Multichannel infrared receiver performance

Multichannel infrared receiver performance

Stephen J. Dunning and Stanley R. Robinson

The performance of an optical receiver designed to detect a target by taking advantage of the target's spec-tral signature is presented. The receiver processes the signal in several narrow frequency bands and is basedupon a statistical model which represents the field in each band as a Gaussian random process whose mo-ments depend upon the target and background characteristics. The optimal Bayes/Neyman-Pearson re-ceiver structure for an M spectral channel, N sequential look receiver is presented, and practical suboptimalreceiver structures are developed. Numerical methods are used to calculate the probability of false alarmand the probability of detection using identical parameters for each processor. The results indicate thatthe approximate receiver structures and the ad hoc receiver structures all have the same performance. Theresults also show that performance depends only upon the difference in the square root of the mean to vari-ance ratios under each hypothesis and the ratio of the variances.

1. Introduction

The use of airborne threat warning receiversemploying scanned ir detectors or staring detector ar-rays has often been proposed as a method of detectingairborne threats such as antiaircraft missiles. Thepracticality of such systems is supported by the well-developed sensor technology acquired during the For-ward Looking Infrared Receiver (FLIR) systems pro-gram. A warning receiver on board an aircraft ideallywould detect a launched airborne threat and cause someform of defensive countermeasure to occur.

As avionics systems become more automatic in theirresponse to the aircraft environment, there is an obviousneed to reduce the false alarm rate of any eventual air-borne threat warning receiver while maintaining ac-ceptable threat detection performance. A large numberof false alarms would generate distrust in the humanoperator and would also unnecessarily expend count-ermeasure resources. A well-designed threat warningreceiver would have to use signal processing techniqueswhich would enable the receiver to distinguish betweenvalid targets and phenomenon such as sun glint, openfires, smokestacks, and other nonhostile thermal sourceswhich could contribute to false alarms.

Stephen J. Dunning is with U.S. Air Force Satellite Control Facility,Data Systems Division, Sunnyvale AFS, California 94086, and S. R.Robinson is with U.S. Air Force Institute of Technology, Departmentof Electrical Engineering, Wright-Patterson AFB, Ohio 45433.

Received 30 March 1978.0003-6935/79/101567-10$00.50/0.Oc 1979 Optical Society of America.

This paper' is based on a recent study 2 which inves-tigated the design and performance of optimal andsuboptimal multichannel ir receivers. We show thatsimple ad hoc linear processors can be used to achievethe same performance as more complex, but optimallyderived, processors. Thus simple receiver processingtechniques may hold promise for some future de-signs.

Our approach is divided into three distinct areas: thestatistical model of ir fields; the receiver structure (in-cluding postdetection processing); and the resultingperformance calculations.

11. Statistical Field and Detector Model

The receivers presented in this paper are designed todetect the presence of a valid target against ir back-ground clutter by processing the output signal from anoptical detector. Before the signal processor structurecan be determined, the statistics of the signals to bedetected must be ascertained. The statistics of theobserved optical field are first examined, and thesestatistics are then used to obtain the statistics for theoutput signal of an ideal power detector.

A. Field Representation

The incident optical field to be detected at a positionr can be described by the scalar field quantity u(r,t),which may represent either the electric or magnetic fieldstrength. For convenience, this field is represented byits complex envelope

U(r,t) = UR(r,t) + jU1 (r,t), (1)

where UR(r,t) is the real part of U(r,t), and U1(r,t) isthe imaginary part of U(r,t). This representation is the

15 May 1979 / Vol. 18, No. 10 / APPLIED OPTICS 1567

same as the quadrature field model used in radar andcommunication systems. The complex envelope isimplicitly defined by the equation

u(r,t) = Re[U(r,t) exp(-j2nfrtt)], (2)

where fo is the optical center frequency, and Re(.) de-notes the real part of the enclosed quantity.3 Becausethe complex envelope U(r,t) is a time varying quantity,it is a valid representation for extremes ranging from theincoherent light due to naturally occurring illuminationto the coherent output of a laser. For the purpose ofthis paper, the complex envelope was used to representthe received fields in the intermediate ir region.

