DoA-LF: A Location Fingerprint Positioning Algorithm with Millimeter-Wave
aa r X i v : . [ c s . I T ] F e b DoA-LF: A Location Fingerprint PositioningAlgorithm with Millimeter-Wave
Zhiqing Wei, Yadong Zhao, Xinyi Liu, Zhiyong Feng
Abstract —Location fingerprint (LF) has been widely appliedin indoor positioning. However, the existing studies on LF mostlyfocus on the fingerprint of WiFi below 6 GHz, bluetooth, ultrawideband (UWB), etc. The LF with millimeter-wave (mmWave)was rarely addressed. Since mmWave has the characteristicsof narrow beam, fast signal attenuation and wide bandwidth,etc., the positioning error can be reduced. In this paper, an LFpositioning method with mmWave is proposed, which is namedas DoA-LF. Besides received signal strength indicator (RSSI) ofaccess points (APs), the fingerprint database contains directionof arrival (DoA) information of APs, which is obtained via DoAestimation. Then the impact of the number of APs, the intervalof reference points (RPs), the channel model of mmWave andthe error of DoA estimation algorithm on positioning error isanalyzed with Cramer-Rao lower bound (CRLB). Finally, theproposed DoA-LF algorithm with mmWave is verified throughsimulations. The simulation results have proved that mmWavecan reduce the positioning error due to the fact that mmWavehas larger path loss exponent and smaller variance of shadowfading compared with low frequency signals. Besides, accurateDoA estimation can reduce the positioning error.
Index Terms —Millimeter-Wave, Location Fingerprint, Direc-tion of Arrival Estimation, Cramer-Rao Lower Bound
I. I
NTRODUCTION
With the development of mobile Internet, the location basedservices and applications such as mobile social networks,online maps and online-to-offline (O2O) services are de-veloping rapidly, which makes the positioning technologiescritical for modern life. The global navigation satellite system(GNSS) consisting of GPS, GLONASS, GALILEO, BDS,etc. is efficient in outdoor environment. However, in indoorenvironment, because satellite signal is blocked by roofs andwalls of buildings, GNSS does not work, which triggers thestudies on indoor positioning.LF positioning plays an important role in complex indoorenvironment because it does not require line-of-sight (LOS)measurement [1]. The scenario of LF positioning is illustratedin Fig. 1. There are five APs in a specific indoor environment.The features of received signals from all APs are measures atreference points (RPs), which constitute LF database. When adevice appears at a spot, it measures the features of all APs tofind the RPs with similar features. Then the estimated locationcan be the mean of the locations of K RPs with most similarfeatures, which is K nearest node (KNN) method.Among LF positioning systems, WiFi fingerprint systemsare most widely studied and applied in indoor positioning. The final version of this paper is published in IEEE Access. ZhiqingWei, Yadong Zhao, Xinyi Liu and Zhiyong Feng are with Beijing Universityof Posts and Telecommunications, email: { weizhiqing, zhaoyadong, xinyi,fengzy } @bupt.edu.cn. Reference Point AP AP AP AP AP K nearest nodes
Positioning viaLocation Fingerprint
K nearest nodes
Fig. 1. The scenario of LF positioning
Except of WiFi fingerprint, UWB fingerprint systems are alsostudied [2][3]. Besides, with the development of bluetoothlow energy (BLE) subsystem, which is “designed for machinetype communication” [4], BLE devices will be more and moredense in buildings. Hence bluetooth LF based positioning isalso applied in indoor positioning [5].The schemes of LF based indoor positioning focus onthe construction of radio map [6][7], the contour-based LFmethod [8], the gradient-based fingerprinting [9], the CSI-based indoor positioning [10][11][12] and machine learningbased LF schemes [13][14], etc. As to the construction ofradio map in LF positioning, Jiang et al. in [6] constructedradio map via crowdsourcing collection. Jung et al. in [7]evaluated the performance of radio map construction methodsin various environments. To provide accurate LF based indoorpositioning, He et al. in [8] developed the contour-basedtrilateration to combine the advantages of trilateration andfingerprinting [8]. Shu et al. in [9] proposed the gradientfingerprinting method to reduce the impact of varying RSSIon the accuracy of LF positioning. LF positioning schemesgenerally adopt RSSI information, however, the channel stateinformation (CSI) contains more information, which can beexploited to improve the accuracy of indoor positioning. Wu et al. in [10] designed the fingerprinting system exploitingCSI to construct the propagation model to improve positioningaccuracy. Wang et al. in [11] and [12] designed