Performance Limits and Geometric Properties
of Array Localization
Abstract
Locationaware networks are of great importance and interest in both civil and military applications. This paper determines the localization accuracy of an agent, which is equipped with an antenna array and localizes itself using wireless measurements with anchor nodes, in a farfield environment. In view of the CramérRao bound, we first derive the localization information for static scenarios and demonstrate that such information is a weighed sum of Fisher information matrices from each anchorantenna measurement pair. Each matrix can be further decomposed into two parts: a distance part with intensity proportional to the squared baseband effective bandwidth of the transmitted signal and a direction part with intensity associated with the normalized anchorantenna visual angle. Moreover, in dynamic scenarios, we show that the Doppler shift contributes additional direction information, with intensity determined by the agent velocity and the root mean squared time duration of the transmitted signal. In addition, two measures are proposed to evaluate the localization performance of wireless networks with different anchoragent and arrayantenna geometries, and both formulae and simulations are provided for typical anchor deployments and antenna arrays.
I Introduction
Localization is of great importance with a wide variety of civil and military applications such as navigation, mobile network services, autonomous vehicles, social networking, and seeking and targeting people [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The global positioning system (GPS) is the most prominent technology to provide locationaware services, but its effectiveness is severely degraded in harsh environments, e.g., in buildings, urban canyons and undergrounds [1, 2, 3]. Localization using a wireless network is a feasible alternative to overcome the GPS limitation since location information can be obtained with the aid of a network that consists of anchor nodes with known position and agent nodes aiming to estimate self positions.
Typically, a localization task is achieved by the radio communications between anchors and agents, which are equipping with a single antenna or an antenna array. By processing the received signals, relevant signal metrics can be extracted for localization, for example, timeofarrival (TOA)[11, 12, 13, 14], timedifferenceofarrival (TDOA)[15, 16, 17], angleofarrival (AOA)[18, 19, 20, 21], and received signal strength (RSS)[22, 23, 24]. Among these signal metrics, TOA and AOA are the two widely used in practice. TOA is a timebased metric obtained via measuring the signal propagation time between the anchor and agent; then the distance measurements translate to location information by trilateration [25]. AOA is a metric characterizing the arriving direction of the signal at the agent, and it can be obtained using an array of antennas and spatial filtering; then the angle measurements translate to location information by triangulation[18]. Techniques using a combination of these signal metrics, such as many hybrid TOA/AOA systems, have also been studied in literature [17, 26].
In practical scenarios, the transceived signals encounter nonideal phenomena such as noise, fading, shadowing, multipath (signal reaches the receiver via multiple paths due to reflection) and nonlineofsight (NLOS) propagation (the first arriving signal does not travel on a straight line)[8], and therefore the location estimates are subject to uncertainty. In the interest of system design and operation, it is important to know the best attainable localization accuracy and the corresponding approaches to achieve such accuracy, which can be rephrased as obtaining the lower bound for localization errors and the achievability result in the language of information theory. For example, in designing the energyefficient locationaware networks, attainable localization accuracy can be a meaningful performance objective to optimize [27, 28, 29, 30, 31, 32, 33]. To evaluate the localization performance in the presence of uncertainty, some studies consider specific systems that employ certain signal metrics extracted from the received waveforms, e.g., the time delay or the angle, and then determine the localization error based on the joint distribution of these metrics[18, 19]. However, the extracted metrics may discard useful information for localization, resulting in suboptimal localization performance. To address this issue, recent studies directly utilize the received waveforms [17, 3] to exploit all relevant information and derive fundamental limits for localization accuracy. For this purpose, the most commonly used tool is the CramérRao lower bound (CRLB) [18, 34, 19, 17, 35] due to its intriguing property in asymptotic statistics[36] in the sense of the Hájek convolution theorem [37] and the HájekLe Cam local asymptotic minimax theorem [38], though some other bounds such as the Barankin bound[39] and ZivZakai bound[40] are also used.
