Fine delay estimation with time integrating correlators
B. V. K. Vijaya Kumar, David Casasent, and Anastasios Goutzoulis
The bias and variance of the delay estimation from a time integrating acoustooptic correlator are considered.A parabolic interpolation of the sampled correlation output is included. General expressions for the biasand variance are derived that include the effect of finite detector element area. Quantitative data are pro-vided for the case of an exponential autocorrelation signal model as functions of various system, signal, andnoise parameters. These results show that the estimate is biased. They also provide guidelines on the delayestimation variance obtainable from a given system and on the design of a system to achieve a desired esti-mation accuracy.
1. IntroductionAcoustooptic devices are being suggested for use in
many new signal processing architectures and for a widevariety of applications. 1 This has occurred because ofthe commercial availability, good reliability, and per-formance of new acoustooptic (AO) cells.1"2 One mostattractive AO signal processing architecture is the timeintegrating correlator.3 ' 4 Such a system is attractivebecause of the large processing gain it provides. Thissame system can also function as a spectrum analyzerwhen the chirp-Z transform algorithm is used.5 Manytime integrating AO processors have been fabricated,and some quantitative data have been published6 onthese systems.
In the correlator version of this system, the locationof the correlation peak defines the time delay betweenthe received and reference signals. In the Fouriertransform version of the system, the locations of theoutput peaks correspond to the input frequenciespresent. However, noise, the random nature of thedata, and the finite size of the individual detector ele-ments degrade the estimation of the location of theoutput correlation or Fourier transform peak(s). In thispaper, we consider the delay estimation accuracy of atime integrating AO correlator when finite area outputdetector elements are used. Our results will later beextended to the case of a time integrating spectrumanalyzer. Very little statistical analysis of AO proces-sors has been published. The only available work has
The authors are with Carnegie-Mellon University, Department ofElectrical Engineering, Pittsburgh, Pennsylvania 15213.
considered spectral analysis systems and has includedeffects such as detector noise and dynamic range.7 Noprior work has considered the parameter estimationaccuracy of optical correlators, effects such as the finitearea of the detector elements and the use of interpola-tion to obtain subpixel parameter estimation accuracy.This paper represents the first such attempt at such ananalysis. Although many error sources exist in an AOprocessor, we consider only the finite detector elementsize, and we show how this effect can be overcome byproper postprocessing. We thus expect that our resultswill be overly optimistic, since only this one error sourceis included. A full analysis of all error sources and theireffect on the parameter estimation accuracy of thesystem are beyond the scope of this initial paper.
In this paper, we provide the first description of thebias and variance of the delay estimation obtainablefrom a time integrating AO correlator. The delay isgenerally estimated by locating the maximum of thecorrelation output. This process provides a delay ac-curacy of D (the size of an output photodetector ele-ment). Better delay estimates are obtainable by usinga nonparametric (the probability density functions ofthe data are not required) second-order parabolic fit orinterpolation of the correlation function.8 In ouranalysis, we include such a postprocessing interpolationtechnique since its use is widely accepted.8-10 Such atechnique allows subpixel delay estimation and/or theuse of fewer detectors to obtain the same delay esti-mation accuracy.
In our analysis, we also include the practical issue ofthe effect of the finite size D of the individual detectorelements in the correlation detector plane. Thisproblem is considerably different from the case that haspreviously been analyzed in the digital literature. 9. 10 Indigital correlators, the detector size D does not arisesince discrete sampled data points are obtained at boththe input and output. More specifically, the outputs
of a digital correlator are sampled versions of continuouscorrelation functions. But the outputs from a timeintegrating AO correlator are continuous outputs dis-cretely sampled by area integration (over nonover-lapping detector areas). Because of this fundamentaldifference, the estimation results of digital correlationscannot be simply transferred to the time integrating AOcorrelator system.
Our major motivation for this analysis is to providea statistical analysis of the effects of various errorsources or system parameters that affect the perfor-mance and design of acoustooptic signal processors.Our initial concern in this paper is the effect of thenonzero detector size D and to determine how the choiceof parameters such as D is affected by various signalparameters (bandwidth, input SNR, time-bandwidthproduct) and system parameters (integration time T1and aperture time of the AO cell TA).
