Systems and method for determining frequency coefficients of signals
A system for determining the frequency coefficients of a one or multi-dimensional signal that is sparse in the frequency domain includes determining the locations of the non-zero frequency coefficients, and then determining values of the coefficients using the determined locations. If N is total number of frequency coefficients across the one or more dimension of the signal, and if R is an upper bound of the number of non-zero ones of these frequency coefficients, the systems requires up to (O(Rlog(R) (N))) samples and has a computation complexity of up to O(Rlog 2 (R) log (N). The system and the processing technique are stable to low-level noise and can exhibit only a small probability of failure. The frequency coefficients can be real and positive or they can be complex numbers.
1. A method for identifying magnitudes of frequency components of a signal f(x) received by a receiver, the method comprising:
obtaining a first plurality of sets of signal samples received by an antenna or a sensor of the receiver, each set comprising a plurality of samples of the signal f(x), a cardinality of each set being greater than an upper bound (R) of a number of non-zero frequency components of the signal f(x) and less than a number of samples N of the signal f(x) according to a Nyquist rate,
wherein obtaining each set of signal samples comprises subsampling the signal f(x) by the receiver, at least one set of signal samples determines a support set S; and
computing by a processor an initial set of frequency coefficients f 0 using the support set S and the first plurality of sets of signal samples, the support set S identifying indices of the non-zero frequency components of the signal f(x).
2. The method of claim 1 , wherein obtaining the first plurality of sets of signal samples comprises accessing from memory T sets of signal samples, wherein:
T is on the order of O(1) or on the order of O(R log R N); and
a t-th set of signal samples, wherein index t ranges from 1 through T, comprises P (t) signal samples, P (t) being greater than R and smaller than N.
3. The method of claim 1 , wherein obtaining the first plurality of sets of signal samples comprises:
determining the number of samples (N) of the signal f(x) according to the Nyquist rate, and obtaining the upper bound (R) of the number of non-zero frequency components of the signal;
selecting a number of sets (T), wherein T is on the order of O( 1 ) or on the order of O(R log R N), and selecting T recovery parameters P (t) , wherein index t ranges from 1 through T;
for each recovery parameter P (t) , sampling the signal f(x) at P (t) distinct locations, providing the first plurality of T sets of signal samples, each sample set having P (t) samples.
4. The method of claim 1 , wherein computing the initial set of frequency coefficients {circumflex over (f)} 0 comprises multiplying the first plurality of sets of signal samples by: (i) a discrete Fourier transform (DFT) matrix F comprising T DFT submatrices, each submatrix having a size P (t) , and index t ranging from 1 through T, and (ii) a binary matrix B based on the support set S.
5. The method of claim 1 , further comprising:
generating a set of frequency coefficients f using a binary matrix B based on the support set S and the initial set of frequency coefficients f 0 .
6. The method of claim 1 , further comprising:
determining that a norm based on a binary matrix B based on the support set S and the initial set of frequency coefficients {circumflex over (f)} 0 is not less than a selected threshold;
obtaining a second plurality of sets of signal samples, each set comprising a plurality of samples of the signal f(x); and
re-computing the initial set of frequency coefficients fo using the support set S and the second plurality of sets of signal samples.
7. The method of claim 6 , further comprising:
determining that a norm based on the binary matrix B and the re-computed initial set of frequency coefficients {circumflex over (f)} 0 is less than or equal to the selected threshold; and
generating a set of frequency coefficients f using the binary matrix B and the re-computed initial set of frequency coefficients f 0 .
8. A system for identifying magnitudes of frequency components of a signal f(x) received by a receiver, the system 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:
obtain a first plurality of sets of signal samples received by an antenna or a sensor of the receiver, each set comprising a plurality of samples of the signal f(x), cardinality of each set being greater than an upper bound (R) of a number of non-zero frequency components of the signal f(x) and less than a number of samples N of the signal f(x) according to a Nyquist rate,
wherein to obtain each set of signal samples the processing unit is configured to subsample the signal f(x) by the receiver, at least one set of signal samples determines a support set S; and
compute an initial set of frequency coefficients f 0 using a support set S and the first plurality of sets of signal samples, the support set S identifying indices of the non-zero frequency components of the signal f(x).
9. The system of claim 8 , wherein to obtain the first plurality of sets of signal samples the processing unit is configured to access from the memory module T sets of signal samples, wherein:
T is on the order of O(1) or on the order of O(R log R N); and
a t-th set of signal samples, wherein index t ranges from 1 through T, comprises P (t) signal samples, P (t) being greater than R and smaller than N.
10. The system of claim 8 , wherein to obtain the first plurality of sets of signal samples the processing unit is configured to:
determine the number of samples (N) of the signal f(x) according to the Nyquist rate, and obtain the upper bound (R) of the number of non-zero frequency components of the signal;
select a number of sets (T), wherein T is on the order of 0 (I) or on the order of 0 (R log R N), and select T recovery parameters P (t) , wherein index t ranges from 1 through T;
for each recovery parameter P (t) , configure a signal sampler to sample the signal f(x) at P (t) distinct locations, providing the first plurality of T sets of signal samples, each sample set having P (t) samples.
11. The system of claim 8 , wherein to compute the initial set of frequency coefficients f 0 the processing unit is programmed to:
multiply the first plurality of sets of signal samples by: (i) a discrete Fourier transform (DFT) matrix F comprising T DFT submatrices, each submatrix having a size P (t) , and index t ranging from 1 through T, and (ii) a binary matrix B based on the support set S.
12. The system of claim 8 , wherein the instructions further program the processing unit to:
generate a set of frequency coefficients {circumflex over (f)} using a binary matrix B based on the support set S and the initial set of frequency coefficients h.
13. The system of claim 8 , wherein the instructions further program the processing unit to:
determine that a norm based on a binary matrix B based on the support set S and the initial set of frequency coefficients f 0 is not less than a selected threshold;
obtain a second plurality of sets of signal samples, each set comprising a plurality of samples of the signal f(x); and
re-compute the initial set of frequency coefficients f 0 using the support set S and the second plurality of sets of signal samples.
14. The system of claim 13 , wherein the instructions further program the processing unit to:
determine that a norm based on the binary matrix B and the re-computed initial set of frequency coefficients {circumflex over (f)} 0 is less than or equal to the selected threshold; and
generate a set of frequency coefficients {circumflex over (f)} using the binary matrix B and the re computed initial set of frequency coefficients {circumflex over (f)} 0 .