NLS using a bounded linear initial search space and a fixed grid with pre-calculated variables
Described herein is NLS using a bounded linear initial search space and a fixed grid with pre-calculated variables. An initial first elevation angle corresponding to a direct reflection from an object and an initial second elevation angle corresponding to a multi-path reflection from the object are received along with a grid comprising pre-calculated variables for each of a plurality of grid points. The initial elevation angle pair is then refined by, for a plurality of iterations, determining a closest grid point to a current elevation angle pair and performing a non-linear least squares iteration on the closest grid point using the pre-calculated variables for the closest grid point to determine a next elevation angle pair. When the iterations have converged, a height of the object is calculated based on a final elevation angle pair.
1 . A method comprising:
receiving an initial elevation angle pair including an initial first elevation angle and an initial second elevation angle, the initial first elevation angle corresponding to a direct path reflection from an object, the initial second elevation angle corresponding to a multipath reflection from the object and a ground;
receiving a grid comprising pre-calculated variables for each of a plurality of grid points, each of the grid points corresponding to a pair of possible first and second elevation angles;
refining the initial elevation angle pair by:
for a first iteration:
determining a first closest grid point to the initial elevation angle pair;
performing a non-linear least squares iteration on the first closest grid point using the pre-calculated variables for the first closest grid point to determine a second elevation angle pair; and
determining a second closest grid point to the second elevation angle pair; and
for each subsequent iteration of one or more subsequent iterations:
performing a non-linear least squares iteration on a current grid point using the pre-calculated variables for the current grid point to determine a next elevation angle pair, the current grid point being the second closest grid point for a first subsequent iteration; and
determining a subsequent closest grid point to the next elevation angle pair, the subsequent closest grid point becoming the current grid point for a next iteration; and
determining, based on the iterations, a final one of the subsequent closest grid points as a final elevation angle pair;
calculating a height of the object based on the final elevation angle pair; and
performing a vehicle function based on the height of the object.
2 . The method of claim 1 , wherein the non-linear least squares iteration uses only linear operations.
3 . The method of claim 1 , wherein the non-linear least squares iteration comprises calculating a Hessian matrix and a gradient vector using the pre-calculated variables and a measurement matrix corresponding to the respective signals.
4 . The method of claim 3 , wherein the non-linear least squares iteration further comprises using the Hessian matrix and the gradient vector to determine an offset from the closest grid point.
5 . The method of claim 4 , wherein the offset comprises a function of the Hessian matrix, the gradient vector, and a damping factor.
6 . The method of claim 5 , wherein the non-linear least squares iteration comprises a line search or a trust region function.
7 . The method of claim 1 , wherein the pre-calculated variables comprise four variables.
8 . The method of claim 7 , wherein the pre-calculated variables comprise:
a first variable based on a first derivative of a steering matrix and the steering matrix;
a second variable based on the steering matrix;
a third variable based on the first derivative of the steering matrix and the steering matrix; and
a fourth variable based on the steering matrix.
9 . The method of claim 1 , wherein one of the pre-calculated variables comprises a pseudoinverse of a steering matrix.
10 . The method of claim 1 , wherein one of the pre-calculated variables comprises a product of a conjugate transpose of a first derivative of a steering matrix and a function that projects a measurement matrix onto an orthogonal complement of a column space of the steering matrix.
11 . The method of claim 10 , wherein another of the pre-calculated variables comprises a product of the conjugate transpose of the first derivative of the steering matrix, the function, and the first derivative of the steering matrix.
12 . The method of claim 1 , wherein one of the pre-calculated variables comprises an inverse of a product of a conjugate transpose of a steering matrix and the steering matrix.
13 . The method of claim 1 , wherein the subsequent iterations are performed until the subsequent closest grid point is the current grid point.
14 . The method of claim 1 , wherein the respective signals are radar signals.
15 . A system comprising:
at least one processor configured to:
receive an initial elevation angle pair including an initial first elevation angle and an initial second elevation angle, the initial first elevation angle corresponding to a direct path reflection from an object, the initial second elevation angle corresponding to a multipath reflection from the object and a ground;
receive a grid comprising pre-calculated variables for each of a plurality of grid points, each of the grid points corresponding to a pair of possible first and second elevation angles;
refine the initial elevation angle pair by:
for a first iteration:
determining a first closest grid point to the initial elevation angle pair;
performing a non-linear least squares iteration on the first closest grid point using the pre-calculated variables for the first closest grid point to determine a second elevation angle pair; and
determining a second closest grid point to the second elevation angle pair; and
for each subsequent iteration of one or more subsequent iterations:
performing a non-linear least squares iteration on a current grid point using the pre-calculated variables for the current grid point to determine a next elevation angle pair, the current grid point being the second closest grid point for a first subsequent iteration; and
determining a subsequent closest grid point to the next elevation angle pair, the subsequent closest grid point becoming the current grid point for a next iteration; and
determine, based on the iterations, a final one of the subsequent closest grid points as a final elevation angle pair;
calculate a height of the object based on the final elevation angle pair; and
perform a vehicle function based on the height of the object.
16 . The system of claim 15 , wherein the non-linear least squares iteration uses only linear operations.
17 . The system of claim 15 , wherein the non-linear least squares iteration comprises calculating a Hessian matrix and a gradient vector using the pre-calculated variables and a measurement matrix corresponding to the respective signals.
18 . The system of claim 15 , wherein the subsequent iterations are performed until the subsequent closest grid point is the current grid point.
19 . The system of claim 15 , wherein at least one of:
one of the pre-calculated variables comprises a pseudoinverse of a steering matrix;
one of the pre-calculated variables comprises a product of a conjugate transpose of a first derivative of the steering matrix and a function that projects a measurement matrix onto an orthogonal complement of a column space of the steering matrix;
one of the pre-calculated variables comprises a product of the conjugate transpose of the first derivative of the steering matrix, the function, and the first derivative of the steering matrix; or
one of the pre-calculated variables comprises an inverse of a product of the conjugate transpose of the steering matrix and the steering matrix.
20 . A non-transitory computer-readable storage media comprising instructions that, when executed, cause at least one processor to:
receive an initial elevation angle pair including an initial first elevation angle and an initial second elevation angle, the initial first elevation angle corresponding to a direct path reflection from an object, the initial second elevation angle corresponding to a multipath reflection from the object and a ground;
receive a grid comprising pre-calculated variables for each of a plurality of grid points, the grid points corresponding to a pair of possible first and second elevation angles;
receive an initial elevation angle pair including an initial first elevation angle and an initial second elevation angle, the initial first elevation angle corresponding to a direct path reflection from an object, the initial second elevation angle corresponding to a multipath reflection from the object and a ground;
refine the initial elevation angle pair by:
for a first iteration:
determining a first closest grid point to the initial elevation angle pair;
performing a non-linear least squares iteration on the first closest grid point using the pre-calculated variables for the first closest grid point to determine a second elevation angle pair; and
determining a second closest grid point to the second elevation angle pair; and
for each subsequent iteration of one or more subsequent iterations:
performing a non-linear least squares iteration on a current grid point using the pre-calculated variables for the current grid point to determine a next elevation angle pair, the current grid point being the second closest grid point for a first subsequent iteration; and
determining a subsequent closest grid point to the next elevation angle pair, the subsequent closest grid point becoming the current grid point for a next iteration; and
determine, based on the iterations, a final one of the subsequent closest grid points as a final elevation angle pair;
calculate a height of the object based on the final elevation angle pair; and
perform a vehicle function based on the height of the object.