It is unreasonable to assume a priori the field that thereceiver would detect. Therefore, it is appropriate tothink of the complex envelope of the field as a randomprocess in both time and space. It is further assumedthat the complex envelope is a sample function of acomplex Gaussian random process. This assumptioncan be partially justified by the application of the cen-tral limit theorem to the received field, where the re-ceived field is due to the sum of a large number of in-dividual fields, each of which is due to the scattering ofnatural light by an independent particle.3 4

For simplicity, the real and imaginary parts of U(r,t)are assumed to be independent and identically dis-tributed. The complete specification is then givenby

E[U(r,t)] = m(r,t), (3)

E[U(r,t)U*(r',t')J = R(r,r',tt'), (4)

where m(i,t) is the mean (ensemble average) of U(i,t),and R (i,i',t,t') is the correlation function. The aboveconditions imply that R (r,r',t,t') is a real function andis twice the correlation function of either the compo-nents UR(f,t) or U1(1,t).

The optical detection problem can now be stated.When a target is present, the received field becomes

U(r,t) = U8 (r,t) + Ub(r,t), (5)

where U8(r,t) is the signal or target field, and Ub(r,t)is the background field. The target hypothesis is de-noted by H1 . If no target is present, denoted by Ho (ornull hypothesis), the received field is only Ub(r,t).Under the assumption that Us (r,t) and Ub (r,t) are in-dependent, the statistics under each hypothesis aregiven by

HO: m(r,t) = mb(r,t)

R(r,r',t,t') = Rb(r,r',t,t'), (6)

HI: m(r,t) = m,(rt) + mb(r,t) (7)R(r,r',t,t') = R(r,r',t,t') + Rb(r,r',t,t').

With the field model described above, it is possibleto describe completely the space-time processing thatshould be accomplished by a receiver and the resultingperformance.5'6 There are three major disadvantagesto the complete space-time processing approach. 6 Theprocessing and performance depend upon m(r,t) andR(r,t',t,t') explicitly, and it is unreasonable to expectthat these quantities are known. Second, the space-

time processing required would typically be much toocomplex to implement even if m(r,t) and R(r,r',t,t')were known. Finally, the processing would require themeasurement of the complex envelope U(r,t), and thisis not possible at most wavelengths using currentlyavailable devices. The exception is, of course, opticalheterodyne receivers which have been used in manyoptical systems. The heterodyne receivers are re-stricted, however, to wavelengths where laser sourcesare available for local oscillators. Since we are inter-ested in detection over a broad range of wavelengths, weno longer consider the heterodyne approach as being aviable candidate.

Although Eqs. (6) and (7) completely specify the fieldstatistics, the determination of m(r,t) and R(r,r',t,t')explicitly is generally quite difficult. Thus, reasonableapproximations were sought which would permit morepractical and realizable signal processor configura-tions.

Two simple but crude parameters which were usedto simplify the field statistics representation were co-herence distance and coherence time, denoted DC andT, respectively. The coherence distance is describedby

R(r,r',t,t') = m(r,t)m(r',t); r - r'l > D, (8)

for all t. It is the distance beyond which samples of thefield are considered to be statistically independent.The resulting coherence cell model for the field is asimple approximation in which it is assumed that theincident field is spatially constant over an area (coher-ence cell) and is statistically independent from the fieldin other cells.

Coherence time is defined as the length of time overwhich the field can be broken into nearly piecewiseconstant independent samples in time and is describedby

R(r,r,t,t + T,) = m(r,t)m(r,t + T,) (9)

for all r. The coherence time/cell model allows the fieldto be decimated in space and time so that the field cannow be considered as a collection of random variablesrather than a random process in space and time.

The field can also be separated into a number ofdisjoint frequency windows. Since the resulting sepa-rated fields are considered to be statistically uncorre-lated, the output field in any spectral window is inde-pendent from the field in any other frequency window.This independence between frequency windows allowsthe coherence time/cell field model developed above tobe applied to the statistical description of the field ineach spectral window, differing only in the momentsrequired to describe each output field.

B. Detector Model

Utilizing the piecewise constant coherence time/cellmodel for each spectral window, an array of detectorsplaced in the measurement plane of the receiver can beconsidered. Each detector is an ideal power detectorwhose active area is matched to the smallest coherencecell expected, where the size of the coherence cell de-

1568 APPLIED OPTICS / Vol. 18, No. 10 / 15 May 1979

pends upon the type of target to be detected. Coher-ence cell size in excess of that required would lead toperformance degradation due to the increase in back-ground noise.

