Cramer-Rao Lower Bounds for Estimation of Doppler Frequency in Emitter Location Systems

Transcription

1 Cramer-Rao Lower Bounds for Estimation of Doppler Frequency in Emitter Location Systems J. Andrew Johnson and Mar L. Fowler Department of Electrical and Computer Engineering State University of New Yor at Binghamton Binghamton, NY Abstract: This paper derives Cramer-Rao bounds on estimates of the Doppler-shifted frequency of a coherent pulse-train intercepted at a single moving antenna. Such estimates are used to locate the emitter that transmitted the pulse train. Coherency from pulse to pulse allows much better frequency accuracy and is considered to be necessary to support accurate emitter location. Although algorithms for estimating the Doppler-shifted frequency of a coherent pulse train have been proposed, previously no results were available for the Cramer-Rao lower bound () for frequency estimation from a coherent pulse train. This paper derives the for estimating the Doppler-shifted frequency of a coherent pulse train as well as for a non-coherent pulse train; a comparison of these two cases is made and the bound is compared to previously published accuracy results. It is shown that a general rule of thumb is that the frequency for coherent pulse trains depends inversely on pulse on-time, number of pulses, variance of pulse times, and the product of signal-to-noise-ratio and sampling frequency SNR F s ; pulse shape and modulation have virtually no impact on the frequency accuracy. For the case that the K intercepted pulses are equally spaced by the pulse repetition interval (PRI), then the decreases as /PRI and as /K 3. It is also shown that roughly the gain in coherent accuracy vs. the non-coherent case is K times the ratio of pulse on-time to PRI; since PRI is typically much larger than pulse on-time, the coherent scenario allows much better frequency estimation accuracy. Index Terms: Emitter Location, Frequency Estimation, Doppler Shift, Cramer-Rao Bound Corresponding Author: Mar Fowler Department of Electrical and Computer Engineering State University of New Yor at Binghamton P. O. Box 6 Binghamton, NY 39-6 Phone: Fax: mfowler@binghamton.edu

2 I. Introduction Passive location of a stationary emitter with unnown frequency from a single moving platform is an important problem that has been investigated recently [] []. The importance of single-platform methods stems from the desire to provide accurate location of emitters from a single aircraft for tactical and/or strategic uses: single aircraft ability provides flexibility over multi-platform methods [9] and therefore there has been much interest in the development and testing of single-platform methods [4], [5]. All the single-platform approaches that have been proposed consist of measuring, at instants within some interval of time, one or more signal features that depend on the emitter s location according to some signal model (i.e., measurement model), and then processing them to estimate the emitter location. For example, the measured signal feature could be the bearing to the emitter [3], [9], the Doppler-shifted frequency of the signal [], [6], [8], the interferometer phase between two spaced antennas on a single platform [], the times-of-arrival of the pulses, etc., or combinations of these [], [7]. Regardless of the feature(s) exploited, the general approach is the same: first estimate the signal features (e.g., frequency, angle, etc.) at a set of time instants and then use a signal model that describes the relationship between the measured signal feature(s) and the emitter location, together with some statistical inference technique (e.g., least-squares, maximum lielihood, etc.) to estimate the emitter s location. Thus, to evaluate the location accuracy of an emitter location method it is first necessary to evaluate the accuracy of the feature(s) estimation. This paper focuses on the specific case of measuring the Doppler-shifted frequency of a coherent pulsetrain signal intercepted at a single moving antenna for use in emitter location. Algorithms for estimating the Doppler-shifted frequency of a coherent pulse train have been proposed [6], [], where coherency from pulse to pulse allows much better frequency accuracy and is considered to be necessary to support the subsequent emitter location processing. Despite the publication of these algorithms for coherent frequency estimation, no results have been available for the Cramer-Rao lower bound () for this case. The only accuracy results

