tsoaposest

Estimate target positions using time-sum-of-arrival (TSOA) measurements

Since R2025a

Syntax

tgtposest = tsoaposest(tsoaest,tsoavar,txpos,rxpos)

tgtposest = tsoaposest(tsoaest,tsoavar,txpos,rxpos,PropagationSpeed=C)

[tgtposest,tgtposcov] = tsoaposest(tsoaest,tsoavar,txpos,rxpos)

Description

Estimate target positions from TSOA measurements combined with anchor positions.

tgtposest = tsoaposest(tsoaest,tsoavar,txpos,rxpos) estimates the position of targets, tgtposest, from time-sum-of-arrival (TSOA) measurements taken at multiple transmit anchor and receive anchor pairs with known positions. Depending on the application scenario, anchors can be cellular base stations, wireless network access points, satellites, ground radar stations, etc., and a target is a scatterer that scatters transmit anchors' signals to receive anchors.

tgtposest = tsoaposest(tsoaest,tsoavar,txpos,rxpos,PropagationSpeed=C) also specifies the signal propagation speed C as a positive scalar. If this syntax is not used, the default value of the signal propagation speed is given by physconst("LightSpeed").

[tgtposest,tgtposcov] = tsoaposest(tsoaest,tsoavar,txpos,rxpos) also returns the estimated target position covariance, tgtposcov.

Examples

Estimate Target Position using MIMO Radar

A distributed MIMO radar system with two transmit anchors and five receive anchors measures the TSOA from each transmit-receive anchor pairs. Then, the radar system applies the two-step WLLS algorithm using the TSOA estimates and anchor positions. The data is loaded from TSOAEstimatorExampleData, whose variables are listed here:

Parameter	Description	Size
tsoaest	TSOA estimates	2-by-5
tsoavar	TSOA estimate variance	2-by-5
txpos	Transmit anchor positions	3-by-2
rxpos	Receive anchor positions	3-by-5
tgtpos	True target position	3-by-1

First, load data from mat-file.

load TSOAEstimatorExampleData

Draw the positions of the transmitters and receivers.

plot3(txpos(1,:),txpos(2,:),txpos(3,:),'.')
a = 50;
axis([-a a -a a -a a])
hold on
plot3(rxpos(1,:),rxpos(2,:),rxpos(3,:),'m.')
plot3(tgtpos(1,:),tgtpos(2,:),tgtpos(3,:),'go')

Estimate and plot the target position.

[tgtposest,tgtposcov] = tsoaposest(tsoaest,tsoavar,txpos,rxpos);
plot3(tgtposest(1,:),tgtposest(2,:),tgtposest(3,:),'.k')
legend('Transmitters','Receivers','Target','Estimated Target')

RMSE of target position estimate.

rmsepos = rmse(tgtposest,tgtpos);
disp(['RMS TSOA positioning error = ',num2str(rmsepos),' meters.'])

RMS TSOA positioning error = 0.1131 meters.

Input Arguments

tsoaest — Time-sum of arrival estimates (sec)

real-valued J-by-L-by-K array

Time-sum of arrival estimates, specified as a real-valued J-by-L-by-K MATLAB^® array, where J is the number of transmitting anchors, L is the number of receiving anchors, and K is the number of TSOA groups. Each array page represents a TSOA group corresponding to the TSOAs measured at the JL anchor pairs for a potential target. In each page, the element in the j^th row and l^th column of tsoaest represents the estimated TSOA from the j^th transmit anchor to the l^th receive anchor. Units are in seconds.

Data Types: single | double

tsoavar — TSOA estimation variance (seconds-squared)

real-valued J-by-L matrix | real-valued J-by-L-by-K array

TSOA estimation variance, specified as a real-valued J-by-L matrix or real-valued J-by-L-by-K array. When tsoavar is a matrix, the variance estimates for JL anchor pairs are the same for all K TSOA groups. When tsoavar is an array, the variance estimates for JL anchor pairs are different for different TSOA groups. When the actual values of tsoavar are unknown, tsoavar is typically chosen to be a matrix with each element chosen to be the same real positive value. This value does not affect the solution tgtposest. Units are in seconds-squared.