In Sec. II, we review the time integrating AO corre-lator, nonparametric parabolic interpolation, and wedefine the problem to be addressed and the notation tobe used. The statistics of the correlation output arethen derived in Sec. III. In Sec. IV, general expressionsfor the mean and variance of our nonparametric delayestimator are then derived. Quantitative data are thenobtained for an exponential signal autocorrelationmodel in Sec. V, where the effects of various signal andsystem parameters on the delay estimation possible areobtained and discussed. A summary and our conclu-sions are then advanced in Sec. VI.
II. Delay EstimationA simplified schematic of the time integrating AO
correlator is shown in Fig. 1. We denote the referencesignal by h(t) and the received signal by g(t). In thissystem, g(t) is fed to an LED or laser diode input pointmodulator. We describe the light distribution leavingPl and incident on P2 byg(t). Lens L, collimates theP, output and uniformly illuminates P2 with g(t). TheAO cell atP2 fed with h(t) has a transmittance h(t + ),where T = XV (where x is the spatial coordinate of P2and v is the velocity of sound in the AO material). Fornotational simplicity, we assume unit velocity v so thatthe space x and time t coordinates can be freely inter-changed in our equations. The light distributionleaving P2 is thus g(t)h(t + T). This pattern is imagedonto P4 through a schlieren filter at P3. The light dis-tribution incident on P4 is thus the correct filteredversion of g(t)h(t + ). The P3 filtering and the fullinput signal modulation applied to P, and P2 providethe desired form for g(t)h)(t + r) as described in detailelsewhere.1
The linear detector array at P4 provides the time in-tegration over T1 of this signal. Including the finitearea D of the detector elements, we write the P4 outputas
1 (i+1/2)D T1 /2C (i- d rl JT g(t)h(t + )dt (1)
where i = -N/2,. .. 0, .. , +N/2 is the index for theN + I detectors with the central detector denoted by i
P1 Li P2 L2 P3 L3 P4
3 ---- -- o B ~ ~ ~ ~ v o i~ ~
9(t)
LaserDiode
h(t)
AO Cell SchlierenImaging
Detector
ArrayFig. 1. Schematic diagram of a time integrating acoustooptic cor-
relator.
= 0. As shown in Eq. (1), the correlation output con-sists of a finite number (N + 1) of samples. In Eq. (1),we assume that all detectors are identical with a uniformsensitivity and with no gaps between detectors. Weconsider the finite detector size and its area integrationas the detector effect of major initial concern. Since weare estimating the correlation peak's location, an errorin this estimate should mainly be due to this detectoreffect. Detector issues, such as detector noise,7 weretreated elsewhere, and other AO correlator error sourceeffects will be the subject of future work. Our prelim-inary simulation results have indicated that nonuniformdetector response effects and spatially different re-sponses between detectors are of less concern than thefinite detector size issue (especially when achievingsubpixel parameter estimation accuracy is our objec-tive). The normalization factors (1D) and (1TI) areincluded to simplify the results. They do not affect thesystem's delay estimation. In Eq. (1), for simplicity,we ignore the effects of the output integration on thenonzero dc values of various terms. The time inte-grating AO correlator can provide the correlation onlyover a limited range TA (the aperture time of the AOcell). We assume 1:1 imaging from P2 to P4 in Fig. 1 andthus relate the number of detectors (N + 1) and thespatial extent (N + 1)D of the output plane to time(through TA) by
TA = (N + 1)D. (2)
No fundamental problem arises with the apparentlydifferent units on both sides of Eq. (2) due to assumingvelocity equal to unity. In practice, T, >> TA, and thusthe attractive feature of the time integrating correlatoris the large integration time-bandwidth product it canprovide.
As our signals, we use
h(t) = s(t) g(t) = s(t + to) + n(t). (3)
In a radar ranging application,1 ' s (t) is the transmittedreference signal. It is reflected off the target to producea received signal that is a delayed version s (t + to) ofh(t), where to is the delay to be estimated, plus additivenoise n(t). The delay is estimated by forming thecross-correlation of g(t) and h(t) as in Eq. (1). We thensearch the correlation plane P4 for the location of thecross-correlation peak.