3 3 previously available were: simulation results from Becer s algorithm [], experimentally obtained accuracy levels reported by corporations developing actual systems [4], [5], and an approximate accuracy analysis []. This paper derives the for estimating the Doppler-shifted frequency of a coherent pulse train and compares the bound to previously published accuracy results; for comparison, the is also derived for a noncoherent pulse train and a simple formula is derived that characterizes the improvement achievable due to exploiting coherency. II. for Frequency Estimation of Coherent Pulse Train A. Signal Model A complex-valued coherent pulse train from an emitter, consisting of K pulses, can be modeled by K j( ωt+ φ) st () e pt ( T), () where p( t) is a single pulse (possibly complex), ω is the carrier frequency,φ is a phase offset, and the are T called the transmitted pulse times. Let T OT be the total pulse on-time and assume that p(t) is time-limited on the interval [, TOT ] with TOT < T T. The signal is received at a moving platform, with initial range d, and radial velocity v. The (noise-free) received signal is model as d rt () s( αt ), () c where α v / c is the signal dilation coefficient from the Doppler effect, and c is the signal propagation rate. Substituting () into () yields that at the receiver the signal appears as K j ωt j ϕ rt () e e pt ( T), (3)

4 4 where ~ ~ ω αω, φ φ ωd / c, ~ p( t) p( αt) ; the will be called the pulse and ( + ) T d c T /α / T times. Furthermore, under the narrowband approximation [3] the time scale factor α has a negligible effect on the pulse so that p(αt) p(t) so the model becomes K j ωt j ϕ rt () e e pt ( T). (4) The received signal is sampled at the Nyquist interval Δ and the samples are assumed to be perturbed by complex zero-mean white Gaussian noise, wn [ ], with variance, resulting in xn [ ] rn [ ] + wn [ ] K jωδn j φ e e p( Δn T ) + w[ n]. (5) From this received x[n] we wish to estimate the parameter vector θ ω φ. (6) T T T K The ey parameter to be estimated is the Doppler frequencyω, and the remaining unnowns are nuisance parameters that must be factored into the analysis (at least until shown irrelevant). It is recognized that the pulse, p() t, is generally not nown, either; however, the will depend on the form of p() t so numerical results for the will depend on the specific pulse shape assumed to have been intercepted. B. Derivation of the General Case FIM and for Doppler Frequency In preparation for computation of the [], the derivatives of the signal with respect to each unnown are found to be ~ r[ n] jδnr[ n] ω ~ r [ n] jr[ n] φ (7) (8)

5 5 T l K j ωδn j ϕ rn [ ] e e δ [ l] p ( Δn T ) j ωδn j ϕ e e p ( Δn T ), l (9) where dp() t p ( Δn T l ). () dt tδn Tl In order to simplify the expressions in the following derivation, define: S p( n ) S n p( n ) S n p( nδ ) () N N N Δ Δ N n N n N n N N * * C Im p( Δn) p ( Δ n) C Im n p( Δn) p ( Δn) () N n N n B Δ SN N p ( n ) (3) n K K ( T ), (4) K K R T R where S is a measure of the pulse power, S is a measure of the pulse temporal centroid, S is a measure of the pulse temporal spread, C and C are time-frequency cross-coupling (or sew [3]) measures, B is a measure of the pulse bandwidth, R is the average of the pulse times, and R is a measure of the temporal spread of the pulse arrival times. We also define I to be the index set where the th received pulse samples are non-zero. The elements of the Fisher Information Matrix (FIM), J, under the signal plus complex WGN assumption are given by

6 6 J ωω * Re r [ n ] r [ n ] n ω ω K ( Δn) p( Δn T ) n I K N ( Δ n+ T ) p( Δn) n (5) N KS KR S KR S, Δ + Δ + where we have assumed that each pulse has N non-zero samples. Continuing in the same fashion gives * J Re r [ n φφ ] r [ n ] n φ φ (6) K N p( Δ n) KNS, n and and J ωφ * Re r [ n ] r [ n ] n ω φ K Δn p( Δn T ) n I Δ + [ KNS NKR S ] (7), * J T Re r [ n ] r [ n ω ] n ω T Δ Δ Δ * Re j n p ( n T ) p ( n T ) n I N Im ( n T ) p ( ) * n p ( n ) n Δ + Δ Δ N Δ C + T C, (8) and

7 7 * J Re r [ n ] r φt [ n ] n φ T * Re jp ( Δn T ) p ( Δn T ) (9) n I N C, and J * Re r [ n ] r [ n ] () TT n T T NS B, and J * Re r [ n ] r [ n ] T, TT T n () where the last result is zero because the pulses are non-overlapping. These matrix elements form the FIM given by N J KS KR S KR S KS KR S C T C C T C C T C Δ KS+ KRS KS C C C Δ C+ T C C SB. () Δ C+ TC C SB Δ C+ T K C C SB Δ + Δ + Δ + Δ + Δ + Δ + K This structure shows that in general there is correlation between the estimates of most of the parameters, except between the pulse times. The algorithm proposed in [6] relies on cross-correlation to estimate the pulse times;