For general wideband systems, the fundamental limits of localization accuracy for a single agent has been obtained in [3] in terms of the CRLB, which are generalized to a cooperative framework with multiple agents [34]. Moreover, it has been shown in [3] that AOA measurements obtained by wideband antenna arrays do not further improve position accuracy beyond that provided by TOA measurements, which implies that it suffices to use TOA measurements in wideband systems. However, the approaches developed for wideband systems are not applicable to commonly used modulationbased communication systems, because those systems modulate a baseband signal onto a carrier frequency with an unknown initial phase[41, 42, 43, 44]. Consequently, unlike wideband systems, not the entire passband signals can be used for TOA measurements due to the unknown phase in modulationbased communication systems. Nevertheless, in farfield environments, by using an antenna array, the carrier phases can be exploited for AOA measurements to improve the localization as widely recognized [8]. These inherent differences from wideband systems call for a comprehensive investigation for the performance limits of localization accuracy in general wireless localization systems with antenna arrays.
In dynamic scenarios that involve moving objects, the localization or tracking accuracy can be characterized by the Posterior CRLB (PCRLB) [45, 46, 47]. Since the derivation of PCRLB relies on multiple snapshots and measurable uncertainties of both observations and hidden states of timevarying locations, the instantaneous localization error of the mobile user is required as an input of PCRLB. To obtain the instantaneous localization error in the dynamic scenario, the Doppler shift may be favored as another source for localization in addition to the TOA and AOA measurements. Most existing research treats the Doppler shift as the frequencyshift solely on the carrier frequency and obtain the performance bounds accordingly[42, 43, 44], whereas the effects of Doppler shift on the baseband signal are simply neglected. Moreover, the accuracy limit of navigation in general wireless networks has only been obtained in the form of block matrices without considering Doppler shift [47]. The effect of Doppler shift to the localization information still remains underexplored.
Upon obtaining the localization accuracy, a natural question arises concerning the optimal geometry when deploying anchors and designing antenna arrays. A few studies have optimized the anchoragent geometry by minimizing the condition number of the visibility matrix [48, 49], while some others focused on a CRLBrelated cost[50, 51, 52, 53], where the anchoragent geometry is optimized in an isotropic source localization. In [54], the arrayantenna geometry was also studied jointly with the anchoragent geometry, and it is proven optimal to place anchors and antennas symmetrically on two circles, respectively. However, existing studies do not provide simple measures to compare two arbitrary geometric structures.
In this paper, we develop a general array localization system with Doppler shifts and use the CRLB to determine the performance limits of localization accuracy in a farfield environment. We also propose two measures to characterize the impact of the anchoragent and arrayantenna geometry on localization accuracy. In particular, we highlight the difference of our model and the wideband model in [3] by introducing the carrier frequency with unknown initial phases in the localization problem. The main contributions are as follows.

We derive the performance limits of localization accuracy for a static agent equipped with an antenna array in terms of the equivalent Fisher information matrix (EFIM). The EFIM can be decomposed as a weighed sum of measuring information matrices, where each matrix contains both distance and direction information with intensities determined by the corresponding anchorantenna measurement pair. Moreover, we show that the direction part can provide dominant information for localization for narrowband signals.

We derive the performance limits of localization accuracy for a moving agent equipped with an antenna array in terms of the EFIM. The Doppler shift is shown to contribute to the direction information with intensity associated with the root mean squared time duration of the transmitted signal, and its contribution to the direction information can be more significant than that of the antenna array.

We propose two measures, i.e., the squared array aperture function and anchor geometric factors, to quantify the effects of anchoragent geometry and arrayantenna geometry on localization, respectively, and give the optimal geometric design of the anchor and antenna locations.
The rest of the paper is organized as follows. In Section II, we describe the system model and formulate the location estimation problem. In Section III, we derive the squared position error bound (SPEB) using EFIM for a static agent, and Section IV generalizes the results for a moving agent. Based on the SPEB and the EFIM, Section V quantifies the impact of the anchoragent and arrayantenna geometries with some examples. Numerical results are given in Section VI, and conclusions are drawn in Section VII.
