IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 1

Range Bias Modeling for Hyperbolic FrequencyModulated Waveforms in Target Tracking

Xiufeng Song, Student Member, IEEE, Peter Willett, Fellow, IEEE, and Shengli Zhou, Senior Member, IEEE

Abstract—Hyperbolic frequency modulated (HFM) waveformsoffer detection performance that is “Doppler insensitive”—relatively unaffected by target range-rate in matched filtering.They have been applied in radar; but more they are particularlyuseful in sonar where the wideband Doppler insensitivity isespecially prized. However, Doppler insensitivity does come handin hand with a range bias, which would significantly degrade thetracking performance. In this paper, we model the range bias asa function of range-rate and system parameters, and then utilizethat to calibrate the measurement equation in very precise targettracking. By doing so, the tracking performance can be improved,particularly for fast maneuvering targets.

Index Terms—Hyperbolic frequency modulation (HFM), linearfrequency modulation (LFM), wideband, narrowband, ambiguityfunction, Doppler tolerant waveform, bias compensation, sonar,radar, tracking, (extended) Kalman filter.

I. INTRODUCTION

A matched filter correlates the transmitted with receivedwaveform to determine whether and where a target appears.It maximizes the output signal-to-noise ratio (SNR) in thepresence of additive white noise in probing a stationary target[1] [2]. However, if the target is moving, a mismatch willoccur. Doppler insensitive (tolerant) waveforms have an espe-cially slowly-decaying thin ridge in their ambiguity functions(AFs), and hence mitigate such range-rate caused detectiondegradation [1] [3]. The concept of Doppler insensitivityclosely relates to target state and system parameters. Forexample, a linear frequency modulated (LFM) waveform isDoppler insensitive for the narrowband transmission scenario[1], but this advantage degrades in the wideband case [4] asshown in Fig. 1(a). And most analyses of Doppler insensitivityassume a constant range-rate; an accelerating target can ‘lose’more SNR than otherwise expected [2].

In this paper, we are interested in the hyperbolic frequencymodulated (HFM) waveform [4]–[12], which is maximallyinsensitive to a constant range-rate and has consequently beenemployed in radar [8] and sonar [4]–[7] for moving targets.

Manuscript received November 20, 2011; revised May 23, 2012; acceptedJune 12, 2012. Date of publication XXX; date of current version XXX. Thiswork was supported by the U.S. Office of Naval Research under GrantsN00014-09-10613 and N00014-10-10412. This work was partially presentedat the IEEE Sensor Array and Multichannel Signal Processing Workshop,Hoboken, NJ, June 2012.

Associate Editor: Douglas Abraham.The authors are with the Department of Electrical and Computer Engineer-

ing, University of Connecticut, 371 Fairfield Way U-2157, Storrs, Connecticut06269, USA (e-mail: xiufeng.song@gmail.com, willett@engr.uconn.edu, andshengli@engr.uconn.edu).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier XXX

The ranging performance of an HFM waveform is determinedby its AF ridge. In the narrowband case, the ridge resemblesa tilted line segment through the origin of the delay-DopplerCartesian coordinates and is centrally symmetric—similar tothat for LFM. That for the wideband case is a tilted curve: it isalso through the origin, however, asymmetrically as depictedin Fig. 1(b). In either case this tilt implies a matched filterpeak that for a moving target becomes displaced from the truerange: this is a bias and is sometimes denoted range-Dopplercoupling [2]–[7]. In brief, the range bias directly relates tothe AF ridge, which is determined by waveform parametersincluding pulse width, frequency support, propagation speed,and modulation [1]–[4].

Classical tracking algorithms such as Kalman and extendedKalman filters expect unbiased measurements [13]. If onedirectly feeds the (biased) extractions to them, tracking per-formance will degrade. In [14]–[17], the authors showed thatconsidering the range bias of a narrowband LFM waveformcould improve the tracking accuracy of a Kalman filter. Thispaper focuses on the application of HFM waveforms—in bothnarrowband and wideband cases—to target tracking, of whichthe significance of range bias has perhaps not been adequatelyappreciated. The precondition is that the range biases canbe analytically modeled, such that the measurement equationshall be properly modified to remove the biases. We refer tothis modification as compensation. In this paper, we make thefollowing contributions:

∙ We find that the range bias of a wideband HFM waveformis nonlinearly dependent on range-rate, and suggest twomethods, respectively based on polynomial regressionand instantaneous frequency, to model it. The former isthrough numerical curve fitting, while the latter obtainsa closed-form function of waveform parameters.

∙ The obtained formulas can be employed to compensatethe range bias in target tracking, and the tracking perfor-mance is thereby significantly improved, particularly forfast moving targets.

HFM is a popular waveform for active acoustic systems,and a comprehensive time-frequency analysis on it can befound in [10] and [11]. HFM waveforms have been used inreverberation suppression [12], target phase extraction [5], andDoppler estimation [6] [7]; this paper dedicates itself to targettracking. Sonar tracking has been investigated in [18] and [19],where a measurement is straightforwardly modeled as the truerange plus a zero-mean white Gaussian noise. As opposed tothem, this paper commences from the system-level, where thewaveform characteristics are incorporated into trackers.

IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 2

range (m)

rang

e−ra

te (

m/s

)

−10 −5 0 5 10 15

−200

−150

−100

−50

0

50

100

150

200

250

300

(a) LFM

range (m)

rang

e−ra

te (

m/s

)

−10 −5 0 5 10 15

−200

−150

−100

−50

0

50

100

150

200

250

300

(b) HFM

Fig. 1. Contour plots of wideband AF for up-sweep (a) LFM and (b) HFMwaveforms, where their frequency span is 4 ∼ 6 kHz, pulse width is 25 ms,and waveform propagation speed is 1500 m/s. The ridge of the HFM isthinner and smoother than that of the LFM; therefore, the former has betterDoppler insensitivity than the latter under the wideband transmission scenario.In addition, the nonlinear property of the AF ridges will introduce nonlinearrange bias for a nonzero range-rate target.

The rest of this paper is as follows: Section II introducesHFM waveforms. Section III models their range bias charac-teristics; Section IV numerically illustrates the bias modelingaccuracy. Section V inserts compensation to target tracking,and several tracking examples are given in VI; conclusionsare drawn after that.