8 8 thus, it is clear from the above FIM that the accuracy of those pulse time estimates will impact the overall accuracy of the Doppler estimate. To proceed further, defining the partitions shown in () as A N J T C C, (3) B allows use of standard results for inverting symmetric partitioned matrices to yield the matrix for Doppler frequency and phase: ωφ, N T ( A CB C ) T CC A N SB, (4) where the fact that matrix B is a scalar (S B) times the identity has been used. Evaluating the form for CC T and substituting into (4) gives with the following definitions ωφ, Δ KS + Δ KR S + KRS Δ KS + KRS N Δ KS + KRS KS S S ( C / S B) S S ( C C / S B) S S ( C / S B). (6) o, (5) Finding the, element of (5) gives the for the Doppler frequency estimate. Using Cramer s rule gives ω S NK S S R S R S S R S ( Δ + Δ + ) ( Δ + ) NKS [( S / S ) ( S / S )] R R ( Δ + ) NK SNR ( C / S B) D R ( )( Δ + ), (7) where SNR S / and

9 9 K D ( S / S ) ( S / S ) R R R ( T R ). (8) K Note that D is a mean-normalized measure of the pulse duration and R is a mean-normalized version of the time spread of the pulse times. These mean-normalized measures ensure that the choice of time origin has no effect on the for the Doppler estimate. C. Special Cases for the FIM and To allow some further insight into this result some special cases are now considered. Note that the values in the upper-right and lower-left corners of the FIM in () depend on the quantities C and C, which are shown in () to be the imaginary part of an expression. Thus, if the pulse p(t) is such that the quantity inside the Im{} in () is purely real then many terms in () become zero and yields a bloc diagonal form given by N J Δ KS+ KRS KS SB. (9) SB SB Δ KS + Δ KRS+ KRS Δ KS+ KRS This bloc diagonal structure gives a simple form and shows that for this special case the estimation of the frequency and phase is independent of the estimation of the pulse times. A sufficient (but not necessary) condition to get the FIM as in (9) is that p(t) be purely real valued, which will mae both C and C in () equal to zero. This holds for a radar pulse train with no phase/frequency modulation on the pulses. Because of the bloc diagonal structure in (9), the for Doppler frequency can be found by inverting the matrix in the upper-left corner of (9) and finding the, element of the result. Using Cramer s rule gives

10 ω S NK S S R S R S S R S ( Δ + Δ + ) ( Δ + ) NK SNR D R ( Δ + ), (3) D S / S S / S ) is a measure of pulse duration. where ( ) ( As mentioned above when we defined the mean-normalized measures D and R, the estimation of frequency is expected to be time-shift invariant. That is, the frequency estimation accuracy is not dependent on the time reference on which T,..., T K and the pulse s starting point are based. Therefore, without loss of generality, we can assume that R (the mean of the pulse times) is zero, and that S (the pulse centroid) is also zero. It should be noted that to get S we are now assuming that the pulse p(t) has support over an interval that extends over negative times as well as positive times; thus, the sums defining the S, S, and S in () must now start at some negative integer. Note that this simplification does affect the estimate of φ, the phase offset, but it is of little consequence since it is a nuisance parameter in the first place. These simplifications, together with the earlier simplification that C C, lead to N J { diag Δ KS + KR S, KS, S B,, S B }, (3) which decouples the phase and frequency estimation. The matrix is now easily found to be diag,,,,, (3) NS Δ K ( S / S) + KR K B B where, as previously defined, K is the number of pulses, N is the number of samples per pulse, B is a measure of pulse bandwidth defined in (3), D is a measure of pulse duration defined after (3), R is a measure of spread of the pulse arrival times defined in (4), and Δ is the sampling interval in seconds. Because of the diagonal nature of this matrix, for this special case the lac of nowledge on the nuisance parameters does not impact the frequency estimate. The for frequency estimation for this special case is