Data Types: single | double

txpos — Transmitting anchor positions (meters)

real-valued Q-by-J matrix

Transmitting anchor positions, specified as a real-valued Q-by-J matrix. You can specify anchor positions in two or three dimensional space. J is the number of transmitting anchors. For two dimensional positions, Q=2, and for three dimensional positions, Q=3. Each column represents the position of a different anchor. The j^th column of txpos denotes the position of the j^th transmitter in Q-dimensional Cartesian space. Units are in meters.

Example: [20 31 –16; 13 64 –44]

Data Types: single | double

rxpos — Receiving anchor positions (meters)

real-valued Q-by-L matrix

Receiving anchor positions, specified as a real-valued Q-by-L matrix. You can specify anchor positions in two or three dimensional space. L is the number of receiving anchors. For two dimensional positions, Q=2, and for three dimensional positions, Q=3. Each column represents the position of a different anchor. The l^th column of rxpos denotes the position of the l^th receiver in Q-dimensional Cartesian space. Units are in meters.

Example: [20 31 –16; 13 64 –44]

Data Types: single | double

C — Signal propagation speed (meters/second)

physconst("LightSpeed") (default) | positive scalar

Signal propagation speed, specified as a positive scalar. Units are in meters/second.

Example: 3e8

Data Types: single | double

Output Arguments

tgtposest — Estimated target position (meters)

real-valued Q-by-K matrix

Estimated target position, returned as a real-valued Q-by-K matrix. tgtposest represents the estimated target positions for K targets. Units are in meters.

Data Types: single | double

tgtposcov — Estimated target position covariance (meters-squared)

real-valued Q-by-Q-by-K array

Estimated target position covariance, returned as a Q-by-Q-by-K array, where each page is a real-valued positive semi-definite symmetric matrix. The function calculates the covariance matrix from the CRLB of the TSOA position estimator. The calculation requires the knowledge of the TSOA estimation variance, tsoavar. Units are in meters-squared.

Data Types: single | double

More About

Algorithm

In the absence of noise, each TSOA measurement defines an ellipse or an ellipsoid where a target must lie in the 2-dimensional or 3-dimensional space. Then, target positions are given by the intersection of multiple ellipses or ellipsoids. The target positions are determined by solving a set of non-linear equations.

In the presence of noise, the target positions can be estimated from more advanced TSOA position estimation algorithms. This function uses a two-step weighted linear least squares (WLLS) algorithm to solve for the target position, tgtposest, from the TSOA measurements, tsoaest, the known transmit anchor positions, tzpos, and the known receive anchor positions, rxpos. The weights used in the WLLS algorithm are calculated from tsoaest and the TSOA estimation variance, tsoavar. The two-step WLLS algorithm approximates the maximum-likelihood estimation which can attain the TSOA position estimation Cramer-Rao lower bound (CRLB) when the TSOA estimation errors are small.

Precision

This function supports single and double precision for input arguments. If the input argument tsoaest is single precision, the outputs are single precision. If the input argument tsoaest is double precision, the outputs are double precision. The precision of the outputs are independent of the precision of the other arguments.

References

[1] Reza Zekavat and R. Michael Buehrer, Handbook of Position Location: Theory, Practice, and Advances, 2019

[2] A. Noroozi, M. A. Sebt, S. M. Hosseini, R. Amiri and M. M. Nayebi, "Closed-Form Solution for Elliptic Localization in Distributed MIMO Radar Systems With Minimum Number of Sensors," in IEEE Transactions on Aerospace and Electronic Systems, vol. 56, no. 4, pp. 3123-3133, Aug. 2020.

[3] M. Malanowski and K. Kulpa, "Two Methods for Target Localization in Multistatic Passive Radar," in IEEE Transactions on Aerospace and Electronic Systems, vol. 48, no. 1, pp. 572-580, Jan. 2012.

[4] P. Stoica and J. Li, "Lecture Notes - Source Localization from Range-Difference Measurements," in IEEE Signal Processing Magazine, vol. 23, no. 6, pp. 63-66, Nov. 2006.

Version History

Introduced in R2025a