Notation: We use upper and lower case boldface to denote matrices and vectors, respectively; and denote the complex conjugate and the firstorder derivative of , respectively; and denote the real and imaginary part of a complex number , respectively; denotes the expectation operator with respect to the random vector ; denotes the Löwner semiorder of matrices which means that is positive semidefinite; , , and denote the trace, the transpose, the conjugate transpose and the inverse of matrix ; denotes a submatrix composed of the rows to and the columns to of its argument; denotes the Euclidean norm of its argument; denotes that and is a negligible number far less than 1, and denotes that for any complex column vector (implying that is positive definite); denotes that ; () and denote the dimensional real (complex) vector space and the set of all complex positive definite matrices, respectively.
The notations of frequentlyused symbols are listed as follows.
9.5em
number of anchors and antennas
position of anchor
position of the reference point
position of the th antenna
distance and direction of the th antenna to the reference point
set of anchors with lineofsight (LOS) and NLOS to agent
array orientation and moving direction of the agent
distance, direction and signal propagation time from anchor to the agent
propagation speed of the signal, speed and relative speed of the agent
reference time and corresponding position of the reference point
initial carrier phase of the transmitted signal from anchor
number of multipath components (MPCs) from anchor to the agent
channel gain, range bias and arrivalangle bias from anchor to the agent via th path
signal propagation time from anchor to the th antenna via th path
observation time interval
the entire and the baseband signal
Fourier Transform of
received waveform at antenna from anchor
effective baseband bandwidth, carrier frequency, and basebandcarrier correlation (BCC)
Fisher information matrix (FIM) and EFIM w.r.t. parameter
ranging direction matrix (RDM) with direction
spectral density of noise
received signaltonoise ratio (SNR) in th path from anchor
visual angle and its angular speed from antenna to anchor
information intensity and path overlap coefficient (POC) from anchor
root mean squared time duration
squared array aperture function (SAAF)
Ii System Model
This section presents a detailed description of the system models and formulates the location estimation problem. Two scenarios are considered in this work: the static scenario in which the agent is stationary, and the dynamic scenario in which the agent is moving.
Iia Static Scenario
Consider a 2D wireless network with anchors and one static agent equipped with a rigid antenna array consisting of elements (see Fig. 1). Anchors have perfect knowledge of their positions, denoted by , where is the set of all anchors. The agent aims to estimate its selfposition based on the received waveforms obtained by its arrayantennas from all anchors, and denotes the position of th antenna in the array where .
The array rigidity implies that it has exactly three degrees of freedom, i.e., translations and rotation, and hence it can be characterized by a predetermined reference point and an orientation . Then, by denoting the distance between the reference point and th antenna by , and the direction (relative to oreintation) from the reference point to th antenna by , we can express the position of each antenna as
(1) 
This work focuses on farfield enviroments, where the distances between anchors and the agent are sufficiently larger than the array dimension so that (i) the angles from each anchor to all arrayantennas are identical and (ii) the channel properties from each anchor to all arrayantennas are identical, e.g., the same SNR. Moreover, the phase differences between received signals in adjacent antennas are assumed to be less than so that there is no periodic phase ambiguity (i.e., the array element spacing is smaller than the signal wavelength). We write the propagation time delay and the direction from anchor to the agent (the reference point) as
(2) 
respectively, where is the propagation speed of the signal.
As for the signal model, we consider that anchor transmits a known signal
(3) 
to the agent, where the signal is formed by the quadrature modulation that consists of the baseband signal (also called the complex envelope) , the carrier wave with central frequency , and the initial carrier phase .^{1}^{1}1The quadrature demodulation requires that the baseband signal be bandlimited by , i.e., for all , where is the Fourier Transform of . In practical modulation systems, the initial carrier phase is usually unknown, and hence we model as an unknown parameter in this work.
Our channel model considers both multipath and NLOS propagation phenomena. Specifically, , where denotes the set of anchors providing LOS signals and for those providing NLOS signals. Together with the transmitted signal given in (3), the received waveform at th antenna from anchor can be written as [3, 43]
(4)  
where and are the amplitude and delay of the th path, respectively, and is the number of MPCs, represents the real observation noise modeled as additive white Gaussian noise (AWGN) with twoside power spectral density , and is the observation time interval. In farfield environments, by geometry (as shown in Fig. 1) the time delays can be written as
(5) 
where and are the arrivalangle bias and range bias of the th path, respectively. In particular, for the first path of a LOS signal, we have
(6) 
and otherwise the range biases are positive and the arrivalangle biases can be between to . Note also that in farfield environments, the multipath parameters and do not depend on the choice of the antenna element .
