Corollary 1. The validity regionsare shown in Fig. 9, where the ones for Problems I, II, III,and Corollary 1 are marked as red, green, blue, and gray,respectively. For better demonstration, we shape theregions by the order of Corollary 1 (gray), Problem III(blue), Problem II (green), and Problem I (red) due to thesize of the regions. It is observed that the borders of thevalidity regions of all three problems are almost the sameand match the analytical one from Corollary 1 precisely.Comparing the validity regions of Problem I and ProblemII, we see that the validity region is independent of noisevariance s2. Comparing the validity regions of Problem Iand Problem III, we see that the validity region isindependent of signal s½n�. Comparing the validity regions

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

β

α

Problem I (red)Problem II (green)Problem III (blue)(42) in Corollary 1 (gray)

Fig. 9. The validity regions for different detection problems with GM

noise. (For interpretation of the references to color in this figure caption,

the reader is referred to the web version of this article.)

−0.5 0 0.5 10.15

0.2

0.25

0.3

0.35

0.4

β

PD

PDTD (simulation)

PDMF using (15) β = 0.42

β = 0.44

Fig. 10. PTDD versus b for the sinusoidal signal detection in GG noise.


obtained from simulation and the analytical one, we seethat the (42) is a sound approximation in determining thevalidity region of the proposed TD.

We then turn to the GG noise. We consider thedetection of the sinusoidal signal (18) in GG noise withs2 ¼ 1, obtain PTD

D for different b by simulation, and showthem in Fig. 10. It can be observed that the validity regionis about fb9b 2 ½0:44,1�g obtained from fb9PTD

D 4PMFD ¼

0:2828g. This is close to the result (43) in Corollary 1,which is fb9b 2 ½0:42,1�g.

5. Conclusion

In this paper, we proposed a low-complexity thresholdsystem based detector to detect any known deterministicsignal embedded in independent unknown non-Gaussiannoise. We assumed that the noise pdf is unimodal andsymmetric, the signal is small compared to the noisevariance, and there are a large number of samples. Theoptimality of the two parts of the proposed detector, thebinary threshold system and the correlator, was proved.

The detection probability and the ROC of the proposed TDwere investigated both analytically and numerically. Fornoises with heavy pdf tails, simulation showed that theperformance of the proposed TD approaches that of theLO detector and the detector in [9], the design of whichneed exact noise pdf information, and is much better thanthe MF. Through a robustness measure, we showed thatthe proposed TD is highly robust to the noise pdf. On theother hand, its robustness to the signal is inferior butcomparable to the LO detector and the detector in [9]. Theimplementation complexity of the proposed detector wasdiscussed and compared with other detection designs.The validity region of the proposed detector was definedand analyzed using the MF as the benchmark.

Appendix A. Proof of Theorem 1

Without loss of generality, we assume that x½n�Z0 andshow that the binarization threshold t¼ s½n�=2 maximizesthe AUC. The case of x½n�o0 can be proved similarly.

Since x½n�Z0, TS1ðtÞ is applied. With the binarizationthreshold t, we define p0ðtÞ and p1ðtÞ as follows:

p0ðtÞ9PðY ½n� ¼ 1;H0Þ ¼

Z 1t

f Xðx;H0Þ dx, ð44Þ

p1ðtÞ9PðY ½n� ¼ 1;H1Þ ¼

Z 1t

f Xðx;H1Þ dx: ð45Þ

Since f Xðx;H0Þ ¼ f W ðwÞ, f Xðx;H1Þ ¼ f W ðw�s½n�Þ and s½n�Z0,f Xðx;H1Þ is a right shift of f Xðx;H0Þ by s½n�. Thus,p0ðtÞrp1ðtÞ for all t’s.

The TS output y½n� has two possibilities y½n� ¼ 0 ory½n� ¼ 1, based on which we will decide on H0 or H1.Therefore the likelihood ratio values of the binary detec-tion problem are as follows:

Lðy½n� ¼ 0Þ ¼Pðy½n� ¼ 0;H1Þ

Pðy½n� ¼ 0;H0Þ¼

1�p1ðtÞ1�p0ðtÞ

,

Lðy½n� ¼ 1Þ ¼Pðy½n� ¼ 1;H1Þ

Pðy½n� ¼ 1;H0Þ¼

p1ðtÞp0ðtÞ

:

Since 0op0rp1o1, we have 0oLðy½n� ¼ 0Þr1rLðy½n� ¼ 1Þ, and the decision rule of the binary detectionproblem is

dðy½n�Þ ¼

H1 if Lðy½n�Þ4g,

H0 if Lðy½n�Þog,

H0 or H1 if Lðy½n�Þ ¼ g:

8><>: ð46Þ

If goLðy½n� ¼ 0Þ, we have PFA ¼ 1 and PD¼1. If Lðy½n� ¼ 0ÞogoLðy½n� ¼ 1Þ, we have PFA ¼ p0ðtÞ and PD ¼ p1ðtÞ. Ifg4Lðy½n� ¼ 1Þ, we have PFA ¼ 0 and PD¼0. With the helpof randomization decision functions, the ROC of thedetection problem is the combination of the segmentfrom ð0;0Þ to ðp0ðtÞ,p1ðtÞÞ and the segment fromðp0ðtÞ,p1ðtÞÞ to (1,1).

Now we show that when t¼ s½n�=2, the AUC is thelargest. Denote the (0,0) point in the PFA�PD square as O,the (1,1) point as I, and the ðp0ðs½n�=2Þ,p1ðs½n�=2ÞÞ point asA, which are shown in Fig. 11. For another tas½n�=2,denote the ðp0ðtÞ,p1ðtÞÞ point as B. It is thus sufficient to

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Pro

babi

lity

of d

etec

tion


ROC using TS1(τ = s [n]/2)

ROC using TS1(τ>s [n]/2)

Auxiliary line CD: parallel to OI passing AAuxiliary line OI: (0,0) to (1,1)

B

A

A = (p0, p1) when using TS1 (τ = s [n]/2)B = (p0, p1) when using TS1 (τ>s [n]/2)B will be between CD and OI,area of Δ OAI>area of Δ OBI

O

D I

C

Fig. 11. ROCs of the TS with different t.


show that the area of nOAI is not smaller than that ofnOBI.

Without loss of generality, assume that t4s½n�=2. Theother case can be proved similarly. Note that p0ðtÞ and

p1ðtÞ are decreasing functions of t. Define Dp09p0ðs½n�=

2Þ�p0ðtÞ ¼R t

s½n�=2 f W ðwÞ dw and Dp19p1ðs½n�=2Þ�p1ðtÞ ¼R ts½n�=2 f W ðw�s½n�Þ dw. Since fW(w) is symmetric at w¼0

and unimodal, we have f W ðw�s½n�ÞZ f W ðwÞ for

s½n�=2owot, and hence Dp1ZDp0. This means that thepoint B is on or under the segment CD in Fig. 11, wheresegment CD includes point A and is parallel to thesegment OI. Thus, the area of nOAI is not smaller thanthat of nOBI.

Appendix B. Proof of Theorem 2

For the binary detection problem with observation y,the optimal test is the LRT, defined as

LðyÞ ¼Pðy;H1Þ

Pðy;H0Þ‘

H1

H0

Z0: ð47Þ

Since entries of y are independent, we have

LðyÞ ¼YN�1

n ¼ 0

Pðy½n�;H1Þ

Pðy½n�;H0Þ¼YN�1

n ¼ 0

Lðy½n�Þ, ð48Þ

where Lðy½n�Þ9Pðy½n�;H1Þ=Pðy½n�;H0Þ.Notice that y½n� only takes 0 or 1, we calculate the

values of Lðy½n�Þ for y½n� ¼ 1 and y½n� ¼ 0, respectively. First,we consider the case of s½n�Z0, for which the optimal TSis TS1 s½n�=2

� . Because f Xðx;H1Þ ¼ f Xðx�s½n�;H0Þ,

f Xðx;H0Þ ¼ f W ðwÞ, and fW(w) is unimodal and symmetricat w¼0, we have

Pðy½n� ¼ 1;H1Þ ¼

Z 1s½n�=2

f Xðx;H1Þ dx¼

Z 1�s½n�=2

f Xðx;H0Þ dx

¼

Z 1�s½n�=2

f W ðwÞ dx¼

Z 10

f W ðwÞ dwþ

Z 0

�s½n�=2f W ðwÞ dw

¼1

2þ

Z s½n�=2

0f W ðwÞ dw,

Pðy½n� ¼ 1;H0Þ ¼

Z 1s½n�=2

f Xðx;H0Þ dx¼

Z 10

f W ðwÞ dw

�

Z s½n�=2

0f W ðwÞ dw¼

1

2�

Z s½n�=2

0f W ðwÞ dw,

Pðy½n� ¼ 0;H1Þ ¼ 1�Pðy½n� ¼ 1;H1Þ ¼1

2�

Z s½n�=2

0f W ðwÞ dw,

Pðy½n� ¼ 0;H0Þ ¼ 1�Pðy½n� ¼ 1;H0Þ ¼1

2þ

Z s½n�=2

0f W ðwÞ dw:

For small signal, i.e., 9s½n�95s, we have

Pðy½n� ¼ 1;H1Þ �1

2þ

s½n�

2f W ð0Þ,

Pðy½n� ¼ 1;H0Þ �1

2�

s½n�

2f W ð0Þ,

Pðy½n� ¼ 0;H1Þ �1

2�

s½n�

2f W ð0Þ,

Pðy½n� ¼ 0;H0Þ �1

2þ

s½n�

2f W ð0Þ, ð49Þ

and thus

ln Lðy½n� ¼ 1Þ � ln

1

2þ

s½n�

2f W ð0Þ

1

2�

s½n�

2f W ð0Þ

0B@

1CA

¼ lnð1þs½n�f W ð0ÞÞ�lnð1�s½n�f W ð0ÞÞ ¼ s½n�f W ð0Þ

�ð�s½n�f W ð0ÞÞþOððs½n�f W ð0ÞÞ2Þ

� 2s½n�f W ð0Þ,


1

2�

s½n�

2f W ð0Þ

1

2þ

s½n�

2f W ð0Þ

0B@

1CA��2s½n�f W ð0Þ: ð50Þ

Now, for the case of s½n�o0, the optimal TS isTS2ðs½n�=2Þ. Similarly, we can show that

Pðy½n� ¼ 1;H1Þ ¼1

2þ

Z �s½n�=2

0f W ðwÞ dw,

Pðy½n� ¼ 1;H0Þ ¼1

2�

Z �s½n�=2

0f W ðwÞ dw,

Pðy½n� ¼ 0;H1Þ ¼ 1�Pðy½n� ¼ 1;H1Þ ¼1

2�

Z �s½n�=2

0f W ðwÞ dw,

Pðy½n� ¼ 0;H0Þ ¼ 1�Pðy½n� ¼ 1;H0Þ ¼1

2þ

Z �s½n�=2

0f W ðwÞ dw:

For small signal, i.e., 9s½n�95s, we have

Pðy½n� ¼ 1;H1Þ �1

2�

s½n�

2f W ð0Þ,

Pðy½n� ¼ 1;H0Þ �1

2þ

s½n�

2f W ð0Þ,

Pðy½n� ¼ 0;H1Þ �1

2þ

s½n�

2f W ð0Þ,

Pðy½n� ¼ 0;H0Þ �1

2�

s½n�

2f W ð0Þ, ð51Þ


and thus


1

2þ

s½n�

2f W ð0Þ

1

2�

s½n�

2f W ð0Þ

0B@

1CA��2s½n�f W ð0Þ,


1

2�

s½n�

2f W ð0Þ

1

2þ

s½n�

2f W ð0Þ

0B@

1CA� 2s½n�f W ð0Þ: ð52Þ

Combining (50) and (52), for any s½n�, we have

ln Lðy½n� ¼ 1Þ � 29s½n�9f W ð0Þ, ln Lðy½n� ¼ 0Þ ��29s½n�9f W ð0Þ:

ð53Þ

Note that the values of ln Lðy½n� ¼ 1Þ and ln Lðy½n� ¼ 0Þ areindependent of x½n� but only depend on the signal 9s½n�9and the value of the noise pdf at 0.

For a given y vector, let D be the number of 1’s in y. Thenumber of 0’s is thus N�D. From (48) and (53), we have

ln LðyÞ � 2X

fn9y½n� ¼ 1g

9s½n�9�X

fn9y½n� ¼ 0g

9s½n�9

0@

1Af W ð0Þ

¼ 2X

fn9y½n� ¼ 1g

9s½n�9�XN�1

n ¼ 0

9s½n�9�X

fn9y½n� ¼ 1g

9s½n�9

0@

1A

24

35f W ð0Þ

¼ 4X

fn9y½n� ¼ 1g

9s½n�9f W ð0Þ�2XN�1

n ¼ 0

9s½n�9f W ð0Þ

¼ 4XN�1

n ¼ 0

9s½n�9y½n�

!f W ð0Þ�2

XN�1

n ¼ 0

9s½n�9f W ð0Þ: ð54Þ

Note that the second term in (54) and f W ð0Þ are constants,independent of the hypotheses and the observation. Thusfrom (47), the optimal test rule becomes

TTDðyÞ9

1

N

XN�1

n ¼ 0

y½n�9s½n�9‘H1

H0

Z,

which shows that the proposed TTDðyÞ is the optimal test

statistic.

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A robust detector of known signal in non-Gaussian noise ...

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