To provide subpixel delay estimation accuracy (tobetter than D), we first search the detector outputs c(i)
to determine the i* on which the peak of c(r) occurs.We exclude from our consideration the rare situationwhere the correlation peak occurs at the edges of twoadjacent detectors. In this case, adjacent c(i) valueswill be similar thus indicating that the peak has oc-curred at the edges of two detectors. We also measurethe c (i* - 1) and c (i* + 1) outputs at detector elementsadjacent to the correlation peak. Without loss of gen-erality, we assume i* = 0. This is also appropriate sinceour concern is to estimate to to an accuracy that is betterthan D. As our nonparametric estimate of to, we ap-proximate the output correlation by the second-orderpolynomial8 :
C(r) = a + bi- + cr 2 , -D < < +D,
where a= c (0),b = [c(l) - c(-1)]/2D, andc = [c(l) + c(-1) - 2c(0)]/2D2 .
To determine to from the three area-sampled corre-lation plane values c(0), c(-1), and c(+1), we set thederivative of Eq. (4) with respect to r equal to zero. Ourto estimate is then
= _ b [c(l)-c(-1)]/2to - = D 5
2c [2c(0) - c(1) - (-1)]
We use this estimator in Eq. (5), because it does notrequire any a priori knowledge of the forms of under-lying probability density functions.10
Ill. Correlation Pattern StatisticsTo evaluate statistically the delay estimation, we
require the expected value and variance of to and thusthe statistics of the correlation output c (i). From Eq.(1), we can assume that the probability density functionof c (i) is a Gaussian random variable. This follows fromcentral limit theorem12 arguments and the fact thateach c (i) sample is the result of the integration ofg(t)h(t + -) over many samples -TI/2 < t < + TI/2.The statistics of c (i) can now be completely specifiedby only its mean and variance.
To complete our statistical model, we assume thats(t) and n(t) are sample realizations from zero-mean,stationary, Gaussian, random processes with autocor-relation functions R () and Rn (r). The expected valueof (i) is then
i 1 (i+1/2)DEjc(i) = R,(,r - to)dr, (6)
D (i- 1/2)D
where the fact that s(t) and n(t) are independent pro-cesses was used.
To obtain the variance var~c (i)}, we first compute thesecond moments of the correlation output:
1 (i+ 1/2)D ((+ 1/2)DElc(i)c(j)l r d r
J(i-1/2)D f(j-1/2)D
1 r T/2. Ti/2X dT'- I dt r dt'T T/2 JT./2X E[s(t + r)s(t' + T')n(t)n(t')
+ s(t + i-)s(t + ')s(t + to)s(t + to)]. (7)
Since s(t) is assumed to be a sample realization froma Gaussian process, the fourth-order moment in Eq. (7)
can be written as the sum of three products of second-order moments.12 Using this and the average value ofc(i) given in Eq. (6), the covariance of c(i) and c(7) canbe shown to be
1 (i+1/2)D f (+1/2)Dcovc(i),c() = - dT I
D2 (i-1/2)D S (j-1/2)D
T -
X d T (TI -It )dt
X [R (t + T - r')R.(t) + R8 (t + T - r')Rs(t)
+ R,(t + r - to)R,(t - T' + to)]. (8)
The covariance in Eq. (8) contains three terms. For D<< Ti (the case when a time integrating correlator isused to its full advantage), the contribution due to thethird term in Eq. (8) is negligible. This follows becauseof the sharp peaked functional form for the correlationfunction Rs (r) and because both factors in terms 1 and2 track as changes, whereas the two factors in term 3do not track (rather they diverge) as r changes. Acomplete analysis shows that only the small fractionDITI of the t values contribute to the integral in thethird term.
The covariance can thus be well approximated by
covc(i),c(j)} - - (TI - ItI)[R.(t) + R(t)]
[SI ~-D D RS(t + + iD -jD)dTI dt. (9)
From Eq. (9), the variance of the ith correlation outputis thus
varlc(i) = ST (T - ItI)[R.(t) + R(t)]
(10)[IfD (D -I rI)R 8(t + T)d-I dt.
Since Eq. (10) is independent of i, the correlation out-puts from all detectors i have identical variances. Wecan also show from (9) that the covariance between c (i)and c(j) is small when i Fd j. To see this, recall thatRn(t) and Rs(t) are sharply peaked around t = 0.Substituting t = 0 into the inner integral in (9) and re-calling that the inner integral term Rs ( + iD - jD) iscentered around = (D - iD), we note that for i = j thepeak occurs in the middle of the integration interval,whereas for i j, the peak occurs toward the edge of theintegration interval [where the triangular weighting (D- I-r 1) reduces its contribution].
