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Implementation of a Directional Beacon-Based Position Location Algorithm in a Signal Processing Framework S. F. A. Shah, Seshan Srirangarajan, and A. H. Tewfik, Fellow, IEEE
Abstract—We present the implementation of a directional beacon-based positioning algorithm using radio frequency signals. This algorithm allows each mobile node to compute its position with respect to a set of reference nodes which are equipped with a rotating directional antenna. The use of directional beacon-based algorithm for position location eliminates the need for strict synchronization between the reference nodes and the mobile node. In contrast to positioning algorithms that rely on signal propagation time and bandwidth, the proposed algorithm depends on the beam-width and rotational speed of the directional antenna. We will show that these parameters can be optimized with a low cost solution that provides good positioning accuracy. The system implementation is based on the GNU Radio software platform and the Universal Software Radio Peripheral as the hardware component. We present an enhanced maximum likelihood method for estimating the received signal amplitude profile. To deal with obstructed line-of-sight scenarios, we do not rely purely on the received signal strength and instead formulate a least squares problem to estimate the line-of-sight component in a multipath environment. These advanced signal processing techniques yield a more accurate estimate of the bearing of the mobile node with respect to each of the reference nodes. We also show that the proposed positioning algorithm is tolerant to errors in timing and synchronization. We demonstrate the ability to obtain mobile node position estimates with sub-meter accuracy by transmitting a narrowband signal of 1 kHz bandwidth in the 2.4-2.5 GHz band. The experimental results show a mean position error of 0.759 m, in a field measuring 55m by 43m, using eight 90∘ rotations of the antenna. Index Terms—Directional beacon, positioning, direction-ofarrival, multipath time-delay estimation, software radio.
I. I NTRODUCTION OSITION location in wireless personal area networks (WPANs) has found a number of applications ranging from commercial and residential (tracking people in assistedliving places and assets in a manufacturing facility) to public safety and military (tracking fire fighters and soldiers during their missions) [1]. The existing localization systems can be broadly classified into two categories. The first category consists of systems
P
Manuscript received September 8, 2008; revised May 18, 2009; accepted December 8, 2009. The associate editor coordinating the review of this paper and approving it for publication was Y. Gong. This research is partially supported by NSF grant CCR-0313224. S. Faisal A. Shah is with COM DEV, Cambridge, ON N1R 7H6, Canada (e-mail:
[email protected]). S. Srirangarajan is with the Intelligent Systems Center, Nanyang Technological University, Singapore (e-mail:
[email protected]). A. H. Tewfik is with the Department of Electrical and Computer Engineering, University of Minnesota, USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TWC.2010.03.081204
which develop a signaling system and support infrastructure focused primarily on the positioning and tracking application. Systems in the second category tend to use existing wireless network infrastructure (or with minimal modifications) to locate a mobile node. The first category systems are designed for a desired positioning accuracy and tend to use expensive, dedicated devices for the purpose. Examples of such systems include global positioning system (GPS). The second category of systems, in the absence of dedicated devices, use smart algorithms to overcome the low accuracy of the measured quantities. Our work is aimed at demonstrating a system that fits in the second category. The position of a node can be determined in a variety of ways [2], such as Angle of Arrival (AOA) [3], [4], [5], Time of Arrival (TOA) [6], Time Difference of arrival (TDOA), or Received Signal Strength (RSS) [1]. Techniques based purely on signal strength are prone to inaccuracies and large variances in position estimates [1], [7]. For TOA/TDOA techniques, the estimation of the line-of-sight (LOS) component in the presence of multipath is extremely challenging [8], [9]. The variance of LOS time-delay estimation and hence positioning accuracy is inversely proportional to the bandwidth of the signals and inversely proportional to the square root of the observation time [10]. Therefore, these techniques need large bandwidths and long observations times which are difficult to meet in a number of real world applications. TOA/TDOA techniques using radio frequency (RF) signals are very sensitive to timing estimation errors. For AOA/DOA techniques, the error in time-delay estimation translates into an error in angle estimation which results in significantly smaller position error. Thus, directionality-based techniques can provide good accuracy with relatively inexpensive hardware. McGillem and Rappaport [4] were one of the earliest to use AOA information for positioning and navigation along with a system implementation. They used infrared beacons with a rotational optical receiving system to obtain angular measurements. Nasipuri presented a directionality based positioning scheme [5] and later proposed a system implementation using rotating optical beacon generators and sensor nodes equipped with photo sensors [11]. The scheme in [5] searches for the maximum of the received signal strength to detect the LOS component. Shah and Tewfik [3] presented an enhanced positioning scheme based on directional beacons using the time of earliest arrival for detecting the LOS component. Although many directionality-based positioning techniques have been
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SHAH et al.: IMPLEMENTATION OF A DIRECTIONAL BEACON-BASED POSITION LOCATION ALGORITHM IN A SIGNAL PROCESSING FRAMEWORK
described in the literature, very few have presented practical system implementations, and to our knowledge none based on RF signals has been presented [12]. Using RF signals allows the opportunity to use the existing transceiver circuitry on wireless devices for positioning. However, the hostile wireless propagation environment introduces distortion in the radio signals and poses unique challenges to the system design. In this paper, we discuss the implementation of a directional beacon-based positioning algorithm using narrowband RF signals. The proposed algorithm relies on direction of arrival (DOA) based triangulation [4] but computes the DOAs in a non-traditional manner. We develop a signal processing framework to work with low-accuracy experimental data, obtained using low-cost hardware, for accurately estimating DOAs. Different from [13], we use rotating directional antenna to compute DOAs resulting in a low-cost solution. Even though the technique needs some modification at the reference nodes in the form of a rotational directional antenna, we show that the mobile nodes do not need hardware modifications. In contrast to positioning algorithms that rely on signal propagation time and bandwidth, the proposed algorithm depends on the width of the beampattern and rotational speed of the directional antenna. Our low-cost implementation using directional antenna and stepper motor provides sub-meter positioning accuracy. Another important advantage of the proposed algorithm is that it does not require any synchronization between the reference nodes and the mobile node. However, the reference nodes are assumed to be synchronized. Using the proposed directional beacon-based algorithm, position estimation error of only 0.5 m results from a 500 msec error in time-delay estimation or synchronization. Instead of relying purely on RSS, we use maximum likelihood (ML)-based amplitude estimation and least squares-based LOS time-delay estimation, to estimate the bearing of the mobile node with respect to each of the reference nodes in the presence of multipath components. We also demonstrate a technique to combine received signal data from multiple transmit antenna rotations to improve estimation accuracy. These techniques allow us to obtain enhanced position estimates with sub-meter accuracy. The paper is organized as follows. Section II introduces the system model and the localization algorithm using directional beacons. In Section III we describe the hardware and software platform used for the prototype implementation. Sections IV and V discuss enhanced ML amplitude estimation and the least squares-based LOS estimation in the presence of multipath, respectively. Experimental set-up and results are presented in Section VI. We conclude with some remarks in Section VII. II. L OCALIZATION P RINCIPLE , S IGNAL M ODEL AND P OSITION L OCATION A LGORITHM A. Localization Principle and System Model We now present the localization principle that forms the basis for the proposed directional beacon-based position location algorithm [3]. We refer to the nodes whose positions are known a priori as the reference nodes (RN) and nodes whose positions are unknown as mobile nodes. Consider a wireless network that contains three reference nodes RN-1, RN-2 and RN-3. RNs can be located at arbitrary but known positions. For
RN-2 (0, L2 )
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(L1, L2 ) RN-1
L1
φ2
φ1 α2
α1
β1
L2 β2
φ3
Q (x,y)
α3
RN-3 (0,0) Fig. 1.