11 ω NK SNR Δ S / S + R. (33) ( ( ) ) There are four ey parts contributing to the denominator: the SNR, a measure of pulse duration (D), a measure of pulse spread (R ), and the total number of samples (NK). If there is only one pulse again assuming that p(t) is such that C C and S then the single pulse time is T and (33) becomes. (34) ( ) ω N SNRΔ S / S Similarly, for the case of a single rectangular pulse of amplitude A with no phase/frequency modulation that starts at t with nown T ; then R R and the FIM in () becomes N N n n S S Δ Δ Δ Δ n n N A N ΔS S Δ J, (35) n n N and the Cramer-Rao bound in (3) becomes 6 ω, (36) Δ SNR N N ( ) where SNR A / ; these are exactly the well-nown results for the FIM and for the estimation of frequency and phase for a complex sinusoid in white Gaussian noise [4]. Thus, the results given here for the pulsed case are consistent with nown results for estimating the frequency of a single complex sinusoid. III. for Frequency Estimation of Non-Coherent Pulse Train As a comparison we consider the case of a non-coherent pulse train from an emitter, consisting of where the phase for each pulse is arbitrary (i.e. no specific relationship), K pulses K jωt jφ st () e e pt ( T). (37)

12 Following the same steps as before yields the received data samples xn [ ] rn [ ] + wn [ ] K jωδn jφ e e p( Δn T ) + w[ n]. (38) The unnown phases give K additional parameters to be estimated; the parameter vector is now θ ω φ φ φk T T T K. (39) This section explores this scenario only for the simplified case where the pulse p(t) is such that (i) the expressions in () evaluate to C C and (ii) the expression for S in () evaluates to S. Results for the more general case could be derived following steps similar to those in Section II-B but since the non-coherent method is nown to not provide sufficient accuracy for location, these results are presented only to allow understanding of the impact of coherency and consideration of the special case is sufficient to achieve this insight. Following steps similar to those for the coherent pulse train case gives N N Jωω Δ KS + KRS J NS J TS l J φφ l l ωφl TT NS B (4) The resulting FIM is then J J J J J T TT l lt l lt l l ω φφ φ φ. (4)

13 3 N J ST S ST S ST K S. (4) SB SB SB Δ KS + KRS ST ST STK This is bloc diagonal, so inverting the (K+) (K+) matrix in the upper-left corner of (4), which can be done using again the standard result for partitioned symmetric matrices gives ω N ( ) K KS KRS S T Δ + (43) KR NK SNRΔ S S ( ) / Comparing the non-coherent case result in (43) to the coherent case results in (33) shows that the noncoherent case result lacs the effect of the pulse spread term R that gets added to Δ D to decrease the for the coherent case. In fact, the K-pulse non-coherent result in (43) is simply /K th the bound for a single pulse; thus, the non-coherent case is equivalent to processing each pulse individually to estimate the frequency and then averaging the K resulting pulse-local estimates together. That is, for the non-coherent pulse train there is no ability to exploit any pulse-to-pulse structure as should be expected. The gain in the coherent case comes from the fact that we can exploit pulse-to-pulse structure because of the fixed phase relationship between the pulses.

14 4 IV. Discussion Note that in all the results given above the depended on NΔ ; to provide proper physical insight it is best that one of the Δs in the Δ term be grouped with the N to form the pulse s total on-time NΔ T OT ; note that for a fixed-length pulse the value of NΔ remains fixed regardless of the value of Δ. In addition, the terms S S, D, and D are in units of samples and it maes sense to multiply those by Δ to convert into a / quantity in units of seconds ; this is done by multiplying by inserting additional Δ that is then neutralized by multiplying by the sampling rate F s. The results obtained above (modified as just mentioned) are summarized in Table. Notice also that all the results in Table depend inversely on T OT SNR F s. This dependence shows that doubling the pulse on-time TOT halves the. It may also appear that doubling the sampling rate F s will also halve the but this is liely not achievable as discussed in the following. One should tae care to remember that increasing F s without increasing the front-end bandwidth will mae the noise samples non-white, so to apply these white-noise results would require a commensurate increase in front-end bandwidth, which will increase the noise power but not increase the signal power. The resulting decrease in SNR perfectly counters the increase in Fs so that the product SNR Fs remains the same. Changes in F s will cause changes in Δ D (and other similar terms) but thos changes will be small for all F s above the Nyquist rate. Thus, we see that there should be very little change as a function of sampling rate, which maes sense in terms of sampling theory that says that sampling faster than the Nyquist rate provides no additional information about the signal. The last column in Table lists definitions used in the results; some definitions used in a given row s result may have been stated in a higher row. Also note that all sums over n are shown without a specific range to allow choice of the starting time of pulse p(t), which affects the value of S. The first row considers the case of a single pulse with zero temporal centroid (S ) where the result shows that the is inversely proportional to S /S, which is a classic measure [3] of a zero-centroid pulse s temporal dispersion. For a zero-centroid pulse the quantity S / S is called the Gabor duration or