the CSI-based fingerprinting schemes via deep learning approach. Thepositioning problem is mathematically a regression problem.Besides, LF positioning methods have large amount of datain the offline database. Thus machine learning approachescan be adopted to solve the positioning problem. Mahfouz et al. in [13] proposed a kernel-based machine learning forLF positioning. Zou et al. applied extreme learning machineto solve indoor positioning problem [14].Besides the design of positioning schemes, the analysis ofpositioning error is also an active area. In [15], Li proposedRSSI based positioning scheme without path loss model andthe Cramer-Rao lower bound (CRLB) was derived to char-acterize the positioning error. Naik et al. in [16] studied theimpact of the height of AP on the positioning accuracy andthe CRLB is correspondingly derived. Jin et al. in [17] havestudied the performance of RSSI based LF positioning schemeusing CRLB. And Hossain et al. in [18] analyzed the CRLBof signal strength difference based LF positioning scheme.Overall, LF positioning schemes are mostly focused on thefingerprint of WiFi, Bluetooth, UWB, etc. To our best knowl-edge, there are rarely studies on LF positioning with mmWave.Because mmWave has the characteristics of narrow beam, fastsignal attenuation and wide bandwidth, etc., mmWave signalscan provide centimeter level ranging accuracy [19][20][21].Besides, with the development of fifth generation mobile com-munication (5G), dense mmWave small cells will be widelydeployed [22], which makes LF positioning with mmWaveto be realizable. In the standard of IEEE 802.11ad, 60 GHzmmWave for multi-gigabit-per-second WiFi is developed [23].Besides, IEEE has also started the standardization of IEEE802.11aj, which is a WLAN system operating at 45 GHzmmWave band [24]. Thus in the near future, LF positioningwith mmWave is practical.In this paper, an LF positioning method with mmWaveis proposed, which may be promising in the era of 5Gwhen APs with mmWave are widely deployed. We utilizethe characteristics of narrow beam and fast signal attenuationof mmWave to reduce positioning error. The proposed LFpositioning scheme is called DoA-LF, because the fingerprintdatabase contains DoA information of APs besides RSSIinformation, which is obtained via DoA estimation. Then weselect K candidate RPs with the most similar features andcalculate their weighted mean through weighted K nearestneighbor (WKNN) algorithm, which is the final estimatedlocation. It is noted that the value of K has an impact onthe positioning error and there exists an optimal K that canminimize the positioning error. With DoA information of APsoperating on mmWave spectrum band, the positioning erroris significantly reduced compared with LF positioning withsignal below 6 GHz. Then the impact of the number of APs,the interval of RPs, the variance of DoA estimation, the pathloss exponent and the shadow fading of mmWave on thepositioning error is analyzed via Cramer-Rao Lower Bound(CRLB). Finally, the analysis results are verified by simulationresults. The key parameters and notations in this paper arelisted in Table I.The rest of this paper is organized as follows. In Section II,system model is introduced. Section III presents the proposedDoA-LF algorithm. In Section IV, CRLB is achieved, whichyields the impact of various parameters on the positioningerror. Simulation results are provided in Section V to verify ourscheme and analysis results. Finally, Section VI summarizesthis paper. TABLE IK EY P ARAMETERS AND N OTATIONS
Symbol Description
P L ( d ) Path loss at a distance dn Path loss exponent X σ Shadow fading factor (ˆ x, ˆ y ) Estimated location coordinates ( x i , y i ) Location coordinates of i th RP s Signal strength vector ϕ Angle vector Q Number of APs M Number of RPs A Test area θ Location of the test point r Location of the estimated point
Pr( r | θ ) Conditional probability d rθ Euclidean distance between r and θd ir Euclidean distance between r and i th AP d iθ Euclidean distance between θ and i th AP s iθ Actual signal strength of i th AP at the test point s ir Received signal strength of the i th AP at point rϕ iθ DoA of i th AP at the test point ϕ ir DoA of the i th AP at point rσ s Variance of received signal strength σ ϕ Variance of DoA estimation J (ˆ θ ) Fisher Information Matrix C ˆ θ Cramer-Rao Lower BoundAP Access pointRP Reference pointTP Test pointLF Location fingerpintRSSI Received signal strength indicator
II. S
YSTEM M ODEL
The 60 GHz mmWave is adopted for multi-gigabit-per-second WiFi [23]. Hence 60 GHz mmWave is adopted inLF positioning. In this section, the channel model of 60 GHzmmWave and the process of LF positioning are presented.