Arrangement of the reference nodes and the coordinate system.
simplicity, we assume that the RNs are located at the corners of a rectangular field as shown in Fig. 1. We further assume that the origin is at RN-3. Now, consider the situation where a mobile node Q joins the network and needs to determine its coordinates (𝑥, 𝑦) with respect to the RNs. We assume that the environment is quasi-static during the position estimation phase of the algorithm. To avoid the difficulties associated with RSS and TOA/TDOA based techniques that were described in Section I, we propose a position location algorithm that relies on triangulation using DOAs. In general, DOAs are estimated using antenna arrays. In this paper we present a novel method for estimating DOAs using low-cost radio transceivers at the RNs and mobile node Q. We next outline the DOA-based triangulation method. Assume that node Q estimates 𝛼1 , 𝛼2 and 𝛼3 as the DOAs of the signal from RN-1, RN-2 and RN-3, respectively. The bearing 𝛽1 of node Q from RN-1 and RN-2 is given by 𝛽1 = 𝛼1 − 𝛼2
(1)
Similarly, 𝛽2 the bearing of node Q from RN-2 and RN-3, is 𝜋 𝛽2 = + 𝛼2 − 𝛼3 . (2) 2 Using simple trigonometry, the coordinates (𝑥, 𝑦) of the node Q can be computed as 𝐿2 cos 𝛼2 cos(𝛽2 − 𝛼2 ) sin 𝛽2 𝐿2 cos 𝛼2 sin(𝛽2 − 𝛼2 ). 𝑦= sin 𝛽2
𝑥=
(3)
The symmetric arrangement of the RNs in Fig. 1 leads to a simple relation in (3). However, the localization principle in (3) is valid for any arrangement of the reference nodes1 . B. DOA Estimation using Directional Beacons Each RN is equipped with a rotating directional antenna for generating directional beacons. The use of directional beacons 1 Mobile
node is not constrained to be inside the triangle formed by 3 RNs.
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 3, MARCH 2010
allows us to compute DOAs by estimating the corresponding times when the directional beam is aligned with node Q. Assume that RNs are synchronized2 and 𝜙1 , 𝜙2 , 𝜙3 represent the initial angular positions of the rotating beam or directional antenna at RN-1, RN-2, RN-3, respectively (refer Fig. 1). If 𝑡1 , 𝑡2 and 𝑡3 represent the times when the rotating beams from RN-1, RN-2 and RN-3, respectively, are aligned with node Q, then the DOAs can be estimated as: 𝛼𝑖 = 𝜙𝑖 + 𝜔𝑡𝑖 ,
𝑖 = 1, 2, 3
(4)
where 𝜔 is the angular speed of the rotating directional beam in rad/s. It is necessary that the transmissions from different RNs are distinguishable at Q. This can be achieved for example by using different frequencies for each beacon. Substituting (4) into (1) and (2) gives 𝛽1 = 𝜙12 − 𝜔(𝑡2 − 𝑡1 ) 𝛽2 = 𝜙23 − 𝜔(𝑡3 − 𝑡2 )
(5)
where 𝜙12 := 𝜙1 − 𝜙2 and 𝜙23 := 𝜋2 + 𝜙2 − 𝜙3 . The angles 𝜙12 and 𝜙23 are the initial angular separation between the directional beams of the RNs. From (5) it is clear that absolute timings are not required as we are dealing with time differences in calculating the bearings. C. Signal Model Assume that the RNs transmit a continuous time signal with complex low-pass equivalent represented by 𝑝(𝑡). Then the low-pass equivalent of the received signal at the mobile node is given by 𝑟(𝑡) = 𝑧(𝜑)𝑝(𝑡) + 𝜈(𝑡) (6) where 𝜈(𝑡) is the additive noise and 𝑧(𝜑) is the amplitude profile of the received signal which is a function of the angular position 𝜑 = 𝜔𝑡 of the antenna. The amplitude profile 𝑧(𝜑) includes the effect of path loss as well as the antenna beampattern. In the presence of reflectors, 𝑧(𝜑) can be modeled as 𝑀 ∑
compare the proposed directional beacon-based position location algorithm with conventional TOA schemes. One striking difference is in the role of 𝑠(𝑛). In TOA techniques, 𝑠(𝑛) refers to the transmitted pulse in the time domain while in the proposed algorithm 𝑠(𝑛) corresponds to the beampattern of the directional antenna. Thus, the antenna beamwidth in our algorithm is analogous to the pulse-width in TOA schemes. In TOA schemes the accuracy of position location depends on the pulse-width (or signal bandwidth in the frequency domain) and the speed of propagation; in our proposed position location algorithm the accuracy depends on the beam-width and the rotational speed 𝜔 of the antenna. We discuss these points further in Section VI-E D. Directional Beacon-based Position Location Algorithm For the system shown in Fig. 1, each RN broadcasts its known position and the initial angular position of its antenna beam. Using this information the mobile node Q executes the following steps of the positioning algorithm to localize itself: Step 1. RN-1 transmits a signal continuously while its antenna is being rotated at a constant angular speed (𝜔). Step 2. The received signal 𝑟(𝑡) at the mobile node consists of multiple copies of the transmit signal due to multipath propagation. Store the discrete-time samples 𝑟(𝑛) of the received signal for further analysis. Step 3. Estimate the amplitude profile 𝑧(𝑛) of the received signal at the mobile node (Section IV). Step 4. Estimate the time shift (𝑡1 ) associated with the LOS signal component in 𝑧(𝑛) due to RN-1 (Section V). Step 5. Repeat steps 1-4 for RN-2 and RN-3 to obtain 𝑡2 and 𝑡3 , respectively. Finally, use (5), (4) and (3) to obtain the coordinates of node Q. III. P ROTOTYPE I MPLEMENTATION
where 𝑠(𝜑) is the true amplitude profile based on the beampattern of the directional antenna. In (7), we assume that there are 𝑀 signal components (the LOS component and reflections) received at the mobile node with amplitudes {𝑎𝑚 }𝑀 𝑚=1 and determined by the angle of departure angular shifts {𝜓𝑚 }𝑀 𝑚=1 of the signal components from the directional antenna. In discrete time, the samples of the received signal can be written as 𝑟(𝑛) = 𝑧(𝑛)𝑝(𝑛) + 𝜈(𝑛) (8)
In this section, we describe the hardware and software implementation of the directional beacon-based position location algorithm. Our implementation is based on a software defined radio (SDR) platform: GNU radio software and its hardware companion, the Universal Software Radio Peripheral (USRP). The SDR set-up allows us the ease and flexibility to design and transmit user-defined waveforms. In addition, the RF transceiver circuitry is fairly inexpensive. The rotating beacons are generated using a directional antenna coupled to a stepper motor. Fig. 2 illustrates the block level set-up at the transmitter (or RN).
where 𝑟(𝑛) = 𝑟(𝑡)∣𝑡=𝑛𝑇𝑠 , 𝑧(𝑛) = 𝑧(𝜑)∣𝜑=𝑛𝜔𝑇𝑠 and 𝑇𝑠 is the sampling period. Thus,
A. Hardware Platform
𝑧(𝜑) =
𝑎𝑚 𝑠(𝜑 − 𝜓𝑚 )
(7)
𝑚=1
𝑧(𝑛) =
𝑀 ∑
𝑎𝑚 𝑠(𝑛 − 𝜏𝑚 )
(9)
𝑚=1
where 𝑠(𝑛) := 𝑠(𝜑)∣𝜑=𝑛𝜔𝑇𝑠 and 𝜏𝑚 := 𝜓𝑚 /𝜔𝑇𝑠 . Based on the signal model for the amplitude profile in (9) we can 2 The
effect of synchronization error will be analyzed in Section VI-E.
