Estimation of Pulse Parameters by Autoconvolution and Least Squares

I. INTRODUCTION

Estimation of Pulse Parameters by Autoconvolution and Least Squares

For the baseband pulse in Fig. 1, the pulse parameter estimation problem is to determine the arrival time D, amplitude A, and width W, from noisy samples of p(t). In Fig. 1, the beginning of the time segment is zero and the arrival time is D. More generally, a time segment begins at a known time T, with respect to a reference. Then the arrival time is T + D.

Y. T. CHAN, Senior Member, IEEE B. H. LEE Royal Military College of Canada R. INKOL, Senior Member, IEEE Defence Research and Development Canada F. CHAN, Member, IEEE Royal Military College of Canada

Let the parameter vector of a baseband pulse be ¹ = [D, A, W]T , where D = arrival time, A = amplitude, and W = width. A method for estimating ¹ for a pulse present in a noisy data segment is given. We start by performing an autoconvolution (AC) of the data segment, and then recording the shift where the AC peaks. By taking half of that shift, from the end of the segment, a point is placed near the middle of the pulse. Next, we partition the pulse at that point and perform two more ACs, (one on the left half of the pulse and another on the right half), and find two more shifts where the ACs peak. These two shifts provide coarse measurements on D and W. A further refinement by least squares (LS) then produces a final estimate for ¹. Simulation results have corroborated the theoretical development and shown that the new estimator performs close to the Cramer-Rao lower bound (CRLB).

Manuscript received February 9, 2007; revised August 23, 2007 and July 9, 2008; released for publication October 29, 2008. IEEE Log No. T-AES/46/1/935948. Refereeing of this contribution was handled by P. Lombardo. Authors’ addresses: Y. T. Chan, B. H. Lee, F. Chan, Dept. of Electrical and Computer Engineering, Royal Military College of Canada, PO Box 17000 Station Forces, Kingston, Ontario K7K 7B4, Canada, E-mail: ([email protected]); R. Inkol, Defence Research and Development Canada, 3701 Carling Ave., Ottawa, Ontario, Canada K1A 0Z4. c 2010 IEEE 0018-9251/10/$26.00 °

Fig. 1. Pulse parameter definitions.

Various applications in radar and electronic warfare (EW) require knowledge of one or more parameters of a pulse. When an EW receiver [1] processes interleaved pulse trains from multiple radars, knowing ¹ = [D, A, W]T for the individual pulses can assist in their deinterleaving [2, 3]. Also, values of D and W contribute to the identification of the radar function and type [1, 4]. Passive localization of an emitter by time-difference-of-arrival (TDOA) [5] is the establishment of several loci on which the emitter can lie. Their intersection is the emitter location. To find a locus, two separate receivers at known locations measure the TDOA of a signal. A hyperbola, with the receiver locations as the foci, is a locus of points whose distance differential between the receivers is proportional to the TDOA. The proportionality constant is the propagation speed of the signal. There are two ways to measure TDOA, and the choice is application dependent. One way to measure TDOA is to relay the received signals to a central location. The location of the peak of the cross-correlation function between a pair of signals is their TDOA. Another way to measure TDOA is by measuring the time-of-arrival (TOA) of the signal at each receiver. Their difference is the TDOA. This is convenient when the signal is a pulse (most probably a carrier modulated pulse) whose D is the TOA. There are also bio-medical examples [7, 8] in which the measurements of D and A can lead to an estimate of the blood pressure of a patient. Past research has concentrated primarily on the estimation of D, since many applications require only

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 46, NO. 1

JANUARY 2010

363

Fig. 2. Graphical AC, tF ¸ tR .

D and not A or W. Yet there is no standard definition for D. In active coherent radar systems, the reference p(t) and echo p(t ¡ ¿ ) are available. The arrival time ¿ is the peak location of the cross-correlation between the reference and the echo [9, 10, 11]. For passive systems where there are no reference signals, D is the time when p(t) reaches a certain threshold on the rising edge. It is necessary that the threshold be dependent on A. If it is not, a p(t) with a high A will have a shorter D than a similar pulse with a lower A. In [12], the threshold is A=2. But this scheme has a wait time since it can find A only after the arrival of the whole pulse. To avoid delays, [13] takes D as the time when the rising edge reaches a maximum slope. Assuming that the rising edge is a linear function of time, [14] measures the rising edge continuously, and D is when it exceeds a fixed threshold. Another D estimator [15] examines the ratios of the successive samples of the rising edge. These ratios are > 1 at the beginning of the rising edge and < 1 at the end. A proper threshold for the ratios can decide D, when the rising edge reaches the midpoint. The following sections give a new approach to pulse parameter estimation for a single pulse. Section II shows that the autoconvolution (AC) of the time segment, which contains the pulse, has a peak at time shift ¿ ¤ . Going back ¿ ¤ =2 from the end of the segment locates a point M, near the pulse center. Partitioning the pulse at M, taking ACs of the two halves, and noting their peak locations, provide additional equations for D and W. However, these only yield coarse estimates. A further refinement by least squares (LS) then gives final estimates of D, A, and W. The simulations results in Section III compare the errors of the autoconvolution-least squares (AC-LS) estimator against the method of [15] and against the Cramer-Rao lower bound (CRLB) bound [16]. The conclusions and discussions are in Section IV. A preliminary version of the paper appeared in [17], where the derivation is valid only for a trapezoidal pulse. The present work generalizes the development to nontrapezoidal pulses and enhances 364