A. 60 GHz mmWave Channel Model
The channel model of 60 GHz mmWave is [25][26]
P L ( d )[ dB ] = P L ( d ) + 10 n log( dd ) + X σ , d > d , (1)where P L ( d ) = 20 log( πd λ ) is free-space path loss at areference distance d , which is generally set as d = 1 m .The path loss exponent is n . X σ is the shadow fading factormodeled by a Gaussian random variable with mean zero andvariance σ s .In DOA-LF algorithm, besides RSSI information, DoAinformation is also needed, which can be obtained via multiplesignal classification (MUSIC) algorithm. Since MUSIC algo-rithm is well known in the area of array signal processing, wedo not introduce MUSIC algorithm in this paper. Readers canrefer to [31] for details. The MUSIC algorithm has lower com-putational complexity compared with other DoA algorithms.However, other DoA algorithms can also be adopted besidesMUSIC algorithm. If a DoA algorithm has small error, theperformance of positioning algorithm can be correspondinglyimproved. StartConstruct offline databasecontaining RSSI and directionfrom APs to each RPEndOnline match throughWKNN algorithm. Getthe estimated coordinate.Obtain RSSI fromAPs to each RP Obtain directionsfrom APs to each RP
Step 1Step 2Step 3
Fig. 2. The proposed DoA-LF algorithm.
B. DoA-LF Positioning Algorithm
LF positioning plays an important role in complex indoorenvironment. LF positioning algorithm consists of two steps,namely, offline database construction and online matching.Weighted K nearest neighbor (WKNN) algorithm is widelyapplied in LF positioning [32][33][34], where the weightedmean of the coordinates of K nearest RPs is calculated as theestimated coordinates. The weight coefficient w i is generallyinversely proportional to the Euclidean distance between theestimated point and i th RP [32][33] in the feature space span-ning by the vectors containing RSSI and DoA information.Hence the estimated coordinates are as follows [32][33]. w i = γd i + ε , (ˆ x, ˆ y ) = K P i =1 w i ( x i , y i ) K P i =1 w i , (2)where d i is the Euclidean distance in the feature space betweenthe measured point and the i th RP. (ˆ x, ˆ y ) are the estimatedcoordinates and ( x i , y i ) are the coordinates of i th RP. γ isnormalized parameter and ε is a small positive number in orderto prevent the denominator being zero.III. T HE P ROPOSED D O A-LF A
LGORITHM
In this section, the characteristics of mmWave, such asnarrow beam and fast attenuation, are exploited to construct anLF positioning algorithm called DoA-LF algorithm. The DoAinformation of APs obtained by MUSIC algorithm is combinedwith the RSSI of APs to construct a new offline database foronline matching. Then WKNN algorithm is adopted to calcu-late the estimated coordinates. Fig. 2 illustrates the process ofproposed DoA-LF algorithm.
AP RP Test Point (TP)Candidate Point DoA-RangeAP1 AP2AP3 AP4
Fig. 3. A scenario of proposed DoA-LF positioning.
Step 1 : Obtain RSSI and DoA information from APs toeach RP. Q APs and M RPs are deployed in a specific area, whichis illustrated in Fig. 3. The RSSIs of APs at each RP aremeasured, which are saved in a vector s i = [ s i , s i , · · · , s Qi ] with s ji , j = 1 , · · · , Q denoting the RSSI of j th AP at i th RP.Meanwhile, the MUSIC algorithm is adopted for DoAestimation. The directions of APs at each RP are measured,which are saved in a vector ϕ i = [ ϕ i , ϕ i , · · · , ϕ Qi ] with ϕ ji , j = 1 , · · · , Q denoting the DoA of j th AP at i th RP.It is noted that the RSSI and DoA information is obtainedsimultaneously. Step 2 : Construct offline database.Combining Q -dimensional RSSI vector and Q -dimensionaldirection vector, we obtain Q -dimensional vector [ s i , ϕ i ] representing the features of i th RP. The vectors of all RPsspan a feature space and construct the offline database forDoA-LF positioning with mmWave. Step 3 : Online matching.Comparing RSSI and direction information of test pointwith the data in offline database, we can find out K nearestRPs whose features are closest to test point. Then we calculatethe estimated coordinates via the weighted mean of K nearestRPs using (2).For example, in Fig. 3, there are APs and RPs inan area. Firstly, an offline database is established, whichconsists of RSSI and DoA information of all RPs. Secondly, K = 4 candidate RPs with the most similar features areselected, which are denoted by the squares in Fig. 3. Finally,the estimated coordinates can be obtained via calculating theweighted mean of the selected K = 4 candidate RPs. IV. A
NALYSIS OF P OSITIONING E RROR
In the above section, the DoA-LF positioning algorithm withmmWave is proposed. In DoA-LF positioning algorithm, thenumber of APs, the interval of RPs, the channel model ofmmWave and the error of DoA estimation have an impact onthe positioning error. In this section, we analyze the impactof these parameters on positioning error, which can provideguideline for the selection of appropriate parameters in DoA-LF algorithm with mmWave.As illustrated in Fig. 3, there are Q APs and M RPs.APs are distributed on the edge of a specific area denotedas area A . RPs are uniform distributed in area A . Thelocation of test point is denoted as θ . DoA-LF algorithmgenerates the estimated point r with conditional probability Pr( r | θ ) . The Euclidean distance between r and θ is d rθ .The conditional probability Pr( r | θ ) and the distance d rθ can represent positioning error. Besides, Pr( r | θ ) is relatedwith d rθ . In LF positioning, one-to-one mapping can beestablished between Q -dimensional feature space and two-dimensional geo-location space. Thus the expression of con-ditional probability Pr( r | θ ) in two-dimensional geo-locationspace equals to the conditional probability Pr( s r , ϕ r | s θ , ϕ θ ) in Q -dimensional feature space, where s θ , ϕ θ , s r and ϕ r arethe RSSI and DoA information of all APs at location θ and r respectively. Therefore the conditional probability Pr( r | θ ) isas follows. Pr( r | θ ) = Pr( s r , ϕ r | s θ , ϕ θ )= Q Y i =1 Pr( s ir , ϕ ir | s iθ , ϕ iθ )= Q Y i =1 Pr( s ir | s iθ ) Q Y i =1 Pr( ϕ ir | ϕ iθ ) . (3)The conditional probability Pr( s ir | s iθ ) denotes that theactual signal strength of i th AP at the test point is s iθ whilethe received signal strength of i th AP is s ir , which is thesignal strength at location r . Pr( s ir | s iθ ) follows Gaussian dis-tribution with mean s iθ and variance σ s [18][37]. Meanwhile, Pr( ϕ ir | ϕ iθ ) denotes the conditional probability that the DoAof i th AP at test point is ϕ iθ while the estimated DoA of i th APis ϕ ir , which is the DoA of i th AP at location r . Pr( ϕ ir | ϕ iθ ) also follows Gaussian distribution with mean ϕ iθ and variance σ ϕ [38][39].According to the path loss model of mmWave in (1),assuming that the transmit power of each AP is P t , the averagesignal strength at θ from i th AP is given by s iθ = P t − P L ( d iθ ) = P t − P L ( d ) − n log( d iθ d ) − X σ , (5)where X σ is a Gaussian random variable with mean zeroand variance σ s . d iθ is the distance between i th AP and thelocation θ .CRLB is an effective tool to estimate the minimum varianceof parameter estimation error [40][41][42]. In this subsection,the lower bound of the variance of DoA-LF algorithm erroris analyzed by CRLB. Assuming that the test point is θ =( x, y ) T , the unbiased estimation value of θ is ˆ θ = (ˆ x, ˆ y ) T , which is the weighted mean of K nearest RPs selected byDoA-LF algorithm. The covariance matrix of ˆ θ is cov (cid:16) ˆ θ (cid:17) = E (cid:16) (ˆ θ − θ )(ˆ θ − θ ) T (cid:17) = (cid:18) var(ˆ x − x ) cov ((ˆ x − x ) , (ˆ y − y ))cov ((ˆ y − y ) , (ˆ x − x )) var(ˆ y − y ) (cid:19) , (6)where E ( α ) is the