Remark 1
Intuitively speaking, when the initial phase is unknown, the time delays or TOA information is completely corrupted in the carrier phase, and hence only the baseband part can be utilized for obtaining the distance information. Nevertheless, the phase differences between different antennas can cancel out the unknown parameter , leading to useful information about but not . Hence, the direction information can be retrieved from the carrier phases of antennas. This constitutes the key difference from the wideband model in [3] which assumes the initial carrier phases be precisely known and consequently both the distance and direction information can be extracted from the carrier phases.
Throughout this paper, we consider the case where there is no a priori knowledge about the parameters, i.e., all unknown parameters are deterministic and nonBayesian approaches are used.
IiB Dynamic Scenario
Built upon the static scenario, we further consider a system model for the dynamic scenario in which the agent is moving at a constant speed along direction throughout the observation time (see Fig. 2). Denote the position of reference point at reference time by , then the position of the th antenna at time can be written as
(7) 
We still consider farfield environments where the angle from each anchor to all antennas and channel properties (e.g., fading gains and multipath delays) remain timeinvariant throughout the observation time.^{2}^{2}2This model is valid when , where is the channel coherence time. Since [55], the preceding condition translates to , which holds for general practical settings, e.g., m/s, GHz and ms for an LTE example, or GHz, ms for a GSM example. Moreover, if we replace with the effective observation time, which is by the timefrequency duality (cf. Assumption 1 for the definition of ), the previous condition can be written as , which usually holds in practice. Then, similar to (4), the received waveform can be expressed as
(8)  
where the timevariant path delay by the Doppler effect is given by
(9) 
in which denotes the relative speed.
Different from the static scenario, we need to introduce two assumptions to simplify the expressions of the main results.
Assumption 1 (Narrowband Signal)
The baseband signal is bandlimited by , i.e., for all . Furthermore, .
Remark 2
In the dynamic scenario we assume the narrowband signal, while in the static scenario only is required for the quadrature demodulation.
Assumption 2 (Balanced Phase)
The baseband signal has a balanced phase, i.e.,
(10) 
where .
Remark 3
Assumption 2 holds for signals of the form , which is typical in communications given a white stationary ergodic source and the same filter used in I–Q two–way modulation, or for signals with constant envelope modulation and random phase uniformly distributed in . Moreover, note that Assumption 2 is only used for obtaining a simplified result, while the information structure for localization does not rely on this assumption.
IiC Location Estimation and Error Bounds
From a statistical inference perspective, a wellformulated estimation problem involves a parameter set, a statistical experiment, and the random variables generated by this experiment. According to the system setting, the parameter vector to be estimated is given by
(11) 
where
(12) 
(13) 
and denotes if and elsewhere. The random variable generated by our statistical experiment is the vector representation of all the received waveforms obtained from the KarhunenLoeve expansion of , and this statistical experiment can be characterized into the loglikelihood function shown as
(14) 
up to an additive constant. Hence, the estimation problem is to estimate the parameter from the observation according to the known parameterized probability distribution in (14). Note that the received waveforms from different anchors can be perfectly separated at the agent due to some implicit multiple access mechanism, but we remark that our estimation problem and thus the error bounds do not depend on the specific mechanism.^{3}^{3}3For example, in both static and dynamic scenarios, for the timedivision mechanism the likelihood function in (14) remains the same, and for the frequencydivision or the codedivision mechanism it suffices to use different downconversion frequencies (for FDMA) or different baseband signals (for CDMA) for the waveforms from different anchors.