For the RF front end of the software radio, we use the USRP boards from Ettus Research.It consists of a daughter board (RFX2400), capable of transmitting and receiving RF signals in the 2.4-2.5 GHz band. USRP also contains a motherboard that includes a USB interface, an FPGA to implement highspeed baseband processing, and analog-to-digital converters (ADCs) and digital-to-analog converters (DACs).
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SHAH et al.: IMPLEMENTATION OF A DIRECTIONAL BEACON-BASED POSITION LOCATION ALGORITHM IN A SIGNAL PROCESSING FRAMEWORK
Rotating Directional Tx Antenna
USB Cable USRP Board 4 MSamples/s
Motor Mechanical Coupling
Stepper Motor
Motor Controller
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a rotation. Using the signal amplitude profile, we estimate the arrival time of LOS component using multipath timedelay estimation which are used to obtain the bearings and eventually the coordinates of node Q. Details of these steps will be provided in Sections IV and V.
GNU Radio
Serial Com HyperTerminal Laptop
Fig. 2. Hardware setup for the transmitter with rotating directional antenna.
The rotating beacons are generated by mounting a directional antenna over the stepper motor. The directional antenna consists of a 16-element linear antenna array from Telex with a main lobe beamwidth of 30∘ . We use a bipolar stepper motor, 8718L-02S, from Lin Engineering which allows us to rotate the antenna with a constant angular speed. The stepper motor is selected to provide a torque of 900 oz-in (6.355 N.m) sufficient to rotate the antenna. We use a narrowband signal with a bandwidth of 1 kHz and set the motor to rotate the antenna at 0.47 rpm (𝜔 = 0.049 rad/s). B. Software Platform The software radio is implemented in an open public license software called GNU radio.GNU radio provides a library of blocks for radio transmission and reception. These blocks are glued together using the Python scripting language. We use gr block and usrp sink c to continuously transmit a single frequency tone. The receiver is implemented using the usrp source c block that captures the data from the RF front-end and writes it to a file. C. Data Collection and Processing Steps 1-4 (from Section II-D) are repeated for each of the RNs and samples of the received signal at node Q are stored to a data file. The first step in storing the data to a file is to downconvert the received RF signal to a complex baseband signal. The two components of the complex signal are passed through the I and Q channels. Each channel path has an ADC sampling at 64MSamples/s. To reduce the burden on the USB interface, the ADC output is downsampled by 32 resulting in a 2MSamples/s data stream. Samples from the I and Q channels are packed together and passed to the GNU radio software over the USB interface resulting in an effective data rate of 4MSamples/s across the USB. Since each data sample is represented by 4 Bytes, the data rate over USB is 16 MB/s. Samples of the received signal are finally written to a file. To keep file sizes manageable, we store only the first 32 samples out of every 6400 samples to the hard disk. The post-processing of the stored data using various signal processing techniques is outlined in Fig. 3. The first step is to separate the data from each of the RNs. We then slice the data to account for multiple antenna rotations where each data slice represents a single rotation of the antenna. We group the sliced data into bursts of 𝑁𝑏 samples and estimate the signal amplitude profile as the directional beam completes
D. Challenges in Prototype Implementation We next outline some of the challenges encountered in the prototype implementation. Solutions to these, based on statistical signal processing techniques, will be presented in Sections IV and V. Due to the data transfer rate limitation over the USB interface, data transfer to the USRP is intermittent resulting in intermittent signal transmission. In addition, the ISM frequency bands are used by many wireless devices resulting in very strong interferers. This necessitates some filtering of the received signal (or the amplitude profile) to remove the effect of interferers. Due to the presence of walls and other objects, which act as reflectors/scatterers, the received signal (and the estimated amplitude profile) consists of multiple copies of the transmit signal arriving via different paths. The aim is to estimate the LOS signal component. In other words, we need to determine scale factors and time delays associated with the LOS and multipath components which together form the received amplitude profile. IV. S IGNAL D ETECTION AND E NHANCED A MPLITUDE E STIMATION A. Signal Detection Using the GNU software radio, we generate a baseband discrete time signal of the form 𝑝(𝑛) = sin(2𝜋𝑓𝑜 𝑛+𝜃) where 𝑓𝑜 ∈ [0, 1] is the normalized digital frequency and 𝜃 is the phase. Samples of the transmit signal are passed to the USRP which in turn produces an analog signal of 1 kHz bandwidth. RFX2400 daughter board on the USRP modulates the analog signal to the 2.4-2.5 GHz RF band. Due to practical constraints, the data transfer over the USB interface to the USRP board is intermittent resulting in intermittent signal transmission. At the receiver, we model this scenario as two ‘hypotheses’: 𝐻1 when the data is successfully transferred to the USRP resulting in a successful transmission (and reception) and 𝐻0 when data transfer to the USRP failed resulting in only noise being received. Thus, the received signal during 𝑙th data burst can be written as 𝐻1 : 𝑟𝑙 (𝑛) = 𝑧 sin(2𝜋𝑓𝑜 𝑛 + 𝜃𝑙 ) + 𝜈𝑙 (𝑛) for 𝑛 = 0, ⋅ ⋅ ⋅ , 𝑁𝑏 − 1 𝐻0 : 𝑟𝑙 (𝑛) = 𝜈𝑙 (𝑛) (10) where 𝑧 represents the amplitude of the received tone that is assumed to be constant over bursts of 𝑁𝑏 samples. This assumption is invariably true for high sampling rates of the order of 106 samples/s. The amplitude 𝑧 of the received signal includes the effect of path loss as well as the antenna beampattern. As a first step we would like to detect whether a given burst of 𝑁𝑏 samples belongs to 𝐻1 or 𝐻0 . The ML estimate of 𝑧 for a given data burst containing 𝑁𝑏 samples is given by [14, pg. 195] 𝑁 −1 𝑏 2 ∑ 𝑟𝑙 (𝑛)𝑒−𝑗2𝜋𝑓𝑜 𝑛 . 𝑧ˆ = (11) 𝑁𝑏 𝑛=0
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Signal Detection
Read received signal data
Fig. 3.
Data slicing for RN separation & multiple antenna rotations
Grouping/ Amplitude Estimation
S/P
Multipath Time Delay Estimation
Obtain Bearing and Coordinates
Nb:1
Post processing of data in a signal processing framework.