The intensity or rate function A(t) of the ith detectoris given by

xi= (t I f U(r,t)I2 dr, (10)

where -q is the quantum efficiency of the detector, hf0is the energy of a photon, Ai is the active area of thedetector, and it is assumed that the intensity of thescalar field in units of power per unit area is given byI U(r,t) 2. The rate function may be defined as theaverage number (ensemble) of photon-electrons ob-served at the output of the detector as a function oftime. 3

Within the constraints of the coherence time/cellmodel and assuming that the detector area Ai is on theorder of the coherence cell and that the time interval inwhich the observation is made is less than the coherencetime, the current output of the ith detector at time t,centered at ri, is given by

yi = qXi(t) = C X jU(rt)I 2 dr ;- q U(ri,t) 2 A,, (11)hf0 J Ai hfo

where q is the charge of an electron.The statistics of the detector output can now be de-

termined. The randomness of the detector output yiis due only to the stochastic nature of the received fields,since the ideal nature of the detector eliminates anydevice noise that would be inherent in a real photode-tector. The ideal nature of the detector serves to sim-plify the analysis and is based upon the contention thatdetector noise need not be considered if the noiselessperformance of the receiver proved to be unacceptable.Using well-known results for the sum of squares of twoGaussian random variables,7 the probability densityfunction (pdf) of the output current of the ith detectoris given by

Yin 0I2?exp- 2 '[1;0 ; =Y m )l0ew h e r e ,0elsewhere,

where I0 (-) is the modified Bessel function of the firstkind of order zero. This pdf is known as the noncentralchi-square density function with two degrees of free-dom. The parameters of this pdf are related to the fieldquantities by

mi= 2- Aim(rit)] (13)

¢i= h A i) 21 %R(ri,ra~,t) -[m(ri,)]21. (14)

It is straightforward to extend Eq. (12) to the jointprobability density function for the detector array.Since the detectors are disjoint in frequency, andtherefore independent, the joint pdf is given by

Mf(y-) = II fi(Yi). (15)

i=1

Equations (12) and (15) also apply to the detectoroutput of a single detector in a particular spectral win-dow, which is scanned across the image plane a coher-ence cell at a time. If the frame time for each coherencecell is on the order of the coherence time, the detectoroutput for each frame time is independent, and the jointpdf is given by Eq. (15), where the index i refers to thetime frame (or coherence cell) in which the measure-ment is made.

C. Validation of Infrared Background Model

To investigate the validity of the detector outputmodel developed above, a literature search was con-ducted to discover any sources of experimental datapertaining to the statistical properties of ir backgrounds.While many sources relating the spectral characteristicsof ir backgrounds were found, only two sources con-cerning the statistical properties of ir backgrounds werediscovered.

The first source of statistical information was foundin Ref. 8. Measurements were made of various back-ground types using a scanning radiometer. The authorsanalyzed the data and developed a statistical model forthe ir background in several spectral windows. Whilethe model proposed in that paper is based upon acombination of Gaussian and Poisson statistics and isdifferent from the model used in this paper, their sta-tistical data presented does support our noncentralchi-square model. The distinctive shapes of theprobability density functions illustrated in the report8

are all possible forms of the noncentral chi-squareprobability density function, dependent upon the spe-cific parameters of the density. Although no quanti-tative error analysis has been completed, it is encour-aging to note how well the data follows the predictionof our model, especially since our statistics were moti-vated by physical considerations rather than an ad hocassumption (as in Ref. 8). Thus the model can be ad-justed to match the histogram data for the same classof background clutter as viewed through spectral win-dows such as 2-3 ,um, 3-4 ,um, and 8-14 ,im.

Another source of ir background statistical data wasfound in a series of reports by the Lockheed Missile andSpace Company (LMSC).9-12 Under the sponsorshipof the Advanced Research Projects Agency and the U.S.Army Missile Command, LMSC began the BackgroundMeasurements Program in which natural ir backgroundwere measured from the air using an ir radiometermounted in a U-2 research aircraft.

In one report, data for one measurement flight wasanalyzed by LMSC and the data presented in the formof histograms for each of six spectral filters. 9 While thehistograms contain data for a combination of differentbackground types that were overflown by the aircraft,comparison of the data with the flight track by LMSCindicated that it was possible to separate the histogramsfor each of four background types from the combinedhistogram. The four background types were high cloud,low cloud, water, and terrain. The experimental his-togram for data from the 0.202-Mm bandwidth filtercentered at 4.5 gm is shown in Fig. 1.


l0,3010I

RAOIANCE IN MICROWATS/CC2-SR-MlICRCN)

Fig. 1. Experimental histogram.

