Systems and methods for training multipath filtering systems
A method for training a neural network to be configured to filter a multipath corrupted signal is provided. The method includes receiving, at the neural network, real or simulated multipath corrupted signal data, and training the neural network on the multipath corrupted signal data using a complex iterated least square thresholding algorithm (CILST) capable of processing both real and complex signals.
1. A method for training a neural network, said method comprising:
receiving, at the neural network, from a signal receiver, real or simulated multipath corrupted signal data;
training the neural network on the multipath corrupted signal data using a complex iterated least square thresholding algorithm (CILST) capable of processing both real and complex signals, wherein the CILST algorithm uses complex arithmetic to process the real and complex signals by:
for each of a plurality of hidden nodes of the neural network:
solving, using the CILST algorithm, a complex linear least-square problem for the hidden node of the neural network to determine a linear fit to the multipath corrupted signal data;
computing, using the CILST algorithm, an error from the linear fit at the hidden node; and
performing, using the CILST algorithm, a new fit; and
fitting, using the CILST algorithm, vector data for the plurality of hidden nodes by solving the complex linear least-square problem and computing a final real result; and
filtering a multipath corrupted signal using the CILST-trained neural network to determine a location of at least one object.
2. A method in accordance with claim 1 , wherein receiving real or simulated multipath corrupted signal data comprises receiving real or simulated multipath corrupted signal data at a complex-valued neural network.
3. A method in accordance with claim 1 , wherein receiving real or simulated multipath corrupted signal data comprises receiving real or simulated multipath corrupted signal data at a neural network that includes an input layer having a plurality of input nodes, a hidden layer having the plurality of hidden nodes, and an output layer having a plurality of output nodes.
4. A method in accordance with claim 1 , wherein solving a complex linear least-square problem for the hidden node comprises solving the complex linear least-square problem for the hidden node in accordance with t −1 (t −1 (ŷ))≈A 1 [{circumflex over (x)},1] for A 1 , where [⋅,1] denotes a matrix whose last column is all 1's, A is a complex matrix of weights from a complex input sample vector x, A j is a jth row of A, and t is an invertible threshold function that produces a positive real output, y, from a complex input.
5. A method in accordance with claim 4 , wherein the invertible threshold function t is defined as
t
(
z
)
=
1
1
+
e
-
z
.
6. A method in accordance with claim 4 , wherein computing an error comprises computing an error according to e 1 =t −1 (ŷ)−z 1 , where z 1 =min{|t(A 1 [{circumflex over (x)},1])|,1}.
7. A method comprising:
receiving, from a signal receiver, a multipath corrupted signal at a computing device;
processing the multipath corrupted signal using a neural network that is trained using a complex iterated least square thresholding algorithm (CILST) capable of processing both real and complex signals, wherein the CILST algorithm uses complex arithmetic to process the real and complex signals by:
for each of a plurality of hidden nodes of the neural network:
solving, using the CILST algorithm, a complex linear least-square problem for the hidden node of the neural network to determine a linear fit to the multipath corrupted signal data;
computing, using the CILST algorithm, an error from the linear fit at the hidden node; and
performing, using the CILST algorithm, a new fit; and
fitting, using the CILST algorithm, vector data for the plurality of hidden nodes by solving the complex linear least-square problem and computing a final real result; and
determining a location of at least one object from the multipath corrupted signal processed using the CILST-trained neural network.
8. A method in accordance with claim 7 , wherein processing the multipath corrupted signal comprises processing the multipath corrupted signal using a complex-valued neural network, and wherein the method further comprises displaying the complex-valued neural network to a user.
9. A method in accordance with claim 7 , wherein processing the multipath corrupted signal comprises processing the multipath corrupted signal using a neural network that includes an input layer having a plurality of input nodes, a hidden layer having the plurality of hidden nodes, and an output layer having a plurality of output nodes.
10. A method in accordance with claim 7 , wherein solving a complex linear least-square problem for the hidden node comprises solving the complex linear least-square problem for the hidden node in accordance with t −1 (t −1 (ŷ))≈A 1 [{circumflex over (x)},1] for A 1 , where [⋅,1] denotes a matrix whose last column is all 1's, A is a complex matrix of weights from a complex input sample vector x, A j is a jth row of A, and t is an invertible threshold function that produces a positive real output, y, from a complex input.
11. A method in accordance with claim 10 , wherein the invertible threshold function t is defined as
t
(
z
)
=
1
1
+
e
-
z
.
12. A method in accordance with claim 10 , wherein computing an error comprises computing an error according to e 1 =t −1 (ŷ)−z 1 , where z 1 =min{|t(A 1 [{circumflex over (x)},1])|,1}.
13. A radar system configured to transmit and receive radar signals to determine a location of at least one object, said radar system comprising a computing device configured to:
receive, from a signal receiver, a multipath corrupted radar signal;
process the multipath corrupted radar signal to generate a filtered radar signal, the multipath corrupted radar signal processed using a neural network that is trained using a complex iterated least square thresholding algorithm (CILST) capable of processing both real and complex signals, wherein the CILST algorithm uses complex arithmetic to process the real and complex signals by:
for each of a plurality of hidden nodes of the neural network:
solving, using the CILST algorithm, a complex linear least-square problem for the hidden node of the neural network to determine a linear fit to the multipath corrupted signal data;
computing, using the CILST algorithm, an error from the linear fit at the hidden node; and
performing, using the CILST algorithm, a new fit; and
fitting, using the CILST algorithm, vector data for the plurality of hidden nodes by solving the complex linear least-square problem and computing a final real result; and
determine a location of the at least one object from the multipath corrupted signal processed using the CILST-trained neural network.
14. A radar system in accordance with claim 13 , wherein the neural network is a complex-valued neural network.
15. A radar system in accordance with claim 13 , wherein the neural network includes an input layer having a plurality of input nodes, a hidden layer having the plurality of hidden nodes, and an output layer having a plurality of output nodes.
16. A radar system in accordance with claim 13 , wherein solving a complex linear least-square problem for the hidden node includes solving the complex linear least-square problem for the hidden node in accordance with t −1 (t −1 (ŷ))≈A 1 [{circumflex over (x)},1] for A 1 , where [⋅,1] denotes a matrix whose last column is all 1's, A is a complex matrix of weights from a complex input sample vector x, A j is a jth row of A, and t is an invertible threshold function that produces a positive real output, y, from a complex input.
17. A radar system in accordance with claim 16 , wherein the invertible threshold function t is defined as
t
(
z
)
=
1
1
+
e
-
z
.