We then use a threshold 𝑍 to decide between the two hy𝐻0
bursts is given by
√
potheses such that 𝑧ˆ ≶ 𝑍. The threshold 𝑍 is empirically set
𝑧ˆ =
𝐻1
to 5-10% of the maximum value of the received signal. The received signal bursts classified under 𝐻1 are used for further processing, as outlined in the next subsection. B. Enhanced Amplitude Estimation It is well known that the ML estimate (11) approaches the minimum variance unbiased estimate as 𝑁𝑏 → ∞. Since the number of samples in a burst is fixed at 𝑁𝑏 3 , we can enhance the estimate by computing the amplitude estimate over multiple consecutive bursts. Let us assume that we have 𝐿 consecutive signal bursts (bursts satisfying 𝐻1 ). The ML estimate in (11) cannot be extended directly for 𝑁𝑏 𝐿 data samples as each burst can have a different phase. Thus we formulate this as the ML amplitude estimation of a constant frequency sinusoid with unknown initial phase. From (10), the transmit signal component under hypothesis 𝐻1 can be expanded as ] [ cos 𝜃𝑙 . 𝑧 sin(2𝜋𝑓𝑜 𝑛 + 𝜃𝑙 ) = 𝑧 [sin 2𝜋𝑓𝑜 𝑛 cos 2𝜋𝑓𝑜 𝑛] sin 𝜃𝑙 𝑇
Let r𝑙 := [𝑟𝑙 (0) ⋅ ⋅ ⋅ 𝑟𝑙 (𝑁𝑏 − 1)] , and ⎡ 0 ⎢ sin 2𝜋𝑓 𝑜 ⎢ H𝑙 := ⎢ .. ⎣ . sin(𝑁𝑏 − 1)2𝜋𝑓𝑜
sin 𝜃𝑙 ]𝑇
𝜽𝑙 := [cos 𝜃𝑙 1 cos 2𝜋𝑓𝑜 .. .
⎤
𝑚=1
where 𝜂(𝑛) represents the estimation error between 𝑧ˆ(𝑛) and the true profile 𝑧(𝑛) given by (9). It is important to mention that the samples of 𝑧ˆ(𝑛) in (16) are spaced at an interval larger than 𝑇𝑠 . Let us assume we have 𝑁 samples of 𝑧ˆ(𝑛) in a single rotation of the antenna that we use to estimate a = [𝑎1 ⋅ ⋅ ⋅ 𝑎𝑀 ]𝑇 and 𝝉 = [𝜏1 ⋅ ⋅ ⋅ 𝜏𝑀 ]𝑇 . We can write (16) in matrix-vector form as (17)
𝑧 (0) ⋅ ⋅ ⋅ 𝑧ˆ(𝑁 − 1)] and S(𝝉 ) is an 𝑁 × 𝑀 matrix where zˆ = [ˆ with [S(𝝉 )]𝑛,𝑚 = 𝑠(𝑛 − 𝜏𝑚 ). The LS error criterion for the model in (17) becomes
cos(𝑁𝑏 − 1)2𝜋𝑓𝑜
for 𝑙 = 1, ⋅ ⋅ ⋅ , 𝐿.
(15)
V. E STIMATING THE LOS C OMPONENT IN M ULTIPATH E NVIRONMENT This section describes the procedure for estimating the LOS component in a multipath environment using the least squares (LS) error criterion. Let 𝑧ˆ(𝑛) represent the estimated amplitude of the 𝑛th received signal burst obtained using the procedure outlined in Section IV. Sample values of 𝑧ˆ(𝑛) corresponding to the parts of the received signal which were classified as 𝐻0 are obtained via linear interpolation of the neighboring samples. Using the multipath model for 𝑧(𝑛) (9), we can write 𝑀 ∑ 𝑧ˆ(𝑛) = 𝑎𝑚 𝑠(𝑛 − 𝜏𝑚 ) + 𝜂(𝑛), (16)
ˆz = S(𝝉 )a + 𝜼,
⎥ ⎥ ⎥, ⎦
then the vector of received signal samples during the 𝑙𝑡ℎ burst is given by r𝑙 = 𝑧H𝑙 𝜽 𝑙 + 𝝂 𝑙
ˆ𝑇 𝜽 ˆ 𝜽 . 𝐿
(12)
Note that 𝜽𝑇𝑙 𝜽 𝑙 = 1. Concatenating these 𝐿 vectors to [ ]𝑇 form r := r𝑇1 ⋅ ⋅ ⋅ r𝑇𝐿 , 𝜽 := 𝑧[𝜽𝑇1 ⋅ ⋅ ⋅ 𝜽𝑇𝐿 ]𝑇 and H := diag(H1 ⋅ ⋅ ⋅ H𝐿 ), we can write (12) as r = H 𝜽 + 𝝂. (13) ∑𝐿 𝑇 2 2 Again observe that 𝜽 𝜽 = 𝑧 𝑙=1 𝜽 𝑙 𝜽 𝑙 = 𝐿𝑧 . Assuming Gaussian noise, the ML estimate of 𝜽 in (13) is given by ( ) ˆ = H𝑇 H −1 H𝑇 r. 𝜽 (14) 𝑇
Due to the invariance property of the ML estimate, the enhanced ML estimate of the amplitude 𝑧 using 𝐿 consecutive 3 In our set-up 𝑁 = 32 since only 32 consecutive samples out of 6400 𝑏 are stored to the disk.
𝐽(𝝉 , a; 𝑛) = ∣∣ˆz − S(𝝉 )a∣∣2 ,
(18)
where ∣∣.∣∣ is the 𝑙2 norm. The minimization of (18) poses two problems [15], [16]: 1) the cost function is non-linear in 𝝉 , and 2) the estimates {ˆ 𝜏𝑚 }𝑀 𝑚=1 can only take values that are integer multiples of the sampling time and hence limit the resolution of the estimate. To overcome these problems we consider the equivalent frequency-domain model of (16) which can be expressed as 𝑧˜(𝑘) = 𝑠˜(𝑘)
𝑀 ∑
𝑎𝑚 𝑒−𝑗2𝜋𝑘𝜏𝑚 /𝑁 + 𝜂˜(𝑘)
(19)
𝑚=1
for 𝑘 = 0, ⋅ ⋅ ⋅ , 𝑁 − 1, ∑𝑁 −1 where 𝑧˜(𝑘) = 𝑛=0 𝑧(𝑛)𝑒−𝑗2𝜋𝑛𝑘/𝑁 represents the 𝑁 -point −1 discrete Fourier transform (DFT) of {𝑧(𝑛)}𝑁 ˜ 𝑛=0 . Similarly, 𝑠 𝑁 −1 is the DFT of {𝑠(𝑛)}𝑛=0 . Re-writing (19) into an equivalent matrix-vector form we obtain ˜ )a + 𝜼, ˜z = G(𝝉
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(20)
SHAH et al.: IMPLEMENTATION OF A DIRECTIONAL BEACON-BASED POSITION LOCATION ALGORITHM IN A SIGNAL PROCESSING FRAMEWORK
˜ ) := S ˜ 𝐷 H(𝝉 ) with z := [˜ 𝑧 (0) ⋅ ⋅ ⋅ 𝑧˜(𝑁 − 1)]𝑇 and G(𝝉 where ˜ H(𝝉 ) being an 𝑁 × 𝑀 matrix of the form [H(𝝉 )]𝑘,𝑚 = ˜ 𝐷 := diag[˜ 𝑒−𝑗2𝜋(𝑘−1)𝜏𝑚 /𝑁 and S 𝑠(0) ⋅ ⋅ ⋅ 𝑠˜(𝑁 − 1)]. Now, the LS error criterion for the frequency-domain model in (20) can be formulated as ˜ )a∣∣2 . (21) 𝐽(𝝉 , a; 𝑘) = ∣∣˜ z − G(𝝉 Since (21) is linear in a for a given 𝝉 , the least squares estimate for a is given by [ ]−1 ˜ ℋ (𝝉 )G(𝝉 ˜ ℋ (𝝉 ) z˜. ˜ ) ˆ= G a G (22)
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(a) Mobile node with unknown co- (b) Reference node equipped with ordinates. directional antenna. Fig. 4.