If the number of sample points is large, the histogramor sample relative frequency plot for each backgroundtype converges to the corresponding pdf for that back-ground. 4 The weighted sum of these pdfs yields a cu-mulative relative frequency plot or histogram. To testthe fit of the model with the experimental data, non-central chi-square probability density functions weregenerated for each background type using parametersestimated from the histogram in the LMSC reports.The pdf's generated for each background type are il-lustrated in Figs. 2-5. These pdf's were then linearlycombined according to the following equation which wasderived through trial and error:

f(y) = 0.08fHc(y) + 0.14fLc(y) + 0.26fw(y) + 0.52fT(Y),

where f(y) is the combined relative frequency function,fHC (y) is the high cloud background pdf, fLC (y) is thelow cloud background pdf, fw(y) is the water back-ground pdf, and fT(Y) is the terrain background pdf.The numerical constants in Eq. (16) are the estimatedfraction of total samples contributed to the combinedrelative frequency function by each type of background.The combined relative frequency plot obtained is il-lustrated in Fig. 6. This sample relative frequency plotclosely approximates the experimental histogram ob-tained by LMSC. Again, no formal error analysis wascompleted since the available data were in graphicalform. However, the pdf match for each backgroundtype was completed in several other wavelength regions,and except for changes in the actual parameter values,the conclusions were the same.2

While the experimental data reviewed here does notconclusively show the validity of the noncentral chi-square model, it does indicate that the model is con-

sistent with the experimental ir background statisticaldata currently available.

Ill. Signal Processor Structures

A signal processor structure developed through theapplication of statistical signal detection techniquesdepends explicitly upon the statistical characteristicsof the signals being detected. The signals to be pro-cessed by the processors developed here are the detectoroutputs under the null and target hypotheses. Bycorrectly processing these signals, it is hoped that thereceiver will distinguish between the presence or ab-sence of a valid target with a high degree of accuracy.This section presents the receiver processor structuresdeveloped by using the noncentral chi-square detectoroutput model to characterize the statistics of the nulland target hypothesis signals.

The basic receiver considered here is a receiver,which, by means of narrowband filters and parallel idealpower detectors, observes M frequency disjoint chan-nels. The reasoning behind this structure lies in thefact that most targets to be detected will have charac-teristic spectral signatures. By properly choosing thespectral channels, the receiver can discriminate againstunwanted sources whose spectral characteristics differfrom those of the desired target.

The receiver processor development and the followinganalysis pertain to a receiver which makes N successivetemporal observations of M detector outputs for anarbitrary coherence cell. The channel outputs duringthe kth observation of the cell would constitute a vectorYk = YklYk2 ... JYkM, where the elements are thechannel detector outputs. Each successive observationis assumed statistically independent and identicallydistributed. The first assumption is based upon thecoherence time/cell model, while the second assumptionserves to simplify the processor structure and lateranalysis by excluding temporal processing of successiveobservations. If the target has known temporal char-acteristics, temporal processing would be advantageous,but at the cost of increased receiver complexity.

(16)

A. Optimal Processor Structure

The totality of all observations made by the receiveris denoted by Y and has the following joint probabilitydensity functions under the null and target hypoth-eses:

I I exp[- (Yki + miO0i IO - 2 a I

Ho: fo(Y) = 1 1X 0; el (Ykis

0; elsewhere,

{N

HI: fl(Y) =

0 ;

Yki 2 0 (17)

LI- T exp 2 (Yki + mil)Ii=i 2, x I~z

X I[ I (Ykimil)1/21;

elsewhere.

Yki 0 (18)


IC8J

DETECTOR HIGH CLOUDMERN=2,0 VRR-0.3

co

0

I.-

12.50 25.00 37.50 50.00RRDIRNCE

ICROWRTTS/(CM2-SR-MICRON)

DETECTOR LOW CLOUDSMERN-10.0 VRR-O.3

.00 12.50 25.00 37.50RROIRNCE

MICROWRTTS/(C112-SR-MICRON)

50.00

Fig. 2. Estimated pdf, high cloud background. Fig. 3. Estimated pdf, low cloud background.

DETECTOR WTERMERN=18.0 VRR=0.05

0

CD*

CO

co

0

0

N

9-

-.00 12.50 25.00 37.50RRDIRNCE

MICROWRTTS/(CM2-SR-MICRON)

DETECTOR I TERRAINMERN=28.0 VRR-0.3

04

Tb oo 12.50 25.00 37.50RRDIRNCE

MICROWRTTS/(CM2-SR-MICRON)s0. 00

Fig. 4. Estimated pdf, water background.

