Method and apparatus of spectral estimation

• US 7,720,644 B2
• Filed: 09/16/2005
• Issued: 05/18/2010
• Est. Priority Date: 09/16/2005
• Status: Active Grant

• Current Assignee: Keysight Technologies, Inc.
• Original Assignee: Agilent Technologies, Inc.
• Inventors: Barford, Lee A.
• Primary Examiner(s): TSAI, CAROL S W
• Application Number:
  US11/229,479
• Publication Number:
  US 20070067167A1
• Time in Patent Office: 1,705 Days
• Field of Search: 702/22, 702/189, 375/224, 375/226, 375/130, 375/141, 375/147, 375/136, 324/450, 324/457, 324/350, 324/357.06, 324/357.01, 324/386, 324/417, 370/463, 370/477, 455/522, 455/517, 704/226
• US Class Current: 702/189
• CPC Class Codes: G06F 17/141 Discrete Fourier transforms
• CPC Class Codes: H04L 43/50 Testing arrangements

Abstract

A spectrum of a set of samples from a data stream of sampled data is estimated until a targeted signal to noise ratio is achieved.

Claims

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 Δ

t_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 f₁ being greater than zero Hertz.

10. The method of spectral estimation of claim 1,

wherein a minimum time step Δ

t_min between selected sets is given by

Δ

t_min=2/f₁/(2|F1+1)

where |F1 is a number of frequencies in a set of frequencies of interest F and f₁ 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.