Experimental setup.
A. LOS Estimation using Multiple Antenna Rotations
ˆ from (22) into (21) reduces the LS error Substituting a function to ˜ ⊥ (𝝉 ) ˜ z∣∣2 , (23) 𝐽(𝝉 ; 𝑘) = ∣∣G [ ]−1 ˜ ⊥ (𝝉 ) = I𝑁 − G(𝝉 ˜ ) G ˜ ℋ (𝝉 )G(𝝉 ˜ ) ˜ ℋ (𝝉 ). where G G The error function in (23) is highly oscillatory or multimodal [15] with closely spaced multiple local minima that makes the minimization extremely difficult. A simple solution, ˆ suggested in [15], [16], is to allow the amplitude estimate a to be complex resulting in a smoother error function. It can be easily shown that the conjugate symmetry associated with ˜ 𝐷 forces the amplitude estimates in (22) to be real. ˜ z and S ˆ to be complex, we need to consider the Thus, to permit a single sided spectrum in the LS error function (21). Hence we define the following using single-sided spectrums: 𝑁 ˜ z𝑝 := [˜ 𝑧 (0) ⋅ ⋅ ⋅ 𝑧˜( − 1)]𝑇 and 2 𝑁 ˜ 𝐷𝑝 := diag[˜ S 𝑠(0) ⋅ ⋅ ⋅ 𝑠˜( − 1)]. 2
Our experimental setup allows us to collect received signal samples at the mobile node for multiple repeated rotations of the antenna. Assuming the environment has not changed significantly between rotations, the data collected during each rotation will result in similar amplitude profiles. If we collect 𝑁 samples in each of the 𝑅 rotations then the received signal vector of 𝑁 𝑅 samples can be modeled as
We can re-write the LS error function in (21) as ˜ 𝑝 (𝝉 )a𝐶 ∣∣2 , 𝐽𝐶 (𝝉 , a𝐶 ; 𝑘) = ∣∣˜ z𝑝 − G
where 𝑧˜(𝑅) (𝑘) and 𝑠˜(𝑅) (𝑘) represents the 𝑁 𝑅-point DFT 𝑅−1 of z(𝑅) and {𝑠(⟨𝑝⟩𝑁 )}𝑁 𝑝=0 , respectively. Transforming (28) into matrix-vector form, we obtain
(24)
˜ 𝑝 (𝝉 ) = S ˜ 𝐷𝑝 H𝑝 (𝝉 ) with H𝑝 (𝝉 ) being an 𝑁/2 × 𝑀 with G matrix of the form [H𝑝 (𝝉 )]𝑘,𝑚 = 𝑒−𝑗2𝜋(𝑘−1)𝜏𝑚 /𝑁 . The LS estimate of the complex amplitudes is given by [ ]−1 ˜ℋ ˜ℋ ˜ ˆ𝐶 = G ˜𝑝 . a G (25) 𝑝 (𝝉 )G𝑝 (𝝉 ) 𝑝 (𝝉 ) z ˆ𝐶 into (24) leads to 𝐽𝐶 (𝝉 ; 𝑘) that is known to Substituting a be a smoother function with far fewer oscillations. However, the minimum of 𝐽𝐶 (𝝉 ; 𝑘) is not the true minimum of 𝐽(𝝉 ; 𝑘). Adding a penalty term proportional to the imaginary part of a𝐶 to 𝐽𝐶 (𝝉 ; 𝑘) allows us to control its smoothness [15]. The modified error function can be written as ′ ′ ′ ˜ ′ (𝝉 )ˆ z𝑝 − G a𝐶 ∣∣2 , (26) 𝐽𝐶 (𝝉 ; 𝑘) = ∣∣˜ 𝑝 ⎤ ⎡ Re{˜ z𝑝 } ′ ˜ ′ (𝝉 ) ⎣Im{˜ ˜ z𝑝 }⎦, where z𝑝 := := G 𝑝 0 ⎤ ⎡ ˜ 𝑝 (𝝉 )} ˜ 𝑝 (𝝉 )} −Im{G [ ] Re{G ′ Re{ˆ a𝐶 } ⎦ ⎣Im{G ˜ ˜ 𝑝 (𝝉 )} ˆ𝐶 := . Re{G𝑝 (𝝉 )} and a Im{ˆ a𝐶 } 0 𝛼𝑝 I𝑀 The penalty term is controlled by 𝛼𝑝 that allows us to tradeoff between the smoothness of 𝐽𝐶 (𝝉 ; 𝑘) and its bias from the global minimum of 𝐽(𝝉 ; 𝑘). We solve for 𝝉ˆ by minimizing ˆ 𝐶 from (25). the error function in (26) after substituting for a
z(𝑅) = S(𝑅) (𝝉 )a + 𝜼,
(27)
where z(𝑅) = [𝑧(0) ⋅[⋅ ⋅ 𝑧(𝑁 𝑅]− 1)]𝑇 and S(𝑅) (𝝉 ) is an 𝑁 𝑅 × 𝑀 matrix with S(𝑅) (𝝉 ) 𝑛,𝑚 = 𝑠(⟨𝑛⟩𝑁 − 𝜏𝑚 ), where ⟨⋅⟩𝑁 represents modulo-by-𝑁 operation. For 𝑀 multipaths, we have 𝝉 = [𝜏1 ⋅ ⋅ ⋅ 𝜏𝑀 ]𝑇 and a = [𝑎1 ⋅ ⋅ ⋅ 𝑎𝑀 ]𝑇 . Similar to (19), the frequency-domain model for the data from 𝑅 rotations of the antenna can be expressed as 𝑧˜(𝑅) (𝑘) = 𝑠˜(𝑅) (𝑘)
𝑀 ∑
𝑎𝑚 𝑒−𝑗2𝜋𝑘𝜏𝑚 /𝑁 𝑅 + 𝜂˜(𝑘)
(28)
𝑚=1
for 𝑘 = 0, ⋅ ⋅ ⋅ , 𝑁 𝑅 − 1
˜ (𝑅) (𝝉 )a + 𝜼, z˜(𝑅) = G
(29)
˜ (𝑅) (𝝉 ) := 𝑧 (𝑅) (0) ⋅ ⋅ ⋅ 𝑧˜(𝑅) (𝑁 𝑅 − 1)]𝑇 , G where ˜z(𝑅) := [˜ (𝑅) (𝑅) (𝑅) (𝑅) (𝑅) ˜ H (𝝉 ) with S ˜ S := diag[˜ 𝑠 (0) ⋅ [⋅ ⋅ 𝑠˜ (𝑁 𝑅 𝐷 𝐷 ] − 1)] and H(𝑅) (𝝉 ) an 𝑁 𝑅 × 𝑀 matrix with H(𝑅) (𝝉 ) 𝑝,𝑚 = 𝑒−𝑗2𝜋(𝑝−1)𝜏𝑚 /𝑁 𝑅 . The LS estimates of a and 𝝉 can be obtained by minimizing an objective function of the form (26). VI. E XPERIMENTAL R ESULTS A. Experimental Setup The experiments were performed indoors in a fieldhouse at the Recreation Center and outdoors at the front plaza of Coffman Memorial Union, both at the University of Minnesota. The indoor test area was a rectangular field measuring 55.14 m by 43 m, while the outdoor test area measured 53.5m by 23.35m. Snapshots of the experimental set-up are shown in Figs. 4(a) and 4(b). The reference nodes were placed at three corners of the rectangular field and the mobile node whose position is to be determined was placed inside the field. The indoor experiments were done under three different channel conditions, which are listed in Table I along with the ¯ corresponding mean position estimation errors (ℰ).