II. HYPERBOLIC FREQUENCY MODULATED WAVEFORMS

A. Definition

Let an HFM waveform based ranging system work in pulsemode, with pulse duration 𝑇 . Define

𝑏 =𝑓1 − 𝑓2𝑓1𝑓2𝑇

, (1)

where 𝑓1 denotes the start frequency of the HFM waveform,while 𝑓2 represents its end; the transmitted analytic signal 𝑠(𝑡)

is written as [4]–[12]

𝑠(𝑡) = 𝐴(𝑡) exp

[𝑗2𝜋

𝑏ln(1 + 𝑏𝑓1𝑡)

], 0 ≤ 𝑡 ≤ 𝑇 (2)

where 𝑗 =√−1, and 𝐴(𝑡) denotes a rectangular envelope—

we shall only treat the rectangular envelope here, but thereis no real loss of generality in doing that. The instantaneousfrequency exhibits the time-varying nature of a nonstationarywaveform, and is defined as

𝑓𝑠(𝑡) =∂

∂𝑡

[1

𝑏ln(1 + 𝑏𝑓1𝑡)

]=

𝑓11 + 𝑏𝑓1𝑡

. (3)

Obviously, 𝑓𝑠(0) = 𝑓1 and 𝑓𝑠(𝑇 ) = 𝑓2. The instantaneousfrequency continuously and monotonically goes from 𝑓1 to 𝑓2within a pulse duration, and does so in a hyperbolic fashion.The bandwidth of 𝑠(𝑡) is loosely but generally recognized tobe 𝐵 = ∣𝑓1 − 𝑓2∣. If 𝑓1 > 𝑓2, 𝑓𝑠(𝑡) is a decreasing functionof time 𝑡, and the corresponding HFM waveform is a down-sweep. If 𝑓2 > 𝑓1, it is an up-sweep. For simplicity, we useHFM− and HFM+ to represent them, respectively.

B. Optimal Modulation PropertyThe wideband performance of an HFM waveform is pre-

cious. Let the range-rate of a target of interest be 𝑣, where 𝑣takes negative values if the target is approaching. Define thewideband compression factor as

𝛼 ≜ 1− 2𝑣

𝑐, (4)

where 𝑐 stands for the waveform propagation speed. Supposethat the initial range of the target is 𝑑0, and hence the collectedwideband reflection is written as [4]–[7]

𝑟𝑤(𝑡, 𝑣) =√𝛼𝑠

(𝛼(𝑡− 𝜏0)

)(5)

=√𝛼𝐴

(𝛼(𝑡− 𝜏0)

)exp

[𝑗2𝜋

𝑏ln(1 + 𝑏𝑓1𝛼

(𝑡− 𝜏0

))]after ignoring noise and propagation loss, where 𝜏0 = 2𝑑0/𝑐.The instantaneous frequency of 𝑟𝑤(𝑡, 𝑣) is expressed as

𝑓𝑤(𝑡, 𝑣) =∂

∂𝑡

[1

𝑏ln(1 + 𝑏𝑓1𝛼

(𝑡− 𝜏0

))]=

𝛼𝑓1

1 + 𝑏𝑓1𝛼(𝑡− 𝜏0

) , (6)

which can be equivalently written as

𝑓𝑤(𝑡, 𝑣) =𝑓1

1 + 𝑏𝑓1(𝑡− 𝜏0 − 𝜏1

) , (7)

where𝜏1 =

1− 1/𝛼

𝑏𝑓1=

2𝑣𝑓2𝑇

𝑐𝛼(𝑓2 − 𝑓1)(8)

is the only term involving range-rate 𝑣. Thus, we obtain

𝑓𝑤(𝑡, 𝑣) = 𝑓𝑠(𝑡− 𝜏0 − 𝜏1). (9)

The contribution of Doppler becomes a ‘delay’, and the mod-ulation characteristic of 𝑟𝑤(𝑡, 𝑣) is intact. This nice property isreferred to as the optimal modulation (or Doppler invariance)law [4] [6]. A waveform that satisfies this will minimize thepeak power loss of a matched filter applied to a slowly movingtarget with constant range-rate, even if the time-bandwidth islarge [4].

IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 3

C. Narrowband, Where HFM Meets LFM

Based on Taylor’s theorem, the instantaneous frequency ofthe HFM waveform can be expanded as

𝑓𝑠(𝑡) = 𝑓𝑠(0) + (𝑏𝑓1𝑡)× ∂𝑓𝑠(𝑡)

∂(𝑏𝑓1𝑡)

∣∣∣𝑏𝑓1𝑡=0

+𝑂((𝑏𝑓1𝑡)

2)

= 𝑓1 +𝑓1𝑓2

⋅ 𝑓1 − 𝑓2𝑇

𝑡+𝑂((𝑏𝑓1𝑡)

2),

(10)

where 𝑂((𝑏𝑓1𝑡)2) incorporates the second and higher orders

of (𝑏𝑓1𝑡). With the narrowband assumption that

𝐵

min{𝑓1, 𝑓2} =∣𝑓2 − 𝑓1∣

min{𝑓1, 𝑓2} ≪ 1, (11)

we have 𝑓1/𝑓2 ⋍ 1 and

∣𝑏𝑓1𝑡∣ =∣∣∣∣ (𝑓1 − 𝑓2)𝑡

𝑓2𝑇

∣∣∣∣ < 𝐵

𝑓2≪ 1. (12)

Hence, the contribution of 𝑂((𝑏𝑓1𝑡)2) is negligible, and this

yields

𝑓𝑠(𝑡) ⋍ 𝑓1 +𝑓1 − 𝑓2

𝑇𝑡 = 𝑓1 + 𝜅𝑡, (13)

where 𝜅 ≜ (𝑓1 − 𝑓2)/𝑇 is termed sweep-rate. Interestingly,the instantaneous frequency of the HFM approximates a linearfunction of 𝑡 under the narrowband scenario; therefore, itsperformance is similar to that of an LFM with the sametime and frequency spans. HFMs are not as popular as LFMsin narrowband systems such as radars, as the latter can beconveniently generated via low cost devices such as voltage-controlled oscillator (VCO) and direct digital synthesizer(DDS) [1] [8].

An LFM approximately satisfies the modulation law inthe narrowband case, but not in the wideband case. If thedispersion ∣𝑇𝐵𝑣∣ is larger than one, its peak amplitude of thematched filter output will be significantly reduced, while themainlobe will be widened at the same time [20].