Based on (14), to derive an error bound for this estimation problem, we recall the notion of FIM defined as
(15) 
The wellknown information inequality asserts that, for any unbiased estimator for , we have [56]. It follows that if is an unbiased estimator for , then
(16) 
The righthand side of (16) is defined as the SPEB, cf. [3, Def. 1]. To avoid inverting the FIM with large dimensions, we also adopt the notion of EFIM in [3, Def. 2], where the EFIM for the first components of is defined as , where the original FIM for is expressed as
(17) 
The EFIM retains all the necessary information to derive the information inequality for the parameter vector , in the sense that according to the Schur complement theory.
Iii Localization Accuracy in the Static Scenario
This section determines the localization accuracy in terms of the SPEB and EFIM in the static scenario, and highlights the role that the knowledge of the phase and array orientation plays in the reduction of localization errors. For notational convenience, we define and adopt the notion of RDM [3, Def. 4] given by
(18) 
Iiia Equivalent Complex Passband Signal Model
For the ease of FIM derivation, one may want to remove the operator and favor the following complex passband signal model
(19) 
where is the complex observation noise with both real and imaginary components following the same distribution as , and all other parameters remain the same as those in (4). Then, the corresponding loglikelihood function becomes
(20)  
up to an additive constant. We next show that in the derivation of the FIM, the complex passband signal model given by (IIIA) and (20) are equivalent to the real passband signal model in (4) and (14).
Proposition 1 (Equivalent Passband Model)
Proof:
See Appendix AA. \qed
Remark 4
In fact, when , we can prove a stronger result than Proposition 1: the statistical experiments given by (4) and (IIIA) are equivalent in terms of a vanishing Le Cam’s distance [57]. As a result, for any loss function and any estimator for in one model, there exists an estimator in the other model which has the identical risk as under any realization of the parameter . We omit the proof here, but point out that the key step is to prove that the random vector obtained via (4) and obtained via (IIIA) are mutual randomizations with the help of the Hilbert transform.
We recall that is a natural condition required by the quadrature demodulation. Hence, in the sequel we will stick to the complex observation model (IIIA) and the loglikelihood function (20).
Before presenting the main results in following sections, we first define a few important metrics.
Definition 1 (Effective Baseband Bandwidth [56] and BasebandCarrier Correlation)
The effective baseband bandwidth and the basebandcarrier correlation (BCC) of are defined respectively as
(21) 
and
(22) 
Definition 2 (Squared Array Aperture Function)
The squared array aperture function (SAAF) for an array is defined as
(23) 
Remark 5
The SAAF is the effective array aperture observed from incident the angle , and fully quantifies the effect of arrayantenna geometry on localization, as will be shown in Section V.
IiiB Case with Known Array Orientation and Initial Phase
We first consider the case where both the array orientation and the initial phase are known. This scenario reduces to the wideband case studied in [3] in the farfield environment. The results are given in the following theorem.
Theorem 1 (Fullknowledge Static EFIM)
When both the array orientation and the initial phase are known, the EFIM for the position is
(24) 
where is the visual angle expressed as
(25) 
and
(26) 
with the pathoverlap coefficient (POC) defined in [3, Thm. 1] and the SNR given by
(27) 
Proof:
See Appendix AB. \qed
Theorem 1 implies that in the full knowledge case, the EFIM for the position is a weighed sum of the RDM from each anchorantenna pair, with direction (i.e., from anchor to the th antenna, cf. Fig. 3) and intensity . Hence, each anchorantenna pair provides distance information for localization, which sums up to the overall localization information. We also have the following observations.

The support of the intensity is , which means that the anchors providing NLOS signal are not useful for localization, for the actual distance and direction are completely corrupted by the first range bias and first arrivalangle bias , respectively.

The intensity depends on the SNR of the first path and the POC , which characterizes the effect of multipath propagation for localization. It is shown in [3] that is determined by the first contiguous cluster (cf. [3, Def. 3]) of the received waveform and does not depend on the path amplitudes , and when the signal of the first coming path from anchor does not overlap with those of other paths. Moreover, the POC is solely determined by the autocorrelation function of the baseband signal , the carrier frequency , and channel parameters and .

The term is the squared effective bandwidth of the entire signal [3], which means that the entire bandwidth can be utilized for localization in the full knowledge case.
