PROCEEDINGS OF THE IEEE, VOL.

55, NO. 4, APRIL 1967

523

Energy Detection of Unknown Deterministic SignalsAbstract-By oinp Shannons sampling form&, the problem of the detectiooofadet~signnlmahite~wise,bymePapofm energy-meosariagde*~tothecoasiderntionofthe~ofthe squares of statiptially indepedent Gaussian variates Wben the signal isabsent,thededsionstatistieh.snceatrPlchCsqaPredistribptioowiththe

sc(t), s,(t) = in-phase

llmnberofdegreesoffreedom~totwicetbetime-bnsdwidthproctpctoftheinpetWhenthesignnlbpresmt,thede&ionstPtistichrsanoocentral

chCsqnuedistribotioawiththesamenumberofdegreesoffreedomanda ~typnrlmeter1equaltother8tioofsignalewrgytotwddnoisespe&ddensily.Siocethencmcentralchisqmredisbibotioahnsaot been tabulated extensively enopgh for OPT prupose, m approximate form was I h i s form replaces the noncentral chi-square with a modified chi-

and quadrature modulation components of s(t) aci= nc(i/W ) aSi = n,(i/ W ) b,, = a,,/,/= bSi =a , J J m aCi = sc(i/ W )

I r o e d .

squarewbosedegrefsoffreedomandthresboMaredeterminedbythemm~trcllity~ermdthepre~degreesOffreed0m.

sets of receiver operating characteristic (Roc) arnes are d r a m for several time-bandwidtb pmduch, pf well as an extended mwogrrun of tbe chisqmre aunlllrtive probability which can be aped for rapid cplealntion of fake alarm and detection p r o b a b i l t i a Related work m energy deteetioo by J. I. Mararm and E . L. Kaplan is

y(t)= input to the energy detector N(m, 02)=a Gaussian variate with mean m and variance 02.

diseassed.

GLOSSARY OF PRINCIPAL SYMBOLS s(t) =signal waveform n(r) = noise waveform No, =two-sided noise power density spectrum E, = signal energy V= normalized test (decision) statistic V; = threshold value of V T = observation time interval, seconds W= bandwidth, cycles persecond sinc x =sin rt.x/n.x a,= n(i/2 W) = noise sample value bi = a J d m = normalized noise sample value ai = s(i/2 = signal sample value Pi = a,/J7iTE& =normalized signal sample value 1=EJN,,, = noncentrality parameter D = modified number of degrees of freedom G =divisor for modified threshold value T,(m, n, r ) = incomplete Toronto function x =central chi-square variate x = noncentral chi-square variate n,(t), ns(t)= in-phase and quadrature modulation components of n(t)

w>

Manuscript received August25,1966; revised December 14,1966, and January 16, 1967. The work reported in this paper was supported in part by the Advanced Research Projects Agency, Department of Defense, under Contract SD-283. The author is with the General Atronics Corporation, Philadelphia, Pa. 11918

I. INTRODUCTION HE DETECTION OF a signal in the presence of noise requires processing which depends upon what is known of the noise characteristics and of the signal. When the noise is Gaussian and the signal has a known form, even with unknown parameters, the appropriate processing includes a matched filter or its correlator equivalent. When the signal has an unknown form, it is sometimes appropriate toconsider the signal as a sample function of a random process. When the signal statistics are known, this knowledge can often be usedto design suitable detectors. In the situationtreated here, so little is known of the signal form that one is reluctant to make unwarranted assumptions about it. However, the signal is considered to be deterministic, although unknown in detail. The spectral region to which it is approximately confined is, however, known. The noise is assumed to be Gaussian and additive with zero mean; the assumption of a deterministic signal means that the inputwith signalpresent is Gaussian but not zero mean. In theabsence of much knowledge concerning the signal, it seems appropriate to use an energy detector to determine the presence of a signal. The energy detector measures the energy in the inputwave over a specific timeinterval. Since only the signal energymatters, and not its form, the results given in this paper apply to any deterministic signal. It is assumed here that the noise has a flat band-limited power density spectrum. By means of a sampling plan, the energy in a finite time sample of the noise can be approximated by the sum of squares of statistically independent random variables having zero means and equal variances. This sum has a chi-square distribution for which extensive tables and a convenient nomogram (to be described later)

