Systems and methods for joint angle-frequency determination

1. A method for determining angle and frequency of transmission from a transmitter, the method comprising:

receiving during a current coherent dwell, sensor data comprising a first plurality of samples, the first plurality of samples comprising N samples collected at a sensor array comprising D sensors, a spacing between a pair of adjacent sensors being Δ;

selecting a second plurality of M samples from the first plurality of samples, M being less than N times D;

determining angle and frequency of transmission of a transmitter located at a range R and an angle θ relative to a plane of the sensor array and transmitting a signal at a frequency f, by minimizing using a solver, an l 1 norm of a product of: (i) a phase functional representing a mapping between time-distance and angle-frequency representations of sensor samples and (ii) the second plurality of M samples, the time-distance representation being based on the spacing Δ, and the angle-frequency representation being based on θ and f.

2. The method of claim 1 , wherein the first plurality of samples corresponds to a single coherent dwell.

3. The method of claim 1 , wherein N is a number of Nyquist intervals in a single coherent dwell.

4. The method of claim 1 , wherein the first plurality of samples comprises:

a first sample, collected by a particular sensor from the D sensors, and associated with a transmission at a first frequency; and

a second sample, collected by that particular sensor, and associated with a transmission at a second, different frequency.

5. The method of claim 1 , wherein selecting the second plurality of M samples from the first plurality of samples comprises filtering the first plurality of samples using a pseudorandom selector.

6. The method of claim 1 , wherein selecting the second plurality of M samples from the first plurality of samples comprises:

ranking the first plurality of samples according to respective signal-to-noise ratios (SNRs) associated with each sample; and

selecting samples corresponding to the M highest ranks.

7. The method of claim 1 , wherein the solver comprises a second order cone programming (SOCP) solver.

8. The method of claim 7 , wherein applying the SOCP solver comprises providing to the SOCP solver a frequency angle solution obtained from samples collected during another coherent dwell prior to the current coherent dwell.

9. A method for determining angle and frequency of transmission from a transmitter, the method comprising:

from a sensor array comprising D sensors, each of which is configured collect a sample during each one of N different sampling periods in one coherent dwell, and a spacing between a pair of adjacent sensors being Δ, obtaining during a current coherent dwell, using a pseudorandom selector, a plurality of M samples, M being less than N times D;

determining angle and frequency of transmission of a transmitter located at a range R and an angle θ relative to a plane of the sensor array and transmitting a signal at a frequency f, by minimizing using a solver, an l 1 norm a product of: (i) a phase functional representing a mapping between time-distance and angle-frequency representations of sensor samples and (ii) the plurality of M samples, the time-distance representation being based on the spacing Δ, and the angle-frequency representation being based on θ and f.

10. The method of claim 9 , wherein obtaining the plurality of M samples comprises:

selecting using the pseudorandom selector at least one of the N sampling periods;

for each selected sampling period, obtaining a sample from at least one of the D sensors; and

at least one of:

(i) selecting at least one of the N sampling periods during each of which a sample is not obtained; and

(ii) during at least one of the selected sampling periods, selecting at least one of the D sensors from which a sample is not obtained.

11. The method of claim 10 , wherein for each selected sampling period, obtaining a sample from at least one of the D sensors comprises rejecting a sample from a sensor if a signal-to-noise ratio (SNR) associated with that sample is less than a specified threshold.

12. The method of claim 9 , wherein the plurality of M samples comprises at least:

a first sample associated with a transmission at a first frequency; and

a second sample associated with a transmission at a second, different frequency.

13. The method of claim 9 , wherein the solver comprises a second order cone programming (SOCP) solver.

14. The method of claim 13 , the processor is further programmed to provide to the SOCP solver a frequency angle solution obtained from samples collected during another coherent dwell prior to the current coherent dwell.

15. The method of claim 13 , wherein applying the SOCP solver comprises providing to the SOCP solver a frequency angle solution obtained from samples collected during another coherent dwell prior to the current coherent dwell.