the AC coarse estimates by an additional LS step so that the errors now approach the CRLB. The current work also contains the derivation of the CRLB for ¹ in Appendix A. Derivations for the CRLB for the estimation of D only are available in [11], [13], and [18]. II.

AUTOCONVOLUTION-LEAST SQUARES ESTIMATION

A. Autoconvolution Estimation Fig. 1 depicts a time segment containing pulse p(t) having the following parameters: S = beginning of pulse; L = duration of the segment; A = full amplitude of the pulse; D = arrival time is the time from the beginning of the segment to the point on the rising edge having an amplitude A=2; f(¢) = f(t ¡ S) is the rising edge monotonic function, f(0) = 0, f(mR ) = A=2, f(tR ) = A; tR = rise time is the time f(¢) takes to rise from 0 to A; tC = duration of portion of pulse having constant amplitude; g(¢) = g(S + tR + tC + tF ¡ t) is the falling edge monotonic function, g(0) = 0, g(mF ) = A=2, g(tF ) = A; W = pulsewidth is the time between the A=2 points on f(¢) and g(¢); tF = fall time is the time g(¢) takes to fall from A to 0; E = end of pulse; B = time between E and L. Given the time segment, the problem is to find D, A, W, in the presence of additive noise. The AC estimator first locates the peak of the AC of the data segment. Referring to the graphical representation of the AC in Fig. 2, it is apparent that for tF ¸ tR , 0 · x · tF , the AC will have a peak at the shift ¿ = 2B + 2tF + tC ¡ x:

(1)

If tF = tR , then the AC is a maximum when p(2L ¡ t) and p(t ¡ ¿ ) overlap, i.e., x = 0. If tF > tR ,

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 46, NO. 1

JANUARY 2010

then x is positive, so as to give the AC a larger value than when the two pulses overlap. To find x, consider the AC value at ¿ given by Z tF C1 (¿ ) = A2 (tC ¡ x) + 2A g(®)d® +2

Z

tF ¡x

tR

f(®)g(® + tF ¡ x ¡ tR )d®:

0

(2)

Setting @C1 (¿ )=@x = 0 gives an equation to find x that maximizes C1 (¿ ). For the first integral in (2), applying Leibniz’s rule [19] gives Z tF @ g(®)d® = g(tF ¡ x): (3) @x tF ¡x

Fig. 3. Graphical AC, tR ¸ tF .

For the second integral, the differentiation is Z tR @ @x

f(®)g(® + tF ¡ x ¡ tR )d®

0

= lim

¢x!0

=

Z

=

Z

Z

tR

0

f(®)[g(® + tF ¡ x ¡ ¢x ¡ tR ) ¡ g(® + tF ¡ x ¡ tR )] d® ¢x

tR

f(®) lim

¢x!0

0 tR

f(®) 0

g(® + tF ¡ x ¡ ¢x ¡ tR ) ¡ g(® + tF ¡ x ¡ tR ) d® ¢x

@fg(® + tF ¡ x ¡ tR )g d®: @x

(4)

Fig. 4. Pulse partitioning at M.

Hence @C1 (¿ ) = ¡A2 + 2Ag(tF ¡ x) @x Z tR @g(® + tF ¡ x ¡ tR ) +2 f(®) d® = 0: @x 0 (5) Solving for x from (5) requires specific knowledge of f(®) and g(®). As an example, suppose the pulse is trapezoidal, so that f(®) = A®=tR and g(®) = A®=tF . Substituting these relationships into (5) and solving gives t ¡t (6) x= F R, tF ¸ tR : 2 If tR ¸ tF , then Fig. 3 shows that C2 (¿ ) will peak at ¿ = 2B + 2tF + tC + x with 2

C2 (¿ ) = A (tC ¡ x) + 2A +2

Z

0

Z

f(®)d®

tR ¡x

tF

(8)

Following the steps from (3)—(5) gives the equation for x that maximizes C2 (¿ ): x=

tR ¡ tF , 2

tR ¸ tF :

operation gives ¿ ¤ , the locations where C1 (¿ ) and C2 (¿ ) peak. Going back from L by a distance of

(7)

tR

g(®)f(® ¡ tF ¡ x + tR )d®:

Fig. 5. AC of left half of pulse.