These pdf's are obtained from Eqs. (12) and (15). Thereceiver must choose between the hypothesis by pro-cessing the received signal described by the aboveprobability density functions. An optimal decision (ineither a Neyman/Pearson or Bayes average risk sense)is made by comparing the likelihood ratio, A(y) =fi(Y)/fo(y), with a threshold and declaring a target if the

Fig. 5. Estimated pdf, terrain background.

threshold is exceeded. 7 An equivalent expression isobtained by computing the logarithm of the likelihoodratio. This results in the statistic z (), which is definedby

N Mz(5) = nA(y) = F ai + byki + lnI[ci(yki)"']

k=1 i=1

- nI1 [ei (Yki) 12 11, (19)

where


0

0

0

C I

co

CDo

0

0000

ri

CD.

0

CD

0

C'

00

0

0

C

DETECTOR I

MW2 = ln(v) - N L ai + '12 ln(ej/ci).

The processing required by this structure is less complexthan that required by the optimal processor and is muchmore practical to implement.

C. Linear Approximate Processor

By approximating the modified Bessel function withits small argument equivalent, the signal processorstructure can be further simplified. The small argu-ment approximation for the modified Bessel functionis given by

Substituting this expression into the optimal processoralgorithm and reducing terms, the processor algorithmbecomes

0. o 12.50 2. 00 37.50 50. 00RRDIRNCE

MICROWRTTS/(CM2-SR-MICRON)

Fig. 6. Relative frequency plot.

ai = 2 n(io/ail) + '/2[(mio/o2zo) - /02

bi = '/2(1/a 2 -/Iy2

Ci = (Mil)1/2/,2l

e = (mo)1/2/a2o.

If the threshold is v, the optimal signal processor algo-rithm is given by

N ME Ibiyki + InIi[C(yki)/21 - nID[ei(yki)"/2 ]) ; W1, (20)

k=1 i=1 Do

whereM

W1 = n(v)-N E ai

and D1 and Do represent the decisions "a valid targetis present" and "no target is present," respectively. Thecomplex processing required may make this structureimpractical for a real time system and also makes per-formance analysis extremely difficult.

B. Nonlinear Approximate Processor

The optimal processor structure may be simplifiedby substituting the large argument approximation forthe modified Bessel function into the optimal signalprocessor algorithm given in Eq. (20). This approxi-mation is given by

I(x) ex/(2,rx)112 ; x >> 1. (21)

Substituting this approximation into the processor al-gorithm and reducing the expression, the new signalprocessor algorithm becomes

N M DE E byki + (ci - ei)(Yki)1/2 W2, (22)

k=1 i=1 Do

where

whereM

W3 = ln(v) - N L aj.

This linear signal processor structure is simpler thaneither the optimal processor or the nonlinear approxi-mate processor and is one of the most elementary signalprocessor structures possible.

D. Ad Hoc Linear Processors

An ad hoc signal processor is one which is obtainedthrough intuition rather than analytical procedures.Two ad hoc linear signal processors are presented hereand will be used for later performance comparison withthe nonlinear approximate processor and the linearapproximate processor. The processor constants usedhere are chosen proportional to the signal mean andinverse to signal fluctuation. Such a choice of constantsmight be motivated by SNR maximization techniquescommonly used in linear diversity receivers for com-munication over fading channels.1 3 The algorithms forthe two ad hoc processors are given by

N M DiZ E [(mil/a? - miol4i)yki] Z W4,

k=1 i=1 Do

N M DiZ E [(mil 1 - miO/u%)Yki] Z W4,

k=1 i=1 Do

where W4 is an arbitrary threshold. The processor de-scribed by Eq. (25) will be referred to as the ad hoc lin-ear processor 1, and the other processor will be desig-nated as the ad hoc linear processor 2.

IV. Signal Processor PerformanceSignal processor performance is characterized by the

probability of the receiver making an error. For theNeyman-Pearson processor structure, performance iscompletely specified by the quantities probability ofdetection (PD) and probability of false alarm (PFA).The probability of detection and the probability of falsealarm are defined by


C)

0

c-

Ur,

T,

a>-)- Cn

CC

CD

Q CD

0o-C-LO-

CD-

,-

N-

0

IO (x) X 2/4; x << 1. (23)

N M DiE E [bi + /4(c0 - eF)Yki] 2 W3,

k=l i=1 Do(24)

(25)

(26)

Table 1. Channel Parameters for Change in Mean

t7o p71 J

8.0 16.0 1.08.0 24.0 1.08.0 32.0 1.0

Table 11. Channel Parameters for Change in Variance

n7o 71 P

8.0 24.0 1.08.0 24.0 1.58.0 24.0 2.0

PD = P(D1|H1) = P[z(j7) > vJH1J, (27)

PFA = P(DiIHo) = P[z(Y) > uIHo]. (28)

These quantities are functions of the threshold v andare normally plotted as a performance curve, PD vs PFA,

known as the receiver operating characteristic or re-ceiver operating curve (ROC). The exact computationof the probability of detection and the probability offalse alarm requires that the probability density func-tion (or distribution function) for the log likelihoodratio, z(y), be known under both hypotheses.