expectation of random variable α . var( α ) isthe variance of α . cov( α, β ) is the covariance of random vari-ables α and β . According to CRLB inequality, the covarianceof ˆ θ satisfies the following inequality [43][44]. cov(ˆ θ ) ≥ (cid:16) J (ˆ θ ) (cid:17) − , (7)where J (ˆ θ ) is fisher information matrix (FIM), which isdefined as follows [41][44]. J (ˆ θ ) = E (cid:18) − ∂ ln f ( r ; θ ) ∂θ∂θ T (cid:19) , (8)where r is the corresponding coordinates of Q -dimensionalobservation features and f ( r ; θ ) is denoted as f ( r ; θ ) = f ( r | θ ) f ( θ ) . (9)According to one-to-one mapping relation between Q -dimensional feature space and two-dimensional geo-locationspace, the value of f ( r | θ ) is provided in (3). Besides, θ isuniformly distributed in area A , hence f ( θ ) = | A | where | A | denotes the area of A . Thus the value of f ( r ; θ ) is f ( r ; θ ) = Q Y i =1 κ exp − ( s ir − s iθ ) σ s − ( ϕ ir − ϕ iθ ) σ ϕ ! , = Q Y i =1 κ exp − (cid:16) n lg( d ir d iθ ) (cid:17) σ s − ( ϕ ir − ϕ iθ ) σ ϕ , (10)where κ = π | A | σ s σ ϕ . d ir is the distance between i th AP andthe location r . lg( ∗ ) is a logarithmic function with base .Therefore the function log f ( r ; θ ) is as follows. log f ( r ; θ )= Q X i =1 log κ exp − (10 n lg( d ir d iθ )) σ s − ( ϕ ir − ϕ iθ ) σ ϕ = Q X i =1 log κ − η (cid:18) log( d ir d iθ ) (cid:19) − ( ϕ ir − ϕ iθ ) σ ϕ ! , (11)where log( ∗ ) is a logarithmic function with base e . The valueof η is η = ( 10 n √ σ s log 10 ) . (12)It is noted that d iθ and ϕ iθ are functions of x and y , whoseexpressions are ∂ log f ( r ; θ ) ∂x = Q X i =1 (cid:18) − η cos ϕ iθ d iθ + η log( d ir d iθ ) 1 − ϕ iθ d iθ − sin ϕ iθ σ ϕ d iθ (cid:19) ,∂ log f ( r ; θ ) ∂x∂y = ∂ log f ( r, θ ) ∂y∂x = Q X i =1 (cid:18) − ηsin ϕ iθ d iθ − η log( d ir d iθ ) sin2 ϕ iθ d iθ + ( ϕ ir − ϕ iθ ) cos ϕ iθ − cosϕ iθ sin ϕ iθ σ ϕ d iθ (cid:19) ,∂ log f ( r, θ ) ∂y = Q X i =1 (cid:18) − ηsin ϕ iθ d iθ + η log( d ir d iθ ) 1 − ϕ iθ d iθ − cos ϕ iθ σ ϕ d iθ (cid:19) . (4) d iθ = q ( x − x i ) + ( y − y i ) ,ϕ iθ = arccos( x i − xd iθ ) = arcsin( y i − yd iθ ) , (13)which are illustrated by Fig. 4.In order to obtain (8), we first derive the first order partialderivatives as follows. ∂ log f ( r ; θ ) ∂x = Q X i =1 η log( d ir d iθ ) x − x i d iθ + ( ϕ ir − ϕ iθ ) sin ϕ iθ σ ϕ d iθ ,∂ log f ( r ; θ ) ∂y = Q X i =1 η log( d ir d iθ ) y − y i d iθ + ( ϕ ir − ϕ iθ ) cos ϕ iθ σ ϕ d iθ . (14)Then the second order derivatives can be derived, which areshown in (4).For DoA-LF algorithm, d ir approximately equals d iθ and ϕ ir approximately equals ϕ iθ . Hence the values of log( d ir d iθ ) and ϕ ir − ϕ iθ approximate zero. The entries of (4) become asfollows. ∂ log f ( r ; θ ) ∂x = Q X i =1 (cid:18) − η cos ϕ iθ d iθ − sin ϕ iθ σ ϕ d iθ (cid:19) ,∂ log f ( r ; θ ) ∂x∂y = ∂ log f ( r ; θ ) ∂y∂x = Q X i =1 ( − η sin 2 ϕ iθ d iθ − sin 2 ϕ iθ σ ϕ d iθ ) ,∂ log f ( r ; θ ) ∂y = Q X i =1 ( − η sin ϕ iθ d iθ − cos ϕ iθ σ ϕ d iθ ) . (15)Then the FIM J (ˆ θ ) can be derived as follows. J (ˆ θ ) = (cid:18) J xx (ˆ θ ) J xy (ˆ θ ) J yx (ˆ θ ) J yy (ˆ θ ) (cid:19) , (16)where the entries of (16) are J xx (ˆ θ ) = − ∂ log f ( r ; θ ) ∂x = Q X i =1 η cos ϕ iθ d iθ + sin ϕ iθ σ ϕ d iθ ,J xy (ˆ θ ) = J yx (ˆ θ ) = Q X i =1 η sin 2 ϕ iθ d iθ + sin 2 ϕ iθ σ ϕ d iθ ,J yy (ˆ θ ) = − ∂ log f ( r ; θ ) ∂y = Q X i =1 η sin ϕ iθ d iθ + cos ϕ iθ σ ϕ d iθ . (17)Therefore (cid:16) J (ˆ θ ) (cid:17) − is (cid:16) J (ˆ θ ) (cid:17) − = 1 (cid:12)(cid:12)(cid:12) J (ˆ θ ) (cid:12)(cid:12)(cid:12) (cid:18) J yy (ˆ θ ) − J yx (ˆ θ ) − J xy (ˆ θ ) J xx (ˆ θ ) (cid:19) , (18)where (cid:12)(cid:12)(cid:12) J (ˆ θ ) (cid:12)(cid:12)(cid:12) is (cid:12)(cid:12)(cid:12) J (ˆ θ ) (cid:12)(cid:12)(cid:12) = J xx (ˆ θ ) J yy (ˆ θ ) − J xy (ˆ θ ) J yx (ˆ θ )= Q X i =1 η (cos ϕ iθ − sin ϕ iθ ) σ ϕ d iθ . (19)Substituting (6) and (18) into (7), we have (cid:18) var(ˆ x − x ) cov ((ˆ x − x ) , (ˆ y − y ))cov ((ˆ y − y ) , (ˆ x − x )) var(ˆ y − y ) (cid:19) ≥ | J (ˆ θ ) | (cid:18) J yy (ˆ θ ) − J yx (ˆ θ ) − J xy (ˆ θ ) J xx (ˆ θ ) (cid:19) . (20)Hence the CRLB C ˆ θ of positioning error is C ˆ θ = var(ˆ x − x ) + var(ˆ y − y ) ≥ J xx (ˆ θ ) + J yy (ˆ θ ) | J (ˆ θ ) | = Q X i =1 d iθ (1 + 2 ησ ϕ )2 η (cos ϕ iθ − sin ϕ iθ ) . (21)According to (21), CRLB is an increasing function of σ ϕ .Besides, CRLB is a decreasing function of η . Because of (12),we have conclusions as follows.1) The variance of positioning error is increasing with thevariance σ s , which means that the positioning error isincreasing with the increase of shadow fading factor.2) The variance of positioning error is decreasing with pathloss exponent n , which means that LF positioning error r i q j ir j x q i x r x i d q y q i y r y y x q i AP Fig. 4. The distance and DoA of θ and r . can be reduced when using the signal with large pathloss exponent, such as mmWave.3) In LF positioning with mmWave, when σ ϕ is small,namely, the error of DoA estimation is small, the po-sitioning error can be reduced.Meanwhile, the variance of positioning error is also im-pacted by the interval of RPs and the number of APs. With thedecreasing of the interval of RPs, the variance of positioningerror is decreasing. However, when the interval is extremelysmall, the computational complexity of DoA-LF algorithm willbe large. Besides, with the increasing of the number of APs,the positioning error can be reduced. However, when the num-ber of APs is extremely large, the computational complexityof DoA-LF algorithm will be intolerable. Simulation resultsin Section V will support our analysis.V. S IMULATION R ESULTS AND A NALYSIS
A. DoA-LF Algorithm
Firstly, the LF positioning scheme with mmWave is com-pared with the traditional LF positioning with 2.4 GHz WiFi.The path loss models of 60 GHz mmWave and 2.4 GHzWiFi under the same indoor NLOS environment are givenas follows [27][28]. P L mmw ( d )[ dB ] = − . . d ) + N (0 , . ,P L . GHz ( d )[ dB ] = − . . d ) + N (0 , . . (22)In order to create an offline database in the training phase, samples of RSSI and DoA estimations at each RP areachieved. Then the average RSSI and DoA estimations at eachRP are recorded in the offline database. The size of indoorenvironment is × and RPs are deployed per The LF positioning is mainly applied in the NLOS environment. x (m) y ( m ) Test PointsEstimated PointsAPs
Fig. 5. A test of DoA-LF positioning algorithm.
Positioning Error (m) C u m u l a t i v e D i s t r i bu t i on F un c t i on Fig. 6. Comparison of CDFs between 60 GHz mmWave and 2.4 GHz WiFiwith different algorithms, where K = 6 for KNN and WKNN. meters. Besides, APs are distributed in the corners of theentire region. The transmit power of each AP is mW. Fig.5 illustrates the intuitive results of DoA-LF algorithm, wherethe estimated points match the test points very well.In order to present the errors of different positioningmethods, the probability of successful positioning within therange of location error, namely, the cumulative distributionfunction (CDF) of LF positioning is simulated. Assume thatthe positioning error is E , which is a random variable. Forany positioning error E = e in the horizontal axis, itscorresponding CDF, namely, the value in vertical axis is theprobability of E < e . Besides, the average positioning errorsof different methods are simulated.Firstly, the impact of path loss models of 2.4 GHz WiFi and60 GHz mmWave on three different LF positioning algorithms,namely, the algorithms of NN, KNN and WKNN, is illustratedin Fig. 6 and Fig. 7. In Fig. 6, the CDF curve of 60 GHzmmWave grows much faster than that of 2.4 GHz WiFi whenusing the same LF positioning algorithm . The underlying The