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TABLE I P OSITION ESTIMATION USING DIRECTIONAL BEACON ALGORITHM UNDER DIFFERENT CHANNEL CONDITIONS († USING SINGLE 90∘ ANTENNA ROTATION ). ℰ¯ †
Indoor: LOS, no scatterers
1.18 m
Indoor: LOS, one stationary reflector near Tx
1.46 m
Indoor: obstructed-LOS, moving objects in the vicinity of Rx
2.18 m
Outdoor: LOS, moving objects in the vicinity of Tx and Rx
3.10 m
4500 4000 Estimated amplitude of the sinusoid
Type of environment
5000
We use the algorithm described in Section V to estimate the LOS and multipath components in the received signal data. For simplicity, we use 𝑀 = 2, i.e., we consider the LOS component and a single reflection or multipath component. In our experimental setup the estimated amplitude profile 𝑧ˆ(𝑛) consists of 𝑁𝑟 = 10000 samples for 90∘ rotation of the directional antenna (𝜑 = 0 to 90∘ ). These samples are taken every 3.2 msec. To keep the required computations at a reasonable level, we downsample 𝑧ˆ(𝑛) and 𝑠(𝑛) by 16 and use 1024-point DFT. For the penalty term in (26), we use 𝛼𝑝 = 104 . We use MATLAB optimization routines (fminunc) to minimize (26) and obtain the time-delay estimate 𝝉ˆ = [ˆ 𝜏1 𝜏ˆ2 ]𝑇 . Substituting 𝝉ˆ in (25) gives us the LS estimate of the complex amplitude ˆ𝐶 = [ˆ a 𝑎𝐶1 𝑎 ˆ𝐶2 ]𝑇 which comprises the LOS and multipath component amplitudes. We use the earliest component (smaller of the 𝜏𝑖 ’s, 𝑖 = 1, 2) as the LOS component which is true for both clear LOS and obstructed LOS scenarios. We repeat this procedure for the received signal from each reference node. D. Position estimation and bound on estimation error variance Based on the estimated LOS component for each RN, we obtain the corresponding time estimates 𝑡𝑖 , 𝑖 = 1, 2, 3. We then use (5), (4) and (3) to estimate the position of the mobile node Q. For further improving the accuracy of the position estimate we use the technique outlined in Section V-A for obtaining the time-delay estimates using multiple 90∘ rotations of the transmitting antenna at each RN. Since this increases the data record length used for estimation we expect the signal-to-noise ratio (SNR) to increase and the estimation error variance to decrease as we add data from multiple antenna rotations. In Fig. 6, we plot the variance of the position estimation error as
2500 2000 1500
500 0
0
10
20
30 40 50 60 70 Instantaneou angle of rotating antenna in degrees
80
30 40 50 60 70 Instantaneou angle of rotating antenna in degrees
80
90
(a) ML amplitude estimation using single burst of 𝑁 = 32 samples 5000 4500 4000 Estimated amplitude of the sinusoid
C. Estimation of the LOS Component
3000
1000
B. Estimation of Amplitude Profile We use (11) to obtain the ML estimate of the amplitude of the sinusoid in a single burst of the received signal. The estimated amplitude profile 𝑧ˆ(𝑛) that corresponds to 𝜑 = 0 to 90∘ is shown in Fig. 5(a). To improve the estimated amplitude profile, we determine instances of consecutive signal bursts and obtain an enhanced ML estimate of the sinusoid amplitude using (15). The enhanced amplitude profile is shown in Fig. 5(b). The use of consecutive bursts to estimate the amplitude reduces the effect of noise as seen by comparing Figs. 5(a) and 5(b).
3500
3500 3000 2500 2000 1500 1000 500 0
0
10
20
90
(b) ML amplitude estimation using multiple consecutive bursts Fig. 5.
Estimated amplitude profile of the received signal from RN-2.
a function of the number of 90∘ antenna rotations. It is seen that with as few as eight 90∘ rotations of the transmit antenna, the estimation error variance goes down by 10 dB. We obtain a lower bound on the variance of the position estimation error (ℰ) as (refer Appendix for details): ( ) 1 var(ℰ) ≥ 𝐾𝑥2 + 𝐾𝑦2 ⋅ 𝜉 (30) 𝐹¯2 𝑁𝑜 /2
This bound is not as tight as the Cramer-Rao lower bound (CRLB) and is rarely attainable, but is mathematically more tractable. Using a single rotation of the antenna, the root mean square value of the position estimation error is 1.457 m. After using eight 90∘ rotations of the antenna the root mean square position error goes down to 0.362 m. E. Effect of Time Delay Estimation, Synchronization and Motor Speed Step Errors on Position Estimation Following the notation used in the Appendix, we can express the estimation error in the 𝑥 and 𝑦 co-ordinates as: ∂𝑔(𝑡2 , 𝑡3 ) ⋅ 𝜔 ⋅ Δ𝑡 = 𝐾𝑥 ⋅ Δ𝑡 ∂𝛼2 ∂ℎ(𝑡2 , 𝑡3 ) Δˆ 𝑦= ⋅ 𝜔 ⋅ Δ𝑡 = 𝐾𝑦 ⋅ Δ𝑡 ∂𝛼2
Δˆ 𝑥=
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SHAH et al.: IMPLEMENTATION OF A DIRECTIONAL BEACON-BASED POSITION LOCATION ALGORITHM IN A SIGNAL PROCESSING FRAMEWORK
12
20
10
Directional Beacon based Method (ω = 0.049 rad/sec) Directional Beacon based Method (ω = 0.098 rad/sec) Directional Beacon based Method (ω = 0.196 rad/sec) TOA based Method
10
10
10
0
10
8
Position estimation error (in meters)
Position estimation error variance (in dB)
From experimental data Analytical lower bound
−10
−20
−30
−40
6
10
4
10
2
10
0
10
−2
−50
−60
1051
10
−4
1
2
3
4
5
6
7
8
Number of rotations of the antenna
Fig. 6. Position estimation error variance as a function of the number of rotations of the antenna. Signal bandwidth is 1 kHz and each antenna rotation takes 32 seconds.
