Separating weak and strong moving targets using the fractional fourier transform

1. A computer-implemented method, comprising:

determining, by a computing system, a rotational parameter a 1 of a strongest signal in a composite echo signal at which an energy of a Wigner Distribution (WD) is maximum for a given step size Δa by computing a peak of a Fractional Fourier Transform (FrFT) at all angles for the step size and choosing an angle that corresponds with the maximum energy such that the strongest signal is a tone when rotated to a 1 ;

after calculating a 1 , rotating by a 1 to an axis of the signal in which the strongest signal is now a tone, by the computing system;

finding a peak along the axis, by the computing system; and

setting the peak to zero, by the computing system, producing a notched composite echo signal with the strongest signal notched out.

2. The computer-implemented method of claim 1 , further comprising:

rotating back to α=0, by the computing system; and

plotting an intermediate time domain signal, by the computing system.

3. The computer-implemented method of claim 1 , further comprising:

for a next strongest signal a 2 , rotating, by the computing system, to a 2 −a 1 , hence performing a next rotation in one step.

4. The computer-implemented method of claim 3 , further comprising:

finding a new strongest peak along an axis of a 2 −a 1 , by the computing system; and

setting the new strongest peak to zero, by the computing system, thereby notching out the next strongest signal from the composite echo signal to search for subsequent lower power targets.

5. The computer-implemented method of claim 4 , wherein successive signals are found by:

rotating, by the computing system, to an axis for an i th strongest signal of K total signals, where i=3, by rotating to a i −a i−1 ; and

finding an i th peak along the axis of a i −a i−1 , by the computing system.

6. The computer-implemented method of claim 5 , further comprising:

setting the i th peak to zero, by the computing system;

incrementing i, by the computing system; and

repeating the process until the K th signal is processed.

7. The computer-implemented method of claim 1 , wherein the FrFT is only performed at all angles a for step size Δa once for the composite echo signal since signals can be notched out from the composite echo signal and a next peak can then be determined without another search.

8. The computer-implemented method of claim 1 , wherein 0.0001≤Δa≤0.01.

9. The computer-implemented method of claim 1 , wherein 1,000 or fewer samples N of the composite echo signal are collected.

10. The computer-implemented method of claim 1 , wherein a Carrier-to-Interference Ratio (CIR) of a weakest signal is between 8-20 decibels lower than the strongest signal and a Signal-to-Clutter Ratio (SCR) of the weakest signal is 5-10 decibels below clutter.

11. A computer program embodied on a non-transitory computer-readable medium, the program configured to cause at least one processor to:

for 0<a<2 and i=1, where a is a rotational parameter in a Wigner Distribution (WD) for a composite echo signal x(t) and i is a current signal:

compute a Fractional Fourier Transform (FrFT) of x(t),

compute a max value for the FrFT of x(t), and

increment a by a step size Δa;

find a peak over all a;

rotate to the peak for signal i;

notch out the peak; and

notch out signal i, producing a notched composite echo signal.

12. The computer program of claim 11 , wherein the program is further configured to cause the at least one processor to:

rotate back to a=0; and

plot an intermediate time domain signal.

13. The computer program of claim 11 , wherein the program is further configured to cause the at least one processor to:

while i≤K, where K is a total number of signals:

increment i such that i=i+1;

rotate for signal i to a i −a i−1 ;

notch out a peak at a i −a i−1 ; and

notch out signal i.

14. The computer program of claim 11 , wherein the FrFT is only performed at all angles a for step size Δa once for a given x(t) since signals can be notched out from x(t) and a next peak can then be determined without another search.

15. The computer program of claim 11 , wherein 0.0001≤Δa≤0.01.

16. The computer program of claim 11 , wherein 1,000 or fewer samples N of x(t) are collected.

17. The computer program of claim 11 , wherein a Carrier-to-Interference Ratio (CIR) of a weakest signal is between 8-20 decibels lower than a strongest signal.

18. The computer program of claim 11 , wherein a Signal-to-Clutter Ratio (SCR) of a weakest signal is 5-10 decibels below clutter.

19. An apparatus, comprising:

memory storing computer program instructions configured to separate targets from a signal; and

at least one processor configured to execute the computer program instructions, wherein the at least one processor, by executing the computer program instructions, is configured to:

for 0<a<2 and i=1, where a is a rotational parameter in a Wigner Distribution (WD) for a composite echo signal x(t) and i is a current signal:

compute a Fractional Fourier Transform (FrFT) of x(t),

compute a max value for the FrFT of x(t), and

increment a by a step size Δa,

find a peak over all a,

while i≤K, where K is a total number of signals:

increment i such that i=i+1,

rotate for signal i to a i −a i−1 ,

notch out a peak at a i −a i−1, and

notch out signal i, wherein

a notched composite echo signal is produced that does not include the K notched out signals.

20. The apparatus of claim 19 , wherein the at least one processor is further configured to:

rotate back to a=0 after processing for signal i=1; and

plot an intermediate time domain signal.

21. The apparatus of claim 19 , wherein the FrFT is only performed at all angles a for step size Δa once for a given x(t) since signals can be notched out from x(t) and a next peak can then be determined without another search.

22. The apparatus of claim 19 , wherein 0.0001≤Δa≤0.01.

23. The apparatus of claim 19 , wherein 1,000 or fewer samples N of x(t) are collected.

24. The apparatus of claim 19 , wherein a Carrier-to-Interference Ratio (CIR) of a weakest signal is between 8-20 decibels lower than a strongest signal.

25. The apparatus of claim 19 , wherein a Signal-to-Clutter Ratio (SCR) of a weakest signal is 5-10 decibels below clutter.