proposed DoA-LF algorithm also adopts WKNN operations. However,in this section, the algorithms of nearest node (NN), K nearest node (KNN)and weight K nearest node (WKNN) denote the LF positioning algorithmswithout DoA estimation, namely, they only use RSSI information. A v e r age P o s i t i on i ng E rr o r ( m ) Fig. 7. Comparison of average positioning errors between 60 GHz mmWaveand 2.4 GHz WiFi with different algorithms, where K = 6 for KNN andWKNN. Number of FPs R SS I ( db m ) Fig. 8. Comparison of RSSI between 60 GHz mmWave and 2.4 GHz WiFiwith the same transmit power of APs. reason is that mmWave has the characteristics of narrow beamand fast attenuation, which can increase the discrimination ofRSSI and DoA information of different RPs and reduce thepositioning error. Besides, the positioning error of 60 GHzmmWave is much higher than that of 2.4 GHz WiFi whenusing the same LF positioning algorithm. Moreover, WKNNalgorithm yields the lowest error and the fastest convergencespeed among the three LF algorithms for 60 GHz mmWave.Fig. 7 shows that 60 GHz mmWave yields less positioningerror compared with 2.4 GHz WiFi. The average positioningerror of 60 GHz mmWave is . less than 2.4 GHz WiFi.Fig. 8 illustrates the comparison of RSSI between 60 GHzmmWave and 2.4 GHz WiFi. In Fig. 8, the RSSI of 2.4GHz WiFi is higher than that of 60 GHz mmWave at thesame RP, which verifies the fact that the attenuation of 60GHz mmWave signal is faster than that of 2.4 GHz signal.Besides, the distribution of RSSIs of mmWave signal ismore concentrated compared with 2.4 GHz signal, which canincrease the discrimination of RSSI to reduce positioning error.Then we analyze the performance of the proposed DoA-LFalgorithm which uses RSSI and DoA information for hybridLF positioning with mmWave. The performance is compared Positioning Error (m) C u m u l a t i v e D i s t r i bu t i on F un c t i on Fig. 9. Comparison of CDFs between DoA-LF algorithm and otheralgorithms in 60 GHz mmWave, where K = 4 for DoA-LF, KNN andWKNN. A v e r age P o s i t i on i ng E rr o r ( m ) Fig. 10. Comparison of average positioning errors between DoA-LFalgorithm and other algorithms in 60 GHz mmWave, where K = 4 for DoA-LF, KNN and WKNN. with other three LF algorithms, which is illustrated in Fig.9 and Fig. 10. Fig. 9 shows that the CDF curve of DoA-LF algorithm grows faster than other algorithms without DoAestimation, which means that the positioning error of DoA-LF algorithm is the lowest. Similarly, as illustrated in Fig.10, DoA-LF algorithm yields less positioning error comparedwith other algorithms. The average positioning error of DoA-LF algorithm is 1.32 meters, which is approximately lessthan WKNN and KNN algorithm without DoA estimation.Finally, we compare the performance of DoA-LF algorithmwith mmWave, WKNN algorithm with 60 GHz mmWave andWKNN algorithm with 2.4 GHz WiFi. As illustrated in Fig.11, the positioning error of DoA-LF algorithm with 60 GHzmmWave is much lower than WKNN algorithm with 60 GHzmmWave and 2.4 GHz WiFi. As illustrated in Fig. 12, theaverage error distance of DoA-LF algorithm is 1.37 meters,which is . less than WKNN with 60 GHz mmWave and . less than WKNN with 2.4 GHz WiFi.The relation between the average positioning error and K isillustrated in Fig. 13 to yield the optimal value of K in WKNNalgorithm. It is noted that if K is small, the randomness is large Positioning Error (m) C u m u l a t i v e D i s t r i bu t i on F un c t i on Fig. 11. Comparison of CDFs between DoA-LF with 60 GHz mmWave,WKNN with 60 GHz mmWave and WKNN with 2.4GHz WiFi, where K = 6 for DoA-LF and WKNN. A v e r age P o s i t i on i ng E rr o r ( m ) Fig. 12. Comparison of average positioning errors between DoA-LF with60 GHz mmWave, WKNN with 60 GHz mmWave and WKNN with 2.4GHzWiFi, where K = 6 for DoA-LF and WKNN.