From this, we see that the covariance of [c(i),c(j)] issmaller than the variance of the correlation coefficients.We have numerically verified this by evaluating thevariances and the covariances for an exponential auto-correlation. In this case, we found that the covarianceswere <10% of the variances for large integration times.Since the covariances are small compared with thevariances and since we have proved that the c(i) areGaussian distributed, we can assume that the correla-tion outputs c(i) are statistically independent. Theresults obtained can be summarized as below:
(1) The correlation outputs c(i) can be modeled asindependent Gaussian random variables;
(2) the mean values of the c(i) are given by Eq. (6);and
(3) the variance of c(i) is independent of i and isgiven in Eq. (10).
We also note from Eq. (10) that the variance of c(i)is independent of the shift to. Thus, under the ap-proximation of a sharply peaked correlation function,the shift to affects only the mean value of the randomvariables c(i). We can also easily see from Eq. (10) thatlarger amounts of input noise cause larger variances inthe correlation outputs, as is expected.
IV. Statistics of the Delay EstimatorFrom Eq. (5), we see that the delay estimator to de-
pends on c(-1), c(0), and c(1). We now use the resultsobtained in Sec. III to derive the mean and variance ofto. These results help us to evaluate the to estimatorfor various detector sizes. The estimate to can be re-written as
Numt =D (11)
Den
where Num = [c(1) - c(-1)]/2 and Den = [2c(0) - c(1)-c(-1)]. Since Num and Den are linear combinationsof Gaussian random variables, they are also Gaussian.Their average values and variances are easily shown tobe
The mean value of to can be seen from Eq. (13) todepend on the detector size D, the integration time TI,and the correlation functions R8 (r) and Rn (T) of thesignal and noise. We see from Eq. (13) that the meanvalue of to is not expected to equal to. This followsbecause to is present in each expectation term in Eq.(13), and there are many such terms in Eq. (13). Forlarge time-bandwidth signals (the case with which weare concerned), the variance of the correlation outputis small and the second term in Eq. (13) is found to bemuch smaller than the first term. In this case, the ex-pected value of to simplifies to
In Eq. (12c), we used the facts that the last term waszero (since the correlation output samples are inde-pendent) and that the first two terms were equal (sinceall correlation variances are equal or independent of i).By var{c(i)} in Eq. (12c), we refer to the variance of anyc(i) sample. (These are all equal as we showed in Sec.III.) Similar results were used in obtaining Eq.(12e).
Since Num and Den are both Gaussian and from Eq.(12e) are uncorrelated, they are statistically indepen-dent. This simplifies evaluation of the mean andvariance of to. We use a Taylor series expansion' 3 ofthe quotient of two random variables (in our caseNum/Den) around (E[Num1/E{Denj) to determine themean of our estimate to:
There is still a difference between the mean Elto} of theestimate to and the true to value. This results in a biasfor the estimator, and we thus expect errors in the ac-curacy of the delay estimate. The bias is best expressedin terms of the fractional bias B:
(15)B to-ElioD
which is normalized with respect to D. This allows usto evaluate more properly the bias for different Dvalues.
Lest confusion arise, we note that the time integratingAO correlator's output contains a bias, but this is avoltage bias present on all output detector elements.The bias we speak of in Eq. (13) has the units of secondsrather than volts, and it affects estimation accuracy ,whereas the other voltage bias does not. If the fullmodulation formulation of the time integrating AOprocessor were included, the effects of the voltage biason our estimator in Eq. (5) can easily be seen to cancel.To simplify this initial treatment, we thus ignore the fulldescription of the AO cell's modulation characteristicsand the effects of voltage bias on the detected out-puts.
Using the general procedure in Ref. 13 for evaluatingthe variance of the quotient of two independent vari-ables, we easily find the variance of to to be
A. Bias of Delay Estimator toWe first evaluate the mean of the correlation output
for our exponential signal autocorrelation model, thenthe mean of our estimator to, and finally the bias B ofto. We consider only the case when to occurs between
= 0 (the center of a detector) and X = D/2 (the edge ofa detector). Similar results can be obtained for -D/2< - < 0 by the symmetry of the problem. For our signalmodel in Eq. (17), we find the mean values of c(0),c(-1), and c(+1) to be
I -D/2Ejc(-1) = - X exp(-aIr - toldT)
D f1 3D/2
FRACTIONAL SHIFT %
Fig. 2. Fractional bias in to as a function of the fractional shift (to/D)for aD = 0.1, 1.0, and 10.0.