(9)

Both (1) and (7) contain unknowns that are dependent on a particular unknown p(t). But the AC

t x ¿¤ = B + tF + C § 2 2 2

(10)

locates the point M in Fig. 4. Partitioning the pulse at M, and taking the AC of the left half results in the peak value of (see Fig. 5) Z tR ³ ´ x 2 tC f(®)d®: (11) C3 (¿ ) = A ¨ ¡ y + 2A 2 2 tR ¡y From (3) and (5), the y that maximizes (11) must satisfy ¡A2 + 2Af(tR ¡ y) = 0 (12) or f(tR ¡ y) =

A : 2

CHAN ET AL.: ESTIMATION OF PULSE PARAMETERS BY AUTOCONVOLUTION AND LEAST SQUARES

(13) 365

Hence the rising edge attains A=2 at the time tR ¡ y = mR , regardless of the shape of the function f(¢). Let t x Pl = C ¨ + y (14) 2 2

amplitude is constant. The AC of the left half pulse in Fig. 5 shows that it will have a zero value starting at ³t x´ Plo = 2 C ¨ (19) + 2tR : 2 2

be the shift where the AC of the left half of the pulse peaks, then ¶ μ ¿¤ : (15) D = L ¡ Pl + 2

Assuming that the rising edge is a straight line, then y = tR =2 from (13), and from (14)

Similarly, an AC of the right half of the pulse in Fig. 4 has a maximum at

Hence from (19) and (20),

t x Pr = C § + z 2 2 with g(tF ¡ z) =

A 2

(16) (17)

(18)

Note that the unknown x cancels in (15) and (18). Note also that the autocorrelation of the pulse in Fig. 1 will give a peak at shift L, which does not contain the pulse parameters. To summarize, the estimation of D and W consists of the following steps. 1) Determine ¿ ¤ by finding the shift corresponding to the peak of the AC of the data segment. 2) Go back ¿ ¤ =2 from the segment end (L) to locate point M, and divide the pulse into left and right halves. 3) Perform an AC on the left half of the pulse and determine Pl , the shift where the AC peaks. Then ˜ = L ¡ (P + ¿ ¤ =2) is the estimate of D. D l 4) Perform an AC on the right half of the pulse, and determine Pr , the shift where the AC peaks. Then ˜ = P + P is the estimate of W. W l r B. Least Squares Estimation The simulation experiments indicate that the mean ˜ and W ˜ are well above the square errors (MSEs) of D CRLB (see Appendix A). This occurs because all of the noise present in the full data segment contributes to errors in the estimation of D. To improve the D estimate, an LS step fits a straight line, which passes ˜ to the rising edge. The LS estimate of through D, D is when the line reaches A=2. By limiting the D estimation over only the rising edge, the LS uses less noisy data. The result is a lower MSE. The same applies for the LS estimation of W. The following describes the LS estimation of A, D, and W. 1) Least Squares for A: To find A, it is necessary to determine tC , the duration over which the pulse 366

tC x tR ¨ + : 2 2 2

(20)

tR = Plo ¡ 2Pl :

i.e., the falling edge attains A=2 at the time tF ¡ z = mF , regardless of the shape of the function g(¢). It follows from Figs. 1 and 4 that W = Pl + Pr :

Pl =

(21)

Similarly, for the right half of the pulse in Fig. 4, the time shift when the AC first reaches zero is ³t x´ Pro = 2 C § (22) + 2tF : 2 2 Assuming that the falling edge is a straight line, then from (16) t x t Pr = C § + F (23) 2 2 2 and tF = Pro ¡ 2Pr :

(24)

Referring to Fig. 4, it is clear that h ³ h ³ t ´i t ´i ¢ rl = L ¡ M + Pl ¡ R · tC · L ¡ M ¡ Pr ¡ F 2 2 ¢

= ru :

(25)

Without loss of generality, let the sampling interval be 1 s so that the samples from the segment length L seconds are x(i) = p(i) + n(i),

i = 0, : : : , L

where n(i) are zero mean, independent Gaussian random variables (RVs) of variance ¾ 2 . Then an estimate for the pulse amplitude is Pru ˆA = i=rl x(i) : ru ¡ rl

(26)

(27)