51 10 15 20 25 30 35 40 45 50 55 60246810121416 K A v e r age P o s i t i on i ng E rr o r ( m ) Fig. 13. The relation between the average positioning error and K . and the average positioning error is large. On the contrary, if K is large, the disturbance from other RPs is correspondinglylarge and the average positioning error is also large. Thus there Inteval of RPs (m) A v e r age P o s i t i on i ng E rr o r ( m ) AP=3AP=4AP=5AP=6 y = x
Fig. 14. The impact of the interval of RPs on average positioning error,where the number of APs varies from 3 to 6. exists an optimal K to minimize the average positioning error,which is illustrated in Fig. 13. Besides, the average positioningerror for any K with mmWave is still smaller than that withlow frequency signal.Overall, we have verified that the positioning error can bereduced significantly when employing DoA-LF positioningalgorithm with mmWave. B. CRLB and Analysis
In this subsection, the simulation results are shown toanalyze the impact of various parameters on the average posi-tioning error of DoA-LF algorithm, which include the numberof APs, the interval of RPs, the path loss exponent, the DoAestimation error and the variance of received signal strength.These simulations and analysis can verify our analysis ofCRLB in section IV. Besides, they can provide a guidelinefor the selection of appropriate parameters in the proposedDoA-LF positioning algorithm.Firstly, we analyze the impact of the interval of RPs onaverage positioning error. Fig. 14 shows that the averagepositioning error is increasing with the increase of the intervalof RPs. The curve y = x is plotted in Fig. 14, where x denotesthe interval of RPs and y denotes the average positioning error.It is noted that when the interval of RPs is too small, theaverage positioning error is larger than the interval of RPs,which is due to the fact that the CRLB has determined that theaverage positioning error cannot be as small as possible. Hencethe interval of RPs does not have to be very small becausethe computational complexity of DoA-LF algorithm will beintolerable in this situation and the average positioning erroris lower bounded by the CRLB. However, when the intervalof RP is larger than a threshold, the average positioning erroris smaller than the interval of RPs.Then we analyze the impact of the number of APs onaverage positioning error. As illustrated in Fig. 15, the po-sitioning error is decreasing with the increase of the numberof APs. Since the increase of the number of APs will enlarge
21 4 6 8 10 12 14 1612345678910
Number of APs A v e r age P o s i t i on i ng E rr o r ( m ) Inteval of RPs=5 mInteval of RPs=6 mInteval of RPs=7 mInteval of RPs=8 m
Fig. 15. The impact of the number of APs on average positioning error,where the interval of RPs varies from 5 to 8 meters.
Positioning Error (m) C u m u l a t i v e D i s t r i bu t i on F un c t i on n=1.0n=2.0n=3.0n=4.0 Fig. 16. The relation between positioning error and path loss exponent. the computational complexity, there is an optimal number ofAPs in order to balance the computational complexity andpositioning error.Finally, we analyze the impact of the variance of DoAestimation, the path loss exponent and the variance of receivedsignal strength on the average positioning error of DoA-LFalgorithm with mmWave. In Fig. 16, the positioning error isdecreasing with the increase of path loss exponent. In Fig. 17,the positioning error is decreasing with the decrease of thevariance of received signal strength. In Fig. 18, the positioningerror is decreasing with the decrease of the variance of DoAestimation. These results support our analysis results of CRLBin Section IV, which can be explained. When the path lossexponent is large, the discrimination of RSSI is correspond-ingly large, which can reduce positioning error. Besides, whenthe variance of signal intensity or DoA estimation is small,the fluctuation of RSSI or DoA estimation is correspondinglysmall, which can also reduce positioning error. However, thepath loss exponent, the variance of received signal strength andthe variance of DoA estimation are not controllable variables
Positioning Error (m) C u m u l a t i v e D i s t r i bu t i on F un c t i on Variance=1.0Variance=2.0Variance=3.0Variance=4.0
Fig. 17. The relation between positioning error and received signal strengthvariance.
Positioning Error C u m u l a t i v e D i s t r i bu t i on F un c t i on DoA Var=1.0DoA Var=2.0DoA Var=3.0DoA Var=4.0
Fig. 18. The relation between positioning error and DoA estimation variance. because they are impacted by the electromagnetic and physicalenvironment. Therefore when the environment is not ideal, wecan properly reduce the interval of RPs or increase the numberof APs to bound positioning error.VI. C
ONCLUSION
In this paper, the DoA-LF positioning algorithm is proposed,which incorporates the features of RSSI and DoA informationinto LF positioning. The characteristics of narrow beam andfast attenuation of mmWave are exploited to reduce positioningerror. The DoA information obtained from MUSIC algorithmis combined with RSSI information to construct a joint offlinedatabase for online matching. Then the K candidate RPs withthe most similar features are selected and their weighted meanis calculated, which is the estimated location. Moreover, theCRLB of DoA-LF positioning algorithm with mmWave isderived. Simulation results show that the positioning errorof DoA-LF algorithm is much smaller than the algorithmswithout DoA estimation. Besides, the positioning error withmmWave is much smaller than that with low frequency signals.Moreover, we have shown that there exists an optimal K tominimize the positioning error via simulation. Finally, simu-lation results show that the positioning error is an increasing function of the shadow fading factor, the interval of RPs andthe error of DoA estimation. And the positioning error is adecreasing function of the path loss exponent and the numberof APs. The study of this paper may provide guideline forindoor positioning in the era of 5G with densely deployedmmWave small cells.A CKNOWLEDGMENT
The authors appreciate editor and anonymous reviewers fortheir precious time and great effort in improving this paper.R
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