D. Discussion

This part clarifies several frequently used terminologies:Doppler insensitivity (tolerance), Doppler invariance, andrange-Doppler coupling. Doppler insensitivity is a descrip-tive terminology and generally refers to a waveform that isrelatively unaffected by the unknown range-rate in matchedfiltering [1] [3]. Insensitivity can be assayed via observingwhether the AF has a slowly-decaying (thin) ridge that goesthrough the origin of the delay-Doppler Cartesian coordinates.For the narrowband case, LFM, HFM, nonlinear frequencymodulation (NLFM), P3, and P4 codes all have some Dopplerinsensitivity [1, ch.5-6]. Doppler invariance is mathematicallyexplicit in (9), and HFM strictly satisfies it. Interestingly, someDoppler insensitive waveforms such as LFM approximatelymeet it in the narrowband case [4] [20]. The one with theDoppler invariance feature is the best Doppler insensitivewaveform [9], and it ‘minimizes the signal losses in the caseof large 𝑇𝐵 and high range-rate’ [4]; however, a Dopplerinsensitive waveform does not necessarily follow the optimalmodulation law. Note that Doppler insensitivity and invariance

range

rang

e-ra

te

true range-rate

true range

extracted range

AF ridge

Fig. 2. An intuitive illustration of range bias with a wideband HFM+. Ifthe range-rate is nonzero, the extracted range from the zero range-rate axis(i.e., the maximum amplitude for a noise-free matched filter matched to zero-Doppler as in (14)) will be biased by the range-rate. The range bias is afunction of range-rate and the AF ridge.

usually refer to a constant range-rate; the extension to aconstant acceleration can be found in [2] [20].

The ridge of the AF for a Doppler insensitive waveform isgenerally tilted, and a direct result is that the range-rate willconfuse the range estimate in matched filtering [1] [3] [6] [20].Such entanglement is termed range-Doppler coupling, whichis essentially a compromise for Doppler insensitivity.

III. RANGE BIAS MODELING

A. Wideband Matched Filtering

Matched filtering is the usual approach for range extraction.Mathematically, the extracted range 𝑑 is obtained via

𝑑0 = argmax𝑑

∣∣∣∣∫ 𝑟𝑤(𝑡, 𝑣)𝑠∗(𝑡− 2𝑑/𝑐)𝑑𝑡

∣∣∣∣ (14)

for a wideband system. An explicit expression of 𝑑0 is unavail-able, and a real system generally infers 𝑑 from matched filtersamples; for increased (interpolated) precision an amplitude-weighted measurement extraction procedure is recommended[21].

The wideband AF (see Fig. 1) enumerates all correlationresults in the range-Doppler plane [2], and the AF ridge isresponsible for the range bias. Based on Fig. 2, the extractedrange 𝑑0 does not necessarily equal the true range 𝑑0 for 𝑣 ∕= 0,and a certain amount of range bias exists. Note that a two-dimensional matched filter bank can be used to alleviate rangebias at a cost in computational complexity.

The previous section shows that a range-rate 𝑣 can mas-querade as time delay, call this offset 𝜏1. However, since theoptimal modulation law does not consider the distortion ofpulse envelope—which may be significant for the widebandcase—the range bias of (14) is not necessarily (𝜏1𝑐/2).

B. Bias Modeling with Polynomial Regression

The range-Doppler ridge of an HFM waveform is a non-linear function of range-rate 𝑣, and hence so is the rangebias. Suppose that the (unknown) true relationship betweenthe range bias and range-rate be Δw

HFM = 𝑔(𝑣). Based on the

IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 4

left

alig

nmen

t

central alignment

matched pulse

t

received pulse

received pulse

T

r T

T r T

Fig. 3. An illustration of pulse alignment for the wideband HFM with apositive range-rate 𝑣. The positive 𝑣 can expand the time width of the receivedsignal, with an increase of 𝑇𝑟 − 𝑇 , so the central alignment requires a delayof the matched pulse with a factor of (𝑇𝑟 − 𝑇 )/2 in range bias modeling.

Weierstrass theorem [22, p.101], for an arbitrary small number𝜀, one can always find a polynomial 𝑃 (𝑣) =

∑𝐾𝑘=0 𝜙𝑘𝑣

𝑘

satisfying

max𝑣∈𝕍

∣𝑃 (𝑣)− 𝑔(𝑣)∣ < 𝜀, (15)

where 𝐾 denotes the order of 𝑃 (𝑣), 𝜙𝑘’s are polynomialcoefficients, and 𝕍 denotes the range-rate support of interest.As a zero range-rate introduces no bias, we have 𝑔(0) = 0,which yields 𝜙0 = 0. In order to find 𝑃 (𝑣), 𝑁 ≥ 𝐾 numericalsamples are required. Let 𝑑𝑛 be the extracted range bias atrange-rate 𝑣𝑛 via (14). Define 𝒅 = [𝑑1, ⋅ ⋅ ⋅ , 𝑑𝑁 ]𝑇 and

𝑽 =

⎡⎢⎣ 𝑣1 𝑣21 ⋅ ⋅ ⋅ 𝑣𝑘1...

.... . .

...𝑣𝑛 𝑣2𝑛 ⋅ ⋅ ⋅ 𝑣𝑘𝑛

⎤⎥⎦ ; (16)

the coefficient vector 𝝓 = [𝜙1, ⋅ ⋅ ⋅ , 𝜙𝐾 ]𝑇 could be estimatedvia least-squares

𝝓 = (𝑽 𝑇𝑽 )−1𝑽 𝑇𝒅. (17)

Hence, the approximated polynomial is 𝑃 (𝑣) =∑𝐾

𝑘=1 𝜙𝑘𝑣𝑘.

To emphasize that since an active system has knowledge ofwaveform parameters, polynomial regression could be per-formed at the processing center.

This method works in both wideband and narrowband cases,and it can reach an arbitrary accuracy with a sufficientlylarge order 𝐾 and a large number of samples. In addition,it is applicable for the situation using shaded windows1,where the instantaneous frequency based approach (in the nextsubsection) may not be accurate because the AF is slightlychanged [1, pp. 61-67]. Two significant drawbacks are that1) the obtained coefficients do not have a clear physicalmeaning, and 2) the regression is computationally inflexibleand has to be performed for each waveform parameter set. Thesecond point limits its application for a system with adaptivewaveform transmission.

1Although range bias modeling with various kinds of weighting windowsis an interesting topic, we will not explore it in this paper.

C. Instantaneous Frequency Based Modeling

Based on (5), the time width of the reflected pulse is𝑇𝑟 = 𝑇/𝛼. If the value of (𝛼− 1) is not negligible, the timewidth of received and matched signals will be significantlydifferent. Informally, the reflected pulse shall be “stretched” if𝑣 is positive, and “shrunk” if 𝑣 is negative.