This section presents the performance characteristicsof the nonlinear approximate processor and the linearapproximate processor only, as the complexity of theoptimal signal processor makes the computation of itsoutput probability density function unrealistic.

An attempt was made to derive analytic performanceexpressions by first determining the joint pdf for theoutput of each signal processor. This appears to beimpossible except for a special case of the linear ap-proximate processor. These results are summarized inthe Appendix.

A. Processor Performance

The lack of analytic expressions with which to com-pare between signal processors necessitated the use ofnumerical methods to achieve a common basis forcomparison. The procedure used here was to write aFortran computer program which directly computed theconditional probability density of the output of theprocessor for a given set of parameters. These densitieswere then numerically integrated over a range ofthreshold values to generate an array of values for theprobability of detection and the probability of falsealarm. These values were in turn plotted as a receiveroperating curve (ROC).

The computation of joint probability density func-tions for multichannel receiver performance evaluationmade use of a fast Fourier transform (FFT) algorithmto compute the transforms of the conditional densitiesof the processor output for each channel. The productof the individual channel transforms was then inversetransformed to obtain the conditional joint processoroutput density functions. These joint density functionswere then numerically integrated as above to obtainvalues for PD and PFA, which were plotted as a receiveroperating curve.

The conditional output density functions for thenonlinear approximate processor and the linear ap-proximate processor were derived from Eqs. (A4) and(A6) and were expressed in terms of the parameters iJo

=mi0/cla) = m21 /cri, and p = U2 /2. Single channelprocessor performance was calculated for changes intarget hypothesis mean and for changes in target hy-pothesis variance. The parameters used to observe theeffects of changes in mean upon processor performanceare listed in Table I. The parameters used to observethe effects of changes in variance upon processor per-formance are listed in Table II.

Typical nonlinear processor receiver performancecurves for change in mean are shown in Fig. 7. For aprobability of false alarm equal to 0.10, the probabilityof detection increases from 0.45 to 0.96 as 'ii goes from16.0 to 32.0 with i70 equal to 8.0. Typical nonlinearprocessor performance curves for change in variance areshown in Fig. 8. As the target hypothesis variance in-creases to twice the null hypothesis variance with con-stant mean to variance ratios, the probability of detec-tion goes from 0.80 to 0.99 for a probability of falsealarm of 0.10. The observed changes in performanceindicate that the receiver is more sensitive to changesin mean than to changes in variance.

To determine the effect of multiple channels uponreceiver performance, performance curves were gener-ated for one, two, and three channel receivers. Threenearly identical channels were used; the channel pa-rameters are listed in Table III. The performancecurves for the individual channels are plotted in Fig. 9.For a probability of false alarm of 0.10, each individual

RECEIVER OPERRTING CURVE03CD

zC)

Ui.

Lii

LL0C

Q

CD

0

C.V IC%. 00 0.25

PROBRILITY0.50 0.75OF FLSE RLRRM

I . 00

Fig. 7. Nonlinear processor ROC, change in mean (m = N = 1).


RECEIVER OPERATING CURVE RECEIVER OPERATING CURVE003

0o

Fig. 8. Nonlinear processor ROC, change in variance (M = N = 1).

1.00

Fig. 9. Individual channel performance curves (N = 1).

RECEIVER OPERATING CURVE003

Th.oo 0.25 0.so 0.75PROBABILITY OF FALSE ALARM

1. 00

Fig. 10. Multichannel nonlinear processor ROCs (N = 1).

Table 111. Multichannel Parameters

Channel n70 n1 p

1 8.0 16.0 1.02 9.0 18.0 1.03 10.0 20.0 1.0


00 .3

C3

z0C)

C..

uJ )CD

CCoC)0

channel has a probability of detection of approximately0.49. Multichannel receiver performance curves for thenonlinear approximate processor are shown in Fig. 10.The probability of detection increases to 0.66 for twochannels and to 0.82 for three channels, an improvementof 67% over single channel performance. Since a similarjoint density function would be obtained for a multilookreceiver, by extension of the multichannel results, onewould also expect receiver performance to improve fora multilook receiver.