In general, the rising and falling edges are not straight lines, and the estimates for tR and tF in (21) and (24) will only be approximations. Their exact values are not critical in forming the bounds of (25). The only requirement is that the summation range in (27) substantially covers the constant amplitude duration, so as to have as many samples as possible to average out the effects of noise. Since a smaller tR or tF will increase the bounds in (25), possibly adding samples from the rising or falling edges, it is better to take, say 4/3 of the tR and tF in (21) and (24) to compute the bounds of (25). 2) Least Squares for D: Consider the equation of a straight line ˜ + i) = ®(D ˜ + i) + ¯, p(D

i = 0, §1, : : : , §Nm

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 46, NO. 1

(28) JANUARY 2010

˜ which models the portion of the rising edge at D. ˜ In (28), p(D + i) denotes a measurement at the time ˜ + i, where D ˜ is the value from (15), when noise D is present. The unknowns ® and ¯ are the slope and intercept of the straight line. The measurements are ˜ since D ˜ is near the mid-point on either side of D, of the rising edge. The integer Nm should be a few samples (say 3) larger than tR =2 (obtained from (21)) to ensure (28) has as many samples on the rising edge as possible. If Nm is too large, (28) will include samples outside the rising edges. However, this should not be a problem since the straight line fitting errors (see (29) below) will be large, and the LS estimator will reject those estimates. For a given N in (28), i = 0, §1, : : : , §N · Nm , (56) ˆ To determine ˆ ¯. in Appendix B gives the estimates ®, the best fitting as a function of N, it is necessary to compute the normalized fitting errors "N =

PN

˜ ˆ 2 ˆ ˜ i=¡N [x(D + i) ¡ ®(D + i) ¡ ¯] 2N + 1

(29)

and select ®ˆ ¤ and ¯ˆ ¤ that give the minimum "N The AC-LS estimator for D then is ˆ A ¡ ¯ˆ ¤ 2 ˆ D= : ®ˆ ¤

(30)

3) Least Squares for W: The straight line for estimating W is ˆ +W ˜ + i) = °(D ˆ +W ˜ + i) + », p(D

i = 0, §1, : : : , §Nm (31)

ˆ from (30), which models the falling edge, with D ˜ W from (18) when noise is present, and Nm slightly larger than tF =2 (obtained from (24)). The unknowns are ° and ³, representing the slope and intercept. Following the same procedure that gives (30), the AC-LS estimation for W is

where

z }| { ˆ ˆ = (D + W) ¡D W ˆ A ˆ ¡ ³¤ z }| { 2 (D + W) = ˆ °¤

(32)

(33)

and °ˆ ¤ and »ˆ¤ are the values that give the smallest normalized fitting errors. C. Estimation Variances Let the noise samples n(i) in (26) have covariance Efn(i)n(j)g = ¾2 ±ij :

(34)

ˆ trapezoidal pulse. Fig. 6. MSE of D,

ˆ is a nonlinear function of three RVs, A, ˆ In (30), D ¤ ˆ ˆ ® , and ¯ . An approximate variance formula for D is [20] à ! à à ! ! ˆ 2 ˆ 2 ˆ 2 @D @D @D 2 2 ˆ var(D) = ¾Aˆ + ¾®ˆ ¤ + ¾¯2ˆ¤ ˆ @ ®ˆ ¤ @A @ ¯ˆ ¤ ˆ¤

+2

ˆ @D ˆ @D ¾ ¤ ˆ¤ : ¤ @ ®ˆ @ ¯ˆ ¤ ®ˆ ¯

(35)

The approximation is valid if the probability mass ˆ concentrates near its mean and the function of D probability density function is smooth near the mean. In (35), the evaluation of the partial derivatives are at ˆ ®ˆ ¤ , and ¯ˆ ¤ . Their variances are the mean values of A, 2 2 2 ¾ ˆ , ¾®ˆ ¤ , and ¾ ˆ ¤ , respectively. The covariance of ®ˆ ¤ A ¯ ˆ with ®ˆ ¤ or ¯ˆ ¤ is and ¯ˆ ¤ is ¾ ¤ ˆ ¤ . The covariance of A ®ˆ ¯

ˆ are independent from zero, since the noise terms in A those in ®ˆ ¤ and ¯ˆ ¤ . The derivations for the variances ˆ ®ˆ ¤ , and ¯ˆ ¤ are in Appendix B. of A, It is of interest to note that when noise is present, ˜ from (15) is also an RV. However, the estimate D ˜ do not affect D ˆ of (30), the estimation errors in D ˜ provided that p(D) is near the middle of the rising edge (preferably at A=2). The LS estimator takes ˜ to perform the LS samples from each side of p(D) fit of a straight line. The LS errors are dependent ˜ In the LS on the number of samples and not p(D). ˜ equations (28), D is a constant and not an RV. When ˜ the signal-to-noise ratio (SNR) is very low, p(D) could be far from A=2 and could lie outside the pulse portion of the data segment, or on the tC portion of the pulse. In experiment 1 (see Section III), this occurred occasionally when the SNR was below 10 dB (see Fig. 6), and resulted in very large errors. z }| { The derivation of the variance of (D + W) in (33) ˆ hence it is not repeated is very similar to that for D,