We introduce an instantaneous frequency based bias model-ing approach in two steps. First, we assume 2𝑣/𝑐 ⋍ 0. There-fore, the two pulses have the same width. If the instantaneousfrequency of the matched signal at a putative delay 𝑑 is equalto that of 𝑟𝑤(𝑡, 𝑣), say

𝑓𝑠(𝑡− 2𝑑/𝑐) = 𝑓𝑤(𝑡, 𝑣), 0 ≤ 𝑡 ≤ 𝑇, (18)

the amplitude of the matched filter output at 𝑑 would bethe maximum. As a consequence, 𝑑 could be the estimate of(14). Second, if the two pulses have different width, perfectalignment within 0 ≤ 𝑡 ≤ 𝑇 does not hold true. Instead ofthe left (pulse-start) or the right (pulse-end) alignment, thecentral (middle of the pulses) is used as shown2 in Fig. 3.Mathematically, we force

𝑓𝑤(𝑡+ (𝑇𝑟 − 𝑇 )/2, 𝑣

)= 𝑓𝑠(𝑡− 2𝑑0/𝑐), (19)

or equivalently,

𝑓𝑤(𝑡, 𝑣) = 𝑓𝑠(𝑡− 2𝑑0/𝑐− (𝑇𝑟 − 𝑇 )/2

)(20)

in bias modeling.Substituting (3) and (6) into (20), we obtain

𝑓1

1 + 𝑏𝑓1(𝑡− 2𝑑0

𝑐 − 𝑇𝑟−𝑇2

) =𝛼𝑓1

1 + 𝑏𝑓1𝛼(𝑡− 𝜏0

) , (21)

or equivalently,

𝛼+ 𝑏𝑓1𝛼(𝑡− 2𝑑0

𝑐− 𝑇𝑟 − 𝑇

2

)= 1 + 𝑏𝑓1𝛼

(𝑡− 2𝑑0

𝑐

). (22)

Subtracting 𝑏𝑓1𝛼𝑡 from both sides, we get

𝛼− 1− 𝛼𝑏𝑓1𝑇𝑟 − 𝑇

2=

2𝛼𝑏𝑓1𝑐

(𝑑0 − 𝑑0), (23)

or equivalently,

−2𝑣 − 𝑏𝑓1𝑣𝑇 = 2𝛼𝑏𝑓1(𝑑0 − 𝑑0). (24)

Define the wideband range bias as ΔwHFM = 𝑑0 − 𝑑0, we get

ΔwHFM =

−2𝑣 − 𝑏𝑓1𝑣𝑇

2𝛼𝑏𝑓1=

𝑣𝑇𝑓2𝛼(𝑓2 − 𝑓1)

− 𝑣𝑇

2𝛼. (25)

where ΔwHFM goes to zero as it should when 𝑣 = 0. It is not

surprising that the first term of ΔwHFM is equal to (𝜏1𝑐/2),

where 𝜏1 is defined in (8), while the second one is exactly(𝑇 − 𝑇𝑟)𝑐/2. Define 𝑓𝑐 = (𝑓1 + 𝑓2)/2, the bias is recast as

ΔwHFM =

𝑣𝑇𝑓𝑐𝛼(𝑓2 − 𝑓1)

+𝑣𝑇 (𝑓2 − 𝑓1)

2𝛼(𝑓2 − 𝑓1)− 𝑣𝑇

2𝛼=

𝑣𝑇𝑓𝑐𝛼(𝑓2 − 𝑓1)

.

(26)

Apparently, the wideband range bias ΔwHFM is a nonlinear

function of range-rate 𝑣 since 𝛼 = 1−2𝑣/𝑐. The nonlinearity is

2We numerically find that the left and right alignments have larger approx-imation errors than the central alignment.

IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 5

due to the nonlinearity of the range-Doppler ridge. Moreover,we see that Δw

HFM is a linear function of 𝑇 : a large 𝑇 willresult in significant bias.

The derivation of ΔwHFM requires perfect detection. Let

𝑅𝑤(𝑣) denote the amplitude of the AF ridge at range-rate𝑣 without the consideration of receiver noise. The power lossratio for an HFM with rectangular envelope can be looselyapproximated as [4]

𝜉𝑤(𝑣) ≜𝑅𝑤(𝑣)

𝑅𝑤(0)⋍ 1− 2∣𝑣∣

𝑐

∣∣∣∣12 +1

𝑏𝑓1𝑇

∣∣∣∣ = 1− ∣𝑣∣2𝑓𝑐𝑐𝐵

, (27)

where 0 ≤ 𝜉𝑤(𝑣) ≤ 1, so that if 𝜉𝑤(𝑣) > 𝜉0 is believed toguarantee a reliable detection, one should ensure that

∣𝑣∣ < 𝑐𝐵(1− 𝜉0)

2𝑓𝑐(28)

when applying (26) in wideband bias modeling.The instantaneous frequency based model has a clear phys-

ical meaning. Furthermore, its computational complexity issignificantly reduced. Nevertheless, if a weighting windowis employed or the pulse envelope is not rectangular, theperformance of this approach will degrade.

D. Special Case: Narrowband and Low Speed Modeling

For low speed target ranging3, the classical narrowbandsignal model assumes that 1) the pulse width variation dueto the range-rate is negligible, and 2) the Doppler uniformlyshifts all the frequency components. Ignoring receiver noiseand propagation loss, the received signal is written as [23]

𝑟𝑛(𝑡, 𝑣) = 𝑠(𝑡− 𝜏0) exp(𝑗2𝜋𝑓𝑑𝑡) (29)

= 𝐴(𝑡− 𝜏0) exp

[𝑗2𝜋

𝑏

(ln(1 + 𝑏𝑓1(𝑡− 𝜏0)

)+ 𝑏𝑓𝑑𝑡

)],

where 𝑓𝑑 denotes the Doppler. One could still employ theinstantaneous frequency approach to obtain the range bias of𝑟𝑛(𝑡, 𝑣). However, it is somewhat complex, as 𝑓𝑑 should beproperly converted to a ‘delay’ within the ln(⋅) function. Asan alternative, we directly simplify (26). Let 𝑣 ≪ 𝑐, and thenwe have 𝛼 = 1 − 2𝑣/𝑐 ≃ 1. Therefore, the range bias of aHFM is approximated as

ΔnHFM ⋍ lim

𝛼→1Δw

HFM =𝑓𝑐𝑇

𝑓2 − 𝑓1𝑣 = 𝜆𝑣, (30)

where 𝜆 ≜ 𝑓𝑐𝑇/(𝑓2 − 𝑓1) is a waveform dependent constant.Clearly, the range bias Δn

HFM is approximately linear withrespect to the range-rate 𝑣. It is interesting to find that Δn

HFM

is exactly the same as that for an LFM with the identicalfrequency and time spans [14]. This again verifies the factthat the HFM and LFM waveforms share similar performancefor the narrowband radar application. The power loss ratiofor a narrowband LFM in matched filtering is approximatelywritten as [4]

𝜉𝑛(𝑣) ⋍ 1− ∣𝑓𝑑∣𝐵

= 1− ∣𝑣∣2𝑓𝑐𝐵𝑐

. (31)

Due to the similarity between the narrowband HFM and LFM,(31) can be used to reliably bound the range-rate for (30).

3Here, a low speed target indicates that 𝑣/𝑐 is very small.