15 5 RMS duration [3] and here has units of samples; the quantity Δ S / S has units of seconds. The division by S normalizes out the effect of pulse power on this measure. Thus, we see that the for the single pulse case depends on two separate measures of pulse duration: pulse on-time T OT and Gabor duration S / S. The second row also considers the single pulse case but now allows that S. The only difference here is that S /S is replaced by D as defined in the second row s Definitions column; note that the second line in the definition for D shows that it subtracts S /S to essentially center time axis to yield a zero-centroid condition. Thus, this result is an obvious and consistent extension of the first row. The third row gives the results for a non-coherent pulse train of K pulses under the simplifying conditions that C C S. This result was derived for the case of R but the following discussion shows that it holds also for R. Note that the definition of R shows that it is the average pulse time, so R means that the entire pulse train is centered in some sense at the time origin. Again we see the same dependence on S /S as in the first row; note that if S the quantity S/S can be replaced by D. Thus, the only difference between a single pulse and a K-pulse non-coherent pulse train is that the is reduced by the division by K; that is, an estimate is obtained from each pulse and then the K estimates are averaged. It is clear then that this result holds even if R. The fourth row gives a simple coherent K-pulse case to compare with the simple non-coherent case in the third row. Here the depends on a new factor, R, that is a measure of the temporal spread of the pulse train in units of seconds. Strictly, R is a mean-square value of the pulse times, but since here the average pulse time is R the value of R is really a variance of the pulse times. Comparing the simplified-case results for non-coherent and coherent pulse trains (rows 3 and 4, respectively) shows that the gain due to coherency comes from the ability to exploit the spread between pulses that is captured by R : more widely spaced pulses increases R which adds to S /S to decrease the. Because Δ S /S is a measure of pulse duration and R is a measure of pulse train duration, the value of R will be much larger than Δ S /S and that leads to a significant decrease in the. The approximation shown relies on this dominance of R.

16 6 The fifth row shows the case when S and R, where comparing to the fourth row shows that this is handled by replacing S /S by D and R by R. Just as D was discussed as a centroid-normalized version of S, R is a centroid-normalized version of R ; that is, R is the variance of the pulse times about the mean pulse time. Finally, the sixth row shows the most general result for the coherent pulse train case, where some of the previously defined quantities are adjusted by the effect of so-called sew measures [3] given by C and C. There are two main effects: (i) the duration as measured by D is reduced from D through the subtraction of sew term in the definition of S S ( C / S B ), and (ii) the effective SNR is reduced by multiplication of the term ( C / S B). As discussed above, though, the impact of D is negligible compared to R so the effect of the sew parameters via D can be ignored, as shown in the first approximation in the sixth row. Thus, C has negligible impact and the only potentially significant impact of the sew is via ( C / S B). However, note that the impact here of C is reduced when the value of B is large; note that the derivatives in the definition of B can be converted into the frequency domain to give a measure of bandwidth []. Thus, phase/frequency modulation will tend to introduce the impact of the sew parameters but the increased bandwidth due to the modulation will temper its impact. In fact, it is possible to show (see the Appendix) that the factor C will be zero if the instantaneous frequency of the pulse p(t) is symmetric (as for a linear FM pulse) and in other cases is expected to be small enough to be negligible; thus, the second approximation in the sixth row is expected to hold for most if not all pulse trains. Comparing all the final results (i.e., after approximations) for the coherent case in Table shows that as a general rule of thumb the depends inversely on: (i) the pulse on-time T OT, (ii) the number of pulses K, (iii) the variance of pulse times R, and (iv) the product SNR F s ; the pulse shape and pulse modulation seem to have little or no impact on the frequency accuracy. As a special case consider the scenario where the intercepted pulses are equally spaced by T PRI, the pulse repetition interval (PRI), then it is easily shown that R TPRI ( K )/and the last approximation in the sixth row of Table can be re-written as