varlio = D2 var [ND-1De I
D2. E2
Numr [varlNumlE2 {Den} |E21Num-
+ var{Den} 2 cov{NumDenj
E21Den} ENum}E{Den}
=D2. varlc(i)j2[2Ejc(0)} - Elc(l)} - Elc(-1)}]
2
+ D2
. 6 var{c(i)j[Ec(1)} -E{c(-1)f2 (16
4[2Ejc(0)} - Elc(l)} - Elc(-1)}]4
From Eqs. (13) and (14), we found that the estimatorto is biased. From Eq. (16), we see that the dependenceof the variance of the estimator to as a function of D andother parameters is not a simple expression. To pursuefurther the statistics of our to estimator and to quantifyour results, we now evaluate these general expressionsfor a specific signal and noise model in the next sec-tion.
V. Quantitative Results and DiscussionTo determine'the behavior of our estimator, we now
evaluate our general expression for the mean and vari-ance for a specific signal and noise model. We then plotthe bias and variance of this estimator as a function ofdifferent signal, noise, and system parameters, and wediscuss our results.
As our signal model, we use an exponential signalcorrelation function
RS(r) = exp(-a I), (17)
where a is the 3-dB bandwidth of the signal. Thismodel is useful14 for signals with low-pass spectra, andit allows us to determine conveniently the effects ofsignal bandwidth on our estimator by simply changinga in our model. In Eq. (17), the signal energy R, (0) isnormalized to unity. This is valid since we are con-cerned with the case when the system's noise can beignored with respect to the correlation plane noise.
- exp(-ato) texp(-aD/ 2 ) -exp(-3aD/2)]= xp-ao)IaD IJ
1 r3D/2Elc(l)} = - exp(-aIr - tol)dr
D fJD/2
[exp(-aD/2) - exp(-3aD/2)1= exp(ato) LaD I
(18a)
(18b)
1 fD1)2Elc(O) = - I exp(-aIr - tol)dr
D ex(-D/2
2 exp(-aD/2) [exp(-ato) + exp(ato)I. (18c)aD aD
From Eq. (18), we see that the means of the correla-tion outputs are independent of the additive input noiselevel and that they depend only on the products aD andato. Substituting Eq. (18) into Eq. (14), we obtain themean of our estimator to:
This also depends only on aD, at 0, and D. To obtainphysical insight into these terms, we recall that l/a isa measure of the correlation peak width. Thus aD isthe number of correlation peaks that can fit within onedetector of width D. The product ato is a normalizedshift or delay (normalized by the reciprocal of the sig-nal's bandwidth). Equations (19) and (14) include onlythe first term in Eq. (13). Both terms in the exact E{to}expression were numerically evaluated for varioussystem and signal parameters. In all cases, the secondterm in Eq. (13) was found to be negligible with respectto the first term. Thus our result in Eq. (19) is quiteaccurate.
Substituting Eq. (19) into Eq. (15), we obtain an ex-pression for the fractional bias B(ato,aD) of our to es-timator as a function of the parameters ato (normalizeddelay) and aD (number of correlation peaks possible perdetector element). In Fig. 2, we show B as a functionof the ratio (ato/aD) = tD for three different valuesof aD. The abscissa tID is the location of the corre-lation peak within the detector element of size D.When tID = 0, this corresponds to the peak lying at thecenter of the detector element, wheres tolD = 0.5 cor-responds to the peak lying at the edge of the detector.We can also provide a different and more instructiveinterpretation for aD for the case of our time integrating
Fig. 3. Worst-case fractional bias and the corresponding fractionalshift as a function of aD.
correlator of Fig. 1. In terms of TA (the aperture timeof the AO cell), D = TAI(N + 1) (assuming 1:1 imagingfrom P2 to P4 in Fig. 1), where aTA is the time-band-width product (TBW) of the AO cell (assuming that thesignal's bandwidth is large enough to utilize fully thebandwidth of the AO cell). Thus aD = 1 correspondsto the case in which the number of detectors equals theTBW of the AO cell (i.e., one detector per resolutionelement of the system). The case aD = 0.1 correspondsto very fine sampling of the correlation function (withthe number N + 1 of detectors equal to ten times theTBW of the AO cell) and aD = 10 corresponds to coarsesampling (with the number N + 1 of detectors equal toone-tenth of the TBW of the system).