We emphasize that (30) requires both the narrowband andlow speed preconditions. It works well for a narrowband radaras (𝑣/𝑐) is almost zero even for a fast maneuvering target.For a narrowband sonar, we find (30) not as good as (26)in range bias fitting if the range-rate is large. In brief, (26)is a safe choice for sonar systems, regardless of bandwidthspecifications.

E. An Application

The range bias characteristic of LFM was used to resolveclose targets [14]. In this part, the idea will be applied towideband HFMs. Suppose that two targets, say 𝐴 and 𝐵, aremoving along the same path. Let 𝑑𝐴0 and 𝑑𝐴0 denote the trueand extracted ranges, respectively, for target 𝐴, while 𝑑𝐵0 and𝑑𝐵0 represent those for target 𝐵. The extracted range differencebetween these two targets is

𝑑𝐴0 − 𝑑𝐵0 =(𝑑𝐴0 +Δw,𝐴

HFM

)− (𝑑𝐵0 +Δw,𝐵

HFM

)(32)

=(𝑑𝐴0 − 𝑑𝐵0

)+

𝑐𝑇𝑓𝑐𝑓2 − 𝑓1

( 𝑣𝐴

𝑐− 2𝑣𝐴− 𝑣

𝐵

𝑐− 2𝑣𝐵

),

where Δw,𝐴HFM and Δw,𝐵

HFM denote the range biases for 𝐴 and𝐵, respectively, while 𝑣

𝐴and 𝑣

𝐵stand for their range-rates.

As∂

∂𝑣

( 𝑣

𝑐− 2𝑣

)=

𝑐

(𝑐− 2𝑣)2> 0, (33)

𝑣𝑐−2𝑣 is an increasing function of 𝑣. This yields

𝑣𝐴

𝑐− 2𝑣𝐴

>𝑣𝐵

𝑐− 2𝑣𝐵

⇐⇒ 𝑣𝐴> 𝑣

𝐵

𝑣𝐴𝑐− 2𝑣

𝐴

<𝑣𝐵

𝑐− 2𝑣𝐵

⇐⇒ 𝑣𝐴 < 𝑣𝐵 ,(34)

where “⇐⇒” means equivalence. If(𝑑𝐴0 −𝑑𝐵0

)and (𝑣

𝐴− 𝑣

𝐵)

share the same sign, a HFM+, of which 𝑓2 > 𝑓1, can induce∣∣𝑑𝐴0 − 𝑑𝐵0∣∣ > ∣∣𝑑𝐴0 − 𝑑𝐵0

∣∣. (35)

Therefore, the extracted difference in the range profile isamplified, and the two targets are more distinguishable, es-pecially when they are close [14]. This situation occurs whena fast object leads a slow one in the same path: for example,launching a fast and small vehicle from a slow underwaterplatform. On the other hand, a HFM− may be useful todistinguish two targets, where the slow one leads the fast one.A possible application is collision avoidance.

IV. NUMERICAL RESULTS FOR BIAS MODELING

A. Wideband Sonar Examples

Three sonar examples will be given to verify the accuracyof wideband bias modeling, where the HFM+ and HFM−

share the same frequency span and pulse width. The waveformpropagation speed is 𝑐 = 1500 m/s (i.e. underwater acoustic).

The first part reveals the dependence of range biases onrange-rate 𝑣, with fixed waveform parameters: 6 kHz ≤𝑓𝑠(𝑡) ≤ 10 kHz and 𝑇 = 25 ms. The extracted andapproximated range biases with polynomial regression andinstantaneous frequency based approaches are shown in Fig.4(a), while the model errors for the HFM+ are depicted in

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−3

−2

−1

0

1

2

3

4

range−rate (m/s)

rang

e bi

ases

(m

)

ExtractedInstantaneous frequencyPolynomial regression

HFM+

HFM−

(a) range biases

−60 −40 −20 0 20 40 60−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

range−rate (m/s)

mod

el e

rror

s fo

r H

FM

+ (

m)

Instantaneous frequencyPolynomial regression

(b) model errors for the HFM+

Fig. 4. Range biases as functions of range-rate for the wideband HFMwaveforms: (a) the extracted and modeled range biases, (b) the model errorsof HFM+. The HFM+ and HFM− share the same frequency span andpulse width. Note that the comparison of model errors for the HFM− issimilar to that for HFM+.

Fig. 4(b). The order of the polynomial is 4. Clearly, the twomethods both have good modeling accuracy.

In the second example, the range-rate, pulse width, andstart frequency are set as 𝑣 = 30 m/s, 𝑇 = 30 ms,and min{𝑓1, 𝑓2} = 6 kHz, respectively, while the end offrequency max{𝑓1, 𝑓2} varies from 10 kHz to 30 kHz. Theextracted and approximated range biases are illustrated inFig. 5(a). The last one uses frequency span and range-rateas 1 kHz ≤ 𝑓𝑠(𝑡) ≤ 2 kHz and 𝑣 = 35 m/s, while pulsewidth 𝑇 varies from 20 ms to 80 ms. The results are in Fig.5(b). Obviously, these approximations are good too.

Based on the observed and analyzed results, the character-istics of wideband range bias for HFMs are that:

∙ The range bias ΔwHFM is a nonlinear function of range-

rate 𝑣, while ∣ΔwHFM∣ is a monotonically increasing

function of ∣𝑣∣. The second point is also physically sound,as a fast moving target usually introduces more bias.

∙ ΔwHFM and 𝑣 are positively correlated for a HFM+, and

they are negatively correlated for a HFM−.

10 15 20 25 30−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

max{f1,f

2} kHz

rang

e bi

ases

(m

)

ExtractedInstantaneous frequency

HFM+

HFM−

(a) bandwidth varying

20 30 40 50 60 70 80−5

−4

−3

−2

−1

0

1

2

3

4

5

pulse width (ms)

rang

e bi

ases

(m

)

ExtractedInstantaneous frequency

HFM+

HFM−

(b) pulse width varying

Fig. 5. Range biases as functions of (a) bandwidth and (b) pulse width forthe wideband HFM waveforms, where HFM+ and HFM− share the samefrequency span and pulse width.

∙ For a given 𝑣, we have ∣ΔwHFM+ ∣ ⋍ ∣Δw

HFM− ∣. This canbe easily perceived as the range biases of HFM+ andHFM− in Figs. 4 and 5 being all symmetric about theline Δw

HFM = 0. Based on (26), we have

∣∣ΔwHFM+

∣∣ = ∣𝑣∣𝑇𝑓𝑐𝛼

⋅ 1

∣𝑓2 − 𝑓1∣ =∣∣Δw

HFM−∣∣,

which is consistent with the observations.∙ Let 𝑣 > 0, we have ∣Δw

HFM(𝑣)∣ > ∣ΔwHFM(−𝑣)∣, which

can be observed from Fig. 4(a). Based on (26), we know

∣∣ΔwHFM

∣∣ = ∣𝑣∣𝑇𝑓𝑐∣𝑓2 − 𝑓1∣ ⋅

𝑐

𝑐− 2𝑣.