524

PROCEEDINGS OF THE IEEE, APRIL 1967

The choice of T as the sampling instant is a matter of convenience; any interval of duration T will serve. Another quantity of interest is the false alarm rate Qo/T,where Qo is the false alarm probability based upon the energy in a sample of duration T. It is known that a sample function, of duration T, of a process which has a bandwidth W (negligible energy outside this band) is described approximately by a set ofsample values 2TW in number. In the case of a low-passprocess, the values are obtained by sampling the process at times 1/2W apart. Inthe case of a relatively narrowband bandpass pro&s, the values are obtained from the in-phase and quadrature modulation components sampled at times 1/ Wapart. The expansion of a finite portion of a process in terms of sampling functions leaves something to be desired from the point of view of precision as well as rigor, although the approximations obtained are often considered satisfactory for engineering purposes. More satisfactory statements concerning the approximation to the energy by a finite sum of squares can be made. A discussion of this point is leftto the Appendix. Here we merely state thatvery good approximations are obtained by 2 TW terms except whenTW= 1. This special case can be treated separately, as we shall show later. Let us start with a low-pass process. With an appropriate choice of time origin, we may expresseach sample of noise 11. DEECTION IN WHITE NOISE : LOWPASS PROCESSES in theform [5] The situation of interest is shown in Fig. 1. The energy detector consists of a squarelaw device followed by a finite time integrator. The outputof the integrator at any time is the energy of the input to the squaring device over the interval T i n the past. The noise prefilter serves to limit the where sinc x = sin nx/z.x, and noise bandwidth; the noise at the input to the squaring device has a band-limited, flat spectral density.SQUARING DEVICE

exist. In the presence of a deterministic signal, the sampling plan yields an approximation to the energy consisting of the sum of squares of random variables, where the sum has a noncentral chi-square distribution. The tabulations of noncentral chi-square are not as extensive nor as convenient as those for central chi-square, making it desirable to use approximations. Grenander, Pollak, and Slepian [l ] havedeveloped a method for obtaining the exact distribution of the energy in a finite time sample of Gaussian noise with a flat, bandlimited power density spectrum. They compare the exact results with the chi-square approximation (in their Section 4.4) for several values oftime-bandwidth product and conclude that the approximation is quite good even for moderate values of time-bandwidth product. Jacobs [2] has obtained expressions for the distribution of energy by using the Karhunen-Loeve expansion. Again, he concludes that the chi-square approximation is a good one for large values of time-bandwidth product. There appears to be little relevant work in finding exact distributions of the energy in a finite time sample of noise plus unknown deterministic signal. An appropriate procedure, then, is to use the sampling plan mentioned above and consider ways to evaluate the noncentral chi-square distribution which results. The closest previous work of which I am aware is that of Marcum [3] and Kaplan [4]; the connection with theseworks will be reviewed later.

v' = (1/NO2)jy 2 ( t ) d t .0

T

(1)

INTEGRATOR

Clearly, each a, is a Gaussian random variable with zero : , which is the variance mean and with the same variance a of n(t); i.e., a ; Using the fact that=

Fig. 1 .

Energy detection.

2NO2W, all i.

(4)

The detection is a test of the following twohypotheses. sinc (2Wt - i) sinc(2Wt - k) dt = 1/2W, i = k 1) Ho : The inputy(t) is noise alone: a) r(t)= 41) = 0, i # k (5) b) E[n(t)]=O c) noise spectal density = No, (two-sided) we may write d) noise bandwidth = W cycles per second. m 2) HI: The inputy(t) is signalplus noise : n2(t)dt = (1/2W) a:. (6) a) v(t)=n(t)+s(t) i = -00 b) E[n(t) s ( t ) ] = s(t). The output of the integrator is denoted by V and we conOver the interval (0, T), n(t) may be approximated by a centrate on a particular interval, say, (0,7), and take the test finite sum of 2TWterms, as follows : statistic as V or any quantity monotonic with V. We shall 2 TW find it convenient to compute the false alarm anddetection n(t) = a, sinc(2Wt - i), 0 t < T. (7) probabilities using the related quantity i= 1

I_

+

I : m

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URKOWITZ : DETECTION OF UNKNOWN DETERMINISTIC SIGNALS

525

Similarly, the energy in a sample of duration T is approximated by 2T W terms of the right-hand side of (6):