Hence, the localization performance for the full knowledge case reduces to the wideband case [3], and AOA measurements obtained by antenna arrays do not further improve position accuracy beyond that provided by TOA measurements.
IiiC Case with Known Array Orientation but Unknown Initial Phase
We now turn to the case in which the orientation is known but not the initial phase . Theorem 2 derives the corresponding localization information.
Theorem 2 (Orientationknown Static EFIM)
When the array orientation is known but the initial phase is unknown, the EFIM for the position is
(28)  
which yields an equivalent expression in terms of the SAAF as
(29)  
Proof:
See Appendix AC. \qed
In farfield environments, we have and thus the following approximation for the EFIM given by (29).
Corollary 1
If , the EFIM for the position in Theorem 2 can be approximated as
(30)  
Note that the expression given by (30) does not depend on the reference point , and hence we can have the following alternative expression of (30) when the array center is chosen as the reference point.
Corollary 2
When the array center is chosen as the reference point, the EFIM in (30) becomes
(31)  
Based on the expression of EFIM given in (31), some observations and insights can be drawn as follows.
IiiC1 Distance Information
the term is the measuring information for distance, with intensity proportional to and direction along the radial angle to anchor , i.e., from anchor to the reference point (see Fig. 3). Hence, this term is the information from TOA and provides localization information with direction towards the agent, and only the baseband signal can contribute to TOA information.
(34) 
IiiC2 Direction Information
the term corresponds to the measuring information for direction, i.e., from AOA, and has a tangent direction to anchor (the direction perpendicular to that connecting anchor and the reference point; see Fig. 3). The intensity of this term consists of two parts. The first part is the visual angle of the effective aperture for th antenna observed from anchor , and the second part is the effective carrier frequency. As a systemlevel interpretation, assuming for simplicity, the intensity scaled by , i.e., , is the squared visual angle normalized by the wavelength. Hence, the AOA information can be retrieved from the effective carrier frequency.
IiiC3 Geometric Interpretation
the EFIM is a weighed sum of measuring information from each anchorantenna pair, where each pair provides information in two orthogonal directions summing up to the entire localization information (depicted as an ellipse in Fig. 3). Note that
(32) 
and we conclude by Theorem 1 and 2 that, in wideband cases [3] the effective bandwidth of the entire signal can be used for obtaining both distance and direction information, while in our model the overall bandwidth is decomposed into two parts, i.e., baseband signal for distance information and carrier frequency for direction information. In particular, AOA information can make significant contributions to localization accuracy beyond that obtained by TOA measurements in our model.
Moreover, the direction of the major axis of ellipse depends on whether . In traditional TOA systems, is comparable to , and hence the distance information dominates since (due to farfield environments). In contrast, in traditional AOA systems, , and hence the direction information dominates. In practice, since holds, the criterion becomes . For example, when MHz, the dominance of the direction information requires m, conforming to the fact that AOA information is more effective in shortdistance localization.
Now we discuss some properties of the BCC , which characterizes the extent how close is the baseband signal to a singlefrequency signal. For example, implies , and thus the entire baseband signal contributes to the AOA information and we cannot extract any TOA information from . On the other hand, when (equivalent to ), the entire baseband signal can be utilized for obtaining the TOA information.
Without loss of generality, we will assume in the following sections, since otherwise we can always substitute the baseband signal and carrier frequency by and .
IiiD Case with Unknown Array Orientation and Initial Phase
We next consider the case in which the array orientation also becomes an unknown parameter to be estimated. Similar to Theorem 2, the EFIM for the position and orientation is derived accordingly.
Theorem 3 (Orientationunknown Static EFIM)
When neither the initial phase nor the array orientation is known, the EFIM for the position and orientation is
(33) 
Proof:
See Appendix AD. \qed
Corollary 3
Theorem 3 claims that the EFIM for the position and orientation is also a weighed matrix sum of measuring information from each anchorantenna pair, and thus the overall localization information in the orientationunknown case possesses a similar structure. Moreover, since is positive semidefinite, the unknown orientation degrades the localization accuracy.