17 7 Coh 6, (44) T K K SNR F T OT ( ) s PRI from which we see that the frequency varies as ( / PRI ) 3 O T and as O( / K ). Thus, for all other parameters fixed, doubling the number of equally spaced pulses decreases the by /8 It is interesting to mae a comparison between the coherent and non-coherent cases, which will be done here for the case of rectangular pulses spaced by the PRI. Expressing the non-coherent case similarly gives, using /K th the result in (36), gives th. Non Coh OT 6 T K SNR F Δ N s ( ) (45) together with (44) and taing that coherent gain in accuracy as the ratio Gain Non Coh PRI, (46) Coh KT T OT where the following approximations have been used: TOT ( N ) Δ and K K. This gives a rule of that that the gain in coherent accuracy is K times the ratio of pulse on-time to PRI; since the PRI is typically much larger than the pulse on-time there is great improvement available when one can exploit the coherency of the pulse train. Numerically computing the for typical values of pulse train parameters shows that it is possible to achieve frequency accuracy much lower than Hz ( Hz is the accuracy that is claimed experimentally in [4] and [5]; the accuracy for the simulation results in [] are on the order of a few Hz). Thus, it is clear that the existing algorithms, while achieving quite good accuracies, are still far from achieving the. Numerical results are given in Table for several typical cases of pulse on-time, pulse shape, PRI, and number of pulses. In all cases the pulse p(t) was real-valued and had a shape given by the Tuey window [5]; the column in Table labeled α is the taper parameter [5] that is used here to control the shape of the pulse and hence the value of Δ D (note that α corresponds to zero taper and so is a rectangular pulse; α corresponds to full taper and has no flat region in the center). In Table the for the coherent case is computed using

18 8 ω, (47) T OT K SNR F s Δ D+ R while for the non-coherent case it is computed using ; (48) ω T OT K SNR F s Δ D the values for the Cramer-Rao bounds are tabulated in square-root form in units of Hz to represent a bound on frequency estimation standard deviation in Hz. In addition, Table lists the computed values for Δ D and R to illustrate the relationship between them; note that R always dominates and therefore the coherent does not depend on Δ D. For comparison, Table 3 presents some simulation accuracy results published in [] side-by-side with the corresponding computed coherent results. It should be noted that the results in [] were given for the case of staggered PRI values within each pulse train, although the computed results given in Table 3 were computed without the stagger for convenience; the stagger was slight and would not significantly change the computed values. Note that the values are much smaller than the reported simulation accuracies; also note that the discrepancy between simulation results and are worse at higher SNR. In fact, in light of the dependence on SNR shown in (47) we would expect the ratio of at 37 db relative to at 3 db to be 5; however, as seen in Table 3 the simulation results exhibit ratios of only.4 to.8. This form of dependence on SNR is characteristic of both coherent and non-coherent frequency estimation methods as well as the classic frequency estimation methods. This indicates that although the algorithm in [] produces very accurate methods that its SNR-behavior does not follow that of the. Looing at the ratio of the at 4 pulses relative to at 6 pulses we expect the ratio to be.87 and the simulation results exhibit ratios from.4 to.; indicating that the algorithm s behavior relative to pulse number follows the behavior more closely than for the SNR-behavior.

19 9 Table : Summary of Results for on Frequency of Pulse Train Case Result ((rad/sec) ) Definitions #: Single Pulse Case with S ( ) ω TOT SNR Fs Δ S / S S p n S n p n ( Δ ) ( Δ) N n N n S n p n SNR S ( Δ ) / N n F sampling rate Δ sampling interval s #: Single Pulse Case ω T OT SNR F s Δ D T OT Pulse On-Time ( / ) ( / ) D S S S S N ( ( ) ) K # of pulses n n S / S p( nδ) S #3: Non- Coherent Case with C C S #4: Coherent Case with C C S R #5: Coherent Case with C C #6: General Coherent Case ( ) ω TOT K SNR Fs Δ S / S ω TOT K SNR F s Δ ( S / S) + R T OT K SNR F R ω TOT K SNR F s Δ D+ R ω T OT s K SNR F R TOT K SNR Fs ( ( C / S B) ) Δ D + R T K SNR F ( C / S B) R T OT OT K SNR F R s s ( ) s * C Im p( Δn) p ( Δn) N n * C Im n p( Δn) p ( Δn) N n K R T T K R K R K K pulse times ( T ) K K R R R ( T ) R T N B p ( nδ) SN K ( / ) ( / ) S S ( C / S B) S S ( C C / S B) o S S ( C / S B) n D S S S S