The same procedures were applied to the linear ap-proximate processor using identical parameters. Thenumerical results indicated that the performance of thelinear approximate processor was identical (i.e., thesame ROC curve) to that of the nonlinear approximateprocessor. This result caused the authors to investigatethe analytical performance expressions. Indeed, it ispossible to demonstrate by a change of a variable thatthe two processors do have identical performance. 2

(Linear and nonlinear processor multichannel perfor-mance is also identical.)

Next, we investigated the performance of the two adhoc processors. The output pdf's are presented in Eqs.(A10), (All), and (A12). The density functions ofprocessor 1 can be expressed in terms of the parametersnlo, 11, and p previously defined. The density functionsof processor 2 can be expressed in terms of the param-eters mi0, mil, and p. For the purpose of performancecalculations the value of mi0 was arbitrarily chosen tobe 2.5. To make the performance calculations for thead hoc linear processor 2 consistent with those for theother processors, mil was determined by the followingequation:

mil = miopnl/n1o), (29)

where nqo, 'il, and p are the channel parameters listed inTables I and II. For identical parameters, the perfor-mances of both ad hoc processors were identical to theperformance observed for both the nonlinear and linearapproximate processors.

V. Conclusions

Based on the results presented, we come to the fol-lowing conclusions:

The noncentral chi-square statistical detector outputmodel appears to be valid. The model has a realisticphysical basis in the complex Gaussian random processused to model the received ir field and is further sup-ported by available experimental ir background sta-tistical data.

Receiver performance equal to that of either thenonlinear approximate processor or the linear approx-imate processor derived from the optical signal pro-cessor can be obtained using a relatively simple ad hoclinear receiver.

Receiver performance depends upon the differencebetween the square roots of the hypothesis mean tovariance ratios [(m,/,yl) 1 /2

- (mO/U2) 1 /2] and the ratioof hypothesis variances (oJo2 ). This result is incontrast to the one obtained from classical signal inGaussian noise analysis. Finally, receiver performancegreatly improves as the number of receiver channels andthe number of looks increase.

0

Appendix: Processor Output Probability Densities

The purpose of this Appendix is to summarize theknown analytical results for the pdf of the output of anumber of different processor structures. Where ap-plicable, these densities were used to compute numer-ically the receiver operating characteristics which arepresented.

A. Nonlinear Approximate Processor

Starting with the nonlinear processor algorithm givenby Eq. (22), the processor output function, z(Y), for asingle channel i is given by

z(y) = biy + (ci - e)y 12 . (Al)

B. Performance Parameter DependenceIn an effort to determine the manner in which re-

ceiver performance depends upon the parameters o,ni, and p, functions of these parameters were testedagainst previous results. The functions tested were71 - fl0 and \/711 - N/712. Again, these choices weremotivated by SNR types arguments, which wouldcommonly be used by designers. The results indicatethat the receiver performance depends upon the dif-ference between the square roots of the hypothesis meanto variance ratios.2 This differs from the classicalGaussian signal case where receiver performance de-pends upon the difference between the hypothesis meanto standard deviation ratios, mila/il - mio/oio. De-termination of receiver performance dependence uponfunctions also involving p was unsuccessful.

Since the detector output y is modeled by a noncentralchi-square random variable, through transformation ofvariables the processor output probability densityfunction can be derived. For any given values of b, ci,and ei there is a one-to-one correspondence between yand z(y), where y 2 0. Completing the square andsolving for y in terms of z yield

y(z) = [(ci - ei)2 bi + Z]1/2 - ( - e)1biJ21bi (A2)

from which the Jacobian of the transformation is found.The Jacobian is given by

dy(z)1 11 (ci - ei) I (A)i dz I Ibi bi [(ci - ei)

2+ biz] 2l

where - indicates the magnitude of the expression.The unconditional single channel processor outputprobability density function can now be expressed by


fh(z) = Jfy[Y(z)]

I ( {[ i+ - ei)2l1/2 - (ci-ei)2 1J ll ~bi j b In--- exp v2~,2~

2 2af~bi --

+ (ci -e )21 1/2 (ci - ei)

x lo I bi ] A/bi _ mi) 1 ' 4

where mi and a7 are hypothesis dependent channelparameters.