Under the hypothesis HI, the test statistic V isf T 2TW

V = (1/No2)J- y 2 ( t ) d t =0

-2 (bi + pi).i= 1

(18)

This statement is supported by the Appendix and (8) may be considered to be obtained as an approximation (valid for large T) after substituting(7) into the left-hand side of (8), or byusing (57) and the statements below it to justify taking 2TW terms of (6). We can see that (8) is No2V, with V here being the test statistic under hypothesis Ho. Let us writeai/,/-2TW

The sumin (18) issaid to have a noncentral chi-square distribution [9], [lo] with 2T W degrees of freedom and anon, given by centrality parameter 1T

2 TW

i =i= 1

82 = (l/No2)Jos 2 ( t ) d t

E,/No2.

(19)

A, the ratio of signal energy to noise spectral density, provides a convenient definition of signal-to-noise ratio.

= bi,

V =i= 1

b?.

111. DETECTION IN WHITE NOISE : BANDPASS PROCESSES If the noise is a bandpass random process, each sample function may be expressed in the form [ 131 (10) (9)n(t) = n,(r) cos a c t - n,(t) sin a c t ,

(20)

Thus, V is the sum ofthe squares of 2TWGaussian random variables, each with zero mean and unity variance. V is said to have a chi-square distribution with 2TW degrees of freedom, for which extensivetables exist [6l-[8]. Now consider the input y(t)when the signal s(t)is present. The segment of signal duration T may be represented by a finite sum of 2TW terms.2 TW

s(t) =i= 1

ai sinc(2Wt - i),

where o,is the reference angular frequency, and n,(t) and n,(t) are, respectively,the in-phase and quadrature modulation components.If n(t) is confined to a frequency band W, n,(t) and n,(t) are low-pass functions whose spectral den4< W/2. If n(t) has a flat sities are confined to the region 1 power density spectrum No2, n,(t) and n,(t) will also have flat powerdensity spectra, each equal to 2NO2over < W/2. Thus, n,(t) and n,(t) each have-TWdegrees of freedom and (11) variance 2NO2 W. Also, to a good approximation, which gets better asT increases, (12) By following the reasoning in the previous section, we get similar series expansions forthe energy in n,(t) and n,(t), as follows :TW

whereai = s(i/2W).

By following the line of reasoning above, we can approximate the signal energy in the interval (0, T) by

joTn:(t)dr = (1/W)i= 1

a : i

Define the coefficient pi by where2TW

Then( 1 / N o 2 ) ~ T s 2d(t t = )0

1 /3?.

5)(1

i= 1

a,, = n,(i/W) aSi= n,(i/W).Define bCiand b,, as follows:

Using (11) and (2), the total input At) with the signal present can be written as:2 TW

fit) =

1 (ai + ai) sinc(2Wt - i).

bci = a c bSi= asi/ Then, under H,,TW T

Jm.(b:ii= 1

i / J m

(24)

(16)

i= 1

V = (l/No2)JonZ(r)dr=

+ bzi).

(25)

Since the variance of any b,, or bSiis unity, the sum on the right-hand side of (25), which is the test statistic I/ under

526

PROCEEDINGS APRIL OF THE IEEE,

1967

ter 3, given by EJNO2,E, being the signal energy.Thus, the results are the same for bandpass processes and for lowpass processes, provided that W is interpreted as the positive frequency bandwidth. That is, in the case of low-pass ins ( t ) = s c ( t ) cos oct- s,(t) sin act. (26) puts, W refers to the interval between zero frequency and sc(t) and s,(t) have frequency components confined to the the upper end of the band. < W/2. Using the sampling formula as above, band AND IV. COMPUTATION OF DETECTION TW FALSE ALARM PROBABILITIES sc(t) = aCi sinc(Wt - i) i= 1 The probability of false alarm Q , for a given threshold TW Vi is given by s,(t) = asi sinc(Wt - i), (27)

H,, has a chi-square distribution with 2TW degrees of freedom. A bandpass signal may be expressedin the form

If1

1

i= 1

whereaci = sc(i/W) The farasi=

Qo = Prob{ V > V;(H,) =Prob{X$,

> Vi}.