Clearly, a positive 𝑣 will lead to larger bias.∙ ∣Δw

HFM∣ is an increasing function of pulse width 𝑇 , buta decreasing function of bandwidth 𝐵.

The nonlinear relation between ΔwHFM and 𝑣 is attributed to

the nonlinearity of the AF ridge.

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−60 −40 −20 0 20 40 60

−6

−4

−2

0

2

4

6

range−rate (m/s)

rang

e bi

as (

m)

ExtractedInstantaneous frequencyPolynomial regression

HFM+

HFM−

(a) range bias

−60 −40 −20 0 20 40 60

−10

−8

−6

−4

−2

0

2

4

6

8

10

range−rate (m/s)

rang

e bi

as (

m)

ExtractedInstantaneous frequencyPolynomial regression

HFM+

HFM−

(b) range bias

Fig. 6. Range biases as functions of range-rate for narrowband HFMwaveforms: (a) frequency span 10 ∼ 11 kHz, (b) frequency span 15 ∼ 16kHz. The HFM+ and HFM− share the same frequency span and pulsewidth.

B. Narrowband Sonar Examples

Two sonar examples will be given to verify the accuracy of(26) for narrowband transmission scenarios. The pulse widthis chosen as 𝑇 = 10 ms, while the frequency spans are10 ∼ 11 kHz and 15 ∼ 16 kHz, respectively. The extractedand approximated range biases with polynomial regression andinstantaneous frequency based approaches are shown in Fig.6. Clearly, the accuracy of those approaches still holds for thenarrowband cases.

V. TARGET TRACKING WITH HFM WAVEFORMS

Although the range-Doppler coupling feature of an HFMhas been recognized for decades [4]–[7], its effect on targettracking is less well known. This section utilizes the obtainedbias models to modify the measurement equations of trackersso as to improve the tracking accuracy. Since we assumethat pulse envelopes are rectangular and that no weightingwindow is used in matched filtering in this paper, only theinstantaneous frequency based results are employed in trackermodification.

A. Tracking without Range Bias CompensationThe tracking problem is investigated in one dimension,

where range and range-rate compose the state vector 𝒙𝑘 =[𝑑𝑘, 𝑣𝑘]

𝑇 at time 𝑘. A discrete white noise acceleration(DWPA) model [13, p. 273] is employed, of which the dynamicand measurement equations are respectively given as

𝒙𝑘+1 = 𝑭𝒙𝑘 + Γ𝑢𝑘 and 𝑧𝑘 = 𝑯𝒙𝑘 + 𝑤𝑘, (36)

where 𝑢𝑘 ∼ 𝒩 (0, 𝜎2𝑢) and 𝑤𝑘 ∼ 𝒩 (0, 𝜎2

𝑤) denote the whiteprocess and measurement noises, respectively,

𝑭 =

[1 𝑇𝑠

0 1

], (37)

Γ =[𝑇 2𝑠 /2 𝑇𝑠

]𝑇, (38)

and 𝑇𝑠 denotes the pulse repetition time. Without range biascompensation, the measurement matrix is written as

𝑯 = [1, 0], (39)

where the extracted range is suboptimally considered as thetrue range plus noise.

The Kalman filtering equations associated with the state andmeasurement models are given by two basic stages: prediction

𝒙𝑘∣𝑘−1 = 𝑭𝒙𝑘−1∣𝑘−1 (40)

𝑷 𝑘∣𝑘−1 = 𝑭𝑷 𝑘−1∣𝑘−1𝑭𝑇 + 𝜎2

𝑣ΓΓ𝑇 (41)

and update:

𝑮𝑘 = 𝑷 𝑘∣𝑘−1𝑯𝑇 [𝑯𝑷 𝑘∣𝑘−1𝑯

𝑇 + 𝜎2𝑤]

−1

= 𝑷 𝑘∣𝑘𝑯𝑇𝜎−2

𝑤 (42)𝒙𝑘∣𝑘 = 𝒙𝑘∣𝑘−1 +𝑮𝑘[𝑧𝑘 −𝑯𝒙𝑘∣𝑘−1] (43)𝑷 𝑘∣𝑘 = [𝑰 −𝑮𝑘𝑯]𝑷 𝑘∣𝑘−1 (44)

where 𝑮𝑘 denotes the Kalman filter gain, and 𝑷 𝑖∣𝑗 stands forprediction or state covariance, where the subscript 𝑖∣𝑗 denotes“of time 𝑖 given measurements up to and including 𝑗”.

B. Tracking with Range Bias CompensationAs for a tracking system with bias compensation, the

dynamic equation is still the same as that in the previoussubsection. The measurement equation, however, is nonlinearand it is modified as

𝑧𝑘 = ℎ(𝒙𝑘) + 𝑤𝑘. (45)

Here ℎ(𝒙𝑘) includes both the true range and range bias.Recalling (26), ℎ(𝒙𝑘) is written as

ℎ(𝒙𝑘) = 𝑑𝑘 +𝑣𝑘𝑇𝑓𝑐

𝛼𝑘(𝑓2 − 𝑓1), (46)

where 𝛼𝑘 = 1 − 2𝑣𝑘/𝑐. An immediate tool for nonlineartracking is the extended Kalman filter. Since the dynamicequation is the same as that for the narrowband, the predictionformulas are identical to (40) and (41). The update equationsare modified as

𝑮𝑘 = 𝑷 𝑘∣𝑘−1𝑯𝑇𝑘 [𝑯𝑘𝑷 𝑘∣𝑘−1𝑯

𝑇𝑘 + 𝜎2

𝑤]−1

= 𝑷 𝑘∣𝑘𝑯𝑇𝑘 𝜎

−2𝑤 (47)

𝒙𝑘∣𝑘 = 𝒙𝑘∣𝑘−1 +𝑮𝑘[𝑧𝑘 − ℎ(𝒙𝑘∣𝑘−1)] (48)𝑷 𝑘∣𝑘 = [𝑰 −𝑮𝑘𝑯𝑘]𝑷 𝑘∣𝑘−1, (49)

IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 8

where 𝑯𝑘 are the first-order derivative of ℎ(⋅) at 𝒙𝑘∣𝑘−1. As∂ℎ(𝒙𝑘)∂𝑑𝑘

= 1, 𝑯𝑘 is written as

𝑯𝑘 = [ 1 ∂ℎ(𝒙)∂𝑣

]∣∣∣𝒙=𝒙𝑘∣𝑘−1

, (50)

where∂ℎ(𝒙)

∂𝑣=

𝑇𝑓𝑐𝛼(𝑓2 − 𝑓1)

+2𝑣𝑇𝑓𝑐

𝑐𝛼2(𝑓2 − 𝑓1)=

𝑇𝑓𝑐𝛼2(𝑓2 − 𝑓1)

. (51)

For the narrowband radar special case, as ℎ(𝒙𝑘) can besimplified as

ℎ(𝒙𝑘) = 𝑑𝑘 + 𝜆𝑣, (52)

the measurement equation is recast as

𝑧𝑘 = ��𝒙𝑘 + 𝑤𝑘, (53)

where �� ≜ [1, 𝜆] is time invariant. Note that (53) sharesthe same form as that for a narrowband LFM based trackingsystem [14]–[17], and the optimal filter is Kalman.