Note that the approximated EFIM for the orientationunknown case still does not depend on the reference point , which seems contradictory to the wideband case in [3]. However, we remark that the invariance of on in farfield enviroments is due to the fact that the AOA information does not rely on , and the term can be neglected in the TOA information.
Iv Localization Accuracy in the Dynamic Scenario
As is shown in the preceding section, the timeinvariant delays between anchorantenna pairs are all the sources of localization information in the static scenario. In this section, we turn to the dynamic scenario where the Doppler effect can be utilized in addition to the TOA and AOA measurements for localization.
Iva Case with Known Orientation and Velocity
We first consider the scenario in which both the velocity and orientation of the antenna array are known. This scenario is relevant in practice, as the agent can obtain its velocity and orientation locally, e.g., by a compass and accelerometer. Note that in the dynamic scenario, Assumptions 1 and 2 are needed to simplify the expressions of the EFIM, as shown in the next theorem.
Theorem 4 (Orientation and Velocityknown Moving EFIM)
The EFIM for the position is
(35) 
where are given by (AE1), (AE) and (AE2), respectively. Furthermore, under Assumption 1 and 2, the EFIM can be approximated as
(36) 
where is the root mean squared time duration of the baseband signal defined as
(37) 
and is the angular speed of the visual angle given by
(38) 
Proof:
See Appendix AE. \qed
Remark 6
By the proof details given in the appendix, we can show that the EFIM in (4) is a tight approximation up to a multiplicative approximation error less than .
Due to the invariance of on the reference point and the reference time , similar to Corollary 2, we can choose the array center to be the reference point and to be the reference time.
Corollary 4
When the array center is chosen as the reference point and , the EFIM in (4) becomes
(39) 
(40) 
(41) 
(42) 
Some observations on the effect of Doppler shift can be drawn from Theorem 4 and Corollay 4 as follows.
IvA1 Intensity Effect
Compared with Theorem 2, there is a new coefficient on the information intensity, which we refer to as the intensity effect of the Doppler shift. Intuitively, the Doppler shift enlarges both the baseband bandwidth and carrier frequency by times, whereas the the SNR is reduced by times. Hence, this new coefficient is obtained by the fact that the EFIM for the position is proportional to SNR times the squared bandwidth. Note that whether the intensity effect does help or harm to localization depends on the array orientation, but this effect is negligible since . With a slight abuse of notations, we will still denote by in the following.
IvA2 Direction Effect
The second effect of Doppler shift on localization is the direction effect, i.e., it provides additional direction information with intensity . This direction information originates from the dependence of the Doppler shift on the direction of the anchor, and faster speed is preferred for accumulating more direction information. Hence, as shown in Fig. 4, the localization information can be decomposed into the distance and direction information, where the latter consists of two parts from AOA measurement and Doppler shifts, respectively. In particular, the Doppler shifts do not affect the distance information for localization.
IvA3 Geometric Interpretation
The geometric interpretations for the new variables and are illustrated in Fig. 4. The variable can be interpreted as the equivalent moving time of the agent from its initial position to the final position, and can be viewed as a synthetic aperture formed by the moving agent, which has the same effect as the real array aperture. Hence, similar to the interpretation that is the visual angle between the reference point and th antenna at any fixed time, is the angular speed of the visual angle formed by the reference points at different time. Moreover, the overall localization information for direction is simply the sum of the synthetic aperture and the real array aperture. In practice, it is likely that the synthetic aperture formed by the moving agent during the observation time is larger than the real array aperture, and thus the Doppler effect can provide considerably more direction information for localization in the dynamic scenarios. Note that since does not depend on the reference time , the localization accuracy for the agent position remains the same at any time, which is consistent to our intuition under the knownvelocity scenario.
IvB Case with Unknown Orientation and Velocity
This subsection will address scenarios in which the array orientation, agent speed or its moving direction (possibly all) is unknown to the agent. We first consider the case where the agent knows its speed but not the array orientation or the moving direction, which is practically relevant as the speed is the only local quantity invariant with translation or rotation among all parameters.
Theorem 5 (Orientation and Directionunknown Speedknown Moving EFIM)
Proof:
Similar to the proof of Theorem 3. \qed