CHAN ET AL.: ESTIMATION OF PULSE PARAMETERS BY AUTOCONVOLUTION AND LEAST SQUARES

367

ˆ is here. From (32), the variance for W z }| { ˆ = var (D + W) +var(D): ˆ var(W) (36) z }| { ˆ is In (36), the covariance between D + W and D zero. Their estimates come from different edges of the pulse, and the noise terms of the rising edge are independent from those of the falling edge. Although not shown, simulation results have confirmed that the ˆ and W ˆ are close to the theoretical values in MSE of D (35) and (36). III. SIMULATION STUDIES This section describes four simulation experiments to evaluate the AC-LS estimator. Each experiment performs 500 independent trials. Each trial involves processing a data segment of length L samples (sampling period being 1 s), which contain a pulse waveform and additive zero mean independent Gaussian noise of variance ¾ 2 . The SNR in dB is SNR = 10 log

A2 ¾2

(37)

and the MSE for the estimates of D, A, W, for example for D, is P500 ˆ 2 ˆ = i=1 [D(i) ¡ D] s2 MSE(D) (38) 500 ˆ where D(i) is the estimate of the ith trial, and D is the true value. Experiment 1. The pulse is a trapezoidal waveform, with, referring to Fig. 1, A = 1, tC = 1850, tR = tF = 100, B = 400, and L = 2500. The rising edge f(t) = At=tR , and the falling edge g(t) = At=tF . Fig. 6 plots the MSE as a function of the SNR for the AC, AC-LS, and sample ratio estimator (SRE) of [15], together with the CRLB (derived in Appendix A). It is clear that the AC-LS estimator has the lowest MSE and approaches the CRLB. The trapezoidal pulse, while it approximates some lowpass signals of practical importance [13, p. 274], is not typical of pulses that are filtered or otherwise band limited. Filtering removes discontinuities in the first derivative of the pulse waveform and increases the pulsewidth. Nevertheless, since the edges of the trapezoidal pulse are linear functions of time, it provides a simple analytic model to verify the development in Section II and the derivations of the CRLB. Figs. 7 and 8 are the MSE plots for the pulse amplitude and width estimates. Again the AC-LS estimates are superior to the AC estimates, and they approach the CRLB. In Fig. 7, there are no amplitude estimates from AC, since it cannot estimate amplitude. The plots also indicate that the AC-LS has an SNR threshold near 8 dB, below which the MSE increases rapidly with decreasing SNR. In contrast, the AC 368

ˆ trapezoidal pulse. Fig. 7. MSE of A,

ˆ trapezoidal pulse. Fig. 8. MSE of W,

estimator exhibits no such behavior. The reason is that ˜ is somewhere on the the AC-LS assumes that p(D) rising edge. At low SNR, in some trials, the AC gives ˜ that is either on the constant amplitude portion p(D) or falls outside the pulse portion of the data segment. When this occurs, the LS estimator of (56) and (28) give an ®ˆ ¤ (i.e., slope of the line) that is near zero ˆ further away or sometimes even negative, moving D from the true value. ˆ A simple remedy is to select D AC-LS , the AC-LS ˆ , the AC estimate, when the SNR is high, and D AC estimate, when the SNR is low. However, this will require knowledge of the SNR, which may not be known. A simple way of implementing this solution ˆ ˆ is to use the relative values of D AC-LS and DAC as a measure of SNR. If they are close, the SNR is high. Otherwise, it is low. The selection criterion then is (ˆ ˆ ¡D ˆ DAC-LS , if jD AC AC-LS j · tR =2 ˆ D= ˆ ˆ ˆ DAC , if jDAC ¡ DAC-LS j > tR =2

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 46, NO. 1

(39) JANUARY 2010

ˆ trapezoidal pulse. using selection criterion Fig. 9. MSE of D, (39).

ˆ raised cosine pulse. Fig. 11. MSE of A,

ˆ raised cosine pulse. Fig. 12. MSE of W,

ˆ raised cosine pulse. Fig. 10. MSE of D,

using tR from (21). Implementing (39) indeed reduces the MSE of the AC-LS at low SNR, as seen in Fig. 9. Experiment 2. The pulse has a raised cosine function for the rising and falling edges, given by [15]: μ ¶ 8 k¼ ¼ 2 > A cos ¡ , k = 0, 1, : : : , tR > > 2tR 2 > > > > < A, k = tR + 1, : : : , tR + tC μ ¶ p(k) = > (k ¡ tR ¡ tF )¼ > > A cos2 , > > 2tF > > : k = tR + tC + 1, : : : , tR + tC + tF

(40) where A = 1, tR = tF = 100, and tC = 2000. The MSE ˆ A, ˆ and W, ˆ results are in Figs. 10, 11, and 12 for D, respectively. The AC-LS MSEs are smaller than those of the AC and SRE, and they are close to the CRLB at SNR ¸ 14 dB. They diverge rapidly from the CRLB for lower SNRs. Applying the selection criterion of (39) should lower the MSE, as demonstrated in Fig. 9, but this has not been done in this experiment.