B. Linear Approximate ProcessorBeginning with the linear approximate processor al-

gorithm given by Eq. (24), the processor output functionfor a single channel i is given by

z(y) = [bi + /4(c2 - e?)Iy. (A5)

Through transformation of variables, the unconditionalsingle channel processor output probability densityfunction is expressed by

h~) 1 (x z/K + i 1 1 zi 1/21 A62IKj I , 2 2 )[ ) ] s (A6)

where Ki = bi + /4(c? - e).Using transform pair 655.1 from the Campbell and

Foster transform table,14 the Fourier transform for anM channel, N look linear approximate processor is givenby

exp ----- J exp m M 2 2u ( iri ifF(f) jfl / L C +4rK2f)} (A7)

(1 +j4 rKi uif) (7

Where j is the imaginary unit . Unfortunately,an inverse transform for Eq. (A7), which would yield anexpression for the general multichannel linear ap-proximate processor output joint density function, couldnot be found except for the special case where eachchannel is assumed to be identically distributed. Underthis assumption Eq. (A7) can be reduced to

F(f) =

exp (-NM 2 ) exp 2 r2(l + j4rKr2f)j

(1 + j4rKg 2f)where the density parameters are no longer channeldependent. Using transform pair 650.0 from theCampbell and Foster table, the processor output jointdensity function for M identically distributed channelsis given by

1 (M *N- m Z 1_ 4KU4Z M2N - 1/2

(2Ka2)MN 2,2 2Ko2 *

X I(M.N)-I [1, m- 1/2 (A9)>IMN)1~ Ka 4 I While the joint density function is in analytic form, thecalculation of the performance parameters PD and PFArequires that this pdf be integrated. This integrationis prohibitively complex, and a search of available in-

tegral tables failed to yield an integral of the requiredform. Even for the simpler structure of the linear ap-proximate processor, the derivation of exact perfor-mance expressions remains prohibitively cumber-some.

C. Ad Hoc Linear ProcessorsThe conditional processor output density functions

for the ad hoc linear processor 1 are obtained from theprocessor algorithm given by Eq. (25) through trans-formation of variables. The resulting single channelconditional processor output density functions are givenby

1j20 ( zi2 -a) in [(zMo 1/21Ho: f-, W 2=exp Iil J '(A10)

12Ki ( 2K; 2 io zm, 1/2 1Hi: f (z) = 2 exp Z Mi 1 I J1 1, /2 (All)

where Ki = (m 1/o4 -miOA4o).Similarly, the single channel conditional processor

output density functions for the ad hoc linear processor2 were derived from the processor algorithm given byEq. (26). The density functions for the ad hoc linearprocessor 2 are of the same form as Eqs. (A10) and (All)except that Ki is now defined by

K, = (mila 2 - mio/CT2 0). (A12)

References1. Portions of this work have been approved for presentation and

publication in the Proceedings of the National Aerospace andElectronics Conferences (NAECON), 16-18 May 1978.

2. S. J. Dunning, "Multichannel Infrared Receiver Performance,"Masters Thesis, Air Force Institute of Technology, Wright-Pat-terson AFB, Ohio 45433 (December 1977), GE/EE/77-15.

3. E. V. Hoversten, "Optical Communication Theory," in LaserHandbook, F. T. Arecchi and E. D. Schultz-Dubois, Eds.(North-Holland, Amsterdam, 1972).

4. A. Papoulis, Probability, Random Variables, and StochasticProcesses (McGraw-Hill, New York, 1965).

5. H. L. Van Trees, Detection, Estimation, and Modulation Theory:Part 3 (Wiley, New York, 1971).

6. S. R. Robinson and J. 0. Gobien, in Proceedings of the IEEE 1977National Aerospace and Electronics Conference (1977), pp.580-587.

7. H. L. Van Trees, Detection, Estimation and Modulation Theory:Part 1 (Wiley, New York, 1968).

8. Y. Itakura et al., Infrared Phys. 14, 17 (1974).9. N. G. Kulgein, Background Measurements Program, Special

Technical Report LMSC-D502556, Lockheed Missile and SpaceCo., Palo Alto, Calif. (May 1976).

10. N. G. Kulgein, Background Measurements Program, SecondSpecial Technical Report LMSC-D506988, Lockheed Missile andSpace Co., Palo Alto, Calif. (September 1976).

11. LMSC, Background Measurements Program, Second Semi-annual Technical Report LMSC-D501944, Lockheed Missile andSpace Co., Palo Alto, Calif. (April 1976).

12. LMSC, Background Measurements Program, Third Semi-annualTechnical Report LMSC-D506926, Lockheed Missile and SpaceCo., Palo Alto, Calif. (September 1976).

13. M. Schwartz, W. R. Bennett, and S. Stein, CommunicationSystems and Techniques (McGraw-Hill, New York, 1966), Chap.10.

14. G. A. Camiipbell ad R. M. Foster, Fourier Integrals for PracticalApplications (Van Nostrand, New York, 1948).


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Multichannel infrared receiver performance