20 Table : Numerical Results for F s MHz and SNR db T PRI (ms) 5 K T OT α Δ D R (µs) Non-Coherent Coherent (µs) (ms) ω /π ω /π (Hz) (Hz).5.7,84.4, , , , , ,

21 Table 3: Comparison to Simulation Results for PRI of approximately ms F s (MHz) 5 5 K SNR Ratio f in (db) ω /π [] π / f ω (Hz) (Hz) f

22 Appendix: Evaluation of C The for the most general coherent pulse train case was shown to depend on the value of C given in (), which is evaluated here using an integral approximation of the summation given by T * OT C Im p( n) p ( n) Im p( t) p * Δ Δ ( t) dt N n N Δ. (49) As discussed in the main body of the paper, C is non-zero only when p(t) is complex, where it can be expressed in polar form or rectangular form, both of which are shown here as [ φ ] [ ] jφ () t p() t A() t e A()cos t () t + ja()sin t φ() t, (5) at () bt () where A(t) is the real-valued envelope function and φ(t) is the real-valued phase function. Then But, using (5) gives TOT C Im a() t + jb() t a () t jb() N Δ T [ ][ t ] OT [ a () t b() t a() t b () t ] dt. N Δ dt. (5) [ φ ] φ [ φ ] a () t A ()cos t () t A() t ()sin t () t [ φ ] φ [ φ ] b () t A ()sin t () t + A() t ()cos t () t (5) and [ φ ] [ φ ] φ [ ] a () tbt () At ()sin () t A ()cos t () t At () ()sin t φ() t [ φ ] [ φ ] φ [ φ ] At () A ()sin t ()cos t () t A() t ()sin t () t [ φ ]{ [ φ ] φ [ φ ]} atb () () t At ()cos () t A ()sin t () t + At () ()cos t () t [ φ ] [ φ ] φ [ φ ] At () A ()sin t ()cos t () t + A() t ()cos t () t (53) which when substituted into (5) gives

23 3 T OT Δ C A t N Recognizing that φ () t ω () t, the instantaneous frequency, we can write i () φ () t dt. (54) T OT () ωi () Δ t dt. (55) C A t N For any signal for whom the instantaneous frequency varies symmetrically around the carrier frequency (i.e., the frequency to be estimated) then (55) implies that C will be zero; this includes linear FM pulses. It is also approximately true for pulses that are subjected to BPSK with a large enough number of phase transitions within each pulse. Although it is difficult to mae a wide-sweeping general conclusion here, it is also liely that most if not all typical phase modulations will give a small value of C because the instantaneous frequency typically varies uniformly (at least approximately) above and below zero. Also, it should be noted that the effect of any non-zero value of C will be deemphasized through the division in ( C / S B). Thus, we conjecture that the parameter C has little effect. References [] Becer, K., An efficient method of passive emitter location, IEEE Transactions on Aerospace and Electronic Systems, AES-8, Oct. 99, pp [] Levanon, N., Interferometry against differential doppler: performance comparison of two emitter location airborne systems, IEE Proceedings, 36, Pt. F, April 989, pp [3] Rao, K. D., and Reddy D. C., A new method for finding electromagnetic emitter location, IEEE Transactions on Aerospace and Electronic Systems, AES-3, Oct. 994, pp [4] Klass, P. J., Tests show RWRs can locate, identify threats, Aviation Wee & Space Technology, Sept., 995, pp [5] Klass, P. J., New RWRs, ESMs locate radars more precisely, Aviation Wee & Space Technology, Sept. 3, 996, pp [6] Czarneci, S. et al., Doppler triangulation transmitter location system, US Patent Nos. 5,874,98 and 5,98,64, issued /3/999 and /9/999.

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