Fig. 13. Input and output for lowpass filtered pulse.

Experiment 3. The pulse is the output of a Butterworth lowpass filter whose input is a rectangular pulse. Fig. 13 shows both the input and output pulse waveforms. The MSEs in Figs. 14, 15, and 16 for ˆ A, ˆ and W, ˆ respectively, follow the same trend as D, the results of experiments 1 and 2. Since the edges

CHAN ET AL.: ESTIMATION OF PULSE PARAMETERS BY AUTOCONVOLUTION AND LEAST SQUARES

369

ˆ filtered pulse. Fig. 14. MSE of D,

ˆ filtered pulse. Fig. 16. MSE of W,

ˆ filtered pulse. Fig. 15. MSE of A,

ˆ trapezoidal pulse with a carrier. Fig. 17. MSE of D,

are not continuous functions of time, the CRLB derivation is based on a trapezoidal approximation of the waveform. Experiment 4. The AC-LS applies to a stand-alone baseband pulse inside a noisy time segment. Pulses in radar and some other applications have carrier modulation. If this is the case, it is then necessary to demodulate prior to pulse parameter estimation. However, it is possible to combine the two steps. Let X(k) be the discrete Fourier transform (DFT) of the data segment x(i), i = 1, : : : , N ¡ 1. Compute the product Z(k) = 2X(k)X(k)U(k) where U(k) =

½

0, 1,

if k · N=2 if k > N=2

(41)

:

(42)

The inverse DFT of Z(k) is z(n) = c(n) + j ²c(n)

(43)

where ²c(n) is the Hilbert transform [21] of c(n). The AC for x(i) is c(n), and jz(n)j is the envelope of the AC of x(i). 370

To verify (41)—(43), this experiment estimates D for a trapezoidal pulse, which has a normalized sinusoidal carrier of 125 Hz and parameters tR = 100 s, tF = 0 s, and B = 0 s. The MSE and CRLB for the corresponding baseband pulse are in Fig. 17. If there are other types of modulation which are also present, such as linear frequency modulation (FM) or binary phase shift keying modulation (BPSK), (43) should still give the AC of the pulse envelope. IV. CONCLUSIONS AND DISCUSSIONS This paper introduces a new estimator for the parameters ¹ = [D, A, W]T of a single baseband pulse present in a noisy data segment. The estimation begins with an AC of the data segment and recording the shift corresponding to the AC peak. Going back from the end of the segment by a distance of half, this shift provides a point near the middle of the pulse. Next the positions of the peaks of the ACs for the left and right ˜ and halves of the pulse provide coarse estimates D ˜ W. A further refinement using LS yields improved estimates for ¹, whose MSE approaches the CRLB.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 46, NO. 1

JANUARY 2010

The symmetric 3 £ 3 Fisher information matrix (FIM) is [16] 1 (48) FIM = 2 F £ FT ¾ where

2 @p(i)

¢¢¢

6 @D 6 6 @p(i) F=6 6 @A 4 @p(i) @W Fig. 18. Pulse parameter definitions.

¢¢¢

3

7 7 7 ¢¢¢ 7: 7 5 ¢¢¢

¢¢¢ ¢¢¢

(49)

Then CRLB = FIM¡1

For pulse modulated carrier waveforms, simultaneous demodulation and AC are possible by squaring the DFT (X(k)) of the segment and by taking the inverse DFT of only the upper half of X(k)X(k). If there is phase modulation, dips in the amplitude of the demodulated pulse will occur at the points where there are discrete phase changes. Again (43) may be able to avoid this demodulation artifact. However, further studies are necessary to corroborate this hypothesis. The AC estimator assumes that the data segment contains a single pulse. If the segment contains a string of pulses, it is necessary to first identify and isolate the individual pulses. APPENDIX A. CRLB FOR PULSE PARAMETER ESTIMATION

(44) The definitions of mR and mF are A , 2

g(mF ) =

A : 2

(45)

The received data is x(i) = p(i) + n(i),

i = 0, : : : , L

(46)

where n(i) is a zero mean Gaussian RV of variance ¾2 . The maximum likelihood (ML) estimate of the ˆ that minimizes vector ¹ = [D, A, W]T is the ¹ ˆ = J(¹)

L X ˆ ¡ x(i)]2 [p(i) i=0

ˆ ˆ is the p(i) with ¹ replaced by ¹. where p(i)

ˆ = CRLB CRLB(D) 1,1 ˆ = CRLB CRLB(A) 2,2

(51)

ˆ = CRLB : CRLB(W) 3,3 The CRLB analysis [16] is valid for only high SNR, when the estimates are near the true values. At low SNR, there are tighter bounds [16] on the MSE. For a given set of pulse parameters, the CRLB is proportional to ¾2 . However, the CRLB will change with pulse parameters values. For example, if tC increases, the CRLB will decrease because a longer tC gives a higher effective SNR. APPENDIX B.