From Fig. 2, we see that the bias of our estimate in-creases as aD increases. This is logical since a largerdetector size D results in averaging of c () over largerintervals to produce c (i), and this introduces more bias.For small aD S 1.0 values (when the number of detec-tors is equal to or greater than the TBW of the AO cell),the bias is small (0.1) and will not present a major1error in our to estimator. However, a large (0.35)maximum bias can result when fewer detectors of largersize D are used. The fractional bias is seen to be neg-ligible when the correlation peak occurs in the center(to/D = 0) or at the edge (toID = 0.5) of a detector ele-ment. However, since the true to value can lie anywherewithin a detector element, we must consider theworst-case bias at the peaks of the curves in Fig. 2.
To better convey the fractional bias that will resultfor different aD values, we computed the maximumworst-case bias for all aD values from 0.1 to 10.0 andplotted these vs aD in Fig. 3. We also include in Fig. 3the location to/D (within a detector element) at whichthe worst-case bias occurs. From Fig. 3, we see that foraD 1 (adequate detector sampling) that the maxi-mum bias is small (0.1) and that this worst-case valueoccurs only when the peak lies about two-thirds of theway between the center and the edge of the detectorelement. As aD is increased (fewer detectors and largerarea detectors), the worst-case bias becomes quite large(0.3 for aD = 10), and these worst-case values occurwhen the peak lies closer to the edge of the detector.
The location of the correlation peak within a detectorelement is not known a priori. Thus curve (1) in Fig.
3 indicates the bias to be expected from a given system.If the signal bandwidth a is known, these data also in-dicate the maximum detector size to use to obtain adesired bias for our to estimator. If D is fixed, thesedata indicate the bias to be expected for a signal of agiven bandwidth. Since o in Eq. (5) underestimatesto when to is positive and overestimates it when to isnegative, we cannot predict the nature of the bias apriori, and thus we cannot correct for it or cancel it.
B. Variance of the Delay Estimator toTo determine the variation to be expected in our es-
timator, we first evaluate the variance of the correlationoutput c(r) for our exponential signal model in Eq. (17).We then evaluate the variance of our o estimator andplot it as a function of various signal and system pa-rameters.
For our signal model in Eq. (17), varfc(i)j is inde-pendent of i and to as seen before, and substituting Eq.(17) into Eq. (10) yields
Tl
varlc(i)) = 3' (TI - I t) [exp(-a I t )
+ Rn (t)] IfD (D- I r ) exp(-a I + t )dr]dt. (20)
To complete the evaluation of varic(i)}, we assume thatthe noise n (t) is a white-noise process with an autocor-relation function Rn(-r) = No3(r). The use of whitenoise is appropriate since a correlator is optimal forsignal detection only in the presence of white noise. InRn (r), the power spectral density of the noise processis No, and the noise energy can be written as aNo (sincethe signal is of bandwidth a and the noise at the outputof the correlation is of the same bandwidth). Since thesignal in Eq. (17) is of unit energy, the input SNR to thecorrelator is SNRi = 1/aNo.
For this noise model, we evaluate Eq. (20) and find
vrCi =DTI X T. T.) X-D D )
X exp[-a(It + rI + ItI)]dT
+ N-D D III exp(-aI r|)drDTJ-D D
2No + exp(-aD) 1 1aDTj [ aD aD
DT, J-T, .
X J ( )exp[-a( t + r I + I t )]dr
2No [ 1 exp(-aD)l
aDTJ aD aD I1 [( 6 6
+ D [4 - + exp(-aD) 2 +-Ia2DT aD aD/I
4a +(a 2{4 exp(-aD) + - [exp(-aD) - 1]
+ 2aN0 1 - 1 + exp(-aD)1} (21)
Many straightforward, but tedious, steps are omittedin the evaluation of the first integral in Eq. (21). We
Fig. 4. Variance of the correlation output as a function of aD for aTI
= 1000 and noise energies of aNo = 0.1, 1.0, and 10.0.
see from Eq. (21) that the variance of the correlationoutputs is a function of three different products,namely, aT, (the integration time-bandwidth product),aNo (the noise energy or the inverse of the input SNR),and aD (the number of correlation peaks that can beaccommodated in a detector of size D).