VI. NUMERICAL RESULTS FOR TARGET TRACKING

This section presents three sonar tracking examples. Tostress the system diversity, three distinct parameter sets areused:

∙ 𝒫1 = {7 kHz ≤ 𝑓 ≤ 11 kHz, 𝑇 = 35 ms, ∣𝑣∣ = 5 m/s};∙ 𝒫2 = {6 kHz ≤ 𝑓 ≤ 8 kHz, 𝑇 = 50 ms, ∣𝑣∣ = 25 m/s};∙ 𝒫3 = {1 kHz ≤ 𝑓 ≤ 2 kHz, 𝑇 = 100ms, ∣𝑣∣ = 35m/s}.

Each set has two waveforms, a HFM+ and a HFM−, whichshare the same frequency and time supports. The target canhave either positive or negative range-rate, with the same ab-solute value. The acoustic propagation speed is 𝑐 = 1500 m/s.The tracking performances with and without bias compensa-tion are depicted in Fig. 7. From those figures, we observethat:

∙ The range tracking accuracy can be improved after biascompensation for either HFM+ or HFM−.

∙ With bias compensation, the range tracking accuracy ofHFM+ is slightly better than that of HFM−; however,neither is uniformly better than the other without com-pensation.

∙ Without bias compensation, the range tracking accuracyfor a positive range-rate is slightly worse than that for anegative one, since the former usually introduces morerange bias as depicted in Section IV A.

References [16] and [17] gave an intuitive explanation of thesecond observation. For a HFM+, the true range and range-rate are negatively correlated in the measurement equation,while those for a HFM− are positively correlated. As therange and range-rate are positively correlated in the dynamicequation, the HFM+ can partially cancel the tracking errorswhile the HFM− cannot. Interestingly, the HFM+ and HFM−

share the same measurement mean square errors but result indifferent tracking performance. This again verifies that obser-vations with the same measurement quality do not necessarilyresult in the same tracking performance [16] [17] [21].

In brief, an up-sweep HFM waveform is a good choice for asonar system from a tracking perspective, as long as its rangebias can be properly modeled and compensated.

VII. CONCLUSIONS

Hyperbolic frequency modulation (HFM) is an often-usedDoppler insensitive waveform for target probing, and itpreserves this property in both narrowband and widebandcases. Doppler insensitivity can mitigate detection degradationagainst a moving target, however, at the cost of introducingsome range bias. In this paper, we have comprehensivelyinvestigated the range bias characteristics of HFM waveforms,and have proffered (26) and (30), the simple approximationsthat work well over a large range of relative signal bandwidths.Significant target tracking improvement can be obtained whenthese formulas are used to compensate the range bias of HFMwaveforms.

Finally, we note that the underwater environment is quitedifferent from that enjoyed by the radar signals in which range-Doppler coupling is usually studied: active sonar surveillancesuffers from sound speed, propagation uncertainties, etc., andin some cases these can be quite large, much larger than theeffects dealt with in this paper. However, our range-Dopplercoupling can be pernicious, an example being the range-ratedependent “noise” offset as opposed to a relatively benignbias from a sound-speed error. In some cases the couplingthat interests us here would be a minor effect; but at shortranges it can be very important.

REFERENCES

[1] N. Levanon and E. Mozeson, Radar Signals. John Wiley & Sons, Inc.,Hoboken, New Jersey, 2004.

[2] E. J. Kelly and R. P. Wishner, “Matched-filter theory for high-velocity,accelerating targets,” IEEE Trans. Military Electron., vol. 9, no. 1, pp.56–69, Jan. 1965.

[3] A. W. Rihaczek, “Doppler-tolerant signal waveforms,” Proc. IEEE,vol. 54, no. 6, pp. 849–857, Jun. 1966.

[4] J. J. Kroszczynski, “Pulse compression by means of linear-periodmodulation,” Proc. IEEE, vol. 57, no. 7, pp. 1260–1266, Jul. 1969.

[5] P. R. Atkins, T. Collins, and K. G. Foote, “Transmit-signal design andprocessing strategies for sonar target phase measurement,” IEEE J. Sel.Topics Signal Process., vol. 1, no. 1, pp. 91–104, Jun. 2007.

[6] Y. Doisy, L. Deruaz, S. P. Beerens, and R. Been, “Target Dopplerestimation using wideband frequency modulated signals,” IEEE Trans.Signal Process., vol. 48, no. 5, pp. 1213–1224, May 2000.

[7] D. W. Tufts, H. Ge, and S. Umesh, “Fast maximum likelihood estimationof signal parameters using the shape of the compressed likelihoodfunction,” IEEE J. Ocean. Eng., vol. 18, no. 4, pp. 388–400, Oct. 1993.

[8] J. J. G. McCue, “Modulation diversity for pulse-compressing radars,”IEEE Trans. Aerosp. Electron. Syst., vol. 13, no. 5, pp. 541–544, Sep.1977.

[9] R. A. Altes and E. L. Titlebaum, “Bat signals as optimally Dopplertolerant waveforms,” J. Acoust. Soc. Amer., vol. 48, no. 4, pp. 1014–1020, Oct. 1970.

[10] A. Papandreou, F. Hlawatsch, and G. F. Boudreaux-Bartels, “The hyper-bolic class of quadratic time-frequency representations, Part I: Constant-Q warping, the hyperbolic paradigm, properties, and members,” IEEETrans. Signal Process., vol. 41, no. 12, pp. 3425–3444, Dec. 1993.

[11] F. Hlawatsch, A. Papandreou-Suppappola, and G. F. Boudreaux-Bartels,“The hyperbolic class of quadratic time-frequency representations, PartII: Subclasses, intersection with the affine and power classes, regularity,and unitarity,” IEEE Trans. Signal Process., vol. 45, no. 2, pp. 303–315,Feb. 1997.

[12] B. K. Newhall, “Continuous reverberation response and comb spectrawaveform design,” IEEE J. Ocean. Eng., vol. 32, no. 2, pp. 524–532,Apr. 2007.