Referring to Fig. 18, the pulse equation is 8 0, 0·t·S > > > > > > > f(t ¡ D + mR ¡ S), S · t · S + tR > < p(t) = A, S + tR · t · E ¡ tF : > > > > > g(D + W + mF ¡ t), E ¡ tF · t · E > > > : 0, t>E

f(mR ) =

and

(50)

ˆ VARIANCE OF D

Let the received data be x(i) = p(i) + n(i)

(52)

where n(i) is a zero mean Gaussian RV of variance ˆ is ¾2 . It follows, from (27), that the variance of A ˆ = Ef(A ˆ ¡ A)2 g = ¾ 2 : var(A)

(53)

Next the noisy set of equations of (28) are · ¸ ® +e y=G ¯ where for i = 0, §1, §2, : : : , §N, 2 ˜ 3 2˜ x(D + N)

6 .. 6 . 6 6 ˜ 6 y = 6 x(D) 6 .. 6 . 4

˜ ¡ N) x(D

7 7 7 7 7, 7 7 7 5

D+N 6 .. 6 . 6 6 ˜ G=6 D

6 6 4

.. .

˜ ¡N D

3

1 .. 7 .7

7 7

17, 7 .. 7 .5

1

2

(54)

˜ + N) n(D

6 .. 6 . 6 6 ˜ 6 e = 6 n(D) 6 .. 6 . 4

˜ ¡ N) n(D

3

7 7 7 7 7: 7 7 7 5

(55) (47)

The LS solution is · ¸ ®ˆ = (GT G)¡1 GT y: ¯ˆ

CHAN ET AL.: ESTIMATION OF PULSE PARAMETERS BY AUTOCONVOLUTION AND LEAST SQUARES

(56)

371

Substituting (54) into (56) gives · ¸ · ¸ ® ®ˆ + (GT G)¡1 GT e: = ¯ ¯ˆ

[8]

(57)

ˆ T is ˆ ¯] Thus the covariance of [®, · ¸ ®ˆ Cov ˆ = (GT G)¡1 GT EfeeT gG(GT G)¡1 G ¯ 2

T

¡1

= ¾ (G G) :

[9]

[10]

(58)

The partial derivatives in (35) are, from (30), ˆ @D 1 = ˆ 2 ®ˆ ¤ @A ˆ ˆ¤ ¡ A ¯ ˆ @D 2 = ®ˆ ¤2 @ ®ˆ ¤ ˆ @D ¡1 = ¤: ˆ ¤ ®ˆ @¯

[11]

(59) [12]

(60) [13]

(61)

Now all the required quantities for the computation of (35) and (36) are available.

[14]

REFERENCES [1]

[2]

[3]

[4]

[5]

[6]

[7]

372

Wiley, R. G. Electronic Intelligence: The Analysis of Radar Signals. Norwood, MA: Artech House, 1982. Maier, N. W. Processing throughput information for radar intercept receivers. IEEE Transactions on Aerospace and Electronic Systems, 34, 1 (June 1998), 84—92. Nishiguchi, N. W., and Kobayashi, M. Improved algorithm for estimating pulse repetition intervals. IEEE Transactions on Aerospace and Electronic Systems, 36, 2 (Apr. 2000), 407—42l. Skolnik, M. L. Radar Handbook. New York: McGraw-Hill, 1970. Torrieri, D. G. Statistical theory of passive localization systems. IEEE Transactions on Aerospace and Electronic Systems, 20, 2 (Mar. 1984), 183—198. Stein, S. Algorithms for ambiguity function processing. IEEE Transactions on Acoustics, Speech and Signal Processing, 29, 3 (June 1981), 588—599. Chua, C. P., and Heneghan, C. Continuous blood pressure estimator using pulse arrival time and photoplethysmogram. 28th Annual International Conference of IEEE Engineering in Medicine and Biology Society, July 2006.