We now consider how varic(i)} varies with these threefactors. From Eq. (21), we see that the variance is in-versely proportional to aT, as expected. To determineits variation with aD and aNo, we plot Eq. (21) vs aD forthree noise energies (aNo = 0.1 or SNRi = 10, aN = 1or SNRj = 1, and aNo = 10 or SNRi = 0.1) in Fig. 4. Anintegration time-bandwidth product of 1000 = aT, waschosen for this case and others. From the data in Fig.4, we see that the variance improves (becomes smaller)as SNRj increases (as expected). We also note that for
. 0_ 10
0 1z
0.1
aD.0.1
1d-e02: 13: 10
(.9c 0.1
0Z 0.01
0.001
0.01 -
Q2 OAto /D
aD 1 (sufficient correlation plane sampling), thevariance is constant. As aD is increased above unity,the variance monotonically decreases. This occursbecause an increasing aD corresponds to larger detec-tors and fewer detectors, and thus we are averaging thecorrelation over a larger area resulting in less variancein the correlation outputs.
Let us now consider the variance of our to estimator.Substituting Eqs. (18) and (21) into Eq. (16) yields aformidable expression for the varlio0 . It will depend onD2, aT,, aNo, aD, and ato since Eqs. (18) and (21) do.Although varlc(i)l is independent of to, varlio0 will de-pend on to, since Elc(i)} does. To include properlydifferent detector sizes, we normalize the variance withrespect to D2, and we consider only the fractional vari-ance (varlto)/D2 of to, which we refer to as Norm[varto}]for simplicity. To determine how Norm[var{to}] is af-fected by different signal and system parameters, wefirst note that it is inversely proportional to aT, as ex-pected. We thus fix aTI = 1000 as before in consideringthe dependence of Norm[var{lo}] on aD, aNo, and ato.In Fig. 5, we plot Norm[varlio}] vs (to/D) (the normal-ized location of the peak within a detector) for aD = 0.1,1.0, and 10 for three different SNRj values (aNo = 0.1,1.0, and 10.0).
From Fig. 5, we see that increasing the noise energyaNo increases the normalized variance of to as expected.For larger aD values, we see that the normalized vari-ance is relatively constant over a larger to/D interval.Thus it will be easier to predict or bound the accuraciesof our estimator for larger aD values, but the bias (Fig.3) of our estimator will be large, and thus its accuracywill be worse for such cases. From Fig. 5, we also notethat the normalized variance increases as the peak oc-curs closer to the edge of the detector element. Recall
aD.1.0
1:aNd0.i2aN, 13:aN. 10
Q001
to/ D
1:&%:.1
2: 1
3:8H.-Os
0.2 0.4
t o /D
(a) (b) (c)
Fig. 5. Normalized variance of o as a function of the fractional shift (to/D) for aT = 1000 and three noise energies aNo = 0.1, 1.0, and 10.0for (a) aD = 0.1, (b) aD = 1.0, and (c) aD = 10.0.
Fig. 6. Normalized variance of to as a function of aD for aT = 1000and aNo = 0.1 for (to/D) = 0.0 and (toID) = 0.10.
from Sec. II that we excluded the case of to occurring atthe edge of a detector element. From Fig. 2, we recallthat the bias is zero when the peak occurs at either thecenter or edge of the detector. Thus our estimator willexhibit the minimum squared error when the true peakis at the center of a detector element. This case cannotbe insured to occur. Comparing Fig. 5, we see that thenormalized variance is small and similar for aD = 1.0and 10.0 and larger for aD = 0.1. Thus, as the bias ofthe estimator is reduced (decreasing aD), the normal-ized variance of the estimator becomes worse. Thechoice aD = 0.1 appears to be the best compromise.'However, the normalized variance values for the threeaD values suggest that a different optimum aD existsthat minimizes the variance.
To examine this, we computed Norm[varlo}] for all,aD values for t/D = 0.0 (peak at the center of a detectorelement) and t/D = 0.1 for aN = 0.1 and aT = 1000.The results in Fig. 6 show that Norm[varto}] decreasesmonotonically and is a minimum at aD 3.0 afterwhich it increases monotonically. From this, we seethat there is an optimum detector size D or aD factorfor which the normalized variance is minimized. Notethat the normalized variance is very small (<0.001).We also found that this same aD _ 3.0 choice is opti-mum for any SNRi and any ato and for all t/D and aT,values.
