Method and apparatus of spectral estimation

1. A computer-implemented method of spectral estimation of a signal represented by a data stream D of sampled data from that signal, the method comprising:

(1) selecting a first set of samples from the data stream D;

(2) initializing a combined set of samples with said first set;

(3) selecting a set of data samples which is not in the combined set of samples;

(4) combining the selected set of data samples with the combined set of samples to create an updated combined set of samples;

(5) determining a measure of signal-to-noise ratio of an estimated spectrum for the updated combined set of samples;

(6) comparing said determined measure to a target measure determined for a target signal-to-noise ratio;

(7) repeating steps (3)-(6) until said determined measure has a predetermined relationship to said target measure; and

(8) providing said estimated spectrum as an output representing said signal.

2. The method of spectral estimation of claim 1 , wherein the target signal to noise ratio is achieved when the determined measure is less than a predetermined limit λ.

3. The method of spectral estimation of claim 1 , wherein the determined measure is proportional to a condition number of a least-squares (L-S) matrix A of a least-squares fit of the combined sample set to the estimated spectrum.

4. The method of spectral estimation of claim 1 ,

wherein the targeted signal to noise ratio is achieved when an upper right triangular factor matrix R of a QR factorization of a least squares matrix (L-S) matrix A of a least squares fit of the combined sample set to an estimated spectrum meets a condition given by

min(abs(diag( R )))>0^1/min(abs(diag( R )))<λ.

where λ is a predetermined limit equal to or greater than approximately 1.1 and less than approximately 1,000.

5. The method of spectral estimation of claim 1 ,

wherein the targeted signal to noise ratio is achieved when an upper right triangular factor matrix R of a QR factorization of a least squares matrix (L-S) matrix A of a least squares fit of the combined sample set to an estimated spectrum meets a condition given by

min(abs(diag( R )))>^max(abs(diag( R )))/min(abs(diag( R )))λ.

where λ is a predetermined limit equal to or greater than approximately 1.1 and less than approximately 1,000.

6. The method of spectral estimation of claim 1 ,

determining an upper right triangular factor matrix R of a QR factorization of a least-squares (L-S) matrix A of a least-squares fit of the combined sample set to an estimated spectrum,

wherein the measure of the signal to noise ratio is a ratio of selected diagonal elements of the QR factorization of the L-S matrix A when the factor matrix R is nonsingular.

7. The method of spectral estimation of claim 6 , wherein the ratio of selected diagonal elements is one or both of a maximum value ratio of an element selected from an element-wise absolute value of a main diagonal of the factor matrix R to a minimum value element selected from the element-wise absolute value of the main diagonal and a ratio that is an inverse of a minimum value element selected from an element-wise absolute value of a main diagonal of the factor matrix R.

8. The method of spectral estimation of claim 7 , wherein the predetermined limit λ is equal to or greater than approximately 1.1 and less than approximately 1,000.

9. The method of spectral estimation of claim 1 ,

wherein selecting other sets of samples uses a minimum time step Δt min between corresponding samples in the selected sets, and wherein selecting other sets of samples comprises taking a quantity of samples equaling at least two times a number of frequencies in a set of frequencies of interest F plus 1 from at least two periods of a lowest frequency of interest f i , the lowest frequency of interest f 1 being greater than zero Hertz.

10. The method of spectral estimation of claim 1 ,

wherein a minimum time step Δt min between selected sets is given by

Δ t min =2/ f 1 /(2| F 1+1)

where |F1 is a number of frequencies in a set of frequencies of interest F and f 1 is a lowest frequency greater than zero Hertz in the set of frequencies of interest F.

11. The method of spectral estimation of claim 1 , further comprising:

computing an estimated spectrum by solving a least-squares fitting problem for the combined set; and

extracting Fourier coefficients of the estimated spectrum from a solution to the least-squares fitting problem.

12. The method of spectral estimation of claim 1 , wherein the data stream D comprises irregularly spaced, time-sampled data.

13. The method of spectral estimation of claim 1 , wherein the samples of the selected sets from the data stream D comprise pairs of values, each pair comprising an explicit timestamp and a sample value.