[15]

[16]

[17]

[18]

[19]

[20]

[21]

Kim, K. L., Chee, Y., Park, J., Kim, J., Lim, Y. K., and Park, K. S. A preliminary study for unconstrained pulse arrival time (PAT) measurements on a chair. 27th Annual International Conference of IEEE Engineering in Medicine and Biology Society, 2005. Helstrom, C. W. Statistical Theory of Signal Detection. Toronto, Canada: Pergamon Press, 1968. Zehavi, E. Estimation of time of arrival for rectangular pulses. IEEE Transactions on Aerospace and Electronic Systems, 20, 6 (Nov. 1984), 742—747. Seidman, L. P. Performance limitations and error calculations for parameter calculations. IEEE Proceedings, 58, 5 (May 1970). Torrieri, D. J. Arrival time estimation by adaptive thresholding. IEEE Transactions on Aerospace and Electronic Systems, 10, 2 (Mar. 1974), 178—184. Torrieri, D. J. Adaptive thresholding systems. IEEE Transactions on Aerospace and Electronic Systems, 13, 3 (May 1977), 273—280. Vojnovic, B. M. Error minimization of sensor pulse signal delay-time measurements. Proceedings of the 23rd International Conference on Microelectronics, May 2002. Ho, K. C., Chan, Y. T., and Inkol, R. J. Pulse arrival time estimation based on pulse sample ratios. IEE Proceedings on Radar Sonar Navigation, 142, 4 (Aug. 1995). Van Trees, H. Detection, Estimation, and Modulation Theory, Part 1. Toronto, Canada: Wiley, 1968. Chan, Y. T., Lee, B. H., Inkol, R., and Chan, F. Estimation of pulse parameters by convolution. IEEE Canadian Conference on Electrical and Computer Engineering, Ottawa, Ontario, Canada, May 2006. Campbell, L. L. Asymptotics of performance of estimators of arrival time. Proceedings of the IEEE International Symposium on Information Theory, Aug. 1998. Thomas, G. B., and Finney, R. L. Calculus and Analytic Geometry. New York: Addison-Wesley, 1992. Papoulis, A. Probability, Random Variables and Stochastic Processes. Toronto, Canada: McGraw-Hill, 1968. Papoulis, A. Signal Analysis. Toronto, Canada: McGraw-Hill, 1977.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 46, NO. 1

JANUARY 2010

Y. T. Chan (SM’80) was born and raised in Hong Kong. He studied electrical engineering in Canada. His Bachelor’s and Master’s degrees are from Queen’s University, and his Ph.D. is from the University of New Brunswick. He was an engineer with Nortel Networks and has been a professor in the Electrical and Computer Engineering Department at the Royal Military College (RMC) of Canada, serving as head of the department from 1994—2000. From 2002—2005, he was a visiting professor at the Electronic Engineering Department of The Chinese University of Hong Kong (CUHK). Presently, he is an emeritus professor at RMC. He was also a visiting professor with the Department of Electronic and Information Engineering at The Hong Kong Polytechnic University. His research interests are in detection, estimation, localization, and tracking. Dr. Chan received the Vice-Chancellor’s Exemplary Teaching Award from CUHK in 2003. He was an Associate Editor of the IEEE Transactions on Signal Processing, the Technical Chair of ICASSP-84, General Chair of ICASSP-91, Vice-Chair of ICASSP-03, and Social Chair of ICASSP-04. He directed a NATO ASI in 1988. He is the author of Wavelet Basics (Kluwer, 1994).

B. Haynes Lee received the B.Eng. degree in electrical engineering from Lakehead University, Thunder Bay, Ontario, Canada, in 1987. From 1989 to 1999 he was a research assistant at the Royal Military College of Canada, Kingston, Ontario, where he is presently at the Department of Electrical and Computer Engineering. His research interests are in digital signal processing and discrete-time queues.

Robert Inkol (M’73–SM’86) (DRDC Ottawa) received his B.Sc. and M.A.Sc. degrees in applied physics and electrical engineering from the University of Waterloo in 1976 and 1978, respectively. Since 1978, he has been with Defence Research and Development Canada, where he is currently a senior scientist. He has also served as an adjunct professor with the Royal Military College. He is responsible for numerous contributions to the application of very large-scale integrated circuit technology and digital signal processing techniques to electronic warfare systems. In addition to having produced numerous publications, Mr. Inkol holds four patents. He has served as a reviewer for various publications and as a Technical Program Committee member for several IEEE conferences. CHAN ET AL.: ESTIMATION OF PULSE PARAMETERS BY AUTOCONVOLUTION AND LEAST SQUARES

373

Fran¸cois Chan received his B.Eng. degree in electrical engineering from McGill University, Montreal, Canada and his M.Sc.A. and Ph.D. degrees from Ecole Polytechnique de Montre´ al, Canada. He is currently an associate professor in the Department of Electrical and Computer Engineering at Royal Military College of Canada, Kingston, Ontario, Canada. He was a visiting researcher at the University of California, Irvine in 2002 and 2005. His research interests include digital communications, wireless communications, and digital signal processing. He has served as Chair of the joint Computer and Communications Chapter of IEEE Kingston since 2008. 374

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 46, NO. 1

JANUARY 2010