[13] Y. Bar-Shalom, X. R. Li, and T. Kirubarajan, Estimation with Applica-tion to Tracking and Navigation. Wiley-Interscience, 2001.

[14] R. J. Fitzgerald, “Effects of range-Doppler coupling on Chirp radartracking accuracy,” IEEE Trans. Aerosp. Electron. Syst., vol. 10, no. 4,pp. 528–532, Jul. 1974.

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0.35

0.4

0.45

0.5

0.55

time(s)

rang

e R

MS

Es

(m)

HFM+ and positive v

HFM+ and negative v

HFM− and positive v

HFM− and negative v

(a) 𝒫1, without bias compensation

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time(s)

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Es

(m)

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HFM− and negative v

HFM+ and positive v

HFM+ and negative v

(b) 𝒫1, with bias compensation

20 25 30 35 40 45 50 55 603.5

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5.5

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e R

MS

Es

(m)

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HFM+ and negative v

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0.3

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e R

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Es

(m)

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HFM− and negative v

HFM+ and positive v

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(d) 𝒫2, with bias compensation

20 25 30 35 40 45 50 55 604

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5

5.5

6

6.5

7

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Es

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0.7

0.75

0.8

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rang

e R

MS

Es

(m)

HFM− and positive v

HFM− and negative v

HFM+ and positive v

HFM+ and negative v

(f) 𝒫3, with bias compensation

Fig. 7. Tracking performance of wideband HFM waveforms with and without bias compensation: (a,c,e) range RMSEs without bias compensation, (b,d,f)range RMSEs with bias compensation. Each curve is obtained with 1000 runs.

IEEE JOURNAL OF OCEANIC ENGINEERING, VOL.XXX, NO.XXX, XXX 10

[15] W. Wong and W. D. Blair, “Steady-state tracking with LFM waveforms,”IEEE Trans. Aerosp. Electron. Syst., vol. 36, no. 2, pp. 701–709, Apr.2000.

[16] X. Song, P. Willett, and S. Zhou, “Posterior Cramer-Rao bounds forDoppler biased multistatic range-only tracking,” in Proc. of Intl. Conf.on Information Fusion, Chicago, IL, Jul. 2011.

[17] ——, “Posterior Cramer-Rao bounds for Doppler biased distributedtracking,” Journal of Advances in Information Fusion, vol. 7, no. 1,pp. 16–27, Jun. 2012.

[18] S. Coraluppi and C. Carthel, “Distributed tracking in multistatic sonar,”IEEE Trans. Aerosp. Electron. Syst., vol. 41, no. 3, pp. 1138–1147, Jul.2005.

[19] W. R. Blanding, P. K. Willett, Y. Bar-Shalom, and R. S. Lynch, “Covertsonar tracking,” in Proc. of IEEE Aerospace Conf., Big Sky, MT, Mar.2005, pp. 2053–2062.

[20] S. A. Kramer, “Doppler and acceleration tolerances of high-gain, wide-band linear FM correlation sonars,” Proc. IEEE, vol. 55, no. 5, pp.627–636, May 1967.

[21] R. Niu, P. Willett, and Y. Bar-Shalom, “Tracking considerations inselection of radar waveform for range and range-rate measurements,”IEEE Trans. Aerosp. Electron. Syst., vol. 38, no. 2, pp. 467–487, Apr.2002.

[22] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differ-ential Equations. Springer, 1994.

[23] L. Weiss, “Wavelets and wideband correlation processing,” IEEE SignalProcess. Mag., vol. 11, no. 1, pp. 13–32, Jan. 1994.

Xiufeng Song (S’08) received the B.S. degree fromXidian University, Xi’an, China, in 2005 and theM.S. degree from Institute of Electronics, ChineseAcademy of Sciences (CAS), Beijing, China, in2008, both in electrical engineering. He is currentlyworking towards the Ph.D. degree with the De-partment of Electrical and Computer Engineering,University of Connecticut, Storrs.

His research interests lie in signal processing,detection, estimation, and tracking.

Peter Willett (F’03) received his BASc (Engineer-ing Science) from the University of Toronto in 1982,and his PhD degree from Princeton University in1986. He has been a faculty member at the Univer-sity of Connecticut ever since, and since 1998 hasbeen a Professor. His primary areas of research havebeen statistical signal processing, detection, machinelearning, data fusion and tracking. He has interests inand has published in the areas of change/abnormalitydetection, optical pattern recognition, communica-tions and industrial/security condition monitoring.

He was editor-in-chief for IEEE Transactions on Aerospace and ElectronicSystems from 2006-2011, and is presently the VP of Publications for theIEEE AESS. In the past he has been associate editor for three activejournals - IEEE Transactions on Aerospace and Electronic Systems (forData Fusion and Target Tracking) and IEEE Transactions on Systems, Man,and Cybernetics, parts A and B. He is also associate editor for the IEEEAES Magazine, associate editor for ISIF’s electronic Journal of Advances inInformation Fusion. He was General Co-Chair (with Stefano Coraluppi) forthe 2006 ISIF/IEEE Fusion Conference in Florence, Italy, Executive Chair ofthe 2008 Fusion Conference in Cologne, and Emeritus Chair for the 2011Fusion Conference in Chicago. He was Program Co-Chair (with EugeneSantos) for the 2003 IEEE Conference on Systems, Man & Cyberneticsin Washington DC, and Program Co-Chair (with Pramod Varshney) for the1999 Fusion Conference in Sunnyvale. He has been a member of the IEEESignal Processing Society’s Sensor-Array & Multichannel (SAM) technicalcommittee since 1997, and both serves on that TC’s SAM conferences’program committees and maintains the SAM website.

Shengli Zhou (SM’11) received the B.S. degreein 1995 and the M.Sc. degree in 1998, from theUniversity of Science and Technology of China(USTC), Hefei, both in electrical engineering andinformation science. He received his Ph.D. degreein electrical engineering from the University ofMinnesota (UMN), Minneapolis, in 2002.

He has been an assistant professor with the De-partment of Electrical and Computer Engineeringat the University of Connecticut (UCONN), Storrs,2003-2009, and now is an associate professor. He

holds a United Technologies Corporation (UTC) Professorship in EngineeringInnovation, 2008-2011. His general research interests lie in the areas ofwireless communications and signal processing. His recent focus is onunderwater acoustic communications and networking.

Dr. Zhou served as an associate editor for IEEE Transactions on WirelessCommunications, Feb. 2005 – Jan. 2007, and IEEE Transactions on SignalProcessing, Oct. 2008 – Oct. 2010. He is now an associate editor forIEEE Journal of Oceanic Engineering. He received the 2007 ONR YoungInvestigator award and the 2007 Presidential Early Career Award for Scientistsand Engineers (PECASE).