where Δ𝑡 is the time-delay estimation error. Thus, the position estimation error is: √ ℰ = (Δˆ 𝑥)2 + (Δˆ 𝑦 )2 (31) On the other hand, for TOA-based techniques, the corresponding position estimation error is given by: √ ℰ𝑇 𝑂𝐴 ≈ (𝑐Δ𝑡)2 + (𝑐Δ𝑡)2 (32) where 𝑐 = 3×108 m/s is the speed of electromagnetic waves in free space. Comparing (31) and (32) it is seen that the position error for directionality-based techniques is proportional to the angular speed of the rotational antenna (𝜔) while for TOA techniques it is proportional to 𝑐. Thus the error due to the proposed algorithm remains nearly 8 orders of magnitude smaller than TOA schemes for similar time-delay estimation errors. This comparison is illustrated in Fig. 7. The directional beacon-based method results in only about 0.5 m error with up to 500 msec error in time-delay estimation for 𝜔 = 0.049 rad/s. Fig. 7 also shows that the time needed for each position estimation in the directional beacon-based method can be reduced significantly by increasing 𝜔 without significant loss in positioning accuracy. Similar analysis is applicable to the synchronization error between RNs. Another source of error in the directional beacon-based algorithm could be changes in the angular speed of the antenna (𝜔). The stepper motor used in our experiments is rated for a step error of ±1.08 arc minutes (or 0.018 deg). It can be shown that a 500 msec error in time-delay estimation is equivalent to a 1.4 deg step error over the duration of one position estimation and the rated step error is close to two orders of magnitude lower. This shows that the step error can be safely ignored as a source of error in our experimental set-up. F. Comparison with existing methods It has been shown that the theoretical accuracy of RSS-based position location system with ultra-wideband transceiver [2] is 1.76m when the target node is 10m away from reference nodes. Since the focus of this paper is to
10
0
0.05
0.1
0.15 0.2 0.25 0.3 0.35 Error in time delay estimation (in sec)
0.4
0.45
0.5
Fig. 7. Comparison of the Directional Beacon and TOA-based position estimation methods in terms of sensitivity to time-delay estimation error. TABLE II C OMPARISON OF THE DIRECTIONAL BEACON AND N ASIPURI -L I [5] ALGORITHMS (I NDOOR LOS WITH NO SCATTERERS ). Position estimation method
Mean position estimation error
Nasipuri-Li algorithm [5]
3.91 m
Directional beacon algorithm
1.18 m
develop a position location system and evaluate its accuracy experimentally, we compare our algorithm to an existing algorithm [5], developed with a similar focus. In [5], the authors estimate node position using DOAs computed based purely on RSS. In our implementation of [5], we first estimate the amplitude profile of the received signal as described in Section IV. In the next step, we apply the time-delay estimation technique of [5] and finally compare the corresponding mean position estimation error with our algorithm. The results are summarized in Table II.
G. Note on practical applicability The prototype implementation of the proposed algorithm establishes its viability. Apart from the use of rotational directional antenna, the proposed position location algorithm can be applied to any wireless network without the need of any costly radio transceiver. To estimate the computational needs of the proposed algorithm, we determine the number of real multiplications at different stages of the algorithm. The parameterized results are presented in Table III with the respective symbols as defined in Sections IV and V. In our experimental setup, 𝑁𝑏 = 32, 𝐿 = 4, 𝑁𝑟 = 10000, 𝑁 = 1024 and 𝑀 = 2. The parameter 𝑛𝑖𝑡𝑒𝑟 (in Table III) represents the number of iterations of the minimization routine and is found to be close to 50. Thus, for our experimental setup the total number of real multiplications is around 1.93 × 109 . Assuming a typical embedded digital signal processor consuming around 13.9nJ per complex multiplication [17], the power consumption at the mobile node is estimated to be 112mW.
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TABLE III C OMPUTATIONAL COST OF THE PROPOSED ALGORITHM IN TERMS OF REAL MULTIPLICATIONS . Algorithm step
Computational complexity
Signal detection
2𝑁𝑟 ) ] [ ( 4 2 4𝐿3 𝑁𝑏 + 8𝐿3 + 2𝐿𝑁𝑟
Enhanced amplitude estimation FFT of estimated amplitude profile Estimation of LOS component
𝑁 log2 𝑁 ( 4𝑛𝑖𝑡𝑒𝑟 𝑁 2 + 𝑁 2 𝑀 + 2𝑁 𝑀 2 ) +𝑁 𝑀 + 𝑁 + 𝑀 3
For the sake of clarity, we assume that the error associated with the estimated coordinates (ˆ 𝑥, 𝑦ˆ) of node Q is due to the error in the estimation of 𝑡2 or 𝜏ˆ2 . In order to obtain a bound on the variance of position estimation error (ℰ) we proceed as follows. The position estimation error ℰ is defined as ℰ 2 := (ˆ 𝑥 − 𝑥)2 + (ˆ 𝑦 − 𝑦)2
where 𝑥ˆ and 𝑦ˆ are the estimated values of 𝑥 and 𝑦 co-ordinates, respectively. Taking expectation of (37) we can write 𝐸(ℰ 2 ) = var(ˆ 𝑥) + var(ˆ 𝑦)
VII. C ONCLUSION The system implementation for a directional beacon-based position location algorithm using RF signals was presented. The algorithm overcomes the limitation of traditional RSS techniques by improving the position estimation accuracy using maximum likelihood amplitude estimation, least squaresbased time-delay estimation and combining data from multiple antenna rotations. Accuracy of the proposed algorithm depends on the width of the antenna beampattern and rotational speed of the directional antenna. The system is suitable for low-cost implementation and has been demonstrated to achieve position estimation with sub-meter accuracy. The robustness of the algorithm/implementation to timing and synchronization errors was also demonstrated. A PPENDIX L OWER BOUND ON POSITION ESTIMATION ERROR Consider the signal model as described by (29). Based on our experimental setup, we use 𝑀 = 2. The vector of parameters to be estimated is 𝝀 := [𝜏1 𝜏2 𝑎2 ]. We ignore 𝑎1 as it can always be normalized to unity. Using Cramer-Rao lower bound (CRLB) we can bound the variance as var(𝜆𝑖 ) ≥ [I−1 (𝝀)]𝑖𝑖
for 𝑖 = 1, 2, 3
(33)
where I(𝝀) is a[3 × 3 Fisher] Information matrix (FIM) with 2 y(𝐿) ,𝝀) . To avoid the matrix inversion [I(𝝀)]𝑖𝑗 = −𝐸 ∂ 𝑝(˜ ∂𝜆𝑖 ∂𝜆𝑗 in (33), we invoke the property that a FIM satisfies [14, p. 65] 1 [I−1 (𝝀)]𝑖𝑖 ≥ . (34) [I(𝝀)]𝑖𝑖 1 , that is easier This leads to a loose bound, var(𝜆𝑖 ) ≥ [I(𝝀)]𝑖𝑖 to compute though difficult to attain, in general. It can be shown that for time-delay estimation, the diagonal elements of the FIM are given by [14]:
[I(𝝀)]𝑖𝑖 =
𝜉 ¯2 𝐹 𝑁𝑜 /2
for 𝑖 = 1, 2
(35)
where 𝜉 is the signal energy, 𝐹¯2 is the mean square bandwidth of the signal and 𝑁𝑜 /2 is the noise variance. Thus, var(𝜏𝑖 ) ≥
1
𝜉 ¯2 𝑁𝑜 /2 𝐹
for 𝑖 = 1, 2.
(36)
Without loss of generality, we can assume that 𝜏ˆ1 < 𝜏ˆ2 . Thus, for the signal received from RN-1 we have 𝑡1 = 𝜏ˆ1 and the same principle applies to the estimation of 𝑡2 and 𝑡3 .
(38)
To compute var(ˆ 𝑥), we assume that the error in 𝑥ˆ is due to the error in the estimation of 𝑡2 . Using (5) and (3) we obtain 𝐿2 := 𝑔(𝑡2 , 𝑡3 ) (39) tan 𝛼2 + cot 𝛼3 since 𝛼2 and 𝛼3 are functions of 𝑡2 and 𝑡3 , respectively. Now, the CRLB for 𝑥 ˆ can be expressed as 𝑥 ˆ=
var(ˆ 𝑥) ≥ 𝐾𝑥2 ⋅ var(𝜏2 ).