C. Case StudiesThere are many uses for the general expressions in
Sec. IV and the specific number data in Figs. 2-6. Withthese data, one can predict the errors being made andthe estimation accuracy (its bias and variance) for agiven system for different types of signal. One can alsouse these data in the design of a system to achieve adesired estimation performance for given signal pa-rameters. Many trade offs exist in any given problem.For example, choosing D = 3/a to minimize vartiol mayresult in an unrealistic or unavailable detector size, AO
cell aperture, or an optical system (between P 2 and P4in Fig. 1) of excessive length. Also, if aD = 3.0 is se-lected, the worst-case fractional bias will be 0.2, and thismay be excessive for certain applications. If thebandwidth of the signal is not known a priori, D canonly be bounded to a range of values. This case willarise in many passive signal processing applications.
As a specific example, let us consider the case whena = 20 MHz and T, = 50 gtsec = TA or aT, = 1000 andaN0 = 0.1 (SNR = 10). We consider the choice of Dso that the worst-case bias is <10 nsec, and the standarddeviation for to/D between 0.0 and 0.1 is <10 nsec. Forour specific system, its time resolution is 50 nsec, andthus our bias and variance requirements correspond to20% of a pixel. Normalizing with respect to D, we haveB* 0.01/D, standard deviation <0.01/D, and aD =20D, where the detector size D is given in microseconds.(This is easily converted to distance by using the ve-locity of sound in the AO cell and the image magnifi-cation in Fig. 1.)
We begin by choosing some D value, computing aD,determining the worst-case B for this D value from Fig.3, and verifying that it is less than our required B *. Ifwe choose D = 0.01, then aD = 0.2, and from Fig. 3 wefind B < 0.1, which satisfies our B* < 0.01/D = 1 goal.Since the maximum D is larger than 0.01, we next tryD = 0.1. For this case, aD = 2.0 and B > 0.1, which doesnot satisfy B* < 0.1. Trying D = 0.05 correspondingto aD = 1.0, we find B < 0.2 = B*. Continuing our it-erations, we find that we require D 0.08 Atsec to satisfythe bias requirement.
Next we consider the effect of D on the variance. ForD = 0.01, aD = 0.2, and from Fig. 6 we see var < 0.01 andthus standard deviation < 0.1, which is easily less thanour required 0.01/D = 1 goal. For D = 0.1, aD = 2, var< 0.001, and the standard deviation again satisfies our0.01/D = 0.1 goal. As D is increased, var decreases.Using the maximum D = 0.08 that satisfies B, we seethat aD = 1.6 easily satisfies our variance requirementand provides nearly the minimum standard deviation.For the given problem, we thus select D = 0.08 busec.
VI. Conclusion
In this paper, we considered the effect of the finitesize D of the detector elements in a time integrating AOcorrelator. The effect of discrete area-sampled corre-lation outputs on the delay estimation was analyzed fora nonparametric interpolation estimator to providesubpixel delay estimates. We found the estimator tobe biased and derived general expressions for its biasand variance.
We found that the bias increased as the detector sizeD was increased and that the bias was independent ofthe noise. For an exponential signal correlation modeland white noise, the worst-case bias occurred at t/D 0.35. For aD < 1, the bias was small (0.1) and forlarger aD (the number of correlation peaks that can beaccommodated in D), the bias was larger (0.3 for aD =10). We also found that the variance decreased as aT,(the integration time-bandwidth product) increasedand that it was a minimum when the true delay occurred
at the center of the detector. The most interesting re-sult observed for our exponential signal model was thatthe minimum variance occurred for aD 3.0 for allaT,, to/D, and input SNR values.
Our results can be used to determine the accuracy ofthe delay estimation for a given AO time integratingcorrelation application or in the design of the system toprovide a desired delay estimation accuracy. Thisinitial parameter estimation performance of an AOsignal processor has considered only the effect of thefinite detector element size D. We thus expect our re-sults to be quite optimistic since the effect of the myriadof other AO system error sources were not included inour initial analysis. When envelope detection of theoutput pattern is required (prior to time integration),we expect an even more complex analysis, since theGaussian nature of the process is no longer a valid as-sumption.
The support of the National Science Foundation(grant ECS-8114344) for this research is gratefully ac-knowledged.
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Interested persons shouldCentre Drive, Foster City