(34)

s,(i/W).

right-hand side of (34) indicates a chi-square vari(28) able with 2TW degrees of freedom. For the same threshold level Vi, the probability of detection Qd is given by

The total input y ( t ) now may be written as

(29) The symbol X ; ~ ~ ( A ) indicates a noncentral chi-square variable with 2TW degrees of freedom and noncentrality Again, using the sampling formula, parameter 3,; in our case jl=EjNo2, and is defined as the signal-to-noise ratio. As mentioned above, extensive tables TW exist for thechi-square distribution, but the noncentral chiy c ( t ) = s c ( t ) nc(t)= (1/W) (aci aci)sinc(Wt - i) i= 1 square has not been as extensively tabulated. ApproxiTW mations were sought which could be used to cover the ranges y,(t) = s&) n,(t) = (1/W) (asi asi) sinc(Wt - i ) . (30) of interest. i= 1 Many approximations have been devised for the nonDefine the coefficients #Ici and Psi as follows : central chi-square. For descriptions of some of these, the interested reader is referred to Owen [ l l ] and to GreenB c i = sc(i/W)/ wood and Hartley [12]. The approximation used in this B s i = ss(i/W)/ (31) paper is the one used by Patnaik [9], [lo]. It was checked against the exact values used by Fix [lo] for 0.01 and 0.05 By following the reasoning of the previous section, we can false alarm probabilities; close agreement was obtained. write The approximationreplaces the noncentral chi-square with TW a central chi-square having a different number of degreesof ( 1 / ~ 0 2 ) J , ~ s ( t ) dt = ( B : i + B:i) = ~ s / ~ * 0 2 (32) freedom and a modified threshold level. If the noncentral i= 1 chi-square variable has 2TW degrees of freedom and nonUnder hypothesis H,, the total bandpass input y ( t ) with centrality parameter A, define a modified number of degrees of freedom D and a threshold divisor G given by the signal presenthas its energy givenby

y ( t ) = [sc(t) nc(t)] cos a c t - [s,(t) = y c ( t ) cos o , t - y,(t) sin o,t.

+

+ n,(t)]

sin act

Qd

= Prob{ V

> VilHl} = PrOb(XyTW(A) > V ; } . (35)

+ +

1 1

+

+

Jm Jm.

JOT

y2(t)d t

= fIoTIY:(t)

D = (2TW G = (2TW

+ 2)/(2TW + 2A) + 2A)/(2TW + A).

(36)

= fIoT[sc(t)

+ n,(t)I2 dtdt.

Then Prob { X ; ~ ~ ( A )> V;} = Prob { X ; > V;/G}.(37)

(32a) The receiver operating characteristic (ROC) curves of Figs. 3 through 6 were drawn forT W> 1, using (34) and (35) apUsing (15),( 1 8 ) , (32), and (32a),we follow the line of reason- Proximated (37). ing of the previous section to show that under hypothesis Figure 2, for W a s obtained in a different way. H,, the test statistic V Referring is to (20), the squared envelope of n (given t ) is by TW Env [n(t)] = n:(t) + n;(t). (38) V = {(bci + B J 2 + (bsi + B c i ) } . (33) i= 1 Similarly, from (26), the squared envelope of the signal s(t) is given by It is seen that V has a noncentral chi-square distribution (39) with 2TW degrees of freedom and a noncentrality parameEnv [s(t)] = s,2(t) + sf(t).

+ i l O T [ s s ( t ) + n,(t)]

1

DETECTION URKOWITZ: SIGNALS DETERMINISTIC UNKNOWN

OF

521

s 0.999999 0.99999 0.99990.9990

9 0 , PROBABILITY OF FALSEALARY

8

P

0.99

0

0 . 9 00.M) 0.70 0.60 0.50 0.40 0.x)

E :p4

0

0.200.10

T

= OnSERVATION

TIME

E.= S I S N A L ENERSV N n = TWO-SIDED NO4S SPECTRAL DENSITY

I I I1

I

Fig. 2. Receiver operating characteristic (ROC) curves.

Fig. 4. Receiver operating characteristic (ROC) curves.0.. PROOABILITY OF FALSE ALARY o0.9999990.99999

x 4 :0.9999

0

8

s

x

-

0.991

p

0.99

E0.90t

a.80