(40)
where 𝐾𝑥 := ∂𝑔(𝑡2 , 𝑡3 )/∂𝑡2 . Similarly, if 𝑦ˆ := ℎ(𝑡2 , 𝑡3 ) and 𝐾𝑦 := ∂ℎ(𝑡2 , 𝑡3 )/∂𝑡2 then using (36) we can bound the mean square value of the estimation error as ( ) 𝐸(ℰ 2 ) ≥ 𝐾𝑥2 + 𝐾𝑦2 ⋅ where
VARIANCE
(37)
𝐾𝑥 =
𝜉 𝑁𝑜 /2
1
𝜉 ¯2 𝑁𝑜 /2 𝐹
(41)
is the SNR. 𝐾𝑥 and 𝐾𝑦 can be written as
∂𝑔(𝑡2 , 𝑡3 ) ∂𝛼2 ⋅ ∂𝛼2 ∂𝑡2
and 𝐾𝑦 =
∂ℎ(𝑡2 , 𝑡3 ) ∂𝛼2 ⋅ . (42) ∂𝛼2 ∂𝑡2
R EFERENCES [1] K. Pahlavan, Xinrong Li, and J. P. Makela, “Indoor geolocation science and technology,” IEEE Commun. Mag., vol. 4, pp. 112–118, Feb. 2002. [2] S. Gezici and H. V. Poor, “Position estimation via ultra-wideband signals,” Proc. IEEE, vol. 97, pp. 386–403, Feb. 2009. [3] S. F. A. Shah and A. H. Tewfik, “Performance analysis of directional beacon based postion location algorithm for UWB systems,” in Proc. IEEE GLOBECOM, vol. 4, pp. 2409–2413, Nov. 2005. [4] C. D. McGillem and T. S. Rappaport, “A beacon navigation method for autonomous vehicles,” IEEE Trans. Veh. Technol., vol. 38, no. 3, pp. 132–139, 1989. [5] A. Nasipuri and K. Li, “A directionality based location discovery scheme for wireless sensor networks,” in Proc. ACM Symposium on Wireless Sensor Network Applications, pp. 105–111, Sept. 2002. [6] J.-Y. Lee and R. A. Scholtz, “Ranging in a dense multipath environment using an UWB radio link,” IEEE J. Sel. Areas Commun., vol. 20, pp. 1677–1683, Dec. 2002. [7] P. Bahl and V. N. Padmanabhan, “RADAR: an in-building RF-based user location and tracking system,” in Proc. INFOCOM 2000, vol. 2, pp. 775–784, Mar. 2000. [8] S. Srirangarajan and A. H. Tewfik, “Localization in wireless sensor networks under non line-of-sight propagation,” in Proc. IEEE GLOBECOM, vol. 6, pp. 3477–3481, Nov. 2005. [9] D. Dardari and et. al., “Ranging with ultrawide bandwidth signals in multipath environments,” Proc. IEEE, vol. 97, pp. 404–426, Feb. 2009. [10] A. H. Quazi, “An overview on the time delay estimate in active and passive systems for target localization,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-29, pp. 527–533, June 1981. [11] A. Nasipuri and R. el Najjar, “Experimental evaluation of an angle based indoor localization system,” in International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, Apr. 2006. [12] Y. Iwasaki, N. Kawaguchi, and Y. Inagaki, “Design, implementation and evaluations of a direction based service system for both indoor and outdoor,” in Proc. Second Intl. Symp. Ubiquitous Comput. Syst., vol. 3598 of Lecture Notes in Computer Science, pp. 20–36, Springer, 2005.
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SHAH et al.: IMPLEMENTATION OF A DIRECTIONAL BEACON-BASED POSITION LOCATION ALGORITHM IN A SIGNAL PROCESSING FRAMEWORK
[13] D. Niculescu and B. Nath, “Ad hoc positioning system (APS) using AoA,” in Proc. IEEE INFOCOM 2003, vol. 3, pp. 1734–1743, Apr. 2003. [14] S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ: Prentice Hall, 1993. [15] D. W. Tufts and R. L. Field, “Least-squares time-delay estimation for transient signals in a multipath environment,” J. Acoust. Soc. Amer, vol. 92, pp. 210–218, July 1992. [16] T. G. Manickam, R. J. Vaccaro, and D. W. Tufts, “A least-squares algorithm for multipath time-delay estimation,” IEEE Trans. Signal Process., vol. 42, pp. 3229–3233, Nov. 1994. [17] L. Wanhammar, DSP Integrated Circuits. Academic Press, 1999. S. Faisal A. Shah received the B.S. degree from NED University of Engineering and Technology, Karachi, Pakistan, in 1998 and the M.S. degree from King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, in 2001, both in electrical engineering. He received the Ph.D. degree in electrical engineering from the University of Minnesota, Minneapolis, MN in 2008. From 2001 to 2004, he was a lecturer in the Department of Electrical Engineering at University of Sharjah, Sharjah, UAE. From 2004 to 2006, he was a graduate research assistant in the Department of Electrical Engineering, University of Minnesota. He has worked as a senior DSP engineer at Azimuth Systems, Acton, MA, from 2008 to 2009. Since August 2009, he has been with COM DEV, Cambridge, Canada, where he is a Systems engineer working on the DSP design for RADARSAT Constellation Mission. His research spans the fields of signal processing and wireless communications, with particular emphasis on OFDMA systems, distributed estimation in wireless sensor networks, adaptive channel estimation and design of low-complexity DSP algorithms.
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Seshan Srirangarajan (S’05-M’08) received the B.E. degree from University of Mumbai, India in 2001, and the M.S. and Ph.D. degrees from University of Minnesota in 2005 and 2008, respectively, all in Electrical Engineering. He is currently a Research Fellow with Intelligent Systems Center at Nanyang Technological University, Singapore. In 2005-06, he was with the Wireless Technologies Group at Honeywell Technology Center, Minneapolis, Minnesota, as an intern. His research interests span wireless communication and signal processing, with a focus on distributed algorithms and event detection in sensor networks, and positioning in wireless networks. Ahmed H Tewfik (F’96) received his B.Sc. degree from Cairo University, Egypt, in 1982 and his M.Sc., E.E. and Sc.D. degrees from MIT, Cambridge, MA, in 1984, 1985 and 1987 respectively. He is the E. F. Johnson professor of Electronic Communications with the Department of Electrical Engineering at the University of Minnesota. He served as a consultant to several companies, and has worked with Texas Instruments and Computing Devices International. From 1997 to 2001, he was the President and CEO of Cognicity, Inc., an entertainment marketing software tools publisher that he co-founded, on partial leave of absence from the University of Minnesota. His current research interests are in genomics and proteomics, audio signal separation, wearable health sensors, brain computing interface and programmable wireless networks. Prof. Tewfik is a Fellow of the IEEE. He was a Distinguished Lecturer of the IEEE Signal Processing Society in 1997-1999. He received the IEEE third Millennium award in 2000. He was elected to the board of governors of the IEEE Signal Processing Society in 2005. He was awarded the E. F. Johnson professorship of Electronic Communications in 1993, a Taylor faculty development award from the Taylor foundation in 1992 and an NSF research initiation award in 1990. Prof. Tewfik delivered plenary lectures at several IEEE and non-IEEE meetings and taught tutorials on bioinformatics, ultrawideband communications, watermarking and wavelets at major IEEE conferences. He was the first Editor-in-Chief of the IEEE S IGNAL P ROCESSING L ETTERS . He is a past associate editor of the IEEE T RANSACTIONS ON S IGNAL P ROCESSING, the IEEE T RANSACTIONS ON M ULTIMEDIA and the IEEE J OURNAL ON S ELECTED T OPICS IN S IGNAL P ROCESSING. He is currently an Associate Editor of the EURASIP J OURNAL ON B IOINFORM . S YST. B IOL.
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