16. A system for determining angle and frequency of transmission from a transmitter, comprising:

a first processor; and

a first memory in electrical communication with the first processor, the first memory comprising instructions which, when executed by a processing unit comprising at least one of the first processor and a second processor, and in electronic communication with a memory module comprising at least one of the first memory and a second memory, program the processing unit to:

receive in the memory module, during a current coherent dwell, sensor data comprising a first plurality of samples, the first plurality of samples comprising N samples collected at a sensor array comprising D sensors, a spacing between a pair of adjacent sensors being Δ;

select a second plurality of M samples from the first plurality of samples, M being less than N times D; and

determine angle and frequency of transmission of a transmitter located at a range R and an angle θ relative to a plane of the sensor array and transmitting a signal at a frequency f, by minimizing an l 1 norm of the second plurality of M samples by solving an underdetermined linear system comprising a product of: (i) a phase functional representing a mapping between time-distance and angle-frequency representations of sensor samples and (ii) the second plurality of M samples, the time-distance representation being based on the spacing Δ, and the angle-frequency representation being based on θ and f.

17. The system of claim 16 , wherein the first plurality of samples corresponds to a single coherent dwell.

18. The system of claim 16 , wherein N is a number of Nyquist intervals in a single coherent dwell.

19. The system of claim 16 , wherein the first plurality of samples comprises:

a first sample, collected by a particular sensor from the D sensors, and associated with a transmission at a first frequency; and

a second sample, collected by that particular sensor, and associated with a transmission at a second, different frequency.

20. The system of claim 16 , wherein for selecting the second plurality of M samples from the first plurality of samples, the processing unit is further programmed to filter the first plurality of samples using a pseudorandom selection.

21. The system of claim 16 , wherein for selecting the second plurality of M samples from the first plurality of samples, the processing unit is further programmed to:

rank the first plurality of samples according to respective signal-to-noise ratios (SNRs) associated with each sample; and

select samples corresponding to the M highest ranks.

22. The system of claim 16 , wherein to solve the underdetermined linear system the processing unit is further programed as a second order cone programming (SOCP) solver.

23. The system of claim 22 , wherein the processing unit is further programmed to provide to the SOCP solver a frequency angle solution obtained from samples collected during another coherent dwell prior to the current coherent dwell.

24. A system for determining angle and frequency of transmission from a transmitter, comprising:

a sensor array comprising D sensors, each of which is configured collect a sample during each one of N different sampling periods in one coherent dwell, a spacing between a pair of adjacent sensors being Δ;

a pseudorandom selector adapted to obtain, during a current coherent dwell, a plurality of M samples, M being less than N times D;

a processor configured for determining angle and frequency of transmission of a transmitter located at a range R and an angle θ relative to a plane of the sensor array and transmitting a signal at a frequency f, by minimizing an l 1 norm by solving an underdetermined linear system comprising a product of: (i) a phase functional representing a mapping between time-distance and angle-frequency representations of sensor samples and (ii) the plurality of M samples, the time-distance representation being based on the spacing Δ, and the angle-frequency representation being based on θ and f.

25. The system of claim 24 , wherein to obtain the plurality of M samples, the pseudorandom selector is adapted to:

select at least one of the N sampling periods;

for each selected sampling period, obtain a sample from at least one of the D sensors; and

at least one of:

(i) select at least one of the N sampling periods during each of which a sample is not obtained; and

(ii) during at least one of the selected sampling periods, select at least one of the D sensors from which a sample is not obtained.

26. The system of claim 25 , wherein to obtain a sample from at least one of the D sensors for each selected sampling period, the pseudorandom selector is further adapted to reject a sample from a sensor if a signal-to-noise ratio (SNR) associated with that sample is less than a specified threshold.

27. The system of claim 24 wherein the plurality of M samples comprises at least:

a first sample associated with a transmission at a first frequency; and

a second sample associated with a transmission at a second, different frequency.

28. The system of claim 24 , wherein to solve the underdetermined linear system the processor is further programed as a second order cone programming (SOCP) solver.